bluuebunny/crossref_metadata_2025 huggingface dataset:
Die Nevanlinna-Charakteristik von algebroiden Funktionen und ihren Ableitungen
Authors: Sato Tomohiko
Published: September 2013

It is well known that, when

f(z)

is an entire function of order

\rho

and

\rho > \infty

, then the limit lim sup

_{r \rightarrow \infty} \frac{T(r,f')}{T(r,f)}

is finite as

r \rightarrow \infty

through all values or outside a set

E

of finite measure. But for

\rho = \infty

, Hayman has shown that the assertion does not hold by constructing an entire function

f(z)

and an exceptional set

E

of even infinite measure. In this paper, we will further extend his result to the case where

f(z)

is an algebroid function of order

\rho = \infty

.
A Modified and a Finite Index Weber Transforms
Authors: Al-Musallam Fadhel , Kim Tuan Vu
Published: September 2013

This paper introduces, by way of constructing, specific finite and infinite integral transforms with Bessel functions

J_\nu

and

Y_\nuä

in their kernels. The infinite transform and its reciprocal look deceptively similar to the known Weber transform and its reciprocal, respectively, but fundamentally differ from them. The new transform enjoys an operational property that makes it useful for applications to some problems in differential equations with non-constant coefficients. The paper gives a characterization of the image of some spaces of square integrable functions with respect to some measure under the infinite and finite transforms.
Existence Result for the One-Dimensional Full Model of Phase Transitions
Authors: Luterotti F. , Stefanelli Ulisse
Published: September 2013

This note deals with a nonlinear system of partial differential equations accounting for phase transition phenomena. The existence of solutions to a Cauchy–Neumann problem is established in the one-dimensional space setting, using a regularization – a priori estimates – passage to limit procedure.
Weighted Hölder Continuity of Hyperbolic Harmonic Bloch functions
Authors: Ren Guangbin , Kähler Uwe
Published: September 2013

Characterizations of weighted Hölder continuity and weighted Lipschitz continuity are obtained for the hyperbolic Bloch functions on the unit ball of

\mathbb R^n

. Similar results are extended to hyperbolic little Bloch and Besov spaces.
Level Sets of Hölder Functions and Hausdorff Measures
Authors: D'Aniello Emma
Published: September 2013

In this paper we investigate some connections between Hausdorff measures, Hölder functions and analytic sets in terms of images of zero-derivative sets and level sets. We characterize in terms of Hausdorff measures and descriptive complexity subsets

M \subseteq \mathbb R

which are(1) the image under some

C^{m, \alpha}

function

f

of the set of points where the derivatives of first

n

orders are zero(2) the set of points where the level sets of some

C^{m, \alpha}

function are perfect(3) the set of points where the level sets of some

C^{m, \alpha}

function are uncountable.
Problem of Functional Extension and Space-Like Surfaces in Minkowski Space
Authors: Grigoryeva E.G. , Klyachin A.A. , Miklyukov V.M.
Published: September 2013

Let

\Xi (x)

be the distribution of convex sets over a domain

D \subset \mathbb R^n

and let

\phi: \partial D \rightarrow \mathbb R

be a function. We consider the existence problem of locally Lipschitz functions

f

defined in the domain

D

so that

f|_{\partial D} = \phi

and

\bigtriangledown f (x) \in \Xi (x)

almost everywhere in

D

. These questions are related to the existence problem for space-like surfaces of arbitrary codimension with prescribed boundary in Minkowski space.
A Leray-Schauder Alternative for Mönch Maps on Closed Subsets of Fréchet Spaces
Authors: Frigon Marlène , O'Regan Donal
Published: September 2013

In this paper, a continuation principle is obtained for maps defined on a closed, convex subset which may have empty interior in a Fréchet space, and satisfying a condition of Mönch type. An application to first order systems of differential equations is presented to illustrate our theory.
Solvability and Galerkin Approximations of a Class of Nonlinear Operator Equations
Authors: Gatica Gabriel N.
Published: September 2013

We generalize the usual Babuska-Brezzi theory to a class of nonlinear variational problems with constraints. The corresponding operator equation has a dual-dual type structure since the nonlinear operator involved has itself a dual structure with a strongly monotone and Lipschitz-continuous main operator. We provide sufficient conditions for the existence and uniqueness of solution of the continuous and Galerkin formulations, and derive a Strang-type estimate for the associated error. An application to the coupling of mixed-FEM and BEM for a nonlinear transmission problem in potential theory is also described.
On Summands of Closed Bounded Convex Sets
Authors: Grzybowski Jerzy , Przybycien H. , Urbanski R.
Published: September 2013

In this paper properties of the Minkowski-Pontryagin subtraction of closed bounded convex sets are investigated (see Propositions 1-3) and four criteria for summands of closed bounded convex sets are given (see Theorems 1-4).
The Finitely Additive Integral of Multifunctions with Closed and Convex Values
Authors: Martellotti Anna , Sambucini Anna Rita
Published: September 2013

We investigate integration with respect to a finitely additive measure of integrands with closed, convex values and we obtain a closedness result for the Aumann integral.
Non-Compact $\lambda$-Hankel Operators
Authors: Martínez-Avendaño Ruben A. , Yuditskii Peter
Published: September 2013

A

\lambda

-Hankel operator

X

is a bounded operator on Hilbert space satisfying the operator equation

S*X–XS = \lambda X

, where

S

is the (unilateral) forward shift and

S*

is its adjoint. We prove that there are non-compact

\lambda

-Hankel operators for

\lambda

a complex number of modulus less than 2, by first exhibiting a way to obtain bounded solutions to the above equation by associating to it a Carleson measure. We then show that an interpolating sequence can be given such that the

