Viability property of jump diffusion processes on manifolds

In this note, we give a necessary and sufficient condition for viability property of diffusion processes with jumps on closed submanifolds of R ^m . Our result is the system is viable in a closed submanifold K iff the coefficients are tangent to K along K if the equation is in the sense of stratonovich integral and the solution jumps from K to K.     

Real phase retrieval from unordered partial frame coefficients

We study the signal recovery from unordered partial phaseless frame coefficients. To this end, we introduce the concepts of m-erasure (almost) phase retrievable frames. We show that with an m-erasure (almost) phase retrievable frame, it is possible to reconstruct (almost) all n-dimensional real signals up to a sign from their arbitrary N ? m unordered phaseless frame coefficients, where N stands for the element number of the frame. We give necessary and sufficient conditions for a frame to be m-erasure (almost) phase retrievable. Moreover, we give an explicit construction of such frames based on prime numbers.     

The Construction of <Emphasis Type="Italic">G</Emphasis>-Invariant Finite Tight Frames

We give a complete classification of the finite tight frames which are G-invariant, i.e., invariant under the unitary action of a group G. This result is constructive, and we use it to consider a number of examples. In particular, we determine the minimum number of generators for a tight frame for the orthogonal polynomials on an n-gon or cube, which is invariant under the symmetries of the weight.     

Large sublattices in subsets of Banach lattices

A classical problem (initially studied by N. Kalton and A. Wilansky) concerns finding closed infinite dimensional subspaces of X / Y, where Y is a subspace of a Banach space X. We study the Banach lattice analogue of this question. For a Banach lattice X, we prove that X / Y contains a closed infinite dimensional sublattice under the following conditions: either (i) Y is a closed infinite codimensional subspace of X, and X is either order continuous or a C(K) space, where K is a compact subset of \({\mathbb {R}}^n\); or (ii) Y is the range of a compact operator.     

Volterra type integral operators with homogeneous kernels in weighted <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

We consider multidimensional integral Volterra type operators with kernels homogeneous of degree (?n); the operators act in L _p -spaces with a submultiplicative weight. For these operators we obtain necessary and sufficient conditions of their invertibility. Besides, we describe the Banach algebra generated by the operators. For this algebra we construct the symbolic calculus, in terms of which we obtain an invertibility criterion of the operators.     

Holomorphic injective extensions of functions in <Emphasis Type="Italic">P</Emphasis>(<Emphasis Type="Italic">K</Emphasis>) and algebra generators

We present necessary and sufficient conditions on planar compacta K and continuous functions f on K in order that f generate the algebras P(K), R(K), A(K) or C(K). We also unveil quite surprisingly simple examples of non-polynomial convex compacta K ? C and f ∈ P(K) with the property that f ∈ P(K) is a homeomorphism of K onto its image, but for which f ^?1 ? P(f(K)). As a consequence, such functions do not admit injective holomorphic extensions to the interior of the polynomial convex hull \(\widehat K\). On the other hand, it is shown that the restriction f*|_G of the Gelfand-transform f* of an injective function f ∈ P(K) is injective on every regular, bounded complementary component G of K. A necessary and sufficient condition in terms of the behaviour of f on the outer boundary of K is given in order that f admit a holomorphic injective extension to \(\widehat K\). We also include some results on the existence of continuous logarithms on punctured compacta containing the origin in their boundary.     

Some connections between frames of radical ideals and frames of <Emphasis Type="Italic">z</Emphasis>-ideals

For any semisimple f-ring A with bounded inversion, we show that the frame of radical ideals of A and the frame of z-ideals of A have isomorphic subfit coreflections. If we assume the Axiom of Choice, then the two coreflections are actually identical. If the f-ring has the property that the sum of two z-ideals is a z-ideal (as in the case of rings of continuous functions), then the epicompletion of the frame of z-ideals is shown to be a dense quotient of the epicompletion of the frame of radical ideals. Baer rings, exchange rings, and normal rings that lie in the class of semisimple f-rings with bounded inversion are shown to have characterizations in terms of frames of z-ideal which are similar to characterizations in terms of frames of radical ideals.     

