Removing some invalid math tests on manual inspection

bench_data/arxiv_math.jsonl

@@ -203,13 +203,6 @@
 {"pdf": "arxiv_math/2503.09581_pg21.pdf", "url": "https://arxiv.org/pdf/2503.09581", "page": 1, "id": "2503.09581_pg21_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\phi_h^{n+1}, \\mu_h^{n+1}) \\in S^h \\times S^h"}
 {"pdf": "arxiv_math/2503.09581_pg21.pdf", "url": "https://arxiv.org/pdf/2503.09581", "page": 1, "id": "2503.09581_pg21_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "h_f = \\frac{L_d}{N_f}"}
 {"pdf": "arxiv_math/2503.09581_pg21.pdf", "url": "https://arxiv.org/pdf/2503.09581", "page": 1, "id": "2503.09581_pg21_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "d_0 : \\overline\\Omega \\to \\mathbb R"}
-{"pdf": "arxiv_math/2503.05742_pg2.pdf", "url": "https://arxiv.org/pdf/2503.05742", "page": 1, "id": "2503.05742_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\texttt{ Solve} \\rightarrow \\texttt{ Estimate } \\rightarrow \\texttt{ Mark } \\rightarrow \\texttt{ Refine}."}
-{"pdf": "arxiv_math/2503.05742_pg2.pdf", "url": "https://arxiv.org/pdf/2503.05742", "page": 1, "id": "2503.05742_pg2_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": ", this means that there is no prior knowledge about the ideal control. Furthermore, the coefficient"}
-{"pdf": "arxiv_math/2503.05742_pg2.pdf", "url": "https://arxiv.org/pdf/2503.05742", "page": 1, "id": "2503.05742_pg2_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\texttt{ Solve} \\rightarrow \\texttt{ Estimate } \\rightarrow \\texttt{ Mark } \\rightarrow \\texttt{ Refine}."}
-{"pdf": "arxiv_math/2503.05742_pg2.pdf", "url": "https://arxiv.org/pdf/2503.05742", "page": 1, "id": "2503.05742_pg2_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": ". The desired control function"}
-{"pdf": "arxiv_math/2503.07895_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07895", "page": 1, "id": "2503.07895_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "z\\dfrac{\\partial}{\\partial z}"}
-{"pdf": "arxiv_math/2503.07895_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07895", "page": 1, "id": "2503.07895_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{D}=g(z)\\dfrac{\\partial}{\\partial z}"}
-{"pdf": "arxiv_math/2503.07895_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07895", "page": 1, "id": "2503.07895_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "f(z)=1/\\left(1-e^z\\right)"}
 {"pdf": "arxiv_math/2503.09424_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09424", "page": 1, "id": "2503.09424_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "i\\in \\{2,\\ldots,n-1\\}"}
 {"pdf": "arxiv_math/2503.09432_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09432", "page": 1, "id": "2503.09432_pg6_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\chi _{2k+1}(f)^2\\leq \\lambda _k(f)\\lambda _{k+1}(f)"}
 {"pdf": "arxiv_math/2503.09432_pg6.pdf", "url": "https://arxiv.org/pdf/2503.09432", "page": 1, "id": "2503.09432_pg6_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "f\\times f:X\\times X\\rightarrow X\\times X"}
@@ -317,15 +310,6 @@
 {"pdf": "arxiv_math/2503.07335_pg33.pdf", "url": "https://arxiv.org/pdf/2503.07335", "page": 1, "id": "2503.07335_pg33_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "Z = \\sum_{i=0}^{i_{\\max}-1}\\sum_{h=1}^{\\omega}Z_{h}^{(i)},"}
 {"pdf": "arxiv_math/2503.07335_pg33.pdf", "url": "https://arxiv.org/pdf/2503.07335", "page": 1, "id": "2503.07335_pg33_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\chi(\\mathcal{G}_{n,d}) = (1+o(1))\\frac{n}{\\alpha(\\mathcal{G}_{n,d})} = (1+o(1))\\frac{d}{2\\log d}"}
 {"pdf": "arxiv_math/2503.07741_pg8.pdf", "url": "https://arxiv.org/pdf/2503.07741", "page": 1, "id": "2503.07741_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\langle \\rho(x_0,t)\\rangle"}
-{"pdf": "arxiv_math/2503.06851_pg17.pdf", "url": "https://arxiv.org/pdf/2503.06851", "page": 1, "id": "2503.06851_pg17_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{P}(Y^{(\\nu)}\\in A | X^{(\\nu)} = x) = p(A|x), \\quad (x\\in \\mathcal{X}, A\\subset L^1([0, L)^d, \\mathcal{X}))."}
-{"pdf": "arxiv_math/2503.06851_pg17.pdf", "url": "https://arxiv.org/pdf/2503.06851", "page": 1, "id": "2503.06851_pg17_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "respectively and satisfy"}
-{"pdf": "arxiv_math/2503.06851_pg17.pdf", "url": "https://arxiv.org/pdf/2503.06851", "page": 1, "id": "2503.06851_pg17_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int_{\\mathscr{M}^T(\\mathcal{X})} \\varepsilon_{\\nu}\\, d\\lambda(\\nu) = \\mathbb{E}\\left(\\frac{1}{L^d} \\int_{[0, L)^d} \\mathbf{d}\\left(T^t X, Y_t\\right) d\\mathbf{m}(t)\\right) +\\delta < \\varepsilon."}
