Periodic solutions of the differential delay equation\dot x(t) = - f(x(t - 1))

For an odd function 1/2(x) defined only on a finite interval, this paper deals with the existence of periodic solutions and the number of simple periodic solutions of the differential delay equation (DDE) . By use of the method of qualitative analysis combined with the constructing of special solutions a series of interesting results are obtained on these problems.Supported by the National Natural Science Foundation of China.     

Existence of exactlyn+1 simple 4-periodic solutions of the differential delay equation\dot x(t) = - f(x(t - 1))

By discussing the zeros of periodic solutions we give in this paper a criterion for the existence of exactlyn+1 simple 4-periodic solutions of the differential delay equation Supported by the Chinese National Foundation for Natural Sciences.     

Removable singular sets for equations of the form\sum {\tfrac{\partial }{{\partial x_i }}a_{ij} (x)\tfrac{{\partial u}}{{\partial x_j }} = f(x,u,\nabla u)}

The following uniformly elliptic equation is considered: $$\sum {\tfrac{\partial }{{\partial x_i }}a_{ij} (x)\tfrac{{\partial u}}{{\partial x_j }} = f(x,u,\nabla u)} , x \in \Omega \subset R^n ,$$ with measurable coefficients. The function f satisfies the condition $$f(x, u, \nabla u) u \geqslant C|u|^{\beta _1 + 1} |\nabla u|^{\beta _1 } , \beta _1 > 0, 0 \leqslant \beta _2 \leqslant 2, \beta _1 + \beta _2 > 1$$ . It is proved that if u(x) is a generalized (in the sense of integral identity) solution in the domain ΩK, where the compactum K has Hausdorff dimension α, and if \(\frac{{2\beta _1 + \beta _2 }}{{\beta _1 + \beta _2 - 1}}< n - \alpha \) , u(x) will be a generalized solution in the domain ω. Moreover, the sufficient removability conditions for the singular set are, in some sense, close to the necessary conditions.     

Limit problems for a Fractional p-La placian as $${ p\to\infty}$$

The purpose of this work is the analysis of the solutions to the following problems related to the fractional p-Laplacian in a Lipschitzian bounded domain \({\Omega \subset \mathbb{R}^N}\),

$$\left\{\begin{array}{lll}-\int_{\mathbb{R}^N}\frac{|u(y)-u(x)|^{p-2}(u(y)-u(x))}{|x-y|^{\alpha p}}\;dy=f(x,u)\;\;&x\in \Omega,\\ u=g(x) &x\in\mathbb{R}^N\setminus \Omega,\end{array}\right.$$

where \({\alpha\in(0,1)}\) and the exponent p goes to infinity. In particular we will analyze the cases:

1. (i)
   \({f=f(x).}\)
    
2. (ii)
   \({f=f(u)=|u|^{\theta(p)-1} u \, {\rm with} \, 0 < \theta(p) < p -1 \, {\rm and} \, \lim_{p\to\infty}\frac{\theta(p)}{p-1}=\Theta < 1 \, {\rm with} \, g \geq 0.}\)
    

We show the convergence of the solutions to certain limit as \({p\to\infty}\) and identify the limit equation. In both cases, the limit problem is closely related to the Infinity Fractional Laplacian:

$$\mathcal{L}_\infty v(x)=\mathcal{L}_\infty^+ v(x)+\mathcal{L}_\infty^- v(x),$$

where

$$\mathcal{L}_\infty^+ v(x)=\sup_{y\in\mathbb{R}^N}\frac{v(y)-v(x)}{|y-x|^\alpha}, \quad \mathcal{L}_\infty^- v(x)=\inf_{y\in\mathbb{R}^N}\frac{v(y)-v(x)}{|y-x|^\alpha}.$$

     

Relaxation of an area-like functional for the function $$\frac{x}{|x|}$$

We compute the relaxation

Forbidden (0,1)-vectors in Hyperplanes of $$\mathbb{R}^{n}$$: The unrestricted case

In this paper, we continue our investigation on “Extremal problems under dimension constraints” introduced [1]. The general problem we deal with in this paper can be formulated as follows. Let be an affine plane of dimension k in . Given determine or estimate .Here we consider and solve the problem in the special case where is a hyperplane in and the “forbidden set” . The same problem is considered for the case, where is a hyperplane passing through the origin, which surprisingly turns out to be more difficult. For this case we have only partial results.AMS Classification: 05C35, 05B30, 52C99     

