Arbitrary Integer Research Articles

On correlation functions of higher-spin currents in arbitrary dimensions d > 3

We revisit the problem of classification and explicit construction of the conformal three-point correlation functions of currents of arbitrary integer spin in arbitrary dimensions. For the conserved currents, we set up the equations for the conservation conditions and solve them completely for some values of spins, confirming the earlier counting of the number of independent structures matching them with the higher-spin cubic vertices in one higher dimension. The general solution for the correlators of conserved currents we delegate to a follow-up work.

Testability of positive integers

Given a set [Formula: see text] of some positive integers, a positive integer [Formula: see text] is [Formula: see text]-testable if there are two disjoint subsets [Formula: see text] and [Formula: see text] of [Formula: see text] such that [Formula: see text]. The testability of the set [Formula: see text] of positive integers not exceeding [Formula: see text], denoted by [Formula: see text], is the minimum integer [Formula: see text] such that there is a set [Formula: see text] with [Formula: see text] positive integers satisfying that each [Formula: see text] is [Formula: see text]-testable. In this short paper, we prove that [Formula: see text] and propose an algorithm to output the representation of an arbitrary positive integer by the form [Formula: see text], where [Formula: see text] and [Formula: see text] are two disjoint sets of some non-negative integers not exceeding [Formula: see text].

Manipulating topological charge of nested skyrmion bags by microwave magnetic fields

Nested skyrmion bags, as magnetic solitons with arbitrary integer topological charges (Q), hold potential for applications in data encoding. A crucial issue is the local manipulation of skyrmions within nested skyrmion bags to control the total Q. In this study, we explore different possible ground states and resonant excitation spectra of nested skyrmion bags through micromagnetic simulations. More importantly, we demonstrate that the manipulation of the Q of nested skyrmion bags can be achieved by using microwave magnetic fields, i.e., the inner, middle, and outer skyrmions within the nested skyrmion bags are selectively excited by using the diverse out-of-plane excitation modes. By calculating the energy of skyrmions, we further analyze the relationship between the annihilation of skyrmions at different positions and the out-of-plane microwave magnetic fields. Our findings present a promising approach for manipulating the Q of nested skyrmion bags, potentially advancing their application in storage and logic devices.

Effective interaction between guest charges immersed in 2D jellium

The model under study is an infinite 2D jellium of pointlike particles with elementary charge e, interacting via the logarithmic potential and in thermal equilibrium at the inverse temperature β. Two cases of the coupling constant Γ≡βe2 are considered: the Debye–Hückel limit Γ→0 and the free-fermion point Γ=2 . In the most general formulation, two guest particles, the one with charge qe (the valence q being an arbitrary integer) and the hard core of radius σ > 0 and the pointlike one with elementary charge e, are immersed in the bulk of the jellium at distance d⩾σ . Two problems are of interest: the asymptotic large-distance behavior of the excess charge density induced in the jellium and the effective interaction between the guest particles. Technically, the induced charge density and the effective interaction are expressed in terms of multi-particle correlations of the pure (translationally invariant) jellium system. It is shown that the separation form of the induced charge density onto its radial and angle parts, observed previously in the limit Γ→0 , is not reproduced at the coupling Γ=2 . Based on an exact expression for the effective interaction between guest particles at Γ=2 , oppositely ( q=0,−1,−2,… ) charged guest particles always attract one another while likely ( q=1,2,… ) charged guest particles repeal one another up to a certain distance d between them and then the mutual attraction takes place up to asymptotically large (finite) distances.

To Study on Number of Subgroups of Finite Abelian Group Zm ⊗ Zn

In this paper, we determine the number of subgroups of group Z_(p^m ) ⊗ Z_(p^n ) which may be cyclic or non –cyclic by using simple number –theoretic formulae. We describe the subgroups of the group and derive a simple formula for the total number s(m,n ) of the subgroups where m, n are arbitrary positive integers. We point out that certain multiplicative properties of related counting functions for finite. Abelian groups are immediate consequences of these formula.

Forms of biisometric operators and biorthogonality

The paper proves two results involving a pair ( A , B ) of P-biisometric or ( m , P ) -biisometric Hilbert-space operators for arbitrary positive integer m and positive operator P. It is shown that if A and B are power bounded and the pair ( A , B ) is ( m , P ) -biisometric for some m, then it is a P-biisometric pair. The important case when P is invertible is treated in detail. It is also shown that if ( A , B ) is P-biisometric, then there are biorthogonal sequence with respect to the inner product 〈 ⋅ ; ⋅ 〉 P = 〈 P ⋅ ; ⋅ 〉 that have a shift-like behavior with respect to this inner product.

