Intersections of Three Nilpotent Subgroups in a Finite Group

Siberian Mathematical Journal - We complete the proof of the theorem that any nilpotent subgroups  $ A $ , $ B $ , and  $ C $ of a finite group  $ G $ satisfy...     

An Adaptive Mesh-Refining Algorithm Allowing for an <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript> Stable <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> Projection onto Courant Finite Element Spaces

Suppose $\cal{S}^1({\cal T})\subset H^1(\Omega)$ is the $P_1$-finite element space of $\cal{T}$-piecewise affine functions based on a regular triangulation $\cal{T}$ of a two-dimensional surface $\Omega$ into triangles. The $L^2$ projection $\Pi$ onto $\cal{S}^1(\cal{T})$ is $H^1$ stable if $\norm{\Pi v}{H^1(\Omega)}\le C\norm{v}{H^1(\Omega)}$ for all $v$ in the Sobolev space $H^1(\Omega)$ and if the bound $C$ does not depend on the mesh-size in $\cal{T}$ or on the dimension of $\cal{S}^1(\cal{T})$. \hskip 1em A red–green–blue refining adaptive algorithm is designed which refines a coarse mesh $\cal{T}_0$ successively such that each triangle is divided into one, two, three, or four subtriangles. This is the newest vertex bisection supplemented with possible red refinements based on a careful initialization. The resulting finite element space allows for an $H^1$ stable $L^2$ projection. The stability bound $C$ depends only on the coarse mesh $\cal{T}_0$ through the number of unknowns, the shapes of the triangles in $\cal{T}_0$, and possible Dirichlet boundary conditions. Our arguments also provide a discrete version $\norm{h_\cal{T}^{-1}\,\Pi v}{L^2(\Omega)}\le C\norm{h_\cal{T}^{-1}\,v}{L^2(\Omega)}$ in $L^2$ norms weighted with the mesh-size $h_\T$.     

A Singular Parabolic Boundary Value Problem in Weighted <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-spaces

We consider the operator ${\cal A}$ formally defined by ${\cal A}u(x)=\alpha(x)\Delta u(x)$ for any $x$ in a sufficiently smooth bounded open set $\Om\subset\R^N$ ($N\ge 1$), where $\alpha\in C(\ov\Omega)$ is a continuous nonnegative function vanishing only on $\partial\Omega$, and such that $1/\alpha$ is integrable in $\Omega$. We prove that the realization $A_p$ of ${\cal A}$, equipped with suitable nonlinear boundary conditions is an m-dissipative operator in suitably weighted $L^p(\Omega)$-spaces in the case where either $(p,N)\in (1,+\infty)\times\{1\}$ or $(p,N)=\{2\}\times\N$. Moreover, we prove that $A_p$ is a densely defined closed operator. We consider nonlinear boundary conditions of the following type: in the one dimensional case we take $\Omega=(0,1)$ and we assume that $u(j)=(-1)^j\beta_j(u(j))$ ($j=0,1$), $\beta_0$ and $\beta_1$ being nondecreasing continuous functions in $\R$ such that $\beta_0(0)=\beta_1(0)=0$; in the $N$-dimensional setting we assume that $(D_{\nu}u)_{|\partial\Omega}=-\beta(u_{|\partial\Omega})$, $\beta$ being a nondecreasing Lipschitz continuous function in $\R$ such that $\beta(0)=0$. Here $\nu$ denotes the unit outward normal to $\partial\Om$.     

On effective σ‐boundedness and σ‐compactness

We prove several dichotomy theorems which extend some known results on σ‐bounded and σ‐compact pointsets. In particular we show that, given a finite number of $\Delta ^{1}_{1}$ equivalence relations $\mathrel {\mathsf {F}}_1,\dots ,\mathrel {\mathsf {F}}_n$, any $\Sigma ^{1}_{1}$ set A of the Baire space either is covered by compact $\Delta ^{1}_{1}$ sets and lightface $\Delta ^{1}_{1}$ equivalence classes of the relations $\mathrel {\mathsf {F}}_i$, or A contains a superperfect subset which is pairwise $\mathrel {\mathsf {F}}_i$‐inequivalent for all i = 1, …, n. Further generalizations to $\Sigma ^{1}_{2}$ sets A are obtained.     

On the 2-part of the ideal class group of the cyclotomic <Emphasis Type="BoldItalic">Z</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-extension over the rationals

Let p be an odd prime number, and $\mbox {\boldmath $\mbox {\boldmath the cyclotomic $\mbox {\boldmath $\mbox {\boldmath -extension over the rationals. We show that the 2-part of the ideal class group of $\mbox {\boldmath $\mbox {\boldmath is trivial for p<500.     

