Infinite Family Research Articles

Heavy states in 3d gravity and 2d CFT

We discuss correlators of light fields in heavy states in AdS3 gravity and holographic 2d CFTs. In the bulk, the propagator of free fields in AdS backgrounds containing a conical defect or a BTZ black hole can be obtained by solving a wave equation, as well as by the method of images. On the boundary, these geometries are sourced by heavy operator insertions, and the propagator is dual to a heavy-light (HHLL) correlator. By matching its expansion in Virasoro blocks to our bulk results, we determine the OPE coefficients of all contributing states in both the s and t channels. In the s channel, these states are excitations of the light field on top of the heavy state, and their OPE coefficients are the amplitudes to create them. The t-channel OPE is dominated by the Virasoro vacuum block, but there is also an infinite family of light two-particle states that contribute to the correlator. The OPE coefficients that couple these states to heavy operators represent their expectation values in heavy backgrounds. We determine them exactly, derive their asymptotic form at large twist, and discuss their behavior near and above the BTZ threshold, where they become thermal one-point functions.

The classifications of o-monomials and of 2-to-1 binomials are equivalent

AbstractWe observe that on the binary finite fields the classification of 2-to-1 binomials is equivalent to the classification of o-monomials, which is a well-studied and elusive problem in finite geometry. This connection implies a complete classification of 2-to-1 binomials $$b=x^d+ux^e$$ b = x d + u x e for a large set of values of (d, e). Further, we show that a number of the known infinite families of 2-to-1 maps can be traced back to o-polynomials or to difference maps of APN maps. We also provide some connections between 2-to-1 maps and hyperovals in non-desarguesian planes.

A branch group in a class of non-contracting weakly regular branch groups

We provide a class of non-contracting groups containing an infinite family of fractal and weakly regular branch groups, and study certain properties including abelianization, just infiniteness, and word problem. We present an example of a branch group in this class and show that it is of exponential growth. It seems this is the first example of a non-contracting branch group constructed explicitly.

On good infinite families of toric codes or the lack thereof

On good infinite families of toric codes or the lack thereof

Integrability meets the charged relativistic particle

We notice an analogy between the motion of a relativistic particle in external homogeneous and time-dependent electromagnetic fields and the Dikii-Eilenberger equation for the Bogoliubov-de Gennes equation. By means of the integrable defocusing nonlinear Schrödinger hierarchy and their solitons appearing in the Gross-Neveu and Nambu-Jona-Lasinio models, we construct an infinite family of solutions of the Lorentz equation with non-vanishing curvature and torsion of the worldline trajectory.

On an infinite family of integral Cayley graphs of Pauli groups

The classical Pauli group can be obtained as the central product of the dihedral group of 8 elements with the cyclic group of order 4. Inspired by this characterization, we introduce the notion of central product of Cayley graphs, which allows to regard the Cayley graph of a central product of groups as a quotient of the Cartesian product of the Cayley graphs of the factor groups. We focus our attention on the Cayley graph Cay(Pn,SPn) of the generalized Pauli group Pn on n-qubits; in fact, Pn may be decomposed as the central product of finite 2-groups, and a suitable choice of the generating set SPn allows us to recognize the structure of central product of graphs in Cay(Pn,SPn).Using this approach, we are able to recursively construct the adjacency matrix of Cay(Pn,SPn) for each n≥1, and to explicitly describe its spectrum and the associated eigenvectors. It turns out that Cay(Pn,SPn) is a (3n+2)-regular bipartite graph on 4n+1 vertices, and it has integral spectrum. This is a highly nontrivial property if one considers that, by choosing as a generating set for P1 the three classical Pauli matrices, one gets the so-called Möbius-Kantor graph, belonging to the class of generalized Petersen graphs, whose spectrum is not integral.