-{"pdf": "arxiv_math/2503.06851_pg17.pdf", "url": "https://arxiv.org/pdf/2503.06851", "page": 1, "id": "2503.06851_pg17_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "I(X; Y) \\geq \\int_{\\mathscr{M}^T(\\mathcal{X})} I(\\nu, p)\\, d\\lambda(\\nu) = \\int_{\\mathscr{M}^T(\\mathcal{X})} I\\left(X^{(\\nu)}; Y^{(\\nu)}\\right) d\\lambda(\\nu)."}
-{"pdf": "arxiv_math/2503.06851_pg17.pdf", "url": "https://arxiv.org/pdf/2503.06851", "page": 1, "id": "2503.06851_pg17_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "we take random variables"}
-{"pdf": "arxiv_math/2503.06851_pg17.pdf", "url": "https://arxiv.org/pdf/2503.06851", "page": 1, "id": "2503.06851_pg17_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "be the regular conditional distribution of"}
-{"pdf": "arxiv_math/2503.06851_pg17.pdf", "url": "https://arxiv.org/pdf/2503.06851", "page": 1, "id": "2503.06851_pg17_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varepsilon_{\\nu} = \\mathbb{E}\\left(\\frac{1}{L^d} \\int_{[0, L)^d} \\mathbf{d}\\left(T^t X^{(\\nu)}, Y^{(\\nu)}_t\\right)d\\mathbf{m}(t)\\right) +\\delta."}
-{"pdf": "arxiv_math/2503.06851_pg17.pdf", "url": "https://arxiv.org/pdf/2503.06851", "page": 1, "id": "2503.06851_pg17_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "p(A|x) = \\mathbb{P}(Y\\in A|X=x), \\quad (x\\in \\mathcal{X}, A\\subset L^1([0, L)^d, \\mathcal{X}))."}
-{"pdf": "arxiv_math/2503.06851_pg17.pdf", "url": "https://arxiv.org/pdf/2503.06851", "page": 1, "id": "2503.06851_pg17_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": ". Suppose that random variables"}
 {"pdf": "arxiv_math/2503.05976_pg3.pdf", "url": "https://arxiv.org/pdf/2503.05976", "page": 1, "id": "2503.05976_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "r \\in \\set{0} \\cup \\N \\cup \\set{\\infty}"}
 {"pdf": "arxiv_math/2503.05976_pg3.pdf", "url": "https://arxiv.org/pdf/2503.05976", "page": 1, "id": "2503.05976_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "z = (z_1, \\dots, z_n)"}
 {"pdf": "arxiv_math/2503.05976_pg3.pdf", "url": "https://arxiv.org/pdf/2503.05976", "page": 1, "id": "2503.05976_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\zeta = (\\zeta_1, \\dots, \\zeta_n)"}
@@ -406,8 +390,6 @@
 {"pdf": "arxiv_math/2503.09208_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09208", "page": 1, "id": "2503.09208_pg18_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "L_I: L^{\\infty}(0,T) \\to \\mathbb{R}"}
 {"pdf": "arxiv_math/2503.09208_pg18.pdf", "url": "https://arxiv.org/pdf/2503.09208", "page": 1, "id": "2503.09208_pg18_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "(p^\\varepsilon, d^\\varepsilon) = G(I^\\varepsilon)"}
 {"pdf": "arxiv_math/2503.06055_pg5.pdf", "url": "https://arxiv.org/pdf/2503.06055", "page": 1, "id": "2503.06055_pg5_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "T_\\tau \\, v \\preceq T_\\sigma \\, v"}
-{"pdf": "arxiv_math/2503.04122_pg13.pdf", "url": "https://arxiv.org/pdf/2503.04122", "page": 1, "id": "2503.04122_pg13_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "g\\in\\{2,3,\\cdots 10\\}"}
-{"pdf": "arxiv_math/2503.04122_pg13.pdf", "url": "https://arxiv.org/pdf/2503.04122", "page": 1, "id": "2503.04122_pg13_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": ". A straightforward computation leads to this initial segment and the corresponding sequence of differences:"}
 {"pdf": "arxiv_math/2503.05685_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05685", "page": 1, "id": "2503.05685_pg4_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi_m(j(E_1), j(E_2))"}
 {"pdf": "arxiv_math/2503.05685_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05685", "page": 1, "id": "2503.05685_pg4_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "m \\gg_\\epsilon( \\# S)^{5+\\epsilon}"}
 {"pdf": "arxiv_math/2503.04567_pg38.pdf", "url": "https://arxiv.org/pdf/2503.04567", "page": 1, "id": "2503.04567_pg38_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\tau_{ij} \\ne \\tau_{ji}"}
@@ -591,16 +573,6 @@
 {"pdf": "arxiv_math/2503.08634_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08634", "page": 1, "id": "2503.08634_pg6_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{E}[h(\\hat{x}_\\eta)]-h^*\\leq \\texttt{Err}_{\\eta}+\\eta M"}
 {"pdf": "arxiv_math/2503.08634_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08634", "page": 1, "id": "2503.08634_pg6_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathbb{E}[h(\\hat{x}_\\eta) + \\eta f(\\hat{x}_\\eta)] -f^*_\\eta \\leq \\texttt{Err}_{\\eta}."}
 {"pdf": "arxiv_math/2503.08634_pg6.pdf", "url": "https://arxiv.org/pdf/2503.08634", "page": 1, "id": "2503.08634_pg6_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "f^*- \\mathbb{E}[ f(\\hat{x}_\\eta)]<M"}
-{"pdf": "arxiv_math/2503.04487_pg20.pdf", "url": "https://arxiv.org/pdf/2503.04487", "page": 1, "id": "2503.04487_pg20_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "have the same number of children, i.e.,"}