A Converse Hawking-Unruh Effect and dS<Superscript>2</Superscript>/CFT Correspondence

Given a local quantum field theory net $ \mathcal{A} $ on the de Sitter spacetime dS ^d , where geodesic observers are thermalized at Gibbons-Hawking temperature, we look for observers that feel to be in a ground state, i.e., particle evolutions with positive generator, providing a sort of converse to the Hawking-Unruh effect. Such positive energy evolutions always exist as noncommutative flows, but have only a partial geometric meaning, yet they map localized observables into localized observables. We characterize the local conformal nets on dS ^d . Only in this case our positive energy evolutions have a complete geometrical meaning. We show that each net has a unique maximal expected conformal subnet, where our evolutions are thus geometrical. In the two-dimensional case, we construct a holographic one-to-one correspondence between local nets $ \mathcal{A} $ on dS ² and local conformal non-isotonic families (pseudonets) $ \mathcal{B} $ on S ¹. The pseudonet $ \mathcal{B} $ gives rise to two local conformal nets $ \mathcal{B}_\pm $ on S ¹, that correspond to the $ \frak{H}_\pm $ horizon components of $ \mathcal{A} $, and to the chiral components of the maximal conformal subnet of $ \mathcal{A} $. In particular, $ \mathcal{A} $ is holographically reconstructed by a single horizon component, namely the pseudonet is a net, iff the translations on $ \frak{H}_\pm $ have positive energy and the translations on $ \frak{H}_\mp $ are trivial. This is the case iff the one-parameter unitary group implementing rotations on dS ² has positive/negative generator. Communicated by Klaus Fredenhagen submitted 07/02/03, accepted: 07/07/03     

No Helly Theorem for Stabbing Translates by Lines in R<Superscript>3</Superscript>

For each $n>2$ we construct a convex body $K\subset {\Bbb R}^3$ and a finite family ${\cal F}$ of disjoint translates of $K$ such that any $n-1$ members ${\cal F}$ admit a line transversal, but ${\cal F}$ has no line transversal.     

On $ \sigma $ -Subnormality of Sylow Subgroups in a Finite Group

Siberian Mathematical Journal - Given an arbitrary partition&nbsp; $ \sigma $ of the set of primes, we give some criteria for the $ \sigma $ -subnormality of a&nbsp;Sylow subgroup of...     

On the Number of Frequency Hypercubes F $ {}^{n}(4;2,2) $

Siberian Mathematical Journal - A&nbsp;frequency $ n $ -cube F $ {}^{n}(4;2,2) $ is an $ n $ -dimensional $ 4 $ -by-…-by- $ 4 $ array filled by&nbsp; $ 0 $ s and&nbsp; $ 1 $ s...     

Schur–Weyl Theory for C*‐algebras

To each irreducible infinite dimensional representation $(\pi ,\mathcal {H})$ of a C*‐algebra $\mathcal {A}$, we associate a collection of irreducible norm‐continuous unitary representations $\pi _{\lambda }^\mathcal {A}$ of its unitary group ${\rm U}(\mathcal {A})$, whose equivalence classes are parameterized by highest weights in the same way as the irreducible bounded unitary representations of the group ${\rm U}_\infty (\mathcal {H}) = {\rm U}(\mathcal {H}) \cap (\mathbf {1} + K(\mathcal {H}))$ are. These are precisely the representations arising in the decomposition of the tensor products $\mathcal {H}^{\otimes n} \otimes (\mathcal {H}^*)^{\otimes m}$ under ${\rm U}(\mathcal {A})$. We show that these representations can be realized by sections of holomorphic line bundles over homogeneous Kähler manifolds on which ${\rm U}(\mathcal {A})$ acts transitively and that the corresponding norm‐closed momentum sets $I_{\pi _\lambda ^\mathcal {A}}^{\bf n} \subseteq {\mathfrak u}(\mathcal {A})^{\prime }$ distinguish inequivalent representations of this type.     

Essentially Invertible Measurable Operators Affiliated to a Semifinite von Neumann Algebra and Commutators

Siberian Mathematical Journal - Suppose that a&nbsp;von&nbsp;Neumann operator algebra&nbsp; $ {\mathcal{M}} $ acts on a&nbsp;Hilbert space&nbsp; $ {\mathcal{H}} $ and $ \tau $...     