Viscosity Approximation Methods for Generalized Modification of the System of Equilibrium Problem and Fixed Point Problems of an Infinite Family of Nonexpansive Mappings

Viscosity Approximation Methods for Generalized Modification of the System of Equilibrium Problem and Fixed Point Problems of an Infinite Family of Nonexpansive Mappings

Solving split equality equilibrium and fixed point problems in Banach spaces

In this paper, we introduce a new algorithm for approximating a common solution of Split Equality Generalised Mixed Equilibrium Problem (SEGMEP) and Split Equality Fixed Point Problem (SEFPP) for two infinite families of closed uniformly Li-Lipschitz continuous and Ki-Lipschitz continuous and uniformly quasi-ϕ-asymptotically nonexpansive mappings (i ∈ N and ϕ is the Lyapunov functional) in Banach spaces. Under standard and mild assumption of monotonicity and lower semicontinuity of the SEGMEP associated mappings, we establish the strong convergence of the scheme without imposing any compactness type conditions on either the operators or the spaces considered. We apply our result to approximate the solution of Split Equality Convex Minimization Problem (SECMP) and Split Equality Variational Inclusion Problem (SEVIP). A numerical example is presented to illustrate the performance and implementability of our method. Our results extend, generalize and complement several related works in the literature.

Graceful Labellings of Variable Windmills Using Skolem-type Sequences

In this paper, we introduce graceful and near graceful labellings of several families of windmills. In particular, we use Skolem-type sequences to prove (near) graceful labellings exist for windmills with \(C_{3}\) and \(C_{4}\) vanes, and infinite families of \(3,5\)-windmills and \(3,6\)-windmills. Furthermore, we offer a new solution showing that the graph obtained from the union of \(t\) 5-cycles with one vertex in common (\(C_{5}^{t}\)) is graceful if and only if \(t \equiv 0, 3 \pmod{4}\) and near graceful when \(t \equiv 1, 2 \pmod{4}\).

Realizing the Asymmetric Index of a Graph

A graph G is asymmetric if its automorphism group is trivial. Asymmetric graphs were introduced by Erd\H{o}s and R\'{e}nyi Erdos [1]. They suggested the problem of starting with an asymmetric graph and removing some number, r , of edges and/or adding some number, s , of edges so that the resulting graph is non-asymmetric. Erd\H{o}s and R\'{e}nyi defined the degree of asymmetry of a graph to be the minimum value of r + s . In this paper, we consider another property that measures how close a given non-asymmetric graph is to being asymmetric. Brewer et al. defined the asymmetric index of a graph G , denoted a i ( G ) , as the minimum of r + s so that the resulting graph G is asymmetric [2]. It is noted that a i ( G ) is only defined for graphs with at least six vertices. We investigate the asymmetric index of both connected and disconnected graphs including paths, cycles, and grids, with the addition of up to two isolated vertices. Furthermore, for a graph in these families G , we determine the number of labeled asymmetric graphs that can be obtained by adding or removing a i ( G ) edges. This leads to the related question: Given a graph G where a i ( G ) = 1 , what is the probability that for a randomly chosen edge e , that G – e will be asymmetric? A graph is called minimally non-asymmetric if this probability is 1 . We give a construction of infinite families of minimally non-asymmetric graphs.

Links in orthoplicial Apollonian packings

In this paper, we establish a connection between Apollonian packings and knot theory. We introduce new representations of links realized in the tangency graph of the regular crystallographic sphere packings. Particularly, we prove that any algebraic link can be realized in the cubic section of the orthoplicial Apollonian packing. We use these representations to improve the upper bound on the ball number of an infinite family of alternating algebraic links. Furthermore, the later allow us to reinterpret the correspondence of rational tangles and rational numbers and to reveal geometrically primitive solutions for the Diophantine equation x4+y4+z4=2t2.

Online class cover problem

In this paper, we study the online class cover problem where a (finite or infinite) family F of geometric objects and a set Pr of red points in Rd are given a prior, and blue points from Rd arrives one after another. Upon the arrival of a blue point, the online algorithm must make an irreversible decision to cover it with objects from F that do not cover any points of Pr. The objective of the problem is to place a minimum number of objects. When F consists of axis-parallel unit squares in R2, we prove that the competitive ratio of any deterministic online algorithm is Ω(log⁡|Pr|), and also propose an O(log⁡|Pr|)-competitive deterministic algorithm for the problem.

