diff --git "a/MNE4T4oBgHgl3EQfiw29/content/tmp_files/load_file.txt" "b/MNE4T4oBgHgl3EQfiw29/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/MNE4T4oBgHgl3EQfiw29/content/tmp_files/load_file.txt" @@ -0,0 +1,760 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf,len=759 +page_content='Springer Nature 2021 LATEX template Density functions of periodic sequences of continuous events Olga Anosova1 and Vitaliy Kurlin1* 1*Computer Science, University of Liverpool, Ashton street, Liverpool, L69 3BX, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Corresponding author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' E-mail(s): vitaliy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='kurlin@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Contributing authors: oanosova@liverpool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='uk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Abstract Periodic Geometry studies isometry invariants of periodic point sets that are also continuous under perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The motivations come from periodic crystals whose structures are determined in a rigid form but any minimal cells can discontinuously change due to small noise in measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' For any integer k ≥ 0, the density function of a periodic set S was previously defined as the fractional volume of all k-fold intersections (within a minimal cell) of balls that have a variable radius t and centers at all points of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' This paper introduces the density functions for periodic sets of points with different initial radii motivated by atomic radii of chemical elements and by continuous events occupying disjoint intervals in time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The contributions are explicit descriptions of the densities for periodic sequences of intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The new densities are strictly stronger and distinguish periodic sequences that have identical densities in the case of zero radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Keywords: computational geometry, periodic set, periodic time series, isometry invariant, density function MSC Classification: 68U05 , 51K05 , 51N20 , 51F30 , 51F20 1 Motivations for the density functions of periodic sets This work substantially extends the previous conference paper [3] in Discrete Geometry and Mathematical Morphology 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The past work explicitly described the density functions for peri- odic sequences of zero-sized points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The new work extends these analytic descriptions to periodic sequences whose points have non-negative radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The proposed extension to the weighted case is motivated by crystallography and materials chem- istry [1] because all chemical elements have differ- ent atomic radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' In dimension 1, the key motiva- tion is the study of periodic time series consisting of continuous and sequential (non-overlapping) events represented by disjoint intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Any such interval [a, b] ⊂ R for a ≤ b is the one-dimensional ball with the center a + b 2 and radius b − a 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The point-set representation of periodic crys- tals is the most fundamental mathematical model for crystalline materials because nuclei of atoms are well-defined physical objects, while chemical bonds are not real sticks or strings but abstractly represent inter-atomic interactions depending on many thresholds for distances and angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Since crystal structures are determined in a rigid form, their most practical equivalence is rigid motion (a composition of translations and rota- tions) or isometry that maintains all inter-point distances and includes also mirror reflections [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Now we introduce the key concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Let Rn be Euclidean space, Z be the set of all integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='05137v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='CG] 12 Jan 2023 Springer Nature 2021 LATEX template 2 Density functions of periodic sequences 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='6 ψkA(t) Radius of Balls ψ0A ψ1A ψ2A ψ3A ψ4A ψ5A ψ6A ψ7A ψ8A 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='6 Σ1nψkA(t) Radius of Balls n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 1 Illustration of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 for the hexagonal lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Left: subregions Uk(t) are covered by k disks for the radii t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='55, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='75, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Right: the densities ψk are above the corresponding densigram of accumulated functions k � i=1 ψi(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='1 (a lattice Λ, a unit cell U, a motif M, a periodic point set S = M + Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' For any linear basis v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , vn of Rn, a lattice is Λ = { n� i=1 civi : ci ∈ Z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The unit cell U(v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , vn) = � n� i=1 civi : ci ∈ [0, 1) � is the par- allelepiped spanned by the basis above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' A motif M ⊂ U is any finite set of points p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , pm ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' A periodic point set [20] is the Minkowski sum S = M + Λ = {u + v | u ∈ M, v ∈ Λ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' ■ In dimension n = 1, a lattice is defined by any non-zero vector v ∈ R, any periodic point set S is a periodic sequence {p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , pm} + vZ with the period v equal to the length of the vector v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 (density functions for periodic sets of points with radii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Let a periodic set S = Λ + M ⊂ Rn have a unit cell U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' For every point p ∈ M, fix a radius r(p) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' For any integer k ≥ 0, let Uk(t) be the region within the cell U covered by exactly k closed balls ¯B(p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' r(p) + t) for t ≥ 0 and all points p ∈ M and their transla- tions by Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The k-th density function ψk[S](t) = Vol[Uk(t)]/Vol[U] is the fractional volume of the k-fold intersections of these balls within U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' ■ The density ψk[S](t) can be interpreted as the probability that a random (uniformly chosen in U) point q is at a maximum distance t to exactly k balls with initial radii r(p) and all centers p ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' For k = 0, the 0-th density ψ0[S](t) mea- sures the fractional volume of the empty space not covered by any expanding balls ¯B(p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' r(p) + t) In the simplest case of radii r(p) = 0, the infi- nite sequence Ψ[S] = {ψk(t)}+∞ k=0 was called in [8, section 3] the density