The purpose of this paper is to establish a connection between various objects such as dynamical $r$-matrices, Lie bialgebroids, and Lagrangian subalgebras. Our method relies on the theory of Dirac structures and Courant algebroids. In particular, we give a new method of classifying dynamical $r$-matrices of simple Lie algebras $𝔤$, and prove that dynamical $r$-matrices are in one-one correspondence with certain Lagrangian subalgebras of $𝔤\oplus 𝔤$.

The paper deals with locally connected continua $X$ in the Euclidean plane. Theorem 1 asserts that there exists a simple closed curve in $X$ that separates two given points $x$, $y$ of $X$ if there is a subset $L$ of $X$ (a point or an arc) with this property. In Theorem 2 the two points $x$, $y$ are replaced by two closed and connected disjoint subsets $A$, $B$. Again – under some additional preconditions – the existence of a simple closed curve disconnecting $A$ and $B$ is stated.

The Leray-Schauder degree is extended to certain multi-valued mappings on separable Hilbert spaces with applications to the existence of weak periodic solutions of discontinuous semilinear wave equations with fixed ends.

We obtain another proof of a Gaussian upper estimate for a gradient of the heat kernel on cofinite covering graphs whose covering transformation group has a polynomial volume growth. It is proved by using the temporal regularity of the discrete heat kernel obtained by Blunck [2] and Christ [3] along with the arguments of Dungey [7] on covering manifolds.

We give sufficient conditions for the discreteness of the spectrum of differential operators of the form $Au=-\mathrm{△}u+\left(\nabla F,\nabla u\right)$ in $L_{\mu }^2\left({\mathbb{R}}^n\right)$ where $d\mu \left(x\right)=e^{-F\left(x\right)}dx$ and for Schrödinger operators in $L^2\left({\mathbb{R}}^n\right)$. Our conditions are also necessary in the case of polynomial coefficients.

Nous donnons ici deux résultats sur le déterminant $\zeta$-régularisé ${det}_{\zeta }A$ d’un opérateur de Schrödinger $A={\Delta }_g+V$ sur une variété compacte $ℳ$. Nous construisons, pour $ℳ=S^1\times S^1$, une suite $(G_n,{\rho }_n,{\Delta }_n)$ où $G_n$ est un graphe fini qui se plonge dans $ℳ$ via ${\rho }_n$ de telle manière que ${\rho }_n\left(G_n\right)$ soit une triangulation de $ℳ$ et où ${\Delta }_n$ est un laplacien discret sur $G_n$ tel que pour tout potentiel $V$ sur $ℳ$, la suite de réels $det({\Delta }_n+V)$ converge après renormalisation vers ${det}_{\zeta }({\Delta }_g+V)$. Enfin, nous donnons sur toute variété riemannienne compacte $(ℳ,g)$ de dimension inférieure ou égale à $3$...

In the framework of jet spaces endowed with a non-linear connection, the special curves of these spaces (h-paths, v-paths, stationary curves and geodesics) which extend the corresponding notions from Riemannian geometry are characterized. The main geometric objects and the paths are described and, in the case when the vertical metric is independent of fiber coordinates, the first two variations of energy and the extended Jacobi field equations are derived.