How fast can we play Tetris greedily with rectangular pieces?

Consider a variant of Tetris played on a board of width w and infinite height, where the pieces are axis-aligned rectangles of arbitrary integer dimensions, the pieces can only be moved before letting them drop, and a row does not disappear once it is full. Suppose we want to follow a greedy strategy: let each rectangle fall where it will end up the lowest given the current state of the board. To do so, we want a data structure which can always suggest a greedy move. In other words, we want a data structure which maintains a set of O(n) rectangles, supports queries which return where to drop the rectangle, and updates which insert a rectangle dropped at a certain position and return the height of the highest point in the updated set of rectangles. We show via a reduction from the Multiphase problem [Pătraşcu, 2010] that on a board of width w=Θ(n), if the OMv conjecture [Henzinger et al., 2015] is true, then both operations cannot be supported in time O(n1/2−ϵ) simultaneously. The reduction also implies polynomial bounds from the 3-SUM conjecture and the APSP conjecture. On the other hand, we show that there is a data structure supporting both operations in O(n1/2log3/2⁡n) time on boards of width nO(1), matching the lower bound up to an no(1) factor.

Bilinear forms with Kloosterman and Gauss sums in function fields

In recent years, there has been a lot of progress in obtaining non-trivial bounds for bilinear forms of Kloosterman sums in Z/mZ for arbitrary integers m. These results have been motivated by a wide variety of applications, such as improved asymptotic formulas for moments of L-functions. However, there has been very little work done in this area in the setting of rational function fields over finite fields. We remedy this and provide a number of new non-trivial bounds for bilinear forms of Kloosterman and Gauss sums in this setting, based on new bounds on the number of solutions to certain modular congruences in Fq[T]. These improve upon some results of Macourt and Shparlinski (2019).

On auxiliary fields and Lagrangians for relativistic wave equations

We address the problem of the existence of a Lagrangian for a given system of linear partial differential equation with constant coefficients. As a subtask, this involves bringing the system into a pre-Lagrangian form, wherein the number of equations matches the number of unknowns. We introduce a class of overdetermined systems, called co-flat, and show that they always admit a pre-Lagrangian form, which can be explicitly constructed by means of auxiliary variables. Moreover, we argue that such systems enjoy pre-Lagrangian formulations without auxiliary variables at all. As an application of our method, we construct new pre-Lagrangian and Lagrangian formulations for free massive fields of arbitrary integer spin. In contrast to the well-known models of Singh and Hagen, our Lagrangians involve much fewer auxiliary fields.

Tip of the Quantum Entropy Cone.

Relations among von Neumann entropies of different parts of an N-partite quantum system have direct impact on our understanding of diverse situations ranging from spin systems to quantum coding theory and black holes. Best formulated in terms of the set Σ_{N}^{*} of possible vectors comprising the entropies of the whole and its parts, the famous strong subaddivity inequality constrains its closure Σ[over ¯]_{N}^{*}, which is a convex cone. Further homogeneous constrained inequalities are also known. In this Letter we provide (nonhomogeneous) inequalities that constrain Σ_{N}^{*} near the apex (the vector of zero entropies) of Σ[over ¯]_{N}^{*}, in particular showing that Σ_{N}^{*} is not a cone for N≥3. Our inequalities apply to vectors with certain entropy constraints saturated and, in particular, they show that while it is always possible to upscale an entropy vector to arbitrary integer multiples it is not always possible to downscale it to arbitrarily small size, thus answering a question posed by Winter.

On Generalized Sums of Six, Seven and Nine Cube

Let u1, u2, u3,・・・ un be integers such that un − un−1 = un−1 − un−2 = ・ ・ ・ = a2 − a1 = d. In this article, the study of sums of cube in arithmetic progression is discussed. In particular, the study develops and introduces some generalized results on sums of six, seven and nine cube for any arbitrary integers in arithmetic sequences. The method of study involves analogy grounded on integer decomposition and factorization. The result in this study will prove the existing results on sums of cubes.

Partition line graphs of multigraphs into two subgraphs with large chromatic numbers

Wang and Yu (2022) prove an enhanced version of the Erdős–Lovász Tihany conjecture for line graphs of multigraphs. That is, let s,t and ℓ be arbitrary integers with t≥s≥3.5ℓ+2, ℓ≥0. If the line graph L(G) of some multigraph G has chromatic number s+t−1>ω(L(G)), then there is a partition (S,T) of the vertex set V(L(G)) such that χ(L(G)[S])≥s and χ(L(G)[T])≥t+ℓ. In this paper, for integers s and t with t≥s≥7, we prove that for each line graph L(G) with χ(L(G))=s+t−1≥ω(L(G)), there is a partition (S,T) of the vertex set V(L(G)) such that χ(L(G)[S])≥s and χ(L(G)[T])≥t+2, which slightly improves the recent result of Wang and Yu (2022) for ℓ=2.

