In two preceding papers [2; 3] the author has studied the algebras of the irreducible representations λ and the classes Ci of a finite group G. Integral representations {λ} and {Ci} of these algebras are derivable from the appropriate multiplication tables [4]. It should be emphasized, however, that the symmetry properties of the two sets of structure constants are not the same, and this leads to somewhat greater complexity in the formulae relating to classes as compared to representations.

We construct a bipartite Euler system in the sense of Howard for Hilbert modular eigenforms of parallel weight two over totally real fields, generalizing works of Bertolini–Darmon, Longo, Nekovar, Pollack–Weston, and others. The construction has direct applications to Iwasawa's main conjectures. For instance, it implies in many cases one divisibility of the associated dihedral or anticyclotomic main conjecture, at the same time reducing the other divisibility to a certain nonvanishing criterion for the associated $p$-adic $L$-functions. It also has applications to cyclotomic main conjectures for Hilbert modular forms over $\text{CM}$ fields via the technique of Skinner and Urban.

Let T(s), s ≧ 0, be a strongly continuous semigroup of bounded operators on a separable Hilbert space . T(s) is said to be quasicyclic if there is a continuum of vectors such that T(s)xt = xs+t for all s, t > 0 and

From Euclid to Hilbert, and beyond, the primitive terms of geometry have been taken as “point,” “line,” etc., while “distance” plays a secondary role. The reversal of this situation is a modern development. Frechet [4], in 1906 first considered the properties of distance which should be formalized. The most significant contributions to the geometric properties of metric spaces have been by Menger [10] and Blumenthal [2; 3].

Let $\mathcal{H}$ and $\mathcal{K}$ be infinite-dimensional separable Hilbert spaces and $\text{Lat}\,\mathcal{H}$ the lattice of all closed subspaces oh $\mathcal{H}$. We describe the general form of pairs of bijective maps $\phi ,\,\psi :\,\text{Lat}\,\mathcal{H}\,\to \,\text{Lat}\,\mathcal{K}$ having the property that for every pair $U,\,V\,\in \,\text{Lat}\,\mathcal{H}$ we have $\mathcal{H}\,=\,U\,\oplus \,V\,\Leftrightarrow \,\mathcal{K}\,=\,\phi \left( U \right)\,\oplus \,\psi \,\left( V \right)$. Then we reformulate this theorem as a description of bijective image equality and kernel equality preserving maps acting on bounded linear idempotent operators. Several known structural results for maps on idempotents are easy consequences.

Soit ℰn l'anneau des germes en 0 des fonctions numériques de classe C∞ dans Rn. On écrira aussi ℰx pour ℰn où x est la variable dans Rn. Désignons par Diff (n) le groupe des germes en 0 des difféomorphismes locaux τ de classe C∞ d'un voisinage de 0 dans Rn sur un voisinage de 0 tels que τ(0) = 0. Soient G un sous-groupe de Lie de Glp(R) et C∞0(Rn, G) l'ensemble des germes en 0 des applications de classe C∞ de Rn dans G.

Hölder's theorem on archimedean groups states:

An ordered (abelian) group G is order isomorphic to an ordered subgroup of the ordered group R of real numbers if and only if it is archimedean.

We comprehend this theorem in the following setting: G is a Z-module and Ris the completion with respect to the open interval topology of the ordered field Q; Qitself is the ordered quotient field of the ordered domain Z.

Rephrasing the situation, we raise the following question: We start with a fully ordered domain A,let Kbe its ordered quotient field. We endow Kwith the open interval topology and consider , the topological completion of K. Is it possible to impose a compatible order structure on and if this can be done, when can we say that an ordered A-module Mis order isomorphic to an ordered A-submodule of ? In Theorem 3.1, we obtain a set of necessary and sufficient conditions for this isomorphism to hold.

Let V be a class of similar algebras. An algebra is V-injective provided and whenever and fis a one-to-one homomorphism from into and g is a homomorphism from into , then there is a homomorphism h from into such that h º f = g. So is injective provided all diagrams of the following sort can be completed.

Dually, is V-projective provided and whenever and f is a homomorphism from onto and g is a homomorphism from into , then there is a homomorphism h from into such that f º h = g. So is projective provided all diagrams of the following sort can be completed:

This usage of the words “projective” and “injective” differs somewhat from the usage current in category theory.

