We extend the notion of diagram spectra of M. A. Mandell, J. P. May, S. Schwede, B. Shipley to general model categories to obtain a stabilization process with respect to any set of objects. Following M. Hovey, we discuss the existence and some properties of diagram spectra, such as monoidal structures, invertibility, idempotency, homotopy invariance and functoriality. We end with some comparisons between categories of spectra associated to different diagrams.

Constancy of some maps into certain f-manifolds is discussed.

Soit $S/k$ une variété de Severi-Brauer définie sur un corps $k$. Soit $f$ un automorphisme de Kummer de $S$, c'est-à-dire un automorphisme d'ordre $n=dim_k(S) +1$ à points fixes isolés. Si $k$ contient une racine primitive $n$--ième de l'unité, alors le corps $k(S)^{f}$ des fonctions rationnelles de $S$ invariantes par $f$ est transcendant pur sur $k$.

Let $S/k$ be a Severi-Brauer variety defined over a field $k$. Let $f$ be a Kummer automorphism of $S$, i.e. an automorphism of order $n=dim_k(S) +1$ with finitely many fixed points. If $k$ contains a primitive $n$-th root of unity, then the field $k(S)^f$ of $f$-invariant rational functions is purely transcendental over $k$.

For a given number $s>-1$ and a multiindex $\alpha\in\Bbb N^{n}$ we give a proof of the following equality: \[ \int_{\left\Vert z\right\Vert <R}z^{\alpha}\overline{z^{\alpha}}\left(R^{2}-\left\Vert z\right\Vert ^{2}\right)^{s}dz=\frac{\pi^{n}\alpha!R^{2(s+|\alpha|+n)}}{\prod_{i=1}^{|\alpha|+n}(s+i)}.\] As a result we receive different properties of the sets defined by the following formula \[ E^{s}(f)=\left\{ z\in\partial\Bbb B^{n}:\:\int_{|\lambda|<1}\left|f(\lambda z)\right|^{2}\left(1-|\lambda|^{2}\right)^{s}d\mathfrak{L}^{2}=\infty\right\} \] for the holomorphic function $f\in\Bbb O(\Bbb B^{n})$.

In this note, we introduce and briefly study a non-binary analogue of the ``classical'' Walsh transform. It is shown that this transform allows to rewrite the definition of normalized epistasis in terms of generalized Walsh coefficients, in a way which is both practical and elegant. Some examples are included, aiming to give a first indication of the strength of this approach.

In this note we solve the Müntz problem for the space $\mathbf{C}(K)$ whenever $K\subset [0,\infty)$ is a countable compact set which satisfies certain additional assumptions and we propose the general case as an open question.

We give a combinatorial proof of a theorem of Gromov, which extends the scope of small cancellation theory to group presentations arising from labelled graphs.

Let $G$ be a finite Dini-smooth domain and $% w=\varphi _{0}(z)$ be the conformal mapping of $G$ onto $D\left( 0,r_{0}\right) :=\{w:\mid w\mid <r_{0}\}$ with the normalization $ \varphi _{0}(z_{0})=0$, $\varphi _{0}^{^{\prime }}(z_{0})=1$, where $z_{0}\in G$. We investigate the approximation properties of the Bieberbach polynomials $ \pi _{n}(z)$, $ n=1,2,3, \cdots $ for the pair $\left( G,z_{0}\right) $ and estimate the error \begin{equation*} \parallel \varphi _{0}-\pi _{n}\parallel _{\overline{G}} := \max \{\mid \varphi _{0}\left( z\right) -\pi _{n}\left( z\right) \mid :z\in \overline{G} \} \end{equation*} in accordance with the geometric parameters of $\overline{G}$.

A commutative semigroup $S$ is subarchimedean if there is an element $z\in S$ such that for every $a\in S$ there exist a positive integer $n$ and $x\in S$ such that $z^n=ax$. Such a semigroup is archimedean if this holds for all ${z\in S}$. A commutative cancellative idempotent-free archimedean semigroup is an $\frak{N}$-semigroup. We study the structure of semigroups in the title as related to $\frak{N}$-semigroups.

The paper presents new approximation and Leray-Schauder type results for multimaps in the class S-KKM.

Given a field $K$, all polynomial maps ${\mathbb S}^{m_1}(K)\times\cdots\times{\mathbb S}^{m_k}(K)\to{\mathbb T}^n(K)$ of generalized tori over $K$ are studied. Furthermore, a full description of holomorphic maps ${\mathbb C}^m\to{\mathbb C}^{n_1}\times\cdots\times{\mathbb C}^{n_k}$ and ${\mathbb C}^{m_1}\times\cdots\times {\mathbb C}^{m_k}\to {\mathbb C}^n$ which restrict to maps ${\mathbb T}^m\to{\mathbb S}^{2n_1-1}\times\cdots\times{\mathbb S}^{2n_k-1}$ and ${\mathbb S}^{2m_1-1}\times\cdots\times{\mathbb S}^{2m_k-1}\to{\mathbb T}^n$ respectively, is presented.

Let $k$ be a field of positive characteristic $p$. First we describe some specific subfunctors of the Burnside functor $ k \otimes_{\mathbb {Z}} B $. We prove next that the restriction of the functor of rational representations $ k \otimes_{\mathbb {Z}} {R}_{\mathbb{Q}} $ to abelian finite $p$-groups, has a unique maximal filtration $$ \qquad k \otimes_{\mathbb {Z}} {R}_{\mathbb{Q}} = \overline{I_{1}} \supseteq \overline{I_{2}} \supseteq \overline{I_{3}} \supseteq \raisebox{0.5ex}{\ldots}$$

We prove that the space \textbf{$L_\Omega ^p\left( R^n\right) $}, where $% \Omega =\left\{ \left( 1+\left\| x\right\| ^2\right) ^s:s>\frac{n(p-1)} 2\right\} $ and $p\in \left] 1,+\infty \right[ $ , is a regular locally $m$-convex algebra. Others results are also obtained.

The present article is part of the program described in Bennett, R. Gramlich, C. Hoffman, S. Shpectorov. Here we study the Phan-theoretic flipflop geometries related to the flip induced by a nondegenerate orthogonal form on a vector space over an arbitrary field of characteristic distinct from two. We obtain amalgam results in the spirit of Phan's theorems for fields that do not admit a quadratic extension and for real closed fields.