Software Support for Arbitrary Precision Arithmetic in Programming Languages

Arbitrary precision arithmetic, also known as bignum arithmetic, is a computational technique that allows programmers to perform arithmetic operations on numbers with significantly higher precision and magnitude than what is typically supported by the built-in numerical data types in programming languages. This technique is especially useful when working with extremely large or extremely precise numbers, such as in cryptography or scientific computations. Arbitrary precision arithmetic has many applications in the areas of Cryptography, Numerical Computation, Statistical Analysis, and High Precision measurements. For example, the calculation of the modulus in the RSA algorithm involves numbers with 1024-bit numbers and higher. An arithmetic calculation involving Multiplication and exponentiation of such numbers using Modulo arithmetic cannot be easily carried out in the existing programming Languages unless special software is provided. We can calculate the mathematical constant Pi to many thousand decimal places using the support provided in Programming Languages. Many programming languages provide built-in support for libraries for arbitrary precision arithmetic. We discuss the support provided in C/C++, Java and Python Languages with examples. Besides Programming Languages, Toolkits like Matlab and Sagemath also are used for scientific computation and special software support provided in these toolkits can enable arbitrary precision arithmetic. Most Programming Languages have support for floating-point arithmetic. We also discuss how arbitrary precision floating point arithmetic. can be supported in C/C++, Java, and Python. In addition, we discuss support for arbitrary precision integer arithmetic in Ruby, Javascript and Matlab and support for arbitrary precision floating point arithmetic in the Perl Language. Finally, we provide an example of computing the constant Pi to many decimal places using the sagemath tool.

Formula omitted]-sequences and 2-step coreness in graphs

An H-sequence of a vertex is defined by iteratively applying H-index to neighbors of the vertex in a graph. It is still not known whether changing the initial values affects the convergence of H-sequences. Our purpose is to investigate the properties of H-sequences initialized with arbitrary positive integers. This paper presents the necessary and sufficient conditions for the convergence of H-sequences. Theoretical results reveal the differences between divergent H-sequences of infinite graphs and finite graphs. Divergent H-sequences of infinite graphs are either periodic or non-periodic, whereas divergent H-sequences of finite graphs must be periodic with a period of 2.Based on H-sequences, we define 2-step coreness on the second-order neighbors of a vertex. A new centrality is proposed by combining classical coreness and 2-step coreness. The new centrality captures a wider range of information in a network. Experimental results show that the proposed centrality makes significant improvements over centralities such as betweenness, H-index and coreness.

On a minimal set of generators for the algebra $H^*(BE_6; \mathbb F_2)$ as a module over the Steenrod algebra and applications

Let $\mathcal P_n \cong H^{*}\big(BE_n; \mathbb F_2 \big)$ be the graded polynomial algebra over the prime field of two elements $\mathbb F_2$, where $E_n$ is an elementary abelian 2-group of rank $n,$ and $BE_n$ is the classifying space of $E_n.$ We study the {\it hit problem}, set up by Frank Peterson, of finding a minimal set of generators for the polynomial algebra $\mathcal P_{n},$ viewed as a module over the mod-2 Steenrod algebra $\mathcal{A}$. This problem remains unsolvable for $n>4,$ even with the aid of computers in the case of $n=5.$ By considering $\mathbb F_2$ as a trivial $\mathcal A$-module, then the hit problem is equivalent to the problem of finding a basis of $\mathbb F_2$-graded vector space $\mathbb F_2 {\otimes}_{\mathcal{A}}\mathcal P_{n}.$ This paper aims to explicitly determine an admissible monomial basis of the $ \mathbb{F}_{2}$-vector space $\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal P_{n}$ in the generic degree $n(2^{r}-1)+2\cdot 2^{r},$ where $r$ is an arbitrary non-negative integer, and in the case of $n=6.$ As an application of these results, we obtain the dimension results for the polynomial algebra $\mathcal P_n$ in degrees $(n-1)\cdot(2^{n+u-1}-1)+\ell\cdot2^{n+u},$ where $u$ is an arbitrary non-negative integer, $\ell =13,$ and $n=7.$ Moreover, for any integer $r>1,$ the behavior of the sixth Singer algebraic transfer in degree $6(2^{r}-1)+2\cdot2^r$ is also discussed at the end of this paper. Here, the Singer algebraic transfer is a homomorphism from the homology of the Steenrod algebra to the subspace of $\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal P_{n}$ consisting of all the $GL_n(\mathbb F_2)$-invariant classes. It is a useful tool in describing the homology groups of the Steenrod algebra, $\text{Tor}^{\mathcal A}_{n, n+*}(\mathbb F_2,\mathbb F_2).$

