The main purpose of this paper is to obtain an explicit Capelli identity relating skew-symmetric matrices under the action of the general linear group $GL_N$. In particular, we give an explicit formula for the skew Capelli element in terms of the trace of powers of a matrix defined by the standard infinitesimal generators of $GL_N$.

AMS 2000 Mathematics subject classification: Primary 17B35. Secondary 17B10; 15A15; 20G05

We establish a generalization of the Widder–Arendt theorem from Laplace transform theory. Given a Banach space $E$, a non-negative Borel measure $\mea$ on the set $\Rplus$ of all non-negative numbers, and an element $\bnd$ of $\R\cup\{-\infty\}$ such that $\natres{-\coefl}$ is $\mea$-integrable for all $\coefl>\bnd$, where $\natres{-\coefl}$ is defined by $\natres{-\coefl}(t)=\exp(-\coefl t)$ for all $t\in\Rplus$, our generalization gives an intrinsic description of functions $\f\colon\Set\to E$ that can be represented as $\f(\coefl)=T(\natres{-\coefl})$ for some bounded linear operator $T\colon\Ma\to E$ and all $\coefl> \bnd$; here $\Ma$ denotes the Lebesgue space based on $\mea$. We use this result to characterize pseudo-resolvents with values in a Banach algebra, satisfying a growth condition of Hille–Yosida type.

AMS 2000 Mathematics subject classification: Primary 44A10; 47A10. Secondary 43A20; 46B22; 46G10; 46J25; 47D06

Two classes of nonlinear operators generalizing the notion of a local operator between ideal function spaces are introduced. The first class, called atomic, contains in particular all the linear shifts, while the second one, called coatomic, contains all the adjoints to former, and, in particular, the conditional expectations. Both classes include local (in particular, Nemytski\v{\i}) operators and are closed with respect to compositions of operators. Basic properties of operators of introduced classes in the Lebesgue spaces of vector-valued functions are studied. It is shown that both classes inherit from Nemytski\v{\i} operators the properties of non-compactness in measure and weak degeneracy, while having different relationships of acting, continuity and boundedness, as well as different convergence properties. Representation results for the operators of both classes are provided. The definitions of the introduced classes as well as the proofs of their properties are based on a purely measure theoretic notion of memory of an operator, also introduced in this paper.

AMS 2000 Mathematics subject classification: Primary 47B38; 47A67; 34K05

For a closed set $E$ contained in the closed unit interval, we show that the big Lipschitz algebra $\varLambda_{\gamma}(E)$ $(0\lt\gamma\lt1)$ is sequentially weak$^{\ast}$ generated by its idempotents if and only if it is weak$^{\ast}$ generated by its idempotents if and only if the little Lipschitz algebra $\lambda_{\gamma}(E)$ is generated by its idempotents, and we describe a class of perfect symmetric sets for which this holds. Moreover, we prove that $\varLambda_1(E)$ is sequentially weak$^{\ast}$ generated by its idempotents if and only if $E$ is of measure zero. Finally, we show that the quotient algebras

$$ \mathcal{A}_{\beta}/\overline{J_{\beta}(E)}^{\text{weak}^{\ast}} $$

of the Beurling algebras need not be weak$^{\ast}$ generated by their idempotents, when $E$ is of measure zero and $\beta\ge\tfrac{1}{2}$.

AMS 2000 Mathematics subject classification: Primary 46J10; 46J30; 26A16; 42A16

We study the geometry of surfaces in $\mathbb{R}^4$ associated to contact with hyperplanes. We list all possible transitions that occur on the parabolic and so-called $A_3$-set, and analyse the configurations of the asymptotic curves and their bifurcations in generic one-parameter families.

AMS 2000 Mathematics subject classification: Primary 58C27. Secondary 53A05

We study one-relator free products in which the relator has free-product length $4$. We find conditions for such presentations to have a Freiheitssatz and classify all non-aspherical presentations under certain conditions.

AMS 2000 Mathematics subject classification: Primary 20F05; 20F06; 20F10

Let $\mathcal{A}$ be a proper independence algebra of finite rank, let $G$ be the group of automorphisms of $\mathcal{A}$, let $a$ be a singular endomorphism and let $a^G$ be the semigroup generated by all the elements $g^{-1}ag$, where $g\in G$. The aim of this paper is to prove that $a^G$ is a semigroup generated by its own idempotents.

AMS 2000 Mathematics subject classification: Primary 20M20; 20M10; 08A35

The paper shows that specializations of finitely generated graded modules are also graded and that many important invariants of graded modules and ideals are preserved by specializations.

AMS 2000 Mathematics subject classification: Primary 13A02

We show that in the classical (fixed-monomer-concentration) Becker–Döring equations truncated at finite cluster size, the slow evolution (metastability) of solutions can be explained in terms of the eigensystem of this linear ordinary differential equation (ODE) system. In particular, for a common choice of coagulation–fragmentation rate constants there is an extremely small non-zero eigenvalue which is isolated from the rest of the spectrum. We give estimates and bounds on the size of this eigenvalue, the gap between it and the second smallest, and the size of the largest eigenvalue. The bounds on the smallest eigenvalue are very sharp when the system size and/or monomer concentration are large enough.

