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Through the application of an evaluation technique based on cyclotomic multiple zeta values recently due to Au, we solve open problems on inverse binomial series that were included in a 2010 analysis textbook by Chen., Comment: a sequel to arXiv:2210.17243v4 and arXiv:2306.04638v3

We introduce and prove evaluations for families of multiple elliptic integrals by solving special types of ordinary and partial differential equations. As an application, we obtain new expressions of Ramanujan-type series of level 4 and associated singular values for the complete elliptic integral $\mathbf K$ with integrals involving $\mathbf K$., Comment: 8 pages

Using cyclotomic multiple zeta values of level $8$, we confirm and generalize several conjectural identities on infinite series with summands involving $\binom{2k}k8^k/(\binom{3k}k\binom{6k}{3k})$. For example, we prove that \[\sum_{k=0}^\infty\frac{(350k-17)\binom{2k}k8^k} {\binom{3k}k\binom{6k}{3k}}=15\sqrt2\,\pi+27\] and \[\sum_{k=1}^\infty\frac{\left\{(5k-1)\left[16\mathsf H_{2k-1}^{(2)}-3\mathsf H_{k-1}^{(2)}\right]-\frac{12(6k-1)}{(2k-1)^2}\right\}\binom{2k}k8^k} {k(2k-1)\binom{3k}k\binom{6k}{3k}}=\frac{\pi^3}{12\sqrt2},\] where $\mathsf H^{(2)}_m$ denotes the second-order harmonic number $\sum_{0

We perform polylogarithmic reductions for several classes of infinite sums motivated by Z.-W. Sun's related works in 2022--2023. For certain choices of parameters, these series can be expressed by cyclotomic multiple zeta values of levels $4$, $5$, $6$, $7$, $8$, $9$, $10$, and $12$. In particular, we obtain closed forms of the series $$\sum_{k=0}^\infty\frac{x_0^k}{(k+1)\binom{3k}k} \ \ \text{and}\ \ \sum_{k=1}^\infty\frac{x_0^k}{k^2\binom{3k}k}$$ for any $x_0\in(-27/4,27/4)$., Comment: 23 pages

We prove and generalize several recent conjectures of Z.-W. Sun surrounding binomial coefficients and harmonic numbers. We show that Sun's series and their analogs can be represented as cyclotomic multiple zeta values of levels $N\in\{4,8,12,16,24\}$, namely Goncharov's multiple polylogarithms evaluated at $N$-th roots of unity., Comment: Addendum to arXiv:2210.17243v3, verifying and generalizing Conjectures 2.2-2.8 in arXiv:2210.07238v8

The hyper-Mahler measures $m_k( 1+x_1+x_2),k\in\mathbb Z_{>1}$ and $m_k( 1+x_1+x_2+x_3),k\in\mathbb Z_{>1}$ are evaluated in closed form via Goncharov-Deligne periods, namely $\mathbb Q$-linear combinations of multiple polylogarithms at cyclotomic points (complex-valued coordinates that are roots of unity). Some infinite series related to these hyper-Mahler measures are also explicitly represented as Goncharov-Deligne periods of levels $1$, $2$, $ 3$, $4$, $6$, $8$, $10$ and $12$., Comment: (v1) i+30 pages, 5 tables. (v2) i+37 pages, 7 tables. Results improved and enriched. Maple and Mathematica worksheets available as ancillary files. (v3) 47 pages, 8 tables. Reformatted and corrected. (v4) 51 pages, 8 tables. Accepted version

Brain-computer interfaces (BCIs), invasive or non-invasive, have projected unparalleled vision and promise for assisting patients in need to better their interaction with the surroundings. Inspired by the BCI-based rehabilitation technologies for nerve-system impairments and amputation, we propose an electromagnetic brain-computer-metasurface (EBCM) paradigm, regulated by human's cognition by brain signals directly and non-invasively. We experimentally show that our EBCM platform can translate human's mind from evoked potentials of P300-based electroencephalography to digital coding information in the electromagnetic domain non-invasively, which can be further processed and transported by an information metasurface in automated and wireless fashions. Directly wireless communications of the human minds are performed between two EBCM operators with accurate text transmissions. Moreover, several other proof-of-concept mind-control schemes are presented using the same EBCM platform, exhibiting flexibly-customized capabilities of information processing and synthesis like visual-beam scanning, wave modulations, and pattern encoding.