\lambda

-Hankel operator associated with the Carleson measure given by the interpolating sequence is non-compact.
On the Minimal Displacement Problem of $\gamma$-Lipschitz Maps and $\gamma$-Lipschitz Retractions onto the Sphere
Authors: Väth Martin
Published: September 2013

We give a general construction in arbitrary normed spaces to produce fixed-point free continuous maps with a large minimal displacement, contractions of the sphere, and retractions onto the sphere such that the corresponding maps have small measures of non-compactness.
Bifurcation of Homoclinic Solutions for Hamiltonian Systems
Authors: Joosten Robert
Published: September 2013

We consider the Hamiltonian system

Ju'(x) + Mu(x) – \bigtriangledown _u F(x,u(x)) = \lambda u(x).

Using variational methods obtained by Stuart on the one hand and by Giacomoni and Jeanjean on the other, we get bifurcation results for homoclinic solutions by imposing conditions on the function

F

. We study both the case where

F

is defined globally with respect to

u

and the case where

F

is defined locally only.
Existence and Asymptotic Behavior of Positive Solutions of a Non-Autonomous Food-Limited Model with Unbounded Delay
Authors: Liu Yuji , Ge Weigao
Published: September 2013

Consider the non-autonomous logistic model

\Delta x_n = p_n x_n \frac{1–x_n–k_n}{1+\lambda x_n–k_n}^r \;\;\;(n≥0)

where

\Delta x_n = x-{n+1} – x_n

, {

p_n

} is a sequence of positive real numbers, {

k_n

} is a sequence of non-negative integers such that {

n–k_n

} is non-decreasing,

\lambda \in [0,1]

, and

r

is the ratio of two odd integers. We obtain new sufficient conditions for the attractivity of the equilibrium

x = 1

of the model and conditions that guarantee the solution to be positive, which improve and generalize some recent results established by Phios and by Zhou and Zhang.
Wiener-Type Tauberian Theorems for Fourier Hyperfunctions
Authors: Pilipović Stevan , Stanković Bogoljub
Published: September 2013

Two Wiener-type Tauberian theorems concerning Fourier hyperfunctions are proved and commented. It is shownt that the shift asymptotics (

S

-asymptotics) of a hyperfunction

f

is determined by the ordinary asymptotics of

(f \star \mathcal K) (x)

as

x \rightarrow \infty

, where

\mathcal K

is Hörmaner’s kernel. Moreover, Wiener-type theorems are used for the asympthotic analysis of solutions to some (pseudo-)differential equations.
Estimates for Quasiconformal Mappings onto Canonical Domains (II)
Authors: Thao Vo Dang
Published: September 2013

In this paper we establish estimates for normal K-quasiconformal mappings

z = g(w)

of any finitely-connected domain in the extended

w

-plane onto the interior or exterior of the unit circle or the extended

z

-plane with

n (≥ 0)

slits on the circles

|z| = R_j (j = 1,...,n)

. The bounds in the estimates for

R_j, |g(w)|

, etc. are explicitly given. They are sharp or asymptotically sharp and deduced mainly from estimates for the inverse mappings of

g

in our previous paper [10] based on Carleman’s and Grötzsch’s inequalities and partly improved here. A generalization of the Schwarz lemma and improvements of some classical inequalities for conformal mappings are shown.
Initial-Mixed Boundary Value Problems for Parabolic Complex Equations of Second Order with Measurable Coefficients
Authors: Wen Guochun
Published: September 2013

In [2], the authors introduced some results on initial-boundary value problems for divergence parabolic equations of second order with measurable coefficients. In [1], the authors considered the first boundary value problem for non-divergence parabolic equations of second order with discontinuous coefficients. In this paper, we consider initial-mixed boundary value problems for non-divergence parabolic complex equations of second order in a multiply connected domain. Firstly, we give a priori estimates of solutions of the above initial-boundary value problems by the method of symmetric extension, and then by using these estimates, the methods of auxiliary functions and parameter extension, we prove the solvability for the foregoing problems. Here the condition (1.3) is weaker than the corresponding one in [1] and [3], i.e. the constant 4/3 in [1] and [3] is replaced by 3/2 in (1.3).
On a Generalized Integro-Differential Equation of Barbashin Type
Authors: Chur-jen Cheng
Published: September 2013

We study a class of integro-differential equations containing multiplication opera-tors, partial integral operators, and ordinary integral operators. Building on the usual identification of real functions of several variables and abstract functions, such integro- differential equations may be reformulated as ordinary differential equations in suitable Banach spaces. We give a representation theorem for the corresponding Cauchy operator and study the (unique) solvability of a general boundary value problem.
On a Property of Harmonic Functions
Authors: Keglević Ivan
Published: September 2013

If we divide the space

\mathbb R^n

into two disjoint areas with one common hypersurface and define a harmonic function in each part of these areas such that their gradients vanish at infinity and the normal components of their gradients are equal on the hypersurface, then for some hypersurfaces such as a circle in

\mathbb R^2

or a hyperplane in

\mathbb R^n

the sum of the tangential components of the gradients is zero. We investigate for which hypersurfaces we have this property and prove that such hypersurfaces in

\mathbb R^2

are only circles and straight lines. We also give an application of this property to an ideal plane flow through a porous surface.
Some Remarks on the Hildebrandt-Graves Theorem
Authors: Vignoli A. , Zabrejko Petr P.
Published: September 2013

