Multiple Sums Research Articles

Multiple Magnetorotons and Spectral Sum Rules in Fractional Quantum Hall Systems.

We study, numerically, the charge neutral excitations (magnetorotons) in fractional quantum Hall systems, concentrating on the two Jain states near quarter filling, ν=2/7 and ν=2/9, and the ν=1/4 Fermi-liquid state itself. In contrast to the ν=1/3 states and the Jain states near half filling, on each of the two Jain states ν=2/7 and ν=2/9 the graviton spectral densities show two, instead of one, magnetoroton peaks. The magnetorotons have spin 2 and have opposite chiralities in the ν=2/7 state and the same chirality in the ν=2/9 state. We also provide a numerical verification of a sum rule relating the guiding center spin s[over ¯] with the spectral densities of the stress tensor.

Repeated integration and explicit formula for the \(n\)-th integral of \(x^m(\ln x)^{m'}\)

Repeated integration is a major topic of integral calculus. In this article, we study repeated integration. In particular, we study repeated integrals and recurrent integrals. For each of these integrals, we develop reduction formulae for both the definite as well as indefinite form. These reduction formulae express these repetitive integrals in terms of single integrals. We also derive a generalization of the fundamental theorem of calculus that expresses a definite integral in terms of an indefinite integral for repeated and recurrent integrals. From the recurrent integral formulae, we derive some partition identities. Then we provide an explicit formula for the \(n\)-th integral of \(x^m(\ln x)^{m'}\) in terms of a shifted multiple harmonic star sum. Additionally, we use this integral to derive new expressions for the harmonic sum and repeated harmonic sum.

Critical points of solutions to a kind of linear elliptic equations in multiply connected domains

In this paper, we mainly study the critical points and critical zero points of solutions u to a kind of linear elliptic equations with nonhomogeneous Dirichlet boundary conditions in a multiply connected domain Ω in ℝ2. Based on the delicate analysis about the distributions of connected components of the super-level sets {x ∈ Ω: u(x) > t} and sub-level sets {x ∈ Ω: u(x) < t} for some t, we obtain the geometric structure of interior critical point sets of u. Precisely, let Ω be a multiply connected domain with the interior boundary γI and the external boundary γE, where u∣γI = ψ1(x), u∣γE = ψ2(x). When ψ1(x) and ψ2(x) have N1 and N2 local maximal points on γI and γE respectively, we deduce that \(\sum\nolimits_{i = 1}^k {{m_i} \le {N_1} + {N_2}} \), where m1,…,mk are the respective multiplicities of interior critical points x1,…,xk of u. In addition, when \({\min _{{\gamma _E}}}{\psi _2}\left( x \right) \ge {\max _{{\gamma _I}}}{\psi _1}\left( x \right)\) and u has only N1 and N2 equal local maxima relative to \(\overline \Omega \) on γI and γe respectively, we develop a new method to show that one of the following three results holds \(\sum\nolimits_{i = 1}^k {{m_i} = {N_1} + {N_2}} \) or \(\sum\nolimits_{i = 1}^k {{m_i} + 1 = {N_1} + {N_2}} \) or \(\sum\nolimits_{i = 1}^k {{m_i} + 2 = {N_1} + {N_2}} \). Moreover, we investigate the geometric structure of interior critical zero points of u. We obtain that the sum of multiplicities of the interior critical zero points of u is less than or equal to the half of the number of its isolated zero points on the boundaries.

A generalization of multiple zeta values. Part 2: Multiple sums

{\bf Abstract:} Multiple zeta values have become of great interest due to their numerous applications in mathematics and physics. In this article, we present a generalization, which we will refer to as multiple sums, where the reciprocals are replaced with arbitrary sequences. We develop formulae to help with manipulating such sums. We develop variation formulae that express the variation of multiple sums in terms of lower order multiple sums. Additionally, we derive a set of partition identities that we use to prove a reduction theorem that expresses multiple sums as a combination of simple sums. We present a variety of applications including applications concerning polynomials and MZVs such as generating functions and expressions for $\zeta(\{2p\}_m)$ and $\zeta^\star(\{2p\}_m)$. Finally, we establish the connection between multiple sums and a type of sums called recurrent sums. By exploiting this connection, we provide additional partition identities for odd and even partitions.

