Finite Sequence Research Articles

Virtualized Delta moves for virtual knots and links

We introduce a local deformation called the virtualized [Formula: see text]-move for virtual knots and links. We prove that the virtualized [Formula: see text]-move is an unknotting operation for virtual knots. Furthermore we give a necessary and sufficient condition for two virtual links to be related by a finite sequence of virtualized [Formula: see text]-moves.

ON THE BINARY SEQUENCE $(1,1,0,1,0^3,1,0^7,1,0^{15},\ldots )$

Abstract Let $\mathbb {F}$ be a field and $(s_0,\ldots ,s_{n-1})$ be a finite sequence of elements of $\mathbb {F}$ . In an earlier paper [G. H. Norton, ‘On the annihilator ideal of an inverse form’, J. Appl. Algebra Engrg. Comm. Comput.28 (2017), 31–78], we used the $\mathbb {F}[x,z]$ submodule $\mathbb {F}[x^{-1},z^{-1}]$ of Macaulay’s inverse system $\mathbb {F}[[x^{-1},z^{-1}]]$ (where z is our homogenising variable) to construct generating forms for the (homogeneous) annihilator ideal of $(s_0,\ldots ,s_{n-1})$ . We also gave an $\mathcal {O}(n^2)$ algorithm to compute a special pair of generating forms of such an annihilator ideal. Here we apply this approach to the sequence r of the title. We obtain special forms generating the annihilator ideal for $(r_0,\ldots ,r_{n-1})$ without polynomial multiplication or division, so that the algorithm becomes linear. In particular, we obtain its linear complexities. We also give additional applications of this approach.

On finite pseudorandom binary sequences: functions from a Hardy field

On finite pseudorandom binary sequences: functions from a Hardy field

Why are inner planets not inclined?

AbstractPoincaré’s work more than one century ago, or Laskar’s numerical simulations from the 1990’s on, have irrevocably impaired the long-held belief that the Solar System should be stable. But mathematical mechanisms explaining this instability have remained mysterious. In 1968, Arnold conjectured the existence of “Arnold diffusion” in celestial mechanics. We prove Arnold’s conjecture in the planetary spatial 4-body problem as well as in the corresponding hierarchical problem (where the bodies are increasingly separated), and show that this diffusion leads, on a long time interval, to some large-scale instability. Along the diffusive orbits, the mutual inclination of the two inner planets is close to $\pi /2$ π / 2 , which hints at why even marginal stability in planetary systems may exist only when inner planets are not inclined. More precisely, consider the normalised angular momentum of the second planet, obtained by rescaling the angular momentum by the square root of its semimajor axis and by an adequate mass factor (its direction and norm give the plane of revolution and the eccentricity of the second planet). It is a vector of the unit 3-ball. We show that any finite sequence in this ball may be realised, up to an arbitrary precision, as a sequence of values of the normalised angular momentum in the 4-body problem. For example, the second planet may flip from prograde nearly horizontal revolutions to retrograde ones. As a consequence of the proof, the non-recurrent set of any finite-order secular normal form accumulates on circular motions – a weak form of a celebrated conjecture of Herman.

Solving Mathematical Programs with Complementarity Constraints Arising in Nonsmooth Optimal Control

AbstractThis paper examines solution methods for mathematical programs with complementarity constraints (MPCC) obtained from the time-discretization of optimal control problems (OCPs) subject to nonsmooth dynamical systems. The MPCC theory and stationarity concepts are reviewed and summarized. The focus is on relaxation-based methods for MPCCs, which solve a (finite) sequence of more regular nonlinear programs (NLP), where a regularization/homotopy parameter is driven to zero. Such methods perform reasonably well on currently available benchmarks. However, these results do not always generalize to MPCCs obtained from nonsmooth OCPs. To provide a more complete picture, this paper introduces a novel benchmark collection of such problems, which we call . The problem set includes 603 different MPCCs and we split it into a few representative subsets to accelerate the testing. We compare different relaxation-based methods, NLP solvers, homotopy parameter update and relaxation parameter steering strategies. Moreover, we check whether the obtained stationary points allow first-order descent directions, which may be the case for some of the weaker MPCC stationarity concepts. In the best case, the Scholtes’ relaxation (SIAM J. Optim. 11, 918–936, 2001) with (Math. Program. 106, 25–57, 2006) as NLP solver manages to solve 73.8% of the problems. This highlights the need for further improvements in algorithms and software for MPCCs.