fingerprint of a periodic point set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' For k = 1 and small t > 0 while all equal-sized balls ¯B(p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' t) remain disjoint, the 1st density ψ1[S](t) increases proportionally to tn but later reaches a maximum and eventually drops back to 0 when all points of Rn are covered of by at least two balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' See the densities ψk, k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , 8 for the square and hexagonal lattices in [8, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The original densities helped find a missing crystal in the Cambridge Structural Database, which was accidentally confused with a slight per- turbation (measured at a different temperature) of another crystal (polymorph) with the same chemical composition, see [8, section 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The new weighted case with radii r(p) ≥ 0 in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 is even more practically important due to different Van der Waals radii, which are individually defined for all chemical elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The key advantage of density functions over other isometry invariants of periodic crystals Springer Nature 2021 LATEX template Density functions of periodic sequences 3 (such as symmetries or conventional representa- tions based on a geometry of a minimal cell) is their continuity under perturbations, see details in section 2 reviewing the related past work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The only limitation is the infinite size of den- sities ψk(t) due to the unbounded parameters: integer index k ≥ 0 and continuous radius t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' We state the following problem in full general- ity to motivate future work on these densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='3 (computation of ψk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Verify if the density functions ψk[S](t) from Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 can be computed in a polynomial time (in the size m of a motif of S) for a fixed dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' ■ The main contribution is the full solution of Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='3 for n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2, and Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='4 efficiently compute all ψk[S](t) depending on infinitely many values of k and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 2 Review of related past work Periodic Geometry was initiated in 2020 by the problem [14, section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='3] to design a computable metric on isometry classes of lattices, which is continuous under perturbations of a lattice basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Though a Voronoi domain is combinatorially unstable under perturbations, its geometric shape was used to introduce two continuous metrics [14, Theorems 2, 4] requiring approximations due to a minimization over infinitely many rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Similar minimizations over rotations or other continuous parameters are required for the com- plete invariant isosets [2, 4] and density functions, which can be practically computed in low dimen- sions [16], whose completeness was proved for generic periodic point sets in R3 [8, Theorem 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The density fingerprint Ψ[S] turned out to be incomplete [8, section 5] in the example below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='1 (periodic sequences S15, Q15 ⊂ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Widdowson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' [20, Appendix B] discussed homometric sets that can be distinguished by the invariant AMD (Average Minimum Distances) and not by diffraction patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The sequences S15 = {0, 1, 3, 4, 5, 7, 9, 10, 12} + 15Z, Q15 = {0, 1, 3, 4, 6, 8, 9, 12, 14} + 15Z have the unit cell [0, 15] shown as a circle in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 2 Circular versions of the periodic sets S15, Q15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' These periodic sequences [9] are obtained as Minkowski sums S15 = U + V + 15Z and Q15 = U − V + 15Z for U = {0, 4, 9}, V = {0, 1, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' ■ For rational-valued periodic sequences, [9, Theorem 4] proved that r-th order invariants (combinations of r-factor products) up to r = 6 are enough to distinguish such sequences up to a shift (a rigid motion of R without reflections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The AMD invariant was extended to the Point- wise Distance Distribution (PDD), whose generic completeness [19, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='4] was proved in any dimension n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' However there are finite sets in R3 [15, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' S4] with the same PDD, which were distinguished by more sophisticated distance-based invariants in [18, appendix C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The subarea of Lattice Geometry devel- oped continuous parameterizations for the moduli spaces of lattices considered up to isometry in dimension two [7, 13] and three [6, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' For 1-periodic sequences of points in Rn, com- plete isometry invariants with continuous and computable metrics appeared in [12], see related results for finite clouds of unlabeled points [11, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 3 The 0-th density function ψ0 This section proves Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 explicitly describ- ing the 0-th density function ψ0[S](t) for any periodic sequence S ⊂ R of disjoint intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' For convenience, scale any periodic sequence S to period 1 so that S is given by points 0 ≤ p1 < · · · < pm < 1 with radii r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , rm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Since the expanding balls in R are growing intervals, volumes of their intersections linearly change with respect to the variable radius t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Hence any density function ψk(t) is piecewise linear and uniquely determined by corner points (aj, bj) where the gradient of ψk(t) changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 4 Density functions of periodic sequences To prepare the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2, we first consider Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='1 for the simple sequence S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='1 (0-th density function ψ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Let the periodic sequence S = � 0, 1 3, 1 2 � + Z have three points p1 = 0, p2 = 1 3, p3 = 1 2 of radii r1 = 1 12, r2 = 0, r3 = 1 12, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 3 shows each point pi and its growing interval Li(t) = [(pi−ri)−t, (pi+ri)+t] of the length 2ri+2t for i = 1, 2, 3 in its own color: red, green, blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' By Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 each density function ψk[S](t) measures a fractional length covered by exactly k intervals within the unit cell [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' We periodicaly map the endpoints of each growing