On <Emphasis Type="Italic">c</Emphasis><Superscript>#</Superscript>-normal subgroups infinite groups

A subgroup H of a finite group G is called a c^#-normal subgroup of G if there exists a normal subgroup K of G such that G = HK and H ∩ K is a CAP-subgroup of G: In this paper, we investigate the influence of fewer c^#-normal subgroups of Sylow p-subgroups on the p-supersolvability, p-nilpotency, and supersolvability of finite groups. We obtain some new sufficient and necessary conditions for a group to be p-supersolvable, p-nilpotent, and supersolvable. Our results improve and extend many known results.     

On the sequential order continuity of the <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">K</Emphasis>)-space

As shown in [1], for each compact Hausdorff space K without isolated points, there exists a compact Hausdorff P′-space X but not an F-space such that C(K) is isometrically Riesz isomorphic to a Riesz subspace of C(X). The proof is technical and depends heavily on some representation theorems. In this paper we give a simple and direct proof without any assumptions on isolated points. Some generalizations of these results are mentioned.     

On some properties of continuous <Emphasis Type="Italic">g</Emphasis>-frames and Riesz-type continuous <Emphasis Type="Italic">g</Emphasis>-frames

In this paper we provide some necessary and sufficient conditions under which, a family of bounded operators is a continuous g-frame (Riesz-type continuous g-frame). Also, we study stability of duals of continuous g-frames.     

K-fusion Frames and the Corresponding Generators for Unitary Systems

Motivated by K-frames and fusion frames, we study K-fusion frames in Hilbert spaces.By the means of operator K, frame operators and quotient operators, several necessary and sufficient conditions for a sequence of closed subspaces and weights to be a K-fusion frame are obtained, and operators preserving K-fusion frames are discussed. In particular, we are interested in the K-fusion frames with the structure of unitary systems. Given a unitary system which has a complete wandering subspace, we give a necessary and sufficient condition for a closed subspace to be a K-fusion frame generator.     

Subspace dual super wavelet and Gabor frames

Due to its potential applications in multiplexing techniques, the study of superframes has interested some researchers. This paper addresses dual super wavelet and Gabor frames in the subspace setting. We obtain a basic-equation characterization for subspace dual super wavelet and Gabor frames. In addition, applying this characterization, we derive a procedure that allows for constructing subspace dual super wavelet frames based on certain subspace dual super Gabor frames, and vice versa. Our results are new even in L²(R;C ^L ) setting.     

On <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Subscript>-factorization of complete bipartite multigraphs

Let λK _m,n be a complete bipartite multigraph with two partite sets having m and n vertices, respectively. A K _p,q -factorization of λK _m,n is a set of edge-disjoint K _p,q -factors of λK _m,n which partition the set of edges of λK _m,n . When p = 1 and q is a prime number, Wang, in his paper [On K _1,q -factorization of complete bipartite graph, Discrete Math., 126: (1994), 359-364], investigated the K _1,q -factorization of K _m,n and gave a sufficient condition for such a factorization to exist. In papers [K _1,k -factorization of complete bipartite graphs, Discrete Math., 259: 301-306 (2002),; K _p,q -factorization of complete bipartite graphs, Sci. China Ser. A-Math., 47: (2004), 473-479], Du and Wang extended Wang’s result to the case that p and q are any positive integers. In this paper, we give a sufficient condition for λK _m,n to have a K _p,q -factorization. As a special case, it is shown that the necessary condition for the K _p,q -factorization of λK _m,n is always sufficient when p : q = k : (k + 1) for any positive integer k.     

Multiplication of conjugacy classes,colligations, and characteristic functions of matrix argument

We extend the classical construction of operator colligations and characteristic functions. Consider the group G of finitary block unitary matrices of order α+∞+···+∞ (m times) and its subgroup K ? U(∞), which consists of block diagonal unitary matrices with the identity block of order α and a matrix u ∈ U(∞) repeated m times. It turns out that there is a natural multiplication on the space G//K of conjugacy classes. We construct “spectral data” of conjugacy classes, which visualize the multiplication and are sufficient for reconstructing a conjugacy class.     

