North-Western European Journal of Mathematics Archives

2020 volume 6
• Given the notion of suborbifold of the second author (based on ideas of Borzellino/Brunsden) and the classical correspondence (up to certain equivalences) between (effective) orbifolds via atlases and effective orbifold groupoids, we analyze which groupoid embeddings correspond to suborbifolds and give classes of suborbifolds naturally leading to groupoid embeddings.
• Given the notion of suborbifold of the second author (based on ideas of Borzellino/Brunsden) and the c Let $\left(\mathcal{H},\left(.,.\right)\right)$ be a Hilbert space and let $\mathcal{L}\left(\mathcal{H}\right)$ be the linear space of bounded operators in $\mathcal{H}$. In this paper, we deal with $\mathcal{L}(\mathcal{H})$-valued function $Q$ that belongs to the generalized Nevanlinna class $\mathcal{N}_{\kappa} (\mathcal{H})$, where $\kappa$ is a non-negative integer. It is the class of functions meromorphic on $C \backslash R$, such that $Q(z)^{*}=Q(\bar{z})$ and the kernel $\mathcal{N}_{Q}\left( z,w \right):=\frac{Q\left( z \right)-{Q\left( w \right)}^{\ast }}{z-\bar{w}}$ has $\kappa$ negative squares. A focus is on the functions $Q \in \mathcal{N}_{\kappa} (\mathcal{H})$ which are holomorphic at $\infty$. A new operator representation of the inverse function $\hat{Q}\left( z \right):=-{Q\left( z \right)}^{-1}$ is obtained under the condition that the derivative at infinity $Q^{'}\left( \infty\right):=\lim\limits_{z\to \infty}{zQ(z)}$ is boundedly invertible operator. It turns out that $\hat{Q}$ is the sum $\hat{Q}=\hat{Q}_{1}+\hat{Q}_{2},\, \, \hat{Q}_{i}\in \mathcal{N}_{\kappa_{i}}\left( \mathcal{H} \right)$ that satisfies $\kappa_{1}+\kappa_{2}=\kappa$. That decomposition enables us to study properties of both functions, $Q$ and $\hat{Q}$, by studying the simple components $\hat{Q}_{1}$ and $\hat{Q}_{2}$.
• We provide a tool how one can view a polynomial on the affine plane of bidegree $(a,b)$ -- by which we mean that its Newton polygon lies in the triangle spanned by $(a,0)$, $(0,b)$ and the origin -- as a curve in a Hirzebruch surface having nice geometric properties. As an application, we study maximal $A_k$-singularities of curves of bidegree $(3,b)$ and find the answer for $b\leq 12$.
• We establish necessary and sufficient conditions for boundedness of composition operators on the most general class of Hilbert spaces of entire Dirichlet series with real frequencies. Depending on whether or not the space being considered contains any nonzero constant function, different criteria for boundedness are developed. Thus, we complete the characterization of bounded composition operators on all known Hilbert spaces of entire Dirichlet series of one variable.
• Hugo Steinhaus (1966a, b) has asked whether inside each acute angled triangle there is a point from which perpendiculars to the sides divide the triangle into three parts of equal areas. In this paper, we prove that $f:\mathbb{D} \rightarrow \mathbb{D}$ is a gyroisometry (hyperbolic isometry) if, and only if it is a continuous mapping that preserves the partition of a gyrotriangle (hyperbolic triangle) asked by Hugo Steinhaus.
2019 volume 5
• We give a new proof of the existence of a surjective symbol whose associated composition operator on $H^2 (\mathbb{D})$ is in all Schatten classes, with the improvement that its approximation numbers can be, in some sense, arbitrarily small. We show, as an application, that, contrary to the $1$-dimensional case, for $N \geq 2$, the behavior of the approximation numbers $a_n = a_n (C_\phi)$, or rather of {$\beta^-_N = \liminf_{n \to \infty} [a_n]^{1/ n^{1/ N}}$ or $\beta^+_N = \limsup_{n \to \infty} [a_n]^{1/ n^{1/ N}}$}, of composition operators on $H^2 (\mathbb{D}^N)$ cannot be determined by the image of the symbol.
