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The main result of this paper is that, under PFA, for every regular space X with F(X) = \omega we have |X| \le w(X)^\omega; in particular, w(X) \le \mathfrak {c} implies |X| \le \mathfrak {c}. This complements numerous prior results that yield consistent examples of even compact Hausdorff spaces X with F(X) = \omega such that w(X) = \mathfrak {c} and |X| = 2^\mathfrak {c}. We also show that regularity cannot be weakened to the Hausdorff property in this result because we can find in ZFC a Hausdorff space X with F(X) = \omega such that w(X) = \mathfrak {c} and |X| = 2^\mathfrak {c}. In fact, this space X has the strongly anti-Urysohn (SAU) property that any two infinite closed sets in X intersect, which is much stronger than F(X) = \omega. Moreover, any non-empty open set in X also has size 2^\mathfrak {c}, and thus our example answers one of the main problems of both Juhász, Soukup, and Szentmiklóssy [Topology Appl. 213 (2016), pp. 8–23] and Juhász, Shelah, Soukup, and Szentmiklóssy [Topology Appl. 323 (2023), Paper No. 108288, 15 pp.] by providing in ZFC a SAU space with no isolated points. [ABSTRACT FROM AUTHOR]

Suppose that K is a compact Hausdorff space, S is a locally compact Hausdorff space and X is a Banach space with Schäffer constant \mathrm {S}(X). In this paper, we prove that if there is a map T from C(K) to C_{0}(S, X) satisfying \begin{equation*} \frac {1}{M} \|f-g\| \leq \|T(f)-T(g)\|\leq M \|f-g\|,\ \forall f, g \in C(K), \end{equation*} with 1\leq M^{2}<\mathrm {S}(X), then there exists a compact subset S_{0} of S and a continuous function \varphi from S_{0} onto K. This theorem on Lipschitz embeddings of C(K) into C_{0}(S, X) is the first nonlinear vector extension of the classical 1966 Holsztyński Theorem. Our result is optimal for many Banach spaces X including the spaces \ell _p^n, \ell _p and L_{p}([0,1]), 1 < p <\infty, n \geq 2, even when T is linear. The motivation to prove this result comes from the fact that it immediately yields a nontrivial lower bound for the C_{0}(S, X)-distortion of the class of all separable Banach spaces whenever S is a scattered space and \mathrm {S}(X)>1, namely \mathrm {S}(X) itself. [ABSTRACT FROM AUTHOR]

In this paper we disprove a conjecture stated in Stratification of minimal surfaces, mean curvature flows, and harmonic maps by Brian White on the equality of two notions of dimension for closed cones. Moreover, we answer in the negative the following question, raised in the same paper. Given a compact family C of closed cones and a set S such that every blow-up of S at every point x ∈ S is contained in some element of C, is it true that the dimension of S is smaller than or equal to the largest dimension of a vector space contained is some element of C? [ABSTRACT FROM AUTHOR]

Inspired by recent papers on twisted K-theory, we consider in this article the question of when a twist R over a locally compact Hausdorff groupoid G (with unit space a CW-complex) admits a twisted vector bundle, and we relate this question to the Brauer group of G. We show that the twists which admit twisted vector bundles give rise to a subgroup of the Brauer group of G. When G is an ètale groupoid, we establish conditions (involving the classifying space BG of G) which imply that a torsion twist R over G admits a twisted vector bundle. [ABSTRACT FROM AUTHOR]

Let Bo(T,τ) be the Borel σ-algebra generated by the topology τ on T. In this paper we show that if K is a Hausdorff compact space, then every subset of K is a Borel set if and only if Bo(C*(K), w* = Bo(C*(K),|⋅|), where w* denotes the weak-star topology and |⋅| is the dual norm with respect to the sup-norm on the space of real-valued continuous functions C(K). Furthermore, we study the topological properties of the Hausdorff compact spaces K such that every subset is a Borel set. In particular, we show that if the axiom of choice holds true, then K is scattered. [ABSTRACT FROM AUTHOR]