Sharp continuity results for the short-time Fourier transform and for localization operators

We completely characterize the boundedness on Wiener amalgam spaces of the short-time Fourier transform (STFT), and on both L ^p and Wiener amalgam spaces of a special class of pseudodifferential operators, called localization operators. Precisely, sufficient conditions for the STFT to be bounded on the Wiener amalgam spaces W(L ^p , L ^q ) are given and their sharpness is shown. Localization operators are treated similarly: using different techniques from those employed in the literature, we relax the known sufficient boundedness conditions for these operators to be bounded on L ^p spaces and prove the optimality of our results. Next, we exhibit sufficient and necessary conditions for such operators to be bounded on Wiener amalgam spaces.     

Inner derivations of simple Lie pencils of rank 1

We prove that simple Lie pencils of rank 1 over an algebraically closed field P of characteristic 0 with operators of left multiplication being derivations are of the form of a sandwich algebra M ₃(U,D′), where U is the subspace of all skew-symmetric matrices in M ₃(P) and D′ is any subspace containing 〈E〉 in the space of all diagonal matrices D in M ₃(P).     

Maximal invariant subspaces for a class of operators

In this note, we characterize maximal invariant subspaces for a class of operators. Let T be a Fredholm operator and \(1-TT^{*}\in\mathcal{S}_{p}\) for some p≥1. It is shown that if M is an invariant subspace for T such that dim?M ? TM<∞, then every maximal invariant subspace of M is of codimension 1 in M. As an immediate consequence, we obtain that if M is a shift invariant subspace of the Bergman space and dim?M ? zM<∞, then every maximal invariant subspace of M is of codimension 1 in M. We also apply the result to translation operators and their invariant subspaces.     

Optimal Reliability Design of <Emphasis Type="Italic">K</Emphasis>-out-of-<Emphasis Type="Italic">N</Emphasis> Systems Subject to Two Kinds of Failure

K-out-of-N systems formed from N identical and independent components are considered in which the components can take two states: 0 (open) or 1 (closed) on command (e.g. Electromagnetic Relays and Solid State Switches). The components are subject to two kinds of failure on command: failure to open and failure to close. A K-out-of-N system is closed if and only if at least K of its components are closed. The system is considered open or closed depending on the states of its components. The optimum system is taken to be that system which maximizes the reliability. This paper finds the optimum K-out-of-N system given a fixed number of components.     

Counting extreme <Emphasis Type="Italic">U</Emphasis><Subscript>1</Subscript> matrices and characterizing quadratic doubly stochastic operators

The U ₁ matrix and extreme U ₁ matrix were successfully used to study quadratic doubly stochastic operators by R. Ganikhodzhaev and F. Shahidi [Linear Algebra Appl., 2010, 432: 24–35], where a necessary condition for a U ₁ matrix to be extreme was given. S. Yang and C. Xu [Linear Algebra Appl., 2013, 438: 3905–3912] gave a necessary and sufficient condition for a symmetric nonnegative matrix to be an extreme U ₁ matrix and investigated the structure of extreme U ₁ matrices. In this paper, we count the number of the permutation equivalence classes of the n × n extreme U ₁ matrices and characterize the structure of the quadratic stochastic operators and the quadratic doubly stochastic operators.     

Clark model in the general situation

For a unitary operator U in a Hilbert space H the family of its unitary perturbations by rank one operators with fixed range is parametrized by a complex parameter γ, ?γ? = 1. Namely, all such unitary perturbations are operators U_γ:= U + (γ ? 1)( ·, b₁)_Hb, where b ∈ H, ∥b∥ = 1, b₁ = U^?1b, ?γ? = 1. For ?γ? < 1, the operators U_γ are contractions with one-dimensional defects.