• If $\mathcal{D}$ is a Reedy category and $\mathcal{M}$ is a model category, the category $\mathcal{M}^{\mathcal{D}}$ of $\mathcal{D}$-diagrams in $\mathcal{M}$ is a model category under the Reedy model category structure. If $\mathcal{C} \to \mathcal{D}$ is a Reedy functor between Reedy categories, then there is an induced functor of diagram categories $\mathcal{M}^{\mathcal{D}} \to \mathcal{M}^{\mathcal{C}}$. Our main result is a characterization of the Reedy functors $\mathcal{C} \to \mathcal{D}$ that induce right or left Quillen functors $\mathcal{M}^{\mathcal{D}} \to \mathcal{M}^{\mathcal{C}}$ for every model category $\mathcal{M}$. We apply these results to various situations, and in particular show that certain important subdiagrams of a fibrant multicosimplicial object are fibrant.
• In this paper, we consider the space of entire functions with normal type growth for a given proximate order and a continuous linear operator from such space into itself which is defined by a partial differential operator of infinite order. We will study corresponding partial differential equations with variable coefficients in the cases of regular singular type and of Korobeı̆nik type.
• An increasing sequence of positive integers $(n_k)$ is said to be Jamison if whenever $T$ is a linear bounded operator on a complex separable Banach space, the following holds : $\sup_{k}||T^{n_k}||<\infty \Rightarrow \sigma_{p}(T)\cap S^1 \text{ is countable}$ In this paper, we study certain perturbations on the set of Jamison sequences and prove a stability result.
• We discuss the evaluation of convolution sums involving the divisor function, $\underset{\substack{ {(l,m)\in\mathbb{N}^{2}} \\ {\alpha\,l+\beta\,m=n} } }{\sum}\sigma(l)\sigma(m)$, for the class of levels $\alpha\beta$ belonging to all natural numbers. The evaluation of convolution sums belonging to this class is achieved by applying modular forms and primitive Dirichlet characters. We illustrate our method with the explicit examples for the levels $\alpha\beta=33$, $40$, $45$, $50$, $54$, and $56$. As a corollary, the known convolution sums for the levels $\alpha\beta=10$, $11$, $12$, $15$, $16$, $18$, $24$, $25$, $27$, $32$ and $36$ are improved when we revisit their evaluations. If the level $\alpha\beta\equiv 0 \pmod{4}$, we determine natural numbers $a,b$ and use the evaluated convolution sums together with other known convolution sums to carry out the number of representations of $n$ by the octonary quadratic forms $a\,\underset{i=1}{\overset{4}{\sum}}x_{i}^{2}+ b\,\underset{i=5}{\overset{8}{\sum}}x_{i}^{2}$. Similarly, if the level $\alpha\beta\equiv 0 \pmod{3}$, we compute natural numbers $c,d$ and make use of the evaluated convolution sums together with other known convolution sums to determine the number of representations of $n$ by the octonary quadratic forms $c\,\underset{i=1}{\overset{2}{\sum}}\,(\,x_{2i-1}^{2}+ x_{2i-1}x_{2i} + x_{2i}^{2}\,) + d\,\underset{i=3}{\overset{4}{\sum}}\,(\,x_{2i-1}^{2}+ x_{2i-1}x_{2i} + x_{2i}^{2}\,)$. In addition, we determine formulae for the number of representations of a positive integer $n$ when $(a,b)=(1,1)$, $(1,3)$, $(1,6)$, $(2,3)$.
