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A Construction of Imprimitive Groups of Rank 4 or 5
Issue:
Volume 9, Issue 6, December 2020
Pages:
175-178
Received:
15 September 2020
Accepted:
23 October 2020
Published:
4 November 2020
Abstract: Let G be a transitive permutation group acting on a finite set Ω. For a point α of Ω, the set of the images of G acting on α is called the orbit of α under G and is denoted by αG, and the set of elements in G which fix α is called the stabilizer of α in G and is denoted by Gα. We can get some new orbits by using the natural action of the stabilizer Gα on Ω, and then we can define the suborbit of G. The suborbits of G on Ω are defined as the orbits of a point stabilizer on Ω. The number of suborbits is called the rank of G and the length of suborbits is called the subdegree of G. For finite primitive groups, the study of the rank and subdegrees of group has a long history. In this paper, we construct a class of imprimitive permutation groups of rank 4 or 5 by using imprimitive action and product action of wreath product, determine the number and the length of the suborbits, and extend the results to imprimitive permutation groups of rank m+1 and 2n+1, where m and n are positive integers.
Abstract: Let G be a transitive permutation group acting on a finite set Ω. For a point α of Ω, the set of the images of G acting on α is called the orbit of α under G and is denoted by αG, and the set of elements in G which fix α is called the stabilizer of α in G and is denoted by Gα. We can get some new orbits by using the natural action of the stabilizer...
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Two-scale Finite Element Discretizations for Semilinear Parabolic Equations
Issue:
Volume 9, Issue 6, December 2020
Pages:
179-186
Received:
5 February 2020
Accepted:
25 September 2020
Published:
16 November 2020
Abstract: In this paper, to reduce the computational cost of solving semilinear parabolic equations on a tensor product domain Ω⊂ℝd with d = 2 or 3, some two-scale finite element discretizations are proposed and analyzed. The time derivative in semilinear parabolic equations is approximated by the backward Euler finite difference scheme. The two-scale finite element method is designed for the space discretization. The idea of the two-scale finite element method is based on an understanding of a finite element solution to an elliptic problem on a tensor product domain. The high frequency parts of the finite element solution can be well captured on some univariate fine grids and the low frequency parts can be approximated on a coarse grid. Thus the two-scale finite element approximation is defined as a linear combination of some standard finite element approximations on some univariate fine grids and a coarse grid satisfying H = O (h1/2), where h and H are the fine and coarse mesh widths, respectively. It is shown theoretically and numerically that the backward Euler two-scale finite element solution not only achieves the same order of accuracy in the H1 (Ω) norm as the backward Euler standard finite element solution, but also reduces the number of degrees of freedom from O(h-d×τ-1) to O(h-((d)+1)/2×τ-1) where τ is the time step. Consequently the backward Euler two-scale finite element method for semilinear parabolic equations is more efficient than the backward Euler standard finite element method.
Abstract: In this paper, to reduce the computational cost of solving semilinear parabolic equations on a tensor product domain Ω⊂ℝd with d = 2 or 3, some two-scale finite element discretizations are proposed and analyzed. The time derivative in semilinear parabolic equations is approximated by the backward Euler finite difference scheme. The two-scale finite...
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Strong Convergence of the Hybrid Halpern Type Proximal Point Algorithm
Issue:
Volume 9, Issue 6, December 2020
Pages:
187-194
Received:
15 May 2020
Accepted:
12 June 2020
Published:
16 November 2020
Abstract: Based on the proximal point algorithm, which is a widely used tool for solving a variety of convex optimization problems, there are many algorithms for finding zeros of maximally monotone operators. The algorithm works by applying successively so-called "resolvent" mappings with errors associated to the original object, and is weakly convergent in Hilbert space. In order to acquiring the strong convergence of the algorithm, in this paper, we construct a hybrid Halpern type proximal point algorithm with errors for approximating the zero of a maximal monotone operator, which is a combination of modified proximal point algorithm raised by Yao and Noor and Halpern inexact proximal point algorithm raised by Zhang, respectively. Then, we prove the strong convergence of our algorithm with weaker assumptions in Hilbert space. Finally, we present a numerical example to show the convergence and the convergence speed, which is not affected but accelerated by the projection in the algorithm. Our work improved and generalized some known results.
Abstract: Based on the proximal point algorithm, which is a widely used tool for solving a variety of convex optimization problems, there are many algorithms for finding zeros of maximally monotone operators. The algorithm works by applying successively so-called "resolvent" mappings with errors associated to the original object, and is weakly convergent in ...
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Uniform Convergence of the Series Expansion of the Multifractional Brownian Motion
Issue:
Volume 9, Issue 6, December 2020
Pages:
195-200
Received:
17 June 2020
Accepted:
16 October 2020
Published:
4 December 2020
Abstract: In this paper we define the multifractional Brownian motion and we give some properties. we study the uniform Convergence of the Serie expansion. After having determined the covariance function, we give in proposition 2 another proof of almost sure uniform convergence on compact K of the series. We will finish by showing that the m.B.f is locally astymptotically self-similar, with field or fractional Brownian field with Hurst exposant H. One of the problem, for application of multifractional Brownian motion, is the regularity of the function. In the filtered white noise model the increments are no more homogeneous as in fractional Brownian field case. It is obvious when we consider the tangent field associated with a function. Still the multifractional function in the previous model is constant and it is not convient for many applications. We show the uniform convergence of the series on K. We deduce from the previous questions the almost sure uniform convergence of the series to a mBm.
Abstract: In this paper we define the multifractional Brownian motion and we give some properties. we study the uniform Convergence of the Serie expansion. After having determined the covariance function, we give in proposition 2 another proof of almost sure uniform convergence on compact K of the series. We will finish by showing that the m.B.f is locally a...
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