arXiv subjects

Jianqiang Zhao

Publications and source records attributed to Jianqiang Zhao.

At least 19 recordsLinked to original sources

arXiv2026

Largest Circle Enclosing Exactly $n$ Interior Lattice Points. II

In a previous article doi.org/10.3390/geometry2030012, the author investigated a class of elementary plane geometry problems closely related to the theme of this work. Here, we prove a weaker version of a previous conjecture by demonstrating that there are infinitely many maximally circlable (MAC) numbers -- positive integers $n$ for which there exists a largest circle enclosing exactly $n$ interior lattice points. Furthermore, by extending numerical computations to $n \le 2700$, we identify two counterexamples to a conjecture in loc cit. regarding the symmetry of the largest circle enclosing a strong MAC number (a MAC number $n$ where $n+1$ is non-MAC). We also propose a potential infinite family of strong MAC numbers derived from Pythagorean triples; the existence of this family would imply the infinity of non-MAC numbers, as conjectured by Zhao. Throughout this paper, we provide extensive data characterizing both MAC and strong MAC numbers alongside their corresponding largest enclosing circles.

math.GM↗

arXiv2026

Rational Approximations for Reciprocals of Multiple Zeta Values and Trivariate Cauchy Numbers

In this paper, we will study a trivariate extension of the Cauchy numbers of both the first kind (also called Gregory coefficients) and the second kind (also called Nörlund numbers) via the Laurent expansion of the reciprocal of any positive integer power (which is called the order) of multiple polylogarithms. In the case of logarithm, we will show by the WZ method that for each order $\ell>1$ some Gregory coefficient of order $\ell$ must vanish, in contrast to the fact that all classical Gregory coefficients are nonzero. We also prove in this higher order logarithm case that the sequence is eventually alternating for each fixed order, a property enjoyed by the classical Gregory coefficients. In the most general setting, we conjecture that these new sequences are all eventually positive, which is supported by strong numerical evidence. Finally, we confirm this conjecture in the special case of polylogarithms and double polylogarithms. As a by product, for each zeta value and double zeta value, we find an infinite family of identities expressing its reciprocal as a sum of a rational number and an improper integral.

math.NT↗

arXiv2026

Unramified Motivic Alternating Multiple Mixed Values

Many variants of the multiple zeta values have been studied in recent years. There are a few among them that remain to be real numbers, such as the multiple mixed values defined by the authors as level-two generalizations, which include both Hoffman's multiple $t$-values and Kaneko--Tsumura's multiple $T$-values. A central question is when such a value descends to level one, that is, when it can be expressed as a $\Q$-linear combination of multiple zeta values. We call these values unramified. In this paper, we further consider the alternating version of the above variants and identify five families of unramified alternating multiple mixed values using the descent theory of Brown and Glanois. We conjecture that all unramified truly alternating multiple mixed values are given in this paper.

math.NT↗

arXiv2026

Effective Euler Sums

Euler sums, also called alternating multiple zeta values (MZVs), have been shown to play important roles in many research areas in mathematics and theoretical physics. Motivated by a similar conjecture for MZVs by Kaneko and Zagier, the author proposed a conjectural isomorphism between the space of finite Euler sums and the space of classical ones modulo $ζ(2)$-products. In this paper, we explicitly construct those finite elements (for which we call effective Euler sums) that correspond to the classical Euler sums under this conjecture in depth one and two. In the appendix, we offer another group of plausible candidates of effective Euler sums when depth is two and weight is even, by extending the heuristic argument of Kaneko and Zagier for the double zeta case. Kina recently obtained independently some the same results in the MZV setting. In particular, using one of his results we prove in the appendix that Kaneko and Zagier's heuristically defined effective double zeta values coincides with ours.

math.NT↗

arXiv2026

Finite and Symmetric Multiple $T$-Values

The multiple $T$-values (MTVs), first studied by Kaneko and Tsumura, are a variation of the multiple zeta values (MZVs) with restricted product structure. Motivated by a deep conjecture of Kaneko and Zagier relating finite MZVs and symmetric MZVs, which was extended to Euler sums by Zhao, we study finite and symmetric multiple $T$-values. In particular, we show that finite MTVs satisfy Hoffman-type duality relations at low height, confirming several conjectures of the second author and discovering new families of identities. In proving relations among symmetric MTVs, our work builds on Xu and Zhao's theory of multiple mixed values, the relations of double zeta values discovered by Gangl, Kaneko and Zagier, and the (weighted) sum formulas of double Euler sums discovered by Berger et al. We then use generating functions derived from the integral structure of MTVs to establish the corresponding relations for finite MTVs. These results aid in the computation of dimensions of the $\Q$-vector space spanned by finite and symmetric MTVs (modulo $ζ(2)$ products), providing strong evidence for an isomorphism between the two spaces.

