An existence‐uniqueness result for the pure binary collisional breakage equation

The study of collision‐induced breakage phenomenon in the particulate process has much current interest. This is an important process arising in many engineering disciplines. In this work, the existence of continuous solution of the pure collisional breakage model is developed beneath some restrictions on the breakage kernels. Furthermore, the mass conservation and uniqueness of solution are investigated in the absence of “shattering transition.” The underlying theory is based on the compactness result of Arzelà‐Ascoli and Banach contraction mapping principle.     

Dunkl generalization of Szász beta‐type operators

The goal in the paper is to advertise Dunkl extension of Szász beta‐type operators. We initiate approximation features via acknowledged Korovkin and weighted Korovkin theorem and obtain the convergence rate from the point of modulus of continuity, second‐order modulus of continuity, the Lipschitz class functions, Peetre's K‐functional, and modulus of weighted continuity by Dunkl generalization of Szász beta‐type operators.     

Carathéodory–Julia type conditions and symmetries of boundary asymptotics for analytic functions on the unit disk

It is shown that the following conditions are equivalent for the generalized Schur class functions at a boundary point t₀ ∈ ??: 1) Carathéodory–Julia type condition of order n; 2) agreeing of asymptotics of the original function from inside and of its continuation by reflection from outside of the unit disk ?? up to order 2n + 1; 3) t₀‐isometry of the coefficients ofthe boundary asymptotics; 4) a certain structured matrix ? constructed from these coefficients being Hermitian. It is also shown that for an arbitrary analytic function, properties 2), 3), 4) are still equivalent to each other and imply 1) (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Isomorphic copies of ℓ1 for m‐homogeneous non‐analytic Bohnenblust–Hille polynomials

We employ a classical result by Toeplitz (1913) and the seminal work by Bohnenblust and Hille on Dirichlet series (1931) to show that the set of continuous m‐homogeneous non‐analytic polynomials on c ₀ contains an isomorphic copy of ℓ₁. Moreover, we can have this copy of ℓ₁ in such a way that every non‐zero element of it fails to be analytic at precisely the same point.     

Bayesian l0‐regularized least squares

Bayesian l₀‐regularized least squares is a variable selection technique for high‐dimensional predictors. The challenge is optimizing a nonconvex objective function via search over model space consisting of all possible predictor combinations. Spike‐and‐slab (aka Bernoulli‐Gaussian) priors are the gold standard for Bayesian variable selection, with a caveat of computational speed and scalability. Single best replacement (SBR) provides a fast scalable alternative. We provide a link between Bayesian regularization and proximal updating, which provides an equivalence between finding a posterior mode and a posterior mean with a different regularization prior. This allows us to use SBR to find the spike‐and‐slab estimator. To illustrate our methodology, we provide simulation evidence and a real data example on the statistical properties and computational efficiency of SBR versus direct posterior sampling using spike‐and‐slab priors. Finally, we conclude with directions for future research.     

Closed convex hulls,contents, and K‐spectral states

Our general result says that the closed convex hull of a set K consists of barycentres of probability contents (i.e., finitely additive set functions) on K. (Here K can be any nonempty subset of any nonempty compact convex set in any real or complex locally convex Hausdorff vector space.) In the equivalent setting of dual spaces, we give a very handy analytic criterion for a linear functional to be in the closed convex hull of a given nonempty point‐wise bounded set K of linear functionals (under some mild additional assumption). This is the notion of a K‐spectral state. Our criterion enhances the Abstract Bochner Theorem for unital commutative Banach *‐algebras (which easily follows from our result), in that it allows us to prescribe the set K on which a representing content should live. The content can be chosen to be a Radon measure if K is weak* compact. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some Results on 1‐Rotational Hamiltonian Cycle Systems

A Hamiltonian cycle system of (briefly, a HCS(v)) is 1‐rotational under a (necessarily binary) group G if it admits G as an automorphism group acting sharply transitively on all but one vertex. We first prove that for any there exists a 3‐perfect 1‐rotational HCS. This allows to get the existence of another infinite class of 3‐perfect (but not Hamiltonian) cycle decompositions of the complete graph. Then we prove that the full automorphism group of a 1‐rotational HCS under G is G itself unless the HCS is the 2‐transitive one. This allows us to give a partial answer to the problem of determining which abstract groups are the full automorphism group of a HCS. Finally, we revisit and simplify by means of a careful group theoretic discussion a formula by Bailey, Ollis, and Preece on the number of inequivalent 1‐rotational HCSs under G. This leads us to a formula counting all 1‐rotational HCSs up to isomorphism. Though this formula heavily depends on some parameters that are hard to compute, it allows us to say that, for any , there are at least nonisomorphic 1‐rotational (and hence symmetric) HCS().     