interval to the unit cell [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' For instance, the interval [− 1 12 − t, 1 12 + t] of the point p1 = 0 ≡ 1 (mod 1) maps to the red intervals [0, 1 12 +t]∪[11 12 − t, 1] shown by solid red lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The same image shows the green interval [1 3 − t, 1 3 + t] by dashed lines and the blue interval [ 5 12 − t, 7 12 + t] by dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' At the moment t = 0, since the starting inter- vals are disjoint, they cover the length l = 2( 1 12 + 0 + 1 12) = 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The non-covered part of [0, 1] has length 1 − 1 3 = 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' So the graph of ψ0(t) at t = 0 starts from the point (0, 2 3), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' At the first critical moment t = 1 24 when the green and blue intervals collide at p = 3 8, only the intervals [1 8, 7 24] ∪ [5 8, 7 8] of total length 5 12 remain uncovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Hence ψ0(t) linearly drops to the point ( 1 12, 5 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' At the next critical moment t = 1 8 when the red and green intervals collide at p = 5 24, only the interval [17 24, 19 24] of length 1 12 remain uncovered, so ψ0(t) continues to (1 8, 1 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The graph of ψ0(t) finally returns to the t-axis at the point (1 6, 0) and remains there for t ≥ 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The piecewise linear behavior of ψ0(t) can be described by specifying the corner points in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 4: � 0, 2 3 � , � 1 24, 5 12 � , �1 8, 1 12 � , �1 6, 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' ■ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 extends Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='1 to any peri- odic sequence S and implies that the 0-th density function ψ0(t) is uniquely determined by the ordered gap lengths between successive intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 (description of ψ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Let a periodic sequence S = {p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , pm} + Z consist of disjoint intervals with centers 0 ≤ p1 < · · · < pm < 1 and radii r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , rm ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Consider the total length l = 2 m � i=1 ri and gaps between successive intervals gi = (pi − ri) − (pi−1 + ri−1), where i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , m and p0 = pm − 1, r0 = rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Put the gaps in increasing order: g[1] ≤ g[2] ≤ · · · ≤ g[m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Then the 0-th density ψ0[S](t) is piecewise linear with the following (unordered) corner points: (0, 1 − l) and � g[i] 2 , 1 − l − i−1 � j=1 g[j] − (m − i + 1)g[i] � for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , m, so the last corner is �g[m] 2 , 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' If any corners are repeated, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' when g[i−1] = g[i], these corners are collapsed into one corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' ■ Proof By Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 the 0-th density function ψ0(t) measures the total length of subintervals in the unit cell [0, 1] that are not covered by any of the grow- ing intervals Li(t) = [pi−ri−t, pi+ri+t], i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' For t = 0, since all initial intervals Li(0) are disjoint, they cover the total length 2 m � i=1 ri = l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Then the graph of ψ0(t) at t = 0 starts from the point (0, 1 − l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' So ψ0(t) linearly decreases from the initial value ψ0(0) = 1 − l except for m critical values of t where one of the gap intervals [pi + ri + t, pi+1 − ri+1−t] between successive growing intervals Li(t) and Li+1(t) shrinks to a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' These critical radii t are ordered according to the gaps g[1] ≤ g[2] ≤ · · · ≤ g[m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The first critical radius is t = 1 2g[1], when a shortest gap interval of the length g[1] is covered by the growing successive intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' At this moment Springer Nature 2021 LATEX template Density functions of periodic sequences 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 3 The sequence S = � 0, 1 3 , 1 2 � + Z has the points of weights 1 12 , 0, 1 12 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The growing intervals around the red point 0 ≡ 1 (mod 1), green point 1 3 , blue point 1 2 have the same color for various radii t, see Examples 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='1, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' t = 1 2g[1], all m growing intervals Li(t) have the total length l + mg[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Then the 0-th density ψ0(t) has the first corner points (0, 1−l) and �g[1] 2 , 1 − l − mg[1] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The second critical radius is t = g[2] 2 , when all intervals Li(t) have the total length l + g[1] + (m − 1)g[2], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' the next corner point is �g[2] 2 , 1 − l − g[1] − (m − 1)g[2] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' If g[1] = g[2], then both corner points coincide, so ψ0(t) will continue from the joint corner point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The above pattern generalizes to the i-th critical radius t = 1 2g[i], when all covered intervals have the total length i−1 � j=1 g[j] (for the fully covered intervals) plus (m − i + 1)g[i] (for the still growing intervals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' For the final critical radius t = g[m] 2 , the whole unit cell [0, 1] is covered by the grown intervals because m � j=1 g[j] = 1 − l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The final corner is ( g[m] 2 , 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' □ Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='3 applies Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 to get ψ0 found for the periodic sequence S in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 1/3Springer Nature 2021 LATEX template 6 Density functions of periodic sequences Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 4 The 0-th density function ψ0(t) for the 1-period sequence S whose points 0, 1 3 , 1 2 have radii 1 12 , 0, 1 12 , respectively, see Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='3 (using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The sequence S = � 0, 1 3, 1 2 � + Z in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='1 with points p1 = 0, p2 = 1 3, p3 = 1 2 of radii r1 = 1 12, r2 = 0, r3 = 1 12, respectively, has l = 2(r1 + r2 + r3) = 1 3 and the initial gaps between successive intervals g1 = p1 − r1 − p3 − r3 = (1 − 1 12) − (1 2 + 1 12) = 1 3, g2 = p2 − r2 − p1 − r1 = (1 3 − 0) − (0 + 1 12) = 1 4, g3 = p3 − r3 − p2 − r2 = (1 2 − 1 12) − (1 3 + 0) = 1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Order the gaps: g[1] = 1 12 < g[2] = 1 4 < g[3] = 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 1 − l = 1 − 1 3 = 2 3, 1 − l − 3g[1] = 2 3 − 3 12 = 5 12, 1 − l − g[1] − 2g[2] = 2 3 − 1 12 − 2 4 = 1 12, 1 − l − g[1] − g[2] − g[3] = 2 3 − 1 12 − 1 4 − 1 3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 