Beckman-Quarles type theorems for mappings from $$ \mathbb{R}^n $$ to $$ \mathbb{C}^n $$

Summary. Let We say that preserves the distance d 0 if for each implies Let A _n denote the set of all positive numbers d such that any map that preserves unit distance preserves also distance d. Let D _n denote the set of all positive numbers d with the property: if and then there exists a finite set S _xy with such that any map that preserves unit distance preserves also the distance between x and y. Obviously, We prove: (1) (2) for n 2 D _n is a dense subset of (2) implies that each mapping f from to (n 2) preserving unit distance preserves all distances, if f is continuous with respect to the product topologies on and      

White Noise Approach Construction of Ф4 4 Quantum to The Fields （Ⅱ）

In this paper a complete proof for the existence of generalized operators satisfying abstract 02 dynamical equations of quantum motions δ^2/δt^2Ф（t, x） ＋ （ △- m^2）Ф（t, x） = -λ ：Ф^3（t, x）, subject to a suitable initial condition, is given under the framework of white noise analysis. Also some important commutation relations related to Ф44 quantum fields are discussed and proved in detail.     

On the Diophantine system $$f(z)\,{=}\,f(x)f(y)\,{=}\,f(u) f(v)$$

We show that the Diophantine system

$$\begin{aligned} f(z)=f(x)f(y)=f(u)f(v) \end{aligned}$$

has infinitely many nontrivial positive integer solutions for \(f(X)=X^2-1\), and infinitely many nontrivial rational solutions for \(f(X)=X^2+b\) with nonzero integer b.

     

The stability of RDDE\ddot x(t) + p_1 (t)\dot x(t) + q_1 (t)x(t) + p_2 (t)\dot x(t - r(t)) + q_2 (t)x(t - r(t)) = O

In this paper, the author considered the stability of zero solution of linear RDDE $$\begin{gathered} \ddot x(t) + p_1 (t)\dot x(t) + q_1 (t)x(t) + p_2 (t)\dot x(t - r(t)) + q_2 (t)x(t - r(t)) = O, \hfill \\ \ddot x(t) + p_1 (t)\dot x(t) + q_1 (t)x(t) + p_2 (t)\dot x(t - r(t)) = O \hfill \\ \end{gathered} $$ using Liapunov-Razumikhin functional and transformations and obtained some sufficient conditions for the stability of Eqs.(1) and (2). These results are suitable both for boundedp _i (t),q _i (t) andr(t).i=1,2.     

On the existence of periodic solutions for the equation\ddot x + f(x)\dot x + g(x) = 0

We establish in this work sufficient conditions for the existence of periodic solutions for the Liénard equation .     

Fibred multilinks and singularities $${f \overline g}$$

In this article we extend Milnor’s fibration theorem to the case of functions of the form with f, g holomorphic, defined on a complex analytic (possibly singular) germ (X, 0). We further refine this fibration theorem by looking not only at the link of , but also at its multi-link structure, which is more subtle. We mostly focus on the case when X has complex dimension two. Our main result (Theorem 4.4) gives in this case the equivalence of the following three statements:

A class of multipliers for $$\mathcal{W}^ \bot $$

Let denote the class of ergodic probability preserving transformations which are disjoint from every weakly mixing system. Let be the class of multipliers for , i.e. ergodic transformations whose all ergodic joinings with any element of are also in . Fix an ergodic rotationT, a mildly mixing actionS of a locally compact second countable groupG and an ergodic cocycle ϕ forT with values inG. The main result of the paper is a sufficient (and also necessary by [LeP] whenG is countable Abelian andS is Bernoullian) condition for the skew product build fromT, ϕ andS to be an element of . Moreover, the self-joinings of such extensions ofT are described with an application to study semisimple extensions of rotations. Dedicated to Hillel Furstenberg on the occasion of his retirement The first-named author was supported in part by CRDF, grant UM1-2546-KH-03. The second-named author was supported in part by KBN grant 1P03A 03826.     

