rss_2.0Pure Mathematics and Applications FeedSciendo RSS Feed for Pure Mathematics and Applications Mathematics and Applications Feed longest increasing subsequence in involutions avoiding 3412 and another pattern<abstract> <title style='display:none'>Abstract</title> <p>In this note, we study the mean length of the longest increasing subsequence of a uniformly sampled involution that avoids the pattern 3412 and another pattern.</p> </abstract>ARTICLEtrue expected number of distinct consecutive patterns in a random permutation<abstract> <title style='display:none'>Abstract</title> <p>Let <italic>π</italic><italic><sub>n</sub></italic> be a uniformly chosen random permutation on [<italic>n</italic>]. Using an analysis of the probability that two overlapping consecutive <italic>k</italic>-permutations are order isomorphic, we show that the expected number of distinct consecutive patterns of all lengths <italic>k</italic> ∈ {1, 2,…, <italic>n</italic>} in <italic>π</italic><italic><sub>n</sub></italic> is <inline-formula> <alternatives> <inline-graphic xmlns:xlink="" xlink:href="graphic/j_puma-2022-0031_eq_001.png"/> <mml:math xmlns:mml="" display="inline"><mml:mrow><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>2</mml:mn></mml:mfrac><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>o</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> <tex-math>{{{n^2}} \over 2}\left( {1 - o\left( 1 \right)} \right)</tex-math> </alternatives> </inline-formula> as <italic>n</italic> → ∞. This exhibits the fact that random permutations pack consecutive patterns near-perfectly.</p> </abstract>ARTICLEtrue of the real and complex eigenvalue problems in the ABS class<abstract> <title style='display:none'>Abstract</title> <p>The class of ABS methods was originally developed for solving systems of linear equations in a finite number of iterations. Later it was shown that methods for solving nonlinear systems of equations, linear programming algorithms, quadratic programming, etc. are also members of the ABS class. In this paper, we show that QR factorization can also be derived from three ABS subclasses, which is an essential step in methods solving eigenvalue problems. Another possibility to use the ABS class in the eigenvalue problems is to transform a matrix to Hessenberg form which is an important initial step when solving eigenvalue problems by QR-type methods. Here we study versions of the plain QR-method which are based on the QR factorization computed by the ABS methods and compare them with the MATLAB eig () function. We also compare the ABS Hessenberg method to the MATLAB hess() function. The preliminary results show the competitiveness of 3 ABS eigenvalue solving methods with the MATLAB eig () function.</p> </abstract>ARTICLEtrue actions and Gelfand duality<abstract> <title style='display:none'>Abstract</title> <p>In this paper we discuss structural Ramsey theory and how it relates to the understanding of extremely amenable groups with an emphasis on the measure theoretic nature of this problem. We discuss the problem within Blass-Ramsey actions on discrete spaces and and more general group actions on compact spaces and explore how we can look at these problems by looking at Gelfand duals of commutative <italic>C</italic><sup>*</sup>-algebras.</p> </abstract>ARTICLEtrue monotone likelihood ratio of stationary probabilities in bonus-malus systems<abstract> <title style='display:none'>Abstract</title> <p>Bonus-malus system is an often used risk management tool in the insurance industry, and it is usually modeled with Markov chains. Under mild conditions it can be stated that the bonus-malus system converges to a unique stationary distribution in the long run. The maximum likelihood ratio property is a well-known statistical concept and we define it for the stationary distribution of a bonus-malus system. For two special cases we could justify it algebraically. For other cases we describe a numerical method with which we can test this property in any case. With the help of the described method, we checked this property for cases that appear in actuarial practice.</p> </abstract>ARTICLEtrue subminimal polynomial and the Cayley-Hamilton theorem<abstract> <title style='display:none'>Abstract</title> <p>A short proof is given for the Cayley-Hamilton theorem using the concept of the subminimal polynomial.</p> </abstract>ARTICLEtrue of Dyck paths associated with numerical semigroups<abstract> <title style='display:none'>Abstract</title> <p>We investigate the relationship between numerical semigroups and Dyck paths discovered by Bras-Amorós and de Mier. More specifically, we consider some classes of Dyck paths and characterize those paths giving rise to numerical semigroups.