This paper is concerned with a generalization of the Hildebrandt-Graves implicit function theorem building on a weaker acting condition for the derivative of the nonlinear operator involved. The abstract theorem is illustrated by means of an application to nonlinear singular integral equations where classical implicit function theorems do not apply.
Problèmes aux limites pour des equations intégro-différentielles de type Barbachine
Authors: Appell Jürgen , Diallo O.W.
Published: September 2013

We discuss existence and uniqueness of solutions of a non-stationary integro-differential equation of Barbashin type, subject to natural boundary conditions. We show that the boundary value problem may be reduced to either a linear Fredholm integral equation, or a system of two linear Volterra integral equations. In the latter case one may apply fixed point principles in K-normed spaces to prove solvability and continuous dependence.
Loewner Interpolation in Matrix Hardy Classes
Authors: Alpay Daniel , Leblond J.
Published: September 2013

We study a Loewner type interpolation problem in the framework of matrix-valued

\mathbb H_2

functions. We establish a necessary and sufficient condition for a matrix-valued function given on an arc of circle to be the trace of such an

\mathbb H_2

function subject to some constraint.
On Associated and Co-Associated Complex Differential Operators
Authors: Heersink R. , Tutschke Wolfgang
Published: September 2013

The paper deals with initial value problems of the form

\frac{\partial u}{\partial t} = \mathcal L u, \ \ \ u = u_0 \ \mathrm {for} \ t = 0

in

[0,T] \times G \subset \mathbb R^+_0 \times \mathbb R^n

where

\mathcal L

is a linear first order differential operator. The desired solutions will be sought in function spaces defined as kernel of a linear differential operator

l

being associated to

\mathcal L

. Mainly two assumptions are required for such initial value problems to be solvable: Firstly, the operators have to be associated, i.e.

lu = 0

implies

l(\mathcal L u) = 0

. Secondly, an interior estimate

\| \mathcal L u \|_{G'} ≤ c(G,G’) \| u \|_G

(with

G' \subset G

) must be true. Moreover, operators

\mathcal L

are investigated possessing a family of associated operators

l_k

(which then are said to be co-associated).

The present paper surveys the use of associated and co-associated differential operators for solving initial value problems of the above (Cauchy–Kovalevskaya) type. Discussing interior estimates as starting point for the construction of related scales of Banach spaces, the paper sets up a possible framework for further generalizations. E.g., that way a theorem of Cauchy–Kovalevskaya type with initial functions satisfying a differential equation of an arbitrary order k (with not necessarily analytic coefficients) is obtained.
Optimal Transportation Flows
Authors: Klötzler Rolf
Published: September 2013

This paper deals with a modification of L. C. Young’s flow concept in application to transportation flow problems. There will be proved necessary and sufficient conditions for an optimal transport and its dual deposit problem.
On the Fundamental Solution of the Operator of Dynamic Linear Thermodiffusion
Authors: Gawinecki Jerzy A. , Ortner Norbert , Wagner Peter
Published: September 2013

The fundamental matrix of the 5-by-5 system of partial differential operators describing linear thermodiffusion inside elastic media is – by a standard procedure – expressible through the fundamental solution of its determinant. This determinant is equal to the square of a wave operator multiplied by the so-called operator of dynamic linear thermodiffusion, which is of the fourth order with respect to the time variable. In this paper, we deduce, by means of a variant of Cagniard–de Hoop’s method, a representation of the fundamental solution of this operator by simple definite integrals. This formula allows the explicit computation of thermal and diffusion effects which result from instantaneous point forces or heat sources.
On a Spatial Generalization of the Complex $\Pi$-Operator
Authors: Gürlebeck Klaus , Kähler Uwe
Published: September 2013

The

\Pi

-operator plays a mayor role in complex analysis, especially in the theory of generalized analytic functions in the sense of Vekua. The present paper deals with a hyper-complex generalization of the complex

\Pi

-operator which turns out to have most of the useful properties of its complex origin such as mapping properties and invertibility. At the end an application of the generalized

\Pi

-operator to the solution of a hypercomplex Beltrami equation will be studied.
On Measures of Non-Compactness in Regular Spaces
Authors: Yerzakova N.A.
Published: September 2013

Previous results on non-compactness obtained in [11–13] are extended to regular spaces of measurable functions, and new criteria for the

\mu

-compactness of sets and operators are proved. An application of the abstract results to elliptic boundary problems is given as well.
Weighted Inequalities for the Fractional Maximal Operator and the Fractional Integral Operator
Authors: Rakotondratsimba Y.
Published: September 2013

A sufficient condition is given on weight functions

u

and

v

on

\mathbb R^n

for which the fractional maximal operator

M_s (0 ≤ s < n)

defined by

(M_sf)(x) = \mathrm {sup}_{Q \ni x} |Q|^{\frac{s}{n}–1} \int_Q | f (y) | dy

or the fractional integral operator

I_s (0 < s < n)

defined by

(I_s,f)(x) = \int_{\mathbb R^n} | x - y |^{s–n} f(y)dy

is bounded from

L^p (\mathbb R^n, vdx)

into

L^q(\mathbb R^n,udx)

for

0 < q < p

with

p> 1

, where

Q

is a cube and n a non-negative integer.
Boundary-Blow-Up Problems in a Fractal Domain
Authors: Matero J.
Published: September 2013

Assume that

\Omega

is a bounded domain in

\mathbb R^N

with

N ≥ 2

, which satisfies a uniform interior and exterior cone condition. We determine uniform a priori lower and upper bounds for the growth of solutions and their gradients, of the problem

