Let $V_k$ be the Weyl module of dimension ${2n\choose k}-{2n\choose k-2}$ for the group $G = \mathrm{Sp}(2n,\mathbb{F})$ arising from the $k$-th fundamental weight of the Lie algebra of $G$. Thus, $V_k$ affords the grassmann embedding of the $k$-th symplectic polar grassmannian of the building associated to $G$. When $\mathrm{char}(\mathbb{F}) = p > 0$ and $n$ is sufficiently large compared with the difference $n-k$, the $G$-module $V_k$ is reducible. In this paper we are mainly interested in the first appearance of reducibility for a given $h := n-k$. It is known that, for given $h$ and $p$, there exists an integer $n(h,p)$ such that $V_k$ is reducible if and only if $n \geq n(h,p)$. Moreover, let $n \geq n(h,p)$ and $R_k$ the largest proper non-trivial submodule of $V_k$. Then $\mathrm{dim}(R_k) = 1$ if $n = n(h,p)$ while $\mathrm{dim}(R_k) > 1$ if $n > n(h,p)$. In this paper we will show how this result can be obtained by an investigation of a certain chain of $G$-submodules of the exterior power $W_k := \wedge^kV$, where $V = V(2n,\mathbb{F})$.

We introduce the notion of generalized invertibility, a weak notion of invertibility that makes sense in the nonstandard hull of a Banach algebra. We show that a standard algebra is the closure of its invertible elements exactly when all the elements of its nonstandard hull are generalized invertible. For commutative algebras, we relate the problem of liftability of homomorphisms to the property of approximately multiplicative functionals being near to multiplicative functionals (AMNM property) and we prove a nonstandard characterization of the latter property. Finally we show that a standard superreflexive Banach algebra has the the AMNM property exactly when its nonstandard hull does.

A subspace arrangement $\mathcal{A}$ in $\mathbb{C}^m$ is a finite set $\{x_0, \dots, x_n \}$ of vector subspaces. The complement space $M(\mathcal{A})$ is $\mathbb{C}^m \setminus \bigcup_{x \in \mathcal{A}} x$. When each subspace is an hyperplane, it is also known as an arrangement of hyperplanes. In that case, it is known that the Poincaré polynomials of $M(\mathcal{A})$ is connected to the Poincaré polynomials of the complements of the deleted arrangement $\mathcal{A}' = \mathcal{A} \setminus \{x_0\}$ and of the restricted arrangement $\mathcal{A}'' = \{ x_0 \cap y \st y \in \mathcal{A}' \}$ by the nice formula \[ Poin(M(\mathcal{A}),t) = Poin(M(\mathcal{A}'),t) + tPoin(M(\mathcal{A}''),t). \] In this paper, we prove that for a subspace arrangement, there is a long exact sequence in cohomology which connects $M(\mathcal{A})$ to $M(\mathcal{A}')$ and $M(\mathcal{A}'')$. Using it, we can extend the above formula to arrangements with a geometric lattice, and to some other specific arrangements.

In this paper we transfer results on embeddings of complex algebraic curves into projective space (which are based on dimension theorems for special divisors) to the case of real algebraic curves with real points.

Let $H(\mathbb B)$ denote the space of all holomorphic functions on the unit ball $\mathbb B$ of $\mathbb C^n$ and $\Re h(z)=\sum_{j=1}^nz_j\frac{\pt h}{\pt z_j}(z)$ the radial derivative of $h.$ Motivated by recent results by S. Li and S. Stević , in this paper we study the boundedness and compactness of the following integral operator $$ L_gf(z)= \int_0^1 \Re f(tz) g(tz)\frac{dt}{t},\quad z\in \mathbb B, $$ between the Hardy space $H^2$ and weighted Bergman spaces.

The main purpose of this paper is to study the mean value properties of a sum analogues to character sums by using the estimate for character sums and elementary methods, and finally give two asymptotic formulae for its fourth power mean over short intervals.

Using representations of quivers of type $\mathbb{A}$, we define an anticyclic cooperad in the category of triangulated categories, which is a categorification of the linear dual of the Diassociative operad.

Let $H$ be a finite dimensional semisimple Hopf algebra, $A$ a differential graded (dg for short) $H$-module algebra. Then the smash product algebra $A\#H$ is a dg algebra. For any dg $A\#H$-module $M$, there is a quasi-isomorphism of dg algebras: $\RHom_A(M,M)\#H\longrightarrow \RHom_{A\#H}(M\ot H,M\ot H)$. This result is applied to $d$-Koszul algebras, Calabi-Yau algebras and AS-Gorenstein dg algebras.

A bounded subset $T$ of a metric space $(X,\rho)$ is said to be remotal (uniquely remotal) if for each $x\in X$ there exists at least one (exactly one) $t\in T$ such that $\rho(x,t)=\sup\{\rho(x,y):y\in T\}.$ Such a point $t$ is called a farthest point to $x$ in $T$. In this paper, we discuss properties of remotal and uniquely remotal sets and, conditions under which remotal and uniquely remotal sets are singleton. The underlying spaces are convex metric spaces or externally convex metric spaces.

Let $A$ be a Banach algebra with a bounded approximate identity. Our first purpose in this paper is to generalize Bekka's results for a certain class of Banach algebras. Let $G$ be an amenable locally compact topological group, and let $A$ be a left Banach $G$-module. Our second purpose, among the other things, is to define certain weak$^*$-closed subspaces of $\mathcal{B}(A,A^*)$ to consider when their weak$^*$-closed subspaces are the range of a bounded projection on $\mathcal{B}(A,A^*)$. Finally, we explore the link between the projections properties and amenability of semigroup algebras.

In this paper,we examine semilinear Neumann problems which at $\pm\infty$ are resonant with respect to two successive eigenvalues (double resonance situation). Using variational methods based on the critical point theory together with Morse theory, we prove two multiplicity results. In the first we obtain two nontrivial solutions and in the second three, two of which have constant sign (one positive, the other negative).

Let $G$ be a non-amenable discrete group. We construct the paradoxical decomposition of $G$, by using the configuration equations under some conditions. This gives a positive answer to a question proposed by G. A. Willis. We find an upper bound for the Tarski number of a given group, by using configurations.

We point out some results on the qualitative theory of three-parameter semiflows by analyzing the stability of the associated linear skew-product semiflow. Extensions of the well-known theorems due to Datko, Pazy, Rolewicz are obtained.