</p> </abstract>ARTICLEtrue Limit of the Non-dictatorship Index<abstract> <title style='display:none'>Abstract</title> <p>In this paper we determine the asymptotic behavior of the Non-dictatorship Index (NDI) introduced in Bednay, Moskalenko and Tasnádi (2019). We show that if <italic>m</italic> denotes the number of alternatives, then as the number of voters tends to infinity the NDI of any anonymous voting rule tends to (<italic>m</italic> − 1)/<italic>m</italic>, which equals the NDI of the constant rule.</p> </abstract>ARTICLEtrue and equilibrium in Banach spaces<abstract> <title style='display:none'>Abstract</title> <p>We establish that the local viability for a convex and compact valued upper Hausdorff continuous map on a convex and compact domain in a Banach space, implies the existence of an equilibrium.</p> </abstract>ARTICLEtrue on the special issue dedicated to the 70th anniversary of prof. Peter Tallos the construction of the elementary trigonometric functions<abstract> <title style='display:none'>Abstract</title> <p>The article discusses the construction of the elementary trigonometric functions. It discusses several approaches, but the main message is that to construct the trigonometric functions one needs to follow the same approach as one should use during the construction of the real exponential function, but one should use complex numbers. The key point is that one must construct the complex square root function to define the trigonometric functions on the binary numbers and then one should use some convexity argument to prove their differentiability. This approach, as the power series approach, based on the intimate relation between the trigonometric and the exponential functions.</p> </abstract>ARTICLEtrue notes on dynamic oligopsonies<abstract> <title style='display:none'>Abstract</title> <p>A special oligopoly model is considered when the firms compete on the factor market, and the used factor volumes determine their outputs. In the <italic>N</italic>firm case conditions are given for the existence of the Nash equilibrium, and in the cooperative case, sufficient conditions are derived for the existence and uniqueness of the joint profit maximizer. In the case of a linear duopoly the dynamic extensions are introduced in both cases based on gradient adjustments. Conditions are given for the local asymptotic stability of the equilibrium and the joint profit maximizer without and with information delays.</p> </abstract>ARTICLEtrue strategies for a linear reverse logistics system<abstract> <title style='display:none'>Abstract</title> <p>The paper investigates an one-product reverse logistics system. The aim is to analyze the functioning of such a system with continuous time and linear cost structure. The costs consist of linear holding costs for both warehouse, and the linear remanufacturing, manufacturing, and disposal costs. The demand and return rates are known deterministic parameters. The problem is solved with the Pontyagin’s maximum principle. Solving the control problem, it is corrected an earlier paper [1].</p> </abstract>ARTICLEtrue of sets in Euclidean space<abstract> <title style='display:none'>Abstract</title> <p>We consider several concepts of computability (recursiveness) for sets in Euclidean space. A list of four ideal properties for such sets is proposed and it is shown in a very elementary way that no notion can satisfy all four desiderata. Most notions introduced here are essentially based on separability of ℝ<italic><sup>n</sup></italic> and this is natural when thinking about operations on an actual digital computer where, in fact, rational numbers are the basis of everything. We enumerate some properties of some naïve but practical notions of recursive sets and contrast these with others, including the widely used and accepted notion of <italic>computable</italic> set developed by Weihrauch, Brattka and others which is based on the “Polish school” notion of a computable real function. We also offer a conjecture about the Mandelbrot set.</p> </abstract>ARTICLEtrue of population systems<abstract> <title style='display:none'>Abstract</title> <p>This is a <italic>selected overview</italic> of a research line initiated and mostly developed by the three authors over the last three decades. Applying the state space paradigm of Mathematical Systems Theory, monitoring means that from the observation (a transform) of an unknown state process, the latter should be recovered. Since most of the dynamic models of population biology are nonlinear, for solving the monitoring problem, tools of nonlinear analysis are applied in different contexts. This approach to monitoring has found different applications ranging from population ecology to radiotherapy, from stock estimation in fisheries to monitoring of solar thermal heating systems.