Let X be a completely regular Hausdorff space, let E and F denote Banach spaces. Let C_b(X,E) denote the space of E-valued bounded continuous functions on X and let \beta be the strict topology on this space. We establish the relationship between nuclear operators T:C_b(X,E)\rightarrow F between the locally convex space (C_b(X,E),\beta) and the Banach space F and their representing operator-valued Borel measures. [ABSTRACT FROM AUTHOR]

This paper concerns the continuity of the solution to the dual Minkowski problem for negative indices. For each q < 0, it is proved that the weak convergence of qth dual curvature measures implies the convergence of the corresponding convex bodies in the Hausdorff metric and that the solution to the dual Minkowski problem is continuous with respect to q. [ABSTRACT FROM AUTHOR]

As a generalization of almost everywhere convergence to vector lattices, unbounded order convergence has garnered much attention. The concept of boundedly uo-complete Banach lattices was introduced by N. Gao and F. Xanthos, and has been studied in recent papers by D. Leung, V. G. Troitsky, and the aforementioned authors. We will prove that a Banach lattice is boundedly uo-complete iff it is monotonically complete. Afterwards, we study completeness-type properties of minimal topologies; minimal topologies are exactly the Hausdorff locally solid topologies in which uo-convergence implies topological convergence. [ABSTRACT FROM AUTHOR]

It is proven that if X is a real strictly convex 2-dimensional space, then there exists \delta >0 such that if K and S are locally compact Hausdorff spaces and T is an isomorphism from C_{0}(K, X) onto C_{0}(S, X) satisfying \begin{equation*} \|T\| \ \|T^{-1}\| \leq \lambda (X)+ \delta, \end{equation*} then K and S are homeomorphic. Here \lambda (X) is the Schäffer constant of X given by \begin{equation*} \lambda (X)=\inf \{\max \{\|x-y\|,\|x+y\|\}\,:\,\|x\|= 1 \text { and } \|y\|= 1\}. \end{equation*} Even for the classical cases X=\ell _p^2, 1

Let Q = {qk}k=1 be a sequence of positive integers with qk = 2 for every k = 1. Then each point x ∈ [0, 1] is attached with an infinite series expansion ... which is called the Cantor series expansion of x. In this paper, we study the shrinking target problems for the system induced by the Cantor series expansion. More precisely, put TQn (x) = q1 · · · qnx-[q1 · · · qnx]; the shrinking target problem in such a nonautonomous system can be formulated as considering the size of the set ... where y is a fixed point in [0, 1] and ϕ : N → (0, 1) is a positive function with ϕ(n) → 0 as n → ∞. It is proved that both the Lebesgue measure and the Hausdorff measure of Ey(ϕ) fulfill a dichotomy law according to the divergence or convergence of certain series. [ABSTRACT FROM AUTHOR]

Two subanalytic subsets of Rn are called s-equivalent at a common point P if the Hausdorff distance between their intersections with the sphere centered at P of radius r vanishes to order > s when r tends to 0. In this paper we prove that every s-equivalence class of a closed semianalytic set contains a semialgebraic representative of the same dimension. In other words any semianalytic set can be locally approximated to any order s by means of a semialgebraic set and hence, by previous results, also by means of an algebraic one. [ABSTRACT FROM AUTHOR]

We prove that for any compact set E ⊂ R2, dimH (E) > 1, there exists x ∈ E such that the Hausdorff dimension of the pinned distance set Δx(E) = {|x − y|: y ∈ E} is no less than min{4/{3 dimH(E) − 2/3, 1}. This answers a question recently raised by Guth, Iosevich, Ou, and Wang, as well as improves results of Keleti and Shmerkin. [ABSTRACT FROM AUTHOR]

