rss_2.0Pure Mathematics and Applications FeedSciendo RSS Feed for Pure Mathematics and Applicationshttps://sciendo.com/journal/PUMAhttps://www.sciendo.comPure Mathematics and Applications 's Coverhttps://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/6005c74be797941b18f26eb3/cover-image.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20220525T123818Z&X-Amz-SignedHeaders=host&X-Amz-Expires=604800&X-Amz-Credential=AKIA6AP2G7AKDOZOEZ7H%2F20220525%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=8544df92ad5345ae999289097e77bd250c41e925a04a432f9cf787ab501dae28200300Enumeration of S-Motzkin paths from left to right and from right to left: a kernel method approachhttps://sciendo.com/article/10.1515/puma-2015-0039<abstract><title style='display:none'>Abstract</title><p>The area of S-Motzkin paths (bijective to ternary trees) is calculated using the kernel method by enumerating these (partial) paths with fixed end-point resp. starting point.</p></abstract>New degenerate Bernoulli, Euler, and Genocchi polynomialshttps://sciendo.com/article/10.1515/puma-2015-0037<abstract><title style='display:none'>Abstract</title><p>We introduce new generalizations of the Bernoulli, Euler, and Genocchi polynomials and numbers based on the Carlitz-Tsallis degenerate exponential function. Also, we present generalizations of some familiar identities and connection between these types of Bernoulli, Euler, and Genocchi polynomials. Moreover, we establish new analogues of the Euler identity for degenerate Bernoulli polynomials and numbers.</p></abstract>Counting Stirling permutations by number of pusheshttps://sciendo.com/article/10.1515/puma-2015-0038<abstract><title style='display:none'>Abstract</title><p>Let 𝒯<sup>(</sup><italic><sup>k</sup></italic><sup>)</sup><italic><sub>n</sub></italic> denote the set of <italic>k</italic>-Stirling permutations having <italic>n</italic> distinct letters. Here, we consider the number of steps required (i.e., <italic>pushes</italic>) to rearrange the letters of a member of 𝒯<sup>(</sup><italic><sup>k</sup></italic><sup>)</sup><italic><sub>n</sub></italic> so that they occur in non-decreasing order. We find recurrences for the joint distribution on 𝒯<sup>(</sup><italic><sup>k</sup></italic><sup>)</sup><italic><sub>n</sub></italic> for the statistics recording the number of levels (i.e., occurrences of equal adjacent letters) and pushes. When <italic>k</italic> = 2, an explicit formula for the ordinary generating function of this distribution is also found. In order to do so, we determine the <italic>LU</italic>-decomposition of a certain infinite matrix having polynomial entries which enables one to compute explicitly the inverse matrix.</p></abstract>Closed trail decompositions on grid graphshttps://sciendo.com/article/10.1515/puma-2015-0040<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
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_puma-2015-0040_eq_001.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msubsup><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mi>t</mml:mi></mml:msubsup><mml:mrow><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>𝒜</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:mrow></mml:math><tex-math>\sum\nolimits_{i = 0}^t {{l_i}} = \left| {E(\mathcal{A}\left( G \right)} \right|</tex-math></alternatives></inline-formula> 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>Finiteness of the criss-cross algorithm for the linear programming problem with s-monotone index selection ruleshttps://sciendo.com/article/10.1515/puma-2015-0041<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>Equivalence classes of permutations modulo descents and left-to-right maximahttps://sciendo.com/article/10.1515/puma-2015-0002<abstract><title style='display:none'>Abstract</title><p> In a recent paper [2], the authors provide enumerating results for equivalence classes of permutations modulo excedances. In this paper we investigate two other equivalence relations based on descents and left-to-right maxima. Enumerating results are presented for permutations, involutions, derangements, cycles and permutations avoiding one pattern of length three.</p></abstract>Enumeration and Automatic Sequenceshttps://sciendo.com/article/10.1515/puma-2015-0008<abstract><title style='display:none'>Abstract</title><p> A sequence (a<sub>n</sub>)n≥0 is k-automatic if there is a finite automaton that, on input n expressed in base k, reaches a state with output a<sub>n</sub>. In this paper I will survey some recent advances concerning enumeration of various aspects of these sequences, such as the recurrence function, and the subword complexity (which counts the number of distinct blocks of length n). </p></abstract>Combinatorial proofs of some Stirling number formulashttps://sciendo.com/article/10.1515/puma-2015-0009<abstract><title style='display:none'>Abstract</title><p> In this note, we provide bijective proofs of some recent identities involving Stirling numbers of the second kind, as previously requested. Our arguments also yield generalizations in terms of a well known q-Stirling number.