AMS 2000 Mathematics subject classification: Primary 34A30; 15A18; 65F15

Let R be a prime ring with extended centroid C, $\rho$ a non-zero right ideal of R and let $f(X_1,\dots,X_t)$ be a polynomial, having no constant term, over C. Suppose that $f(X_1,\dots,X_t)$ is not central-valued on RC. We denote by $f(\rho)$ the additive subgroup of RC> generated by all elements $f(x_1,\dots,x_t)$ for $x_i\in\rho$. The main goals of this note are to prove two results concerning the extension properties of finiteness conditions as follows.

(I) If $f(\rho)$ spans a non-zero finite-dimensional $C$-subspace of $RC$, then $\dim_CRC$ is finite.

(II) If $f(\rho)\ne0$ and is a finite set, then $R$ itself is a finite ring.

AMS 2000 Mathematics subject classification: Primary 16N60; 16R50

Let $v$ be a henselian valuation of a field $K$ with value group $G$, let $\bar{v}$ be the (unique) extension of $v$ to a fixed algebraic closure $\bar{K}$ of $K$ and let $(\tilde{K},\tilde{v})$ be a completion of $(K,v)$. For $\alpha\in\bar{K}\setminus K$, let $M(\alpha,K)$ denote the set $\{\bar{v}(\alpha-\beta):\beta\in\bar{K},\ [K(\beta):K] \lt [K(\alpha):K]\}$. It is known that $M(\alpha,K)$ has an upper bound in $\bar{G}$ if and only if $[K(\alpha):K]=[\tilde{K}(\alpha):\tilde{K}]$, and that the supremum of $M(\alpha,K)$, which is denoted by $\delta_{K}(\alpha)$ (usually referred to as the main invariant of $\alpha$), satisfies a principle similar to the Krasner principle. Moreover, each complete discrete rank 1 valued field $(K,v)$ has the property that $\delta_{K}(\alpha)\in M(\alpha,K)$ for every $\alpha\in\bar{K}\setminus K$. In this paper the authors give a characterization of all those henselian valued fields $(K,v)$ which have the property mentioned above.

AMS 2000 Mathematics subject classification: Primary 12J10; 12J25; 13A18

We characterize the Bloch space and the Besov spaces of harmonic functions on the open unit disc $D$ by using the following oscillation:

$$ \sup_\{\beta(z,w)\ltr\}(1-|z|^2)^{\alpha}(1-|w|^2)^{\beta}\biggl|\frac{\hat{D}^{(n-1)}h(z)-\hat{D}^{(n-1)}h(w)}{z-w}\biggr|, $$

where $\alpha+\beta=n$, $\alpha,\beta\in\mathbb{R}$ and $\displaystyle{\hat{D}^{(n)}=(\partial^{n}/\partial^{n}z+\partial^{n}/\partial^{n}\bar{z})}$.

AMS 2000 Mathematics subject classification: Primary 46E15

Let $G$ be a periodic residually finite group containing a nilpotent subgroup $A$ such that $C_G(A)$ is finite. We show that if $\langle A,A^g\rangle$ is finite for any $g\in G$, then $G$ is locally finite.

AMS 2000 Mathematics subject classification: Primary 20F50

Each permutation representation of a finite group $G$ can be used to pull cohomology classes back from a symmetric group to $G$. We study the ring generated by all classes that arise in this fashion, describing its variety in terms of the subgroup structure of $G$.

We also investigate the effect of restricting to special types of permutation representations, such as $\mathrm{GL}_n(\mathbb{F}_p)$ acting on flags of subspaces.

AMS 2000 Mathematics subject classification: Primary 20J06

This paper gives a number which is used to determine the component number of links from their associated planar graphs. In particular, we use this number to determine the component numbers of links whose associated planar graphs are fans, wheels and 2-sums of graphs.

AMS 2000 Mathematics subject classification: Primary 05C10

This paper concerns perturbations of smooth vector fields on $\mathbb{T}^n$ (constant if $n\geq3$) with zeroth-order $C^\infty$ and Gevrey $G^\sigma$, $\sigma\geq1$, pseudodifferential operators. Simultaneous resonance is introduced and simultaneous resonant normal forms are exhibited (via conjugation with an elliptic pseudodifferential operator) under optimal simultaneous Diophantine conditions outside the resonances. In the $C^\infty$ category the results are complete, while in the Gevrey category the effect of the loss of the Gevrey regularity of the conjugating operators due to Diophantine conditions is encountered. The normal forms are used to study global hypoellipticity in $C^\infty$ and Gevrey $G^\sigma$. Finally, the exceptional sets associated with the simultaneous Diophantine conditions are studied. A generalized Hausdorff dimension is used to give precise estimates of the ‘size’ of different exceptional sets, including some inhomogeneous examples.

AMS 2000 Mathematics subject classification: Primary 37C15; 11J13. Secondary 58J40; 11J20; 35H05