A Stochastic Analysis of Some Two‐Person Sports

We consider two‐person sports where each rally is initiated by a server, the other player (the receiver) becoming the server when he/she wins a rally. Historically, these sports used a scoring based on the side‐out scoring system, in which points are only scored by the server. Recently, however, some federations have switched to the rally‐point scoring system in which a point is scored on every rally. As various authors before us, we study how much this change affects the game. Our approach is based on a rally‐level analysis of the process through which, besides the well‐known probability distribution of the scores, we also obtain the distribution of the number of rallies. This yields a comprehensive knowledge of the process at hand, and allows for an in‐depth comparison of both scoring systems. In particular, our results help to explain why the transition from one scoring system to the other has more important implications than those predicted from game‐winning probabilities alone. Some of our findings are quite surprising, and unattainable through Monte Carlo experiments. Our results are of high practical relevance to international federations and local tournament organizers alike, and also open the way to efficient estimation of the rally‐winning probabilities.     

Minimal extensions of Π01 classes

A minimal extension of a Π⁰₁ class P is a Π⁰₁ class Q such that P ? Q, Q – P is infinite, and for any Π⁰₁ class R, if P ? R ? Q, then either R – P is finite or Q – R is finite; Q is a nontrivial minimal extension of P if in addition P and Q′ have the same Cantor‐Bendixson derivative. We show that for any class P which has a single limit point A, and that point of degree ≤ 0 , P admits a nontrivial minimal extension. We also show that as long as P is infinite, then P does not admit any decidable nontrivial minimal extension Q. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

(p,q)‐Generalization of Szász–Mirakyan operators

In this paper, we introduce new modifications of Szász–Mirakyan operators based on (p,q)‐integers. We first give a recurrence relation for the moments of new operators and present explicit formula for the moments and central moments up to order 4. Some approximation properties of new operators are explored: the uniform convergence over bounded and unbounded intervals is established, direct approximation properties of the operators in terms of the moduli of smoothness is obtained and Voronovskaya theorem is presented. For the particular case p = 1, the previous results for q‐Sz ász–Mirakyan operators are captured. Copyright © 2015 John Wiley & Sons, Ltd.     

Gâteaux derivatives and their applications to approximation in Lorentz spaces Γp,w

We establish the formulas of the left‐ and right‐hand Gâteaux derivatives in the Lorentz spaces Γ_p,w = {f: ∫₀^α (f **)^p w < ∞}, where 1 ≤ p < ∞, w is a nonnegative locally integrable weight function and f ** is a maximal function of the decreasing rearrangement f * of a measurable function f on (0, α), 0 < α ≤ ∞. We also find a general form of any supporting functional for each function from Γ_p,w , and the necessary and sufficient conditions for which a spherical element of Γ_p,w is a smooth point of the unit ball in Γ_p,w . We show that strict convexity of the Lorentz spaces Γ_p,w is equivalent to 1 < p < ∞ and to the condition ∫₀^∞ w = ∞. Finally we apply the obtained characterizations to studies the best approximation elements for each function f ∈ Γ_p,w from any convex set K ? Γ_p,w (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Turánnical hypergraphs

This paper is motivated by the question of how global and dense restriction sets in results from extremal combinatorics can be replaced by less global and sparser ones. The result we consider here as an example is Turán's theorem, which deals with graphs G = ([n],E) such that no member of the restriction set \begin{align*}\mathcal {R}\end{align*} = \begin{align*}\left( {\begin{array}{*{20}c} {[n]} \\ r \\ \end{array} } \right)\end{align*} induces a copy of K_r. Firstly, we examine what happens when this restriction set is replaced by \begin{align*}\mathcal {R}\end{align*} = {X∈ \begin{align*}\left( {\begin{array}{*{20}c} {[n]} \\ r \\ \end{array} } \right)\end{align*}: X ∩ [m]≠??}. That is, we determine the maximal number of edges in an n ‐vertex such that no K_r hits a given vertex set. Secondly, we consider sparse random restriction sets. An r ‐uniform hypergraph \begin{align*}\mathcal R\end{align*} on vertex set [n] is called Turánnical (respectively ε ‐Turánnical), if for any graph G on [n] with more edges than the Turán number t_r(n) (respectively (1 + ε)t_r(n) ), no hyperedge of \begin{align*}\mathcal {R}\end{align*} induces a copy of K_r in G. We determine the thresholds for random r ‐uniform hypergraphs to be Turánnical and to be ε ‐Turánnical. Thirdly, we transfer this result to sparse random graphs, using techniques recently developed by Schacht [Extremal results for random discrete structures] to prove the Kohayakawa‐?uczak‐Rödl Conjecture on Turán's theorem in random graphs.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

Stancu‐type generalization of Dunkl analogue of Szász–Kantorovich operators

The purpose of the paper is to introduce Stancu‐type linear positive operators generated by Dunkl generalization of exponential function. We present approximation properties with the help of well‐known Korovkin‐type theorem and weighted Korovkin‐type theorem and also acquire the rate of convergence in terms of classical modulus of continuity, the class of Lipschitz functions, Peetre's K‐functional, and second‐order modulus of continuity by Dunkl analogue of Szász operators. Copyright © 2015 John Wiley & Sons, Ltd.     