We carry out quantitative studies on the Green operator $ \hat{\mathscr G}$ associated with the Born equation, an integral equation that models electromagnetic scattering, building the strong stability of the evolution semigroup $\{\exp(-i\tau\hat{\mathscr G})|\tau\geq0\} $ on polynomial compactness and the Arendt-Batty-Lyubich-V\~u theorem. The strongly-stable evolution semigroup inspires our proposal of a non-perturbative method to solve the light scattering problem and improve the Born approximation., Comment: (v1) 22 pages, 3 TikZ figures. Continuation of arXiv:1007.4375v3. Updated from Chapter 5 and Appendix D in https://www.proquest.com/docview/634514288?accountid=13151 (v2) 22 pages, 3 TikZ figures. Accepted version with updated references

Through algebraic manipulations on Wro\'nskian matrices whose entries are reducible to Bessel moments, we present a new analytic proof of the quadratic relations conjectured by Broadhurst and Roberts, along with some generalizations. In the Wro\'nskian framework, we reinterpret the de Rham intersection pairing through polynomial coefficients in Vanhove's differential operators, and compute the Betti intersection pairing via linear sum rules for on-shell and off-shell Feynman diagrams at threshold momenta. From the ideal generated by Broadhurst--Roberts quadratic relations, we derive new non-linear sum rules for on-shell Feynman diagrams, including an infinite family of determinant identities that are compatible with Deligne's conjectures for critical values of motivic $L$-functions., Comment: (v1) i+59 pages. Continuation of arXiv:1711.01829v2, reinterpretation and generalization of Fres\'an-Sabbah-Yu (arXiv:2005.11525v2, arXiv:2006.02702v1). (v2) i+55 pages. Notations upgraded and ancillary file updated. (v3) i+55 pages. For the penultimate paragraph in Sect. 4.2, there are minor notational changes and related updates in the ancillary file WronskianAlgebra_v3.nb

We establish two geometric inequalities, respectively, for harmonic functions in exterior Dirichlet problems, and for Green's functions in interior Dirichlet problems, where the boundary surfaces are smooth and convex. Both inequalities involve integrals over the mean curvature and the Gaussian curvature on an equipotential surface, and the normal derivative of the harmonic potential thereupon. These inequalities generalize a geometric conservation law for equipotential curves in dimension two, and offer solutions to two free boundary problems in three-dimensional electrostatics., Comment: 13 pages; (v2) References updated; paragraph after Theorem 3.4 rewritten. (v3) Revised according to referees' reports

Let $U(\boldsymbol r),\boldsymbol r\in\Omega\subset \mathbb R^2$ be a harmonic function that solves an exterior Dirichlet problem. If all the level sets of $U(\boldsymbol r),\boldsymbol r\in\Omega$ are smooth Jordan curves, then there are several geometric inequalities that correlate the curvature $\kappa(\boldsymbol r) $ with the magnitude of gradient $ |\nabla U(\boldsymbol r)|$ on each level set ("equipotential curve"). One of such inequalities is $ \langle [\kappa(\boldsymbol r)-\langle\kappa(\boldsymbol r)\rangle][|\nabla U(\boldsymbol r)|-\langle |\nabla U(\boldsymbol r)|\rangle]\rangle\geq0$, where $ \langle \cdot\rangle$ denotes average over a level set, weighted by the arc length of the Jordan curve. We prove such a geometric inequality by constructing an entropy for each level set $U(\boldsymbol r)=\varphi $, and showing that such an entropy is convex in $\varphi$. The geometric inequality for $\kappa(\boldsymbol r) $ and $ |\nabla U(\boldsymbol r)|$ then follows from the convexity and monotonicity of our entropy formula. A few other geometric relations for equipotential curves are also built on a convexity argument., Comment: (v1) 12 pages, 1 TikZ figure; (v2) 14 pages, 1 TikZ figure. New results added. Typos corrected; (v3) 14 pages, 1 table, 1 TikZ figure. Revised according to reviewer's comments

