Sum Problem Research Articles

Solving the subset sum problem with a nonideal biological computer

We consider the solution of the subset sum problem based on a parallel computer consisting of self-propelled biological agents moving in a nanostructured network that encodes the computing task in its geometry. We develop an approximate analytical method to analyze the effects of small errors in the nonideal junctions composing the computing network by using a Gaussian confidence interval approximation of the multinomial distribution. We concretely evaluate the probability distribution for error-induced paths and determine the minimal number of agents required to obtain a proper solution. We finally validate our theoretical results with exact numerical simulations of the subset sum problem for different set sizes and error probabilities, and discuss the scalability of the nonideal problem using realistic experimental error probabilities.

On the subset sum problem for finite fields

Let G be the additive group of a finite field. J. Li and D. Wan determined the exact number of solutions of the subset sum problem over G, by giving an explicit formula for the number of subsets of G of prescribed size whose elements sum up to a given element of G. They also determined a closed-form expression for the case where the subsets are required to contain only nonzero elements. In this paper we give an alternative proof of the two formulas. Our argument is purely combinatorial, as in the original proof by Li and Wan, but follows a different and somehow more “natural” approach. We also indicate some new connections with coding theory and combinatorial designs.

Design and fabrication of networks for bacterial computing

Non-deterministic polynomial (NP-) complete problems, whose number of possible solutions grows exponentially with the number of variables, require by necessity massively parallel computation. Because sequential computers, such as solid state-based ones, can solve only small instances of these problems within a reasonable time frame, parallel computation using motile biological agents in nano- and micro-scale networks has been proposed as an alternative computational paradigm. Previous work demonstrated that protein molecular motors-driven cytoskeletal filaments are able to solve a small instance of an NP complete problem, i.e. the subset sum problem, embedded in a network. Autonomously moving bacteria are interesting alternatives to these motor driven filaments for solving such problems, because they are easier to operate with, and have the possible advantage of biological cell division. Before scaling up to large computational networks, bacterial motility behaviour in various geometrical structures has to be characterised, the stochastic traffic splitting in the junctions of computation devices has to be optimized, and the computational error rates have to be minimized. In this work, test structures and junctions have been designed, fabricated, tested, and optimized, leading to specific design rules and fabrication flowcharts, resulting in correctly functioning bio-computation networks.

Researches on summation problem in Dnipropetrovsk State University

Researches on summation problem in Dnipropetrovsk State University.

Approximate query processing over static sets and sliding windows

Indexing of static and dynamic sets is fundamental to a large set of applications such as information retrieval and caching. Denoting the characteristic vector of the set by B, we consider the problem of encoding sets and multisets to support approximate versions of the operations rank(i) (i.e., computing ∑j≤iB[j]) and select(i) (i.e., finding min⁡{p|rank(p)≥i}) queries. We study multiple types of approximations (allowing an error in the query or the result) and present lower bounds and succinct data structures for several variants of the problem. We also extend our model to sliding windows, in which we process a stream of elements and compute suffix sums. This is a generalization of the window summation problem that allows the user to specify the window size at query time. Here, we provide an algorithm that supports updates and queries in constant time while requiring just (1+o(1)) factor more space than the fixed-window summation algorithms.

The monodromy problem for hyperelliptic curves

We study the Dynkin diagrams associated to the monodromy of direct sums of polynomials. The monodromy problem asks under which conditions on a polynomial, the monodromy of a vanishing cycle generates the whole homology of a regular fiber. We consider the case y4+g(x), which is a generalization of the results of Christopher and Mardešić about the monodromy problem for hyperelliptic curves. Moreover, we solve the monodromy problem for direct sums of fourth degree polynomials.

A Novel Path-Following-Method-Based Polynomial Fuzzy Control Design.

