We are interested in proving the convergence of Monte Carlo approximations for vortex equations in bounded domains of $\mathbb{R}^2$ with Neumann’s condition on the boundary. This work is the first step towards justifying theoretically some numerical algorithms for Navier–Stokes equations in bounded domains with no-slip conditions.

We prove that the vortex equation has a unique solution in an appropriate energy space and can be interpreted from a probabilistic point of view through a nonlinear reflected process with space-time random births on the boundary of the domain.

Next, we approximate the solution $w$ of this vortex equation by the weighted empirical measure of interacting diffusive particles with normal reflecting boundary conditions and space-time random births on the boundary. The weights are related to the initial data and to the Neumann condition. We prove a trajectorial propagation-of-chaos result for these systems of interacting particles. We can deduce a simple stochastic particle algorithm to simulate $w$.

AMS 2000 Mathematics subject classification: Primary 60K35; 76D05

Let $k$ be a field, and let $\alpha$ and $\alpha'$ be two algebraic numbers conjugate over $k$. We prove a result which implies that if $L\subset k(\alpha,\alpha')$ is an abelian or Hamiltonian extension of $k$, then $[L:k]\leq[k(\alpha):k]$. This is related to a certain question concerning the degree of an algebraic number and the degree of a quotient of its two conjugates provided that the quotient is a root of unity, which was raised (and answered) earlier by Cantor. Moreover, we introduce a new notion of the non-torsion power of an algebraic number and prove that a monic polynomial in $X$—irreducible over a real field and having $m$ roots of equal modulus, at least one of which is real—is a polynomial in $X^m$.

AMS 2000 Mathematics subject classification: Primary 11R04; 11R20; 11R32; 12F10

In answer to Jarchow’s 1981 text, we recently characterized when $C_{\textrm{c}}(X)$ is a $df$-space, finding along the way attractive analytic characterizations of when the Tychonov space $X$ is pseudocompact. Analogues now reveal how exquisitely Warner boundedness lies between these two notions. To illustrate, $X$ is pseudocompact, $X$ is Warner bounded or $C_{\textrm{c}}(X)$ is a $df$-space if and only if for each sequence $(\mu_{n})_{n}\subset C_{\textrm{c}}(X)'$ there exists a sequence $(\varepsilon_{n})_{n}\subset(0,1]$ such that $(\varepsilon_{n}\mu_{n})_{n}$ is weakly bounded, is strongly bounded or is equicontinuous, respectively. Our characterizations and proofs add to and simplify Warner’s.

AMS 2000 Mathematics subject classification: Primary 46A08; 46A30; 54C35

In this paper we study the eigenvalues associated with a positive eigenfunction of a quasilinear elliptic problem with an operator that is not necessarily bounded. For that, we use the bifurcation theory and obtain the existence of positive solutions for a range of values of the bifurcation parameter.

AMS 2000 Mathematics subject classification: Primary 35J60; 35J25. Secondary 35D05

It is shown that a condition on the order of a meromorphic function in a result of A. A. Gol’dberg cannot be relaxed.

AMS 2000 Mathematics subject classification: Primary 30D35

This paper introduces a family of area-type integrals over cones. These are used to investigate non-tangential boundary behaviour of harmonic functions on a half-space, extending results of Stein and Brossard.

AMS 2000 Mathematics subject classification: Primary 31B25

We establish various martingale inequalities in a rearrangement-invariant (RI) Banach function space. If $X$ is an RI space that is not too small, we associate with it RI spaces $\mathcal{H}_p(X)$ $(1\leq p\lt\infty)$ and $K(X)$, and discuss martingale inequalities in these spaces. One of our results is as follows. Let $1\leq p\lt\infty$, let $f=(f_n)$ be an $L_p$-bounded martingale, and let $|f|^p=g+h$ be the Doob decomposition of the submartingale $|f|^p=(|f_n|^p)$ into a martingale $g=(g_n)$ and a predictable non-decreasing process $h=(h_n)$ with $h_0=0$. Then, in the case where $1\ltp\lt\infty$, we obtain the inequalities

$$ \|h_{\infty}^{1/p}\|_X\leq2\|f_{\infty}\|_{\mathcal{H}_p(X)}\quad\text{and}\quad \Big\|\sup_n|g_n|^{1/p}\Big\|_X\leq4\|f_{\infty}\|_{\mathcal{H}_p(X)}, $$

and, in the case where $p=1$, we obtain the inequalities

$$ \|h_{\infty}\|_X\leq\sup_{n\in\mathbb{Z}_{+}}\|f_n\|_{K(X)}\quad\text{and}\quad \sup_{n\in\mathbb{Z}_{+}}\|g_n\|_X\leq2\sup_{n\in\mathbb{Z}_{+}}\|f_n\|_{K(X)}. $$

For some specific choices of $X$, we can give explicit expressions for $\mathcal{H}_p(X)$ and $K(X)$. For example, $\mathcal{H}_1(L_1)=L\log L$, $\mathcal{H}_p(L_{p,\infty})=L_{p,1}$, and so on. Furthermore, if the Boyd indices of $X$ satisfy $0\lt\alpha_{X}\leq\beta_{X}\lt1/p$ (respectively, $0\lt\alpha_{X}$), then $\mathcal{H}_p(X)=X$ (respectively, $K(X)=X$). In any case, $\mathcal{H}_p(X)$ is embedded in $K(X)$, and $K(X)$ is embedded in $X$.