\Delta u(x) = f(u(x)) (x \in \Omega)

with boundary blow-up, where

f(t) = e^t

or

f(t) = t^p

with

p \in (1,+\infty)

. The boundary estimates imply existence and uniqueness of a solution of the above problem. For

f(t) = t^p

with

p \in (1,+\infty)

the solution is positive. These results are used to construct a solution of the problem when

\Omega \subset \mathbb R^2

is the von Koch snowflake domain.
On Optimal Regularization Methods for the Backward Heat Equation
Authors: Tautenhahn Ulrich , Schröter Torsten
Published: September 2013

In this paper we consider different regularization methods for solving the heat equation

u_1 + Au = 0 (0 ≤ t < T)

backward in time, where

A : H \to H

is a linear (unbounded) operator in a Hilbert space

H

with norm

\| \cdot \|

and

z^{\delta}

are the available (noisy) data for

u(T)

with

\| z^{\delta} - u(T)\| ≤ \delta

. Assuming

\|u{0}\| ≤ E

we consider different regularized solutions

q^{\delta}_{\alpha} (t)

for

u(t)

and discuss the question how to choose the regularization parameter

\alpha = \alpha (\delta, E, t)

in order to obtain optimal estimates sup

\| q^{\delta}_{\alpha} (t) - u(t)\| ≤ E^{1 – \frac{t}{T} \delta \frac{t}{T}}

where the supremum is taken over

z^{\delta} \in H, \|u(0)\| ≤ E

and

\|z^{\delta} - u(T)\| ≤ \delta

.
Representation and Approximation of the Solution of an Initial Value Problem for a First Order Differential Equation in Banach Spaces
Authors: Gavrilyuk I.P. , Makarov V.L.
Published: September 2013

An initial value problem

x(0) = x_0

for the first order differential equation

x(t) + Ax(t) = g(t)

with an unbounded operator coefficient

A

in a Banach space is considered. Using the Cayley transform we give explicit formulas for the solution of this problem in case the operator

A

is strongly positive. On the basis of these formulas we propose numerical algorithms for the approximate solution of the initial value problem and give error estimates. The main property of these algorithms is the following: the accuracy of the approximate solutions depends automatically on the "smoothness" of the initial data (the initial vector ro and the right-hand side

g

).
Fourier Multipliers between Weighted Anisotropic Function Spaces. Part I: Besov Spaces
Authors: Dintelmann P.
Published: September 2013

We determine classes

M(B^{s_0}_{p_0, q_0}(w_0),B^{s_1}_{p_1, q_1} (w_1))

of Fourier multipliers between weighted anisotropic Besov spaces

B^{s_0}_{p_0, q_0}(w_0)

and

B^{s_1}_{p_1, q_1}(w_1)

where

p_0 ≤ 1

and

w_0, w_1

are weight functions of, polynomial growth. To this end we use a discrete characterization of the function spaces akin to the

\varphi

-transform of Frazier and Jawerth which leads to a unified approach to the multiplier problem. In this way widely generalized versions of known results of Bui, Johnson, Peetre and others are obtained from a single theorem.
Radial Symmetry for an Electrostatic, a Capillarity and some Fully Nonlinear Overdetermined Problems on Exterior Domains
Authors: Reichel Wolfgang
Published: September 2013

We consider two physically motivated problems: (1) Suppose the surface of a body in

\mathbb R^2

or

\mathbb R^3

is charged with a constant density. If the induced single-layer potential is constant inside the body, does it have to be a ball? (2) Suppose a straight solid cylinder of unknown cross-section is dipped into a large plain liquid reservoir. If the liquid rises to the same height on the cylinder wall, does the cylinder necessarily have circular cross-section? Both questions are answered with yes, and both problems are shown to be of the type

\mathrm {div} (g(|\triangledown u |)\triangledown u) + f(u, | \triangledown u|) = 0 \ \mathrm {in} \ \Omega, \ \ u = \mathrm {const}, \frac{\partial u}{\partial v} = \mathrm {const} \ \mathrm {on} \ \partial \Omega, \ u = 0 \ \mathrm {at} \ \infty

where

\partial_u f ≤ 0

and

\Omega = \mathbb R^N \ \bar{G}

is the connected exterior of the smooth bounded domain

G

. The overdetermined nature of this possibly degenerate boundary value problem forces

\Omega

to be radial. This is shown by a variant of the Alexandroff-Serrin method of moving hyperplanes, as recently developed for exterior domains by the author in [19]. The results extend to Monge-Ampere equations.
On the Convergence of the Goerisch Method for Self-Adjoint Eigenvalue Problems with Arbitrary Spectrum
Authors: Mertins U.
Published: September 2013

It was shown recently in [13] that the Goerisch method provides upper and lower bounds to eigenvalues of variationally posed self-adjoint eigenvalue problems with arbitrary spectrum. In the present paper the approximation of eigenelements is established. In addition, the convergence of the eigenvalue and eigenelement approximations is shown in a pure func-tional analytic procedure. A numerical example is given where the curve veering phenomenon occurs.
Second Order Sufficient Optimality Conditions for a Nonlinear Elliptic Boundary Control Problem
Authors: Casas Eduardo , Tröltzsch Fredi , Unger Andreas
Published: September 2013

In this paper sufficient second order optimality conditions are established for opti-mal control problems governed by a linear elliptic equation with nonlinear boundary condition, where pointwise constraints on the control are given. The second order condition requires coercivity of the Lagrange function on a suitable subspace together with first order sufficient conditions on a certain set of strongly active points.
A Generalization of the Weierstrass Theorem
Authors: Drwalewska A.
Published: September 2013

