Generalizing Pisier’s idea, we introduce a Hilbertian matrix cross normed space associated with a pair of symmetric normed ideals. When the two ideals coincide, we show that our construction gives an operator space if and only if the ideal is the Schatten class. In general, a pair of symmetric normed ideals that are not necessarily the Schatten class may give rise to an operator space. We study the space of completely bounded mappings between the matrix cross normed spaces obtained in this way and show that the multiplicator norm naturally appears as the completely bounded norm.

We study conformal metrics on the unit ball of Euclidean space. We prove an extension of a theorem originally due to Gerasch on the broadly accessibility of the boundary points of a domain quasiconformally equivalent to a ball. We also show that our result is close to optimal. Our abstract approach leads to new results also for the boundary behavior of (quasi)conformal mappings.

We extend to the multivariate noncommutative context the descriptions of a “once-stripped” probability measure in terms of Jacobi parameters, orthogonal polynomials, and the moment generating function. The corresponding map $Φ$ on states was introduced previously by Belinschi and Nica. We then relate these constructions to the c-free probability theory, which is a version of free probability for algebras with two states, introduced by Bożejko, Leinert, and Speicher. This theory includes the free and Boolean probability theories as extreme cases. The main objects in the paper are the analogs of the Appell polynomial families in the two state context. They arise as fixed points of the transformation which takes a polynomial family to the associated polynomial family (in several variables), and their orthogonality is also related to the map $Φ$ above. In addition, we prove recursions, generating functions, and factorization and martingale properties for these polynomials, and describe the c-free version of the Kailath–Segall polynomials, their combinatorics, and Hilbert space representations.

Let $R$ be a commutative Noetherian ring. The notion of regular sequences with respect to a Serre class of $R$-modules is introduced and some of their essential properties are given. Then in the local case, we explore a theory of Cohen–Macaulayness with respect to Serre classes.

A characterization of Banach spaces possessing the Radon–Nikodým property is given in terms of finitely additive interval functions. We prove that a Banach space $X$ has the RNP if and only if each $X$-valued finitely additive interval function possessing absolutely continuous variational measure is a variational Henstock integral of an $X$-valued function. Due to that characterization several $X$-valued set functions that are only finitely additive can be represented as integrals.

For each variety in positive characteristic, there is a series of canonically defined blowups, called F-blowups. We are interested in the question of whether the ($e+1$)th blowup dominates the $e$th, locally or globally. It is shown that the answer is affirmative (globally for any $e$) when the given variety is F-pure. As a corollary, we obtain some result on the stability of the sequence of F-blowups. We also give a sufficient condition for local domination.

We consider the Hardy–Littlewood approach to the Twin prime problem, which uses a certain exponential sum over prime numbers. We propose a conjecture on the behaviour of the exponential sum in short intervals of the argument. We first show that this conjecture implies the Twin prime conjecture. We then prove that an analogous conjecture is true for exponential sums over integers without small prime factors.

Applying Ribaucour transformations, we construct two new 3-parameter families of complete surfaces, immersed in $H^3$, with constant mean curvature 1 and infinitely many embedded horosphere type ends. Each surface of the first family is locally associated to an Enneper cousin. It has one irregular end and infinitely many regular ends asymptotic to horospheres. The surfaces of the second family are locally associated to a catenoid cousin. Each surface of this family has infinitely many embedded horosphere type ends and one regular end with infinite total curvature.

We show that a Banach space with numerical index one cannot enjoy good convexity or smoothness properties unless it is one-dimensional. For instance, it has no WLUR points in its unit ball, its norm is not Fréchet smooth and its dual norm is neither smooth nor strictly convex. Actually, these results also hold if the space has the (strictly weaker) alternative Daugavet property. We construct a (noncomplete) strictly convex predual of an infinite-dimensional $L_1$ space (which satisfies a property called lushness which implies numerical index $1$). On the other hand, we show that a lush real Banach space is neither strictly convex nor smooth, unless it is one-dimensional. Therefore, a rich subspace of the real space $C[0,1]$ is neither strictly convex nor smooth. In particular, if a subspace $X$ of the real space $C[0,1]$ is smooth or strictly convex, then $C[0,1]/X$ contains a copy of $C[0,1]$. Finally, we prove that the dual of any lush infinite-dimensional real space contains a copy of $\ell_1$.

We compute the low dimensional String bordism groups $\widetilde{\Omega}_{k}^{\mathit{String}}BE_8$ and $\widetilde{\Omega }_{k}^{\mathit{String}}(BE_8\times BE_8)$ using a combination of Adams spectral sequences together with comparisons to the Spin bordism cases.

In this article, we study the fractional Marcinkiewicz integral with variable kernel defined by $$\mu_{\Omega,\alpha}(f)(x)=\bigg(\int_{0}^{\infty}\bigg| {\int_{|x-y|\leq t}}\frac{\Omega(x,x-y)}{|x-y|^{n-1}}f(y)\,dy\bigg|^{2} \frac{dt}{t^{3-\alpha}}\bigg)^{1/2}, %\\ %\quad0<\alpha\leq2. $$where $0<\alpha\leq2$. We first prove that $\mu_{\Omega,\alpha}$ is bounded from $L^{{2n}/{n+\alpha}}(\mathbb{R}^n)$ to $L^2(\mathbb{R}^n)$ without any smoothness assumption on the kernel $\Omega$. Then we show that, if the kernel $\Omega$ satisfies a class of Dini condition, $\mu_{\Omega,\alpha}$ is bounded from $H^p(\mathbb{R}^n)$ ($p \le1$) to $H^q(\mathbb{R}^n)$, where $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{2n}$. As corollary of the above results, we obtain the $L^p-L^q$ ($1< p<2$) boundedness of this fractional Marcinkiewicz integral.

