We construct quasi-isomorphisms of A-infinity algebras by using oriented planar trees and give an analog of Hodge decompositions for A-infinity algebras.

Let $\Sigma \subset \mathbb{P}^4$ be an integral hypersurface of degree $s$ with a $(s-2)$-uple plane. We show that the degrees of smooth surfaces $S \subset \Sigma$ with $q(S)=0$ are bounded by a function of $s$. We also show that if $S \subset \mathbb{P}^4$ is a smooth surface with $q(S)=0$ and if $S$ lies on a quartic hypersurface $\Sigma$ such that $\dim(\Sing(\Sigma))=2$, then $\deg(S) \leq 40$.

Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem. Our main results are the asymptotic formulas for the integral of the cube and the fourth power of $\Delta(x)$. The exponents that we obtain in the error terms, namely $\beta = {\sfrac{7}{5}}$ and $\gamma = {\sfrac{23}{12}}$, respectively, are new. They improve on the values $\beta = {\sfrac{47}{28}}, \gamma = {\sfrac{45}{23}}$, due to K.-M. Tsang. A result on integrals of $\Delta^3(x)$ and $\Delta^4(x)$ in short intervals is also proved.

In the regular class of weak Hilbert spaces we exhibit a complex space which is not isomorphic to its complex conjugate. Thus there exists a real weak Hilbert space with at least two non-isomorphic complex structures.

The zone modulus is a conformally invariant functional over the space of two-component links embedded in $\mathbf{R}^3$ or $\mathbf{S}^3$. It is a positive real number and its lower bound is $1.$ Its main property is that the zone modulus of a non-split link is greater than $(1 + \sqrt{2})^2.$ In this paper, we will show that the only non-split links with modulus equal to $(1 + \sqrt{2})^2$ are the Clifford links, that is, the conformal images of the standard geometric Hopf link.

Let $\mathcal H$ be a complex separable Hilbert space and let ${\mathcal L}({\mathcal H})$ denote the collection of bounded linear operators on ${\mathcal H}$. In this paper, we show that if $T=A^{(n_1)}_{1}\oplus A^{(n_2)}_{2}\oplus\dots\oplus A^{(n_k)}_{k}$, where $A_{i}\not\sim A_{j}$ for $1\leq i\neq j\leq k$, and ${\mathcal A}'(A_{i})/\rad{\mathcal A}'(A_{i})$ is commutative, $K_{0}({\mathcal A}'(A_{i}))\cong Z$ for $i=1,2,\dots, k$, and for any positive integer $n$ and minimal idempotent $P\in {\mathcal A}'(T^{(n)})$, ${\mathcal A}'(T^{(n)}|_{P{\mathcal H}^{(n)}})/\rad{\mathcal A}'(T^{(n)}|_{P{\mathcal H}^{(n)}})$ is commutative, then $T$ is a stably finitely decomposable operator and has a stably unique (SI) decomposition up to similarity. Moreover, we give a similarity classification of the operators which satisfy the above conditions by using the $K_{0}$-group of the commutant algebra as an invariant.

If $p,q$ are idempotents in a Banach algebra $A$ and if $p+q-1$ is invertible, then the Kovarik formula provides an idempotent $k(p,q)$ such that $pA=k(p,q)A$ and $Aq=Ak(p,q)$. We study the existence of such an element in a more general situation. We first show that $p+q-1$ is invertible if and only if $k(p,q)$ and $k(q,p)$ both exist. Then we deduce a local parametrization of the set of idempotents from this equivalence. Finally, we consider a polynomial parametrization first introduced by Holmes and we answer a question raised at the end of his paper.

We give some conditions under which an "into" isometric mapping from the unit sphere of an $\mathcal{L}^{\infty}(\Gamma)$-type space (in particular, the atomic $AM$-space) to the unit sphere of some Banach space can be (real) linearly extended.

We investigate the value distribution of the index of visible points with congruence constraints by estimating all moments of the index twisted by an arbitrary Dirichlet character with respect to a fixed modulus.

We characterize bounded and compact weighted composition operators acting between weighted Bergman spaces and between Hardy spaces. Our results use certain integral transforms that generalize the Berezin transform. We also estimate the essential norms of these operators. As applications, we characterize bounded and compact pointwise multiplication operators between weighted Bergman spaces and estimate their essential norms.

A complete description of the modulus of an outer function on the real line is well known. Indeed, this characterization is considered as one of the classical results of the theory of Hardy spaces. However, a satisfactory characterization of the argument of an outer function on the real line is not available yet. In this paper, we define some classes of real functions which can serve as the argument of an outer function. In particular, for any $0 < p \leq \infty$, an increasing bi-Lipschitz function is the argument of an outer function in $H^p(\mathbb{R})$.