Categorical resolutions of filtered schemes

We give an alternative proof of the theorem by Kuznetsov and Lunts, stating that any separated scheme of finite type over a field of characteristic zero admits a categorical resolution of singularities. Their construction makes use of the fact that every variety (over a field of characteristic zero) can be resolved by a finite sequence of blow-ups along smooth centres. We merely require the existence of (projective) resolutions. To accomplish this we put the \mathcal{A} -spaces of Kuznetsov and Lunts in a different light, viewing them instead as schemes endowed with finite filtrations. The categorical resolution is then constructed by gluing together differential graded categories obtained from a hypercube of finite length filtered schemes.

A Study of Cunningham Bounds Through Rogue Primes

A sequence of prime numbers {p, 2p+1, 4p+3,...,2n-1(p+1)-1} is called a Cunningham chain. These are finite sequences of prime numbers, for which each element is a Sophie Germain prime. It is conjectured that there are arbitrarily large such Cunningham chains, and these chains form an essential part of the study of Sophie Germain primes. In this paper, we aim to significantly improve existing bounds for the length of Cunningham chains by considering their behaviour in the framework of what we will define as rogueness.

Learn2Extend: Extending Sequences by Retaining their Statistical Properties with Mixture Models

This paper addresses the challenge of extending general finite sequences of real numbers within a subinterval of the real line, maintaining their inherent statistical properties by employing machine learning. Our focus lies on preserving the gap distribution and pair correlation function of these point sets. Leveraging advancements in deep learning applied to point processes, this paper explores the use of an auto-regressive Sequence Extension Mixture Model (SEMM) for extending finite sequences, by estimating directly the conditional density, instead of the intensity function. We perform comparative experiments on multiple types of point processes, including Poisson, locally attractive, and locally repelling sequences, and we perform a case study on the prediction of Riemann ζ function zeroes. The results indicate that the proposed mixture model outperforms traditional neural network architectures in sequence extension with the retention of statistical properties. Given this motivation, we showcase the capabilities of a mixture model to extend sequences, maintaining specific statistical properties, i.e. the gap distribution, and pair correlation indicators.

Matrix characterisation of the hierarchy FiC n of bivaluated logics

In this paper, we define the family FiC n := { Fi C n } n ∈ ω of logics by means of semantics of bivaluations. This family is a variant of the family of paraconsistent bivaluated Ci u n -logics, defined and studied by J. Ciuciura. We will show that every logic in FiC n can be alternatively defined by means of finite matrices. This result arises from the characterisation of the truth-values of the involved matrices (relative to each Fi C n -logic) as being specific finite sequences of elements of the set 2 := { 0 , 1 } . Moreover, we will show in this paper that such characterisation is related to the well-known standard Fibonacci Sequence, which is presented here by means of its binary expansion.

Randomized complexity of mean computation and the adaption problem

Recently the adaption problem of Information-Based Complexity (IBC) for linear problems in the randomized setting was solved in Heinrich (2024) [8]. Several papers treating further aspects of this problem followed. However, all examples obtained so far were vector-valued. In this paper we settle the scalar-valued case. We study the complexity of mean computation in finite dimensional sequence spaces with mixed LpN norms. We determine the n-th minimal errors in the randomized adaptive and non-adaptive settings. It turns out that among the problems considered there are examples where adaptive and non-adaptive n-th minimal errors deviate by a power of n. The gap can be (up to log factors) of the order n1/4. We also show how to turn such results into infinite dimensional examples with suitable deviation for all n simultaneously.