The well-known Weierstrass theorem stating that a real-valued continuous function

f

on a compact set

K \subset \mathbb R

attains its maximum on

K

is generalized. Namely, the space of real numbers is replaced by a set

Y

with arbitrary preference relation

p

(in place of the inequality ≤), and the assumption of continuity of

f

is replaced by its monotonic semicontinuity (with respect to the relation

p

).
On the Existence of Connecting Orbits
Authors: Shu-Xiang Yu
Published: September 2013

Two existence criteria of orbits connecting a pair of critical points of planar differ-ential equations are given.
Lifting Theorem as a Special Case of Abstract Interpolation Problem
Authors: Kupin Stanislas
Published: September 2013

Using properties of the de Branges-Rovnyak spaces we include the classical lifting problem into the general scheme of the abstract interpolation problem.
On the Green Function of the Landau Operator and its Properties Related to Point Interactions
Authors: Geyler V.A. , Demidov V.V.
Published: September 2013

The Green function

G

of the Schrödinger operator with a magnetic field (i.e. the Landau operator)

H

is studied. Two representations of

G

are used, namely in form of an integral and of a series. The space-variable asymptotics as well as the energetic ones are obtained. The analytical and asymptotical properties of

G

we obtain are related to point perturbations of

H

.
Heat Semi-Group and Function Spaces on Symmetric Spaces of Non-Compact Type
Authors: Skrzypczak Leszek
Published: September 2013

Besov-Triebel scales of function spaces defined on symmetric spaces of non-compact type are investigated. We prove an atomic decomposition theorem for the function spaces and give their characterization in terms of heat semigroup. In consequence we can describe the spectrum of the Laplace-Beltrami operator in these spaces and improve the generalized Riemann-Lebesgue lemma for the spherical Fourier transform.
On a Class of Multilinear Operator Equations
Authors: Janno Jaan , von Wolfersdorf Lothar
Published: September 2013

By means of contraction principle in a Banach space

E

with a scale of norms

\| \cdot \|_{\sigma} (\sigma ≥ 0)

existence, uniqueness and stability of solutions are proved for a general class of operator equations

u + G_0u + G_1u = g

including multilinear ones where

G_0, G_1 \in (E \to E)

are some operators. The theorems are applicable to equations with operators of generalized convolution type.
$\varepsilon k^0$-Subdifferentia1s of Convex Functions
Authors: Henkel E.-Ch.
Published: September 2013

The paper as a contribution to convex analysis in ordered linear topological spaces. For any convex function

f

from a Banach space

X

into a partially ordered one

Y

endowed with a convex cone

K

some properties of the

\varepsilon k^0

-subdifferential

\partial ^\leq_{\varepsilon k^0}f(x)

of

f

are examined. The non-emptyness of

\partial ^\leq_{\varepsilon k^0}f(x)

is proved, whenever

Y

is a normal order complete vector lattice and

f

belongs to the class of functions which are continuous and convex with respect to the cone

K

. For the real-valued case Bronsted and Rockafellar have proved that the set of subgradients of a lower semicontinuous function f on a Banach space

X

is dense in the set of

\varepsilon

-subgradients [21]. We deduce a similar result for a class of

\varepsilon k^0

-subdifferentials of functions which takes values in an ordered linear topological space

Y

.
A Survey on Huygens’ Principle
Authors: Belger M. , Schimming Rainer , Wünsch V.
Published: September 2013

There are two classes of physical wave phenomena: with (e.g. water waves) or without (e.g. flash or bang in space) after effects. The second class is said to obey Huygens’ principle. The mathematical formulation concerns Cauchy’s initial value problem to a given linear hyperbolic differential equation, and is generalized to arbitrary dimensions as well as to curved spacetimes, i.e. Lorentzian manifolds. The original conjecture that every Huygens-type equation is transformable to the wave equation in Minkowski spacetime was refuted by counter-examples found by K. L. Steilmacher and by P. Günther. Since then, many results accumulated, but a general characterization of the equations which satisfy Huygens’ principle is not yet known. Some classes of examples show interesting relations to other branches of physics or mathematics: the new higher spinor field equations of Buchdahl and Wünsch solve the long-standing inconsistency problem, Huygensian wave equations on symmetric spaces are treated by means of Lie-theoretical methods, far-reaching generalizations of the Stellmacher-Lagnese examples are related to Coxeter groups and to integrable dynamical systems. The present paper surveys the research on Huygens’ principle - from Hadamard up to recent results.
An Explicit Determination of the Non-self-adjoint Wave Equations that Satisfy Huygens’ Principle on Petrov Type III Background Space-times
Authors: Anderson W.G. , McLenaghan Raymond G. , Walton Tom G.
Published: September 2013

It is shown that the validity of Huygens’ principle for the non-self-adjoint wave equation on a Petrov type Ill space-time implies that the space-time is conformally related to one in which every repeated null vector field of the Weyl tensor is recurrent. It is further shown that, given a certain mild assumption imposed on the covariant derivative of the Weyl curvature spinor, there are no Petrov type III space-times on which the non-self-adjoint scalar wave equation satisfies Huygens’ Principle.
On Huygens’ Principle for the Hodge-de Rham Equations with Lorentzian Gauge
Authors: Wünsch V.
Published: September 2013

In an arbitrary curved space-time the Hodge-de Rham equations with Lorentzian gauge are studied. Using the spinor calculus and propositions on the curvature tensors, espe-cially on Hall’s canonical forms of Ricci tensors, some properties of the tail terms with respect to second order differential operators are proved. Finally, all Huygens’ operators are explicitly determined.
Some Remarks on Geodesic and Curvature Preserving Mappings
Authors: Belger M. , Beyer Klaus
Published: September 2013