AMS 2000 Mathematics subject classification: Primary 60G42; 60G46. Secondary 46E30

Let $A$ be a unital $C^*$-algebra and for each $n\in\mathbb{N}$ let $M_n$ be the $n\times n$ matrix algebra over $\mathbb{C}$. In this paper we shall give a necessary and sufficient condition that there is a unital $C^*$-algebra $B$ satisfying $A\not\cong B$ but for which $A\otimes M_n\cong B\otimes M_n$ for some $n\in\mathbb{N}\setminus\{1\}$. Also, we shall give some examples of unital $C^*$-algebras satisfying the above property.

AMS 2000 Mathematics subject classification: Primary 46L05

In this work, the behaviour of solutions for the Dirichlet problem of the non-local equation

$$ u_t=\varDelta(\kappa(u))+\frac{\lambda f(u)}{(\int_{\varOmega}f(u)\,\mathrm{d}x)^p},\quad \varOmega\subset\mathbb{R}^N,\quad N=1,2, $$

is studied, mainly for the case where $f(s)=\mathrm{e}^{\kappa(s)}$. More precisely, the interplay of exponent $p$ of the non-local term and spatial dimension $N$ is investigated with regard to the existence and non-existence of solutions of the associated steady-state problem as well as the global existence and finite-time blow-up of the time-dependent solutions $u(x,t)$. The asymptotic stability of the steady-state solutions is also studied.

AMS 2000 Mathematics subject classification: Primary 35K60. Secondary 35B40

We study the differential geometry of hypersurfaces in hyperbolic space. As an application of the theory of Lagrangian singularities, we investigate the contact of hypersurfaces with families of hyperspheres or equidistant hyperplanes.

AMS 2000 Mathematics subject classification: Primary 53A35. Secondary 58C27

We determine when the $*$-double of a finite-dimensional complex algebra is $*$-finite, $*$-tame and $*$-wild.

AMS 2000 Mathematics subject classification: Primary 46K10; 16G60

It is well known that $\varOmega^2S^{2n+1}$ is approximated by $\textrm{Rat}_{k}(\mathbb{C}P^{n})$, the space of based holomorphic maps of degree $k$ from $S^2$ to $\mathbb{C}P^{n}$. In this paper we construct a space $G_{k}^{n}$ which is an analogue of $\textrm{Rat}_{k}(\mathbb{C}P^{n})$, and prove that under the natural map $j_k:G_{k}^{n}\to\varOmega^2S^{2n}$, $G_{k}^{n}$ approximates $\varOmega^2S^{2n}$.

AMS 2000 Mathematics subject classification: Primary 55P35

For $q$ a non-negative integer, we introduce the internal $q$-homology of crossed modules and we obtain in the case $q=0$ the homology of crossed modules. In the particular case of considering a group as a crossed module we obtain that its internal $q$-homology is the homology of the group with coefficients in the ring of the integers modulo $q$.

The second internal $q$-homology of crossed modules coincides with the invariant introduced by Grandjeán and López, that is, the kernel of the universal $q$-central extension. Finally, we relate the internal $q$-homology of a crossed module to the homology of its classifying space with coefficients in the ring of the integers modulo $q$.

AMS 2000 Mathematics subject classification: Primary 18G50; 20J05. Secondary 18G30

Let $X$ be a Banach space and let $Y$ be a closed subspace of a Banach space $Z$. The following theorem is proved. Assume that $X^*$ or $Z^*$ has the approximation property. If there exists a bounded linear extension operator from $Y^*$ to $Z^*$, then any bounded linear operator $T:X\rightarrow Y$ is nuclear whenever $T$ is nuclear from $X$ to $Z$. The particular case of the theorem with $Z=Y^{**}$ is due to Grothendieck and Oja and Reinov. Numerous applications are presented. For instance, it is shown that a bounded linear operator $T$ from an arbitrary Banach space $X$ to an $\mathcal{L}_\infty$-space $Y$ is nuclear whenever $T$ is nuclear from $X$ to some Banach space $Z$ containing $Y$ as a subspace.