Let $I_0$ and $K_0$ be modified Bessel functions of the zeroth order. We use Vanhove's differential operators for Feynman integrals to derive upper bounds for dimensions of the $\mathbb Q$-vector space spanned by certain sequences of Bessel moments \[ \left\{\left.\int_0^\infty [I_0(t)]^a[K_0(t)]^b t^{2k+1}\mathrm{d}\, t\right|k\in\mathbb Z_{\geq0}\right\},\]where $a$ and $b$ are fixed non-negative integers. For $ a\in\mathbb Z\cap[1,b)$, our upper bound for the $ \mathbb Q$-linear dimension is $\lfloor (a+b-1)/2\rfloor$, which improves the Borwein-Salvy bound $\lfloor (a+b+1)/2\rfloor$. Our new upper bound $\lfloor (a+b-1)/2\rfloor$ is not sharp for $ a=2,b=6$, due to an exceptional $ \mathbb Q$-linear relation $\int_0^\infty [I_0(t)]^2[K_0(t)]^6 t\mathrm{d}\, t=72\int_0^\infty [I_0(t)]^2[K_0(t)]^6 t^{3}\mathrm{d}\, t$, which is provable by integrating modular forms., Comment: (v1) i+21 pages. Simplification and extension of some results in Section 5 of arXiv:1706.08308; (v2) i+29 pages; (v3) 20 pages, accepted version

We characterize the meaning of words with language-independent numerical fingerprints, through a mathematical analysis of recurring patterns in texts. Approximating texts by Markov processes on a long-range time scale, we are able to extract topics, discover synonyms, and sketch semantic fields from a particular document of moderate length, without consulting external knowledge-base or thesaurus. Our Markov semantic model allows us to represent each topical concept by a low-dimensional vector, interpretable as algebraic invariants in succinct statistical operations on the document, targeting local environments of individual words. These language-independent semantic representations enable a robot reader to both understand short texts in a given language (automated question-answering) and match medium-length texts across different languages (automated word translation). Our semantic fingerprints quantify local meaning of words in 14 representative languages across 5 major language families, suggesting a universal and cost-effective mechanism by which human languages are processed at the semantic level. Our protocols and source codes are publicly available on https://github.com/yajun-zhou/linguae-naturalis-principia-mathematica, Comment: Main text (12 pages, 7 figures); Software Manual (ii+262 pages, 16 figures, 12 tables, available as two ancillary files). Revised according to reviewers' comments

One encounters iterated elliptic integrals in the study of Hall effect devices, as a result of conformal mappings of Schwarz--Christoffel type. Some of these double elliptic integrals possess amazing symmetries with regard to the physical parameters of the underlying Hall effect devices. We give a unified mathematical treatment of such symmetric double integrals, in the context of Hall effect devices with three and four contacts., Comment: 8 pages

This article reports on some recent progresses in Bessel moments, which represent a class of Feynman diagrams in 2-dimensional quantum field theory. Many challenging mathematical problems on these Bessel moments have been formulated as a vast set of conjectures, by David Broadhurst and collaborators, who work at the intersection of high energy physics, number theory and algebraic geometry. We present the main ideas behind our verifications of several such conjectures, which revolve around linear and non-linear sum rules of Bessel moments, as well as relations between individual Feynman diagrams and critical values of modular $L$-functions., Comment: (v1) 25 pages, 2 tables, 2 TikZ figures. An expository survey of recent progresses (arXiv:1706.01068, arXiv:1706.08308, arXiv:1708.02857, arXiv:1711.01829, arXiv:1801.02182) in Bessel moments and Broadhurst's conjectures (arXiv:1604.03057); (v2) Minor clarifications in Sect. 2.3 and updates in references; (v3) 25+3 pages. Fig. 2 corrected, erratum at the end