We ask for the converse of Gauss’ theorema egregium. Because in general isocurved manifolds are not isometric we ask stronger for isocurved, geodesic equivalent manifolds. For these we give a local criterion from which there follows that two-dimensional manifolds

\overline{\mathcal M}^2

and of that type essentially are isometric, or both are Euclidean with an affine mapping in the ordinary sense.
Remark on the Normal Forms of Diversors of Second Order Differential Equations of Normal Hyperbolic Type
Authors: Burkhardt M.
Published: September 2013

With respect to the monograph of P. Günther "Huygens’ Principle and Hyperbolic Equations" this paper contains a supplement to diversors of second order differential equations of normal hyperbolic type [3: Chapter IV]. We construct a "normal form" of a diversor and consider the coefficients of this form in a certain neighbourhood of the characteristic backward conoid

C–(\xi)

of a point

C

.
On Affine Connections whose Holonomy is a Tensor Representation
Authors: Schwachhöfer L.
Published: September 2013

In 1955, Berger [4] gave a list of irreducible reductive representations which can occur as the holonomy of a torsion-free affine connection. While this list was stated to be complete in the case of metric connections, the situation in the general case remained unclear. The (non-metric) representations which are missing from this list are called exotic. In recent years, it has been determined that exotic holonomies do exist. Thus, Berger’s classification is yet to be completed in the non-metric case. In this paper, we investigate certain holonomy representations of reductive Lie groups whose semi-simple part is not simple.
On the Capillary Surface in a Wide Circular Tube
Authors: Miersemann Erich
Published: September 2013

An explicit asymptotic formula for the wide circular tube is proved. The leading term defines the capillary surface over the half plane with the given boundary contact angle. An explicit estimate of the remainder uniform with respect to the boundary contact angle and the radius of the tube is given. This uniform behaviour is caused by the strong non-linearity of the problem and has no counterpart for linear problems. The proof of the main result is based on a maximum principle and on a mapping which brings the right parameter of development onto the right place of the equation.
A Use of Ideal Decomposition in the Computer Algebra of Tensor Expressions
Authors: Fiedler Bernold
Published: September 2013

Let

I

be a left ideal of a group ring

\mathbb C[G]

of a finite group

G

, for which a decomposition

I = \oplus ^m_{k=1} I_k

into minimal left ideals

I_k

is given. We present an algorithm, which determines a decomposition of the left ideal

I \cdot a, a \in \mathbb C[G]

, into minimal left ideals and a corresponding set of primitive orthogonal idempotents by means of a computer. The algorithm is motivated by the computer algebra of tensor expressions. Several aspects of the connection between left ideals of the group ring

\mathbb C[S_r]

of a symmetric group

S_r

, their decomposition and the reduction of tensor expressions are discussed.
Distribution Approximations for Nonlinear Functionals of Weakly Correlated Random Processes
Authors: vom Scheidt J. , Mehlhose S. , Wunderlich R.
Published: September 2013

In this paper nonlinear functionals of weakly correlated processes with correlation length

\epsilon > 0

are investigated. Expansions of moments and distribution densities of nonlinear functionals with respect to

\epsilon

up to terms of order

o(\epsilon)

are considered. For the case of a single nonlinear functional a shorter proof than in [8] is given. The results are applied to cigenvalues of random matrices which are obtained by application of the Ritz method to random differential operators. Using the expansion formulas as to e approximations of the density functions of the matrix cigenvalues can be found. In addition to [7] not only first order approximations (exact up to terms of order

O(\epsilon)

) but also second order approximations (exact up to terms of order

o(\epsilon)

) are investigated. These approximations are compared with estimations from Monte-Carlo simulation.
Hardy Inequalities for Overdetermined Classes of Functions
Authors: Kufner Alois , Simader Christian G.
Published: September 2013

Conditions on weights

w_0

and

w_k

are given for the

k

-th order Hardy inequality

(\int^1_0 |u(t)|^q w_0(t)dt^{1/q} ≤ c(\int^1_0 |u^{(k)}(t)|^p w_k(t) dt)^{1/p}

to hold for two special classes of functions

u

satisfying

2k

and

k + 1

boundary conditions, respectively. The conditions are sufficient and partially also necessary. For one class, a hypothesis is formulated describing necessary and sufficient conditions on

w_0

and

w_k

.
A Uniform Attractor for a Non-Autonomous Generalized Navier–Stokes Equation
Authors: Gazzola Filippo , Pata Vittorino
Published: September 2013

A global existence and uniqueness result of a weak solution for a generalized non-autonomous Navier–Stokes equation is given, independently of the dimension

n ≥ 2

of the space. Furthermore, the family of processes associated to the equation is shown to-possess a weak uniform attractor with respect to a large class of non-autonomous forcing terms.
Hardy Spaces on Products of Halfplanes and Prediction of Homogeneous Random Fields
Authors: Cheng R. , Klotz L.
Published: September 2013

Using the fractional linear map from the unit disk onto the upper halfplane we transfer some facts about Hardy spaces on polydisks to Hardy spaces on products of halfplanes. The results are applied to the prediction theory of homogeneous random fields on

\mathbb R^n

. In particular, we give a characterization of regular and weakly (strongly) commutative

L^2(\mu)

spaces.
Convergence Results for Discrete Trigonometric Collocation Methods with Product Integration in Hölder-Zygmund Spaces
Authors: Schroderus L.
Published: September 2013