math.NT↗

arXiv2026

Are Production Cloud Skills Adequately Tested? Measuring and Governing Skill Test Adequacy in Practice

Cloud platforms increasingly deliver reusable Cloud Skills that guide AI agents through multi-step resource operations, user choices, validation, and recovery. Existing Skill evaluation primarily measures whether a Skill improves task success, but passing the available testcases does not reveal which behaviors specified by the Skill remain untested. We introduce Skill Test Adequacy, a scenario-conditioned criterion that evaluates a test suite against the complete set of operational test obligations specified by a Skill. Given a Skill package and normalized testcases containing a prompt, an initial resource state, and expected user decisions, the assessment determines whether each obligation is exercised by at least one testcase scenario; the resulting records provide both a suite-level score and explicit test gaps. We operationalize the criterion through parallel obligation proposals, disagreement-preserving aggregation, testcase-level status proposals, expert review, and source-grounded recommendations. Alibaba Cloud deploys this process as a mandatory gate before task-success evaluation and subsequent release checks. Among 157 initial assessments recorded before gate-driven remediation, 57 (36.3%) fall below the mandatory 80% gate and 76 (48.4%) remain below the recommended 90% level. The process also produces 132 reports containing 639 obligation-level recommendations, with a median of four per Skill. Finally, we release SkillAdeqBench, an exploratory subset of the reviewed records for studying automatic adequacy assessment. Skill Test Adequacy complements task-success evaluation by making the untested scope of production Cloud Skills explicit.

cs.SE↗

arXiv2026

Ramified and Unramified Motivic Multiple $t$-, $T$- and $S$-Values

In this paper, we consider several variants of motivic multiple zeta values of level two by restricting the summation indices to fixed parity patterns. These variants include Hoffman's multiple $t$-values, Kaneko and Tsumura's multiple $T$-values, and the multiple $S$-values previously studied by the authors. By applying the descent theory of Brown and Glanois to the motivic versions of these values, we derive criteria for determining when they are ramified or unramified. Assuming Grothendieck's period conjecture, our results partially confirm a conjecture by Kaneko and Tsumura regarding the unramified nature of multiple $T$-values of depth less than four. We obtain similar results for motivic multiple $S$-values. Furthermore, we generalize a result of Charlton to broader families of unramified multiple $t$-values with unit components. Finally, we propose several open problems for future research.

math.NT↗

arXiv2026

Unramified Motivic Multiple Mixed Values

The multiple mixed values (MMVs) are level two variants of multiple zeta values produced by restricting the summation indices to fixed parity patterns. One particularly interesting problem is to determine exactly when such values are actually in level one, namely, expressible in terms of multiple zeta values. To solve this completely is beyond our current knowledge since it calls for new ideas from transcendental number theory. However, much progress has been made on the motivic level. Previously, using the descent theory developed by Brown et al. we have tackled this problem for a few special classes of MMVs with regular parity patterns among the summation indices, including Hoffman's multiple $t$-values, Kaneko and Tsumura's multiple $T$-values, and our own multiple $S$-values. In this paper, we turn to the general case and determine completely all the unramified motivic MMVs of depth less than four as well as a few families of unbounded depths. At the end of the paper, we will present some general conjectures to describe the unramified MMVs at all depths greater than three.

math.AG↗

arXiv2026

Beyond Semantics: Uncovering the Physics of Fakes via Universal Physical Descriptors for Cross-Modal Synthetic Detection

