Related papers: Identifying Maximal Non-Redundant Integer Cone Gen…

The number of non-negative integer matrices with given row and column sums appears in a variety of problems in mathematics and statistics but no closed-form expression for it is known, so we rely on approximations of various kinds. Here we…

This paper tackles the reduction of redundant repeating generation that is often observed in RNN-based encoder-decoder models. Our basic idea is to jointly estimate the upper-bound frequency of each target vocabulary in the encoder and…

Let $d(G)$ be the smallest cardinality of a generating set of a finite group $G.$ We give a complete classification of the finite groups with the property that, whenever $ \langle x_1, \dots, x_{d(G)} \rangle = \langle y_1, \dots, y_{d(G)}…

A subset $\{g_1, \ldots , g_d\}$ of a finite group $G$ invariably generates $G$ if the set $\{g_1^{x_1}, \ldots, g_d^{x_d}\}$ generates $G$ for every choice of $x_i \in G$. The Chebotarev invariant $C(G)$ of $G$ is the expected value of the…

Power system operators must ensure that dispatch decisions remain feasible in case of grid outages or contingencies to prevent cascading failures and ensure reliable operation. However, checking the feasibility of all $N - k$ contingencies…

The integer division of a numerator n by a divisor d gives a quotient q and a remainder r. Optimizing compilers accelerate software by replacing the division of n by d with the division of c * n (or c * n + c) by m for convenient integers c…

The integration of intermittent renewable energy sources into distribution networks introduces significant uncertainties and fluctuations, challenging their operational security, stability, and efficiency. This paper considers robust…

Fix a prime $p$ and an integer $m$ with $p> m \geq 2$. Define the family of finite groups \[ G_n :=SL_m (\mathbb{Z}/p^{n}\mathbb{Z}) \] for $n=1,2,... $. We will prove that there exist two positive constants $C$ and $d$ such that for any…

For a positive integer $N$, we define the N-rank of a non singular integer $d\times d$ matrix $A$ to be the maximum integer $r$ such that there exists a minor of order $r$ whose determinant is not divisible by $N$. Given a positive integer…

We present an algorithm to find an integral vector in the polyhedral cone $\Gamma=\{X | \textbf{A}X \leq \textbf{0}\}$, without assuming the explicit knowledge of $\textbf{A}$. About the polyhedral cone, $\Gamma$, it is only given that, (i)…

We consider the following problem: Given a matrix A, find minimal subsets of columns of A with cardinality no larger than a given bound that are linear dependent or nearly so. This problem arises in various forms in optimization, electrical…

Does a given system of linear equations with nonnegative constraints have an integer solution? This is a fundamental question in many areas. In statistics this problem arises in data security problems for contingency table data and also is…

Erd\H{o}s and West (Discrete Mathematics'85) considered the class of $n$ vertex intersection graphs which have a {\em $d$-dimensional} {\em $t$-representation}, that is, each vertex of a graph in the class has an associated set consisting…

Sets of $d\times d$ matrices sharing a common invariant cone enjoy special properties, which are widely used in applications. However, finding this cone or even proving its existence/non-existence is hard. This problem is known to be…

In quasi-Monte Carlo methods, generating high-dimensional low discrepancy sequences by generator matrices is a popular and efficient approach. Historically, constructing or finding such generator matrices has been a hard problem. In…

Edge-girth-regular graphs (abbreviated as $egr$ graphs) are a class of highly regular graphs. More specifically, for integers $v$, $k$, $g$ and $\lambda$ an $egr(v,k,g,\lambda)$ graph is a $k$-regular graph with girth $g$ on $v$ vertices…

Let $r_5(N)$ be the largest cardinality of a set in $\{1,\ldots,N\}$ which does not contain $5$ elements in arithmetic progression. Then there exists a constant $c\in (0,1)$ such that \[r_5(N)\ll \frac{N}{\exp((\log\log N)^{c})}.\] Our work…

We investigate the problem of determining a set S of k indistinguishable integers in the range [1,n]. The algorithm is allowed to query an integer $q\in [1,n]$, and receive a response comparing this integer to an integer randomly chosen…

This paper introduces a new hyper-parameter for Retrieval-Augmented Generation (RAG) systems called Context Window Utilization. RAG systems enhance generative models by incorporating relevant information retrieved from external knowledge…

Many mathematical optimization algorithms fail to sufficiently explore the solution space of high-dimensional nonlinear optimization problems due to the curse of dimensionality. This paper proposes generative models as a complement to…