In the present paper a general form of integral limit laws for additive functions is obtained. Our limit law contains Kubilius’ results [5] on his class H. In the proof we make use of characteristic functions (Fourier transforms), which reduces our problem to finding asymptotic formulas for sums of multiplicative functions. This requires an extension of previous results in order to enable us to take into consideration the parameter of the characteristic function in question. We call this extension a parametric mean value theorem for multiplicative functions and its proof is analytic on the line of [4].

The non-archimedean power series spaces, ${{A}_{1}}(a)\,\text{and}\,{{A}_{\infty }}(b)$, are the best known and most important examples of non-archimedean nuclear Fréchet spaces. We prove that the range of every continuous linear map from ${{A}_{p}}(a)\,\text{to}\,{{A}_{q}}(b)$ has a Schauder basis if either $p\,=\,1$ or $p\,=\,\infty $ and the set ${{M}_{b,a}}$ of all bounded limit points of the double sequence (${{({{b}_{i}}/{{a}_{j}})}_{i,j\in \mathbb{N}}}$ is bounded. It follows that every complemented subspace of a power series space ${{A}_{p}}(a)$ has a Schauder basis if either $p\,=\,1$ or $p\,=\,\infty $ and the set ${{M}_{a,a}}$ is bounded.

The notions of monotone bases and bases of subspaces are well known in a normed linear space setting and have obvious extensions to pseudo-metrizable linear topological spaces. In this paper, these notions are extended to arbitrary linear topological spaces. The principal result gives a list of properties that are equivalent to a sequence (Mi) of complete subspaces being an e-Schauder basis of subspaces for the closed linear span of . A corollary of this theorem is the fact that an e-Schauder basis for a dense subspace of a linear topological space is an e-Schauder basis for the whole space.

It is the purpose of this note to prove the following theorem: Let ƒ: G → R be a non-constant continuous function with G a locally compact connected topological group and with R the real numbers. Let C = ƒ(G) and suppose that F: C × C → C is a junction such that

Then ƒ is monotone and open and F is continuous.

Let $\left( T,\,M \right)$ be a complete local (Noetherian) ring such that $\dim\,T\,\ge \,2$ and $\left| T \right|\,=\,\left| T/M \right|$ and let ${{\left\{ {{p}_{i}} \right\}}_{i\in \Im }}$ be a collection of elements of $T$ indexed by a set $\mathcal{J}$ so that $\left| \mathcal{J} \right|\,<\,\left| T \right|$. For each $i\,\in \,\mathcal{J}$, let ${{C}_{i}}:=\left\{ {{Q}_{i1}},...,{{Q}_{i{{n}_{i}}}} \right\}$ be a set of nonmaximal prime ideals containing ${{p}_{i}}$ such that the ${{Q}_{ij}}$ are incomparable and ${{p}_{i}}\in {{Q}_{jk}}$ if and only if $i\,=\,j$. We provide necessary and sufficient conditions so that $T$ is the $\mathbf{m}$-adic completion of a local unique factorization domain $\left( A,\,\mathbf{m} \right)$, and for each $i\,\in \,\mathcal{J}$, there exists a unit ${{t}_{i}}$ of $T$ so that ${{p}_{i}}{{t}_{i}}\in A$ and ${{C}_{i}}$ is the set of prime ideals $Q$ of $T$ that are maximal with respect to the condition that $Q\cap A={{p}_{i}}{{t}_{i}}A$.

We then use this result to construct a (nonexcellent) unique factorization domain containing many ideals for which tight closure and completion do not commute. As another application, we construct a unique factorization domain $A$ most of whose formal fibers are geometrically regular.

In [6] we have constructed certain normal subgroups G7 of the elementary subgroup GR of the Chevalley group G(L, R) over R corresponding to a finite dimensional simple Lie algebra L over the complex field, where R is a commutative ring with identity. The method employed was to augment somewhat the generators of the elementary subgroup EI of G corresponding to an ideal I of the underlying Chevalley algebra LR;EI is thus the group generated by all xr(t) in G having the property that ter ⊂ I. In [6, § 5] we noted that in general EI actually had to be enlarged for a normal subgroup of GR to be obtained.

This note is an addendum to my earlier paper [8]. The class of adjoint abelian operators discussed there was small because the compatibility relation between the operator and the duality map was too restrictive. (In effect, the relation is appropriate for Hilbert space, but ill-suited for other Banach spaces where the unit ball is not round.) However, the techniques introduced in [8] permit us to readily obtain a spectral theory (of the Dunford type) for a wider class of operators on Banach spaces, as we shall show.

