The space of permutation pseudographs is a probabilistic model of 2-regular pseudographs on n vertices, where a pseudograph is produced by choosing a permutation σ of {1,2,…, n} uniformly at random and taking the n edges {i,σ(i)}. We prove several contiguity results involving permutation pseudographs (contiguity is a kind of asymptotic equivalence of sequences of probability spaces). Namely, we show that a random 4-regular pseudograph is contiguous with the sum of two permutation pseudographs, the sum of a permutation pseudograph and a random Hamilton cycle, and the sum of a permutation pseudograph and a random 2-regular pseudograph. (The sum of two random pseudograph spaces is defined by choosing a pseudograph from each space independently and taking the union of the edges of the two pseudographs.) All these results are proved simultaneously by working in a general setting, where each cycle of the permutation is given a nonnegative constant multiplicative weight. A further contiguity result is proved involving the union of a weighted permutation pseudograph and a random regular graph of arbitrary degree. All corresponding results for simple graphs are obtained as corollaries.

In ‘Automorphisms of Dowling lattices and related geometries’, J. Bonin constructed the automorphism group A of a Dowling lattice as the image of a certain semidirect product, A = θ(K [rtimes] H). In this work we find necessary and sufficient conditions for this quotient to be the semidirect product A = θ(K) [rtimes] θ(H). In addition, we include a construction of A that lends itself to computation more readily than that found in Bonin's work.

Let $G$ be a Lie groupoid over $M$ such that the target-source map from $G$ to $M\times M$ is proper. We show that, if$\mathcal{O}$ is an orbit of finite type (i.e. which admits a proper function with finitely many critical points), thenthe restriction $G|_\mathcal{U}$ of $G$ to some neighbourhood $\mathcal{U}$ of $\mathcal{O}$ in $M$ is isomorphic to asimilar restriction of the action groupoid for the linear action of the transitive groupoid $G|_\mathcal{O}$ on thenormal bundle $N\mathcal{O}$. The proof uses a deformation argument based on a cohomology vanishing theorem, along with aslice theorem which is derived from a new result on submersions with a fibre of finite type.

AMS 2000 Mathematics subject classification: Primary 58H05. Secondary 57R99

The integrable model corresponding to the $\mathcal{N}=2$ supersymmetric SU$(N)$ gauge theory with matter in thesymmetric representation is constructed. It is a spin chain model, whose key feature is a new twisted monodromycondition.

AMS 2000 Mathematics subject classification: Primary 53D30; 53D20. Secondary 37J35

Cet article est le résultat d'une relecture de la démonstration de la conjecture{\fontencoding{U}\fontfamily{lasy}\selectfont(\kern-0.20em(\kern.175em}faiblement admissibleimplique admissible{\fontencoding{U}\fontfamily{lasy}\selectfont\kern.175em)\kern-0.20em)}par J.-M. Fontaine et l'auteur. On donne une version renforcée de l'un des ingrédients(le {\fontencoding{U}\fontfamily{lasy}\selectfont(\kern-0.20em(\kern.175em}lemme fondamental{\fontencoding{U}\fontfamily{lasy}\selectfont\kern.175em)\kern-0.20em)})de cette démonstration, ce qui nous amène à introduire un corps noncommutatif gigantesque $\mathfrak{C}$, de centre $\textbf{Q}_p$ et contenant un corps $C$ algébriquementclos et complet pour la norme $p$-adique comme sous-corps commutatif maximal. Un peu{\fontencoding{U}\fontfamily{lasy}\selectfont(\kern-0.20em(\kern.175em}d'algèbre linéaire{\fontencoding{U}\fontfamily{lasy}\selectfont\kern.175em)\kern-0.20em)}sur ce corps mène à l'introduction de la catégorie des Espaces (avec un E majuscule)de Banach de dimension finie, et la conjecture {\fontencoding{U}\fontfamily{lasy}\selectfont(\kern-0.20em(\kern.175em}faiblementadmissible implique admissible{\fontencoding{U}\fontfamily{lasy}\selectfont\kern.175em)\kern-0.20em)} seréduit alors à un calcul de dimensions d'objets de cette catégorie.

