Desch‐Schappacher perturbation of one‐parameter semigroups on locally convex spaces

We prove a Desch‐Schappacher type perturbation theorem for strongly continuous and locally equicontinuous one‐parameter semigroups which are defined on a sequentially complete locally convex space.     

An anisotropic area-preserving flow for convex plane curves

This paper will deal with an anisotropic area-preserving flow which keeps the convexity of the evolving curve and the limiting curve converges to a homothety of a symmetric smooth strictly convex plane curve.     

The behavior of differential forms under purely inseparable extensions

Let F be a field of characteristic 2. In this paper we give a complete computation of the kernel of the homomorphism $H_2^{m+1}(F)?H_2^{m+1}(L)$ induced by scalar extension, where $L/F$ is a purely inseparable extension (of any degree), $H_2^{m+1}(F)$ is the cokernel of the Artin–Schreier operator $?:{\mathrm{\Omega }}_F^m?{\mathrm{\Omega }}_F^m/\mathrm{d}{\mathrm{\Omega }}_F^{m?1}$ given by: $x\frac{\mathrm{d}{x}_1}{x_1}\wedge ?\wedge \frac{\mathrm{d}{x}_m}{x_m}?(x^2?x)\frac{\mathrm{d}{x}_1}{x_1}\wedge ?\wedge \frac{\mathrm{d}{x}_m}{x_m}+\mathrm{d}{\mathrm{\Omega }}_F^{m?1}$, where ${\mathrm{\Omega }}_F^m$ is the space of absolute m-differential forms over F and d is the differential operator. Other related results are included.     

Characteristic parameter sets and limits of circulant Hermitian polygon transformations

Polygon transformations based on taking the apices of similar triangles constructed on the sides of an initial polygon are analyzed as well as the limit polygons obtained by iteratively applying such transformations. In contrast to other approaches, this is done with respect to two construction parameters representing a base angle and an apex perpendicular subdivision ratio. Furthermore, a combined transformation leading to circulant Hermitian matrices is proposed, which eliminates the rotational effect of the basic transformation. A finite set of characteristic parameter subdomains is derived for which the sequence converges to specific eigenpolygons. Otherwise, limit polygons turn out to be linear combinations of up to three eigenpolygons. This leads to a full classification of circulant Hermitian similar triangles based polygon transformations and their limit polygons. As a byproduct classical results as Napoleon’s theorem and the Petr-Douglas-Neumann theorem can be easily deduced.     

Bijective linear maps on semimodules spanned by Boolean matrices of fixed rank

Let M_m,n(B) be the semimodule of all m×n Boolean matrices where B is the Boolean algebra with two elements. Let k be a positive integer such that 2?k?min(m,n). Let B(m,n,k) denote the subsemimodule of M_m,n(B) spanned by the set of all rank k matrices. We show that if T is a bijective linear mapping on B(m,n,k), then there exist permutation matrices P and Q such that T(A)=PAQ for all A∈B(m,n,k) or m=n and T(A)=PA^tQ for all A∈B(m,n,k). This result follows from a more general theorem we prove concerning the structure of linear mappings on B(m,n,k) that preserve both the weight of each matrix and rank one matrices of weight k². Here the weight of a Boolean matrix is the number of its nonzero entries.     

Riemann theta functions periodic wave solutions and rational characteristics for the nonlinear equations

In this paper, based on a multidimensional Riemann theta function, a lucid and straightforward generalization of the Hirota-Riemann method is presented to explicitly construct multiperiodic Riemann theta functions periodic wave solutions for nonlinear equations such as the Caudrey-Dodd-Gibbon-Sawada-Kotera equation and (2+1)-dimensional breaking soliton equation. Among these periodic waves, the one-periodic waves are well-known cnoidal waves, their surface pattern is one-dimensional, and often they are used as one-dimensional models of periodic waves. The two-periodic waves are a direct generalization of one-periodic waves, their surface pattern is two-dimensional so that they have two independent spatial periods in two independent horizontal directions. A limiting procedure is presented to analyze in detail, asymptotic behavior of the multiperiodic waves and the relations between the periodic wave solutions and soliton solutions are rigorously established. This generalized Hirota-Riemann method can also be demonstrated on a class variety of nonlinear difference equations such as Toeplitz lattice equation.     

On impulsive functional differential inclusions with Hille-Yosida operators in Banach spaces

We consider a semilinear functional differential inclusion with infinite delay and impulse characteristics in a Banach space assuming that its linear part is a non-densely defined Hille-Yosida operator. We assume that the multivalued nonlinearity of upper Carathèodory or almost lower semicontinuous type satisfies a regularity condition expressed in terms of the measures of noncompactness. We apply the theory of integrated semigroups and the theory of condensing multivalued maps to obtain local and global existence results. The application to an optimization problem for an impulsive feedback control system is given.     

A global bifurcation index for set-valued perturbations of Fredholm operators

In the paper the homotopy invariant detecting global bifurcations of solutions to multi-parameter equations involving general set-valued perturbations of linear Fredholm operators of non-negative index is introduced. Some applications to the existence problems for differential inclusions are provided.     

Quasi-Banach spaces of almost universal disposition

We show that for each p∈(0,1]$p\in (0,1]$ there exists a separable p  -Banach space _Gp${\mathbb{G}}_p$ of almost universal disposition, that is, having the following extension property: for each ε>0$\epsilon >0$ and each isometric embedding g:X→Y$g:X\to Y$, where Y is a finite-dimensional p-Banach space and X   is a subspace of _Gp${\mathbb{G}}_p$, there is an ε  -isometry f:Y→_Gp$f:Y\to {\mathbb{G}}_p$ such that x=f(g(x))$x=f(g(x))$ for all x∈X$x\in X$.     

A “multigrid” extension of theFFT for the numerical inversion of Fourier and Laplace Transforms

An algorithm for accurate numerical inversion of slowly convergent Fourier and Laplace Transforms is studied. It makes use of several equidistant grids with the same number of points, covering different symmetric intervals of the time and frequency axes. Typically, the number of operations per computed function value is about twice as large as for an ordinary FFT. The distribution of points is, however, for many applications much more adequate because, globally, the union of the grids is an approximately equidistant point set on a logarithmic scale.Dedicated to Gene H. Golub on the occasion of his 60'th birthday