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In this paper, an ancient Chinese algorithm is used to estimate the length constant of a long cooling fin, and an approximate solution formulation is obtained. The obtained results show that this method is a simple but promising method without any requirement for advanced calculus. [ABSTRACT FROM AUTHOR]

This paper deals with undiscounted infinite stage two-person zero-sum stochastic games with finite state and action spaces. It was recently shown that such games possess a value. But in general there are no optimal strategies. We prove that for each player there exists a nonempty set of easy initial states, i.e. starting states for which the player possesses an optimal stationary strategy. This result is proved with the aid of facts derived by Bewley and Kohlberg for the limit discount equation for stochastic games. [ABSTRACT FROM AUTHOR]

Among several number triangles that exist in mathematics, Pascal's triangle is the most fascinating and prominent structure possessing numerous properties. In this paper, we will introduce a number triangle consisting of positive integers arranged in ascending to descending order so that the entries in each row would be mirror images with respect to its middle number. Using this simple number triangle, we have proved six amusing results which interestingly depend only as functions of its centred numbers. Moreover some of the concepts and results discussed in this paper have connection with Tamil literary work and triangular numbers. [ABSTRACT FROM AUTHOR]

A one-dimensional deformed Heisenberg algebra [X, P] = i f (P) is studied. We answer the question: for what function of deformation f (P) does there exist a nonzero minimal uncertainty in position (minimal length)? We also find an explicit expression for the minimal length in the case of an arbitrary function of deformation. [ABSTRACT FROM AUTHOR]

In this paper, we prove some results on uniqueness of meromorphic functions with three weighted sharing values. The results in this paper improve those given by H. X. Yi, I. Lahiri, T. C. Alzahary and H. X. Yi and other authors. [ABSTRACT FROM AUTHOR]

In this paper, robust regulation of stable infinite-dimensional plants with transfer functions in the H 8 -algebra is considered. The reference and disturbance signals are allowed to have an infinite number of poles on the imaginary axis, which makes it possible to consider e.g. arbitrary periodic signals. A controller depending on a positive scalar e and design parameters K k is proposed. The design parameters are matrices over H 8 , and by the main result of the paper, if the design parameters satisfy a certain condition, the proposed controller is a robust regulator for sufficiently small values of e. The only knowledge needed to verify this condition is the so-called -stability of the plant and values of the plant transfer function at the poles of the reference and disturbance signals. The analysis in this paper is in the frequency domain and it is based on the fractional representation approach. [ABSTRACT FROM AUTHOR]

Comparing the large number of research papers on the ordered weighted averaging (OWA) operator, the researches on relative quantifier are relatively rare so far. In the present paper, based on the quantifier guided aggregation method with OWA operator which was proposed by Yager [“Quantifier guided aggregation using OWA operators”, Int. J. Intell. Syst. , 11, pp. 49–73, 1996], a generating function representation method for regular increasing monotone (RIM) quantifiers is proposed. We extend the the properties of OWA operator to the RIM quantifier which is represented with a monotone function instead of the OWA weighting vector. A class of parameterized equidifferent RIM quantifier which has minimum variance generating function is proposed and its properties are also analyzed. The equidifferent RIM quantifier is consistent with its orness level for any aggregated elements, which can be used to represent the decision maker's preference. [ABSTRACT FROM AUTHOR]

In this paper, we present upper and lower linear bounds for the Shannon function for length of checking tests for repetition-free functions in the basis {0, 1, &, ∨, ¬}. The research was supported by Russian Foundation for Basic Research, grants 04-01-00359 and 05-01-01000. [ABSTRACT FROM AUTHOR]

In view of the importance of aggregate density to the mechanical properties of concrete, it is necessary to determine the aggregate density at any point in concrete or any cross-section of a concrete element Once the aggregate density is known, some other mechanical properties of the concrete can be determined based on the theory of micro-mechanics. This paper attempts to simulate aggregate distribution in plain concrete taking into account of wall effect. Based on the study of the characteristics of the simulated aggregate distributions, a model of aggregate density is developed and its statistics are derived. Simulation results show that the average aggregate density of concrete in both two and three dimensions converges asymptotically to a constant when the number of simulations tends to infinity. Using mathematical regression, the aggregate density as a function of distance from the boundary and the thickness of the boundary layers can be expressed analytically in terms of the aggregate area or volume fraction of the concrete. The experimental verification of the developed model of aggregate density is also presented in the paper. [ABSTRACT FROM AUTHOR]