AMS 2000 Mathematics subject classification: Primary 11S

Let $X$ be a compact surface such that $Y\hookrightarrow X$ as a separating, strictly pseudoconvex,real hypersurface;

$$X\setminus Y=X_+\sqcup X_-,$$

where $X_{+}$ ($X_-$) is the strictly pseudoconvex (pseudoconcave) component of the complement. Suppose further that$X_-$ contains a positively embedded, compact curve $Z$. Under cohomological hypotheses on $(X_-,Z)$ we show that if$\dbarb'$ is a sufficiently small, embeddable deformation of the CR-structure on $Y$, then

$$\mathrm{R-ind}(\bar{\partial}_{b},\bar{\partial}_{b}')\geq-[\dim H^{0,2}(X_-)+\dim H^0(Z,\mathcal{O}_Z)].$$

This implies that the set of small, embeddable deformations of the CR-structure on $Y$ is closed, in the$\mathcal{C}^{\infty}$-topology on the set of all deformations.

AMS 2000 Mathematics subject classification: Primary 32V15; 32V30. Secondary 32G07; 32W10; 32W05

Segre proved that a smooth cubic surface over $Q$ is unirational if and only if it has a rational point.We prove that the result also holds for cubic hypersurfaces over any field, including finite fields.

AMS 2000 Mathematics subject classification: Primary 14G05; 14G15. Secondary 11G25; 11D25

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

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

For Q the variance of some centred Gaussian random vector in a separable Banach space it is shown that, necessarily, Q factors through $\ell^2$ as a product of 2-summing operators. This factorization condition is sufficient when the Banach space is of Gaussian type 2. The stochastic integral of a deterministic family of operators with respect to a Q-Wiener process is shown to exist under a continuity condition involving the 2-summing norm. A Langevin equation

$$ \rd\bm{Z}_t+\sLa\bm{Z}_t\,\rd t=\rd\bm{B}_t, $$

with values in a separable Banach space, is studied. The operator $\sLa$ is closed and densely defined. A weak solution $(\bm{Z}_t,\bm{B}_t)$, where $\bm{Z}_t$ is centred, Gaussian and stationary, while $\bm{B}_t$ is a Q-Wiener process, is given when $\ri\sLa$ and $\ri\sLa^*$ generate $C_0$ groups and the resolvent of $\sLa$ is uniformly bounded on the imaginary axis. Both $\bm{Z}_t$ and $\bm{B}_t$ are stochastic integrals with respect to a spectral Q-Wiener process.

AMS 2000 Mathematics subject classification: Primary 60G15. Secondary 46E40; 47B10; 47D03; 60H10

The cusp density of a hyperbolic 3-manifold is the ratio of the largest possible volume in a set of cusps with disjoint interiors to the volume in the manifold. It is known that all cusp densities fall in the interval $[0,0.853\dots]$. It is shown that the cusp densities of finite-volume orientable hyperbolic 3-manifolds are dense in this interval.

AMS 2000 Mathematics subject classification: Primary 57M50

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

This paper provides mathematical analysis of optical tomography in a situation when the examined object, for example the human brain, is strongly scattering with non-scattering inclusions. Light propagation in biological tissue is often modelled by the diffusion approximation of the radiative transfer equation. To be justified, the diffusion approximation demands that the medium is strongly scattering. Naturally, this is not true for non-scattering inclusions, for which some other model is needed. This is found through geometrical optics. Combination of the two models leads to an elliptic partial differential equation with boundary conditions on the outer boundary as well as on the boundaries of the non-scattering regions. The well-posedness of this forward problem is the main concern of this work.

