On rectifiable measures in Carnot groups: Marstrand–Mattila rectifiability criterion

Strong singularity, disjointness, and strong finite additivity of finitely additive measures

We provide a sufficient geometric condition for \mathbb R^n to be countably (\mu,m) rectifiable of class \mathcal C^{1,\alpha} (using the terminology of Federer), where \mu is a Radon measure having positive lower density and finite upper density \mu almost everywhere. Our condition involves integrals of certain many-point interaction functions (discrete curvatures) which measure flatness of simplexes spanned by the parameters.

We refine a recent result on the structure of measures satisfying a linear partial differential equation 𝓐μ = σ, μ, σ are Radon measures, considering the measure μ(x) = g(x)d x + μus (x̃)(μs (x̄) + d x̄) where x = (x̃,x̄) ∈ ℝ k × ℝ d−k , μus is a uniformly singular measure in x̃ 0 and the measure μs is a singular measure. We proved that for μus -a.e. x̃ 0 the range of the Radon-Nykodim derivative f ~ ( x ~ 0 ) = d μ u s d | μ u s | ( x ~ 0 ) $\begin{array}{} \tilde{f}(\tilde{{\bf x}}_0) = \frac{d \mu_{us}}{d | \mu_{us}|}(\tilde{{\bf x}}_0) \end{array}$ is in the set ∩ ξ̃ ∈P̃ 𝓚er A P̃ ( ξ ) and, if μs is different to zero, for μs -a.e. x̄ 0 the range of the Radon-Nykodim derivative f ¯ ( x ¯ 0 ) = d μ s d | μ s | ( x ¯ 0 ) $\begin{array}{} \bar{f}(\bar{{\bf x}}_0) = \frac{d \mu_{s}}{d | \mu_{s}|}(\bar{{\bf x}}_0) \end{array}$ is in the set ∪ ξ̄ ∈P̄ 𝓚er A P̄ ( ξ ) where P̃ × P̄ = P is a manifold determined by the main symbol A P = A P̃ ⋅ A P̄ of the operator 𝓐.

Eggs in PG(4 n−1, q), q even, containing a pseudo-pointed conic

A normal Hall subgroup $N$ of a group $G$ is a normal subgroup with its order coprime with its index. Schur-Zassenhaus theorem states that every normal Hall subgroup has a complement subgroup, that is a set of coset representatives H which also forms a subgroup of G. In this paper, we present a framework to test isomorphism of groups with at least one normal Hall subgroup, when groups are given as multiplication tables. To establish the framework, we first observe that a proof of Schur-Zassenhaus theorem is constructive, and formulate a necessary and sufficient condition for testing isomorphism in terms of the associated actions of the semidirect products, and isomorphisms of the normal parts and complement parts. We then focus on the case when the normal subgroup is abelian. Utilizing basic facts of representation theory of finite groups and a technique by Le Gall [STACS 2009], we first get an efficient isomorphism testing algorithm when the complement has bounded number of generators. For the case when the complement subgroup is elementary abelian, which does not necessarily have bounded number of generators, we obtain a polynomial time isomorphism testing algorithm by reducing to generalized code isomorphism problem. A solution to the latter can be obtained by a mild extension of the singly exponential (in the number of coordinates) time algorithm for code isomorphism problem developed recently by Babai. Enroute to obtaining the above reduction, we study the following computational problem in representation theory of finite groups: given two representations rho and tau of a group $H$ over Z_p^d , p a prime, determine if there exists an automorphism phi:H -> H, such that the induced representation rho_phi=rho o phi and tau are equivalent, in time poly(|H|,p^d).

On groups with an interval isomorphic to a projective geometry

Erdős conjectured that for any set A ⊆ ℕ with positive lower asymptotic density, there are infinite sets B;C ⊆ ℕ such that B + C ⊆ A. We verify Erdős’ conjecture in the case where A has Banach density exceeding ½ . As a consequence, we prove that, for A ⊆ ℕ with positive Banach density (amuch weaker assumption than positive lower density), we can find infinite B;C ⊆ ℕ such that B+C is contained in the union of A and a translate of A. Both of the aforementioned results are generalized to arbitrary countable amenable groups. We also provide a positive solution to Erdős’ conjecture for subsets of the natural numbers that are pseudorandom.

Some rigidity results for II1 factors arising from wreath products of property (T) groups

In this paper we study normal subgroups of Kleinian groups as well as discrepancy groups (d-groups), that are Kleinian groups for which the exponent of convergence is strictly less than the Hausdorff dimension of the limit set. We show that the limit set of a d-group always contains a range of fractal subsets, each containing the set of radial limit points and having Hausdorff dimension strictly less than the Hausdorff dimension of the whole limit set. We then consider normal subgroups $G$ of an arbitrary non-elementary Kleinian group $H$, and show that the exponent of convergence of $G$ is bounded from below by half of the exponent of convergene of $H$. Finally, we give a discussion of various examples of d-groups.