A generalization of multiple zeta values. Part 1: Recurrent sums

Multiple zeta star values have become a central concept in number theory with a wide variety of applications. In this article, we propose a generalization, which we will refer to as recurrent sums, where the reciprocals are replaced by arbitrary sequences. We introduce a toolbox of formulas for the manipulation of such sums. We begin by developing variation formulas that allow the variation of a recurrent sum of order $m$ to be expressed in terms of lower order recurrent sums. We then proceed to derive theorems (which we will call inversion formulas) which show how to interchange the order of summation in a multitude of ways. Later, we introduce a set of new partition identities in order to then prove a reduction theorem which permits the expression of a recurrent sum in terms of a combination of non-recurrent sums. Finally, we use these theorems to derive new results for multiple zeta star values and recurrent sums of powers.

Mixing second- and third-order nonlinear interactions in nanophotonic lithium-niobate waveguides

In this paper, we have investigated the interplay between the second and third-order nonlinearities in lithium-niobate waveguides with strong waveguide dispersion using uniform and linearly-chirped poling patterns at input powers in the pico-joule range. We have implemented the accurate unidirectional pulse propagation model to take into account all the possible nonlinear interactions inside these structures. In particular, the poling period has been designed to quasi-phase-match single and multiple sum- and difference-frequency generation processes. We have shown how the poling period can be used as an additional degree of freedom to tailor the output spectra of chip-based nonlinear waveguides in an unprecedented way.

Zonotopic interval estimation for discrete-time Takagi–Sugeno fuzzy systems with a delayed nonquadratic framework

This paper investigates the delayed nonquadratic zonotopic interval estimation for discrete-time Takagi–Sugeno (T-S) fuzzy systems. Due to the improvements made on traditional quadratic methods, the estimation accuracy is enhanced and the conservatism is efficiently reduced. Firstly, based on multiple sums and multisets of delays, an overall nonquadratic zonotopic estimation framework with tuneable delayed terms is proposed. A general observer gain computation method with L∞ performance is derived in terms of matrix inequalities shortly afterward. The design conditions are relaxed by selecting a group of specific multisets of delays that the observer gain acquisition problem is transformed into solving a set of linear matrix inequalities (LMIs). Furthermore, the rules for selecting appropriate multisets of delays, which limit the computational complexity and improve the design freedom, are analyzed. The feasibility, effectiveness, and practicability of the proposed method are illustrated by two examples finally.

Tooth instance segmentation from cone-beam CT images through point-based detection and Gaussian disentanglement

Individual tooth segmentation and identification from cone-beam computed tomography images are preoperative prerequisites for orthodontic treatments. Instance segmentation methods using convolutional neural networks have demonstrated ground-breaking results on individual tooth segmentation tasks, and are used in various medical imaging applications. While point-based detection networks achieve superior results on dental images, it is still a challenging task to distinguish adjacent teeth because of their similar topologies and proximate nature. In this study, we propose a point-based tooth localization network that effectively disentangles each individual tooth based on a Gaussian disentanglement objective function. The proposed network first performs heatmap regression accompanied by box regression for all the anatomical teeth. A novel Gaussian disentanglement penalty is employed by minimizing the sum of the pixel-wise multiplication of the heatmaps for all adjacent teeth pairs. Subsequently, individual tooth segmentation is performed by converting a pixel-wise labeling task to a distance map regression task to minimize false positives in adjacent regions of the teeth. Experimental results demonstrate that the proposed algorithm outperforms state-of-the-art approaches by increasing the average precision of detection by 9.1%, which results in a high performance in terms of individual tooth segmentation. The primary significance of the proposed method is two-fold: (1) the introduction of a point-based tooth detection framework that does not require additional classification and (2) the design of a novel loss function that effectively separates Gaussian distributions based on heatmap responses in the point-based detection framework.