In this paper convergence results with respect to Hölder-Zygmund norms - including also maximum norm error estimates - are derived for the fully discrete trigonometric collocation method presented earlier by Saranen and Vainikko for solution of boundary integral equations on smooth closed curves. Approximation of the integral operator is based on product integration for which the explicit Fourier representation of the main part is not needed, and still the convergence of arbitrarily high rate for smooth solutions can be achieved. Saranen and Vainikko have given their analysis with respect to Sobolev norms yielding results that do not imply pointwise error estimates of optimal order. In this work the approach is based on the use of Hölder-Zygmund norms, and the optimal order maximum norm estimates are accomplished.
About Strongly Fejér Monotone Mappings and Their Relaxations
Authors: Schott Dieter
Published: September 2013

We consider the general class of strongly Fejér monotone mappings and some of their basic properties. These properties are useful for a convergence theory of corresponding iterative methods which are widely used to solve convex problems. Especially, we study the relation between these mappings and their relaxations
On the Local Property of Factored Fourier Series
Authors: Bor H.
Published: September 2013

In this paper two theorems on

|\bar{N}, p_n; \delta |_k

-summability methods have been proved. These theorems include some known results.
Nonlinear Geometric Optics for Shock Waves Part II: System Case
Authors: Wang Ya-Guang
Published: September 2013

In this paper we investigate the nonlinear geometric optics of a stable shock wave perturbed by high frequency oscillations for quasilinear hyperbolic conservation laws in one space variable. We obtain the existence of the oscillatory shock wave and its leading profiles, which are solutions to a boundary value problem of integro.differential systems. Furthermore, the asymptotic properties of the oscillatory shock wave as well as the shock front are justified.
A Semilinear Elliptic Equation with Dirac Measure as Right-Hand Side
Authors: Spielmann R.
Published: September 2013

We investigate solutions to the problem

\Delta u = \lambda eû + m \delta \ \ \ \mathrm {in} \ \mathcal D' (\Omega)

u=g \ \ \ \mathrm {a.e. on} \ \partial \Omega,

where

\delta

is the Dirac measure and

\lambda, m

are real parameters,

m > 0

. We discuss the existence and uniqueness of solutions in dependence of these parameters. For the homogeneous Dirichlet problem in a ball we give multiplicity results.
A Priori Estimates for the Solution of Convection-Diffusion Problems and Interpolation on Shishkin Meshes
Authors: Dobrowolski M. , Roos H.-G.
Published: September 2013

The solution of singularly perturbed convection-diffusion problems can be split into a regular and a singular part containing the boundary layer terms. In dimensions

n = 1

and

n = 2

, sharp estimates of the derivatives of both parts up to order 2 are given. The results are applied to estimate the interpolation error for the solution on Shishkin meshes for piecewise bilinear finite elements on rectangles and piecewise linear elements on triangles. Using the anisotropic interpolation theory it is proved that the interpolation problem on Shishkin meshes is quasi-optimal in

L_{\infty}

and in the energy norm.
An Example of Blowup for a Degenerate Parabolic Equation with a Nonlinear Boundary Condition
Authors: Chipot Michel , Filo Ján
Published: September 2013

In this paper, a nonlinear parabolic equation of the form

u_t = (a(u_x))_x

for

x \in (0, 1), t > 0, a(u_x) = |u_x|^{p–2}u_x

if

u_x ≥ \eta > 0, 1 < p < 2

, with nonlinear boundary condition

a(u_x r(1, t)) = |u|^{q–2} u(l,t)

is considered. It is proved that if

qp - 3p + 2 > 0

, then the solutions blow up in finite time. Moreover, estimates on the blowup profile (in

x

) and the blowup rate (in

t

) for

x = 1

are derived.
Hausdorff and Fractal Dimension Estimates for Invariant Sets of Non-Injective Maps
Authors: Boichenko V.A. , Franz A. , Leonov G.A. , Reitmann Volker
Published: September 2013

In this paper we are concerned with upper bounds for the Hausdorff and fractal dimensions of negatively invariant sets of maps on Riemannian manifolds. We consider a special class of non-injective maps, for which we introduce a factor describing the "degree of non-injectivity". This factor can be included in the Hausdorif dimension estimates of Douady-Oesterlé type [2, 7, 10] and in fractal dimension estimates [5, 13, 15] in order to weaken the condition to the singular values of the tangent map. In a number of cases we get better upper dimension estimates.
Strong Duality for Transportation Flow Problems
Authors: Klötzler Rolf
Published: September 2013

This paper is a supplement and correction to the author’s article "Optimal transportation flows" [2]. By new methods the existence of optimal transportation flows and the strong duality to deposit problems is proved.
Essential Properties of $L^{\infty}$-functions
Authors: Felgenhauer U. , Wagner M.
Published: September 2013

The paper deals with local characteristics of

L^{\infty}

-elements given as equivalence classes of measurable, essentially bounded functions

f: \mathbb R^m \to \mathbb R

. Besides of essential lower and upper limit functions we introduce a new set-valued map carrying the information on a class, the essential limit set at a point, and analyze their main properties. Criteria for qualifying the continuity of function representatives are appended. The results can be applied e.g. in control theory to intcrprete "almost everywhere" conditions.
Fixed Point Theory for Weakly Contractive Maps with Applications to Operator Inclusions in Banach Spaces Relative to the Weak Topology
Authors: O'Regan Donal
Published: September 2013

A variety of fixed point results is presented for weakly contractive multi-valued maps. Our theory will then be applied to establish general existence principles for operator inclusions in Banach spaces relative to the weak topology.
Degree Theory for Variational Inequalities in Complementary Systems
Authors: Khidirov Yu.E.
Published: September 2013