The rapid advancement of AI generated content (AIGC) has blurred the boundaries between real and synthetic images, exposing the limitations of existing deepfake detectors that often overfit to specific generative models. This adaptability crisis calls for a fundamental reexamination of the intrinsic physical characteristics that distinguish natural from AI-generated images. In this paper, we address two critical research questions: (1) What physical features can stably and robustly discriminate AI generated images across diverse datasets and generative architectures? (2) Can these objective pixel-level features be integrated into multimodal models like CLIP to enhance detection performance while mitigating the unreliability of language-based information? To answer these questions, we conduct a comprehensive exploration of 15 physical features across more than 20 datasets generated by various GANs and diffusion models. We propose a novel feature selection algorithm that identifies five core physical features including Laplacian variance, Sobel statistics, and residual noise variance that exhibit consistent discriminative power across all tested datasets. These features are then converted into text encoded values and integrated with semantic captions to guide image text representation learning in CLIP. Extensive experiments demonstrate that our method achieves state-of-the-art performance on multiple Genimage benchmarks, with near-perfect accuracy (99.8%) on datasets such as Wukong and SDv1.4. By bridging pixel level authenticity with semantic understanding, this work pioneers the use of physically grounded features for trustworthy vision language modeling and opens new directions for mitigating hallucinations and textual inaccuracies in large multimodal models.

cs.CV↗

arXiv2025

The Largest Circle Enclosing $n$ Lattice Points

In this paper, we propose a class of elementary plane geometry problems closely related to the title of this paper. Here, a circle is the 1-dimensional curve bounding a disk. For any nonnegative integer, a circle is called $n$-enclosing if it contains exactly $n$ lattice points on the $xy$-plane in its interior. The main questions are when the largest $n$-enclosing circle exists and what the largest radius is. We study the small integer cases by hand and extend to all $n<1100$ with the aid of a computer. We find that frequently such a circle does not exist, e.g., when $n=5,6$. We then show a few general results on these circles including some regularities among their radii and an easy criterion to determine exactly when largest $n$-enclosing circles exist. Further, from numerical evidence, we conjecture that the set of integers whose largest enclosing circles exist is infinite, and so is its complementary in the set of nonnegative integers. Throughout this paper we present more mysteries/problems/conjectures than answers/solutions/theorems. In particular, we list many conjectures and some unsolved problems including possible higher dimensional generalizations at the end of the last two sections.

math.GM↗

arXiv2025

Some Variants of Apéry-Type Series and Level Four Colored Multiple Zeta Values

In this paper, we study Apéry-type series involving the central binomial coefficients \begin{align*} \sum_{n_1>\cdots>n_d>0} \frac1{4^{n_1}}\binom{2n_1}{n_1} \frac{1}{n_1^{s_1}\cdots n_d^{s_d}} \end{align*} and its variations where the summation indices may have mixed parities and some or all ``$>$'' are replaced by ``$\ge$'', as long as the series are defined. These sums have naturally appeared in the calculation of massive Feynman integrals by the work of Jegerlehner, Kalmykov and Veretin. We show that all these sums can be expressed as $\mathbb Q$-linear combinations of the real and/or imaginary parts of the colored multiple zeta values at level four, i.e., special values of multiple polylogarithms at fourth roots of unity. We also show that the corresponding series where ${\binom{2n_1}{n_1}}/4^{n_1}$ is replaced by ${\binom{2n_1}{n_1}}^2/16^{n_1}$ can be expressed in a similar way except for a possible extra factor of $1/π$.

math.NT↗

arXiv2025

Alternating Multiple Mixed Values: Regularization, Special Values, Parity and Dimension Conjectures

In this paper, we define and study a variant of multiple zeta values (MZVs) of level four, called alternating multiple mixed values or alternating multiple $M$-values (AMMVs), forming a $\Q[i]$-subspace of the colored MZVs of level four. This variant includes the alternating version of Hoffman's multiple $t$-values, Kaneko-Tsumura's multiple $T$-values, and the multiple $S$-values studied by the authors previously as special cases. We exhibit nice properties similar to the ordinary MZVs such as the generalized duality, integral shuffle and series stuffle relations. After setting up the algebraic framework we derive the regularized double shuffle relations of the AMMVs by adopting the machinery from color MZVs of level four. As an important application, we prove a parity result for AMMVs previously conjectured by us. We also investigate several alternating multiple $S$- and $T$-values by establishing some explicit relations of integrals involving arctangent function. At the end, we compute the dimensions of a few interesting subspaces of AMMVs for weight less than 9. Supported by theoretical and numerical evidence aided by numerical and symbolic computation, we formulate a few conjectures concerning the dimensions of the above-mentioned subspaces of AMMVs. These conjectures hint at a few very rich but previously overlooked algebraic and geometric structures associated with these vector spaces.