-{"pdf": "arxiv_math/2503.04487_pg20.pdf", "url": "https://arxiv.org/pdf/2503.04487", "page": 1, "id": "2503.04487_pg20_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "(often referred to as the Thue--Morse substitution). We note that both"}
-{"pdf": "arxiv_math/2503.04487_pg20.pdf", "url": "https://arxiv.org/pdf/2503.04487", "page": 1, "id": "2503.04487_pg20_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "is the alphabet of the substitution,"}
-{"pdf": "arxiv_math/2503.04487_pg20.pdf", "url": "https://arxiv.org/pdf/2503.04487", "page": 1, "id": "2503.04487_pg20_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "(Case 1) or a chain of letters that all have images of length"}
-{"pdf": "arxiv_math/2503.04487_pg20.pdf", "url": "https://arxiv.org/pdf/2503.04487", "page": 1, "id": "2503.04487_pg20_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "is a non-negative integer,"}
-{"pdf": "arxiv_math/2503.04487_pg20.pdf", "url": "https://arxiv.org/pdf/2503.04487", "page": 1, "id": "2503.04487_pg20_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": ". The corresponding Dumont--Thomas numeration system (over"}
-{"pdf": "arxiv_math/2503.04487_pg20.pdf", "url": "https://arxiv.org/pdf/2503.04487", "page": 1, "id": "2503.04487_pg20_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "and the choice of final letter. If we let"}
-{"pdf": "arxiv_math/2503.04487_pg20.pdf", "url": "https://arxiv.org/pdf/2503.04487", "page": 1, "id": "2503.04487_pg20_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "(Case 2). In all other cases, no iterated image of"}
-{"pdf": "arxiv_math/2503.04487_pg20.pdf", "url": "https://arxiv.org/pdf/2503.04487", "page": 1, "id": "2503.04487_pg20_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": ". Applying our result gives the substitution"}
-{"pdf": "arxiv_math/2503.04487_pg20.pdf", "url": "https://arxiv.org/pdf/2503.04487", "page": 1, "id": "2503.04487_pg20_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "). Thus the claim is proved. Now that we have proven that"}
 {"pdf": "arxiv_math/2503.04881_pg25.pdf", "url": "https://arxiv.org/pdf/2503.04881", "page": 1, "id": "2503.04881_pg25_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "U_{\\rm EF}=U_{\\rm EF,0}"}
 {"pdf": "arxiv_math/2503.04881_pg25.pdf", "url": "https://arxiv.org/pdf/2503.04881", "page": 1, "id": "2503.04881_pg25_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "V_{\\rm EF}=V_{\\rm EF,0}"}
 {"pdf": "arxiv_math/2503.04612_pg8.pdf", "url": "https://arxiv.org/pdf/2503.04612", "page": 1, "id": "2503.04612_pg8_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\psi(\\omega_0) \\coloneqq \\log b(\\omega_0)"}
@@ -1450,7 +1422,6 @@
 {"pdf": "arxiv_math/2503.04590_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04590", "page": 1, "id": "2503.04590_pg7_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "T(u)\\equiv f(u)- P_{\\Phi(u)}(f(u)-\\alpha u)"}
 {"pdf": "arxiv_math/2503.04590_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04590", "page": 1, "id": "2503.04590_pg7_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\dot{V}(u) \\leq -K.\\big(V(u)\\big)^p \\quad \\forall u\\in U\\setminus \\{u^*\\},"}
 {"pdf": "arxiv_math/2503.04590_pg7.pdf", "url": "https://arxiv.org/pdf/2503.04590", "page": 1, "id": "2503.04590_pg7_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "f(u)- P_{\\Phi(u)}(f(u)-\\alpha u)=0"}
-{"pdf": "arxiv_math/2503.05643_pg19.pdf", "url": "https://arxiv.org/pdf/2503.05643", "page": 1, "id": "2503.05643_pg19_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "contains at most two 1s in each row and column. If a row (or column) contains two 1s, each of them appears in different adjacent blocks. All other entries in the matrix are zeros. In the corresponding graphical invariants, the symbol"}
 {"pdf": "arxiv_math/2503.07166_pg4.pdf", "url": "https://arxiv.org/pdf/2503.07166", "page": 1, "id": "2503.07166_pg4_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "(e^+)^2 - (e^-)^2 = e^+ + e^-"}
 {"pdf": "arxiv_math/2503.04415_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04415", "page": 1, "id": "2503.04415_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma\\in[0,\\frac{1-\\gamma}{2})"}
 {"pdf": "arxiv_math/2503.04415_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04415", "page": 1, "id": "2503.04415_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "N = \\left\\lfloor \\frac{1}{\\gamma} \\right\\rfloor"}
@@ -1474,10 +1445,6 @@
 {"pdf": "arxiv_math/2503.06555_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06555", "page": 1, "id": "2503.06555_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\cdot,\\cdot)_{\\omega}"}