This paper discusses the estimation of average treatment effects in observational causal inferences. By employing a working propensity score and two working regression models for treatment and control groups, Robins et al. (1994, 1995) introduced the augmented inverse probability weighting (AIPW) method for estimation of average treatment effects, which extends the inverse probability weighting (IPW) method of Horvitz and Thompson (1952); the AIPW estimators are locally efficient and doubly robust. In this paper, we study a hybrid of the empirical likelihood method and the method of moments by employing three estimating functions, which can generate estimators for average treatment effects that are locally efficient and doubly robust. The proposed estimators of average treatment effects are efficient for the given choice of three estimating functions when the working propensity score is correctly specified, and thus are more efficient than the AIPW estimators. In addition, we consider a regression method for estimation of the average treatment effects when working regression models for both the treatment and control groups are correctly specified; the asymptotic variance of the resulting estimator is no greater than the semiparametric variance bound characterized by the theory of Robins et al. (1994, 1995). Finally, we present a simulation study to compare the finite-sample performance of various methods with respect to bias, efficiency, and robustness to model misspecification. [ABSTRACT FROM AUTHOR]

The maximal covering location problem has been shown to be a useful tool in siting emergency services. In this paper we expand the model along two dimensions - workload capacities on facilities and the allocation or multiple levels of backup or prioritized service for all demand points. In emergency service facility location decisions such as ambulance sitting, when all of a facility's resources are needed to meet each call for service and the demand cannot be queued, the need for a backup unit may be required. This need is especially significant in areas of high demand. These areas also will often result in excessive workload for some facilities. Effective siting decisions, therefore, must address both the need for a backup response facility for each demand point and a reasonable limit on each facility's workload. In this paper, we develop a model which captures these concerns as well as present an efficient solution procedure using Lagrangian relaxation. Results of extensive computational experiments are presented to demonstrate the viability of the approach. [ABSTRACT FROM AUTHOR]

We consider the so-called Encrypted Davies-Meyer (EDM) construction, which turns a permutation P on {0,1}n into a function from {0,1}n to {0,1}n defined as P(P(x)⊕x). A similar construction using two independent permutations, namely P′(P(x)⊕x), was previously analyzed by Cogliati and Seurin (Advances in cryptology—CRYPTO 2016 (Proceedings, Part I). LNCS, vol 9814, pp. 121-149, 2016) who showed that when P and P′ are secret and random, then any black-box adversary needs at least roughly 22n/3 queries to distinguish the construction from a uniformly random function from {0,1}n to {0,1}n. In this paper, we focus on the single-permutation variant of the construction. Our main result is that the PRF-security of the single-permutation EDM construction is also (at least) roughly 22n/3, in the sense that any black-box adversary needs at least this number of queries to distinguish the construction from a uniformly random function. This yields the first PRP-to-PRF conversion method which uses a single permutation, does not shrink the original domain nor range of the permutation, and provides security beyond the birthday bound. [ABSTRACT FROM AUTHOR]

The article reports on the discussion of the study concerning the learning of mathematics. The discussion contains arguments regarding the paper written by professor Stone about functions. One of the arguments that is given attention to, is the concept of function being connected with the primitive concepts of pure logic, nearly unanalyzable and must be grasped by a psychological act of direct perception. This argument is also related to the child's development as the notion of grasping and perception are described to occur at some point in the development.