The additive index of polynomials over finite fields

In this paper we introduce the additive analogue of the index of a polynomial over finite fields. We show that every polynomial P(x)∈Fq[x] can be expressed uniquely in its additive index form such that P(x)=f(L(x))+M(x) where L(x),M(x) are p-linearized polynomials over Fq, deg⁡(M)<deg⁡(L), L(x) splits completely over Fq and L is of the maximal degree. As applications, we study several problems in the theory of polynomials over finite fields in terms of their additive indices, such as value set sizes, bounds on multiplicative character sums, and characterization of permutation polynomials.

Retracted] On Multiplicative Topological Invariants of Magnesium Iodide Structure

In recent times, the applications of graph theory in molecular and chemical structure research have far exceeded human expectations and have grown exponentially. In this paper, we have determined the multiplicative Zagreb indices, multiplicative hyper‐Zagreb indices, multiplicative universal Zagreb indices, sum and product connectivity of multiplicative indices, multiplicative atom‐bond connectivity index, and multiplicative geometric‐arithmetic index of a famous crystalline structure, magnesium iodide (MgI2).

Nov kombinatorni dokaz identiteta sa višestrukim binomnim koeficijentima

Introduction/purpose: In this paper a new combinatorial proof of an already existing multiple sum with multiple binomial coefficients is given. The derived identity is related to the Fibonacci numbers. Methods: Combinatorial reasoning is used to obtain the results. Results: The already known identity was obtained by using a new combinatorial reasoning. Conclusions: The new combinatorial reasoning led to the solution of the already existing identity.

Treatment of wastewater by slow sand process with polystyrene

The ability of our project to treat the sewage treatment plant (STP) water by using slow sand filter (SSF) to remove the fine suspended particles, microbes, minerals and flocs present in effluents. So the generation and disposal of STP effluent is high. It will damage the environment in the case of presence fine minerals and create bad odour to the environment. Therefore, treatment of effluent is essential. Somewhere it is a better and substitute technique for the removal of microbes, minerals in water by means of lively coagulant mediators as polystyrene and to take away the ruthless aroma via a eucalyptus greenery. In this process we enhance grinded eucalyptus paste and well detached polystyrene in directive to attain our results in respectable. SSF have been exposed to efficiently advance the water superiority. Through pleasing multiple sum of testes from engaged sample it displays that the water is polluted and not proper for irrigation resolves. So directed simple physical and chemical parameters of pH, Turbidity, Conductivity, COD, BOD, DO Tests. In the SSF technique the polystyrene is in one layer which is acts as a filter media. In this education was carried out SSF filter as a main research work and we achieved the Turbidity, TSS, BOD, Oil & Grease and COD as15.5 NTU, 97.85%, 84.67%, 98.8% and 97.65% correspondingly. It is originating that SSF with polystyrene process the finest near of water quality was better-quality and its well suitable for irrigation purposes.

Euler-type sums involving multiple harmonic sums and binomial coefficients

Abstract In this paper, we mainly show that generalized Euler-type sums of multiple harmonic sums with reciprocal binomial coefficients can be expressed in terms of rational linear combinations of products of classical multiple zeta values (MZVs) and multiple harmonic star sums (MHSSs). Furthermore, applying the stuffle relations, we prove that the Euler-type sums involving products of generalized harmonic numbers and reciprocal binomial coefficients can be evaluated by MZVs and MHSSs.

