For a probability measure μ on a real separable Hilbert space H, we are interested in “volume-based” approximations of the d-dimensional least squares error of μ, i.e., least squares error with respect to a best fit d-dimensional affine subspace. Such approximations are given by averaging real-valued multivariate functions which are typically scalings of squared (d+1)-volumes of (d+1)-simplices in H. Specifically, we show that such averages are comparable to the square of the d-dimensional least squares error of μ, where the comparison depends on a simple quantitative geometric property of μ. This result is a higher dimensional generalization of the elementary fact that the double integral of the squared distances between points is proportional to the variance of μ. We relate our work to two recent algorithms, one for clustering affine subspaces and the other for Monte-Carlo singular value decomposition based on volume sampling.

We study a Dirichlet problem involving the weak Laplacian on the Sierpiński gasket, and we prove the existence of at least two distinct nontrivial weak solutions using Ekeland’s Variational Principle and standard tools in critical point theory combined with corresponding variational techniques.

We prove the following generalization of the ham sandwich theorem, conjectured by Imre Bárány. Given a positive integer k and d nice measures μ1,μ2,…,μd in ℝd such that μi(ℝd)=k for all i, there is a partition of ℝd into k interior-disjoint convex parts C1,C2,…,Ck such that μi (Cj)=1for all i,j. If k=2 , this gives the ham sandwich theorem. This result was proved independently by R. N. Karasev.

We prove that, under certain conditions, uniform weak mixing (to zero) of the bounded sequences in Banach space implies uniform weak mixing of their tensor product. Moreover, we prove that ergodicity of tensor product of the sequences in Banach space implies their weak mixing. As applications of the results obtained, we prove that the tensor product of uniquely E-weak mixing C*-dynamical systems is also uniquely E-weak mixing.

The paper deals with the following problem: characterize Tichonov spaces X whose realcompactification υX is a Lindelöf Σ-space. There are many situations (both in topology and functional analysis) where Lindelöf Σ (even K-analytic) spaces υX appear. For example, if E is a locally convex space in the class 𝔊 in sense of Cascales and Orihuela (𝔊 includes among others (LM ) -spaces and (DF ) -spaces), then υ(E′,σ(E′,E))is K-analytic and E is web-bounded. This provides a general fact (due to Cascales–Kakol–Saxon): if E∈𝔊, then σ(E′,E)is K-analytic if and only if σ(E′,E)is Lindelöf. We prove a corresponding result for spaces Cp (X)of continuous real-valued maps on X endowed with the pointwise topology: υX is a Lindelöf Σ-space if and only if X is strongly web-bounding if and only if Cp (X)is web-bounded. Hence the weak* dual of Cp (X)is a Lindelöf Σ-space if and only if Cp (X)is web-bounded and has countable tightness. Applications are provided. For example, every E∈𝔊 is covered by a family {Aα :α∈Ω} of bounded sets for some nonempty set Ω⊂ℕℕ.

We discuss the problem of the regularity-in-time of the map t ↦ Tt ∊ Lp(ℝd, ℝd; σ), where Tt is a transport map (optimal or not) from a reference measure σ to a measure μt which lies along an absolutely continuous curve t ↦ μt in the space (). We prove that in most cases such a map is no more than 1/p-Hölder continuous.

We define an infinite class of fractals, called horizontally and vertically blocked labyrinth fractals, which are dendrites and special Sierpiński carpets. Between any two points in the fractal there is a unique arc α; the length of α is infinite and the set of points where no tangent to α exists is dense in α.

Start with a compact set K ⊂ R d . This has a random number of daughter sets, each of which is a (rotated and scaled) copy of K and all of which are inside K. The random mechanism for producing daughter sets is used independently on each of the daughter sets to produce the second generation of sets, and so on, repeatedly. The random fractal set F is the limit, as n goes to ∞, of the union of the nth generation sets. In addition, K has a (suitable, random) mass which is divided randomly between the daughter sets, and this random division of mass is also repeated independently, indefinitely. This division of mass will correspond to a random self-similar measure on F. The multifractal spectrum of this measure is studied here. Our main contributions are dealing with the geometry of realisations in R d and drawing systematically on known results for general branching processes. In this way we generalise considerably the results of Arbeiter and Patzschke (1996) and Patzschke (1997).