Using a generalized Dunkl translation, we obtain an analog of Theorem 5.2 in Younis' paper for the Dunkl transform for functions satisfying the (δ, γ)-Dunkl Lipschitz condition in the space L² (ℝ, |x|2α+1dx). [ABSTRACT FROM AUTHOR]

In this paper, we characterize the family of quotients of internally quasicontinuous functions. Moreover, we study cardinal invariants related to quotients in the case of internally quasicontinuous functions and the complement of this family. [ABSTRACT FROM AUTHOR]

In this paper, a unique and more distinctive encryption algorithm is proposed. This is based on the complexity of highly nonlinear S box in Flesnelet domain. The nonlinear pattern is transformed further to enhance the confusion in the dummy data using Fresnelet technique. The security level of the encrypted image boosts using the algebra of Galois field in Fresnelet domain. At first level, the Fresnelet transform is used to propagate the given information with desired wavelength at specified distance. It decomposes given secret data into four complex subbands. These complex sub-bands are separated into two components of real subband data and imaginary subband data. At second level, the net subband data, produced at the first level, is deteriorated to non-linear diffused pattern using the unique S-box defined on the Galois field . In the diffusion process, the permuted image is substituted via dynamic algebraic S-box substitution. We prove through various analysis techniques that the proposed scheme enhances the cipher security level, extensively. [ABSTRACT FROM AUTHOR]

In this paper we isolate a property for forcing notions, the *-Prikry condition, that is similar to the Prikry con- dition but that is topological: A forcing P satisfies it iff for every p ∈ P and for every open dense D ⊆ P, there are n ∈ ω and q ≤∗ p such that for any r ≤ q with l(r) = l(q) + n, r ∈ D, for some length notion l. This is implicit in many proofs in literature. We prove this for the tree Prikry forcing and the long extender Prikry forcing. [ABSTRACT FROM AUTHOR]

This paper contains a new elementary proof of the Fundamental Theorem of Calculus for the Lebesgue integral. The hardest part of our proof simply concerns the convergence in L1 of a certain sequence of step functions, and we prove it using only basic elements from Lebesgue integration theory. [ABSTRACT FROM AUTHOR]

In this paper we establish interrelations between fuzzy direct revelation axiom (FDRA), fuzzy transitive-closure coherence axiom (FTCCA), fuzzy consistent-closure coherence axiom (FCCCA) and fuzzy intermediate congruence axiom (FICA). We also establish their relationships with weak fuzzy congruence axiom (WFCA), strong fuzzy congruence axiom (SFCA) and weak axiom of fuzzy revealed preference (WAFRP). Condition for equivalence of fuzzy Arrow axiom (FAA) and weak fuzzy congruence axiom (WFCA) on arbitrary domain is also given. [ABSTRACT FROM AUTHOR]

In this paper, we considered the following second order multi-valued iterative equation λ1F(x)+λ2F²(x)=G(x), and the decreasing solution for decreasing G which may be multi-valued at endpoints is given. [ABSTRACT FROM AUTHOR]

Inspired by a theorem of Marcinkiewicz [J. Marcinkiewicz, On a class of functions and their Fourier series, C. R. Soc. Sci. Varsovie, 26:71-77, 1934. Reprinted in: J. Marcinkiewicz, Collected Papers (A. Zygmund (Ed.)), PaństwoweWydawnictwo Naukowe,Warsaw, 1964] stating that the maximum of the absolute values of real Fourier coefficients a and b of a function of bounded p-variation $$ \left( {p \geqslant 1} \right) $$ on an interval [0 , 1] is of order O( n) as n → ∞, we compute the Fourier coefficients of the linear fractional stable motion (LFSM) and of the closely related Riemann-Liouville (RL) process and investigate the rate of their decay. [ABSTRACT FROM AUTHOR]

In this paper we establish two theta function identities with four parameters by the theory of theta functions. Using these identities we introduce common generalizations of Hirschhorn-Garvan-Borwein cubic theta functions, and also re-derive the quintuple product identity, one of Ramanujan’s identities, Winquist’s identity and many other interesting identities. [ABSTRACT FROM AUTHOR]

In this paper we show that the commutator generated by the weighted Lipschitz function and the singular integral operator with rough kernel satisfying suitable Dini condition is bounded on weighted Lebesgue spaces. [ABSTRACT FROM AUTHOR]

