General implicit variational inclusion problems based on A-maximal (m)-relaxed monotonicity (AMRM) frameworks

A general framework for an algorithmic procedure based on the variational convergence of operator sequences involving A-maximal (m)-relaxed monotone (AMRM) mappings in a Hilbert space setting is developed, and then it is applied to approximating the solution of a general class of nonlinear implicit inclusion problems involving A-maximal (m)-relaxed monotone mappings. Furthermore, some specializations of interest on existence theorems and corresponding approximation solvability theorems on H-maximal monotone mappings are included that may include several other results for general variational inclusion problems on general maximal monotonicity in the literature.     

Pronormal subgroups of a direct product of groups

We give criteria to characterize abnormal, pronormal and locally pronormal subgroups of a direct product of two finite groups $A\times B$, under hypotheses of solvability for at least one of the factors, either A or B.     

Algebraic properties of compatible Poisson brackets

We discuss algebraic properties of a pencil generated by two compatible Poisson tensors A(x) and B(x). From the algebraic viewpoint this amounts to studying the properties of a pair of skew-symmetric bilinear forms A and B defined on a finite-dimensional vector space. We describe the Lie group G _P of linear automorphisms of the pencil P = {A + λB}. In particular, we obtain an explicit formula for the dimension of G _P and discuss some other algebraic properties such as solvability and Levi-Malcev decomposition.     

On the solvability of a group with permutable subgroups

The solvability of a finite group G = AB is established under the assumption that the subgroups A and B are solvable and the Carter subgroups of A commute with the Carter subgroups of B.     

Essential dimension of algebraic groups,including bad characteristic

Let A and G be finite groups of relatively prime orders and assume that A acts on G via automorphisms. We study how certain conditions on G imply its solvability when we assume the existence of a unique A-invariant Sylow p-subgroup for p equal to 2 or 3.     

On the global solvability in Gevrey classes on the n-dimensional torus

Let P be a linear partial differential operator with coefficients in the Gevrey class G^s(Tⁿ), where Tⁿ is the n-dimensional torus and s?1. We prove a necessary condition for the s-global solvability of P on Tⁿ. We also apply this result to give a complete characterization for the s-global solvability for a class of formally self-adjoint operators with nonconstant coefficients.     

L p solvability of divergence type parabolic and elliptic systems with partially BMO coefficients

We prove the ${{\mathcal{H}}^{1}_{p,q}}$ solvability of second order systems in divergence form with leading coefficients A ^???? only measurable in (t, x ¹) and having small BMO (bounded mean oscillation) semi-norms in the other variables. In addition, we assume one of the following conditions is satisfied: (i) A ¹¹ is measurable in t and has a small BMO semi-norm in the other variables; (ii) A ¹¹ is measurable in x ¹ and has a small BMO semi-norm in the other variables. The corresponding results for the Cauchy problem and elliptic systems are also established. Some of our results are new even for scalar equations. Using the results for systems in the whole space, we obtain the solvability of systems on a half space and Lipschitz domain with either the Dirichlet boundary condition or the conormal derivative boundary condition.     

Long-time behavior of the difference solutions to the 2D GKS equation

We study the long-time behavior of the finite difference solution to the generalized Kuramoto-Sivashinsky equation in two space dimensions with periodic boundary conditions. The unique solvability of numerical solution is shown. It is proved that there exists a global attractor of the discrete dynamical system and the upper semicontinuity d(A_h,τ,A)→0. Finally, we obtain the long-time stability and convergence of the difference scheme. Our results show that the difference scheme can effectively simulate the infinite dimensional dynamical systems.     

Diagonalization modulo a class of Orlicz ideals

We consider a class of norm ideals CG which are non-commutative analogues of Orlicz spaces and which satisfy a previously introduced condition called (QK). We give a spectral condition which is necessary and sufficient for a commuting tuple of self-adjoint operators A=(A₁,…,An) to be simultaneously diagonalizable modulo CG.     