A novel smart hybrid multimorph piezoelectric spherical shell cloak for broadband near-perfect underwater acoustic camouflage applications

Non-visual auditory camouflage plays a major role in the art of underwater deception. In this work, a hybrid active/semi-active omnidirectional cloaking shell structure composed of alternate complementary piezoelectric and smart viscoelastic (PZT/SVE) actuator layers is proposed that can effectively conceal a three dimensional underwater macroscopic object from broadband incident sound waves. The smart hybrid structure incorporates a finite sequence of fully active parallel-connected multimorph PZT constraining layers inter-stacked with semi-active SVE core layers both of which are collaboratively operative in the framework of a Particle Swarm Optimized (PSO) multiple-input multiple-output active damping control (MIMO-ADC) scheme. The elasto-acoustic modeling of the problem is conducted by coupling the spatial state space methodology based on the classical three-dimensional exact piezoelasticity theory with the wave equations for the inner and outer acoustic domains. The acoustic cloaking performance of proposed configuration is evaluated for four distinct classes of highly functional SVE interlayer materials with tunable (field-dependent) rheological properties, namely, magnetorheological elastomer (MRE), shape memory polymer (SMP), electrorheological fluid (ERF), and magnetorheological shear thickening polishing fluid (MRSTPF). Extensive numerical results reveal significant broadband reductions of the far-field backscattering amplitude in the (f∞θ=π,kexRex)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left|{f}_{\\infty }\\left(\ heta =\\pi ,{k}_{\ ext{ex}}{R}_{\ ext{ex }}\\right)\\right|)$$\\end{document} as well as the percentage error of external cloaked field (%Err)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\%\ ext{Err})$$\\end{document} by incorporating a sufficient number of smart multimorph PZT/SVE material layers. Furthermore, it is concluded that comparable low frequency acoustic cloaking effects is possible without expenditure of any external energy just by employing the entirely inactive MRSTPF-based cloak as an alternative to the semiactive or fully active multimorph PZT/SVE cloaks. The outcome of proposed study can advantageously serve as the first step towards practical development and experimental implementation of future high performance smart acoustic cloaking devices with expanded broadband near-perfect omnidirectional invisibility for three dimensional objects of diverse geometries.

A weak theory of building blocks

AbstractWe apply the mereological concept of parthood to the coding of finite sequences. We propose a first‐order theory in which coding finite sequences is intuitive and transparent. We compare this theory with Robinson arithmetic, adjunctive set theory and weak theories of finite strings and finite trees using interpretability.

Finite Sequences Representing Expected Order Statistics

Characterizations of finite sequences β1<⋯<βn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta _{1}<\\cdots <\\beta _{n}$$\\end{document} representing expected values of order statistics from a random sample of size n are given. As a by-product, a characterization of binomial mixtures, when the mixing random variable is supported in the open interval (0, 1), is presented; this enables the exact description of the convex hull of the open binomial curve, as well as the open moment curve.

CANTOR’S THEOREM MAY FAIL FOR FINITARY PARTITIONS

Abstract A partition is finitary if all its members are finite. For a set A, $\mathscr {B}(A)$ denotes the set of all finitary partitions of A. It is shown consistent with $\mathsf {ZF}$ (without the axiom of choice) that there exist an infinite set A and a surjection from A onto $\mathscr {B}(A)$ . On the other hand, we prove in $\mathsf {ZF}$ some theorems concerning $\mathscr {B}(A)$ for infinite sets A, among which are the following: (1) If there is a finitary partition of A without singleton blocks, then there are no surjections from A onto $\mathscr {B}(A)$ and no finite-to-one functions from $\mathscr {B}(A)$ to A. (2) For all $n\in \omega $ , $|A^n|&lt;|\mathscr {B}(A)|$ . (3) $|\mathscr {B}(A)|\neq |\mathrm {seq}(A)|$ , where $\mathrm {seq}(A)$ is the set of all finite sequences of elements of A.

