On the size and structure of certain subsequence sum set

Abstract The set of sums of two squares plays a significant role in number theory. We establish the existence of several rich monochromatic configurations in the natural numbers by exploiting algebraic structures induced by the set of sums of two squares. The proofs rely on algebraic properties arising from the induced structures on the Stone–Čech compactification of the natural numbers.

Abstract We provide an observability inequality in terms of a measurable set for general Gevrey‐regular functions. As an application, we establish an observability estimate from a measurable set for sums of Laplace eigenfunctions in a compact and connected boundaryless Riemannian manifold that belongs to the Gevrey class. The estimate has an explicit dependence on the maximal eigenvalue.

Increasing reliability and reducing disruptions in supply networks are of increasing importance; for example, power outages in electricity distribution networks cost $35–$50 billion annually in the United States. Motivated by the operational constraints of such networks and their rapid adoption of decentralized paradigms and self-healing components, we introduce the minimum reconnection time (MRT) problem, which models reliability metrics such as the System Average Interruption Duration Index (SAIDI). MRT seeks to reduce outage time after network disruptions by programming reconnection times of different edges (i.e., switches), ensuring that the operating network is acyclic. We show that MRT is NP-hard and is a special case of the well-known (weighted) minimum sum set cover (MSSC) problem. We develop the theory of kernel-based randomized rounding approaches to give a tight polynomial-time approximation for MSSC, improving the state-of-the-art approximation factor for these instances. Further, motivated by the reliability incentive structure for utility companies and operational energy losses in distribution networks, we study minimizing energy losses and reliability metrics such as SAIDI and reconnection times simultaneously. Optimizing for any single objective at a time can create unfair duration of expected outage for industrial and residential areas. We, therefore, propose local search over spanning trees to balance these multiple objectives. We computationally validate our reconfiguration methods on the National Renewable Energy Laboratory Synthetic Models for Advanced, Realistic Testing: Distribution Systems and Scenarios Greensboro synthetic network and show that this improves equity by a factor of four across industrial and residential areas. History: Accepted by Erwin Pesch, Area Editor for Heuristic Search & Approximation Algorithms. Funding: C. Hettle and D. Molzahn's research was partially supported by the National Science Foundation [Grant NSF 2112533], received by the Georgia Institute of Technology. S. Gupta’s research is supported by the National Science Foundation CAREER [Grant 2239824], received by the Massachusetts Institute of Technology. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0295 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0295 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

The determination of tungsten oxidation states and W-ligand bond lengths for pyranopterin tungsten enzymes can be negatively impacted by Fourier series termination effects and photodamage/photoreduction in the X-ray beam. As a result, a new set of bond valence sum (BVS) parameters have been derived from bond length data on W(+4) and W(+6) model compounds that were obtained from X-ray crystallography. These new W enzyme-specific BVS parameters have been used in the analysis of pyranopterin tungsten enzyme structural data. The results of this analysis indicate that there are potential issues with the enzyme crystal structures, including the number of ligating atoms to the tungsten atom, the W-ligand bond lengths, and the W oxidation state. We conclude that a BVS analysis of crystallographic and EXAFS structural data will help address these issues, and EXAFS should be more routinely employed in the determination of pyranopterin tungsten enzyme active site structures due to the increased accuracy of this technique for the determination of W-ligand bond distances.

For an undirected graph G, a zero-sum flow is an assignment of nonzero integer weights to the edges such that each vertex has a zero-sum, namely the sum of all incident edge weights with each vertex is zero. This concept is an undirected analog of nowhere-zero flows for directed graphs. We study a more general one, namely constant-sum A-flows, which gives edge weights using nonzero elements of an additive Abelian group A and requires each vertex to have a constant-sum instead. In particular, we focus on two special cases: A=Zk, the finite cyclic group of integer congruence modulo k, and A=Z, the infinite cyclic group of integers. The constant sum under a constant-sum A-flow is called an index of G for short, and the set of all possible constant sums (indices) of G is called the constant sum spectrum. It is denoted by Ik(G) and I(G) for A=Zk and A=Z, respectively. The zero-sum flows and constant-sum group flows for regular graphs regarding cases Z and Zk have been studied extensively in the literature over the years. In this article, we study the constant sum spectrum of nearly regular graphs such as wheel graphs Wn and fan graphs Fn in particular. We completely determine the constant-sum spectrum of fan graphs and wheel graphs concerning Zk and Z, respectively. Some open problems will be mentioned in the concluding remarks.

