Well‐posedness of time‐space fractional stochastic evolution equations driven by α‐stable noise

This paper is devoted to the well‐posedness for time‐space fractional Ginzburg‐Landau equation and time‐space fractional Navier‐Stokes equations by α‐stable noise. The spatial regularity and the temporal regularity of the nonlocal stochastic convolution are firstly established, and then the existence and uniqueness of the global mild solution are obtained by the Banach fixed point theorem and Mittag‐Leffler functions, respectively. Numerical simulations for time‐space fractional Ginzburg‐Landau equation are provided to verify the analysis results.     

Increasing η ‐representable degrees

In this paper we prove that any Δ₃⁰ degree has an increasing η ‐representation. Therefore, there is an increasing η ‐representable set without a strong η ‐representation (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Local well‐posedness and zero‐α limit for the Euler‐α equations

In this paper, we prove the local‐in‐time existence and a blow‐up criterion of solutions in the Besov spaces for the Euler‐α equations of inviscid incompressible fluid flows in . We also establish the convergence rate of the solutions of the Euler‐α equations to the corresponding solutions of the Euler equations as the regularization parameter α approaches 0 in . Copyright © 2016 John Wiley & Sons, Ltd.     

The Hankel Convolution and the Zemanian Spaces βμ and β'μ

In this paper we give structure theorems for the elements of the Zemanian spaces β'_μ and β'_μ,a' Also, bounded sets and convergent sequences in β'_μ,a' are characterized through representations as derivatives of measurable functions. Finally, we analyze the Hankel convolution on the above spaces.     

Characterization of g∞,σ‐integral operators

The application of the general tensor norms theory of Defant and Floret to the ideal of (p, σ)‐absolutely continuous operators of Matter, 0 < σ < 1, 1 ≤ p < ∞ leads to the study of g_p′,σ‐nuclear and g_p′,σ‐integral operators. Characterizations of such operators has been obtained previously in the case p > 1. In this paper we characterize the g_∞,σ‐nuclear and g_∞,σ‐integral operators by the existence of factorizations of some special kinds. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On initial data problem for a periodic two‐component μ‐Hunter–Saxton system

This paper is concerned with a periodic two‐component μ‐Hunter–Saxton system. We prove that the solution map of the Cauchy problem of the μ‐Hunter–Saxton system is not uniformly continuous in , s > 5/2. Copyright © 2014 John Wiley & Sons, Ltd.     

Pointwise convergence along a tangential curve for the fractional Schrödinger equation with 0 < m < 1

In this article, we study the pointwise convergence problem about solution to the fractional Schrödinger equation with 0 < m < 1 along a tangential curve and estimate the capacitary dimension of the divergence set. We extend the results of Cho and Shiraki (2021) for the case m > 1 to the case 0 < m < 1, which is sharp up to the endpoint.     

Interior C1, γ‐regularity for weak solutions of nonlinear second orderelliptic systems

The interior C^{0, γ}‐regularity for the first gradient of a weak solution to a class of nonlinear second order elliptic systems is proved under the assumption that oscillations of coefficients are controlled by the ellipticity constant. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Random intersection graphs when m=ω(n): An equivalence theorem relating the evolution of the G(n, m, p) and G(n, p) models

When the random intersection graph G(n, m, p) proposed by Karoński, Scheinerman, and Singer‐Cohen [Combin Probab Comput 8 (1999), 131–159] is compared with the independent‐edge G(n, p), the evolutions are different under some values of m and equivalent under others. In particular, when m=n^α and α>6, the total variation distance between the graph random variables has limit 0. ©2000 John Wiley & Sons, Inc. Random Struct. Alg., 16, 156–176, 2000     

Time decay rate for two 3D magnetohydrodynamics‐ α models

We consider the long time behavior of solutions to the magnetohydrodynamics‐ α model in three spatial dimensions. Time decay rate in L²‐norm of the solution is obtained. Similar results for a generalized Leray‐ α‐magnetohydrodynamics model are also established. As a by‐product, an optimal time decay rate for the Navier–Stokes‐ α model is achieved. Copyright © 2013 John Wiley & Sons, Ltd.     