Let K and S be locally compact Hausdorff spaces and let T be a real linear isomorphism of an extremely regular subspace A of C_{0}(K) into C_{0}(S) satisfying \|T \| \ \|T^{-1}\|<2. Put L=T \ \|T^{-1}\| and pick 0 \leq \varepsilon <1 such that \|T \| \ \|T^{-1}\|=1+\varepsilon. We prove that there exist a subset S_0 of S, a proper map \varphi of S_{0} onto K and a map \lambda :S_{0} \to \{-1, 1\} such that \begin{equation*} |Lf(s)-\lambda (s) f(\varphi (s))| \leq \varepsilon \|f\|, \ \forall s \in S_{0} \ \text {and} \ f \in A. \end{equation*} For \varepsilon < 1/2 the maps \varphi and \lambda are continuous, and this approximation of L by such a weighted composition operator improves the well-known Holsztyński theorem [Studia Math. 26 (1966), pp. 133–136], the case where \varepsilon =0, K and S are compact and A=C_{0}(K). The approximation of L in the general case where 0 \leq \varepsilon <1 is an extension of a Benyamini's result [Proc. Amer. Math. Soc. 83 (1981), pp. 479–485] which allowed us to strengthen a classical Jarosz's theorem [Proc. Amer. Math. Soc. 90 (1984), pp. 373–377] by proving that K is a continuous image not just of a subset of S but of a locally compact subset of S. Moreover, when K is compact, it is a continuous image not just a closed subset of S but of a compact subset of S, namely the set S_0 having the above properties. [ABSTRACT FROM AUTHOR]

Let C_b(X) denote the Banach space of all bounded continuous real-valued functions on a completely regular Hausdorff space X. We characterize nuclear tight operators T:C_b(X)\rightarrow E in terms of their representing Radon vector measures. As an application, it is shown that some natural kernel operators on C_b(X) are tight and nuclear. [ABSTRACT FROM AUTHOR]

It is proven that if X is a Banach space, K and S are locally compact Hausdorff spaces and there exists an (M, L)-quasi isometry T from C_{0}(K,X) onto C_{0}(S, X), then K and S are homeomorphic whenever 1 \leq M^{2}< S(X), where S(X) denotes the Schäffer constant of X, and L \geq 0. As a consequence, we show that the first nonlinear extension of Banach-Stone theorem for C_{0}(K, X) spaces obtained by Jarosz in 1989 can be extended to infinite-dimensional spaces X, thus reinforcing a 1991 conjecture of Jarosz himself on \epsilon-bi-Lipschitz surjective maps between Banach spaces. Our theorem is optimal when X is the classical space \ell _p, 2 \leq p< \infty. [ABSTRACT FROM AUTHOR]

Assume that X, Y are compact Hausdorff spaces. Let C(X)_+, C(Y)_+ be the sets of all positive real functions on X,Y respectively. Let T be a surjective map from a point separating multiplicative subgroup G\subset C(X)_+ onto a point separating multiplicative subgroup H\subset C(Y)_+ satisfying T1=1 and \begin{equation*} \frac {1}{M}\|f\cdot g^{-1}\|\leq \|Tf\cdot (Tg)^{-1}\|\leq M \|f\cdot g^{-1}\| \end{equation*} for all f, g\in G and for some constant M\geq 1. Then there exists a homeomorphism \tau from the Šilov boundary \partial H of H onto the Šilov boundary \partial G of G such that \begin{equation*} \frac {f(\tau (y))}{M^2}\leq (Tf)(y)\leq M^2\cdot f(\tau (y)) \end{equation*} for all f\in G and all y\in \partial H. The constant M^2 in the inequality above is optimal. A similar argument yields a generalization of the classical Banach-Stone Theorem to isometries defined from an additive subgroup of C_0(X) into C_0(Y) for locally compact Hausdorff spaces X and Y. [ABSTRACT FROM AUTHOR]

It is well known that every non-isolated point in a compact Hausdorff space is the accumulation point of a discrete subset. Answering a question raised by Z. Szentmiklossy and the first author, we show that this statement fails for countably compact regular spaces, and even for W-bounded regular spaces. In fact, there are K-bounded counterexamples for every infinite cardinal K. The proof makes essential use of the so-called strong colorings that were invented by the second author. [ABSTRACT FROM AUTHOR]

Given a Banach space we consider the \sigma-ideal of all of its subsets which are covered by countably many hyperplanes and investigate its standard cardinal characteristics as the additivity, the covering number, the uniformity, the cofinality. We determine their values for separable Banach spaces, and approximate them for nonseparable Banach spaces. The remaining questions reduce to deciding if the following can be proved in ZFC (i.e. Zermelo-Fraenkel set theory with the axiom of choice) for every nonseparable Banach space X: (1) X can be covered by \omega _1-many of its hyperplanes; (2) All subsets of X of cardinalities less than \operatorname {cf}([\operatorname {dens}(X)]^\omega) can be covered by countably many hyperplanes. We prove (1) and (2) for all Banach spaces in many well-investigated classes and that they are consistent with any possible size of the continuum. (1) is related to the problem whether every compact Hausdorff space which has small diagonal is metrizable and (2) to large cardinals. [ABSTRACT FROM AUTHOR]