We show that the conjecture of Kannan, Lovász, and Simonovits on isoperimetric properties of convex bodies and log-concave measures is true for log-concave measures of the form ρ(∣x∣B) dx on ℝn and ρ(t,∣x∣B) dx on ℝ1+n, where ∣x∣B is the norm associated to any convex body B already satisfying the conjecture. In particular, the conjecture holds for convex bodies of revolution.

We consider the spectra of the Laplacians of two sequences of fractal graphs in the context of the general theory introduced by Sabot in 2003. For the sequence of graphs associated with the pentagasket, we give a description of the eigenvalues in terms of the iteration of a map from (ℂ2)3 to itself. For the sequence of graphs introduced in a previous paper by the author, we show that the results found therein can be related to Sabot's theory.

We study conical density properties of general Borel measures on Euclidean spaces. Our results are analogous to the previously known result on the upper density properties of Hausdorff and packing-type measures.

It is well known that an exponential real function, which is Lebesgue measurable (Baire measurable, respectively) or bounded on a set of positive Lebesgue measure (of the second category with the Baire property, respectively), is continuous. Here we consider bounded on ‘big’ set solutions of an equation generalizing the exponential equation as well as the Goła̧b–Schinzel equation. Moreover, we unify results into a more general and abstract case.

On the boundary of a Galton-Watson tree we can define the visibility measure by splitting mass equally between the children of each vertex, and the branching measure by splitting unit mass equally between all vertices in the nth generation and then letting n go to infinity. The multifractal structure of each of these measures is well studied. In this paper we address the question of a joint multifractal spectrum, i.e. we ask for the Hausdorff dimension of the boundary points which simultaneously have an unusual local dimension for both these measures. The resulting two-parameter spectrum exhibits a number of surprising new features, among them the emergence of a swallowtail-shaped spectrum for the visibility measure in the presence of a nontrivial condition on the branching measure.

We find the almost-sure Hausdorff and box-counting dimensions of random subsets of self-affine fractals obtained by selecting subsets at each stage of the hierarchical construction in a statistically self-similar manner.

Given a metrizable locally convex-solid Riesz space of measurable functions we provide a procedure to construct a minimal Fréchet (function) lattice containing it, called its Fatou completion. As an application, we obtain that the Fatou completion of the space L1(ν) of integrable functions with respect to a Fréchet-space-valued measure ν is the space L1w(ν) of scalarly ν-integrable functions. Further consequences are also given.

A characterization of Banach spaces possessing the weak Radon–Nikodým property is given in terms of finitely additive interval functions. Due to that characterization several Banach space valued set functions that are only finitely additive can be represented as integrals.

Some classical examples in vector integration due to Phillips, Hagler and Talagrand are revisited from the point of view of the Birkhoff and McShane integrals.

Let Xi be transient βi-stable processes on ℝdi, i=1,2. Assume further that X1 and X2 are independent. We shall find the exact Hausdorff measure function for the product sets R1(1)×R2(1), where . The result of Hu generalizes [Some fractal sets determined by stable processes, Probab. Theory Related Fields 100 (1994), 205–225].

A reverse iterated function system is defined as a family of expansive maps {T1,T2,…,Tm} on a uniformly discrete set . An invariant set is defined to be a nonempty set satisfying F=⋃ j=1mTj(F). A computation method for the dimension of the invariant set is given and some questions asked by Strichartz are answered.

Let Cr[0,t] be the function space of the vector-valued continuous paths x:[0,t]→ℝr and define Xt:Cr[0,t]→ℝ(n+1)r by Xt(x)=(x(0),x(t1),…,x(tn)), where 0<t1<⋯<tn=t. In this paper, using a simple formula for the conditional expectations of the functions on Cr[0,t] given Xt, we evaluate the conditional analytic Feynman integral Eanfq[Ft∣Xt] of Ft given by where θ(s,⋅) are the Fourier–Stieltjes transforms of the complex Borel measures on ℝr, and provide an inversion formula for Eanfq[Ft∣Xt]. Then we present an existence theorem for the solution of an integral equation including the integral equation which is formally equivalent to the Schrödinger differential equation. We show that the solution can be expressed by Eanfq[Ft∣Xt] and a probability distribution on ℝr when Xt(x)=(x(0),x(t)).