In this paper, we construct a bivariate tensor product generalization of Kantorovich-type Bernstein-Stancu-Schurer operators based on the concept of (p, q)-integers. We obtain moments and central moments of these oper...
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In this paper, we construct a bivariate tensor product generalization of Kantorovich-type Bernstein-Stancu-Schurer operators based on the concept of (p, q)-integers. We obtain moments and central moments of these operators, give the rate of convergence by using the complete modulus of continuity for the bivariate case and estimate a convergence theorem for the lipschitz continuous functions. We also give some graphs and numerical examples to illustrate the convergence properties of these operators to certain functions.
In this paper, we introduce a new kind of Kantorovich-type Bernstein-Stancu-Schurer operators based on the concept of (p, q)-integers. We investigate statistical approximation properties and establish a local approxim...
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In this paper, we introduce a new kind of Kantorovich-type Bernstein-Stancu-Schurer operators based on the concept of (p, q)-integers. We investigate statistical approximation properties and establish a local approximation theorem, we also give a convergence theorem for the lipschitz continuous functions. Finally, we give some graphics and numerical examples to illustrate the convergence properties of operators to some functions. (C) 2015 Elsevier Inc. All rights reserved.
We define the tangential derivative, a notion of directional derivative which is invariant under diffeomorphisms. In particular this notion is independent of local charts and thus well-defined for functions defined on...
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We define the tangential derivative, a notion of directional derivative which is invariant under diffeomorphisms. In particular this notion is independent of local charts and thus well-defined for functions defined on a differentiable manifold. This notion is stronger than the classical directional derivative and equivalent to the latter for lipschitz continuous functions. We characterize also the pairs of tangentially differentiable functions for which the chain rule holds. (C) 2015 Elsevier Inc. All rights reserved.
It is known that a p-adic, locally lipschitzcontinuous semi-algebraic function, is piecewise lipschitzcontinuous, where finitely many pieces suffice and the pieces can be taken semi-algebraic. We prove that if the f...
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It is known that a p-adic, locally lipschitzcontinuous semi-algebraic function, is piecewise lipschitzcontinuous, where finitely many pieces suffice and the pieces can be taken semi-algebraic. We prove that if the function has locally lipschitz constant 1, then it is also piecewise lipschitzcontinuous with the same lipschitz constant 1 (again, with finitely many pieces). We do this by proving the following fine preparation results for p-adic semi-algebraic functions in one variable. Any such function can be well approximated by a monomial with fractional exponent such that moreover the derivative of the monomial is an approximation of the derivative of the function. We also prove these results in parameterized versions and in the subanalytic setting.
We prove that a (globally) subanalytic function f : X subset of Q(p)(n) -> Q(p) which is locally lipschitzcontinuous with some constant C is piecewise (globally on each piece) lipschitzcontinuous with possibly so...
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We prove that a (globally) subanalytic function f : X subset of Q(p)(n) -> Q(p) which is locally lipschitzcontinuous with some constant C is piecewise (globally on each piece) lipschitzcontinuous with possibly some other constant, where the pieces can be taken to be subanalytic. We also prove the analogous result for a subanalytic family of functions f(y) : X(y) subset of Q(p)(n) -> Q(p) depending on p-adic parameters. The statements also hold in a semi-algebraic set-up and also in a finite field extension of Q(p). These results are p-adic analogues of results of K. Kurdyka over the real numbers. To encompass the total disconnectedness of p-adic fields, we need to introduce new methods adapted to the p-adic situation.
The paper focuses at the estimates for the rate of convergence of the q-Bernstein polynomials (0 < q < 1) in the complex plane. In particular, a generalization of previously known results on the possibility of a...
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The paper focuses at the estimates for the rate of convergence of the q-Bernstein polynomials (0 < q < 1) in the complex plane. In particular, a generalization of previously known results on the possibility of analytic continuation of the limit function and an elaboration of the theorem by Wang and Meng is presented. (C) 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Making extensive use of small transfinite topological dimension trind, we ascribe to every metric space X an ordinal number (or -1 or Omega) tHD(X), and we call it the transfinite Hausdorff dimension of X. This ordina...
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Making extensive use of small transfinite topological dimension trind, we ascribe to every metric space X an ordinal number (or -1 or Omega) tHD(X), and we call it the transfinite Hausdorff dimension of X. This ordinal number shares many common features with Hausdorff dimension. It is monotone with respect to subspaces. it is invariant under bi-lipschitz maps (but in general not under homeomorphisms), in fact like Hausdorff dimension, it does not increase under lipschitz maps, and it also satisfies the intermediate dimension property (Theorem 2.7). The primary goal of transfinite Hausdorff dimension is to classify metric spaces with infinite Hausdorff dimension. Indeed, if tHD(X) >= omega(0), then HD(X) = +infinity. We prove that tHD(X) <= omega(1) for every separable metric space X, and, as our main theorem, we show that for every ordinal number alpha < omega(1) there exists a compact metric space X(alpha) (a subspace of the Hilbert space l(2)) with tHD(X(alpha)) = alpha and which is a topological Cantor set, thus of topological dimension 0. In our proof we develop a metric version of Smirnov topological spaces and we establish several properties of transfinite Hausdorff dimension, including its relations with classical Hausdorff dimension. (C) 2009 Elsevier B.V. All rights reserved.
An interval method for a class of min-max-min problems is described in this paper, in which the objective functions are lipschitzcontinuous. The convergence of algorithm is proved, numerical results from a Turbo B im...
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An interval method for a class of min-max-min problems is described in this paper, in which the objective functions are lipschitzcontinuous. The convergence of algorithm is proved, numerical results from a Turbo B implementation of the algorithm are presented. The method can get both the optimal value and all global solutions of min-max-min problem. Theoretical analysis and numerical tests indicate the algorithm is stable and reliable. (C) 2007 Elsevier Inc. All rights reserved.
Lagrange once made a claim having the consequence that a smooth function f has a local minimum at a point if all the directional derivatives of fat that point are nonnegative. That the Lagrange claim is wrong was show...
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Lagrange once made a claim having the consequence that a smooth function f has a local minimum at a point if all the directional derivatives of fat that point are nonnegative. That the Lagrange claim is wrong was shown by a counterexample given by Peano. In this note, we show that an extended claim of Lagrange is right. We show that, if all the lower directional derivatives of a locally lipschitz function f at a point are positive, then f has a strict minimum at that point.
By using the technique of factoring weakly compact operators through reflexive Banach spaces we prove that a class of ordinary differential equations with lipschitzcontinuous perturbations has a strong solution when ...
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By using the technique of factoring weakly compact operators through reflexive Banach spaces we prove that a class of ordinary differential equations with lipschitzcontinuous perturbations has a strong solution when the problem is governed by a closed linear operator generating a strongly continuous semigroup of compact operators.
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