Let C(X,E) be the linear space of all continuous functions on a compact Hausdorff space X with values in a locally convex space E. We characterize maps T:C(X,E) → C(Y,E) which satisfy Ran(TF-TG) ⊂ Ran(F-G) for all F,G ∈ C(X,E). [ABSTRACT FROM AUTHOR]

We provide a nonlinear version of an extension of the classical Holsztyński theorem due to Jarosz (1984) concerning the into isomorphisms of spaces of continuous functions. More precisely, supposing that K and S are locally compact Hausdorff spaces, we prove that if there exists a map T from an extremely regular subspace A of C0(K) to C0(S) satisfying 1/M || ƒ-g || −L ≤ || T(ƒ) − T(g) || ≤ M || ƒ-g || + L ∀ ƒ, g ∈ A, with 1 ≤ M2 < 2 and L ≥ 0, then there exist a subset S0 of S and a proper mapping φ of S0 onto K. We show that φ is not only continuous in the obvious case when K is compact, but also in the case when M2 < 4/3, where S0 can be taken locally compact. In the Lipschitz case, that is, when L = 0, and K and S are intervals of ordinal numbers, our result improves some others by Procházka and Sánchez-González (2017) concerning the Lipschitz embeddings between C(K) spaces and also solves a problem raised by them. [ABSTRACT FROM AUTHOR]

We introduce the concept of Banach-Stone subsets of C0(K) spaces. This allows us to unify and improve several extensions of the classical theorem due to Banach (1933) and Stone (1937). More precisely, we prove that if K and S are locally compact Hausdorff spaces, A and B are Banach-Stone subsets of C0(K) and C0(S), respectively, and there exists a map T from A to B (not necessarily injective) with image θ-dense in B for some θ > 0 such that 1/M‖f - g‖ - L ≤ ‖T(f) - T(g)‖ ≤ M‖f - g‖ + L, for every f,g ∊ A, then K and S are homeomorphic whenever L ≥ 0 and M < √2. As an application of this more general theorem concerning the quasi-isometries T on subsets of C0(K) spaces, we show that certain quasiisometries on C(1)0 (K) spaces also determine the locally compact subspaces K of the real line ℚ with no isolated points. In turn, this result enables us to prove a unification and improvement of some theorems of Cambern, Pathak, and Vasavada for the first time to the nonlinear case. [ABSTRACT FROM AUTHOR]

We show that every limit point of a Zariski dense discrete subgroup Γ of the isometry group of a symmetric space of noncompact type is conical if and only if Γ is convex cocompact. [ABSTRACT FROM AUTHOR]

The almost sure Hausdorff dimension of the limsup set of randomly distributed rectangles in a product of Ahlfors regular metric spaces is computed in terms of the singular value function of the rectangles. [ABSTRACT FROM AUTHOR]

We study topological properties of the Gromov-Hausdorff metric on the set of isometry classes of nonnegatively curved 2-spheres. [ABSTRACT FROM AUTHOR]

A topological space is said to be sequential if every subspace closed under taking limits of convergent sequences is closed. We consider Banach spaces with weak*-sequential dual ball. In particular, we show that if X is a Banach space with weak*-sequentially compact dual ball and Y ⊂ X is a subspace such that Y and X/Y have weak*-sequential dual ball, then X has weak*-sequential dual ball. As an application we obtain that the Johnson- Lindenstrauss space JL2 and C(K) for K a scattered compact space of countable height are examples of Banach spaces with weak*-sequential dual ball. These results provide a negative solution to a question of A. Plichko, who asked whether the dual ball of a Banach space is weak*-angelic whenever it is weak*-sequential. [ABSTRACT FROM AUTHOR]