We prove Laporta's conjecture\begin{align*}&\int_0^\infty\frac{\mathrm d\, x_1}{x_1}\int_0^\infty\frac{\mathrm d\, x_2}{x_2}\int_0^\infty\frac{\mathrm d\, x_3}{x_3}\int_0^\infty\frac{\mathrm d\, x_4}{x_4}\frac{1}{\left(1+\sum^4_{k=1}x_k\right)\left(1+\sum^4_{k=1}\frac{1}{x_{k}} \right)-1}\\={}&\frac43 \int_{0}^\pi\mathrm d\, \phi_1 \int_{0}^\pi\mathrm d\, \phi_2\int_{0}^\pi\mathrm d\, \phi_3 \int_{0}^\pi\mathrm d\, \phi_4\frac{1}{4-\sum_{k=1}^4\cos \phi_k}, \end{align*} which relates the 4-loop sunrise diagram in 2-dimensional quantum field theory to Watson's integral for 4-dimensional hypercubic lattice. We also establish several related integral identities proposed by Laporta, including a reduction of the 4-loop sunrise diagram to special values of Euler's gamma function and generalized hypergeometric series:\begin{align*} \frac{4 \pi ^{5/2}}{\sqrt{3}}\left\{ \frac{\sqrt{3} }{2^6 }\left[\frac{\Gamma \left(\frac{1}{3}\right)}{\sqrt{\pi}}\right]^9\, _4F_3\left(\left. \begin{array}{c}\frac{1}{6},\frac{1}{3},\frac{1}{3},\frac{1}{2}\\[4pt]\frac{2}{3},\frac{5}{6},\frac{5}{6}\end{array} \right|1\right)-\frac{2^{4}}{3}\left[\frac{\sqrt{\pi}}{\Gamma \left(\frac{1}{3}\right)}\right]^9\, _4F_3\left(\left. \begin{array}{c}\frac{1}{2},\frac{2}{3},\frac{2}{3},\frac{5}{6}\\[4pt]\frac{7}{6},\frac{7}{6},\frac{4}{3}\end{array} \right|1\right) \right\}. \end{align*}, Comment: i+28 pages

Drawing on Vanhove's contributions to mixed Hodge structures for Feynman integrals in two-di\-men\-sion\-al quantum field theory, we compute two families of determinants whose entries are Bessel moments. Via explicit factorizations of certain Wro\'nskian determinants, we verify two recent conjectures proposed by Broadhurst and Mellit, concerning determinants of arbitrary sizes. With some extensions to our methods, we also relate two more determinants of Broadhurst--Mellit to the logarithmic Mahler measures of certain polynomials., Comment: i+32 pages. Proof of two conjectures proposed by Broadhurst-Mellit (arXiv:1604.03057), using Vanhove's theory (arXiv:1401.6438). Evaluation of vacuum determinants via Mahler measures, in a new Section 5

Let $ p_n(x)=\int_0^\infty J_0(xt)[J_0(t)]^n xt\,\mathrm{d}\, t$ be Kluyver's probability density for $n$-step uniform random walks in the Euclidean plane. Through connection to a similar problem in 2-dimensional quantum field theory, we evaluate the third-order derivative $ p_5'''(0^{+})$ in closed form, thereby giving a new proof for a conjecture of J. M. Borwein. By further analogies to Feynman diagrams in quantum field theory, we demonstrate that $ p_n(x),0\leq x\leq 1$ admits a uniformly convergent Maclaurin expansion for all odd integers $ n\geq5$, thus settling another conjecture of Borwein., Comment: (v1) 16 pages, 5 TikZ figures. Proof of Borwein's sum rule for ramble integrals $W_n(s)$. An addendum to arXiv:1706.08308 (v2) 17 pages, with updates in references and Theorem 5.1

Using contour deformations and integrations over modular forms, we compute certain Bessel moments arising from diagrammatic expansions in two-dimensional quantum field theory. We evaluate these Feynman integrals as either explicit constants or critical values of modular $L$-series, and verify several recent conjectures of Broadhurst., Comment: i+37 pages. Simplification and generalization of Bloch-Kerr-Vanhove (arXiv:1406.2664) and Samart (arXiv:1511.07947). Verification of several conjectures by Broadhurst (arXiv:1604.03057) on critical values of modular L-series. Revised according to referees' reports

Using Hilbert transforms, we establish two families of sum rules involving Bessel moments, which are integrals associated with Feynman diagrams in two-dimensional quantum field theory. With these linear relations among Bessel moments, we verify and generalize two conjectures by Bailey-Borwein-Broadhurst-Glasser and Broadhurst-Mellit., Comment: (v1) 9 pages. Verification and generalization of the integrality for Crandall numbers (OEIS A262961) (v2) 10 pages. Minor corrections and additions. Revised according to referee's report

We show that the following double integral \[\int_{0}^\pi {\rm d}x \int_0^x {\rm d}y \frac{1}{\sqrt{1-\smash[b]{p}\cos x}\sqrt{1+\smash[b]{q\cos y}}}\]remains invariant as one trades the parameters $p$ and $q$ for $p'=\sqrt{1-p^2}$ and $q'=\sqrt{1-q^2}$ respectively. This invariance property is suggested from symmetry considerations in the operating characterstics of a semiconductor Hall-effect device.