A fundamental question raised by M. Kac in 1966 is: Must two isospectral planar domains necessarily be isometric? Following a long history of investigation C. Gordon, D. L. Webb and S. Wolpert in 1992 finally proved that the answer is no. By using the idea of transposition maps one can construct a wide class of planar domains with piecewise continuous boundaries which are isospectral but non-isometric. In this note we study the Kac question in relation to domains with fractal boundaries and by following...

In recent ten years, there has been much concentration and increased research activities on Hamilton’s Ricci flow evolving on a Riemannian metric and Perelman’s functional. In this paper, we extend Perelman’s functional approach to include logarithmic curvature corrections induced by quantum effects. Many interesting consequences are revealed.

In this paper we consider a chain of strings with fixed end points coupled with nearest neighbour interaction potential of exponential type, i.e.$$\left\{\begin{array}{cc}{\varphi }_{tt}^i-{\varphi }_{xx}^i=exp({\varphi }^{i+1}-{\varphi }^i)-exp({\varphi }^i-\varphi i-1)\hfill & 0\<x\<\pi ,t\in ℝ,i\in \mathbb{Z}\left(TC\right)\hfill \\ {\varphi }^i(0,t)={\varphi }^i(\pi ,t)=0\hfill & \forall t,i.\hfill \end{array}\right.$$We consider the case of “closed chains” i.e. ${\varphi }^{i+N}={\varphi }^i\forall i\in \mathbb{Z}$ and some $N\in \mathbb{N}$ and look for solutions which are peirodic in time. The existence of periodic solutions for the dual problem is proved in Orlicz space setting.

In this paper we consider a chain of strings with fixed end points coupled with nearest neighbour interaction potential of exponential type, i.e.$$\left\{\begin{array}{c}{\varphi }_{tt}^i-{\varphi }_{xx}^i=exp({\varphi }^{i+1}-{\varphi }^i)-exp({\varphi }^i-\varphi i-1)0<x<\pi ,t\in \mathrm{I}\mathrm{R},\mathrm{i}\in \mathrm{Z}\mathrm{Z}\left(\mathrm{TC}\right){\varphi }^{\mathrm{i}}(0,\mathrm{t})={\varphi }^{\mathrm{i}}(\pi ,\mathrm{t})=0\forall \mathrm{t},\mathrm{i}.\hfill \end{array}\right.$$ We consider the case of “closed chains" i.e.${\varphi }^{i+N}={\varphi }^i\forall i\in ZZ$ and some $N\in IN$ and look for solutions which are peirodic in time. The existence of periodic solutions for the dual problem is proved in Orlicz space setting.

The purpose of this paper is to study the existence of periodic solutions for the non-autonomous second order Hamiltonian system $$\left\{\begin{array}{c}\ddot{u}\left(t\right)=\nabla F(t,u\left(t\right)),\text{a.e.}t\in [0,T],\hfill \\ u\left(0\right)-u\left(T\right)=\dot{u}\left(0\right)-\dot{u}\left(T\right)=0.\hfill \end{array}\right.$$ Some new existence theorems are obtained by the least action principle.

In this note we provide a probabilistic proof that Poisson and/or Dirichlet problems in an ellipsoid in Rd, that have polynomial data, also have polynomial solutions. Our proofs use basic stochastic calculus. The existing proofs are based on famous lemma by E. Fisher which we do not use, and present a simple martingale proof of it as well.

Let $𝒫{ℬ}_m$ be the category of all principal fibred bundles with $m$-dimensional bases and their principal bundle homomorphisms covering embeddings. We introduce the concept of the so called $(r,m)$-systems and describe all gauge bundle functors on $𝒫{ℬ}_m$ of order $r$ by means of the $(r,m)$-systems. Next we present several interesting examples of fiber product preserving gauge bundle functors on $𝒫{ℬ}_m$ of order $r$. Finally, we introduce the concept of product preserving $(r,m)$-systems and describe all fiber product preserving gauge...

Let Y → M be a fibred manifold with m-dimensional base and n-dimensional fibres. Let r, m,n be positive integers. We present a construction $B^r$ of rth order holonomic connections $B^r(\Gamma ,\nabla ):Y\to J^{r}Y$ on Y → M from general connections Γ:Y → J¹Y on Y → M by means of torsion free classical linear connections ∇ on M. Then we prove that any construction B of rth order holonomic connections $B(\Gamma ,\nabla ):Y\to J^{r}Y$ on Y → M from general connections Γ:Y → J¹Y on Y → M by means of torsion free classical linear connections ∇ on M is equal to $B^r$. Applying...