Factorization of composition of reciprocal polynomials with monomials

A polynomial f(x) is referred to as reciprocal if it satisfies f(x)=±xdeg⁡ff(1/x). It is established that if f(x)∈Q[x] is reciprocal and irreducible over Q, and r>0 is an odd integer, then f(xr) has an irreducible reciprocal factor over Q. Furthermore, if f(x) has a root on the unit circle, and r>0 is an arbitrary integer, then every irreducible factor of f(xr) is reciprocal. This generalizes a related result of Filaseta and Meade for reciprocal 0,1-polynomials. As an outcome of our methods, we obtain some information on the irreducible factors over Q of f(xr) for an arbitrary f(x)∈Q[x].

Bessel–Gauss Beams of Arbitrary Integer Order: Propagation Profile, Coherence Properties, and Quality Factor

We present a novel approach to generate Bessel–Gauss modes of arbitrary integer order and well-defined optical angular momentum in a gradient index medium of transverse parabolic profile. The propagation and coherence properties, as well as the quality factor, are studied using algebraic techniques that are widely used in quantum mechanics. It is found that imposing the well-defined optical angular momentum condition, the Lie group SU(1,1) comes to light as a characteristic symmetry of the Bessel–Gauss beams.

Dynamics of general higher-order rogue waves in the two-component long wave–short wave model of Newell type

An integrable two component long-wave–short-wave (2-LS) model of Newell type governing the resonant interaction between two capillary waves with a common gravity wave is considered. The general higher order rogue wave (RW) solutions of this system are obtained by employing the bilinear KP hierarchy reduction method. Our study explores three different families of RWs depending upon the root structure of an algebraic equation characterizing them. Those are the Nth-order bounded rogue waves, mixed bounded (N1,N2)th-order rogue waves and degradable bounded (N¯1,N¯2)th-order rogue waves, respectively, where N,N1,N2 are arbitrary positive integer and N¯1,N¯2 are arbitrary non-negative integers. It is interesting to note that the fundamental rogue wave in one short wave (SW) component can attain a peak amplitude as high as more than six times the background amplitude while it is three times the background amplitude in the other SW component. We identify that the Nth-order bounded rogue waves are composed of N(N+1)2 fundamental rogue waves, the mixed bounded (N1,N2)th-order rogue waves comprise N1th-order bounded rogue waves mixing with another type N2th-order bounded rogue waves existing in different state, and the degradable bounded (N¯1,N¯2)th-order rogue waves consist of N¯12+N¯22−N¯1(N¯2−1) fundamental rogue waves. Non-trivial super rouge waves are reported.

Analysis of two fully discrete spectral volume schemes for hyperbolic equations

AbstractIn this paper, we analyze two classes of fully discrete spectral volume schemes (SV) for solving the one‐dimensional scalar hyperbolic equation. These two schemes are constructed by using the forward Euler (EU) method or the second‐order Runge–Kutta (RK2) method in time‐discretization, and by letting a piecewise kth degree( is an arbitrary integer) polynomial satisfy the local conservation law in each control volume designed by subdividing the underlying mesh with Gauss–Legendre points (LSV) or right‐Radau points (RRSV). We prove that for the EU‐SV schemes, the weak(2) stability holds and the norm errors converge with optimal orders , provided that the CFL condition is satisfied. While for the RK2‐SV schemes, the weak(4) stability holds and the norm errors converge with optimal orders , provided that the CFL condition is satisfied. Here and are, respectively, the spacial and temporal mesh sizes and the constant is independent of and . Our theoretical findings have been justified by several numerical experiments.

Dynamics of skyrmion bags driven by spin wave

Skyrmion bags are spin textures with arbitrary integer topological charge numbers, holding promise for utilization in high-density racetrack memory devices. In this study, we investigate the dynamics of skyrmion bags driven by propagating spin waves in a magnetic nanostrip, employing micromagnetic simulations. Our findings reveal that propagating spin waves can induce a stable movement of skyrmion bags, irrespective of their topological charge numbers. The amplitude and frequency of the spin wave, alongside material parameters such as damping constant and saturation magnetization, exhibit discernible effects on the behavior of skyrmion bags. These outcomes have the potential to advance the development of racetrack memory based on skyrmion bags.