ψ0(t) has the corner points (0, 1 − l) = � 0, 2 3 � , �1 2g[1], 1 − l − 3g[1] � = � 1 24, 5 12 � , �1 2g[2], 1 − l − g[1] − 2g[2] � = �1 8, 1 12 � , �1 2g[3], 1 − l − g[1] − g[2] − g[3] � = �1 6, 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' See the graph of the 0-th density ψ0(t) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' ■ By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 any 0-th density function ψ0(t) is uniquely determined by the (unordered) set of gap lengths between successive intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Hence we can re-order these intervals with- out changing ψ0(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' For instance, the periodic sequence Q = {0, 1 2, 2 3} + Z with points 0, 1 2, 2 3 of weights 1 12, 1 12, 0 has the same set ordered gaps g[1] = 1 12, d[2] = 1 3, d[3] = 1 2 as the periodic sequence S = � 0, 1 3, 1 2 � + Z in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The above sequences S, Q are related by the mirror reflection t �→ 1 − t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' One can eas- ily construct many non-isometric sequences with ψ0[S](t) = ψ0[Q](t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' For any 1 ≤ i ≤ m − 3, the sequences Sm,i = {0, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , i + 2, i + 4, i + 5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , m + 2} + (m + 2)Z have the same interval lengths d[1] = · · · = d[m−2] = 1, d[m−1] = d[m] = 2 but are not related by isometry (translations and reflections in R) because the intervals of length 2 are separated by i−1 intervals of length 1 in Sm,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 4 The 1st density function ψ1 This section proves Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 explicitly describ- ing the 1st density ψ1[S](t) for any periodic sequence S of disjoint intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' To prepare the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2, Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='1 finds ψ1[S] for the sequence S from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='1 (ψ1 for S = � 0, 1 3, 1 2 � + Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The 1st density function ψ1(t) can be obtained as a sum of the three trapezoid functions ηR, ηG, ηB, each measuring the length of a region covered by a single interval of one color, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' At the initial moment t = 0, the red intervals [0, 1 12] ∪ [11 12, 1] have the total length ηR(0) = 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' These red intervals [0, 1 12 + t] ∪ [11 12 − t, 1] for t ∈ [0, 1 8] grow until they touch the green interval [ 7 24, 3 8] and have the total length ηR(1 8) = 1 6 + 2 8 = Springer Nature 2021 LATEX template Density functions of periodic sequences 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 5 The trapezoid functions ηR, ηG, ηB and the 1st density function ψ1(t) for the 1-period sequence S whose points 0, 1 3 , 1 2 have radii 1 12 , 0, 1 12 , see Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 5 12 in the second picture of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' So the graph of the red length ηR(t) linearly grows with gradient 2 from the point (0, 1 6) to the corner point (1 8, 5 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' For t ∈ [1 8, 1 6], the left red interval is shrink- ing at the same rate (due to the overlapping green interval) as the right red interval continues to grow until t = 1 6, when it touches the blue interval [1 4, 3 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Hence the graph of ηR(t) remains constant for t ∈ [1 8, 1 6] up to the corner point (1 6, 5 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' After that, the graph of ηR(t) linearly decreases (with gradient −2) until all red intervals are fully covered by the green and blue intervals at moment t = 3 8, see the 6th picture in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Hence the trapezoid function ηR has the piece- wise linear graph through the corner points (0, 1 6), (1 8, 5 12), (1 6, 5 12), (3 8, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' After that, ηR(t) = 0 remains constant for t ≥ 3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 5 shows the graphs of ηR, ηG, ηB and ψ1 = ηR + ηG + ηB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' ■ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 extends Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='1 and proves that any ψ1(t) is a sum of trapezoid functions whose corners are explicitly described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' We con- sider any index i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , m (of a point pi or a gap gi) modulo m so that m + 1 ≡ 1 (mod m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 (description of ψ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Let a periodic sequence S = {p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , pm} + Z consist of disjoint intervals with centers 0 ≤ p1 < · · · < pm < 1 and radii r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , rm ≥ 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Consider the gaps gi = (pi−ri)−(pi−1+ri−1), where i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , m and p0 = pm − 1, r0 = rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Then the 1st density ψ1(t) is the sum of m trapezoid functions ηi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , m, with the corners (0, 2ri), �gi 2 , g + 2ri � , �gi+1 2 , g + 2ri � , �gi + gi+1 2 + ri, 0 � , where g = min{gi, gi+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Hence ψ1(t) is determined by the unordered set of unordered pairs (gi, gi+1), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' ■ Proof The 1st density ψ1(t) equals the total length of subregions covered by exactly one of the intervals RSpringer Nature 2021 LATEX template 8 Density functions of periodic sequences Li(t) = [pi − ri − t, pi + ri + t], i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , m, where all intervals are taken modulo 1 within [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Hence ψ1(t) is the sum of the functions η1i, each measuring the length of the subinterval of Li(t) not covered by other intervals Lj(t), j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , m}−{i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Since the initial intervals Li(0) are disjoint, each function η1i(t) starts from the value η1i(0) = 2ri and linearly grows (with gradient 2) up to ηi(1 2g) = 2ri+g, where g = min{gi, gi+1}, when the growing interval Li(t) of the length 2ri+2t = 2ri+g touches its closest neighboring interval Li±1(t) with a shortest gap g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' If (say) gi < gi+1, then the subinterval covered only by Li(t) is shrinking on the left and is grow- ing at the same rate on the right until Li(t) touches the growing interval Li+1(t) on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' During this growth, when t is between 1 2gi and 1 2gi+1, the trapezoid function ηi(t) = g remains constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' If gi = gi+1, this horizontal line collapses to one point in the graph of ηi(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' For t ≥ max{gi, gi+1}, the subinterval covered only by Li(t) is shrinking on both sides until the neighboring intervals Li±1(t) meet at a mid-point between their initial closest endpoints pi−1 + ri−1 and pi+1 − ri+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' This meeting time