Group‐divisible designs with block size k having k + 1 groups,for k = 4, 5

We investigate the spectrum for k‐GDDs having k + 1 groups, where k = 4 or 5. We take advantage of new constructions introduced by R. S. Rees (Two new direct product‐type constructions for resolvable group‐divisible designs, J Combin Designs, 1 (1993), 15–26) to construct many new designs. For example, we show that a resolvable 4‐GDD of type g⁵ exists if and only if g ≡ 0 mod 12 and that a resolvable 5‐GDD of type g⁶ exists if and only if g ≡ 0 mod 20. We also show that a 4‐GDD of type g⁴m¹ exists (with m > 0) if and only if g ≡ m ≡ 0 mod 3 and 0 < m ≤ 3g/2, except possibly when (g,m) = (9,3) or (18,6), and that a 5‐GDD of type g⁵m¹ exists (with m > 0) if and only if g ≡ m ≡ 0 mod 4 and 0 < m ≤ 4g/3, with 32 possible exceptions. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 363–386, 2000     

The improved split‐step θ methods for stochastic differential equation

Two improved split‐step θ methods, which, respectively, named split‐step composite θ method and modified split‐step θ‐Milstein method, are proposed for numerically solving stochastic differential equation of Itô type. The stability and convergence of these methods are investigated in the mean‐square sense. Moreover, an approach to improve the numerical stability is illustrated by choices of parameters of these two methods. Some numerical examples show the accordance between the theoretical and numerical results. Further numerical tests exhibit not only the Hamiltonian‐preserving property of the improved split‐step θ methods for a stochastic differential system but also the positivity‐preserving property of the modified split‐step θ‐Milstein method for the Cox–Ingersoll–Ross model. Copyright © 2013 John Wiley & Sons, Ltd.     

Stability of the fractional Volterra integro‐differential equation by means of ψ‐Hilfer operator

In this paper, using the Riemann‐Liouville fractional integral with respect to another function and the ψ?Hilfer fractional derivative, we propose a fractional Volterra integral equation and the fractional Volterra integro‐differential equation. In this sense, for this new fractional Volterra integro‐differential equation, we study the Ulam‐Hyers stability and, also, the fractional Volterra integral equation in the Banach space, by means of the Banach fixed‐point theorem. As an application, we present the Ulam‐Hyers stability using the α‐resolvent operator in the Sobolev space .     

α-Stable characterization of Banach spaces (1 < α < 2)

This work characterizes some subclasses of α-stable (0 < α < 1) Banach spaces in terms of the extendibility to Radon laws of certain α-stable cylinder measures. These result extend the work of S. Chobanian and V. Tarieladze (J. Multivar. Anal.7, 183–203 (1977)). For these spaces it is shown that every Radon stable measure is the continuous image of a stable measure on a suitable L_β space with β = α(1 − α)⁻¹. The latter result extends some work of Garling (Ann. Probab.4, 600–611 (1976)) and Jain (Proceedings, Symposia in Pure Math. XXXI, p. 55–65, Amer. Math. Soc., Providence, R.I.).     

Functional impulsive differential equation of order α∈(1,2) with nonlocal initial and integral boundary conditions

The main work is related to show the existence and uniqueness of solution for the fractional impulsive differential equation of order α∈(1,2) with an integral boundary condition and finite delay. Using the application of the Banach and Sadovaskii fixed‐point theorems, we obtain the main results. An example is presented at the end to verify the results of the paper. Copyright © 2016 John Wiley & Sons, Ltd.     

On asymptotic properties of multidimensional α ‐stable densities

A class of multidimensional α ‐stable distributions is considered. The Poisson spectral measure of each distribution is assumed to be absolutely continuous with respect to the surface Lebesgue measure. The author concentrates his attention on the asymptotic behavior of the α ‐stable densities s (x) as |x | →∞and |x | → 0. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The spectrum of BSA(v, 3, λ; α) with α = 2,3

In this article, a kind of auxiliary design BSA^* for constructing BSAs is introduced and studied. Two powerful recursive constructions on BSAs from 3‐IGDDs and BSA^*s are exploited. Finally, the necessary and sufficient conditions for the existence of a BSA(v, 3, λ; α) with α = 2, 3 are established. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 61–76, 2007     

Global well‐posedness of the Cauchy problem for certain magnetohydrodynamic‐α models

This paper is devoted to study the Cauchy problem for certain incompressible magnetohydrodynamics‐α model. In the Sobolev space with fractional index s>1, we proved the local solutions for any initial data, and global solutions for small initial data. Furthermore, we also prove that as α→0, the MHD‐α model reduces to the MHD equations, and the solutions of the MHD‐α model converge to a pair of solutions for the MHD equations. Copyright © 2010 John Wiley & Sons, Ltd.     

Admissible perturbations of α‐ψ‐pseudocontractive operators: convergence theorems

We prove some convergence theorems for α‐ψ‐pseudocontractive operators in real Hilbert spaces, by using the concept of admissible perturbation. Our results extend and complement some theorems in the existing literature. Copyright © 2016 John Wiley & Sons, Ltd.