Algunos procesos matemáticos utilizados en la antigua civilización egipcia

The objective of this document is to analyze some mathematical procedures used in ancient Egyptian civilization. It directs its interest towards the form of addition used as the fundamental basis for the calculation of various operations, and towards the form of determining the decimal expression corresponding to a given fraction. In this sense, pay attention to the use of addition as a fundamental accessory operation for the calculation of multiplications and divisions. Likewise, it shows some procedures used to determine the decimal expression corresponding to a given fraction, aided by obtaining unit fractions. All this was specified through a documentary research supported by a content analysis of a series of theoretical-referential and procedural aspects necessary to justify and illustrate the process of obtaining results requested by the problem statements that call for the use of unit fractions and fractions. to the calculation of results of multiplications and divisions. Among the findings, it stands out that the Mathematics of ancient Egypt was basically additive and was based on the decomposition of any natural number as the sum of positive multiples of 2, pointing its procedures to the fact of multiplying or dividing by 2. Likewise, the possibility of obtaining decimal expressions built with unit fraction formats whose numerators are always equal to 1 and their denominators are natural greater than 1. It is concluded that these and other contents such as the Decimal Numbering System have constituted relevant contributions to the current mathematical foundations.

Extremal problems for multigraphs

An (n,s,q)-graph is an n-vertex multigraph in which every s-set of vertices spans at most q edges. Turán-type questions on the maximum of the sum of the edge multiplicities in such multigraphs have been studied since the 1990s. More recently, Mubayi and Terry (2019) [13] posed the problem of determining the maximum of the product of the edge multiplicities in (n,s,q)-graphs. We give a general lower bound construction for this problem for many pairs (s,q), which we conjecture is asymptotically best possible. We prove various general cases of our conjecture, and in particular we settle a conjecture of Mubayi and Terry on the (s,q)=(4,6a+3) case of the problem (for a≥2); this in turn answers a question of Alon. We also determine the asymptotic behaviour of the problem for ‘sparse’ multigraphs (i.e. when q≤2(s2)). Finally we introduce some tools that are likely to be useful for attacking the problem in general.

On the stability of a multiplicative type sum form functional equation

On the stability of a multiplicative type sum form functional equation

Non-vanishing of certain cyclotomic multiple harmonic sums and application to the non-vanishing of certain p-adic cyclotomic multiple zeta values

We define and apply a method to study the non-vanishing of p-adic cyclotomic multiple zeta values. We prove the non-vanishing of certain cyclotomic multiple harmonic sums, and, via a formula proved in another paper, which expresses certain cyclotomic multiple harmonic sums as infinite sums of products of p-adic cyclotomic multiple zeta values, this implies the non-vanishing of certain p-adic cyclotomic multiple zeta values.

On Some Properties of Multiplicative Topological Indices in Silicon-Carbon

The use of graph theory can be visualized in nanochemistry, computer networks, Google maps, and molecular graph which are common areas to elaborate application of this subject. In nanochemistry, a numeric number (topological index) is used to estimate the biological, physical, and structural properties of chemical compounds that are associated with the chemical graph. In this paper, we compute the first and second multiplicative Zagreb indices ( M 1 G and ( M 1 G )), generalized multiplicative geometric arithmetic index ( GA α II G ), and multiplicative sum connectivity and multiplicative product connectivity indices ( SCII G and PCII G ) of SiC 4 − I m , n and SiC 4 − II m , n .

On a generalization of Jacobi sums

We prove an estimate for multi-variable multiplicative character sums over affine subspaces of Akn, which generalizes the well known estimates for both classical Jacobi sums and one-variable polynomial multiplicative character sums.

Advancing family business research through modeling nonlinear relationships: Comparing PLS-SEM and multiple regression

While nonlinear relationships play an important role in explaining distinct family business behaviors and outcomes, researchers rarely consider them in their theoretical and statistical models. To address this concern, this article introduces partial least squares structural equation modeling (PLS-SEM) as a suitable means for estimating nonlinear effects in latent variable models and describes its advantages vis-à-vis multiple (sum scores) regression. We conceptually compare and empirically illustrate the two methods by means of a family business research model. Based on our discussions, we provide family business researchers with a checklist of best practice recommendations when applying PLS-SEM. The article adds new methodological instruments to the family business researchers’ toolbox that enable them to explain and explore the mutual and often nonlinear interactions between family and business. Thereby, this research contributes to more rigorous and meaningful family business science.