is t = 1 2(pi+1 −ri+1 −pi−1 −ri−1) = 1 2(gi +2ri +gi+1), which is also illustrated by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' So the trapezoid function ηi has the corners (0, 2ri), �gi 2 , 2ri + g � , �gi+1 2 , 2ri + g � , �gi + gi+1 2 + ri, 0 � as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' □ Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='3 applies Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 to get ψ1 found for the periodic sequence S in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='3 (using Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 for ψ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The sequence S = � 0, 1 3, 1 2 � + Z in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='1 with points p1 = 0, p2 = 1 3, p3 = 1 2 of radii r1 = 1 12, r2 = 0, r3 = 1 12, respectively, has the initial gaps between successive intervals g1 = 1 3, g2 = 1 4, g3 = 1 12, see all the computations in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Case (R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' In Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 for the trapezoid func- tion ηR = η1 measuring the fractional length covered only by the red interval, we set k = 1 and i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Then ri = 1 12, gi = 1 3 and gi+1 = 1 4, so gi + gi+1 2 + ri = 1 2 �1 3 + 1 4 � + 1 12 = 3 8, g = min{gi, gi+1} = 1 4, g + 2ri = 1 4 + 2 12 = 5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Then ηR = η1 has the following corner points: (0, 2ri) = � 0, 1 6 � , �gi 2 , g + 2ri � = �1 6, 5 12 � , �gi+1 2 , g + 2ri � = �1 8, 5 12 � , �gi + gi+1 2 + ri, 0 � = �3 8, 0 � , where the two middle corners are accidentally swapped due to gi > gi+1 but they define the same trapezoid function as in the first picture of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Case (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' In Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 for the trapezoid func- tion ηG = η2 measuring the fractional length covered only by the green interval, we set k = 1 and i = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Then ri = 0, gi = 1 4 and gi+1 = 1 12, so gi + gi+1 2 + ri = 1 2 �1 4 + 1 12 � + 0 = 1 6, g = min{gi, gi+1} = 1 12, g + 2ri = 1 12 + 0 = 1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Then ηG = η2 has the following corner points exactly as shown in the second picture of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 5: (0, 2ri) = (0, 0) , �gi 2 , g + 2ri � = �1 8, 1 12 � , �gi+1 2 , g + 2ri � = � 1 24, 5 12 � , �gi + gi+1 2 + ri, 0 � = �1 6, 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Case (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' In Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 for the trapezoid func- tion ηB = η3 measuring the fractional length covered only by the blue interval, we set k = 1 and Springer Nature 2021 LATEX template Density functions of periodic sequences 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 6 The distances g, s, g′ between line intervals used in the proofs of Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2, shown here for k = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' i = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Then ri = 1 12, gi = 1 12 and gi+1 = 1 3, so gi + gi+1 2 + ri = 1 2 � 1 12 + 1 3 � + 1 12 = 7 24, g = min{gi, gi+1} = 1 12, g + 2ri = 1 12 + 2 12 = 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Then ηB = η3 has the following corner points: (0, 2ri) = � 0, 1 6 � , �gi 2 , g + 2ri � = � 1 24, 1 4 � , �gi+1 2 , g + 2ri � = �1 6, 1 4 � , �gi + gi+1 2 + ri, 0 � = � 7 24, 0 � exactly as shown in the third picture of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' ■ 5 Higher density functions ψk This section proves Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 describing the k- th density function ψk[S](t) for any k ≥ 2 and a periodic sequence S of disjoint intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' To prepare the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2, Exam- ple 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='1 computes ψ2[S] for S from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='1 (ψ2 for S = � 0, 1 3, 1 2 � + Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The density ψ2(t) can be found as the sum of the trape- zoid functions ηGB, ηBR, ηRG, each measuring the length of a double intersection, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' For the green interval [1 3 −t, 1 3 +t] and the blue interval [ 5 12 − t, 7 12 + t], the graph of the function ηGB(t) is piecewise linear and starts at the point ( 1 24, 0) because these intervals touch at t = 1 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The green-blue intersection [ 5 12 −t, 1 3 +t] grows until t = 1 6, when the resulting interval [1 4, 1 2] touches the red interval on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' At the same time, the graph of ηGB(t) is linearly growing (with gradient 2) to the corner (1 6, 1 4), see Fig, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' For t ∈ [1 6, 7 24], the green-blue intersection interval becomes shorter on the left, but grows at the same rate on the right until t = 7 24 when [1 8, 5 8] touches the red interval [5 8, 1] on the right, see the 5th picture in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' So the graph of ηGB(t) remains constant up to the point ( 7 24, 1 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' For t ∈ [ 7 24, 5 12] the green-blue intersection interval is shortening from both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' So the graph of ηGB(t) linearly decreases (with gradient −2) and returns to the t-axis at the corner ( 5 12, 0), then remains constant ηGB(t) = 0 for t ≥ 5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 7 shows all trapezoid functions for double intersections and ψ2 = ηGB + ηBR + ηRG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' ■ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 (description of ψk for k ≥ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Let a periodic sequence S = {p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , pm} + Z consist of disjoint intervals with centers 0 ≤ p1 < · · · < pm < 1 and radii r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , rm ≥ 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Consider the gaps gi = (pi − ri) − (pi−1 + ri−1) between the successive intervals of S, where i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , m and p0 = pm − 1, r0 = rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' For k ≥ 2, the density function ψk(t) equals the sum of m trapezoid functions ηk,i(t), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , m, each having the following corner points: �s 2, 0 � , �g + s 2 , g � , �s + g′ 2 , g � , �g + s + g′ 2 , 0 � , +K P i+k-1 1+1 distance distance CSpringer Nature 2021 LATEX template 10 Density functions of periodic sequences Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 7 The trapezoid functions ηGB, ηBR, ηRG and the 2nd density function ψ2(t) for the 1-period sequence S whose points 0, 1 3 , 1 2 have radii 1 12 , 0, 1 12 , see Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' where g, g′ are the minimum and maximum values in the pair {gi + 2ri, gi+k + 2ri+k−1}, and s = i+k−1 � j=i+1 gj + 2 i+k−2 � j=i+1 rj, so s = gi+1 for k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Hence ψk(t) is determined by the unordered set of the ordered tuples (g, s, g′), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' ■ Proof The k-th density function ψk(t) measures the total fractional length of