Abstract Stone Duality (ASD) is a direct axiomatisation of general topology, in contrast to the traditional and all other contemporary approaches, which rely on a prior notion of discrete set, type or object of a topos. ASD reconciles mathematical and computational viewpoints, providing an inherently computable calculus that does not sacrifice key properties of real analysis such as compactness of the closed interval. Previous theories of recursive analysis failed to do this because they were based on points; ASD succeeds because, like locale theory and formal topology, it is founded on the algebra of open subspaces. ASD is presented as a lambda calculus, of which we provide a self-contained summary, as the foundational background has been investigated in earlier work. The core of the paper constructs the real line using two-sided Dedekind cuts. We show that the closed interval is compact and overt, where these concepts are defined using quantifiers. Further topics, such as the Intermediate Value Theorem, are presented in a separate paper that builds on this one. The interval domain plays an important foundational role. However, we see intervals as generalised Dedekind cuts, which underly the construction of the real line, not as sets or pairs of real numbers. We make a thorough study of arithmetic, in which our operations are more complicated than Moore's, because we work constructively, and we also consider back-to-front (Kaucher) intervals. Finally, we compare ASD with other systems of constructive and computable topology and analysis. 1. Introduction 757 2. Cuts and intervals 762 3. Topology as lambda calculus 770 4. The ASD lambda calculus 774 5. The monadic principle 782 6. Dedekind cuts 788 7. The interval domain in ASD 793 8. The real line as a space in ASD 799 9. Dedekind completeness 803 10. Open, compact and overt intervals 807 11. Arithmetic 812 12. Multiplication 816 13. Reciprocals and roots 821 14. Axiomatic completeness 824 15. Recursive analysis 828 16. Conclusion 832. [ABSTRACT FROM AUTHOR]

The paper offers a matrix-based logic (relevant matrix quantum physics) for propositions which seems suitable as an underlying logic for empirical sciences and especially for quantum physics. This logic is motivated by two criteria which serve to clean derivations of classical logic from superfluous redundancies and uninformative complexities. It distinguishes those valid derivations (inferences) of classical logic which contain superfluous redundancies and complexities and are in this sense "irrelevant" from those which are "relevant" or "nonredundant" in the sense of allowing only the most informative consequences in the derivations. The latter derivations are strictly valid in RMQ, whereas the former are only materially valid. RMQ is a decidable matrix calculus which possesses a semantics and has the finite model property. It is shown in the paper how RMQ by its strictly valid derivations can avoid the difficulties with commensurability, distributivity, and Bell's inequalities when it is applied to quantum physics. [ABSTRACT FROM AUTHOR]

Temlyakov considered the optimal recovery on the classes of functions with bounded mixed derivative in the L p metrics and gave the upper estimates of the optimal recovery errors. In this paper, we determine the asymptotic orders of the optimal recovery in Sobolev spaces by standard information, i.e., function values, and give the nearly optimal algorithms which attain the asymptotic orders of the optimal recovery. [ABSTRACT FROM AUTHOR]

We consider a problem of finding optimal contracts in continuous time, when the agent’s actions are unobservable by the principal, who pays the agent with a one-time payoff at the end of the contract. We fully solve the case of quadratic cost and separable utility, for general utility functions. The optimal contract is, in general, a nonlinear function of the final outcome only, while in the previously solved cases, for exponential and linear utility functions, the optimal contract is linear in the final output value. In a specific example we compute, the first-best principal’s utility is infinite, while it becomes finite with hidden action, which is increasing in value of the output. In the second part of the paper we formulate a general mathematical theory for the problem. We apply the stochastic maximum principle to give necessary conditions for optimal contracts. Sufficient conditions are hard to establish, but we suggest a way to check sufficiency using non-convex optimization. [ABSTRACT FROM AUTHOR]

Pure Pattern Type Systems (P2TS) combine the frameworks and capabilities of rewriting and ?-calculus within a unified setting. Their type systems, which are adapted from Barendregt's ?-cube, are especially interesting from a logical point of view. Until now, strong normalisation, which is an essential property for logical soundness, has only been conjectured: in this paper, we give a positive answer for the simply-typed system and the dependently-typed system.The proof is based on a translation of terms and types from P2TSinto the ?-calculus. First, we deal with untyped terms, ensuring that reductions are faithfully mimicked in the ?-calculus. For this, we rely on an original encoding of the pattern matching capability of P2TSinto the System F?.Then we show how to translate types: the expressive power of System F? is needed in order to fully reproduce the original typing judgments of P2TS. We prove that the encoding is correct with respect to reductions and typing, and we conclude with the strong normalisation of simply-typed P2TSterms. The strong normalisation with dependent types is in turn obtained by an intermediate translation into simply-typed terms. [ABSTRACT FROM AUTHOR]