AMS 2000 Mathematics subject classification: Primary 35Q60; 35R30

The uniform stability of the zero solution and the asymptotic behaviour of all solutions of the neutral delay differential equation

$$ [x(t)-P(t)x(t-\tau)]'+Q(t)x(t-\sigma)=0,\quad t\ge t_0, $$

are investigated, where $\tau,\sigma\in(0,\infty)$, $P\in C([t_0,\infty),\mathbb{R})$, and $Q\in C([t_0,\infty), [0,\infty))$. The obtained sufficient conditions improve the existing results in the literature.

AMS 2000 Mathematics subject classification: Primary 34K20; 34K15; 34K40

We prove that spaces with an uncountable $\omega$-independent family fail the Kunen–Shelah property. Actually, if $\{x_i\}_{i\in I}$ is an uncountable $\omega$-independent family, there exists an uncountable subset $J\subset I$ such that $x_j\notin\overline{\co}(\{x_i\}_{i\in J\setminus\{j\}})$ for every $j\in J$. This improves a previous result due to Sersouri, namely that every uncountable $\omega$-independent family contains a convex right-separated subfamily.

AMS 2000 Mathematics subject classification: Primary 46B20; 46B26

In this paper, we establish the existence of infinitely many solutions to a Neumann problem involving the p-Laplacian and with discontinuous nonlinearities. The technical approach is mainly based on a very recent result on critical points for possibly non-smooth functionals in a Banach space due to Marano and Motreanu, namely Theorem 1.1 in a paper that is to appear in the journal J. Diff. Eqns (see Theorem 2.3 in the body of this paper). Some applications are presented.

AMS 2000 Mathematics subject classification: Primary 35A15; 35J65; 35R05

As a consequence of the main result of the paper we obtain that every 2-local isometry of the $C^*$-algebra $B(H)$ of all bounded linear operators on a separable infinite-dimensional Hilbert space $H$ is an isometry. We have a similar statement concerning the isometries of any extension of the algebra of all compact operators by a separable commutative $C^*$-algebra. Therefore, on those $C^*$-algebras the isometries are completely determined by their local actions on the two-point subsets of the underlying algebras.

AMS 2000 Mathematics subject classification: Primary 47B49

Complete positivity of ‘atomically extensible’ bounded linear operators between $C^*$-algebras is characterized in terms of positivity of a bilinear form on certain finite-rank operators. In the case of an elementary operator on a $C^*$-algebra, the approach leads us to characterize k-positivity of the operator in terms of positivity of a quadratic form on a subset of the dual space of the algebra and in terms of a certain inequality involving factorial states of finite type I.

As an application we characterize those $C^*$-algebras where every k-positive elementary operator on the algebra is completely positive. They are either k-subhomogeneous or k-subhomogeneous by antiliminal. We also give a dual approach to the metric operator space introduced by Arveson.

AMS 2000 Mathematics subject classification: Primary 46L05. Secondary 47B47; 47B65

Under an extra hypothesis satisfied in every known case, we show that the Euler class of an orientable odd-dimensional Poincaré duality group over any ring has order at most two. We construct groups that are of type FL over the complex numbers but are not FL over the rationals. We construct group algebras over fields for which $K_0$ contains torsion, and construct non-free stably free modules for the group algebras of certain virtually free groups.

AMS 2000 Mathematics subject classification: Primary 19A31. Secondary 16S34; 20J05; 55N15; 57P10

A cotorsion theory is defined as a pair of classes Ext-orthogonal to each other. We give a hereditary condition (HC) which is satisfied by the (flat, cotorsion) cotorsion theory and give properties satisfied by arbitrary cotorsion theories with an HC. Given a cotorsion theory with an HC, we consider the class of all modules having a special precover with respect to the first class in the cotorsion theory and show that this class is closed under extensions. We then raise the question of whether this class is resolving or coresolving.

AMS 2000 Mathematics subject classification: Primary 18G15. Secondary 18E40