On some relationships between biorthogonal systems,linear topologies and the weak‐AFPP

In this paper, we obtain new results for the weak‐AFPP in abstract spaces by exploiting biorthogonal systems techniques. Firstly, we investigate the strong‐AFPP on countably infinite dimensional Hausdorff locally convex spaces. Spaces of this class are shown to be sequentially complete iff they have the hereditary FPP for totally bounded, closed convex sets. This might open a research line for the analysis of weak‐AFPP in such frames. In connection, we provide a simple criterion for the containement of ?₁‐sequences in terms of strongly‐equicontinuous biorthogonal systems. We then establish a few results concerning the existence of Hausdorff finer vector topologies on abstract spaces having as prescribed condition the existence of such systems. The proofs are based on methods of Peck and Porta concerning building of finer vector topologies, and a classical construction of Singer which allows us to prove under rather natural conditions the existence of equicontinuous biorthogonal systems in metrizable locally convex spaces. These results are compatible with the failure of the weak‐AFPP. We also study the inverse problem by proving that every infinite dimensional vector space admits a (non‐locally convex) Hausdorff vector topology which is complete, non‐metrizable and is compatible with a bounded Hamel Schauder basis. It is shown further that such a topology has the ‐AFPP, where is the linear span of coefficient functionals associated to a Hamel basis. Finally, inspired by a result of Shapiro, we observe that if X is a non‐locally convex F‐space with an absolute basis, then the weak‐AFPP is equivalent to the fact that every bounded convex subset of X is compact.     

A (3+1)‐dimensional Painlevé integrable model obtained by deformation

To find some non‐trivial higher‐dimensional integrable models (especially in (3+1) dimensions) is one of the most important problems in non‐linear physics. An efficient deformation method to obtain higher‐dimensional integrable models is proposed. Starting from (2+1)‐dimensional linear wave equation, a (3+1)‐dimensional non‐trivial non‐linear equation is obtained by using a non‐invertible deformation relation. Further, the Painlevé integrability of the resulting model is also proved. Copyright © 2002 John Wiley & Sons, Ltd.     

Turán problems for integer‐weighted graphs

A multigraph is (k,r)‐dense if every k‐set spans at most r edges. What is the maximum number of edges ex_?(n,k,r) in a (k,r)‐dense multigraph on n vertices? We determine the maximum possible weight of such graphs for almost all k and r (e.g., for all r>k³) by determining a constant m=m(k,r) and showing that ex_?(n,k,r)=m +O(n), thus giving a generalization of Turán's theorem. We find exact answers in many cases, even when negative integer weights are also allowed. In fact, our main result is to determine the maximum weight of (k,r)‐dense n‐vertex multigraphs with arbitrary integer weights with an O(n) error term. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 195–225, 2002     

Ultrametrics, Banach’s fixed point theorem and the Riordan group

We interpret the reciprocation process in as a fixed point problem related to contractive functions for certain adequate ultrametric spaces. This allows us to give a dynamical interpretation of certain arithmetical triangles introduced herein. Later we recognize, as a special case of our construction, the so-called Riordan group which is a device used in combinatorics. In this manner we give a new and alternative way to construct the proper Riordan arrays. Our point of view allows us to give a natural metric on the Riordan group turning this group into a topological group. This construction allows us to recognize a countable descending chain of normal subgroups.     

Family of Pearson discrete distributions generated by the univariate hypergeometric function 3F2(α1, α2, α3; γ1, γ2; λ)

In this work we present a study of the Pearson discrete distributions generated by the hypergeometric function ₃F₂(α₁, α₂, α₃;γ₁, γ₂; λ), a univariate extension of the Gaussian hypergeometric function, through a constructive methodology. We start from the polynomial coefficients of the difference equation that lead to such a function as a solution. Immediately after, we obtain the generating probability function and the differential equation that it satisfies, valid for any admissible values of the parameters. We also obtain the differential equations that satisfy the cumulants generating function, moments generating function and characteristic function, From this point on, we obtain a relation in recurrences between the moments about the origin, allowing us to create an equation system for estimating the parameters by the moment method. We also establish a classification of all possible distributions of such type and conclude with a summation theorem that allows us study some distributions belonging to this family. © 1997 by John Wiley & Sons, Ltd.     

A generalization of Turán's theorem

In this paper, we obtain an asymptotic generalization of Turán's theorem. We prove that if all the non‐trivial eigenvalues of a d‐regular graph G on n vertices are sufficiently small, then the largest K_t‐free subgraph of G contains approximately (t ? 2)/(t ? 1)‐fraction of its edges. Turán's theorem corresponds to the case d = n ? 1. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Zero‐Hopf bifurcation in the FitzHugh–Nagumo system

We characterize the values of the parameters for which a zero‐Hopf equilibrium point takes place at the singular points, namely, O (the origin), P₊, and P_? in the FitzHugh–Nagumo system. We find two two‐parameter families of the FitzHugh–Nagumo system for which the equilibrium point at the origin is a zero‐Hopf equilibrium. For these two families, we prove the existence of a periodic orbit bifurcating from the zero‐Hopf equilibrium point O. We prove that there exist three two‐parameter families of the FitzHugh–Nagumo system for which the equilibrium point at P₊ and at P_? is a zero‐Hopf equilibrium point. For one of these families, we prove the existence of one, two, or three periodic orbits starting at P₊ and P_?. Copyright © 2014 John Wiley & Sons, Ltd.