 {"pdf": "arxiv_math/2503.06555_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06555", "page": 1, "id": "2503.06555_pg3_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "W^{m,2}(\\omega)=H^{m}(\\omega)"}
 {"pdf": "arxiv_math/2503.06555_pg3.pdf", "url": "https://arxiv.org/pdf/2503.06555", "page": 1, "id": "2503.06555_pg3_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\|\\cdot\\|_{0,\\infty,\\omega}"}
-{"pdf": "arxiv_math/2503.06889_pg12.pdf", "url": "https://arxiv.org/pdf/2503.06889", "page": 1, "id": "2503.06889_pg12_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "0.3, 0.5, 0.5, 0.5, 0.5"}
-{"pdf": "arxiv_math/2503.06889_pg12.pdf", "url": "https://arxiv.org/pdf/2503.06889", "page": 1, "id": "2503.06889_pg12_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "0.3, 0.3, 0.3, 0.5, 0.5"}
-{"pdf": "arxiv_math/2503.06889_pg12.pdf", "url": "https://arxiv.org/pdf/2503.06889", "page": 1, "id": "2503.06889_pg12_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "0.5, 0.5, 0.5, 0.5, 0.5"}
-{"pdf": "arxiv_math/2503.06889_pg12.pdf", "url": "https://arxiv.org/pdf/2503.06889", "page": 1, "id": "2503.06889_pg12_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "0.3, 0.3, 0.5, 0.5, 0.5"}
 {"pdf": "arxiv_math/2503.06102_pg16.pdf", "url": "https://arxiv.org/pdf/2503.06102", "page": 1, "id": "2503.06102_pg16_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\Phi_K(D_{2k}) = \\Phi_{T_{2,2h+1}}(D_{2k}) = t_{a_{2k+1}(D_{2k})}"}
 {"pdf": "arxiv_math/2503.08261_pg14.pdf", "url": "https://arxiv.org/pdf/2503.08261", "page": 1, "id": "2503.08261_pg14_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\psi(t)= \\begin{cases} 1 &\\text{ if } t\\geq 1,\\\\ 0&\\text{ if }t\\leq 0. \\end{cases}"}
 {"pdf": "arxiv_math/2503.08261_pg14.pdf", "url": "https://arxiv.org/pdf/2503.08261", "page": 1, "id": "2503.08261_pg14_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\underline{u}\\leq \\overline{u}"}
@@ -1684,16 +1651,6 @@
 {"pdf": "arxiv_math/2503.06710_pg5.pdf", "url": "https://arxiv.org/pdf/2503.06710", "page": 1, "id": "2503.06710_pg5_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\varphi=\\varphi^\\xi=\\left[\\varphi^{\\xi,+},\\varphi^{\\xi,-}\\right]^\\top"}
 {"pdf": "arxiv_math/2503.06710_pg5.pdf", "url": "https://arxiv.org/pdf/2503.06710", "page": 1, "id": "2503.06710_pg5_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "z\\in\\Sigma_{\\lambda_1,\\lambda_2,\\Phi}."}
 {"pdf": "arxiv_math/2503.06710_pg5.pdf", "url": "https://arxiv.org/pdf/2503.06710", "page": 1, "id": "2503.06710_pg5_math_011", "type": "math", "max_diffs": 0, "checked": null, "math": "|\\sin2\\pi(\\theta+n\\Phi)|<\\exp(-|n|^{\\frac{1}{2\\tau}})"}
-{"pdf": "arxiv_math/2503.04725_pg19.pdf", "url": "https://arxiv.org/pdf/2503.04725", "page": 1, "id": "2503.04725_pg19_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": ", because it matters less if the sequence starts at the beginning of a sequence or not if there are many"}
-{"pdf": "arxiv_math/2503.04725_pg19.pdf", "url": "https://arxiv.org/pdf/2503.04725", "page": 1, "id": "2503.04725_pg19_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "being a sequence of tokens. However, as discussed in the main text, the"}
-{"pdf": "arxiv_math/2503.04725_pg19.pdf", "url": "https://arxiv.org/pdf/2503.04725", "page": 1, "id": "2503.04725_pg19_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "-tuple of tokens with"}
-{"pdf": "arxiv_math/2503.04725_pg19.pdf", "url": "https://arxiv.org/pdf/2503.04725", "page": 1, "id": "2503.04725_pg19_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "prior tokens to conditional on. Therefore, we focusing on reducing the bias for small"}
-{"pdf": "arxiv_math/2503.04725_pg19.pdf", "url": "https://arxiv.org/pdf/2503.04725", "page": 1, "id": "2503.04725_pg19_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": ". Then, the entropy of the distribution can be estimated naively as \\begin{equation} \\hat{H}^{\\text{na\\\"ive}}(Y_{1:i}) = - \\sum_{y_{1:i}} \\frac{n_{y_{1:i}}}{N}\\log \\frac{n_{y_{1:i}}}{N} = \\log N - \\frac{1}{N}\\sum_{y_{1:i}} n_{y_{1:i}} \\log n_{y_{1:i}}, \\end{equation} where the summation runs over all possible combination of tokens"}
-{"pdf": "arxiv_math/2503.04725_pg19.pdf", "url": "https://arxiv.org/pdf/2503.04725", "page": 1, "id": "2503.04725_pg19_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "starts at the beginning of a sentence, but LLMs model distributions conditioned on BOS token. To mitigate this issue, we use"}
-{"pdf": "arxiv_math/2503.04725_pg19.pdf", "url": "https://arxiv.org/pdf/2503.04725", "page": 1, "id": "2503.04725_pg19_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "term suffers from an additional bias---we cannot guarantee that"}