Attainable bounds for algebraic connectivity and maximally connected regular graphs

AbstractWe derive attainable upper bounds on the algebraic connectivity (spectral gap) of a regular graph in terms of its diameter and girth. This bound agrees with the well‐known Alon–Boppana–Friedman bound for graphs of even diameter, but is an improvement for graphs of odd diameter. For the girth bound, we show that only Moore graphs can attain it, and these only exist for well‐known special cases. For the diameter bound, we use a combination of stochastic algorithms and exhaustive search to find graphs which attain it. For 3‐regular graphs, we find attainable graphs for all diameters up to and including (the case of is open). These graphs are extremely rare and also have high girth; for example, we found exactly 45 distinct cubic graphs on 44 vertices attaining the upper bound when ; all have girth 8. We also exhibit several infinite families attaining the upper bound with respect to diameter or girth. In particular, when is a power of prime, we construct a ‐regular graph having diameter 4 and girth 6 which attains the upper bound with respect to diameter.

Infinite Families of Congruences Modulo 5 for the Number of Irreducible Characters of the Alternating Groups

In 1919, Ramanujan proved red three congruences for the partition function p(n) which denotes the number of partitions of n. The partition function p(n) can be understood as the number of irreducible characters of the symmetric group Sn . Recently, Nath and Sellers, and Xia established a number of congruences modulo 2, 3, and 5 for the numbers of spin characters of S ̂ n and A ̂ n , where S ̂ n and A ̂ n are the double covering group of Sn and the double covering group of the alternating group An , respectively. Motivated by their work, we establish infinite families of congruences modulo 5 for a(n), which denotes the number of irreducible characters of the alternating group An . In particular, we prove some strange congruences modulo 5 for a(n). For example, we prove that for k ≥ 0 , a ( 95 × 73 2 k + 1 24 ) ≡ − 3 k ( mod 5 ) .

Principal Landau determinants

We reformulate the Landau analysis of Feynman integrals with the aim of advancing the state of the art in modern particle-physics computations. We contribute new algorithms for computing Landau singularities, using tools from polyhedral geometry and symbolic/numerical elimination. Inspired by the work of Gelfand, Kapranov, and Zelevinsky (GKZ) on generalized Euler integrals, we define the principal Landau determinant of a Feynman diagram. We illustrate with a number of examples that this algebraic formalism allows to compute many components of the Landau singular locus. We adapt the GKZ framework by carefully specializing Euler integrals to Feynman integrals. For instance, ultraviolet and infrared singularities are detected as irreducible components of an incidence variety, which project dominantly to the kinematic space. We compute principal Landau determinants for the infinite families of one-loop and banana diagrams with different mass configurations, and for a range of cutting-edge Standard Model processes. Our algorithms build on the Julia package Landau.jl and are implemented in the new open-source package PLD.jl available at https://mathrepo.mis.mpg.de/PLD/. Program summaryProgram title:PLD.jlCPC Library link to program files:https://doi.org/10.17632/7h5644mm4n.1Developer's repository link:https://mathrepo.mis.mpg.de/PLD/Licensing provisions: Creative Commons by 4.0Programming language:JuliaSupplementary material: The repository includes the source code with documentation (PLD_code.zip), a jupyter notebook tutorial providing installation and usage instructions (PLD_notebook.zip), a database containing the output of our algorithm on 114 examples of Feynman integrals (PLD_database.zip).Nature of problem: A fundamental challenge in scattering amplitude is to determine the values of complexified kinematic invariants for which an amplitude can develop singularities. Bjorken, Landau, and Nakanishi wrote a system of polynomial constraints, nowadays known as the Landau equations. This project aims to rigorously revisit the Landau analysis of the singularity locus of Feynman integrals with a practical view towards explicit computations.Solution method: We define the principal Landau determinant (PLD), which is a variety inspired by the work of Gelfand, Kapranov, and Zelevinsky (GKZ). We conjecture that it provides a subset of the singularity locus, and we implement effective algorithms to compute its defining equation explicitly.