• Let $X$ be a Banach space and $K$ an absolutely convex, weak$^\ast$-compact subset of $X^\ast$. We study consequences of $K$ having a large or undefined Szlenk index and subsequently derive a number of related results concerning basic sequences and universal operators. We show that if $X$ has a countable Szlenk index then $X$ admits a subspace $Y$ such that $Y$ has a basis and the Szlenk indices of $Y$ are comparable to the Szlenk indices of $X$. If $X$ is separable, then $X$ also admits subspace $Z$ such that the quotient $X/Z$ has a basis and the Szlenk indices of $X/Z$ are comparable to the Szlenk indices of $X$. We also show that for a given ordinal $\xi$ the class of operators whose Szlenk index is not an ordinal less than or equal to $\xi$ admits a universal element if and only if $\xi<\omega_1$; W.B. Johnson's theorem that the formal identity map from $\ell_1$ to $\ell_\infty$ is a universal non-compact operator is then obtained as a corollary. Stronger results are obtained for operators having separable codomain.
2018 volume 4
• We study in this paper the functional equation $\alpha \mathbf{u}(t)+\mathcal{C}\star(\chi \mathbf{u})(t)=\mathbf{f}(t)$ where $\alpha\in\mathbb{C}^{d\times d}$, $\mathbf{u},\mathbf{f}:\mathbb{R}\rightarrow\mathbb{C}^d$, $\mathbf{u}$ being unknown. The term $\mathcal{C}\star(\chi \mathbf{u})(t)$ denotes the discrete convolution of an almost zero matricial mapping $\mathcal{C}$ with discrete support together with the product of $\mathbf{u}$ and the characteristic function $\chi$ of a fixed segment. This equation combines some aspects of recurrence equations and/or delayed functional equations, so that we may construct a matricial based framework to solve it. We investigate existence, unicity and determination of the solution to this equation. In order to do this, we use some new results about linear independence of monomial words in matrix algebras.
• By a recent work of Gran-Kadjo-Vercruysse, the category of cocommutative Hopf algebras over a field of characteristic zero is semi-abelian. In this paper, we explore some properties of this category, in particular we show that its abelian core is the category of commutative and cocommutative Hopf algebras.
• We investigate the inverse problem of simultaneously estimating the state and the spatial diffusion coefficient for an age-structured population model. The time evolution of the population is supposed to be known on a subdomain in space and age. We generalize to the infinite dimensional setting an adaptive observer originally proposed for finite dimensional systems.
• We discuss comparison principles, the asymptotic behaviour, and the occurrence of blow up phenomena for nonlinear parabolic problems involving the $p$--Laplacian operator of the form $\left\{\begin{array}{ll} \partial_t u=\Delta_p u+f(t,x,u)&\mbox{in}\ \Omega\ \mbox{ for }\ t>0,\\ \sigma \partial_t u+|\nabla u|^{p-2}\partial_\nu u=0&\mbox{on}\ \partial\Omega\ \mbox{ for }\ t>0,\\ u(0,\cdot)=u_0 &\mbox{in}\ \overline{\Omega},\\ \end{array}\right.$ where $\Omega$ is a bounded domain of ${\mathbb R}^N$ with Lipschitz boundary, and where $\Delta_p u:={\mathrm div}\, \left(|\nabla u|^{p-2}\nabla u\right)$ is the $p$--{\itshape Laplacian} operator for $p>1$. As for the {\itshape dynamical} time lateral boundary condition $\sigma \partial_t u+|\nabla u|^{p-2}\partial_\nu u=0$ the coefficient $\sigma$ is assumed to be a nonnegative constant. In particular, the asymptotic behaviour in the large for the parameter dependent nonlinearity $f(\cdot,\cdot,u)=\lambda|u|^{q-2}u$ will be investigated by means of the evolution of associated norms.
• Let $X$ be a hyperkähler variety, and assume $X$ has a non-symplectic automorphism $\sigma$ of order $>\frac{1}{2}\dim X$. Bloch's conjecture predicts that the quotient $X/<\sigma>$ should have trivial Chow group of $0$--cycles. We verify this for Fano varieties of lines on certain special cubic fourfolds having an order $3$ non-symplectic automorphism.