The Rice probability density function has received considerable attention for its various important technical and scientific applications. One of the more attractive techniques for extracting the distribution parameters, and possibly the most frequently applied, relies on the maximization of the likelihood function for a given set of experimentally determined samples, and many applications are documented in the literature. This paper offers a mathematical analysis which demonstrates that, subject to conditions universally verified in physical systems, an absolute maximum exists, and it is the unique point internal to the domain of existence which zeroes the gradient of the likelihood function. In all previous results, the presence of additional maxima, which are possibly larger than the one that had numerically been found, could not be excluded. We can incidentally state that this pa- per demonstrates that all previous results based on numerically finding a maximum indeed corresponded to absolute maxima. The mathematical derivations offered here are also suggestive of actions capable of improving the insight into the maximum-likelihood technique and its numerical implementation. [ABSTRACT FROM AUTHOR]

In this paper, we study conditions for the existence of a function satisfying constraints on its value at each point and on its derivative. [ABSTRACT FROM AUTHOR]

This paper considers the problem of approximating an integrable function by piecewise linear functions that keep the integral and positivity of the original function. [ABSTRACT FROM AUTHOR]

We will show in this paper that if λ is very close to 1, then can be attained, where M is a compact–manifold with boundary. This result gives a counter–example to the conjecture of de Figueiredo and Ruf in their paper titled "On an inequality by Trudinger and Moser and related elliptic equations" ( Comm. Pure. Appl. Math., 55, 135–152, 2002). [ABSTRACT FROM AUTHOR]

In this paper, we apply the dual approach developed by A. Chambolle for the Rudin-Osher-Fatemi model to regularization functionals with higher order derivatives. We emphasize the linear algebra point of view by consequently using matrix-vector notation. Numerical examples demonstrate the differences between various second order regularization approaches. [ABSTRACT FROM AUTHOR]

The present paper continues a series of papers of the author (some of them are written in collaboration), in which the Yoneda algebras are calculated for several families of algebras of dihedral and semidihedral type (in K. Erdmann’s classification). In the paper, the Yoneda algebras are described (in terms of quivers with relations) for the algebras of semidihedral type that form the family SD(3 $$\mathcal{K}$$ ). Bibliography: 10 titles. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the determination of the entire set of Pareto solutions of location problems involving Q general criteria. These criteria include median, center, or centdian objective functions as particular instances. We characterize the set of Pareto solutions of all these multicriteria problems for any polyhedral gauge. An efficient algorithm is developed for the planar case and its complexity is established. Extensions to the nonconvex case are also considered. The proposed approach is more general than previously published approaches to multicriteria location problems. [ABSTRACT FROM AUTHOR]

Boolean functions are widely used because they can be used to precisely describe logical circuits. Properties of Boolean functions with respect to their applications to cryptography have been studied, but relationship between Boolean functions are rarely studied. This paper studies the independence of Boolean functions, gives necessary and sufficient conditions for judging whether two given Boolean functions are independent. Some enumeration formulae are given. [ABSTRACT FROM AUTHOR]

Abstract In the present paper the well-known Gdels Churchs argument concerning the undecidability of logic (of the first order functional calculus) is exhibited in a way which seems to be philosophically interestingfi The natural numbers are not used. (Neither Chinese Theorem nor other specifically mathematical tricks are applied.) Only elementary logic and very simple set-theoretical constructions are put into the proof. Instead of the arithmetization I use the theory of concatenation (formalized by Alfred Tarski). This theory proves to be an appropriate tool. The decidability is defined directly as the property of graphical discernibility of formulas. [ABSTRACT FROM AUTHOR]