Let [Formula: see text] be an odd prime, [Formula: see text] be the set of quadratic residues modulo [Formula: see text], and [Formula: see text] be the set of quadratic non-residues modulo [Formula: see text]. In this paper, we observe that the problem of finding the number of solutions of [Formula: see text] over [Formula: see text], where [Formula: see text] is equivalent to the problem of counting the multiplicity (see Definition 1) of [Formula: see text], [Formula: see text], or zero in the set of sums of [Formula: see text] quadratic residues, which are not necessarily distinct, depending on whether [Formula: see text] is a quadratic residue, a non-residue, or zero, respectively. Thus, we count the multiplicity of [Formula: see text], [Formula: see text], or zero in the set of sums of [Formula: see text] quadratic residues using a graphical approach. In addition, we compute the number of times the sum of [Formula: see text] squares and [Formula: see text] non-squares over [Formula: see text] yields a square, non-square, or zero. Furthermore, we obtain an explicit formula for the sum of [Formula: see text] distinct quadratic residues (non-residues) modulo [Formula: see text], where [Formula: see text]. The results obtained in this study are useful in the factorization of the polynomial [Formula: see text] over [Formula: see text], where [Formula: see text] is a quadratic residue modulo [Formula: see text], and are also helpful in developing time-efficient computer programs.

This research paper pioneers an innovative extensions of antimultigroup theory by seemingly integrating the concept of cuts and comultiset, thereby revolutionizing the field. Notably, we demonstrate that the root sets of antimultigroup sums and differences are subgroups, uncovering a profound connection. Furthermore, we establish that if is a complete sub-antimultigroup of such that all the counts in are factors of their corresponding counts in . Finally, we prove that the cuts of antimultigroup unions and intersections also form subgroups, further enriching our understanding of these complex structures.

An approach to the approximation and analysis of the CT density signal component associated with intravascular radiocontrast agent (RCA) based on computed tomography angiography (CTA) images of the abdominal aorta is presented. The aim of the work is to study the possibility of extracting and analyzing the RCA-induced component in the lumen and wall of the abdominal aorta on the CTA image. A functionality for describing one-dimensional and two-dimensional distribution of the CTA as a set of sums of sigmoid of a special type is proposed. The nonlinear least squares method with Levenberg – Marquardt optimization is used for approximation. The algorithm is tested on an open data set consisting of 594 CTA images. Data preparation is performed using specialized software Slicer 3D. The results demonstrate the absence of statistically significant differences in the CTA density values between the original images and the approximation results (p> 0.05, paired Wilcoxon test). The sensitivity of the model to different distributions of the RCA in the area of aneurysm, thrombosis and origin of the main arteries is demonstrated. Sensitivity is defined as the presence of statistically significant differences in the calculated parameters of the model for the area of homogeneous and non-homogeneous distribution of the RCA within each of the CT studies. The values of the root-mean-square approximation error for the specified areas do not differ statistically significantly and are unimodally distributed (p > 0.7) within a single CT study. The proposed approach can be useful for personalizing CTA, developing algorithms for processing CTA data, synthesizing non-contrast CT data, and training artificial intelligence algorithms.

The goal of discrete tomography is to reconstruct an unknown function $f$ via a given set of line sums. In addition to requiring accurate reconstructions, it is favourable to be able to perform the task in a timely manner. This is complicated by the presence of ghosts, which allow many solutions to exist in general. In this paper we consider the case of a function $f : A \to \mathbb{R}$ where $A$ is a finite grid in $\mathbb{Z}^3$. Previous work has shown that in the two-dimensional case it is possible to determine all solutions in parameterized form in linear time (with respect to the number of directions and the grid size) regardless of whether the solution is unique. In this work, we show that a similar linear method exists in three dimensions under the condition of nonproportionality. We show that the condition of nonproportionality is fulfilled in the case of three-dimensional boundary ghosts.