Working out degree theory for the investigation of finite-dimensional variational inequalities with continuous mappings in the usual way. All properties typical for a topologiocal degree are proved. Then the

K

-degree is generalized by the Galerkin procedure for some class

S_A(X)

of monotone-like operators in complementary systems. On the basis of our theory some new results concerning solvability of variational inequalities in complementary systems are proved. These results make it possible to obtain new facts on solvability of variational inequalities as well as operator equations with stongly nonlinear differential operators.
On the Solvability of Linear Differential Equations with Unbounded Operators in Banach Spaces
Authors: Barkova E.A. , Zabrejko P. P.
Published: September 2013

This article deals with some solvability results on the Cauchy problem for linear differential equations with unbounded operators. The main result consists in the description of the set of initial data for which the corresponding solutions are represented by means of the classical exponential formula in the stationary case, and by means of the Peano matriciant formula in the non-stationary case. In this connection a new generalization of Gelfand’s lemma about analytic vectors of the generator of a strongly continuous group is proved.
The 2-Dimensional Dirichiet Problem in an External Domain with Cuts
Authors: Krutitskii P.A.
Published: September 2013

The Dirichlet problem for the Laplace equation in an external connected plane region with cuts is studied. The existence of a classical solution is proved by potential theory. The problem is reduced to a Fredhoim equation of the second kind, which is uniquely solvable. Consequently, the solution can be computed by standard codes. The solvability of the Dirichlet problem in an internal domain with cuts is proved with the help of a conformal mapping.
On the Geometric Structure of Minimal Dilations on Hilbert $C*$-Modules
Authors: Popovici D.
Published: September 2013

We build a unitary extension for an isometry on a Hilbert $C*-module and then, with this extension help, we obtain the minimal unitary dilation for an adjointable contraction starting from one of-its isometric dilations. Having as a starting point a result of B. Sz.-Nagy and C. Foias regarding the geometric structure of the minimal unitary dilations for Hubert space contractions we prove that this structure maintains itself on Hilbert modules. Finally, we present a necessary and sufficient condition on the minimal isometric dilation in order to admits a Wold-type decomposition, condition which also assures the complementability of the residual part space of the minimal unitary dilation.
Spectral Asymptotics for Variational Fractals
Authors: Posta G.
Published: September 2013

In this paper a generalization of a classic result of H. Weyl concerning the asymptotics of the spectrum of the Laplace operator is proved for variational fractals. Physically we are studying the density of states for the diffusion trough a fractal media. A variational fractal is a couple

(K, \mathcal E)

where

K

is a self-similar fractal and

\mathcal E

is an energy form with some similarity properties connected with those of

K

. In this class we can find some of the most widely studied families of fractals as nested fractals, p.c.f. fractals, the Sierpinski carpet etc., as well as some regular self-similar Euclidean domains. We will see that if

r(x)

is the number of eigenvalues associated with

\mathcal E

smaller than

x

, then

r(x) ~ x^{\nu / 2}

, where

\nu

is the intrinsic dimension of

(K, \mathcal E)

.
An Analysis of the Block Structure of Certain Subclasses of $j_{qq}$-Inner Functions
Authors: Fritzsche B. , Kirstein B. , Müller K.
Published: September 2013

This paper is aimed at characterizing distinguished subclasses of

j_{qq}

-inner functions in terms of connected pairs of matrix-valued Hardy class functions. Moreover, an inverse problem for a given pair of matrix-valued Carathéodory functions is discussed.
The Hausdorff Nearest Circle to a Convex Compact Set in the Plane
Authors: Ginchev I. , Hoffmann Armin
Published: September 2013

The problem of finding the nearest in the Hausdorff metric circle to a non-empty convex compact set

T

in the plane is considered from geometrical point of view. The consideration is based on the equivalence of this problem with the Chebyshevian best approximation of

2\pi

-periodic functions by trigonometric polynomials of first order, whence it follows that the Hausdorif nearest circle to a convex compact set in the plane exists and is unique. It can be characterized by a geometric Chebyshevian alternance. As a consequence, in the particular case of a polygon the centre of the circle is described as an intersection of a midline between some two vertices and a bisectrix of some two sides. In the general case, geometrical algorithms corresponding to the one and the four point exchange Remez algorithms are described. They assure correspondingly linear and superlinear convergence. Following the idea, in the case of a polygon to get the exact solution in finite number of steps, a modified two-point exchange algorithm is suggested and illustrated by a numerical example. An application is given to estimate the Hausdorif distance between an arbitrary convex set and its Hausdorif nearest circle. The considered problem arises as a practical problem by measuring and pattern recognition in the production of circular machine parts.
Impulsive Controllability and Optimization Problems Lagrange’s Method and Applications
Authors: Nenov Svetoslav I.
Published: September 2013

the present paper, a class of optimization problems to impulsive control of smooth dynamical processes is investigated. Necessary and sufficiently conditions for existence of optimal impulsive controllability of the initial-value problem for a dynamical system are obtained. The derived results are applied to the analysis of some classical problems from population dynamics.
Oscillation Theorems for Certain Class of Nonlinear Difference Equations
Authors: Thandapani E. , Ramuppillai L.
Published: September 2013

Some new oscillation results for certain class of forced nonlinear difference equations of the form

\Delta (a_n + p_n x_{n–k})) + q_n f(x_{n+1–l}) = e_n \ \ \ (n \in \mathbb N_0)

are established. Examples which dwell upon the importance of the results are also given.