A Dedekind Criterion over Valued Fields

Siberian Mathematical Journal - Let $ (K,\nu) $ be an arbitrary-rank valued field, let&nbsp; $ R_{\nu} $ be the valuation ring of&nbsp; $ (K,\nu) $ , and let&nbsp; $ K(\alpha)/K $ be...     

On Finite Groups Factorizable by Weakly Subnormal Subgroups

Siberian Mathematical Journal - Let $ G $ be a group. A&nbsp;subgroup $ H $ is weakly subnormal in&nbsp; $ G $ if $ H=\langle A,B\rangle $ for some subnormal subgroup&nbsp; $ A $ and...     

Typical <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>-Dimensions of Measures

For a probability measure μ on a subset of , the lower and upper L^q-dimensions of order are defined by We study the typical behaviour (in the sense of Baire’s category) of the L^q-dimensions and . We prove that a typical measure μ is as irregular as possible: for all q ≥ 1, the lower L^q-dimension attains the smallest possible value and the upper L^q-dimension attains the largest possible value.     

On <Emphasis Type="Italic">N</Emphasis>-Summations,II

Given any R-semimodule M equipped with a semitopology we construct an N-protosummation for M. If satisfies certain properties, then a similar construction leads to an unconditional N-summation for M, that is an N-summation for M equipped with the trivial prenorm MD over the N-summation (D^N,_D) for D. Conversely any N-protosummation on M gives rise to a topology . If both and satisfy a certain separation property, then and form a Galois connection. Dedicated to my friend and collegue Nico Pumplün on the occasion of his 70th birthdayMathematics Subject Classifications (2000) 16Y60, 54A05.     

<Emphasis Type="Italic">X</Emphasis>-Trees and Weighted Quartet Systems

In this note, we consider a finite set X and maps W from the set $ \mathcal{S}_{2|2} (X) $ of all 2, 2- splits of X into $ \mathbb{R}_{\geq 0} $. We show that such a map W is induced, in a canonical way, by a binary X-tree for which a positive length $ \mathcal{l} (e) $ is associated to every inner edge e if and only if (i) exactly two of the three numbers W(ab|cd),W(ac|bd), and W(ad|cb) vanish, for any four distinct elements a, b, c, d in X, (ii) $ a \neq d \quad\mathrm{and}\quad W (ab|xc) + W(ax|cd) = W(ab|cd) $ holds for all a, b, c, d, x in X with #{a, b, c, x} = #{b, c, d, x} = 4 and $ W(ab|cx),W(ax|cd) $ > 0, and (iii) $ W (ab|uv) \geq \quad \mathrm{min} (W(ab|uw), W(ab|vw)) $ holds for any five distinct elements a, b, u, v, w in X. Possible generalizations regarding arbitrary $ \mathbb{R} $-trees and applications regarding tree-reconstruction algorithms are indicated.AMS Subject Classification: 05C05, 92D15, 92B05.     

Cohomology with compact supports for cohomologically<Emphasis Type="Italic">q</Emphasis>-convex spaces

We show that if X is a complex space of dimension n which is cohomologically q-convex (resp., cohomologically q-complete), then $ H^{i}_{c} (X, \mathbb{C}) $ is a finite dimensional vector space (resp., vanishes) for $ i \leq \nu_{q}(X)-q $, where the number $ \nu_{q}(X) $ depends on the nature of singularities of X and equals n if X is smooth.     

A Semilattice of Degrees of Computable Metrics

Siberian Mathematical Journal - Under study is the ordering $ {\mathcal{CM}}_{c}({\mathbf{X}}) $ of $ c $ -degrees of computable metrics on a&nbsp;Polish space&nbsp; $ {\mathbf{X}} $ with...     

Cyclotomic Quotients of Two Conjugates of an Algebraic Number

Siberian Mathematical Journal - Let $ \alpha $ be an algebraic number of degree $ d\geq 2 $ . We consider the set $ E(\alpha) $ of positive integers&nbsp; $ n $ such that the primitive $ n $ th...     

Dimension of the Torelli group for Out(F<Subscript>n</Subscript>)

Let be the kernel of the natural map Out(F_n)→GL_n(ℤ). We use combinatorial Morse theory to prove that has an Eilenberg–MacLane space which is (2n-4)-dimensional and that is not finitely generated (n≥3). In particular, this shows that the cohomological dimension of is equal to 2n-4 and recovers the result of Krstić–McCool that is not finitely presented. We also give a new proof of the fact, due to Magnus, that is finitely generated.