A note on the exceptional set for sums of unlike powers of primes

We study weighted sums of free identically distributed self-adjoint random variables with weights chosen randomly from the unit sphere and show that the Kolmogorov distance between the distribution of such a weighted sum and Wigner’s semicircle law is of order n-12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n^{-\\frac{1}{2}}$$\\end{document} with high probability. Replacing the Kolmogorov distance by a weaker pseudometric, we obtain a rate of convergence of order n-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n^{-1}$$\\end{document}, thus providing a free analog of the Klartag-Sodin result in classical probability theory. Moreover, we show that our ideas generalize to the setting of sums of free non-identically distributed bounded self-adjoint random variables leading to a new rate of convergence in the free central limit theorem.

Let [Formula: see text] be a group with a metric invariant under left and right translations, and let [Formula: see text] be the ball of radius [Formula: see text] around the identity. A [Formula: see text]-metric approximate subgroup is a symmetric subset [Formula: see text] of [Formula: see text] such that the pairwise product set [Formula: see text] is covered by at most [Formula: see text] translates of [Formula: see text]. This notion was introduced in [T. Tao, Product set estimates for noncommutative groups, Combinatorica, 28(5) (2008) 547–594, doi:10.1007/s00493-008-2271-7; T. Tao, Metric entropy analogues of sum set theory (2014), https://terrytao.wordpress.com/2014/03/19/metric-entropy-analogues-of-sum-set-theory/] along with the version for discrete groups (approximate subgroups). In [E. Hrushovski, Stable group theory and approximate subgroups, J. Amer. Math. Soc. 25(1) (2012) 189–243, doi:10.1090/S0894-0347-2011-00708-X], it was shown for the discrete case that, at the asymptotic limit of [Formula: see text] finite but large, the “approximateness” (or need for more than one translate) can be attributed to a canonically associated Lie group. Here we prove an analogous result in the metric setting, under a certain finite covering assumption on [Formula: see text] replacing finiteness. In particular, if [Formula: see text] has bounded exponent, we show that any [Formula: see text]-metric approximate subgroup is close to a [Formula: see text]-metric approximate subgroup for an appropriate [Formula: see text].

Two problems on subset sums

We prove that for each u ⩾ 2 the class of all single-valued Σ0 ucomputable numberings of any infinite family of total functions is effectively infinite and the class of all its Σ0 u-1-computable numberings is generated by the downward closure with respect to the reducibility of the set of all infinite direct sums of uniformly Σ0 u-1-computable sequences of its single-valued numberings. It is established that if u > 2, then the class of all Σ0 u-computable numberings of any infinite family is generated by infinite direct sums of uniformly Σ0 u-computable and uniformly Σ0 u-minimal sequences of its numberings.

For a given sequence [Formula: see text] of nonnegative integers, let [Formula: see text] be the set of all finite subsequence sums of [Formula: see text]. [Formula: see text] is called complete if [Formula: see text] contains all sufficiently large integers. A real number [Formula: see text] is called as an infinite diadical fraction (briefly i.d.f.) if the digit 1 appears infinitely many times in the binary representation of [Formula: see text]. Hegyvári conjectured that [Formula: see text] is complete if [Formula: see text] or [Formula: see text] is i.d.f. and [Formula: see text], where [Formula: see text] is a sequence of integers. In this paper, we give a partial result of Hegyvári’s conjecture.

