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Current Volume:

The current volume is the volume of 2019.

• 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.

Acknowledgements

This journal owes much to the following persons: Denis Bitouzé for LaTex support, Jean-Jacques Derycke for printing service, Sébastien Huart for computer engineering issues and Aurore Smets for secretarial tasks.

Supports

The department of Mathematics of Lille 1 provides financial support which ensures that this journal stays open access. The French Mathematical Society (SMF), the Dutch Mathematical Society (KWG), the Luxembourg Mathematical Society (SML) and the Fields Institute support this project, of a mathematics journal by a non-profit publisher, making research available free of charge for readers and authors.

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