We study the possibility of constructing a Frobenius manifold for the standard Landau–Ginzburg model of odd-dimensional quadrics Q2n+1 and matching it with the Frobenius manifold attached to the quantum cohomology of these quadrics. Namely, we show that the initial conditions of the quantum cohomology Frobenius manifold of the quadric can be obtained from the suitably modified standard Landau–Ginzburg model.

Let Λ be a finite-dimensional algebra and G be a finite group whose elements act on Λ as algebra automorphisms. Assume that Λ has a complete set E of primitive orthogonal idempotents, closed under the action of a Sylow p-subgroup S ≤ G. If the action of S on E is free, we show that the skew group algebra Λ G and Λ have the same finitistic dimension, and have the same strong global dimension if the fixed algebra ΛS is a direct summand of the ΛS-bimodule Λ. Using a homological characterization of piecewise hereditary algebras proved by Happel and Zacharia, we deduce a criterion for Λ G to be piecewise hereditary.

(0.1)

\begin{equation}\label{eq:0.1}\left\{\begin{array}{ll}\displaystyle-\Delta_{\mathbb{H}^{N}}u=|v|^{p-1}v x, \\\displaystyle-\Delta_{\mathbb{H}^{N}}v=|u|^{q-1}u, \\\end{array}\right.\end{equation}

If A(z) belongs to the Bergman space , then the differential equation f″+A(z)f=0 is Blaschke-oscillatory, meaning that the zero sequence of every nontrivial solution satisfies the Blaschke condition. Conversely, if A(z) is analytic in the unit disc such that the differential equation is Blaschke-oscillatory, then A(z) almost belongs to . It is demonstrated that certain “nice” Blaschke sequences can be zero sequences of solutions in both cases when A ∈ or A ∉ . In addition, no condition regarding only the number of zeros of solutions is sufficient to guarantee that A ∈ .

Let A be a domain over an algebraically closed field with Gelfand–Kirillov dimension in the interval [2,3). We prove that if A has two locally nilpotent skew derivations satisfying some natural conditions, then A must be one of five algebras. All five algebras are Noetherian, finitely generated, and have Gelfand–Kirillov dimension equal to 2. We also obtain some results comparing the Gelfand–Kirillov dimension of an algebra to its subring of invariants under a locally nilpotent skew derivation.

We consider quasi-Einstein metrics in the framework of contact metric manifolds and prove some rigidity results. First, we show that any quasi-Einstein Sasakian metric is Einstein. Next, we prove that any complete K-contact manifold with quasi-Einstein metric is compact Einstein and Sasakian. To this end, we extend these results for (κ, μ)-spaces.

This work defines a new algebraic structure, to be called an alternative Clifford algebra associated to a given quadratic form. I explored its representations, particularly concentrating on connections to the well-understood octonion algebras. I finished by suggesting directions for future research.

We define a pseudometric on the set of all unbounded subsets of a metric space. The Kolmogorov quotient of this pseudometric space is a complete metric space. The definition of the pseudometric is guided by the principle that two unbounded subsets have distance 0 whenever they stay sublinearly close. Based on this pseudometric we introduce and study a general concept of boundaries of metric spaces. Such a boundary is the closure of a subset in the Kolmogorov quotient determined by an arbitrarily chosen family of unbounded subsets. Our interest lies in those boundaries which we get by choosing unbounded cyclic sub(semi)groups of a finitely generated group (or more general of a compactly generated, locally compact Hausdorff group). We show that these boundaries are quasi-isometric invariants and determine them in the case of nilpotent groups as a disjoint union of certain spheres (or projective spaces). In addition we apply this concept to vertex-transitive graphs with polynomial growth and to random walks on nilpotent groups.

The following Chen's bi-harmonic conjecture made in 1991 is well-known and stays open: The only bi-harmonic submanifolds of Euclidean spaces are the minimal ones. In this paper, we prove that the bi-harmonic conjecture is true for bi-harmonic hypersurfaces with three distinct principal curvatures of a Euclidean space of arbitrary dimension.

Given the pair (P, η) of (0,2) tensors, where η defines a volume element, we consider a new variational problem varying η only. We then have Einstein metrics and slant immersions as critical points of the 1st variation. We may characterize Ricci flat metrics and Lagrangian submanifolds as stable critical points of our variational problem.

We apply appropriate maximum principles in order to obtain characterization results concerning complete linear Weingarten hypersurfaces with bounded mean curvature in the hyperbolic space. By supposing a suitable restriction on the norm of the traceless part of the second fundamental form, we show that such a hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder, when its scalar curvature is positive, or to a spherical cylinder, when its scalar curvature is negative. Related to the compact case, we also establish a rigidity result.

Let A be an invertible operator on a complex Banach space X. For a given α ≥ 0, we define the class $\mathcal{D}$Aα(ℤ) (resp. $\mathcal{D}$Aα (ℤ+)) of all bounded linear operators T on X for which there exists a constant CT>0, such that

$\begin{equation*}\Vert A^{n}TA^{-n}\Vert \leq C_{T}\left( 1+\left\vertn\right\vert \right) ^{\alpha },\end{equation*}$

$\mathcal{D}$

$\mathcal{D}$

$\mathcal{D}$

$\mathcal{D}$

1. Let r(n) denote the number of representations of the natural number n as the sum of one square and three fifth powers of positive integers. A formal use of the circle method predicts the asymptotic relation(1)

$\begin{equation*}r(n) = \frac{\Gamma(\frac32)\Gamma(\frac65)^3}{\Gamma(\frac{11}{10})} {\mathfrak s}(n) {n}^\frac1{10} (1 + o(1)) \qquad (n\to\infty).\end{equation*}$

${\mathfrak s}$

An algebra A is said to be directly finite if each left-invertible element in the (conditional) unitization of A is right invertible. We show that the reduced group C*-algebra of a unimodular group is directly finite, extending known results for the discrete case. We also investigate the corresponding problem for algebras of p-pseudofunctions, showing that these algebras are directly finite if G is amenable and unimodular, or unimodular with the Kunze–Stein property. An exposition is also given of how existing results from the literature imply that L1(G) is not directly finite when G is the affine group of either the real or complex line.

Let X and Y be infinite-dimensional complex Banach spaces, and ${\mathcal B}$(X) (resp. ${\mathcal B}$(Y)) be the algebra of all bounded linear operators on X (resp. on Y). For an operator T ∈ ${\mathcal B}$(X) and a vector x ∈ X, let σT(x) denote the local spectrum of T at x. For two nonzero vectors x0 ∈X and y0 ∈ Y, we show that a map ϕ from ${\mathcal B}$(X) onto ${\mathcal B}$(Y) satisfies

$\sigma_{\varphi(T)\varphi(S)}(y_0)~=~\sigma_{TS}(x_0),~(T,~S\in{\mathcal B}(X)),$

${\mathcal B}$