Asymptotic behavior and representation of solutions to a Volterra kind of equation with a singular kernel

Semigroup Forum - Let A be a densely defined closed, linear $$\omega$$ -sectorial operator of angle $$\theta \in [0,\frac{\pi }{2})$$ on a Banach space X, for some $$\omega \in \mathbb...     

Modified Alternating $$\vec{k}$$–generators

This paper deals with a class of pseudorandom bit generators – modified alternating –generators. This class is constructed similarly to the class of alternating step generators. Three subclasses of are distinguished, namely linear, mixed and nonlinear generators. The main attention is devoted to the subclass of linear and mixed generators generating periodic sequences with maximal period lengths. A necessary and sufficient condition for all sequences generated by the linear generators of to be with maximal period lengths is formulated. Such sequences have good statistical properties, such as distribution of zeroes and ones, and large linear complexity. Two methods of cryptanalysis of the proposed generators are given. Finally, three new classes of modified alternating –generators, designed especially to be more secure, are presented.     

Spectral Shorted Operators

If $$\mathcal{H}$$ is a Hilbert space, $$\mathcal{S}$$ is a closed subspace of $$\mathcal{H},$$ and A is a positive bounded linear operator on $$\mathcal{H},$$ the spectral shorted operator $$\rho \left( {\mathcal{S},\mathcal{A}} \right)$$ is defined as the infimum of the sequence $$\sum (\mathcal{S},A^n )^{1/n} ,$$ where denotes $$\sum \left( {\mathcal{S},B} \right)$$ the shorted operator of B to $$\mathcal{S}.$$ We characterize the left spectral resolution of $$\rho \left( {\mathcal{S},\mathcal{A}} \right)$$ and show several properties of this operator, particularly in the case that dim $${\mathcal{S} = 1.}$$ We use these results to generalize the concept of Kolmogorov complexity for the infinite dimensional case and for non invertible operators.     

Linear Maps Preserving Generalized Invertibility

Let H be an infinite-dimensional complex separable Hilbert space and the algebra of all bounded linear operators on H. Let be a bijective continuous unital linear map preserving generalized invertibility in both directions. Then the ideal of all compact operators is invariant under ϕ and the induced linear map on the Calkin algebra is either an automorphism or an antiautomorphism.     

Vector Variational Inequalities and the (S)<Subscript>+</Subscript> Condition

Let $${\cal Z}$$ and X be Hausdorff real topological vector spaces and let $${\cal L}_b(X,{\cal Z})$$ be the space of continuous linear mappings from X into $${\cal Z}$$ equipped with the topology of bounded convergence. In this paper, we define the (S)₊ condition for operators from a nonempty subset of X into $${\cal L}_b(X,{\cal Z})$$ and derive some existence results for vector variational inequalities with operators of the class (S)₊. Some applications to vector complementarity problems are given.     

On the decomposability of linear combinations of Bernoulli polynomials

Monatshefte für Mathematik - In the present paper we describe the complete decomposition (over $$\mathbb {C}$$ ) of linear combinations of the form $$\begin{aligned} R_n(x)=B_n(x)+cB_{n-2}(x)...     

Closedness properties of internal relations V: Linear Mal’tsev conditions

As J. W. Snow showed, every linear Mal’tsev condition on a variety of universal algebras, is equivalent to a relational condition on . Using slightly different relational reformulations of linear Mal’tsev conditions, we develop a purely categorical approach to these conditions. Received August 10, 2006; accepted in final form January 23, 2007.     

Solutions of the analogues of time-dependent Schrödinger equations corresponding to a pair of $$H^{3+2}$$ Hamiltonian systems

Theoretical and Mathematical Physics - We construct joint $$2\times2$$ matrix solutions of the scalar linear evolution equations $$\Psi'_{s_k}=H^{3+2}_{s_k}(s_1,s_2, x_1,x_2, \partial/\partial...     

Projectively equivariant quantization and symbol on supercircle S1∣3

Czechoslovak Mathematical Journal - Let $${{\cal D}_{\lambda ,\mu }}$$ be the space of linear differential operators on weighted densities from $${{\cal F}_\lambda }$$ to $${{\cal F}_\mu }$$ as...     