• We continue our study of solutions to linear parabolic partial differential equations (PDEs) by means of an asymptotic method that is based on approximate Green functions. A substantial part of this method is devoted to constructing the approximate Green function. In this paper, we approximate the Green function (or heat kernel) by asymptotically developing it in a small parameter other than time. While the method is general, in order to better illustrate it, we concentrate on the $\lambda$-SABR partial differential equation (PDE for short), which we study in detail. The $\lambda$SABR PDE is a particular evolution PDE that arises in applications to stochastic volatility models (Hagan, Kumar, Lesniewski, and Woodward, Wilmott Magazine, 2002). Concretely, we study the generation and approximation of several semi-groups associated to the SABR PDE, some of which are non-standard because their generators are not uniformly elliptic and have unbounded coefficients. These type of generators appear also in the study of quasi-linear evolution equations. For some of the resulting semi-groups, we obtain explicit formulas by using a general technique based on solvable Lie groups that we develop in this paper. We thus obtain a simple, explicit approximation for the solution of the $\lambda$-SABR PDE and we prove explicit error bounds. In view of the potential applications, we have tried to make our paper as self-contained as reasonably possible.
• We summarize our findings in the analysis of adaptive finite element methods for the efficient discretization of control constrained optimal control problems. We particularly focus on convergence of the adaptive method, i.e we show that the sequence of adaptively generated discrete solutions converges to the true solution. The result covers the variational discretization (Hinze) as well as control discretizations with piecewise discontinuous finite elements. Moreover, the presented theory can be applied to a large class of state equations, to boundary control and boundary observation.
• The classical Hopf invariant is defined for a map $f\colon S^r \to X$. Here we define hcat' which is some kind of Hopf invariant built with a construction in Ganea's style, valid for maps not only on spheres but more generally on a relative suspension' $f\colon \Sigma_A W \to X$. We study the relation between this invariant and the sectional category and the relative category of a map. In particular, for $\iota_X\colon A\to X$ being the `restriction' of $f$ on $A$, we have ${\rm relcat}~\iota_X \leq {\rm hcat}~f \leq {\rm relcat}~ \iota_X +1$ and ${\rm relcat}~f \leq {\rm hcat}~f$.
2017 volume 3
• We consider a multidimensional SDE with a Gaussian noise and a drift vector being a vector function of bounded variation. We prove the existence of generalized derivative of the solution with respect to the initial conditions and represent the derivative as a solution of a linear SDE with coefficients depending on the initial process. The obtained representation is a natural generalization of the expression for the derivative in the smooth case. In the proof we use the results on continuous additive functionals.
• We study the structure of the space of coarse Lipschitz maps between Banach spaces. In particular we introduce the notion of norm attaining coarse Lipschitz maps. We extend to the case of norm attaining coarse Lipschitz equivalences, a result of G. Godefroy on Lipschitz equivalences. This leads us to include the non separable versions of classical results on the stability of the existence of asymptotically uniformly smooth norms under Lipschitz or coarse Lipschitz equivalences.
• We characterize the limited operators by differentiability of convex continuous functions. Given Banach spaces $Y$ and $X$ and a linear continuous continuous operator $T: Y\longrightarrow X$, we prove that $T$ is a limited operator if and only if, for every convex continuous function $f: X \longrightarrow \mathbb{R}$ and every point $y\in Y$, $f\circ T$ is Fréchet differentiable at $y \in Y$ whenever $f$ is Gâteaux differentiable at $T(y)\in X$.
• We provide further techniques to study the Dolbeault and Bott-Chern cohomologies of deformations of solvmanifolds by means of finite-dimensional complexes. By these techniques, we can compute the Dolbeault and Bott-Chern cohomologies of some complex solvmanifolds, and we also get explicit examples, showing in particular that either the $\partial\overline{\partial}$-Lemma or the property that the Hodge and Frölicher spectral sequence degenerates at the first level are not closed under deformations.