This article presents a novel path-following-method-based polynomial fuzzy control design. By examining the stabilization problem, the nonconvex stabilization criterion represented in terms of bilinear sum-of-squares (SOS) constraints is proposed to complement the existing convex stabilization criteria. Based on the polynomial Lyapunov function and considering the operation domain, the stabilization control is designed with a systematic region of attraction (ROA) analysis method. Since the proposed stabilization criterion remains in nonconvex form, the conservativeness caused by the transformation from nonconvex (bilinear SOS) constraints into convex (SOS) constraints can be avoided. Moreover, the restriction on the Lyapunov function candidates for the convex transformation in the literature does not exist in the proposed nonconvex stabilization criterion. The stabilization analysis for polynomial fuzzy control systems is concerned with the double fuzzy summation problem that can be treated as the copositivity problem. Therefore, the SOS-based copositive relaxation technique is applied for the proposed stabilization criterion. Since the proposed nonconvex stabilization criterion is represented in terms of bilinear SOS constraints, the path-following method is employed for solving the bilinear SOS problem. Finally, design examples are provided to demonstrate that the proposed nonconvex stabilization criterion complements the existing convex stabilization criteria.

Number Partitioning With Grover’s Algorithm in Central Spin Systems

Numerous conceptually important quantum algorithms rely on a black-box device known as an oracle, which is typically difficult to construct without knowing the answer to the problem that the algorithm is intended to solve. A notable example is Grover's search algorithm. Here we propose a Grover search for solutions to a class of NP-complete decision problems known as subset sum problems, including the special case of number partitioning. Each problem instance is encoded in the couplings of a set of qubits to a central spin or boson, which enables a realization of the oracle without knowledge of the solution. The algorithm provides a quantum speedup across a known phase transition in the computational complexity of the partition problem, and we identify signatures of the phase transition in the simulated performance. Whereas the naive implementation of our algorithm requires a spectral resolution that scales exponentially with system size for NP-complete problems, we also present a recursive algorithm that enables scalability. We propose and analyze implementation schemes with cold atoms, including Rydberg-atom and cavity-QED platforms.

Two problems of binomial sums involving harmonic numbers

Two open problems recently proposed by Xi and Luo (Adv. Differ. Equ. 2021:38, 2021) are resolved by evaluating explicitly three binomial sums involving harmonic numbers, that are realized mainly by utilizing the generating function method and symmetric functions.

On a Certain Quadratic Character Sums of Ternary Symmetry Polynomials mod p

In this article, we are using the elementary methods and the properties of the classical Gauss sums to study the calculating problem of a certain quadratic character sums of a ternary symmetry polynomials modulo p and obtain some interesting identities for them.

Bi-Objective Competition Pricing Model for Component Web Service Economy

In the web service marketplace, component-based economy has been proposed for describing participants' behavioral patterns. Composite service networks combine multiple composite services required by various service consumers. With each composite service as a product, web services comprise heterogeneous products. In this study, the pricing behavior of networked individual service providers is investigated. With the objective of service survival or high profitability, service providers compete both on the single-service and service-network levels. Using examples, several mild assumptions are formulated and analyzed. Then, a bi-objective optimization model is proposed based on these assumptions, which attempts to maintain a reasonable effectiveness-fairness trade-off from the individual service providers' perspective. The NP-completeness of the single-objective version is demonstrated by transforming the problem into a subset sum problem, which highlights the challenge of obtaining a pareto set for the bi-objective model. Finally, to validate the proposed model, numerical experimentation and case study are conducted, and both the bi-objective and many-objective versions of the problem are discussed.

Putting fractions together.

Learning fractions is a critical step in children’s mathematical development. However, many children struggle with learning fractions, especially fraction arithmetic. In this article, we propose a general framework for integrating understanding of individual fractions and fraction arithmetic, and we use the framework to generate interventions intended to improve understanding of both individual fractions and fraction addition. The framework, Putting Fractions Together (PFT), emphasizes that both individual fractions and sums of fractions are composed of unit fractions and can be represented by concatenating them (putting them together). To illustrate, both “3/9” and “2/9 + 1/9” can be represented by concatenating three 1/9s; similarly, 2/9 + 1/8 can be represented by concatenating two 1/9s and one 1/8. Interventions based on the PFT framework were tested in 2 experiments with fourth, fifth, and sixth grade children. The interventions led to improved performance on number line estimation and magnitude comparison tasks involving individual fractions and sums of fractions with equal and unequal denominators. Especially large improvements were observed on relatively difficult unequal-denominator fraction sum problems. The findings suggest that viewing individual nonunit fractions and sums of fractions as concatenations of unit fractions provides a sound conceptual foundation for improving children’s knowledge of both. We discuss implications of the research for teaching and learning fractions, children’s numerical development, and mathematics education in general. (PsycInfo Database Record (c) 2021 APA, all rights reserved)