k-fold intersections among m intervals Li(t) = [pi − ri − t, pi + ri + t], i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Now we visualize all such intervals Li(t) in the line R without mapping them modulo 1 to the unit cell [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Since all radii ri ≥ 0, only k successive inter- vals can contribute to k-fold intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' So a k-fold intersection of growing intervals emerges only when two intervals Li(t) and Li+k−1(t) overlap because their intersection should be also covered by all the intermediate intervals Li(t), Li+1(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , Li+k−1(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Then the density ψk(t) equals the sum of the m trapezoid functions ηk,i, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , m, each equal to the length of the k-fold intersection ∩i+k−1 j=i Lj(t) not covered by other intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Then ηk,i(t) remains 0 until the first critical moment t when 2t equals the distance between the points pi + ri and pi+k−1 − ri+k−1 in R, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 6, so 2t = i+k−1 � j=i+1 gj + 2 i+k−2 � j=i+1 rj = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Hence t = s 2 and ( s 2, 0) is the first corner point of ηk,i(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' At t = s 2, the interval of the k-fold intersection ∩i+k−1 j=i Lj(t) starts expanding on both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Hence ηk,i(t) starts increasing (with gradient 2) until the k-fold intersection touches one of the neighboring intervals Li−1(t) or Li+k(t) on the left or on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The left interval Li−1(t) touches the k-fold inter- section ∩i+k−1 j=i Lj(t) when 2t equals the distance from pi−1 + ri−1 (the right endpoint of Li−1) to pi+k−1 − ri+k−1 (the left endpoint of Li+k−1), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 6, so 2t = i+k−1 � j=i gj + 2 i+k−2 � j=i rj = gi + 2ri + s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The right interval Li+k−1(t′) touches the k-fold intersection ∩i+k−1 j=i Lj(t′) when 2t′ equals the distance from pi + ri (the right endpoint of Li) to pi+k − ri+k (the left endpoint of Li+k), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 6, so 2t′ = i+k � j=i+1 gj + 2 i+k−1 � j=i+1 rj = s + gi+k + 2ri+k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' If (say) gi + 2ri = g < g′ = gi+k + 2ri+k−1, the k-fold intersection ∩i+k−1 j=i Lj(t) first touches Li−1 at the earlier moment t before reaching Li+k(t′) at the later moment t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' At the earlier moment, ηk,i(t) equals 2(t − s 2) = gi + 2ri = g and has the corner (g + s 2 , g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' After that, the k-fold intersection is shrinking on the left and is expanding at the same rate on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' So the function ηk,i(t) = g remains constant until the k-fold intersection touches the right interval Li+k(t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' GB BR RGSpringer Nature 2021 LATEX template Density functions of periodic sequences 11 At this later moment t′ = s + gi+k 2 + ri+k−1 = g′, ηk,i(t′) still equals g and has the corner (s + g′ 2 , g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' If gi + 2ri = g′ > g = gi+k + 2ri+k−1, the grow- ing intervals Li−1(t) and Li+k−1(t) touch the k-fold intersection ∩i+k−1 j=i Lj(t) in the opposite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' How- ever, the above arguments lead to the same corners (g + s 2 , g) and (s + g′ 2 , g) of ηk,i(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' If g = g′, the two corners collapse to one corner in the graph of ηk,i(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The k-fold intersection ∩i+k−1 j=i Lj(t) becomes fully covered when the intervals Li−1(t), Li+k(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' At this moment, 2t equals the distance from pi−1 + ri−1 (the right endpoint of Li−1) to pi+k − ri+k (the left end- point of Li+k), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 6, so 2t = i+k � j=i gj +2 i+k−1 � j=i rj = gi + 2ri + s + gi+k + 2ri+k−1 = g + s + g′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The graph of ηk,i(t) has the final corner �g + s + g′ 2 , 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' □ Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='3 applies Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 to get ψ2 found for the periodic sequence S in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='3 (using Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 for ψ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The sequence S = � 0, 1 3, 1 2 � + Z in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='1 with points p1 = 0, p2 = 1 3, p3 = 1 2 of radii r1 = 1 12, r2 = 0, r3 = 1 12, respectively, has the initial gaps g1 = 1 3, g2 = 1 4, g3 = 1 12, see Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' In Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2, the 2nd density function ψ2[S](t) is expressed as a sum of the trapezoid functions computed via their corners below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Case (GB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' For the function ηGB measuring the double intersections of the green and blue intervals centered at p2 = pi and p3 = pi+k−1, we set k = 2 and i = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Then we have the radii ri = 0 and ri+1 = 1 12, the gaps gi = 1 4, gi+1 = 1 12, gi+2 = 1 3, and the sum s = gi+1 = 1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The pair {gi + 2ri, gi+2 + 2ri+1} = �1 4 + 0, 1 3 + 2 12 � has the minimum value g = 1 4 and maximum value g′ = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Then η2,2[S](t) = ηGB has the following corners as expected in the top picture of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 7: �s 2, 0 � = � 1 24, 0 � , �g + s 2 , g � = �1 2 �1 4 + 1 12 � , 1 4 � = �1 6, 1 4 � , �s + g′ 2 , g � = �1 2 � 1 12 + 1 2 � , 1 4 � = � 7 24, 1 4 � , �g + s + g′ 2 , 0 � = �1 2(1 4 + 1 12 + 1 2), 0 � = � 5 12, 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Case (BR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' For the trapezoid function ηBR mea- suring the double intersections of the blue and red intervals centered at p3 = pi and p1 = pi+k−1, we set k = 2 and i = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Then we have the radii ri = 1 12 = ri+1, the gaps gi = 1 12, gi+1 = 1 3, gi+2 = 1 4, and s = gi+1 = 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The pair {gi + 2ri, gi+2 + 2ri+1} = � 1 12 + 2 12, 1 4 + 2 12 � has the minimum g = 1 4 and maximum g′ = 5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Then η2,3[S](t) = ηBR has the following corners as expected in the second picture of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 7: �s 2, 0 � = �1 6, 0 � , �g + s 2 , g � = �1 2 �1 4 + 1 3 � , 1 4 � = � 7 24, 1 4 � , �s + g′ 2 , g � = �1 2 �1 3 + 5 12 � , 1 4 � = �3 8, 1 4 � , �g + s + g′ 2 , 0 � = �1 2(1 4 + 1 3 + 5 12), 0 � = �1 2, 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Case (RG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' For the trapezoid function ηRG mea- suring the double intersections of the red and green intervals centered at p1 = pi and p2 = pi+k−1, we set k = 2 and i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Then we have the radii ri = 1 12 and ri+1 = 0, the gaps gi = 1 3, Springer Nature 2021 LATEX