We introduce a notion of an ε-non-expansive map and study the problem of uniform approximation of such a map by a non-expansive map. We apply then obtained results to show the following Hyers-Ulam stability of ε-isometries: Let X be a Hausdorff compact space and Y be a metric compact space. Let F : C(X) → C(Y) be an ε-isometry. Then there is an isometry H: C(X) → C(Y) such that ... If in addition for every proper closed subset S ⊂ Y there is an f ∈ C(X) with |F(f)(z)| < ‖F(f)‖ - 3.5ε for every z ∈ S, then H can be chosen linear. This assertion does not hold for the ‖p norm with 1 < p < ∞. [ABSTRACT FROM AUTHOR]

It is shown that there exists a nonseparable completely metrizable bounded metric space (X, d) such that the hyperspace CL(X) of the nonempty closed subsets of X endowed with the Hausdorff metric Hd is not completely metrizable. [ABSTRACT FROM AUTHOR]

We prove that a semiregular topological space X is completely regular if and only if its topology is generated by a normal quasi-uniformity. This characterization implies that each regular paratopological group is completely regular. This resolves an old problem in the theory of paratopological groups, which stood open for about 60 years. Also we define a natural uniformity on each paratopological group and using this uniformity prove that each (first countable) Hausdorff paratopological group is functionally Hausdorff (and submetrizable). This resolves another two known open problems in the theory of paratopological groups. [ABSTRACT FROM AUTHOR]

We present an alternative proof of a result of Kenig and Toro (2006), which states that if Ω ⊂ ℝn+1 is a 2-sided NTA domain, with Ahlfors-David regular boundary, and the log of the Poisson kernel associated to Ω as well as the log of the Poisson kernel associated to Ωext are in VMO, then the outer unit normal ν is in VMO. Our proof exploits the usual jump relation formula for the non-tangential limit of the gradient of the single layer potential. We are also able to relax the assumptions of Kenig and Toro in the case that the pole for the Poisson kernel is finite: in this case, we assume only that ∂Ω is uniformly rectifiable, and that ∂Ω coincides with the measure theoretic boundary of Ω a.e. with respect to Hausdorff Hn measure. [ABSTRACT FROM AUTHOR]

We give quadratic bounds on the dimension of the space of conjugacy classes of subgroups of SLn(R) that are limits under conjugacy of the diagonal subgroup. We give the first explicit examples of abelian n - 1- dimensional subgroups of SLn(R) which are not such a limit, and show that all such abelian groups are limits of the diagonal group iff n ≤ 4. [ABSTRACT FROM AUTHOR]

It is shown that the isoperimetric deficit of a convex polygon P admits a lower bound in terms of the variance of the radii of P, the area of P, and the variance of the barycentric angles of P. The proof involves circulant matrix theory and a Taylor expansion of the deficit on a compact manifold. [ABSTRACT FROM AUTHOR]

We characterize sets of probabilities, Π, on a measure space (Ω,F), with the following representation property: for every measurable set of Borel probabilities, A, on a complete separable metric space, (M,d), there exists a measurable X : Ω → M with A = {X(P) : P ∈ Π}. If Π has this representation property, then: if Kn → K0 is a sequence of compact sets of probabilities on M, there exists a sequence of measurable functions, Xn : Ω → M such that Xn(Π) ≡ Kn and for all P ∈ Π, P({ω : Xn(ω) → X0(ω)}) = 1; if the Kn are convex as well as compact, there exists a jointly measurable (K, ω) |→ H(K, ω) such that H(Kn,Π) ≡ Kn and for all P ∈ Π, P({ω : H(Kn, ω) → H(K0, ω)}) = 1. [ABSTRACT FROM AUTHOR]

Let W, X, Y and Z be Dedekind complete Riesz spaces. For A ε Lr(Y,Z) and B ε Lr(W,X) let MA,B be the two-sided multiplication operator from Lr(X, Y ) into Lr(W, Z) defined by MA,B(T) = ATB. We show that for every 0 ⩽ A0 ε Lrn (Y,Z), ǀMA,B,Bǀ(T) = MA0,ǀBǀ(T) holds for all B ε Lr(W,X) and all T ε Lrn(X, Y ). Furthermore, if W, X, Y and Z are Dedekind complete Banach lattices such that X and Y have order continuous norms, then ǀMA,Bǀ = MǀAǀ, ǀBǀ for all A ε Lr(Y,Z) and all B ε Lr(W,X). Our results generalize the related results of Synnatzschke and Wickstead, respectively. [ABSTRACT FROM AUTHOR]

