Search results for "Hausdorff space"
showing 10 items of 64 documents
A common extension of Arhangel'skii's Theorem and the Hajnal-Juhasz inequality
2019
AbstractWe present a result about $G_{\unicode[STIX]{x1D6FF}}$ covers of a Hausdorff space that implies various known cardinal inequalities, including the following two fundamental results in the theory of cardinal invariants in topology: $|X|\leqslant 2^{L(X)\unicode[STIX]{x1D712}(X)}$ (Arhangel’skiĭ) and $|X|\leqslant 2^{c(X)\unicode[STIX]{x1D712}(X)}$ (Hajnal–Juhász). This solves a question that goes back to Bell, Ginsburg and Woods’s 1978 paper (M. Bell, J.N. Ginsburg and R.G. Woods, Cardinal inequalities for topological spaces involving the weak Lindelöf number, Pacific J. Math. 79(1978), 37–45) and is mentioned in Hodel’s survey on Arhangel’skiĭ’s Theorem (R. Hodel, Arhangel’skii’s so…
A new definition of well-behaved discrimination functions
2009
Abstract A discrimination function shows the probability or degree with which stimuli are discriminated from each other when presented in pairs. In a previous publication [Kujala, J.V., & Dzhafarov, E.N. (2008). On minima of discrimination functions. Journal of Mathematical Psychology , 52 , 116–127] we introduced a condition under which the conformity of a discrimination function with the law of Regular Minimality (which says, essentially, that “being least discriminable from” is a symmetric relation) implies the constancy of the function’s minima (i.e., the same level of discriminability of every stimulus from the stimulus least discriminable from it). This condition, referred to as “well…
Convergence for varying measures in the topological case
2023
In this paper convergence theorems for sequences of scalar, vector and multivalued Pettis integrable functions on a topological measure space are proved for varying measures vaguely convergent.
Conical upper density theorems and porosity of measures
2008
Abstract We study how measures with finite lower density are distributed around ( n − m ) -planes in small balls in R n . We also discuss relations between conical upper density theorems and porosity. Our results may be applied to a large collection of Hausdorff and packing type measures.
Dimension gap under conformal mappings
2012
Abstract We give an estimate for the Hausdorff gauge dimension of the boundary of a simply connected planar domain under p -integrability of the hyperbolic metric, p > 1 . This estimate does not degenerate when p tends to one; for p = 1 the boundary can even have positive area. The same phenomenon is extended to general planar domains in terms of the quasihyperbolic metric. We also give an example which shows that our estimates are essentially sharp.
Sets versus trial sequences, Hausdorff versus von Mises: “Pure” mathematics prevails in the foundations of probability around 1920
2010
Abstract The paper discusses the tension which occurred between the notions of set (with measure) and (trial-) sequence (or—to a certain degree—between nondenumerable and denumerable sets) when used in the foundations of probability theory around 1920. The main mathematical point was the logical need for measures in order to describe general nondiscrete distributions, which had been tentatively introduced before (1919) based on von Mises’s notion of the “Kollektiv.” In the background there was a tension between the standpoints of pure mathematics and “real world probability” (in the words of J.L. Doob) at the time. The discussion and publication in English translation (in Appendix ) of two …
New applications of extremely regular function spaces
2017
Let $L$ be an infinite locally compact Hausdorff topological space. We show that extremely regular subspaces of $C_0(L)$ have very strong diameter $2$ properties and, for every real number $\varepsilon$ with $0<\varepsilon<1$, contain an $\varepsilon$-isometric copy of $c_0$. If $L$ does not contain isolated points they even have the Daugavet property, and thus contain an asymptotically isometric copy of $\ell_1$.
Planar Sobolev homeomorphisms and Hausdorff dimension distortion
2011
We investigate how planar Sobolev-Orlicz homeomorphisms map sets of Hausdorff dimension less than two. With the correct gauge functions the generalized Hausdorff measures of the image sets are shown to be zero.
Generalized Hausdorff dimension distortion in Euclidean spaces under Sobolev mappings
2010
Abstract We investigate how the integrability of the derivatives of Orlicz–Sobolev mappings defined on open subsets of R n affect the sizes of the images of sets of Hausdorff dimension less than n. We measure the sizes of the image sets in terms of generalized Hausdorff measures.
Dimension gap under Sobolev mappings
2015
Abstract We prove an essentially sharp estimate in terms of generalized Hausdorff measures for the images of boundaries of Holder domains under continuous Sobolev mappings, satisfying suitable Orlicz–Sobolev conditions. This estimate marks a dimension gap, which was first observed in [2] for conformal mappings.