Pointwise estimates for relative fundamental solutions of heat equations in $${\mathbb{R} \times \mathbb{C}}$$

Let be a subharmonic, nonharmonic polynomial and a parameter. Define , a closed, densely defined operator on . If and , we solve the heat equations , u(0,z) = f(z) and , . We write the solutions via heat semigroups and show that the solutions can be written as integrals against distributional kernels. We prove that the kernels are C ^∞ off of the diagonal {(s, z, w) : s = 0 and z = w} and find pointwise bounds for the kernels and their derivatives.      

Tauberian conditions for the $$(C, \alpha )$$ integrability of functions

For a real-valued continuous function f(x) on \([0,\infty )\), we define

$$\begin{aligned} s(x)=\int _{0}^{x} f(u)du\quad \text {and}\quad \sigma _{\alpha } (x)= \int _{0}^{x}\left( 1-\frac{u}{x}\right) ^{\alpha }f(u)du \end{aligned}$$

for \(x>0\). We say that \(\int _{0}^{\infty } f(u)du\) is \((C, \alpha )\) integrable to L for some \(\alpha >-1\) if the limit \(\lim _{x \rightarrow \infty } \sigma _{\alpha } (x)=L\) exists. It is known that \(\lim _{x \rightarrow \infty } s(x) =L\) implies \(\lim _{x \rightarrow \infty }\sigma _{\alpha } (x) =L\) for all \(\alpha >-1\). The aim of this paper is twofold. First, we introduce some new Tauberian conditions for the \((C, \alpha )\) integrability method under which the converse implication is satisfied, and improve classical Tauberian theorems for the \((C,\alpha )\) integrability method. Next we give short proofs of some classical Tauberian theorems as special cases of some of our results.

     

On the Cauchy Problem for $$\mathop {2b}\limits^ \to$$ -Parabolic Systems with Growing Coefficients

For -parabolic dissipative systems and systems with growing coefficients as |x| in the presence of degeneracies in the initial hyperplane, we investigate the fundamental matrix of solutions and the solvability of the Cauchy problem.     

Markov subshifts and partial representation of $$\mathbb{F}_{\text n}$$

In this paper we fix a set * of positive elements of the free group (e. g. the set of finite words occurring in a Markov subshift) as well as n partial isometries on a Hilbert space H. Based on these we define a map S : which we prove to be a partial representation of on H under certain conditions studied by Matsumoto.*Supported by Capes.     

The Abelian category of quotients of $$\mathcal{L}\mathfrak{F}$$ -spaces

We construct the category of quotients of -spaces and we show that it is Abelian. This answers a question of L. Waelbroeck from 1990.     

On the functional equation $${\varvec{f}}({\varvec{x}})+{\varvec{f}}({\varvec{y}})=\mathbf{max} \{{\varvec{f}}({\varvec{xy}}),{\varvec{f}}({\varvec{xy}}^{-{\varvec{1}}})\}$$ on groups

We analyse the functional equation

$$\begin{aligned} f(x)+f(y)=\max \{f(xy),f(xy^{-1})\} \end{aligned}$$

for a function \(f:G\rightarrow \mathbb R\) where G is a group. Without further assumption it characterises the absolute value of additive functions. In addition \(\{z\in G\mid f(z)=0\}\) is a normal subgroup of G with abelian factor group.

     

\mathcal{P}\mathcal{A}{\text{-isomorphisms}} of inverse semigroups

A partial automorphism of a semigroup S is any isomorphism between its subsemigroups, and the set all partial automorphisms of S with respect to composition is an inverse monoid called the partial automorphism monoid of S. Two semigroups are said to be if their partial automorphism monoids are isomorphic. A class of semigroups is called if it contains every semigroup to some semigroup from Although the class of all inverse semigroups is not we prove that the class of inverse semigroups, in which no maximal isolated subgroup is a direct product of an involution-free periodic group and the two-element cyclic group, is It follows that the class of all combinatorial inverse semigroups (those with no nontrivial subgroups) is A semigroup is called if it is isomorphic or antiisomorphic to any semigroup that is to it. We show that combinatorial inverse semigroups which are either shortly connected [5] or quasi-archimedean [10] are To Ralph McKenzieReceived April 15, 2004; accepted in final form October 7, 2004.