ARITHMETIC PROPERTIES OF AN ANALOGUE OF t-CORE PARTITIONS

Abstract An integer partition of a positive integer n is called t-core if none of its hook lengths is divisible by t. Gireesh et al. [‘A new analogue of t-core partitions’, Acta Arith.199 (2021), 33–53] introduced an analogue $\overline {a}_t(n)$ of the t-core partition function. They obtained multiplicative formulae and arithmetic identities for $\overline {a}_t(n)$ where $t \in \{3,4,5,8\}$ and studied the arithmetic density of $\overline {a}_t(n)$ modulo $p_i^{\,j}$ where $t=p_1^{a_1}\cdots p_m^{a_m}$ and $p_i\geq 5$ are primes. Bandyopadhyay and Baruah [‘Arithmetic identities for some analogs of the 5-core partition function’, J. Integer Seq.27 (2024), Article no. 24.4.5] proved new arithmetic identities satisfied by $\overline {a}_5(n)$ . We study the arithmetic densities of $\overline {a}_t(n)$ modulo arbitrary powers of 2 and 3 for $t=3^\alpha m$ where $\gcd (m,6)$ =1. Also, employing a result of Ono and Taguchi [‘2-adic properties of certain modular forms and their applications to arithmetic functions’, Int. J. Number Theory1 (2005), 75–101] on the nilpotency of Hecke operators, we prove an infinite family of congruences for $\overline {a}_3(n)$ modulo arbitrary powers of 2.

Another Infinite Family of Binary Cyclic Codes With Best Parameters Known

Another Infinite Family of Binary Cyclic Codes With Best Parameters Known

N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 supersymmetry in the twistor description of higher-spin holography

We study the holographic duality between higher-spin (HS) gravity in 4d and free vector models in 3d, with special attention to the role of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 supersymmetry (SUSY). For the type-A bosonic bulk theory, dual to spin-0 fields on the boundary, there exists a twistor-space description; this maps both single-trace boundary operators and linearized bulk fields to spacetime-independent twistor functions, whose HS-algebra products compute all boundary correlators. Here, we extend this description to the type-B bosonic theory (dual to spin-1/2 fields on the boundary), and to the supersymmetric theory containing both. A key role is played by boundary bilocals, which in type-A are dual to the Didenko-Vasiliev 1/2-BPS “black hole”. We extend this to an infinite family of linearized 1/2-BPS “black hole” solutions. Remarkably, the full supersymmetric theory (along with the SUSY generators) fits in the same space of twistor functions as the type-A theory. Instead of two sets of bosonic bulk fields, the formalism sees one set of linearized fields, but with both types of boundary data allowed.

Semisymmetric Graphs of Order 2 p3 with Valency p2

A simple undirected regular graph is said to be semisymmetric if it is edge-transitive but not vertex-transitive. For a semisymmetric graph [Formula: see text] of order [Formula: see text], [Formula: see text] a prime, it is well known that [Formula: see text] is bipartite with two biparts having equal size. The complete classification of such graphs has been given for the full automorphism group [Formula: see text] acting unfaithfully on at least one bipart of [Formula: see text], which shows that there is only one infinite family of such graphs with valency [Formula: see text]. The graphs of this kind have been determined when [Formula: see text] acts faithfully and primitively on at least one bipart of [Formula: see text], and thus there is only one remaining case for classifying such graphs of valency [Formula: see text], [Formula: see text] acting faithfully and imprimitively on both biparts of [Formula: see text], which is dealt with in this paper. As a result, there is only one infinite family of semisymmetric graphs of order [Formula: see text] with valency [Formula: see text].

Linear Codes Constructed from Two Weakly Regular Plateaued Functions with Index (p - 1)/2.

Linear codes are the most important family of codes in cryptography and coding theory. Some codes only have a few weights and are widely used in many areas, such as authentication codes, secret sharing schemes and strongly regular graphs. By setting p≡1(mod4), we constructed an infinite family of linear codes using two distinct weakly regular unbalanced (and balanced) plateaued functions with index (p-1)/2. Their weight distributions were completely determined by applying exponential sums and Walsh transform. As a result, most of our constructed codes have a few nonzero weights and are minimal.