</p></abstract>Recursions for the flag-excedance number in colored permutations groupshttps://sciendo.com/article/10.1515/puma-2015-0005<abstract><title style='display:none'>Abstract</title><p> The excedance number for S<sub>n</sub> is known to have an Eulerian distribution. Nevertheless, the classical proof uses descents rather than excedances. We present a direct proof based on a recursion which uses only excedances and extend it to the flag-excedance parameter defined on the group of colored permutations Gr<sub>,n</sub> = ℤ<sub>r</sub> ≀ S<sub>n</sub>. We have also computed the distribution of a variant of the flag-excedance number, and show that its enumeration uses the Stirling number of the second kind. Moreover, we show that the generating function of the flag-excedance number defined on ℤ<sub>r</sub> ≀ S<sub>n</sub> is symmetric, and its variant is log-concave on ℤ<sub>r</sub> ≀ S<sub>n</sub>.. </p></abstract>Sampling parts of random integer partitions: a probabilistic and asymptotic analysishttps://sciendo.com/article/10.1515/puma-2015-0007<abstract><title style='display:none'>Abstract</title><p> Let λ be a partition of the positive integer n, selected uniformly at random among all such partitions. Corteel et al. (1999) proposed three different procedures of sampling parts of λ at random. They obtained limiting distributions of the multiplicity μ<sub>n</sub> = μ<sub>n</sub>(λ) of the randomly-chosen part as n → ∞. The asymptotic behavior of the part size σ<sub>n</sub> = σ<sub>n</sub>(λ), under these sampling conditions, was found by Fristedt (1993) and Mutafchiev (2014). All these results motivated us to study the relationship between the size and the multiplicity of a randomly-selected part of a random partition. We describe it obtaining the joint limiting distributions of (μ<sub>n</sub>; σ<sub>n</sub>), as n → ∞, for all these three sampling procedures. It turns out that different sampling plans lead to different limiting distributions for (μ<sub>n</sub>; σ<sub>n</sub>). Our results generalize those obtained earlier and confirm the known expressions for the marginal limiting distributions of μ<sub>n</sub> and σ<sub>n</sub>. </p></abstract>A recursive structure of sand pile model and its applicationshttps://sciendo.com/article/10.1515/puma-2015-0006<abstract><title style='display:none'>Abstract</title><p> The Sand Pile Model (SPM) and its generalization, the Ice Pile Model (IPM), originate from physics and have various applications in the description of the evolution of granular systems. In this article, we deal with the enumeration and the exhaustive generation of the accessible configuration of the system. Our work is based on a new recursive decomposition theorem for SPM configurations using the notion of staircase bases. Based on this theorem, we provide a recursive formula for the enumeration of SPM(n) and a constant amortized time (CAT) algorithm for the generation of all SPM(n) configurations. The extension of the same approach to the Ice Pile Model is also discussed. </p></abstract>A generating function for bit strings with no Grand Dyck pattern matchinghttps://sciendo.com/article/10.1515/puma-2015-0003<abstract><title style='display:none'>Abstract</title><p> We study the construction and the enumeration of bit strings, or binary words in {0, 1}*, having more 1’s than 0’s and avoiding a set of Grand Dyck patterns which form a cross-bifix-free set. We give a particular jumping and marked succession rule which describes the growth of such words according to the number of 1’s. Then, we give the enumeration of the class by means of generating function. </p></abstract>Forewordhttps://sciendo.com/article/10.1515/puma-2015-0010The dominance order for permutationshttps://sciendo.com/article/10.1515/puma-2015-0004<abstract><title style='display:none'>Abstract</title><p> We define an order relation over S<sub>n</sub> considering the Robinson-Schensted bijection and the dominance order over Young tableaux. This order relation makes S<sub>n</sub>(k k - 1...3 2 1) -the set of permutations of length n that avoid the pattern k k - 1...3 2 1, k ≤ n- a principal filter in Sn. We study in detail these order relations on Sn(321) and Sn(4321), finding order-isomorphisms between these sets and sets of lattice paths. </p></abstract>The optimal rate of return for defined contribution pension systems in a stochastic frameworkhttps://sciendo.com/article/10.1515/puma-2015-0023<abstract><title style='display:none'>Abstract</title><p>This paper deals with the problem of the optimal rate of return to be paid by a defined contribution pension system to its participants’ savings, namely the rate that achieves the goal of the most favorable returns on their contributions jointly with the sustainability of the pension system.</p><p>We consider defined contribution pension systems provided with a funded component, and for their study we use the “theory of the logical sustainability of pension systems” already developed in several previous works. In this paper, we focus on pension systems in a demographically stable state, whereas the productivity of the active participants and the financial rate of return on the pension system’s fund, rates that constitute the “ingredients” of the optimal rate of return on contributions, are modeled by two stochastic processes.