• It is known for quite some time that the extension theorems play an important role in the homogenization of the periodic (heterogeneous) mediums. However, the construction of such extension operators depends on a reflection technique but for the functions in $H^{l,r}(\Omega_p^\varepsilon)$ $(l>2)$ this reflection technique is not so straightforward, and would lead to a rather cumbersome anaylsis. In this work, we will give a short overview of some extension operators mapping from $L^{r}(S;H^{l,r}(\Omega_p^\varepsilon))\cap H^{1,r}(S; H^{l,s}(\Omega_p^\varepsilon)^*) \to L^{r}(S;H^{l,r}(\Omega))\cap H^{1,r}(S; H^{l,s}(\Omega)^*)$ using a much simpler approach. This note also generalizes the previously known results to Lipschitz domains and for any $r\in \mathbb{N}$ such that (s.t.) $\frac{1}{r}+\frac{1}{s}=1$.
• An MSTD set is a finite set of integers with more sums than differences. It is proved that, for infinitely many postivie integers $k$, there are infinitely many affinely inequivalent MSTD sets of cardinality $k$. There are several related open problems.
• In this paper we prove that for any simplicial set $B$, there is a Quillen equivalence between the covariant model structure on $\mathbf{S}/B$ and a certain localization of the projective model structure on the category of simplicial presheaves on the simplex category $\Delta/B$ of $B$. We extend this result to give a new Quillen equivalence between this covariant model structure and the projective model structure on the category of simplicial presheaves on the simplicial category $\mathfrak{C}[B]$. We study the relationship with Lurie's straightening theorem. Along the way we also prove some results on localizations of simplicial categories and quasi-categories.
2016 volume 2
• In this paper iteration stable (STIT) tessellations of the $d$-dimensional Euclidean space are considered. By a careful analysis of the capacity functional an alternative proof is given for the fact that STIT tessellations are mixing.
• The invariant $\Theta$ is the simplest $3$--manifold invariant defined by counting graph configurations. It is actually an invariant of rational homology $3$--spheres $M$ equipped with a combing $X$ over the complement of a point, where a combing is a homotopy class of nowhere vanishing vector fields. The invariant $\Theta(M,X)$ is the sum of $6 \lambda(M)$ and $\frac{p_1(X)}{4}$, where $\lambda$ denotes the Casson-Walker invariant, and $p_1$ is an invariant of combings, which is an extension of a first relative Pontrjagin class, and which is simply related to a Gompf invariant $\theta_G$. In Lescop (2015), we proved a combinatorial formula for the $\Theta$--invariant in terms of decorated Heegaard diagrams. In this article, we study the variations of the invariants $p_1$ or $\theta_G$ when the decorations of the Heegaard diagrams that define the combings change, independently. Then we prove that the formula of Lescop (2015) defines an invariant of combed once punctured rational homology $3$--spheres without referring to configuration spaces. Finally, we prove that this invariant is the sum of $6 \lambda(M)$ and $\frac{p_1(X)}{4}$ for integer homology $3$--spheres, by proving surgery formulae both for the combinatorial invariant and for $p_1$.
• We study an impartial avoidance game introduced by Anderson and Harary. The game is played by two players who alternately select previously unselected elements of a finite group. The first player who cannot select an element without making the set of jointly-selected elements into a generating set for the group loses the game. We develop criteria on the maximal subgroups that determine the nim-numbers of these games and use our criteria to study our game for several families of groups, including nilpotent, sporadic, and symmetric groups.
• We study composition operators acting between $\mathcal{N}_p$-spaces in the unit ball in $\mathbb{C}^n$. We obtain characterizations of the boundedness and compactness of $C_{\varphi}:\mathcal{N}_p\longrightarrow\mathcal{N}_q$ for $p, q>0$.
• In 2002, Fatiha Alabau, Piermarco Cannarsa and Vilmos Komornik investigated the extent of asymptotic stability of the null solution for weakly coupled partially damped equations of the second order in time. The main point is that the damping operator acts only on the first component and, whenever it is bounded, the coupling is not strong enough to produce an exponential decay in the energy space associated to the conservative part of the system. As a consequence, for initial data in the energy space, the rate of decay is not exponential. Due to the nature of the result it seems at first sight impossible to obtain the asymptotic stability result by the classical Liapunov method. Surprisingly enough, this turns out to be possible and we exhibit, under some compatibility conditions on the operators, an explicit class of Liapunov functions which allows to do 3 different things: \\ 1) When the problem is reduced to a stable finite dimensional space, we recover the exponential decay by a single differential inequality and we estimate the logarithmic decrement of the solutions with worst (slowest) decay. The estimate is optimal at least for some values of the parameters. \\ 2) We explain the form of the stability result obtained by the previous authors when the coupling operator is a multiple of the identity, so that the decay is not exponential.\\ 3) We obtain new exponential decay results when the coupling operator is strong enough (in particular unbounded). The estimate is again sharp for some solutions.