The subset sum game revisited

We discuss a game theoretic variant of the subset sum problem, in which two players compete for a common resource represented by a knapsack. Each player owns a private set of items, players pack items alternately, and each player either wants to maximize the total weight of his own items packed into the knapsack or to minimize the total weight of the items of the other player. We show that finding the best packing strategy against a hostile or a selfish adversary is PSPACE-complete, and that against these adversaries the optimal reachable item weight for a player cannot be approximated within any constant factor (unless P=NP). The game becomes easier when the adversary is short-sighted and plays greedily: finding the best packing strategy against a greedy adversary is NP-complete in the weak sense. This variant forms one of the rare examples of pseudo-polynomially solvable problems that have a PTAS, but do not allow an FPTAS (unless P=NP).

On a shifted convolution sum problem

Let f be a holomorphic newform of prime level p and trivial nebentypus. For p1+ε≪M≪p3−ε, and 0<|u|≪M/p we prove that∑m=1∞λf(m)λf(m+pu)F(mM)≪p1/4+εM3/4 where F is a compactly supported smooth bump function.

Генерация второй гармоники-суммарной частоты в тонком сферическом слое. I. Аналитическое решение

The problem of the simultaneous second–harmonic and sum–frequency generation by two coherent plane electromagnetic waves with elliptical polarizations and equal frequencies in a thin spherical layer is solved in the Rayleigh–Gans–Debye model. The second–order dielectric susceptibility tensor is chosen in a form containing four independent components (including one chiral component). Fundamental differences are shown between the problem statements as well as between solutions to the problem of second–order nonlinear generation by several coherent electromagnetic waves and the problem of sum–frequency generation in the limiting case of equality of the frequencies of the incident waves. Combinations of parameters are determined at which the spatial distribution of the radiation generated as a result of incidence of two plane waves coincides with the distribution of second–harmonic radiation generated by one plane wave. The limiting cases of the obtained solution are considered: small and large radii of the spherical particle. In these cases the contributions of the chiral and non-chiral anisotropy coefficients to the generated radiation are determined.

Universal Nonlinear Spiking Neural P Systems with Delays and Weights on Synapses.

The nonlinear spiking neural P systems (NSNP systems) are new types of computation models, in which the state of neurons is represented by real numbers, and nonlinear spiking rules handle the neuron's firing. In this work, in order to improve computing performance, the weights and delays are introduced to the NSNP system, and universal nonlinear spiking neural P systems with delays and weights on synapses (NSNP-DW) are proposed. Weights are treated as multiplicative constants by which the number of spikes is increased when transiting across synapses, and delays take into account the speed at which the synapses between neurons transmit information. As a distributed parallel computing model, the Turing universality of the NSNP-DW system as number generating and accepting devices is proven. 47 and 43 neurons are sufficient for constructing two small universal NSNP-DW systems. The NSNP-DW system solving the Subset Sum problem is also presented in this work.

Modular arithmetic and subset sum problem: a state-of-art technique in information security issues towards smart vehicular system

Modular arithmetic and subset sum problem: a state-of-art technique in information security issues towards smart vehicular system

Modular arithmetic and subset sum problem: a state-of-art technique in information security issues towards smart vehicular system

Modular arithmetic and subset sum problem: a state-of-art technique in information security issues towards smart vehicular system

Voronoï summation for GLn: collusion between level and modulus

We investigate the Voronoi summation problem for ${\rm GL}_n$ in the level aspect for $n\geq 2$. Of particular interest are those primes at which the level and modulus are jointly ramified - a common occurrence in analytic number theory when using techniques such as the Petersson trace formula. Building on previous legacies, our formula stands as the most general of its kind; in particular we extend the results of Ichino-Templier. We also give (classical) refinements of our formula and study the $p$-adic generalisations of the Hankel transform.

On generic complexity of the subset sum problem in monoids and groups of integer matrix of order two

Generic-case approach to algorithmic problems was suggested by A. Miasnikov, I. Kapovich, P. Schupp and V. Shpilrain in 2003. This approach studies behavior of an algo-rithm on typical (almost all) inputs and ignores the rest of inputs. In this paper, we prove that the subset sum problems for the monoid of integer positive unimodular matrices of the second order, the special linear group of the second order, and the modular group are generically solvable in polynomial time.