We prove that for the cases $X=\mathcal{C}(K)$ ($K$infinite) and$X$=$L$_{1}(μ) (μ σ-finite and atomless) it holds that every slice of the unit ball of the$N$-fold symmetric tensor product of$X$has diameter two. In fact, we prove more general results for weak neighborhoods relative to the unit ball. As a consequence, we deduce that the spaces of$N$-homogeneous polynomials on those classical Banach spaces have no points of Fréchet differentiability.

We study the geometry of Hilbert schemes of points on abelian surfaces and Beauville’s generalized Kummer varieties in positive characteristics. The main result is that, in characteristic two, the addition map from the Hilbert scheme of two points to the abelian surface is a quasifibration such that all fibers are nonsmooth. In particular, the corresponding generalized Kummer surface is nonsmooth, and minimally elliptic singularities occur in the supersingular case. We unravel the structure of the singularities in dependence of$p$-rank and$a$-number of the abelian surface. To do so, we establish a McKay Correspondence for Artin’s wild involutions on surfaces. Along the line, we find examples of canonical singularities that are not rational singularities.

In this paper, a random graph process {$G$($t$)}_{$t$≥1}is studied and its degree sequence is analyzed. Let {$W$_{$t$}}_{$t$≥1}be an i.i.d. sequence. The graph process is defined so that, at each integer time$t$, a new vertex with$W$_{$t$}edges attached to it, is added to the graph. The new edges added at time$t$are then preferentially connected to older vertices, i.e., conditionally on$G$($t$-1), the probability that a given edge of vertex$t$is connected to vertex$i$is proportional to$d$_{$i$}($t$-1)+δ, where$d$_{$i$}($t$-1) is the degree of vertex$i$at time$t$-1, independently of the other edges. The main result is that the asymptotical degree sequence for this process is a power law with exponent τ=min{τ_{W},τ_{P}}, where τ_{W}is the power-law exponent of the initial degrees {$W$_{$t$}}_{$t$≥1}and τ_{P}the exponent predicted by pure preferential attachment. This result extends previous work by Cooper and Frieze.

Nonself-adjoint, non-dissipative perturbations of possibly unbounded self-adjoint operators with real purely singular spectrum are considered under an additional assumption that the characteristic function of the operator possesses a scalar multiple. Using a functional model of a nonself-adjoint operator (a generalization of a Sz.-Nagy–Foiaş model for dissipative operators) as a principle tool, spectral properties of such operators are investigated. A class of operators with almost Hermitian spectrum (the latter being a part of the real singular spectrum) is characterized in terms of existence of the so-called weak outer annihilator which generalizes the classical Cayley identity to the case of nonself-adjoint operators in Hilbert space. A similar result is proved in the self-adjoint case, characterizing the condition of absence of the absolutely continuous spectral subspace in terms of the existence of weak outer annihilation. An application to the rank-one nonself-adjoint Friedrichs model is given.

We consider several modified versions of the Positivstellensatz for global analytic functions that involve infinite sums of squares and/or positive semidefinite analytic functions. We obtain a general local-global criterion which localizes the obstruction to have such a global result. This criterion allows us to get completely satisfactory results for analytic curves, normal analytic surfaces and real coherent analytic sets whose connected components are all compact.

Let$X$be a complex manifold and let$f$:$X$→ℂ^{$p$}be a holomorphic mapping defining a complete intersection. We prove that the iterated Mellin transform of the residue integral associated with$f$has an analytic continuation to a neighborhood of the origin in ℂ^{$p$}.

In this paper we introduce Coleff–Herrera residue currents defined by systems of c-holomorphic functions and prove a Lelong–Poincaré and a Cauchy-type formula as well as the transformation law for these currents.

Given a domain Ω⊂ℂ^{$n$}, the Lempert function is a functional on the space $\text{Hol}(\mathbb{D},\Omega)$ of analytic disks with values in Ω, depending on a set of poles in Ω. We generalize its definition to the case where poles have multiplicities given by local indicators (in the sense of Rashkovskii) to obtain a function which still dominates the corresponding Green function, behaves relatively well under limits, and is monotonic with respect to the local indicators. In particular, this is an improvement over the previous generalization used by the same authors to find an example of a set of poles in the bidisk so that the (usual) Green and Lempert functions differ.

Stein’s higher Riesz transforms are translation invariant operators on$L$^{2}($R$^{$n$}) built from multipliers whose restrictions to the unit sphere are eigenfunctions of the Laplace–Beltrami operators. In this article, generalizing Stein’s higher Riesz transforms, we construct a family of translation invariant operators by using discrete series representations for hyperboloids associated to the indefinite quadratic form of signature ($p$,$q$). We prove that these operators extend to$L$^{$r$}-bounded operators for 1<$r$<∞ if the parameter of the discrete series representations is generic.

