In this paper we prove the equivalence of two conjectures on linear systems through fat points on a generic $K3$ surface. The first conjecture is exactly as Segre conjecture on the projective plane. Whereas the second characterizes such linear system and can be compared to the Gimigliano-Harbourne-Hirschowitz conjecture.

We construct a BN-pair for subgroups of Chevalley groups, which are fixed under some distinguished automorphisms. This generalizes the standard approach for the twisted groups of type $^2A_\ell$, $^2D_\ell$ and $^2E_6$ to groups which are not necessarily quasi-split.

This paper deals with a spectral problem for the Laplacian stemming from a parabolic problem in a bounded domain under a dynamical boundary condition. As a distinctive feature the eigenvalue parameter appears here also in the boundary condition: $$ \begin{cases} \,-\Delta u=\lambda u&\text{ in }\Omega,\\ \,\partial_\nu u=\lambda\sigma u&\text{ on }\partial\Omega. \end{cases} $$ By variational techniques the resulting eigenvalue sequence can be compared with the spectra under Dirichlet or Neumann boundary conditions and with the spectrum of the Steklov problem in order to get upper bounds for the spectral growth. For continuous positive $\sigma$, the growth order is determined and upper and lower bounds for the leading asymptotic coefficient are obtained. Moreover, the exact asymptotic behavior of the eigenvalue sequence is determined in the one--dimensional case.

Let $G$ be a compact connected semisimple Lie group, $G^{\mathbb{C}}$ its complexification and let $\ P$ be a parabolic subgroup of $G^{C}$. Let $ P=L.R_{u}(P)$ be the Levi decomposition of $P$, where $L$ is the Levi component of $P$ and $R_{u}(P)$ is the unipotent part of $P$. The group $L$ acts by the adjoint representation on the successive quotients of the central series \begin{equation*} \mathfrak{u}(\mathfrak{p})\,=\,\mathfrak{u}^{(0)}(\mathfrak{p})\,\supset \, \mathfrak{u}^{(1)}(\mathfrak{p})\,\,\supset \,\cdots \,\supset \,\mathfrak{u} ^{(i)}(\mathfrak{p})\,\supset \,\cdots \,\supset \,\mathfrak{u}^{(r-1)}( \mathfrak{p})\supset \mathfrak{u}^{(r)}(\mathfrak{p})\,=\,0\,, \end{equation*}% where $\mathfrak{u}(\mathfrak{p})$ is the Lie algebra of $R_{u}(P)$. We determine for each $0\leq i\leq r-1$ the irreducible components $ V_{i}^{(n_{1},\text{ }...,\text{ }n_{\nu })}$ of the adjoint action of $L$ on $\mathfrak{u}^{(i)}(\mathfrak{p})/\mathfrak{u}^{(i+1)}(\mathfrak{p})$.

We show that binary circular words of length $n$ avoiding $7/3^+$ powers exist for every sufficiently large $n$. This is not the case for binary circular words avoiding $k^+$ powers with $k<7/3$.

In our earlier paper [{\bf 10}], for a foundation $*-$semigroup $S$ with an identity and with a Borel-measurable weight function $w\leq 1$, we proved that on the unit ball of ${\cal P} (S,w)$, the cone of $w-$bounded continuous positive definite functions on $S$, the weak topology coincides with the compact open topology. In the present paper, through some $C^*-$algebras techniques, we shall extend this result to the unit ball of the Fourier-Stieltjes algebra ${\cal F} (S,w)$ of a foundation semigroup $S$ with a Borel measurable weight function $w$. Indeed, we shall establish our conjecture in [{\bf 10}] even in the more general setting of the Fourier-Stieltjes algebra ${\cal F}(S,w)$ for any Borel measurable weight function $w$. It should be noted that the family of foundation semigroups is quite extensive, for which locally compact groups and discrete semigroups are elementary examples. For further examples we refer to Appendix B of [{\bf 13}].

We show that on any locally conformal Kähler (l.c.K.) manifold $(M,J,g)$ with parallel Lee form the unit anti-Lee vector field is harmonic and minimal and determines a harmonic map into the unit tangent bundle. Moreover, the canonical distribution locally generated by the Lee and anti-Lee vector fields is also harmonic and minimal when seen as a map from $(M,g)$ with values in the Grassmannian $G^{or}_2(M)$ endowed with the Sasaki metric. As a particular case of l.c.K. manifolds, we look at locally conformal hyperkähler manifolds and show that, if the Lee form is parallel (that is always the case if the manifold is compact), the naturally associated three- and four-dimensional distributions are harmonic and minimal when regarded as maps with values in appropriate Grassmannians. As for l.c.K. manifolds without parallel Lee form, we consider the Tricerri metric on an Inoue surface and prove that the unit Lee and anti-Lee vector fields are harmonic and minimal and the canonical distribution is critical for the energy functional, but when seen as a map with values in $G^{or}_2(M)$ it is minimal, but not harmonic.

Convolution structure for Jacobi series allows end point summability of Fourier-Jacobi expansions to lead an approximation of function by a linear combination of Jacobi polynomials. Thus, using Ces$\grave a$ro summability of some orders $>1$ at $x=1,$ we prove a result of approximation of functions on $[-1,1]$ by operators involving Jacobi polynomials. Precisely, we pick up functions from a Lebesgue integrable space and then study its representation by Jacobi polynomials under different conditions.

We discuss a formula that the dimension of the rational cohomology gives a lower bound for the L-S category of a rationally elliptic space.

The purpose of this paper is to prove an analogue of A. Weil's explicit formula for a fundamental class of functions, i.e. the class of meromorphic functions that have an Euler sum representation and satisfy certain a functional equation. The advance of this explicit formula is that it enlarges the class of allowed test functions, from the class of functions with bounded Jordan variation to the class of functions of $\phi $-bounded variation. A condition posed to the test function at zero is also reconsidered.

We use a multidimensional extension of Bailey's transform to derive two very general $q$-generating functions, which are $q$-analogues of a paper by Exton. These expressions are then specialized to give more practical formulae, which are $q$-analogues of generating relations for Karlssons generalized Kampé de Fériet function. A number of examples are given including $q$-Laguerre polynomials of two variables.

Lower and upper bounds for the size of the smallest maximal partial $t$-spreads in $\mathrm{PG}(n,q)$ are presented. In some cases these bounds are sharp.

An abelian surface $A$ over a field $K$ has potential quaternionic multiplication if the ring $\End _{\bar K}(A)$ of geometric endomorphisms of $A$ is an order in an indefinite rational division quaternion algebra. In this brief note, we study the possible structures of the ring of endomorphisms of these surfaces and we provide explicit examples of Jacobians of curves of genus two which show that our result is sharp.

In this paper, we consider a tiling generated by a Pisot unit number of degree $d \geq 3$ which has a finite expansible property. We compute the states of a finite automaton which recognizes the boundary of the central tile. We also prove in the case $d=3$ that the interior of each tile is simply connected.