Introduction.Classical least squares linear prediction theory is concerned with a stationary stochastic process (SSP); that is, with a family X(n) (n = 0, 1, -1,...) of complex-valued random variables on a probability space (Q, P), that have zero means, Gaussian joint distributions, and finite covariances (,X(ri), X(m)} depending only on n -m.One accomplishment of the theory is an analytical characterization of those SSPs for which some X(n) differs from its conditional expectation relative to those X(m) with m < n, the so-called nondeterministic SSPs.A second feature of the classical theory is an analytical characterization of the nullness or nonnullness of the remote past, that is, of the intersection (over all integers n) of the closed spans in L2(ß, P) of families, {X(m) \ m^n}.It is the generalization of these two results, replacing the integers with an arbitrary discrete Abelian group, to which this paper is directed.The classical characterizations referred to are given in terms of the spectral measure of the SSP; that is, the unique regular Borel measure p^tO in the unit circle G, satisfying (We shall represent G in the form {x | 0^ x < 2n}, with addition modulo 2-rr and the appropriate topology; the character x" of G is given by Xn(x) = einx.)Since the joint distributions are Gaussian, taking conditional expectations relative to a family of random variables is the same as taking orthogonal projections in L2(Q, P) onto the closed span of the family.(See [2, Chapter 3].) Hence the classical analytical criterion for nondeterminacy is given by the following version of Szegö's Theorem: Theorem 1.1.Let /j.> 0 be a finite regular Borel measure in the unit circle G; let dp/do be the Radon-Nikodym derivative of its absolutely continuous part relative to Lebesgue measure a in G;for each k let Pk be the orthogonal projection of L2(/i) onto the closed span of all \m with m^k.Then for all n, the norm being that in L2(p).

For graph mappings constructed from contact mappings of arbitrary two-step Carnot groups, conditions for the correct formulation of minimal surfaces’ problem are found. A suitable notion of the (sub-Riemannian) area functional increment is introduced, differentiability of this functional is proved, and necessary minimality conditions are deduced. They are also expressed in terms of sub-Riemaninan mean curvature.&#x0D;

We give a complete description of the second Fox subgroup G ∩ (1 + Δ2(G)Δ(H)) relative to a given normal subgroup H of an arbitrary finitely generated group G.

The Burnside ring A{G) of an arbitrary compact group G is defined in analogy to torn Dieck's definition for a compact Lie group.A topology is introduced on the set of conjugacy classes of closed subgroups, and a theorem of Dress on the topology of the prime spectrum of A(G) for profinite G is improved and extended to this general setting.This paper is condensed from part of my doctoral thesis [5], written under the supervision of Professor T. torn Dieck, to whom I am indebted for encouragement and advice.1.The Burnside ring and its additive structure.Suppose G is a compact group, and define a G-manifold as a differentiable manifold M together with a continuous left G-action such that, for each g G G, the translation g : M - M is a diffeomorphism.Examples are real G-representations and manifolds invariantly imbedded in such.Theorem 1.1.If G is a compact group, M a compact effective G-manifold, then G is a Lie group and M is a differentiable G-manifold.Proof.Since M is compact and locally connected, it has finitely many connected components, say M = Ji"A/,.For 1 < / < n, set G, = {g G GI g(M) = A/,}.This is a subgroup of G, and, since for any w, G M, the map /: G- M via f(g) = g-m is continuous, while G, = f~x(M), G, is clopen in G. Thus H = n"G, is a clopen normal subgroup.Now set H = {h E H\im G M, h-m = m], so that H is a normal subgroup of H and H/H acts effectively on A/,.Then since M is connected, Montgomery-Zippin [7, p. 208, Theorem 2] shows /////, to be a Lie group.Hence H/C\"xHi is again a Lie group.But since G operates effectively.Pi?//, = {h G //|Vw G M, h m = m) = {e}, so that H is a Lie group, and with it G since it is open in G.That M is then a differentiable G-manifold follows from [7, p. 208, Theorem 3].Now for any G-space X, let t(X) = {g G G|Vx G X,g x = x) be that part of G which acts trivially.Then for a G-manifold M, the G-action is derived from an effective G/i(A/)-action via the projection G-G/i(A/).Hence G/t(M) is a Lie group.

We study the properties of sets Σ having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets Σ ⊂ ℝ2 satisfying the inequality maxy∈M dist (y,Σ) ≤ r for a given compact set M ⊂ ℝ2 and some given r &gt; 0. Such sets play the role of shortest possible pipelines arriving at a distance at most r to every point of M, where M is the set of customers of the pipeline. We describe the set of minimizers for M a circumference of radius R &gt; 0 for the case when r &lt; R ∕ 4 .98, thus proving the conjecture of Miranda, Paolini and Stepanov for this particular case. Moreover we show that when M is the boundary of a smooth convex set with minimal radius of curvature R, then every minimizer Σ has similar structure for r &lt; R ∕ 5. Additionaly, we prove a similar statement for local minimizers.

The aim of this note is to prove that any compact metric space can be made connected at a minimal cost, where the cost is taken to be the one-dimensional Hausdorff measure.