###### The Logical Sustainability Theory for pension systems: the discrete-time model in a stochastic framework under variable mortality

## Abstract

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.

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.

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 β indicator, are also proved.

###### Controlling a demographic wave in defined contribution pension systems

## Abstract

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.

###### Duplex selections, equilibrium points, and viability tubes

## Abstract

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.

###### The optimal rate of return for defined contribution pension systems in a stochastic framework

## Abstract

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.

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.

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.

###### Personalised Hiking Time Estimation

## Abstract

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.

###### A simple model of production and trade in an oligopolistic market: back to basics

## Abstract

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.

###### Enumeration of small Wilf classes avoiding 1324 and two other 4-letter patterns

## Abstract

Recently, it has been determined that there are 242 Wilf classes of triples of 4-letter permutation patterns by showing that there are 32 non-singleton Wilf classes. Moreover, the generating function for each triple lying in a non-singleton Wilf class has been explicitly determined. In this paper, toward the goal of enumerating avoiders for the singleton Wilf classes, we obtain the generating function for all but one of the triples containing 1324. (The exceptional triple is conjectured to be intractable.) Our methods are both combinatorial and analytic, including generating trees, recurrence relations, and decompositions by left-right maxima. Sometimes this leads to an algebraic equation for the generating function, sometimes to a functional equation or a multi-index recurrence amenable to the kernel method.

###### Enumeration of small Wilf classes avoiding 1342 and two other 4-letter patterns

## Abstract

This paper is one of a series whose goal is to enumerate the avoiders, in the sense of classical pattern avoidance, for each triple of 4-letter patterns. There are 317 symmetry classes of triples of 4-letter patterns, avoiders of 267 of which have already been enumerated. Here we enumerate avoiders for all small Wilf classes that have a representative triple containing the pattern 1342, giving 40 new enumerations and leaving only 13 classes still to be enumerated. In all but one case, we obtain an explicit algebraic generating function that is rational or of degree 2. The remaining one is shown to be algebraic of degree 3. We use standard methods, usually involving detailed consideration of the left to right maxima, and sometimes the initial letters, to obtain an algebraic or functional equation for the generating function.

###### Forests and pattern-avoiding permutations modulo pure descents

## Abstract

We investigate an equivalence relation on permutations based on the pure descent statistic. Generating functions are given for the number of equivalence classes for the set of all permutations, and the sets of permutations avoiding exactly one pattern of length three. As a byproduct, we exhibit a permutation set in one-to-one correspondence with forests of ordered binary trees, which provides a new combinatorial class enumerated by the single-source directed animals on the square lattice. Furthermore, bivariate generating functions for these sets are given according to various statistics.

###### A generalization of André-Jeannin’s symmetric identity

## Abstract

In 1997, Richard André-Jeannin obtained a symmetric identity involving the reciprocal of the Horadam numbers *W _{n}*, defined by a three-term recurrence

*W*

_{n}_{+2}=

*P W*

_{n}_{+1}

*− QW*with constant coefficients. In this paper, we extend this identity to sequences

_{n}*{a*

_{n}}_{n}_{∈ℕ}satisfying a three-term recurrence

*a*

_{n}_{+2}=

*p*

_{n}_{+1}

*a*

_{n}_{+1}+

*q*

_{n}_{+1}

*a*with arbitrary coefficients. Then, we specialize such an identity to several

_{n}*q*-polynomials of combinatorial interest, such as the

*q*-Fibonacci,

*q*-Lucas,

*q*-Pell,

*q*-Jacobsthal,

*q*-Chebyshev and

*q*-Morgan-Voyce polynomials.