</p> </abstract>ARTICLEtrue of the criss-cross algorithm for the linear programming problem with s-monotone index selection rules<abstract> <title style='display:none'>Abstract</title> <p>The traditional criss-cross algorithm for the linear programming problem is shown to be finite when <bold>s</bold>-monotone index selection rules are used. The set of <bold>s</bold>-monotone index selection rules, among others, include the Last In First Out (LIFO) and the Most Often Selected Variable rule (MOSV). The advantage of applying the <bold>s</bold>-monotone index selection rule is the flexibility it provides in selecting the pivot element while still preserving the guarantee for finiteness. Such flexibility may be used to improve the numerical stability of the algorithm.</p> </abstract>ARTICLEtrue trail decompositions on grid graphs<abstract> <title style='display:none'>Abstract</title> <p>Let <italic>G</italic> = <italic>G</italic>(<italic>n, m</italic>) be a rectangular solid grid graph and 𝒜(<italic>G</italic>) be a minimum length Eulerian augmentation of <italic>G</italic>. Let <italic>l</italic><sub>0</sub>, . . ., <italic>l<sub>t</sub></italic> ∈ ℕ such that ∑<sup>t</sup><italic><sub>i</sub></italic><sub>=0</sub> <italic>l<sub>i</sub></italic> = <italic>|E</italic>(𝒜(<italic>G</italic>)<italic>|,</italic> where 2(<italic>n</italic> + <italic>m</italic>) <italic>≤ l<sub>i</sub></italic> = 2<italic>k<sub>i</sub></italic>. In this paper, we exhibit a constructive procedure providing an edge-disjoint decomposition of 𝒜 (<italic>G</italic>) into closed trails <italic>T</italic><sub>0</sub>, . . ., <italic>T<sub>t</sub></italic> such that <italic>|E</italic>(<italic>T<sub>i</sub></italic>)<italic>|</italic> = <italic>l<sub>i</sub></italic>.</p> </abstract>ARTICLEtrue actuarial mathematical model for a new pension philosophy. An application to the accountant pension fund<abstract> <title style='display:none'>Abstract</title> <p>This paper adapts an actuarial mathematical model, built for the Italian public pension system, based on the law proposal 3035/2009 to the Accountant Pension Fund (CNPADC). The aim is to introduce a new philosophy pension highly correlated with the concept of adequacy for an ambitious social welfare; using the logic of the 3035/2009 proposal, which guarantees a minimum threshold for the replacement rate of the direct pension, this study provides a rigorous actuarial mathematical model that explains a sort of rate of contribution at a tendential equilibrium, in a pay-as-you-go pension system. This model reveals for which parameters it is possible to intervene to maintain the standard of living in retirement.</p> </abstract>ARTICLEtrue Logical Sustainability Theory for pension systems: the discrete-time model in a stochastic framework under variable mortality<abstract> <title style='display:none'>Abstract</title> <p>The aim of this work is to provide the logical sustainability model for defined contribution pension systems (see [1], [2]) in the discrete framework under stochastic financial rate of the pension system fund and stochastic productivity of the active participants. In addition, the model is developed in the assumption of variable mortality tables.</p> <p>Under these assumptions, the evolution equations of the fundamental state variables, the pension liability and the fund, are provided. In this very general discrete framework, the necessary and sufficient condition of the pension system sustainability, and all the other basic results of the logical sustainability theory, are proved.</p> <p>In addition, in this work new results on the efficiency of the rule for the stabilization over time of the level of the unfunded pension liability with respect to wages, level that is defined as <italic>β</italic> indicator, are also proved.</p> </abstract>ARTICLEtrue bit frequency in Fibonacci words<abstract> <title style='display:none'>Abstract</title> <p>It is known that binary words containing no <italic>k</italic> consecutive 1s are enumerated by <italic>k</italic>-step Fibonacci numbers. In this note we discuss the expected value of a random bit in a random word of length <italic>n</italic> having this property.</p> </abstract>ARTICLEtrue