Nilpotent Lie groups with stepwise square integrable representations were recently investigated by J.A. Wolf. We give an alternative approach to these representations by relating them to the stratifications of the duals of nilpotent Lie algebras, thus proving that they correspond to a subset with relative Hausdorff topology, dense interior, and total Plancherel measure in the unitary dual of the Lie group under consideration. [ABSTRACT FROM AUTHOR]

Denote by [0,ω1) the set of countable ordinals, equipped with the order topology, let L0 be the disjoint union of the compact ordinal intervals [0, α] for α countable, and consider the Banach spaces C0[0,ω1) and C0(L0) consisting of all scalar-valued, continuous functions which are defined on the locally compact Hausdorff spaces [0,ω1) and L0, respectively, and which vanish eventually. Our main result states that a bounded, linear operator T between any pair of these two Banach spaces fixes an isomorphic copy of C0(L0) if and only if the identity operator on C0(L0) factors through T, if and only if the Szlenk index of T is uncountable. This implies that the set ...C0(L0)(C0(L0)) of C0(L0)-strictly singular operators on C0(L0) is the unique maximal ideal of the Banach algebra B(C0(L0)) of all bounded, linear operators on C0(L0), and that SC0(L0) (C0[0,ω1) is the second-largest proper ideal of B(C0[0,ω1)) . Moreover, it follows that the Banach space C0(L0) is primary and complementably homogeneous. [ABSTRACT FROM AUTHOR]

The rath symmetric product of a metric space is the set of its nonempty subsets with cardinality at most ra, equipped with the Hausdorff metric. We prove that every symmetric product of the line is an absolute Lipschitz retract and admits a bi-Lipschitz embedding into a Euclidean space of sufficiently high dimension. [ABSTRACT FROM AUTHOR]

We consider C*-algebras associated with stable and unstable equivalence in hyperbolic dynamical systems known as Smale spaces. These systems include shifts of finite type, in which case these C*-algebras are both AF-algebras. These algebras have fundamental representations on a single Hilbert space (subject to a choice of periodic points) which have a number of special properties. In particular, the product between any element of the first algebra with one from the second is compact. In addition, there is a single unitary operator which implements actions on both. Here, under the hypothesis that the system is mixing, we show that the (semi-finite) traces on these algebras may be obtained through a limiting process and the usual operator trace. [ABSTRACT FROM AUTHOR]

We show that any countably generated von Neumann algebra with specified normal faithful state can arise as the tail algebra of a quantum exchangeable sequence of noncommutative random variables. We also characterize the cases when the state corresponds to a limit of convex combinations of free product states. [ABSTRACT FROM AUTHOR]

We study different conditions which turn out to be equivalent to equicontinuity for a transitive compact Hausdorff flow with a general group action. Among them are a notion of "regional" equicontinuity, also known as the "Furstenberg" condition, and the condition that every point of the phase space is almost automorphic. Then we study relations on the phase space arising from dynamical properties, among them the regionally proximal relation and two relations introduced by Veech. We generalize Veech's results for minimal actions of non-Abelian groups preserving a probability measure with respect to the regionally proximal relation. We provide proofs in the framework of dynamical systems rather than harmonic analysis as given by Veech. [ABSTRACT FROM AUTHOR]