</p><p>We show that the decisional rule defining the optimal rate of return on contributions is optimal in the sense that it is effective in terms of sustainability, and also efficient in the sense that if the system pays to its participants’ contributions a rate of return that is either higher or lower than the one provided by the rule, then the pension system becomes unsustainable or overcapitalized, respectively.</p></abstract>Controlling a demographic wave in defined contribution pension systemshttps://sciendo.com/article/10.1515/puma-2015-0001<abstract><title style='display:none'>Abstract</title><p>In several developed countries, the baby boomers will come to retire in the next decades. This problem will threaten the sustainability and the intergenerational equity of mandatory pay-as-you-go pension systems because they will have to drain the “demographic wave” of retirees with a relatively small number of contributors. In this paper, we give the operating method developed on the basis of a general principle, which a defined contribution pension system, in a state of stable sustainability, should adopt to control these issues in the presence of a demographic wave. In the theoretical profile, our approach breaks and overcomes the classical juxtaposition between funded and pay-as-you-go pension schemes, carrying out the integration of the two financial methods.</p></abstract>Personalised Hiking Time Estimationhttps://sciendo.com/article/10.1515/puma-2015-0017<abstract><title style='display:none'>Abstract</title><p>There are numerous attempts to estimate hiking time since the age of the ancient Roman Empire, the new digital era calls for more precise and exact solutions to be implemented in mobile applications. The importance of the topic lies in the fact that route planning algorithms and shortest path problems apply time estimations as cost functions. Our intention is to design a hiking time estimation method that accounts for terrain circumstances as well as personal factors, while the level of accuracy and the simplicity of the algorithm should enable the solution to be utilised in the practice. We refine Tobler’s earlier results to estimate a relation between terrain steepness and hiker’s velocity. Later we use fitted curve to design our novel, personalised hiking time estimation method.</p></abstract>A simple model of production and trade in an oligopolistic market: back to basicshttps://sciendo.com/article/10.1515/puma-2015-0022<abstract><title style='display:none'>Abstract</title><p>We provide a two good model of oligopolistic production and trade with one good being commodity money. There is the usual demand function of the consumers for the produced good that producer-sellers face. Each seller is a budget constrained preference maximizer and derives utility (or satisfaction) from consuming bundles comprising commodity money and the produced good. We define a competitive equilibrium strategy profile and a Cournotian equilibrium and show that under our assumptions both exist. We further show that at a competitive equilibrium strategy profile, each seller maximizes profits given his own consumption of the produced good and the price of the produced good, the latter being determined by the inverse demand function. Similarly we show that at a Cournotian the sellers are at a Cournot equilibrium given their own consumption of the produced good. Assuming sufficient differentiability of the cost functions we show that at a competitive equilibrium each seller either sets price equal to marginal cost or exhausts his capacity of production; at a Cournotian equilibrium each seller either sets marginal revenue equal to marginal cost or exhausts his capacity of production. We also study the evolution of Cournotian strategies as the sellers and buyers are replicated. As the number of buyers and sellers go to infinity any sequence of interior symmetric Cournotian equilibrium strategies admits a convergent subsequence, which converges to an interior symmetric competitive equilibrium strategy. In a final section we discuss the Bertrand Edgeworth price setting game and show that a Bertrand Edgeworth equilibrium must be a derived from a competitive equilibrium price. Here we show that if at a symmetric competitive equilibrium, the sellers consume positive quantities of the produced good then the competitive equilibrium cannot be a Bertrand Edgeworth equilibrium. Thus, if at all symmetric competitive equilibria the sellers consume positive amounts of the produced good, then a Bertrand Edgeworth equilibrium simply does not exist.</p></abstract>Duplex selections, equilibrium points, and viability tubeshttps://sciendo.com/article/10.1515/puma-2015-0011<abstract><title style='display:none'>Abstract</title><p>Existence of viable trajectories to nonautonomous differential inclusions are proven for time-dependent viability tubes. In the convex case we prove a double-selection theorem and a new Scorza-Dragoni type lemma. Our result also provides a new and palpable proof for the equilibrium form of Kakutani’s fixed point theorem.</p></abstract>An actuarial mathematical model for a new pension philosophy. An application to the accountant pension fundhttps://sciendo.com/article/10.1515/puma-2015-0036<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>en-us-1