相关论文: Two determinant evaluations in Sun's conjectures i…

We consider the Izergin-Korepin determinant [1] together with another determinant which was invented by Kuperberg [2]. He used these determinants to prove a formula for the total number of half-turn symmetric alternating sign matrices of…

We develop an effective version of Kronecker's Theorem on the splitting of polynomials, based on asymptotic arguments proposed by the Chudnovsky brothers, coming from Hermite-Pad\'e approximation. In conjunction with Honda's proof of the…

In this work, we establish modular parameterizations for two general formulas for $\frac{1}{\pi}$ that subsume conjectural Ramanujan type formulas due to Z.-W. Sun, which have remained open since 2011. As an application of this, in a…

In this paper we provide an identity between determinant and generalized matrix function. Also, a criterion of positive semi-definite matrices affirming the permanent dominant conjecture is given. As a consequence, infinitely many infinite…

We prove some supercongruence and divisibility results on sums involving Domb numbers, which confirm four conjectures of Z.-W. Sun and Z.-H. Sun. For instance, by using a transformation formula due to Chan and Zudilin, we show that for any…

We study the maximum absolute value of the determinant of matrices with entries in the set of $\ell$-th roots of unity; this is a generalization of $D$-optimal designs and Hadamard's maximal determinant problem, which involves $\pm 1$…

This note presents explicit formulae for the exponentials of a wide variety of matrices which are 4x4, anti-Hermitian. Easily verifiable conditions characterizing when such matrices admit one of three minimal polynomials are also given.…

We derive an identity for the determinant of the sum of two $n\times n$ matrices, $A$ and $B$, whose entries are defined via contour integrals. Specifically, we consider $A(i,j)=\frac{1}{2\pi\mathrm{i}}\oint_0…

In [Boltje,Hartmann: Permutation resolutions for Specht modules, J. Algebraic Combin. 34 (2011), 141-162], a chain complex was constructed in a combinatorial way which conjecturally is a resolution of the (dual of the) integral Specht…

Rational solutions for the Painlev\'e IV equation are investigated by Hirota bilinear formalism. It is shown that the solutions in one hierarchy are expressed by 3-reduced Schur functions, and those in another two hierarchies by Casorati…

We prove that the symmetric-cube coefficients $A_n=(-27)^n[z^n]\,_2F_1(1/3,1/3;1;z)^3$ satisfy the supercongruence $A(mp)\equiv A(m) \bmod p^4$ for every prime $p\geq 5$ and every $m\geq 1$. The proof rests on three ingredients: (i) the…

We prove that the inverse of a positive-definite matrix can be approximated by a weighted-sum of a small number of matrix exponentials. Combining this with a previous result [OSV12], we establish an equivalence between matrix inversion and…

We prove that the following pointwise inequality holds \begin{equation*} -\Delta u \ge \sqrt\frac{2}{(p+1)-c_n} |x|^{\frac{a}{2}} u^{\frac{p+1}{2}} + \frac{2}{n-4} \frac{|\nabla u|^2}{u} \ \ \text{in}\ \ \mathbb{R}^n \end{equation*} where…

Let $Hilb ^{p(t)}(P^n)$ be the Hilbert scheme of closed subschemes of $P^n$ with Hilbert polynomial $p(t) \in Q[t]$, and let $W:= \overline{W(\underline{b};\underline{a};r)}$ be the closure of the locus in $Hilb ^{p(t)}(P^n)$ of…

Computing the determinant of a matrix with the univariate and multivariate polynomial entries arises frequently in the scientific computing and engineering fields. In this paper, an effective algorithm is presented for computing the…

In this paper we investigate Donkin's $(p,r)$-Filtration Conjecture, and present two proofs of the "if" direction of the statement when $p\geq 2h-2$. One proof involves the investigation of when the tensor product between the Steinberg…

We prove seven of the Rogers-Ramanujan type identities modulo $12$ that were conjectured by Kanade and Russell. Included among these seven are the two original modulo $12$ identities, in which the products have asymmetric congruence…

We give the criterion for the irreducibility, the Schur irreducibility and the indecomposability of the set of two $n\times n$ matrices $\Lambda_n$ and $A_n$ in terms of the subalgebra associated with the "support" of the matrix $A_n$,…

Two conjectures, posed by Finch-Smith, Harrington, and Wong in a paper published in Integers in $2023$, are proven. Given a monic biquadratic polynomial $f(x) = x^4 + cx^2 + e$, we prove a formula for the sum of its distinct outputs modulo…

Let $p$ be an odd prime and let $(\frac{\cdot}p)$ be the Legendre symbol. In this paper, we study the determinant $$\det\left[\left(\frac{j^2-k^2}p\right)+\left(\frac{jk}p\right)w\right]_{\delta\le j,k\le (p-1)/2}$$ with $\delta\in\{0,1\}$.…