Plenty of hyperbolicity on a class of linear homogeneous jerk differential equations

Aequationes mathematicae - We consider $$3\times 3$$ partially hyperbolic linear differential systems over an ergodic flow $$X^t$$ and derived from the linear homogeneous differential equation...     

The Laws of Large Numbers Associated with the Linear Self-attracting Diffusion Driven by Fractional Brownian Motion and Applications

Journal of Theoretical Probability - Let $$B^H$$ be a fractional Brownian motion with Hurst index $$\frac{1}{2}\le H&lt;1$$ . In this paper, we consider the linear self-attracting diffusion...     

Description of the Structure of Irregularity Sets of Linear Differential Systems with a Linear Parameter

Differential Equations - We consider the class of linear parametric differential systems $$\dot {x}=\mu A(t)x $$ defined on the half-line $$t\geq 0 $$ , where $$\mu \in \mathbb {R} $$ is a...     

On sums of polynomial-type exceptional units in $$\varvec{\mathbb {Z}}/\varvec{n\mathbb {Z}}$$Z/nZ

A unit u in a commutative ring with unity R is called exceptional if $$u-1$$ is also a unit. We introduce the notion of a polynomial version of this (abbreviated as $$f\hbox {-exunits}$$) for any $$f(X) \in \mathbb {Z}[X]$$. We find the number of representations of a non-zero element of $$\mathbb {Z}/n\mathbb {Z}$$ as a sum of two f-exunits for an infinite family of polynomials f of each degree $$\ge 1$$. We also derive the exact formulae for certain infinite families of linear and quadratic polynomials. This generalizes a result proved by Sander (J Number Theory 159:1–6, 2016).     

Newforms of squarefree level and theta series

We show that the cuspidal part of the span of the theta series associated to maximal integral lattices on a definite quaternion algebra over is precisely the space of newforms. Furthermore, using the Eichler commutation relation and an idea coming out of the Yoshida lifting, we show how to express each newform as an explicit linear combination of such theta series. The coefficients of these linear combinations come from cuspidal eigenvectors of Brandt matrices.     

Reverse Inequalities for Subelliptic Functions

Mathematical Notes - We study a wedge $$\mathscr{K}(A)$$ of solutions of the inequality $$A(u) \ge 0$$ , where $$A$$ is a linear elliptic operator of order $$2m$$ . For the elements of the wedge,...     

Proof of the Jacobian Conjecture in the Two-Dimensional Case and Global Isochronous Centers of Polynomial Hamiltonian Differential Systems

Differential Equations - It is proved that, up to a nonsingular linear transformation, the real Hamiltonian system $$\dot {x}=-y-P(x) $$ , $$\dot {y}=x+(y+P(x))P^{\prime }(x) $$ , where $$P(x) $$...     

Maximum distance separable poset codes

We derive the Singleton bound for poset codes and define the MDS poset codes as linear codes which attain the Singleton bound. In this paper, we study the basic properties of MDS poset codes. First, we introduce the concept of I-perfect codes and describe the MDS poset codes in terms of I-perfect codes. Next, we study the weight distribution of an MDS poset code and show that the weight distribution of an MDS poset code is completely determined. Finally, we prove the duality theorem which states that a linear code C is an MDS -code if and only if is an MDS -code, where is the dual code of C and is the dual poset of      

Perron Conditions for Pointwise and Global Exponential Dichotomy of Linear Skew-Product Flows

The aim of this paper is to give necessary and sufficient conditions for pointwise and uniform exponential dichotomy of linear skew-product flows. We shall obtain that the pointwise exponential dichotomy of a linear skew-product flow is equivalent to the pointwise admissibility of the pair As a consequence, we prove that a linear skew-product flow on is uniformly exponentially dichotomic if and only if the pair is uniformly admissible for .     

Bauer–Furuta invariants under $${\mathbb{Z}_2}$$ -actions

S. Bauer and M. Furuta defined a stable cohomotopy refinement of the Seiberg–Witten invariants. In this paper, we prove a vanishing theorem of Bauer–Furuta invariants for 4-manifolds with smooth -actions. As an application, we give a constraint on smooth -actions on homotopy K3#K3, and construct a nonsmoothable locally linear -action on K3#K3. We also construct a nonsmoothable locally linear -action on K3.