-{"pdf": "arxiv_math/2503.04725_pg19.pdf", "url": "https://arxiv.org/pdf/2503.04725", "page": 1, "id": "2503.04725_pg19_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat{H}^{\\text{na\\\"ive}}(Y_{1:i}) = - \\sum_{y_{1:i}} \\frac{n_{y_{1:i}}}{N}\\log \\frac{n_{y_{1:i}}}{N} = \\log N - \\frac{1}{N}\\sum_{y_{1:i}} n_{y_{1:i}} \\log n_{y_{1:i}},"}
-{"pdf": "arxiv_math/2503.04725_pg19.pdf", "url": "https://arxiv.org/pdf/2503.04725", "page": 1, "id": "2503.04725_pg19_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "is small, we can iterate over the dataset and construct a histogram for the"}
-{"pdf": "arxiv_math/2503.04725_pg19.pdf", "url": "https://arxiv.org/pdf/2503.04725", "page": 1, "id": "2503.04725_pg19_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "and the total number of samples with"}
 {"pdf": "arxiv_math/2503.07281_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07281", "page": 1, "id": "2503.07281_pg12_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "% H^1_{\\Theta}=K^1_{\\Theta}\\oplus \\Theta H^1"}
 {"pdf": "arxiv_math/2503.07281_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07281", "page": 1, "id": "2503.07281_pg12_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "H^1_\\Theta/(K^1_{\\Theta}\\oplus \\Theta H^1 )"}
 {"pdf": "arxiv_math/2503.07281_pg12.pdf", "url": "https://arxiv.org/pdf/2503.07281", "page": 1, "id": "2503.07281_pg12_math_017", "type": "math", "max_diffs": 0, "checked": null, "math": "K^1_{\\Theta}\\oplus \\Theta H^1 \\subsetneq H^1_{\\Theta},"}
@@ -1854,9 +1811,6 @@
 {"pdf": "arxiv_math/2503.08498_pg10.pdf", "url": "https://arxiv.org/pdf/2503.08498", "page": 1, "id": "2503.08498_pg10_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "R(z)=\\frac{z^d}{ \\beta z^d -\\frac{\\alpha}{d}}"}
 {"pdf": "arxiv_math/2503.08498_pg10.pdf", "url": "https://arxiv.org/pdf/2503.08498", "page": 1, "id": "2503.08498_pg10_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "c^d = \\frac{\\alpha}{d \\beta}"}
 {"pdf": "arxiv_math/2503.08498_pg10.pdf", "url": "https://arxiv.org/pdf/2503.08498", "page": 1, "id": "2503.08498_pg10_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "N_R(z)=\\frac{1}{3}z(z^3+2)"}
-{"pdf": "arxiv_math/2503.07628_pg5.pdf", "url": "https://arxiv.org/pdf/2503.07628", "page": 1, "id": "2503.07628_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "is a constant depends upon the modeling and material parameters. This property implies that the tensor-valued function"}
-{"pdf": "arxiv_math/2503.07628_pg5.pdf", "url": "https://arxiv.org/pdf/2503.07628", "page": 1, "id": "2503.07628_pg5_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "is the body force term and"}
-{"pdf": "arxiv_math/2503.07628_pg5.pdf", "url": "https://arxiv.org/pdf/2503.07628", "page": 1, "id": "2503.07628_pg5_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "is coercive, i.e. there exists constant"}
 {"pdf": "arxiv_math/2503.03762_pg1.pdf", "url": "https://arxiv.org/pdf/2503.03762", "page": 1, "id": "2503.03762_pg1_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\begin{split} T_{\\Lambda}: &\\left(c_{1,0}, c_{1,1}, \\ldots, c_{1, m_1-1} ; c_{2,0}, c_{2,1}, \\ldots, c_{2, m_2-1} ; \\ldots ; c_{\\ell, 0}, c_{\\ell, 1}, \\ldots, c_{\\ell, m_{\\ell}-1}\\right)\\mapsto\\\\ &\\left(\\lambda_1 c_{1, m_1-1},c_{1,0}, \\ldots, c_{1, m_1-2} ; \\lambda_2 c_{2, m_2-1}, c_{2,0}, \\ldots, c_{2, m_2-2} ; \\ldots ; \\lambda_{\\ell} c_{\\ell, m_{\\ell}-1}, c_{\\ell, 0}, \\ldots, c_{\\ell, m_{\\ell}-2}\\right). \\end{split}"}
 {"pdf": "arxiv_math/2503.03762_pg1.pdf", "url": "https://arxiv.org/pdf/2503.03762", "page": 1, "id": "2503.03762_pg1_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left\\{\\mathbf{g}_1, \\mathbf{g}_2, \\ldots, \\mathbf{g}_\\rho\\right\\} \\subseteq V"}
 {"pdf": "arxiv_math/2503.03762_pg1.pdf", "url": "https://arxiv.org/pdf/2503.03762", "page": 1, "id": "2503.03762_pg1_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "0 \\neq \\lambda_i \\in \\mathbb{F}_q"}
@@ -2182,16 +2136,6 @@
 {"pdf": "arxiv_math/2503.07421_pg26.pdf", "url": "https://arxiv.org/pdf/2503.07421", "page": 1, "id": "2503.07421_pg26_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "H^h(l^h)=H(l^*,l^h)=H(l)= cov(l) - 2\\pi\\sum_{i\\in E} l_i,"}
 {"pdf": "arxiv_math/2503.07421_pg26.pdf", "url": "https://arxiv.org/pdf/2503.07421", "page": 1, "id": "2503.07421_pg26_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{\\partial H^h}{\\partial l_i} = - K_i, \\quad e_i\\in E^h."}
 {"pdf": "arxiv_math/2503.07421_pg26.pdf", "url": "https://arxiv.org/pdf/2503.07421", "page": 1, "id": "2503.07421_pg26_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "(l_{23}, l_{24}, l_{34})=l^h_{\\sigma} \\in \\R^3"}
-{"pdf": "arxiv_math/2503.09264_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09264", "page": 1, "id": "2503.09264_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": ". We obtain the \\textit{conilpotent filtration}"}