In this article we develop a unifying theory of many years of work by a number of researchers. Especially, we establish general sufficient conditions for maps between algebras of bounded continuous functions on locally compact Hausdorff spaces to be almost composition or almost weighted composition operators, which extend the main results of many previous works on this subject. The following are some typical results. Let T : A → B be a surjective map between two function algebras on locally compact Hausdorff spaces X and Y with Choquet boundaries ΔA ⊂ X and ΔB ⊂ Y. If ||Tf Tg|| = ||fg|| and there is an ε, 0 . ε < 2/3, so that the peripheral spectrum σπ(Tf Tg) is contained in an ε ||fg||-neighborhood of σπ(fg) for all f , A and all g , A with ||g|| = 1, then there is a continuous function α: ΔB → {}1} and a homeomorphism ƒÕ: ΔB → ΔA so that |(Tf)(y).α(y) f(ƒÕ(y))| . 2ε |f(ƒÕ(y))| for each f , A and every y , ΔB. Therefore, T is an almost weighted composition operator on ΔB. If ||Tf Tg|| = ||fg||, there are ε, 0 . ε < 1, and ƒÅ, 0 . ƒÅ < 1, so that d(σπ(Tf Tg), σπ(fg)) . ε ||fg||, while σπ(Tf) is contained in an ƒÅ- neighborhood of σπ(f) for all f , A and all g , A with ||g|| = 1. Then |(Tf)(y) . f(ƒÕ(y))| . (ε + ƒÅ) |f(ƒÕ(y))| for each y , ΔB and every f , A; i.e. T is an almost composition operator on ΔB. If σπ(Tf Tg) → σπ(fg) (or σπ(fg) → σπ(Tf Tg)), and d(σπ(Tg), σπ(g)) . ƒÅ for some ƒÅ, 0 . ƒÅ < 1, for all f , A and all g , A with ||g|| = 1, then (Tf)(y) = f(ƒÕ(y)) for each f , A and every y , ΔB. Consequently, T is a composition operator on ΔB and, therefore, an algebra isomorphism. [ABSTRACT FROM AUTHOR]

We prove a boundary Harnack type inequality for nonnegative solutions to singular equations of p-parabolic type, 2n/(n + 1) < p < 2, in a time-independent cylinder whose base is C1,1-regular. Simple examples show, using the corresponding estimates valid for the heat equation as a point of reference, that this type of inequality cannot, in general, be expected to hold in the degenerate case (2 < p < ∞). [ABSTRACT FROM AUTHOR]

The uniform box product problem is a weakening of the wellknown box product problem, which asks whether box products of certain compact spaces are normal or even paracompact. Using uniformities, a new topology on products is defined between the box and Tychonov topologies. This new product, called the uniform box product, is an extension of the sup metric to powers of compact spaces. We investigate a certain non-metrizable compact space whose uniform box product, in ZFC, is normal, countably paracompact, and collectionwise Hausdorff. [ABSTRACT FROM AUTHOR]

We investigate contrasting behaviours emerging when studying foliations on non-metrisable manifolds. [ABSTRACT FROM AUTHOR]

We construct a locally compact Hausdorff topology on the path space of a directed graph E and identify its boundary-path space ?E as the spectrum of a commutative C*-subalgebra DE of C*(E). We then show that ?E is homeomorphic to a subset of the infinite-path space of any desingularisation F of E. Drinen and Tomforde showed that we can realise C*(E) as a full corner of C*(F), and we deduce that DE is isomorphic to a corner of DF . Lastly, we show that this isomorphism implements the homeomorphism between the boundary-path spaces. [ABSTRACT FROM AUTHOR]

We show that the supremum for c real of the Hausdorff dimension of the Julia set of the polynomial z ↦ zd + c (d is an even natural number) is greater than 2d/(d + 1). [ABSTRACT FROM AUTHOR]

We show that there is a point on a computable arc that does not belong to any computable rectifiable curve. We also show that there is a point on a computable rectifiable curve with computable length that does not belong to any computable arc. [ABSTRACT FROM AUTHOR]

We investigate the set of biaccessible points for connected polynomial Julia sets of arbitrary degrees d ≥ 2. We prove that the Hausdorff dimension of the set of external angles corresponding to biaccessible points is less than 1, unless the Julia set is an interval. This strengthens theorems of Stanislav Smirnov and Anna Zdunik: they proved that the same set of external angles has zero 1-dimensional measure. [ABSTRACT FROM AUTHOR]

Given a compact subset F of R2, the visible part VθF of F from direction θ is the set of x in F such that the half-line from x in direction θ intersects F only at x. It is suggested that if dimH F ≥ 1, then dimH VθF = 1 for almost all θ, where dimH denotes Hausdorff dimension. We confirm this when F is a self-similar set satisfying the convex open set condition and such that the orthogonal projection of F onto every line is an interval. In particular the underlying similarities may involve arbitrary rotations and F need not be connected. [ABSTRACT FROM AUTHOR]