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Mathematics and statistics journals publish papers on the theory and application of mathematics, statistics, and probability. Most mathematics journals have a broad scope that encompasses most mathematical fields. These commonly include logic and foundations, algebra and number theory, analysis (including differential equations, functional analysis and operator theory), geometry, topology, combinatorics, probability and statistics, numerical analysis and computation theory, mathematical physics, etc.

Mathematics and Statistics

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For a continuous and positive function w(λ), λ > 0 and μ a positive measure on (0, ∞) we consider the following integral transform

D w , μ t : = 0 w λ λ + t 1 d μ λ ,

where the integral is assumed to exist for t > 0.

We show among others that D(w, μ) is operator convex on (0, ∞). From this we derive that, if f : [0, ∞) → R is an operator monotone function on [0, ∞), then the function [f(0) -f(t)] t -1 is operator convex on (0, ∞). Also, if f : [0, ∞) → R is an operator convex function on [0, ∞), then the function f 0 + f + 0 t f t t 2 is operator convex on (0, ∞). Some lower and upper bounds for the Jensen’s difference

D w , μ A + D w , μ B 2 D w , μ A + B 2

under some natural assumptions for the positive operators A and B are given. Examples for power, exponential and logarithmic functions are also provided.

Open access

Problem 2 of Welsh’s 1976 text Matroid Theory, asking for criteria telling when two families of sets have a common transversal, is solved.

Another unsolved problem in the text Matroid Theory, on whether the “join” of two non-decreasing submodular functions is submodular, is answered in the negative. This resolves an issue first raised by Pym and Perfect in 1970.

Open access

With distributed computing and mobile applications becoming ever more prevalent, synchronizing diverging replicas of the same data is a common problem. Reconciliation – bringing two replicas of the same data structure as close as possible without overriding local changes – is investigated in an algebraic model. Our approach is to consider two sequences of simple commands that describe the changes in the replicas compared to the original structure, and then determine the maximal subsequences of each that can be propagated to the other. The proposed command set is shown to be functionally complete, and an update detection algorithm is presented which produces a command sequence transforming the original data structure into the replica while traversing both simultaneously. Syntactical characterization is provided in terms of a rewriting system for semantically equivalent command sequences. Algebraic properties of sequence pairs that are applicable to the same data structure are investigated. Based on these results the reconciliation problem is shown to have a unique maximal solution. In addition, syntactical properties of the maximal solution allow for an efficient algorithm that produces it.

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This manuscript deals with the global existence and asymptotic behavior of solutions for a Kirchhoff beam equation with internal damping. The existence of solutions is obtained by using the Faedo-Galerkin method. Exponential stability is proved by applying Nakao’s theorem.

Open access

We consider hypersphere x = x(u, v, w) in the four dimensional Euclidean space. We calculate the Gauss map, and the curvatures of it. Moreover, we compute the second Laplace-Beltrami operator the hypersphere satisfying ΔIIx = Ax, where A ϵ Mat (4,4).

Open access

In this paper, we show a Marcinkiewicz type interpolation theorem for Orlicz spaces. As an application, we obtain an existence result for a parabolic equation in divergence form.

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Extending Blaschke and Lebesgue’s classical result in the Euclidean plane, it has been recently proved in spherical and the hyperbolic cases, as well, that Reuleaux triangles have the minimal area among convex domains of constant width D. We prove an essentially optimal stability version of this statement in each of the three types of surfaces of constant curvature. In addition, we summarize the fundamental properties of convex bodies of constant width in spaces of constant curvature, and provide a characterization in the hyperbolic case in terms of horospheres.

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Let P be a set of n points in general position in the plane. Let R be a set of points disjoint from P such that for every x, y € P the line through x and y contains a point in R. We show that if R < 3 2 n  and P R is contained in a cubic curve c in the plane, then P has a special property with respect to the natural group structure on c. That is, P is contained in a coset of a subgroup H of c of cardinality at most |R|.

We use the same approach to show a similar result in the case where each of B and G is a set of n points in general position in the plane and every line through a point in B and a point in G passes through a point in R. This provides a partial answer to a problem of Karasev.

The bound R < 3 2 n is best possible at least for part of our results. Our extremal constructions provide a counterexample to an old conjecture attributed to Jamison about point sets that determine few directions.

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Let E, G be Fréchet spaces and F be a complete locally convex space. It is observed that the existence of a continuous linear not almost bounded operator T on E into F factoring through G causes the existence of a common nuclear Köthe subspace of the triple (E, G, F). If, in addition, F has the property (y), then (E, G, F) has a common nuclear Köthe quotient.

Open access

In this paper we study the sum p x   τ ( n p ) , where τ ( n ) denotes the number of divisors of n, and {np } is a sequence of integers indexed by primes. Under certain assumptions we show that the aforementioned sum is   x  as  x   . As an application, we consider the case where the sequence is given by the Fourier coefficients of a modular form.

Open access