A new incomplete data model, the trunsored model, in lifetime analysis is introduced. This model can be regarded as a unified model of the censored and truncated models. Using the model, we can not only estimate the ratio of the fragile population to the mixed fragile and durable populations, but also test a hypothesis that the ratio is equal to a prescribed value. A central point of the paper is that such a test can easily be realized through the newly introduced trunsored model, because it has been difficult to do such a hypothesis test under only the framework of censored and truncated models. Therefore, the relationship of the trunsored model to the censored and truncated models is clarified because the trunsored model unifies the censored and truncated models. The paper also shows how to obtain the estimates of the parameters in lifetime estimation, and corresponding confidence intervals for the fragile population. Typical examples applied to electronic board failures. and to breast cancer data, for lifetime estimation are demonstrated, and successfully worked using the trunsored model. [ABSTRACT FROM AUTHOR]

In this paper, a model of a linear multilevel programming problem with dominated objective functions (LMPPD(l)) is proposed, where multiple reactions of the lower levels do not lead to any uncertainty in the upper-level decision making. Under the assumption that the constrained set is nonempty and bounded, a necessary optimality condition is obtained. Two types of geometric properties of the solution sets are studied. It is demonstrated that the feasible set of LMPPD(l) is neither necessarily composed of faces of the constrained set nor necessarily connected. These properties are different from the existing theoretical results for linear multilevel programming problems. [ABSTRACT FROM AUTHOR]

This paper deals with the numerical solution of optimal control problems for ODEs. The methods considered here rely on some standard optimization code to solve a discretized version of the control problem under consideration. We aim to make available to the optimization software not only the discrete objective functional, but also its gradient. The objective gradient can be computed either from forward (sensitivity) information or backward (adjoint) information. The purpose of this paper is to discuss various ways of adjoint computation. It will be shown both theoretically and numerically that methods based on the continuous adjoint equation require a careful choice of both the integrator and gradient assembly formulas in order to obtain a gradient consistent with the discretized control problem. Particular attention is given to automatic differentiation techniques which generate automatically a suitable integrator. [ABSTRACT FROM AUTHOR]

We study second order sufficient optimality conditions (SSC) for optimal control problems with control appearing linearly. Specifically, time-optimal bang-bang controls will be investigated. In [N. P. Osmolovskii, Sov. Phys. Dokl., 33 (1988), pp. 883-885; Theory of HigherOrder Conditions in Optimal Control, Doctor of Sci. thesis, Moscow, 1988 (in Russian); Russian J. Math. Phys., 2 (1995), pp. 487-516; Russian J. Math. Phys., 5 (1997), pp. 373-388; Proceedings of the Conference "Calculus of Variations and Optimal Control," Chapman & Hall/CRC, Boca Raton, FL, 2000, pp. 198-216; A. A. Milyutin and N. P. Osmolovskii, Calculus of Variations and Optimal Control, Transl. Math. Monogr. 180, AMS, Providence, RI, 1998], SSC have been developed in terms of the positive definiteness of a quadratic form on a critical cone or subspace. No systematical numerical methods for verifying SSC are to be found in these papers. In the present paper, we study explicit representations of the critical subspace. This leads to an easily implementable test for SSC in the case of a bang-bang control with one or two switching points. In general, we show that the quadratic form can be simplified by a transformation that uses a solution to a linear matrix differential equation. Particular conditions even allow us to convert the quadratic form to perfect squares. Three numerical examples demonstrate the numerical viability of the proposed tests for SSC. [ABSTRACT FROM AUTHOR]

For the problem of minimizing the sum of Euclidean norms (MSN), most existing quadratically convergent algorithms require a strict complementarity assumption. However, this assumption is not satisfied for a number of MSN problems. In this paper, we present a globally and quadratically convergent algorithm for the MSN problem. In particular, the quadratic convergence result is obtained without assuming strict complementarity. Examples without strictly complementary solutions are given to show that our algorithm can indeed achieve quadratic convergence. Preliminary numerical results are reported. [ABSTRACT FROM AUTHOR]

