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We prove a stratification result for certain families of $n$-dimensional (complete algebraic) multiplicative character sums. The character sums we consider are sums of products of $r$ multiplicative characters evaluated at rational…
Sum-rank codes have wide applications in multishot network coding, distributed storage and the construction of space-time codes. Asymptotically good sequences of linearized algebraic geometry sum-rank codes, exceeding the…
Sum-networks are networks where all the terminals demand the sum of the symbols generated at the sources. It has been shown that for any finite set/co-finite set of prime numbers, there exists a sum-network which has a vector linear…
Complex polynomial optimization has recently gained more and more attention in both theory and practice. In this paper, we study the optimization of a real-valued general conjugate complex form over various popular constraint sets including…
Automatic abstractive text summarization is an important and challenging research topic of natural language processing. Among many widely used languages, the Chinese language has a special property that a Chinese character contains rich…
The main goal of this paper is to present the application of structural sums, mathematical objects originating from the computational materials science, in construction of a feature space vector of 2D random composites simulated by…
Modular composition is the problem of computing the coefficient vector of the polynomial $f(g(x)) \bmod h(x)$, given as input the coefficient vectors of univariate polynomials $f$, $g$, and $h$ over an underlying field $\mathbb{F}$. While…
This paper presents a quantum algorithm for efficiently computing partial sums and specific weighted partial sums of quantum state amplitudes. Computation of partial sums has important applications, including numerical integration,…
Abelian codes and complementary dual codes form important classes of linear codes that have been extensively studied due to their rich algebraic structures and wide applications. In this paper, a family of abelian codes with complementary…
In this work, we present a new Vector Space Model (VSM) of speech utterances for the task of spoken dialect identification. Generally, DID systems are built using two sets of features that are extracted from speech utterances; acoustic and…
For a nonempty finite set $A$ of positive integers, let $\gcd\left(A\right)$ denote the greatest common divisor of the elements of $A$. Let $f\left(n\right)$ and $\Phi\left(n\right)$ denote, respectively, the number of subsets $A$ of…
We obtain a new upper bound for binary sums with multiplicative characters over variables belong to some sets, having small additive doubling.
There is an increased interest in solving complex constrained problems where part of the input is not given as facts but received as raw sensor data such as images or speech. We will use "visual sudoku" as a prototype problem, where the…
Fitting geometric models onto outlier contaminated data is provably intractable. Many computer vision systems rely on random sampling heuristics to solve robust fitting, which do not provide optimality guarantees and error bounds. It is…
In this paper, we initiate a study of a new problem termed function computation on the reconciled data, which generalizes a set reconciliation problem in the literature. Assume a distributed data storage system with two users $A$ and $B$.…
The challenge of automatically synthesizing high-quality vector fonts, particularly for writing systems (e.g., Chinese) consisting of huge amounts of complex glyphs, remains unsolved. Existing font synthesis techniques fall into two…
A few elementary estimates of a basic character sum over the prime numbers are derived here. These estimates are nontrivial for character sums modulo large q. In addition, an omega result for character sums over the primes is also included.
We give an efficient algorithm to evaluate a certain class of exponential sums, namely the periodic, quadratic, multivariate half Gauss sums. We show that these exponential sums become $\#\mathsf{P}$-hard to compute when we omit either the…
Given a boolean n by n matrix A we consider arithmetic circuits for computing the transformation x->Ax over different semirings. Namely, we study three circuit models: monotone OR-circuits, monotone SUM-circuits (addition of non-negative…
In this paper, we extend the work of (Abbondati et al., 2024) on decoding simultaneous rational number codes by addressing two important scenarios: multiplicities and the presence of bad primes (divisors of denominators). First, we…