math.NT↗

arXiv2025

New Proofs of the Explicit Formulas of Arakawa--Kaneko Zeta Values and Kaneko--Tsumura $η$- and $ψ$- Values

In this paper, we establish some new identities of integrals involving multiple polylogarithm functions and their level two analogues in terms of Hurwitz-type multiple zeta (star) values. Using these identities, we provide new proofs of the explicit formulas of Arakawa--Kaneko zeta values, Kaneko--Tsumura $η$- and $ψ$-values, and also give a formula for double $T$-values.

math.NT↗

arXiv2024

Apéry-Like Sums and Colored Multiple Zeta Values

In this article we shall survey some recent progress on the study of Apéry-like sums which are multiple variable generalizations of the two sums Apéry used in his famous proof of the irrationality of $ζ(2)$ and $ζ(3)$. We only allow the central binomial coefficients to appear in these infinite sums but they can appear either on the numerator or on the denominator. Special values of both types are closely related to the colored multiple zeta values and have played important roles in the calculations of the $\eps$-expansion of multiloop Feynman diagrams. We will summarize several different approaches to computing these sums and prove a few conjectural identities of Z.-W. Sun as corollaries along the way.

math.NT↗

arXiv2024

Parametric Apéry-type Series and Hurwitz-type Multiple Zeta Values

In this paper, we extend the main results of a 2024 \emph{Advances in Applied Mathematics} paper \cite{XuZhao2021c} about Apéry-type series involving central binomial coefficients and the multiple ($t-$)harmonic sums to parametric Apéry-type series involving parametric binomial coefficients and Hurwitz-type multiple harmonic (star) sums. In particular, we will establish many explicit relations between parametric Apéry-type series involving one or two parametric binomial coefficients and Hurwitz-type multiple zeta values (with $r$-variables) by using the method of iterated integrals.

math.NT↗

arXiv2024

Reciprocal Hyperbolic Series of Ramanujan Type

This paper presents an approach to summing a few families of infinite series involving hyperbolic functions, some of which were first studied by Ramanujan. The key idea is based on their contour integral representations and residue computations with the help of some well-known results of Eisenstein series given by Ramanujan, Berndt et al. As our main results, several series involving hyperbolic functions are evaluated and expressed in terms of $z={}_2F_1(1/2,1/2;1;x)$ and $z'=dz/dx$. When a certain parameter in these series is equal to $π$ the series are expressed in closed forms in terms of some special values of the Gamma function. Moreover, many new illustrative examples are presented.

math.NT↗

arXiv2024

Berndt-Type Integrals and Series Associated with Ramanujan and Jacobi Elliptic Functions

In this paper, we evaluate in closed forms two families of infinite integrals containing hyperbolic and trigonometric functions in their integrands. We call them Berndt-type integrals since he initiated the study of similar integrals. We first establish explicit evaluations of four classes of hyperbolic sums by special values of the Gamma function, by two completely different approaches, which extend those sums considered by Ramanujan and Zucker previously. We discover the first by refining two results of Ramanujan concerning some $q$-series. For the second we compare both the Fourier series expansions and the Maclaurin series expansions of a few Jacobi elliptic functions. Next, by contour integrations we convert two families of Berndt-type integrals to the above hyperbolic sums, all of which can be evaluated in closed forms. We then discover explicit formulas for one of the two families. Throughout the paper we present many examples which enable us to formulate a conjectural explicit formula for the other family of the Berndt-type integrals at the end.

math.NT↗

arXiv2024

Finite and Symmetric Euler Sums and Finite and Symmetric (Alternating) Multiple $T$-Values

In this paper, we will study finite multiple $T$-values (MTVs) and their alternating versions, which are level two and level four variations of finite multiple zeta values, respectively. We will first provide some structural results for level two finite multiple zeta values (i.e., finite Euler sums) for small weights, guided by the author's previous conjecture that the finite Euler sum space of weight, $w$, is isomorphic to a quotient Euler sum space of weight, $w$. Then, by utilizing some well-known properties of the classical alternating MTVs, we will derive a few important $\Q$-linear relations among the finite alternating MTVs, including the reversal, linear shuffle, and sum relations. We then compute the upper bound for the dimension of the $\Q$-span of finite (alternating) MTVs for some small weights by rigorously using the newly discovered relations, numerically aided by computers.

math.NT↗