We compute local Gromov–Witten invariants of cubic surfaces at all genera. We use a deformation a of cubic surface to a nef toric surface and the deformation invariance of Gromov–Witten invariants.

A recent approach of D. S. Lubinsky yields universality in random matrix theory and fine zero spacing of orthogonal polynomials under very mild hypothesis on the weight function, provided the support of the generating measure μ is [-1,1]. This paper provides a method with which analogous results can be proven on general compact subsets of$R$. Both universality and fine zero spacing involves the equilibrium measure of the support of μ. The method is based on taking polynomial inverse images, by which results can be transferred from [-1,1] to a system of intervals, and then to general sets.

We are studying the Diophantine exponent μ_{$n$,$l$}defined for integers 1≤$l$<$n$and a vector α∈ℝ^{$n$}by letting $$\mu_{n,l}=\sup\{\mu\geq0: 0 < \Vert\underline{x}\cdot\alpha\Vert<H(\underline{x})^{-\mu}\ \text{for infinitely many}\ \underline{x}\in\mathcal{C}_{n,l}\cap\mathbb{Z}^n\},$$ where $\cdot$ is the scalar product, $\|\cdot\|$ denotes the distance to the nearest integer and $\mathcal{C}_{n,l}$ is the generalised cone consisting of all vectors with the height attained among the first$l$coordinates. We show that the exponent takes all values in the interval [$l$+1,∞), with the value$n$attained for almost all α. We calculate the Hausdorff dimension of the set of vectors α with μ_{$n$,$l$}(α)=μ for μ≥$n$. Finally, letting$w$_{$n$}denote the exponent obtained by removing the restrictions on $\underline{x}$ , we show that there are vectors α for which the gaps in the increasing sequence μ_{$n$,1}(α)≤...≤μ_{$n$,$n$-1}(α)≤$w$_{$n$}(α) can be chosen to be arbitrary.

We consider Fourier integral operators with symbols in modulation spaces and non-smooth phase functions whose second orders of derivatives belong to certain types of modulation space. We prove continuity and Schatten–von Neumann properties of such operators when acting on$L$^{2}.

So-called exceptional isomorphisms of low-dimensional spinor groups are directly worked out for the terminal two cases of rank five and six (in the sense of quadratic modules) at the level of special Clifford groups with norm characters as group schemes over a ring. Explicit formulas of identifications are available on big cells, restrictions to which being justified by scheme theory.

We present new results concerning the solvability, or lack of thereof, in the Cauchy problem for the $\bar\partial$ operator with initial values assigned on a weakly pseudoconvex hypersurface, and provide illustrative examples.

We give several characterizations of those sequences of holomorphic self-maps {φ_{$n$}}_{$n$≥1}of the unit disk for which there exists a function$F$in the unit ball $\mathcal{B}=\{f\in H^{\infty}: \|f\|_\infty\leq1\}$ of$H$^{∞}such that the orbit {$F$∘φ_{$n$}:$n$∈ℕ} is locally uniformly dense in $\mathcal{B}$ . Such a function$F$is said to be a $\mathcal{B}$ -universal function. One of our conditions is stated in terms of the hyperbolic derivatives of the functions φ_{$n$}. As a consequence we will see that if φ_{$n$}is the$n$th iterate of a map φ of $\mathbb{D}$ into $\mathbb{D}$ , then {φ_{$n$}}_{$n$≥1}admits a $\mathcal{B}$ -universal function if and only if φ is a parabolic or hyperbolic automorphism of $\mathbb{D}$ . We show that whenever there exists a $\mathcal{B}$ -universal function, then this function can be chosen to be a Blaschke product. Further, if there is a $\mathcal{B}$ -universal function, we show that there exist uniformly closed subspaces consisting entirely of universal functions.

We introduce a basis of the Orlik–Solomon algebra labeled by chambers, the so called chamber basis. We consider structure constants of the Orlik–Solomon algebra with respect to the chamber basis and prove that these structure constants recover D. Cohen’s minimal complex from the Aomoto complex.

In the present paper we prove, that in the real normed space$X$, having at least three dimensions, the Nordlander’s conjecture about the modulus of convexity of the space$X$is true, i.e. from the validity of Day’s inequality for a fixed real number from the interval (0,2), follows that$X$is an inner product space.

In this paper, we compare some deterministic and probabilistic techniques in the study of upper bounds in problems related to certain mean square discrepancie with respect to balls in the$d$-dimensional unit torus, and show that the quality of these techniques depends in an intricate way on the dimension$d$under consideration.