This paper explores the relationship between convexity and sum sets. In particular, we show that elementary number theoretical methods, principally the application of a squeezing principle, can be augmented with the Elekes-Szabó Theorem in order to give new information. Namely, if we let $A \subset \mathbb R$, we prove that there exist $a,a' \in A$ such that\[\left | \frac{(aA+1)^{(2)}(a'A+1)^{(2)}}{(aA+1)^{(2)}(a'A+1)} \right | \gtrsim |A|^{31/12}.\]We are also able to prove that\[\max \{|A+ A-A|, |A^2+A^2-A^2|, |A^3 + A^3 - A^3|\} \gtrsim |A|^{19/12}.\]Both of these bounds are improvements of recent results and takes advantage of computer algebra to tackle some of the computations.

In this paper, let $ q $ be a power of a prime, we construct several classes of new projective three-weight or four-weight linear codes over $ \mathbb{F}_q $ from the defining sets construction, and determine their weight distributions by using additive character sums. Especially, these codes are suitable for applications in secret sharing schemes. Furthermore, basing on two classes of projective three-weight codes, we construct some $ s $-sum sets for any odd $ s>1 $.

<p style='text-indent:20px;'>In this paper, for any odd prime <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula> and an integer <inline-formula><tex-math id="M2">\begin{document}$ m\ge 3 $\end{document}</tex-math></inline-formula>, several classes of linear codes with <inline-formula><tex-math id="M3">\begin{document}$ t $\end{document}</tex-math></inline-formula>-weight <inline-formula><tex-math id="M4">\begin{document}$ (t = 3,5,7) $\end{document}</tex-math></inline-formula> are obtained based on some defining sets, and then their complete weight enumerators are determined explicitly by employing Gauss sums and quadratic character sums. Especially for <inline-formula><tex-math id="M5">\begin{document}$ m = 3 $\end{document}</tex-math></inline-formula>, a class of MDS codes with parameters <inline-formula><tex-math id="M6">\begin{document}$ [p,3,p-2] $\end{document}</tex-math></inline-formula> are obtained. Furthermore, some of these codes can be suitable for applications in secret sharing schemes and <inline-formula><tex-math id="M7">\begin{document}$ s $\end{document}</tex-math></inline-formula>-sum sets for any odd <inline-formula><tex-math id="M8">\begin{document}$ s>1 $\end{document}</tex-math></inline-formula>.</p>

Abstract The minimum linear ordering problem (MLOP) generalizes well-known combinatorial optimization problems such as minimum linear arrangement and minimum sum set cover. MLOP seeks to minimize an aggregated cost $$f(\cdot )$$ f ( · ) due to an ordering $$\sigma $$ σ of the items (say [n]), i.e., $$\min _{\sigma } \sum _{i\in [n]} f(E_{i,\sigma })$$ min σ ∑ i ∈ [ n ] f ( E i , σ ) , where $$E_{i,\sigma }$$ E i , σ is the set of items mapped by $$\sigma $$ σ to indices [i]. Despite an extensive literature on MLOP variants and approximations for these, it was unclear whether the graphic matroid MLOP was NP-hard. We settle this question through non-trivial reductions from mininimum latency vertex cover and minimum sum vertex cover problems. We further propose a new combinatorial algorithm for approximating monotone submodular MLOP, using the theory of principal partitions. This is in contrast to the rounding algorithm by Iwata et al. (in: APPROX, 2012), using Lovász extension of submodular functions. We show a $$(2-\frac{1+\ell _{f}}{1+|E|})$$ ( 2 - 1 + ℓ f 1 + | E | ) -approximation for monotone submodular MLOP where $$\ell _{f}=\frac{f(E)}{\max _{x\in E}f(\{x\})}$$ ℓ f = f ( E ) max x ∈ E f ( { x } ) satisfies $$1 \le \ell _f \le |E|$$ 1 ≤ ℓ f ≤ | E | . Our theory provides new approximation bounds for special cases of the problem, in particular a $$(2-\frac{1+r(E)}{1+|E|})$$ ( 2 - 1 + r ( E ) 1 + | E | ) -approximation for the matroid MLOP, where $$f = r$$ f = r is the rank function of a matroid. We further show that minimum latency vertex cover is $$\frac{4}{3}$$ 4 3 -approximable, by which we also lower bound the integrality gap of its natural LP relaxation, which might be of independent interest.