2015 volume 1
• A metric compact space $M$ is seen as the closure of the union of a sequence $(M_n)$ of finite $\epsilon_n$-dense subsets. Extending to $M$ (up to a vanishing uniform distance) Banach-space valued Lipschitz functions defined on $M_n$, or defining linear continuous near-extension operators for real-valued Lipschitz functions on $M_n$, uniformly on $n$ is shown to be equivalent to the bounded approximation property for the Lipschitz-free space $\mathcal{F}(M)$ over $M$. Several consequences are spelled out.
• This tutorial paper presents a survey of results, both classical and new, linking inner functions and operator theory.Topics discussed include invariant subspaces, universal operators, Hankel and Toeplitz operators, model spaces, truncated Toeplitz operators, restricted shifts, numerical ranges, and interpolation.
• This subordination principle states roughly: if a property is true for Hardy spaces in some kind of domains in ${\mathbb{C}}^{n}$ then it is also true for the Bergman spaces of the same kind of domains in ${\mathbb{C}}^{n-1}.$ We give applications of this principle to Bergman-Carleson measures, interpolating sequences for Bergman spaces, $A^{p}$ Corona theorem and characterization of the zeros set of Bergman-Nevanlinna class. These applications give precise results for bounded strictly-pseudo convex domains and bounded convex domains of finite type in ${\mathbb{C}}^{n}.$
• Let $d\geq 2$, $A \subset \mathbb{Z}^d$ be finite and not contained in a translate of any hyperplane, and $q \in \mathbb{Z}$ such that $|q| \geq 2$. We show $$|A+ q \cdot A| \geq (|q|+d+1)|A| - O(1).$$
• We characterise smooth curve in a smooth cubic threefold whose blow-ups produce a weak-Fano threefold. These are curves $C$ of genus $g$ and degree $d$, such that (i) $2(d-5) \le g$ and $d\le 6$; (ii) $C$ does not admit a 3-secant line in the cubic threefold. Among the list of ten possible such types $(g,d)$, two yield Sarkisov links that are birational selfmaps of the cubic threefold, namely $(g,d) = (0,5)$ and $(2,6)$. Using the link associated with a curve of type $(2,6)$, we are able to produce the first example of a pseudo-automorphism with dynamical degree greater than $1$ on a smooth threefold with Picard number $3$. We also prove that the group of birational selfmaps of any smooth cubic threefold contains elements contracting surfaces birational to any given ruled surface.
• Cet article est consacré à l'étude détaillée du vaste projet éditorial de Borel dans les années 1920 et 1930 autour des probabilités et de leurs applications. Après avoir rappelé quelques éléments sur la biographie de Borel et décrit la mise en place du projet, nous examinons les acteurs qui y ont participé pour mieux cerner le réseau que Borel a mis en place pour arriver à ses fins. Enfin, dans une troisième partie les fascicules du Traité sont aussi examinés individuellement afin de dessiner le contour du domaine probabiliste tel que Borel le concevait, dont nous montrons qu'il est en fait déjà obsolète au moment où la publication s'achève.
• We are interested here in a birth-and-growth process where germs are born according to a Poisson point process with intensity measure invariant under space translations. The germs can be born in free space and then start growing until occupying the available space. In order to consider various ways of growing, we describe the crystals at each time through their geometrical properties. In this general framework, the crystallization process can be characterized by the random field giving for a point in the state space the first time this point is reached by a crystal. We prove under general conditions that this random field is mixing in the sense of ergodic theory and obtain estimates for the coefficient of absolute regularity.

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