A Note on a Composition of Two Random Integral Mappings β and Some Examples

Abstract

The method of random integral representation, that is, the method of representing a given probability measure as the probability distribution of some random integral, was quite successful in the past few decades. In this note, we show that a composition of two random integral mappings ^β is again a random integral mapping. We illustrate our results with some examples.     

A Note on Large Deviations for the Stable Marriage of?Poisson and Lebesgue with Random Appetites

Let Ξ⊂ℝ ^d be a set of centers chosen according to a Poisson point process in ℝ ^d . Let ψ be an allocation of ℝ ^d to Ξ in the sense of the Gale–Shapley marriage problem, with the additional feature that every center ξ∈Ξ has an appetite given by a nonnegative random variable α. Generalizing some previous results, we study large deviations for the distance of a typical point x∈ℝ ^d to its center ψ(x)∈Ξ, subject to some restrictions on the moments of α.     

Bolzano’s Argument for the Existence of Substances: a Formalization with Two Types of Predication

The topic of our analysis is the argument for the existence of substances given by Bernard Bolzano in Athanasia (1827), where he essentially employs two ontological categories: substance and adherence. Bolzano considers the real and conditioned Inbegriff of all adherences, which are wirklich and nicht selbst bestehen. He claims that the formed collection is dependent on something external and non-adherential, which therefore is a substance. Bolzano’s argumentation turns out to be structurally similar to his argument for the existence of God from Lehrbuch der Religionswissenschaft (1834), but in each of these reasonings, we find different plausible interpretations of the key concept “Inbegriff”. The latter argumentation refers to the mereological totality of existentially conditioned objects. We propose the explication of the Bolzanian Inbegriff of all adherences using two types of predication: we consider its extension as composed of certain intensional counterparts of adherences. In our approach, we use a fragment of the theory of abstract objects formulated by E. Zalta (1983), describing two different relations between individuals and properties: extensional exemplification and intensional encoding. We put our reconstruction in a wider context of Bolzano’s ontology, formulating the needed axioms with two primitive predicates of second order ... is an adherence, ... is conditioned by something real as well as the conditionally introduced first order predicate constant \(\mathfrak {In}\) for Inbegriff of all adherential ideas. Finally, we sketch a model for our theory.     

A Note on Two’s Company: “The Humbug of Many Logical Values”

The present note offers a proof for separating the truth-values of an arbitrary finitely many valued ?ukasiewicz logic by making use of Gray codes.     

A historical teaching module on ‘the unreasonable effectiveness of mathematics’: Boolean algebra and Shannon circuits

This article describes the design and implementation of a historical teaching module, where Danish upper secondary students studied primary source material from the mathematicians R W Hamming, G Boole and C E Shannon. The module is an example of a so-called HAPh-module, which focuses on elements of the history, application and philosophy of mathematics. A previous version was presented at the History and Pedagogy of Mathematics (HPM) meeting in Deajeon, Korea in 2012.     

A Note on Erdös and Kac’s Identity: Boundary Crossing Probabilities of Brownian Motion Over Constant Boundaries

Methodology and Computing in Applied Probability - The finite Markov chain imbedding technique is an emerging approach for calculating boundary crossing probabilities for high-dimensional Brownian...     

Two refinements of Frink’s metrization theorem and fixed point results for Lipschitzian mappings on quasimetric spaces

Aequationes mathematicae - Quasimetric spaces have been an object of thorough investigation since Frink’s paper appeared in 1937 and various generalisations of the axioms of metric spaces are...     

An improvement on the Tracy and Chen model ‘A generalized model for weight restrictions in DEA’

Recently Tracy and Chen presented a parametric DEA model (PDEA) to assess relative efficiency in the presence of a generalized form of linear weight restrictions. This paper proposes a modification to the PDEA model that avoids the need to resort to searching algorithms to estimate efficiency, and assures that the correct efficiency scores are obtained in a single stage using mathematical programming solvers. The results of this model and the results of Tracy and Chen's PDEA model are compared using the examples reported in their paper. The results confirm the superiority of the model proposed in this paper.     

A note on the paper ‘Ultra discrete permanent and the consistency of max plus linear equations’

This work is concerned with the consistency conditions for the equations in max plus algebra. The three classes of the max plus linear equations presented in the paper Shinzawa [Ultra discrete permanent and the consistency of max plus linear equations, Linear Algebra Appl. 506 (2016) 445–477] can be equivalently converted into the corresponding system of inequalities. The equivalence relation between ultra discrete permanent and maximum cycle mean is suggested. Thus, the necessary and sufficient conditions for solvability of the equations are obtained using the maximum cycle mean in this note.     

A Note on C1,α Estimates for Solutions of Fully Nonlinear Elliptic Equations and Obstacle Problems

We deal with C^{1,α} interior estimates for solutions of fully nonlinear equation F(D²u, Du, x) = f(x) with the bounded gradient Du and a bounded f(x). Based on these estimates we obtain the existence of strong solutions of the obstacle problem for fully nonlinear elliptic equations under natural structure conditions.     

A Note on the Paper “Linear Complementarity Problems Over Symmetric Cones: Characterization of Qb-Transformations and Existence Results”

In this note, we correct a mistake in the paper (López et al., J Optim Theory Appl 159(3):741–768, 2013).     

A Remark on Lower Bounds for the Chromatic Numbers of Spaces of Small Dimension with Metrics ℓ1 and ℓ2

Mathematical Notes - A particular class of estimates related to the Nelson–Erd?s–Hadwiger problem is studied. For two types of spaces, Euclidean and spaces with metric ?1,...     

Problem with analogs of the Frankl’ condition on a characteristic and the degeneration segment for an equation of mixed type with a singular coefficient

For the equation (sign y)|y| ^m u _xx +u _yy ?m(2y)^?1 u _y = 0, where m > 0, considered in some mixed domain, we prove existence and uniqueness theorems for the solution of the boundary value problem with an analog of the Frankl’ condition on a characteristic and on the degeneration segment of the equation.     

A response to Tisdell on ‘Critical perspectives of the “new” difference equation solution method’ of Rivera-Figueroa and Rivera-Rebolledo

ABSTRACT

Recently, Christopher C. Tisdell questions some claims and results that appear in our article [Rivera-Figueroa A, Rivera-Rebolledo JM. A new method to solve the second-order linear difference equations with constant coefficients. Int J Math Educ Sci Technol. 2016;47(4):636–649.] regarding the novelty and simplicity of a method to solve second-order linear difference equations. He argues that our method was not new because the method already appeared in the mathematical literature, neither was simpler than existing ones. After carefully analysing and comparing the references quoted by Tisdell, we confirm that our method differs from that quoted by Tisdell. Moreover, our approach is simpler because it avoids the uniqueness theorem and the method of variation of parameters, in this context, it offers some clear didactic and epistemological advantages.     

Stirling’s Formula for Gamma Functions,Bounds for Ratios of Gamma Functions,Beta Functions and Percentiles of a Studentized Sample Mean: A Synthesis with New Results

In the context of Stirling’s formula for gamma functions and bounds for ratios of gamma functions, this work has a threefold purpose: (1) Outline recently published literature; (2) Synthesize techniques and results from Bhattacharjee and Mukhopadhyay (Commun Stat, Theory & Methods 39:1046–1053, 2010) and Mukhopadhyay (Commun Stat, Theory & Methods 40:1283–1297, 2011) which have gone perhaps unnoticed by some recent researchers; and (3) Incorporate new results for beta functions and useful bounds for the percentiles of a Studentized sample mean obtained from a normal distribution. This synthesized review may help in gaining a wider perspective about this area.