template 12 Density functions of periodic sequences gi+1 = 1 4, gi+2 = 1 12, and s = gi+1 = 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The pair {gi + 2ri, gi+2 + 2ri+1} = �1 3 + 2 12, 1 12 + 0 � has the minimum g = 1 12 and maximum g′ = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Then η2,1[S](t) = ηRG has the following corners: �s 2, 0 � = �1 8, 0 � , �g + s 2 , g � = �1 2 � 1 12 + 1 4 � , 1 12 � = �1 6, 1 12 � , �s + g′ 2 , g � = �1 2 �1 4 + 1 2 � , 1 12 � = �3 8, 1 12 � , �g + s + g′ 2 , 0 � = �1 2( 1 12 + 1 4 + 1 2), 0 � = � 5 12, 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' as expected in the third picture of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' ■ 6 Properties of new densities This section proves the periodicity of the sequence ψk with respect to the index k ≥ 0 in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2, which was a bit unexpected from original Defini- tion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' We start with the simpler example for the familiar 3-point sequence in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='1 (periodicity of ψk in the index k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Let the periodic sequence S = � 0, 1 3, 1 2 � +Z have three points p1 = 0, p2 = 1 3, p3 = 1 2 of radii r1 = 1 12, r2 = 0, r3 = 1 12, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The ini- tial intervals L1(0) = [− 1 12, 1 12], L2(0) = [ 1 3, 1 3], L3(0) = [ 5 12, 7 12] have the 0-fold intersection mea- sured by ψ0(0) = 2 3 and the 1-fold intersection measured by ψ1(0) = 1 3, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' By the time t = 1 2 the initial intervals will grow to L1( 1 2) = [− 7 12, 7 12], L2( 1 2) = [− 1 6, 5 6], L3( 1 2) = [− 1 12, 13 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The grown intervals at the radius t = 1 2 have the 3-fold intersection [− 1 12, 7 12] of the length ψ3( 1 2) = 2 3, which coincides with ψ0(0) = 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' With the extra interval L4( 1 2) = [ 5 12, 19 12] cen- tered at p4 = 1, the 4-fold intersection is L1 ∩ L2 ∩ L3 ∩ L4 = [ 5 12, 7 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' With the extra inter- val L5( 1 2) = [ 5 6, 11 6 ] centered at p5 = 4 3, the 4-fold intersection L2 ∩ L3 ∩ L4 ∩ L5 is the single point 5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' With the extra interval L6( 1 2) = [ 11 12, 13 12] centered at p6 = 3 2, the 4-fold intersection is L3∩L4∩L5∩L6 = [ 11 12, 13 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Hence the total length of the 4-fold intersection at t = 1 2 is ψ4( 1 2) = 1 3, which coincides with ψ1(0) = 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' For the larger t = 1, the six grown intervals L1(1) = � −13 12, 13 12 � , L2(1) = � −2 3, 4 3 � , L3(1) = � − 7 12, 19 12 � , L4(1) = � − 1 12, 25 12 � , L5(1) = �1 3, 7 3 � , L6(1) = � 5 12, 31 12 � have the 6-fold intersection � 5 12, 13 12 � of length ψ6(1) = 2 3 coinciding with ψ0(0) = ψ3( 1 2) = 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' ■ Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 proves that the coincidences in Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='1 are not accidental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The periodicity of ψk with respect to k is illustrated by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 (periodicity of ψk in the index k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The density functions ψk[S] of a periodic sequence S = {p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , pm} + Z consist of disjoint intervals with centers 0 ≤ p1 < · · · < pm < 1 and radii r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , rm ≥ 0, respectively, satisfy the periodicity ψk+m(t + 1 2) = ψk(t) for any k ≥ 0 and t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' ■ Proof Since the initial intervals are disjoint, for k ≥ 0, any (k +m)-fold intersection involves k +m successive intervals Li(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , Li+k+m−1(t) centered around the points of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Then we can find an interval [x, x + 1] covering exactly m of these initial intervals of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' By collapsing [x, x+1] to the point x, any (k+m)- fold intersection of k + m intervals grown by a radius r ≥ 1 2 becomes a k-fold intersection of k intervals Springer Nature 2021 LATEX template Density functions of periodic sequences 13 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 8 The densities ψk, k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , 9 for the 1-period sequence S whose points 0, 1 3 , 1 2 have radii 1 12 , 0, 1 12 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The densities ψ0, ψ1, ψ2 are described in Examples 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='1, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='1 and determine all other densities by periodicity in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' grown by t = r− 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Both k-fold and (k+m)-fold inter- sections within any unit cell have the same fractional length, so ψk+m(t + 1 2) = ψk(t) for any t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' □ The symmetry ψm−k( 1 2 − t) = ψk(t) for k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , [ m 2 ], and t ∈ [0, 1 2] from [3, Theorem 8] no longer holds for points with different radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' For example, ψ1(t) ̸= ψ2( 1 2 − t) for the periodic sequence S = � 0, 1 3, 1 2 � + Z, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 5, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' If all points have the same radius r, [3, Theorem 8] implies the symmetry after replacing t by t + 2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The main results of [3] implied that all den- sity functions cannot distinguish the non-isometric sequences S15 = {0, 1, 3, 4, 5, 7, 9, 10, 12} + 15Z and Q15 = {0, 1, 3, 4, 6, 8, 9, 12, 14}+15Z of points with zero radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='3 shows that the den- sities for sequences with non-zero radii are strictly stronger and distinguish the sequences S15 ̸∼= Q15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='3 (ψk for S15, Q15 with neighbor radii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' For any point p in a periodic sequence S ⊂ R, define its neighbor radius as the half-distance to a closest neighbor of p within the sequence S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' This choice of radii respects the isometry in the sense that periodic sequences S, Q with zero-sized radii are isometric if and only if S, Q with neighbor radii are isometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 9 shows that the densi- ties ψk for k ≥ 2 distinguish the non-isometric sequences S15 and Q15 scaled down by factor 15 to the unit cell [0, 1], see Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' ■ Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='4 (computation of ψk(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Let S, Q ⊂ R be periodic sequences with at most m motif points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' For k ≥ 1, one can draw the graph of the k-th density function ψk[S] in time O(m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' One can check in time O(m3) if Ψ[S] = Ψ[Q].