Given an initial graph $G$, one may apply a rule $\R$ to $G$ which replaces certain vertices of $G$ with other graphs called replacement graphs to obtain a new graph $\R(G)$. By iterating this procedure on each resulting graph, a sequence of graphs $\{\R^n(G)\}$ is obtained. When the graphs in this sequence are normalized to have diameter one, questions of convergence can be investigated. Sufficient conditions for convergence in the Gromov-Hausdorff metric were given by J. Previte, M. Previte, and M. Vanderschoot for such normalized sequences of graphs when the replacement rule $\R$ has more than one replacement graph. M. Previte and H.S. Yang showed that under these conditions, the limits of such sequences have topological dimension one. In this paper, we compute the box and Hausdorff dimensions of limit spaces of normalized sequences of iterated vertex replacements when there is more than one replacement graph. Since the limit spaces have topological dimension one and typically have Hausdorff (and box) dimension greater than one, they are fractals. Finally, we give examples of vertex replacement rules that yield fractals.

Let $\mathcal H$ be a Hilbert space. Given a bounded positive definite operator $S$ on $\mathcal H$, and a bounded sequence $\mathbf{c} = \{c_k \}_{k \in \mathbb N}$ of nonnegative real numbers, the pair $(S, \mathbf{c})$ is frame admissible, if there exists a frame $\{ f_k \}_{k \in \mathbb{N}} $ on $\mathcal H$ with frame operator $S$, such that $\|f_k \|^2 = c_k$, $k \in \mathbb {N}$. We relate the existence of such frames with the Schur-Horn theorem of majorization, and give a reformulation of the extended version of Schur-Horn theorem, due to A. Neumann. We use this to get necessary conditions (and to generalize known sufficient conditions) for a pair $(S, \mathbf{c})$ to be frame admissible.

The closure under extensions of a class of objects in an abelian category is often an important property of that class. Recently the closure of such classes under transfinite extensions (both direct and inverse) has begun to play an important role in several areas of mathematics, for example, in Quillen's theory of model categories and in the theory of cotorsion pairs. In this paper we prove that several important classes are closed under transfinite extensions.

Let $\psi$ be a given function defined on a Riemannian space. Under what conditions does there exist a compact starshaped hypersurface $M$ for which $\psi$, when evaluated on $M$, coincides with the $m$-th elementary symmetric function of principal curvatures of $M$ for a given $m$? The corresponding existence and uniqueness problems in Euclidean space have been investigated by several authors in the mid 1980s. Recently, conditions for existence were established in elliptic space and, most recently, for hyperbolic space. However, the uniqueness problem has remained open. In this paper we investigate the problem of uniqueness in hyperbolic space and show that uniqueness (up to a geometrically trivial transformation) holds under the same conditions under which existence was established.

In this paper we completely characterize finite rank semicommutator or commutator of two Toeplitz operators with bounded harmonic symbols on the Bergman space. We show that if the product of two Toeplitz operators with bounded harmonic symbols has finite rank, then one of the Toeplitz operators must be zero.

The size of the coefficients of cyclotomic polynomials is a problem that has been well-studied. This paper investigates the following generalization: suppose $f(x)\in\mathbb{Z}[x]$ is a divisor of $x^n-1$, so that $f(x)$ is the product of the cyclotomic polynomials corresponding to some of the divisors of $n$. We ask about the largest coefficient in absolute value over all such divisors $f(x)$ of $x^n-1$, obtaining a fairly tight estimate for the maximal order of this function.

As an application of a more general result on singular Riemannian foliations we prove that variationally complete actions on nonnegatively curved manifolds are hyperpolar.

A characterization of $L_h^2$-domains of holomorphy in the class of Hartogs domains in $\mathbb{C}^2$ is given.

In the context of spaces of homogeneous type, we introduce and develop some new function spaces of Morrey-Campanato type. The new function spaces are defined by variants of maximal functions associated with generalized approximations to the identity, and they generalize the classical Morrey-Campanato spaces. We show that the John-Nirenberg inequality holds on these spaces. We also establish the endpoint boundedness of fractional integrals.

For any bounded regular domain $\Omega$ of a real analytic Riemannian manifold $M$ we denote by $\lambda_{k}(\Omega)$ the $k$-th eigenvalue of the Dirichlet Laplacian of $\Omega$. In this paper, we consider $\lambda_k$ as a functional on the set of domains of fixed volume in $M$. We introduce and investigate a natural notion of critical domain for this functional. In particular, we obtain necessary and sufficient conditions for a domain to be critical, locally minimizing or locally maximizing for $\lambda_k$. These results rely on Hadamard type variational formulae that we establish in this general setting.

As an application, we obtain a characterization of critical domains of the trace of the heat kernel under Dirichlet boundary conditions.

We study differential equations satisfied by modular forms of two variables associated to $\Gamma_1\times \Gamma_2$, where $\Gamma_i$ ($i=1,2$) are genus zero subgroups of $SL_2(\R)$ commensurable with $SL_2(\Z)$, e.g., $\Gamma_0(N)$ or $\Gamma_0(N)^*$ for some $N$. In some examples, these differential equations are realized as the Picard-Fuchs differential equations of families of $K3$ surfaces with large Picard numbers, e.g., $19, 18, 17, 16$. Our method rediscovers some of the Lian-Yau examples of "modular relations" involving power series solutions to the second and the third order differential equations of Fuchsian type in \cite{LY1}, \cite{LY2}.