-{"pdf": "arxiv_math/2503.09264_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09264", "page": 1, "id": "2503.09264_pg5_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "is conilpotent. Similar definitions can be made for comodules over"}
-{"pdf": "arxiv_math/2503.09264_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09264", "page": 1, "id": "2503.09264_pg5_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "to be the dual of the group algebra"}
-{"pdf": "arxiv_math/2503.09264_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09264", "page": 1, "id": "2503.09264_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "be an augmented coalgebra over a field"}
-{"pdf": "arxiv_math/2503.09264_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09264", "page": 1, "id": "2503.09264_pg5_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "be the reduced coproduct. Then we call"}
-{"pdf": "arxiv_math/2503.09264_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09264", "page": 1, "id": "2503.09264_pg5_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "in the profinite topology of"}
-{"pdf": "arxiv_math/2503.09264_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09264", "page": 1, "id": "2503.09264_pg5_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": ". For a profinite group, we define"}
-{"pdf": "arxiv_math/2503.09264_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09264", "page": 1, "id": "2503.09264_pg5_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "is a bigraded algebra and for a graded comodule"}
-{"pdf": "arxiv_math/2503.09264_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09264", "page": 1, "id": "2503.09264_pg5_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "is a bigraded coalgebra. Additionally, if"}
-{"pdf": "arxiv_math/2503.09264_pg5.pdf", "url": "https://arxiv.org/pdf/2503.09264", "page": 1, "id": "2503.09264_pg5_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "we define its group coalgebra"}
 {"pdf": "arxiv_math/2503.09478_pg23.pdf", "url": "https://arxiv.org/pdf/2503.09478", "page": 1, "id": "2503.09478_pg23_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\xi_k = \\|\\boldsymbol{x}_k-\\boldsymbol{x}_*\\|,\\quad f(k) = -\\ln \\xi_k."}
 {"pdf": "arxiv_math/2503.09478_pg23.pdf", "url": "https://arxiv.org/pdf/2503.09478", "page": 1, "id": "2503.09478_pg23_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lim_{k\\to\\infty}\\frac{\\xi_{k+1}}{\\xi_k^q}=Q_q, \\quad q>1,\\quad 0<Q_q<\\infty."}
 {"pdf": "arxiv_math/2503.09478_pg23.pdf", "url": "https://arxiv.org/pdf/2503.09478", "page": 1, "id": "2503.09478_pg23_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sum_{j=k}^\\infty\\frac{|d(j)|}{q^{\\,j+1}} \\le M\\sum_{j=k}^\\infty\\frac{1}{q^{\\,j+1}},"}
@@ -2229,16 +2173,6 @@
 {"pdf": "arxiv_math/2503.08244_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08244", "page": 1, "id": "2503.08244_pg9_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\omega \\in \\Sigma_\\vartheta"}
 {"pdf": "arxiv_math/2503.08244_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08244", "page": 1, "id": "2503.08244_pg9_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\int_{\\Sigma_\\vartheta} \\mu^{(2)} \\left( \\left( T^{(2)}_\\omega\\right)^{-1} (A) \\right) \\, d\\mathbb{P} (\\omega) = \\mu^{(2)}(A),"}
 {"pdf": "arxiv_math/2503.08244_pg9.pdf", "url": "https://arxiv.org/pdf/2503.08244", "page": 1, "id": "2503.08244_pg9_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sigma \\omega \\coloneqq (\\omega_{i+1})_{i\\in\\mathbb{N}}"}
-{"pdf": "arxiv_math/2503.07530_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07530", "page": 1, "id": "2503.07530_pg7_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "and the fact that for every"}
-{"pdf": "arxiv_math/2503.07530_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07530", "page": 1, "id": "2503.07530_pg7_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lim_{n \\rightarrow \\infty} \\sup_{x \\geq \\varepsilon n} \\left| \\frac{\\P(\\overline{W}_{n} \\in (x,x+1])}{ n \\P(\\overline{X}_{1} \\in (x,x+1])} -1 \\right|=0,"}
-{"pdf": "arxiv_math/2503.07530_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07530", "page": 1, "id": "2503.07530_pg7_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sup_{x \\geq \\varepsilon n} \\sup_{|u| \\leq \\eta_{n}} \\left| \\frac{L(x+u)}{L(x)}-1\\right|\\to 0."}
-{"pdf": "arxiv_math/2503.07530_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07530", "page": 1, "id": "2503.07530_pg7_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": ". This will indeed imply that"}