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' ■ Proof To draw the graph of ψk[S] or evaluate the k- th density function ψk[S](t) at any radius t, we first use the periodicity from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 to reduce k to the range 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' In time O(m log m) we put the points from a unit cell U (scaled to [0, 1] for conve- nience) in the increasing (cyclic) order p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' In time O(m) we compute the gaps gi = (pi−ri)−(pi−1+ ri−1) between successive intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' For k = 0, we put the gaps in the increasing order g[1] ≤ · · · ≤ g[m] in time O(m log m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 in time O(m2), we write down the O(m) corner points whose horizontal coordinates are the critical radii where ψ0(t) can change its gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' We evaluate ψ0 at every critical radius t by sum- ming up the values of m trapezoid functions at t, which needs O(m2) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' It remains to plot the points at all 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='8 psi 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='6 psi_1 psi_2 psi_3 psi_4 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='4 - p psi_5 psi_6 psi_7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 psi_8 psi_9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='5 TSpringer Nature 2021 LATEX template 14 Density functions of periodic sequences Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 9 The densities ψk, k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , 10, distinguish (already for k ≥ 2) the sequences (scaled down by period 15) S15 = {0, 1, 3, 4, 5, 7, 9, 10, 12} + 15Z (top) and Q15 = {0, 1, 3, 4, 6, 8, 9, 12, 14} + 15Z (bottom), where the radius ri of any point is the half-distance to its closest neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' These sequences with zero radii have identical ψk for all k, see [3, Example 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' O(m) critical radii t and connect the successive points by straight lines, so the total time is O(m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' For any larger fixed index k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , m, in time O(m2) we write down all O(m) corner points from Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2, which leads to the graph of ψk(t) similarly to the above argument for k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' To decide if the infinite sequences of density func- tions coincide: Ψ[S] = Ψ[Q], by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 it suffices to check only if O(m) density functions coincide: ψk[S](t) = ψk[Q](t) for k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , [ m 2 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' To check if two piecewise linear functions coincide, it remains to compare their values at all O(m) critical radii t from the corner points in Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Since these values were found in time O(m2) above, the total time for k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' , [ m 2 ] is O(m3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' □ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='75 psi_0 psi_1 psi_2 psi_3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='50 K psi_4 psi psi_5 psi_6 psi_7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='25 psi_8 psi_ 9 psi_10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='6 T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='75 psi_ 0 psi_1 psi_2 psi_3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='50 K psi_4 psi_5 psi_6 psi_7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='25 psi_8 psi_ 9 psi_10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='6 TSpringer Nature 2021 LATEX template Density functions of periodic sequences 15 All previous examples show densities with a single local maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' However, the new R code [5] helped us discover the opposite examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 10 For the periodic sequence S = � 0, 1 8 , 1 4 , 3 4 � + Z whose all points have radii 0, the 2nd density ψ2[S](t) has the local minimum at t = 1 4 between two local maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='5 (densities with multiple maxima).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 10 shows a simple 4-point sequence S whose 2nd density ψ2[S] has two local maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 11 and 12 show more complicated sequences whose density functions have more than two maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' ■ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 11 For the sequence S = � 0, 1 81 , 1 27 , 1 9 , 1 3 � +Z whose all points have radii 0, ψ2[S] equal to the sum of the shown five trapezoid functions has three maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 12 For the sequence S = � 0, 1 64 , 1 16 , 1 8 , 1 4 , 3 4 � + Z whose all points have radii 0, ψ3[S] has 5 local maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' 7 Conclusions and future work In comparison with the past work [3], the key contributions of this paper are the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 extends density functions ψk to any periodic sets of points with radii ri ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 explicitly describe all ψk for any periodic sequence S of points with radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The descriptions of ψk allowed us to justify the periodicity of ψk in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 and a quadratic algorithm computing any ψk in Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' The code [5] helped us distinguish S15 ̸∼= Q15 in Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='3 and find sequences whose densities have multiple local maxima in Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Here are the open problems for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Verify if density functions ψk[S](t) for small values of k distinguish all non-isometric periodic point sets S ⊂ Rn at least with radii 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Characterize the periodic sequences S ⊂ R whose all density functions ψk for k ≥ 1 have a unique local maximum, not as in Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' Similar to Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2, analytically describe the density function ψk[S] for periodic point sets S ⊂ Rn in higher dimensions n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' This research was supported by the grants of the UK Engineering Physical Sciences Research 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='75 psi_0 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='50 psi_1 Isd psi_2 psi_3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='5 t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='08 eta_2_1 eta_2_2 eta_2_3 .' metadata={'source': 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+page_content=' We thank all reviewers for their time and helpful advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' References [1] Anosova, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=', Kurlin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=': Introduction to periodic geometry and topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content=' arxiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} +page_content='02749 (2021) [2] Anosova, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE4T4oBgHgl3EQfiw29/content/2301.05137v1.pdf'} 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