A duality system for the operator T is an ordered sextuple

(i) T is a bounded linear operator mapping the Banach space B into B,

(ii) ϕ is a duality map from B to B*. Thus, for x ∊ B, ϕ(x) = x* ∊ B*, where ‖x‖ = ‖x*‖ and x*(x) = ‖x‖2. The existence of ϕ follows easily from the Hahn-Banach Theorem.

Soit R un anneau et soit Gop un sous groupe de Aut (R). Formons l'anneau de groupe à gauche (“skew group ring”) RSG, en imposant, sur le module libre à gauche avec une base G, la multiplication rg = grg. (On écrit Gop au lieu de G car, si G < Inn (R) = le groupe d'automorphismes intérieurs, la condition g–rg = rg; implique que le homomorphisme naturel est vraiment un anti-homomorphisme).

Nous considérons le cas où R = MnF, F un corps, MnF l'anneau de matrices, d'ordre n, et Gop un sous-groupe (généralement) fini, de PGL(n, F) = Inn (R). Ce travail est consacré à l'étude de la simplicité de RSG.

The Banach space of bounded continuous real or complexvalued functions on a topological space X is denoted C(X). An averaging operator for an onto continuous function ϕ : X → Y is a bounded linear projection of C(X) onto the subspace ﹛ƒ ∈ C(X) : f is constant on each set ϕ -1(y) for y ∈ Y﹜. The projection constant p(ϕ) for an onto continuous map ϕ is the lower bound for the norms of all averaging operators for ϕ ﹛p(ϕ) = ∞ if there is no averaging operator for ϕ).

This article aims to establish fractional Sobolev trace inequalities, logarithmic Sobolev trace inequalities, and Hardy trace inequalities associated with time-space fractional heat equations. The key steps involve establishing dedicated estimates for the fractional heat kernel, regularity estimates for the solution of the time-space fractional equations, and characterizing the norm of $\dot {W}^{\nu /2}_p(\mathbb {R}^n)$ in terms of the solution $u(x,t)$. Additionally, fractional logarithmic Gagliardo–Nirenberg inequalities are proven, leading to $L^p-$logarithmic Sobolev inequalities for $\dot {W}^{\nu /2}_{p}(\mathbb R^{n})$. As a byproduct, Sobolev affine trace-type inequalities for $\dot {H}^{-\nu /2}(\mathbb {R}^n)$ and local Sobolev-type trace inequalities for $Q_{\nu /2}(\mathbb {R}^n)$ are established.

There are infinitely many triplets of primes $p,\,q,\,r$ such that the arithmetic means of any two of them, $\frac{p+q}{2},\,\frac{p+r}{2},\,\frac{q+r}{2}$ are also primes. We give an asymptotic formula for the number of such triplets up to a limit. The more involved problem of asking that in addition to the above the arithmetic mean of all three of them, $\frac{p+q+r}{3}$ is also prime seems to be out of reach. We show by combining the Hardy-Littlewood method with the sieve method that there are quite a few triplets for which six of the seven entries are primes and the last is almost prime.

The main theme of this paper is to study $\tau $-tilting subcategories in an abelian category $\mathscr {A}$ with enough projective objects. We introduce the notion of $\tau $-cotorsion torsion triples and investigate a bijection between the collection of $\tau $-cotorsion torsion triples in $\mathscr {A}$ and the collection of support $\tau $-tilting subcategories of $\mathscr {A}$, generalizing the bijection by Bauer, Botnan, Oppermann, and Steen between the collection of cotorsion torsion triples and the collection of tilting subcategories of $\mathscr {A}$. General definitions and results are exemplified using persistent modules. If $\mathscr {A}=\mathrm{Mod}\mbox {-}R$, where R is a unitary associative ring, we characterize all support $\tau $-tilting (resp. all support $\tau ^-$-tilting) subcategories of $\mathrm{Mod}\mbox {-}R$ in terms of finendo quasitilting (resp. quasicotilting) modules. As a result, it will be shown that every silting module (resp. every cosilting module) induces a support $\tau $-tilting (resp. support $\tau ^{-}$-tilting) subcategory of $\mathrm{Mod}\mbox {-}R$. We also study the theory in $\mathrm {Rep}(Q, \mathscr {A})$, where Q is a finite and acyclic quiver. In particular, we give an algorithm to construct support $\tau $-tilting subcategories in $\mathrm {Rep}(Q, \mathscr {A})$ from certain support $\tau $-tilting subcategories of $\mathscr {A}$.