Let $G$ be a group, let $A$ be an Abelian group, and let $n$ be an integer such that $n\backslash ge\; -1$. In the paper, the sets $\backslash Phi\_n(G,A)$ of functions from $G$ into $A$ of degree not greater than $n$ are studied. In essence, these sets were introduced by Logachev, Sal'nikov, and Yashchenko. We describe all cases in which any function from $G$ into $A$ is of bounded (or not necessarily bounded) finite degree. Moreover, it is shown that if $G$ is finite, then the study of the set $\backslash Phi\_n(G,A)$ is reduced to that of the set $\backslash Phi\_n(G/O^p(G),A\_p)$ for primes $p$ dividing $|G/G\text{'}|$. Here $O^p(G)$ stands for the $p$-coradical of the group $G$, $A\_p$ for the $p$-component of $A$, and $G\text{'}$ for the commutator subgroup of $G$. [ABSTRACT FROM AUTHOR]

The concept of a strict extended partial order (SEPO) has turned out to be very useful in explaining (resp. rationalizing) non-binary choice functions. The present paper provides a general account of the concept of extended binary relations, i.e. relations between subsets and elements of a given universal set of alternatives. In particular, we define the concept of a weak extended partial order (WEPO) and show how it can be used in order to represent rankings of Opportunity sets that display a ‘preference for opportunities’. We also clarify the relationship between SEPOs and WEPOs, which involves a non-trivial condition, called ‘strict properness’. Several characterizations of strict (and weak) properness are provided, based on which we argue for properness as an appropriate condition demarcating ‘choice based’ preference. [ABSTRACT FROM AUTHOR]

In this paper we study the point-objective problem of locating in R2 a facility serving a finite number of customers to minimize the travel time, or the distance, to each customer. Travel times, or distances, are measured by going in the directions of some given vectors which means that, under some conditions, are evaluated by a norm function. An algorithm is proposed to find all the quasi-efficient points and all the efficient points (alternately or strictly), for any given set of travelling directions. Consequently, the problem of efficiency is addressed in a general framework. [ABSTRACT FROM AUTHOR]

The paper studies projective stationary sets. The Projective Stationary Reflection Principle is the statement that every projective stationary set contains an increasing continuous --chain of length É1. It is shown that, if Martin's Maximum holds, then the Projective Stationary Reflection Principle holds. Also, this principle is equivalent to the Strong Reflection Principle. The paper shows that the saturation of the nonstationary ideal on É1 is equivalent to a certain kind of reflection. [ABSTRACT FROM AUTHOR]

This paper presents a quantitative discussion on the stability of the second-order periodic difference equation, which characterizes the discrete periodic time-varying system. The periodic parameter discrete system, which corresponds to Mathieu's equation in the continuous system, is represented by a second-order linear difference equation with a small parameter E in the varying term. The parameter E plays an important role in the determination of the stability. By applying McLachlan's method, the expression for the boundary between the stability and the instability can be determined analytically with regard to the parameters contained in the equation. For the case where the periodic parameter of the difference equation is represented as a sum of two even functions, the boundary between stability and instability is determined. The stability can be analyzed in a similar way for the case where the periodic parameter is represented as a sum of N even functions in Fourier series. [ABSTRACT FROM AUTHOR]

With the arrival of the nineteenth century, a process of change guided the treatment of three basic elements in the development of mathematics: rigour, the arithmetization and the clarification of the concept of function, categorised as the most important tool in the development of the mathematical analysis. In this paper we will show how several prominent mathematicians contributed greatly to the development of these basic elements that allowed the solid underpinning of mathematics and the consideration of mathematics as an axiomatic way of thinking in which anyone can deduce valid conclusions from certain types of premises. This nineteenth century stage shares, possibly with the Heroic Age of Ancient Greece, the most revolutionary period in all history of mathematics. [ABSTRACT FROM AUTHOR]

Hausman and Shapiro gave an estimate for the number of practical numbers to be 0(x ⁄(logx)β ) for every positive β<1/2 ( 1/log2 -1)². In this paper, we generalize Hausman and Shapiro bound by proving the number of t-practical numbers n (n ≤x ), ( t ≥1 ) to be 0(x ⁄(logx)β ) for every positive β<1/2 ( 1/log2 -1)², and 1≤t≤exp.(( logx ) δ1 ) for any δ1 satisfying 0< δ1 <1-1(1 +√2 β) log2 . We mean by the t-practical number n, the number in which every integer 1≤m≤tn is of the form m= ∑ d/n cd d, 0 ≤ cd≤ t [ABSTRACT FROM AUTHOR]