Optimal control for reachability of Markov jump switching Boolean control networks subject to output trackability

This article aims at designing the finite horizon optimal control sequence (FHOCS) for ensuring the reachability of Markov jump switching Boolean control networks (MJSBCNs) subject to output trackability. The key step in solving such a problem is to construct a Luenberger-like observer (LLO). Our approach includes a few steps as follows. We firstly propose the formulation of the optimal control problem (OCP) of maximising the probability of finite-time reachability subject to output trackability. Then, by converting the dynamics of MJSBCNs into an equivalent expectation system, an LLO is established for MJSBCNs. In this setup, a Luenberger-like observation matrix is constructed, which is an effect tool to solve the finite-time reachability problem. Finally, an LLO method is developed to find the FHOCS by maximising the T -step Luenberger-like observation matrix. In a biological example, we show the effectiveness of our developed methods.

An Analysis of Sequence and Series

This article provides a comprehensive exploration of sequences and series, fundamental concepts in mathematics with wide-ranging applications. Starting with an introduction to sequences, including arithmetic and geometric sequences, the discussion extends to finite sequences and n-tuple sequences. Arithmetic and geometric series are explored, with formulas for their sums. The article delves into the properties of sequences and series, discussing convergence, divergence, and absolute convergence. Various types of sequences, such as bounded and monotonic sequences, are defined, and the Cauchy Criterion is introduced as a criterion for the existence of the limit of a sequence. Elementary facts about series, including absolute convergence and the number e as the sum of a series, are covered. Additionally, the concept of sequences of functions is introduced. The article concludes by emphasizing the significance of sequences and series in fields like higher mathematics, art, science, technology, and finance, showcasing their crucial role in decision-making, financial analysis, and risk assessment across diverse disciplines.

Human intelligence for authors, reviewers and editors using artificial intelligence

We call artificial intelligence any machine that processes information with some purpose, complying with the logical rules of Turing's computation described more than 70 years ago. These machines work with instructions called algorithms, a finite and well-defined sequence of information processing implemented by automata (computers) or any digital technology to optimize a process. This means that the purpose of artificial intelligence is optimization

Erdős-Ko-Rado theorem for bounded multisets

Let k,m,n be positive integers with k⩾2. A k-multiset of [n]m is a collection of k integers from the set {1,2,…,n} in which the integers can appear more than once but at most m times. A family of such k-multisets is called an intersecting family if every pair of k-multisets from the family have non-empty intersection. A finite sequence of real numbers (a1,a2,…,an) is said to be unimodal if there is some k∈{1,2,…,n}, such that a1⩽a2⩽…⩽ak−1⩽ak⩾ak+1⩾…⩾an. Given m,n,k, denote Ck,ℓ as the coefficient of xk in the generating function (∑i=1mxi)ℓ, where 1⩽ℓ⩽n. In this paper, we first show that the sequence of (Ck,1,Ck,2,…,Ck,n) is unimodal. Then we use this as a tool to prove that the intersecting family in which every k-multiset contains a fixed element attains the maximum cardinality for n⩾k+⌈k/m⌉. In the special case when m=1 and m=∞, our result gives rise to the famous Erdős-Ko-Rado Theorem and an unbounded multiset version for this problem given by Meagher and Purdy [11], respectively. The main result in this paper can be viewed as a bounded multiset version of the Erdős-Ko-Rado Theorem.

Laplacian eigenvalues and eigenspaces of cographs generated by finite sequence

Laplacian eigenvalues and eigenspaces of cographs generated by finite sequence

Application of Genetic Algorithms for Periodicity Recognition and Finite Sequences Sorting

Unpredictable strings are sequences of data with complex and erratic behavior, which makes them an object of interest in various scientific fields. Unpredictable strings related to chaos theory was investigated using a genetic algorithm. This paper presents a new genetic algorithm for converting large binary sequences into their periodic form. The MakePeriod method is also presented, which is aimed at optimizing the search for such periodic sequences, which significantly reduces the number of generations to achieve the result of the problem under consideration. The analysis of the deviation of a nonperiodic sequence from its considered periodic transformation was carried out, and methods of crossover and mutation were investigated. The proposed algorithm and its associated conclusions can be applied to processing large sequences and different values of the period, and also emphasize the importance of choosing the right methods of crossover and mutation when applying genetic algorithms to this task.