-{"pdf": "arxiv_math/2503.07530_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07530", "page": 1, "id": "2503.07530_pg7_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\sup_{x \\geq \\varepsilon n} \\sup_{|u| \\leq \\eta_{n}} \\left| \\frac{L(x+u)}{L(x)}-1\\right|\\to 0."}
-{"pdf": "arxiv_math/2503.07530_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07530", "page": 1, "id": "2503.07530_pg7_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": "\\frac{\\overline{W}_{n}}{\\widehat{b}_{n}} = \\frac{X_{1}+ \\cdots+X_{n}-b_{n}}{a_{n}} \\cdot \\frac{a_{n}}{\\widehat{b}_{n}}+ \\frac{b_{n}+\\gamma n}{\\widehat{b}_{n}},"}
-{"pdf": "arxiv_math/2503.07530_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07530", "page": 1, "id": "2503.07530_pg7_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "which implies tightness since"}
-{"pdf": "arxiv_math/2503.07530_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07530", "page": 1, "id": "2503.07530_pg7_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "and that condition (3.3) there holds also with"}
-{"pdf": "arxiv_math/2503.07530_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07530", "page": 1, "id": "2503.07530_pg7_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "d_{\\mathrm{TV}}\\left((\\overline{X}_i^{(n)}:1\\leq i\\leq n-1),(\\overline{X}_i:1\\leq i\\leq n-1)\\right) \\to 0,"}
-{"pdf": "arxiv_math/2503.07530_pg7.pdf", "url": "https://arxiv.org/pdf/2503.07530", "page": 1, "id": "2503.07530_pg7_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "\\lim_{n \\rightarrow \\infty} \\sup_{x \\geq \\varepsilon n} \\left| \\frac{ \\P({X}_{1} \\in(x-m_{n},x-m_{n}+1])}{ \\P({X}_{1} \\in(x-\\gamma,x-\\gamma+1])} -1 \\right|=0,"}
 {"pdf": "arxiv_math/2503.08176_pg20.pdf", "url": "https://arxiv.org/pdf/2503.08176", "page": 1, "id": "2503.08176_pg20_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{A}_{3}^{\\circ}(3,1,2)"}
 {"pdf": "arxiv_math/2503.08176_pg20.pdf", "url": "https://arxiv.org/pdf/2503.08176", "page": 1, "id": "2503.08176_pg20_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{A}_{3}^{*}(3,1,2)"}
 {"pdf": "arxiv_math/2503.08176_pg20.pdf", "url": "https://arxiv.org/pdf/2503.08176", "page": 1, "id": "2503.08176_pg20_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "(k_{1},k_{2})\\in \\mathcal{A}_{2}^{*}(3,1,2)"}
@@ -2647,11 +2581,6 @@
 {"pdf": "arxiv_math/2503.05873_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05873", "page": 1, "id": "2503.05873_pg4_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mathcal{V} \\in \\{0,1\\}"}
 {"pdf": "arxiv_math/2503.05873_pg4.pdf", "url": "https://arxiv.org/pdf/2503.05873", "page": 1, "id": "2503.05873_pg4_math_009", "type": "math", "max_diffs": 0, "checked": null, "math": "\\hat{\\underline{V}}_{\\mathcal{L}}(\\underline{Y}_{\\mathcal{L}})"}
 {"pdf": "arxiv_math/2503.07310_pg2.pdf", "url": "https://arxiv.org/pdf/2503.07310", "page": 1, "id": "2503.07310_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "x\\in \\mathcal{X}\\subseteq \\mathbb{R}^{n_x}"}
-{"pdf": "arxiv_math/2503.08266_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08266", "page": 1, "id": "2503.08266_pg4_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "is the inverse of a (fictitious) mass matrix. The solutions to the equations of motion resulting from this Hamiltonian can be used as a preliminary step in the HMC algorithm. The time reversal invariance of Hamiltonian systems guarantees that the detailed balance holds. In general, the equations of motions are solved with a numerical integrator, and detailed balance is satisfied only in the limit of time step"}
-{"pdf": "arxiv_math/2503.08266_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08266", "page": 1, "id": "2503.08266_pg4_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": ". In the condition of detailed balance, the kinetic energy arising from the Metropolis acceptance probability and that coming from the Maxwell--Boltzmann term in the {\\it a priori} probability"}
-{"pdf": "arxiv_math/2503.08266_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08266", "page": 1, "id": "2503.08266_pg4_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "as the momenta associated to the weights"}
-{"pdf": "arxiv_math/2503.08266_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08266", "page": 1, "id": "2503.08266_pg4_math_008", "type": "math", "max_diffs": 0, "checked": null, "math": ", we make use of a modified HMC algorithm that damps fluctuations in the direction opposite to"}
-{"pdf": "arxiv_math/2503.08266_pg4.pdf", "url": "https://arxiv.org/pdf/2503.08266", "page": 1, "id": "2503.08266_pg4_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": "intermediate pivot points. Although this method allows for more flexible connections, it is computationally demanding, requiring multiple training runs to optimize the pivot locations. Furthermore, the number of required pivots can grow significantly, particularly in less overparameterized settings, making the approach increasingly impractical in such regimes. As an alternative to generate trajectories from"}
