Related papers: A generalized truncated logarithm
Let $\mathfrak g$ be a simple Lie algebra over an algebraically closed field $\bf k$ of characteristic zero and $\bf G$ its adjoint group. Let $\mathfrak q$ be a biparabolic subalgebra of $\mathfrak g$. The algebra $Sy(\mathfrak q)$ of…
Let $A \cong k\langle X \rangle / I$ be an associative algebra. A finite word over alphabet $X$ is $I${\it-reducible} if its image in $A$ is a $k$-linear combination of length-lexicographically lesser words. An {\it obstruction} in a…
Learning with a {\it convex loss} function has been a dominating paradigm for many years. It remains an interesting question how non-convex loss functions help improve the generalization of learning with broad applicability. In this paper,…
In this study we introduce a second type of higher order generalised geometric polynomials. This we achieve by examining the generalised stirling numbers $S(n; k;\alpha;\beta;\gamma)$ [Hsu & Shiue,1998] for some negative arguments. We study…
In this paper, we study sums of shifted products $\sum\limits_{n \leq x} F(n) G(n-h)$ for any $|h| \leq x/2$ and arithmetic functions $F=f*1$ and $G=g*1$, with $f$ and $g$ small. We obtain asymptotic formula for different orders of…
We introduce a triangular array $\widehat{\sf L}^{(\alpha)}$ of 5-variable homogeneous polynomials that enumerate Laguerre digraphs (digraphs in which each vertex has out-degree 0 or 1 and in-degree 0 or 1) with separate weights for peaks,…
This paper investigates the generalized beta-logarithmic matrix function (GBLMF),which combines the extended beta matrix function and the logarithmic mean. The study establishes essential properties of this function, including functional…
We prove strong estimates for averages of shifted convolution sums consisting of quadratic twists of $\mathrm{GL}_{2}$ $L$-functions. The key input involves the circle method together with standard tools such as Vorono\u{\i}, quadratic…
We investigate the intertwining of Laguerre processes of parameter $\alpha$ in different dimensions. We introduce a Feller kernel that depends on $\alpha $ and intertwines the $\alpha$-Laguerre process in $N+1$ dimensions and that in $N$…
We estimate the truncated double trigonometric series $\sum_{n=0}^{N}\sum_{m=0}^{M}a_{mn} {e}^{2\pi \imath \left(m x+n y\right)}$, $a_{mn} \in\mathbb{C}$, in Lebesgue spaces with mixed norms in terms of the $p^{th}-q^{th}$ power finite…
We study the asymptotic zero distribution of the rescaled Laguerre polynomials, $\displaystyle L_n^{(\alpha_n)}(nz)$, with the parameter $\alpha_n$ varying in such a way that $\displaystyle \lim_{n\rightarrow \infty}\alpha_n/n=-1$. The…
Given a smooth function $f$, we develop a general approach to turn Monte Carlo samples with expectation $m$ into an unbiased estimate of $f(m)$. Specifically, we develop estimators that are based on randomly truncating the Taylor series…
We study relatively affine actions of a diagonalizable group $G$ on locally noetherian schemes. In particular, we generalize Luna's fundamental lemma when applied to a diagonalizable group: we obtain criteria for a $G$-equivariant morphism…
An approximation algorithm is proposed to transform truncated QCD (or QED) series for observables. The approximation is a modification of the Baker-Gammel approximants, and is independent of the renormalization scale (RScl) $\mu$ -- the…
In this paper, we present a new generalization of the Lucas numbers by matrix representation using Genaralized Lucas Polynomials. We give some properties of this new generalization and some relations between the generalized order-k Lucas…
We consider a $(q,y)$-analogue of Laguerre polynomials $L^{(\alpha)}_n(x;y;q)$ for integral $\alpha\geq -1$, which turns out to be a rescaled version of Al-Salam--Chihara polynomials. A combinatorial interpretation for the $(q,y)$-Laguerre…
Over the past few years considerable progress has been made on the resummation of double-logarithmically enhanced threshold (large-x) and high-energy (small-x) higher-order contributions to the splitting functions for parton and…
In this paper, we introduce various covering number bounds for linear function classes, each subject to different constraints on input and matrix norms. These bounds are contingent on the rank of each class of matrices. We then apply these…
In this paper, we will prove the non-trivial bound for the weighted average version of shifted convolution sum for $GL(3)\times GL(2)$, i.e. for any $\epsilon >0$ and $X^{1/4+\delta} \leq H \leq X$ with $\delta >0$, \[…
The linearization coefficient $\mathcal{L}(L_{n_1}(x)\dots L_{n_k}(x))$ of classical Laguerre polynomials $L_n(x)$ is known to be equal to the number of $(n_1,\dots,n_k)$-derangements, which are permutations with a certain condition.…