We adapt a number-theoretic technique of Yu to prove a purely analytic theorem: if $f\in L^1({\mathbb R}) \cap L^2({\mathbb R})$ is nonnegative and supported on an interval of length $I$, then the supremum of $f\ast f$ is at least $0.631 \|f\|_1^2/I$. This improves the previous bound of $0.591389 \|f\|_1^2/I$. Consequently, we improve the known bounds on several related number-theoretic problems. For a set $A\subseteq\{1,2,\dots,n\}$, let $g$ be the maximum multiplicity of any element of the multiset $\{a_1+a_2 : a_i\in A\}$. Our main corollary is the inequality $g n > 0.631 |A|^2$, which holds uniformly for all $g$, $n$, and $A$.

In this paper, we prove that a binary definite quadratic form over $\mathbf{F}_q [t]$, where $q$ is odd, is completely determined up to equivalence by the polynomials it represents up to degree $3m-2$, where $m$ is the degree of its discriminant. We also characterize, when $q>13$, all the definite binary forms over $\mathbf{F}_q [t]$ that have class number one.

Given a ring $R$, two classes $\mathcal A$ and $\mathcal B$ of $R$-modules are said to form a cotorsion pair $(\mathcal A, \mathcal B)$ in $\operatorname{Mod}R$ if $\mathcal A=\operatorname {Ker}\operatorname{Ext}^1_R(-,\mathcal B)$ and $\mathcal B=\operatorname{Ker}\operatorname{Ext}^1_R(\mathcal A,-)$. We investigate relative homological dimensions in cotorsion pairs. This can be applied to study the big and the little finitistic dimension of $R$. We show that $\operatorname{Findim} R<\infty$ if and only if the following dimensions are finite for some cotorsion pair $(\mathcal A, \mathcal B)$ in $\operatorname{Mod}R$: the relative projective dimension of $\mathcal A$ with respect to itself, and the $\mathcal A$-resolution dimension of the category $\mathcal P$ of all $R$-modules of finite projective dimension. Moreover, we obtain an analogous result for $\operatorname{findim} R$, and we characterize when $\operatorname{Findim} R=\operatorname{findim} R.$

We establish contractions of the unitary irreducible representations of a compact semisimple Lie group to the unitary irreducible representations of a Heisenberg group by means of Berezin quantization.

Let $\mathcal{H}$ be a function space on a compact space $K$. If $\mathcal{H}$ is not simplicial, we can ask at which points of $K$ there exist unique maximal representing measures. We shall call the set of such points the set of simpliciality. The aim of this paper is to examine topological, algebraic and measure-theoretic properties of the set of simpliciality. We shall also define and investigate sets of points enjoying other simplicial-like properties.

We are concerned with Hardy and BMO spaces of operator-valued functions analytic in the unit disk of $\mathbb{C}$. In the case of the Hardy space, we involve the atomic decomposition since the usual argument in the scalar setting is not suitable. Several properties (the Garsia-norm equivalent theorem, Carleson measure, and so on) of BMOA spaces are extended to the operator-valued setting. Then, the operator-valued $\mathrm{H}^1$-BMOA duality theorem is proved. Finally, by the $\mathrm{H}^1$-BMOA duality we present the Lusin area integral and Littlewood-Paley $g$-function characterizations of the operator-valued analytic Hardy space.

Let $(R,m)$ be a local ring. We study the question of when there exists a positive integer $h$ such that for all prime ideals $P\subseteq R$, the symbolic power $P^{(hn)}$ is contained in $P^n$, for all $n\geq1$. We show that such an $h$ exists when $R$ is a reduced isolated singularity such that $R$ either contains a field of positive characteristic and $R$ is $F$-finite or $R$ is essentially of finite type over a field of characteristic zero.

Using Väisälä’s metric duality on joinability of sets, we show, among other things, that in $\mathbb{R}^3$ if the complementary domains of a surface are LLC, they are also uniform. As an application, we show that an Ahlfors regular topological sphere that admits a quasiconformal reflection is quasisymmetrically equivalent to the standard sphere.

A semi-regular sequence in $S=k[x_1, \ldots, x_n]$ is a sequence of polynomials $f_1, \ldots, f_r$ of degrees $d_1, \ldots, d_r$ which satisfy a certain generic condition. Suppose that $I\subset S$ is generated by such a semi-regular sequence and let $\rho$ be the Castelnuovo--Mumford regularity of $S/I$. We show that a minimal free resolution of $S/I$ is isomorphic to the Koszul complex on $f_1, \ldots, f_r$ in degrees $\le\rho-2$. If a common numerical condition is satisfied, then this isomorphism also holds in degree $\rho-1$. Therefore, the Betti diagram of $S/I$ and the Betti diagram of the Koszul complex always agree in rows $\le\rho-2$; we can sometimes determine that they also agree in row $\rho-1$. We also give a partial converse, that if the Betti diagram of $S/I$ agrees with the diagram of the Koszul complex except in possibly the last two rows, then~$I$ can be generated by a (not necessarily minimal) semi-regular sequence.