 {"pdf": "arxiv_math/2503.04646_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04646", "page": 1, "id": "2503.04646_pg2_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "K \\subset \\{\\mu \\in P(X \\times Y): \\mu_X^* \\in P(X) \\text{ is the } X\\text{-marginal of } \\mu\\}"}
 {"pdf": "arxiv_math/2503.04646_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04646", "page": 1, "id": "2503.04646_pg2_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\mu^* \\in P(X\\times Y)"}
 {"pdf": "arxiv_math/2503.04646_pg2.pdf", "url": "https://arxiv.org/pdf/2503.04646", "page": 1, "id": "2503.04646_pg2_math_002", "type": "math", "max_diffs": 0, "checked": null, "math": "K \\subset P(X\\times Y)"}
@@ -2849,16 +2778,6 @@
 {"pdf": "arxiv_math/2503.05610_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05610", "page": 1, "id": "2503.05610_pg5_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi_\\zeta^k(x_1)\\leq\\phi_\\zeta^k(\\gamma_m)\\leq\\phi_\\zeta^k(x_2), \\forall k"}
 {"pdf": "arxiv_math/2503.05610_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05610", "page": 1, "id": "2503.05610_pg5_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "\\phi_\\zeta^m(x_j)\\to\\zeta, j=1,2"}
 {"pdf": "arxiv_math/2503.05610_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05610", "page": 1, "id": "2503.05610_pg5_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "x_1, x_2 \\in \\sigma(\\Delta_n)"}
-{"pdf": "arxiv_math/2503.08799_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08799", "page": 1, "id": "2503.08799_pg3_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "not being defined on the whole"}
-{"pdf": "arxiv_math/2503.08799_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08799", "page": 1, "id": "2503.08799_pg3_math_003", "type": "math", "max_diffs": 0, "checked": null, "math": "where the left value is the result of the Riemann integral computed in"}
-{"pdf": "arxiv_math/2503.08799_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08799", "page": 1, "id": "2503.08799_pg3_math_004", "type": "math", "max_diffs": 0, "checked": null, "math": "\\left( \\int_{[a, b]} f \\right)^{M} = \\left( \\int_{[a, b]} g \\right)^{N} \\! \\! \\! \\!,"}
-{"pdf": "arxiv_math/2503.08799_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08799", "page": 1, "id": "2503.08799_pg3_math_005", "type": "math", "max_diffs": 0, "checked": null, "math": "we have a bounded real-valued function defined on some rectangle"}
-{"pdf": "arxiv_math/2503.08799_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08799", "page": 1, "id": "2503.08799_pg3_math_006", "type": "math", "max_diffs": 0, "checked": null, "math": "is unique except in a Lebesgue measure zero set: if in"}
-{"pdf": "arxiv_math/2503.08799_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08799", "page": 1, "id": "2503.08799_pg3_math_007", "type": "math", "max_diffs": 0, "checked": null, "math": ", then there exists some measure zero set"}
-{"pdf": "arxiv_math/2503.08799_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08799", "page": 1, "id": "2503.08799_pg3_math_010", "type": "math", "max_diffs": 0, "checked": null, "math": ", and the right one is the result of the integral computed in"}
-{"pdf": "arxiv_math/2503.08799_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08799", "page": 1, "id": "2503.08799_pg3_math_012", "type": "math", "max_diffs": 0, "checked": null, "math": "is a \\emph{Boolean algebra} if"}
-{"pdf": "arxiv_math/2503.08799_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08799", "page": 1, "id": "2503.08799_pg3_math_013", "type": "math", "max_diffs": 0, "checked": null, "math": "is a bounded function. Then,"}
-{"pdf": "arxiv_math/2503.08799_pg3.pdf", "url": "https://arxiv.org/pdf/2503.08799", "page": 1, "id": "2503.08799_pg3_math_014", "type": "math", "max_diffs": 0, "checked": null, "math": "is another Riemann integrable function on"}
 {"pdf": "arxiv_math/2503.07030_pg8.pdf", "url": "https://arxiv.org/pdf/2503.07030", "page": 1, "id": "2503.07030_pg8_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "F(\\Phi^{T_F},\\mathbf{u},\\mathbf{y},\\mathbf{p})=0"}
 {"pdf": "arxiv_math/2503.07030_pg8.pdf", "url": "https://arxiv.org/pdf/2503.07030", "page": 1, "id": "2503.07030_pg8_math_001", "type": "math", "max_diffs": 0, "checked": null, "math": "\\widehat{\\sigma}_\\alpha (u^k,y^k,\\mathbf{p})"}
 {"pdf": "arxiv_math/2503.05218_pg5.pdf", "url": "https://arxiv.org/pdf/2503.05218", "page": 1, "id": "2503.05218_pg5_math_000", "type": "math", "max_diffs": 0, "checked": null, "math": "(\\alpha,\\beta,\\gamma)\\in R_2"}