A Semifield Plane of Odd Order Admitting an Autotopism Subgroup Isomorphic to <Emphasis Type="Italic">A</Emphasis><Subscript>5</Subscript>

We develop an approach to constructing and classifying semifield projective planes with the use of a spread set. The famous conjecture is discussed on the solvability of the full collineation group of a finite semifield nondesarguesian plane. We construct a matrix representation of a spread set of a semifield plane of odd order admitting an autotopism subgroup isomorphic to the alternating group A₅ and find a series of semifield planes of odd order not admitting A₅.     

Morita context of weak Hopf algebras

Let H be a finite-dimensional weak Hopf algebra and A a left H-module algebra with its invariant subalgebra A~H.We prove that the smash product A#H is an A-ring with a grouplike character, and give a criterion for A#H to be Frobenius over A. Using the theory of A-rings, we mainly construct a Morita context connecting the smash product A#H and the invariant subalgebra A~H , which generalizes the corresponding results obtained by Cohen, Fischman and Montgomery.     

On character amenability of Banach algebras

We continue our work [E. Kaniuth, A.T. Lau, J. Pym, On φ-amenability of Banach algebras, Math. Proc. Cambridge Philos. Soc. 144 (2008) 85-96] in the study of amenability of a Banach algebra A defined with respect to a character φ of A. Various necessary and sufficient conditions of a global and a pointwise nature are found for a Banach algebra to possess a φ-mean of norm 1. We also completely determine the size of the set of φ-means for a separable weakly sequentially complete Banach algebra A with no φ-mean in A itself. A number of illustrative examples are discussed.     

Ranges of nonlinear asymptotically linear operators

This paper deals with the solvability of the equation A(u) ? S(u) = f, where A is a continuous self-adjoint operator defined on a real Hilbert space H with values in H, the null-space of A is nontrivial, and N is a nonlinear completely continuous perturbation. Sufficient, and necessary-sufficient conditions are given for the equation to be solvable. Abstract theorems are applied to solving boundary value problems for partial differential equations.     

Necessary solvability conditions of systems of linear extremal equations

Systems of linear equations of the form A?X = B?X and of the form A?X = A?Y over the structure based on linearly ordered commutative group (G, ?, ≤) where the role of ⊕ plays the maximum are treated. Necessary solvability conditions are derived using known results concerning eigenvectors of matrices in such structures. In the special case of idempotent, increasing matrices A and B a condition is given which is necessary and sufficient for the existence of a non-trivial solution.     

A not so simple local multiplier algebra

We construct an AF-algebra A such that its local multiplier algebra Mloc(A) does not agree with Mloc(Mloc(A)), thus answering a question raised by G.K. Pedersen in 1978.     

An implicit-function theorem for inclusions

We consider the question of the solvability of an inclusion F(x, σ) ∈ A, i.e., of determining a mapping (implicit function) σ ? x(σ) defined on a set such that F(x(σ), σ) ∈ A for any σ from this set. Results of this kind play a key role in the different branches of analysis and, especially, in the theory of extremal problems, where they are the main tool for deriving conditions for an extremum.     

Linear equations over commutative rings

A generalized rank (McCoy rank) of a matrix with entries in a commutative ring R with identity is discussed. Some necessary and sufficient conditions for the solvability of the linear equation Ax = b are derived, where x, b are vectors and A is a matrix with entries in either a Noetherian full quotient ring or a zero dimensional ring.     

Weighted integral inequalities for conjugate A-harmonic tensors

We first prove a local weighted integral inequality for conjugate A-harmonic tensors. Then, as an application of our local result, we prove a global weighted integral inequality for conjugate A-harmonic tensors in L^s(μ)-averaging domains, which can be considered as a generalization of the classical result. Finally, we give applications of the above results to quasiregular mappings.     

Conditions for unique solvability of the matrix equation AX + X*B = C

Conditions for the unique solvability of the matrix equation AX + X*B = C are formulated in terms of the eigenvalues and the Kronecker structure of the matrix pencil A + ??B* associated with this equation.