AMS 2000 Mathematics subject classification: Primary 46B20; 46B28; 47B10

Let $R$ be a semiprime ring with $U_{\trm{s}}$ its maximal symmetric ring of quotients and let $\rho_1$ and $\rho_2$ be two right ideals of $R$. We show that $\ell_R(\rho_1)=\ell_R(\rho_2)$ if and only if $\rho_1$ and $\rho_2$ satisfy the same differential identities with coefficients in $U_{\trm{s}}$, where $\ell_R(\rho_i)$ denotes the left annihilator of $\rho_i$ in $R$. This gives a generalization of several previous results in this area.

AMS 2000 Mathematics subject classification: Primary 6R50. Secondary 16N60

It is known that, for generic $q$, the $\mathcal{H}$-invariant prime ideals in $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$ are generated by quantum minors (see S. Launois, Les idéaux premiers invariants de $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$, J. Alg., in press). In this paper, $m$ and $p$ being given, we construct an algorithm which computes a generating set of quantum minors for each $\mathcal{H}$-invariant prime ideal in $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$. We also describe, in the general case, an explicit generating set of quantum minors for some particular $\mathcal{H}$-invariant prime ideals in $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$. In particular, if $(Y_{i,\alpha})_{(i,\alpha)\in[[1,m]]\times[[1,p]]}$ denotes the matrix of the canonical generators of $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$, we prove that, if $u\geq3$, the ideal in $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$ generated by $Y_{1,p}$ and the $u\times u$ quantum minors is prime. This result allows Lenagan and Rigal to show that the quantum determinantal factor rings of $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$ are maximal orders (see T. H. Lenagan and L. Rigal, Proc. Edinb. Math. Soc.46 (2003), 513–529).

AMS 2000 Mathematics subject classification: Primary 16P40. Secondary 16W35; 20G42

We obtain inverse factorial-series solutions of second-order linear difference equations with a singularity of rank one at infinity. It is shown that the Borel plane of these series is relatively simple, and that in certain cases the asymptotic expansions incorporate simple resurgence properties. Two examples are included. The second example is the large $a$ asymptotics of the hypergeometric function ${}_2F_1(a,b;c;x)$.

AMS 2000 Mathematics subject classification: Primary 34E05; 39A11. Secondary 33C05

The notion of an atomic operator between spaces of measurable functions was introduced in 2002 in a paper by Drakhlin, Ponosov and Stepanov in order to provide a reasonable generalization of local operators useful for applications. It has been shown that, roughly speaking, atomic operators amount to compositions of local operators with shifts. A natural problem is then when a continuous-in-measure atomic operator can be represented as a composition of a Nemytskiiˇ (composition) operator generated by a Carathéodory function, and a shift operator. In this paper we will show that the answer to this question is inherently related to the possibility of extending an atomic operator with continuity from a space of functions measurable with respect to some $\sigma$-algebra to a larger space of functions measurable with respect to a larger $\sigma$-algebra, as well as to the possibility of extending any $\sigma$-homomorphism from a smaller-measure algebra to a $\sigma$-homomorphism on a larger-measure algebra. We characterize precisely the condition on the respective $\sigma$-algebras which provides such possibilities and induces the positive answer to the above representation problem.

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

Let $u$ be a solution of a generalized Cauchy–Riemann system in $\mathbb{R}^n$. Suppose that $|u|\le1$ in the unit ball and $|u|\le\varepsilon$ on some closed set $E$. Classical results say that if $E$ is a set of positive Lebesgue measure, then $|u|\le C\varepsilon^\alpha$ on any compact subset of the unit ball. In the present work the same estimate is proved provided that $E$ is a subset of a hyperplane and the (capacitary) dimension of $E$ is greater than $n-2$. The proof gives control of constants $C$ and $\alpha$.

AMS 2000 Mathematics subject classification: Primary 31B35. Secondary 35B35; 35J45

New spaces of Lipschitz type on metric-measure spaces are introduced and they are shown to be just the well-known Besov spaces or Triebel–Lizorkin spaces when the smooth index is less than 1. These theorems also hold in the setting of spaces of homogeneous type, which include Euclidean spaces, Riemannian manifolds and some self-similar fractals. Moreover, the relationships amongst these Lipschitz-type spaces, Hajłasz–Sobolev spaces, Korevaar–Schoen–Sobolev spaces, Newtonian Sobolev space and Cheeger–Sobolev spaces on metric-measure spaces are clarified, showing that they are the same space with equivalence of norms. Furthermore, a Sobolev embedding theorem, namely that the Lipschitz-type spaces with large orders of smoothness can be embedded in Lipschitz spaces, is proved. For metric-measure spaces with heat kernels, a Hardy–Littlewood–Sobolev theorem is establish, and hence it is proved that Lipschitz-type spaces with small orders of smoothness can be embedded in Lebesgue spaces.

AMS 2000 Mathematics subject classification: Primary 42B35. Secondary 46E35; 58J35; 43A99