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1
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On a discrete version of the antipodal theorem

100%
Oleszkiewicz K.
Fundamenta Mathematicae
|
1996
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tom 151
|
nr 2
189-194
EN
The classical theorem of Borsuk and Ulam [2] says that for any continuous mapping $f: S^k → ℝ^k$ there exists a point $x ∈ S^k$ such that f(-x) = f(x). In this note a discrete version of the antipodal theorem is proved in which $S^k$ is replaced by the set of vertices of a high-dimensional cube equipped with Hamming's metric. In place of equality we obtain some optimal estimates of $inf_x ||f(x)-f(-x)||$ which were previously known (as far as the author knows) only for f linear (cf. [1]).
2
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On the Signatures of Torus Knots

64%
Borodzik M., Oleszkiewicz K.
Bulletin of the Polish Academy of Sciences. Mathematics
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2010
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tom 58
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nr 2
167-177
EN
We study properties of the signature function of the torus knot $T_{p,q}$. First we provide a very elementary proof of the formula for the integral of the signature over the circle. We also obtain a closed formula for the Tristram-Levine signature of a torus knot in terms of Dedekind sums.
3
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On Measure Concentration of Vector-Valued Maps

64%
Ledoux M., Oleszkiewicz K.
Bulletin of the Polish Academy of Sciences. Mathematics
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2007
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tom 55
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nr 3
261-278
EN
We study concentration properties for vector-valued maps. In particular, we describe inequalities which capture the exact dimensional behavior of Lipschitz maps with values in $ℝ^{k}$. To this end, we study in particular a domination principle for projections which might be of independent interest. We further compare our conclusions with earlier results by Pinelis in the Gaussian case, and discuss extensions to the infinite-dimensional setting.
4
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Small ball probability estimates in terms of width

64%
Latała R., Oleszkiewicz K.
Studia Mathematica
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2005
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tom 169
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nr 3
305-314
EN
A certain inequality conjectured by Vershynin is studied. It is proved that for any symmetric convex body K ⊆ ℝⁿ with inradius w and γₙ(K) ≤ 1/2 we have $γₙ(sK) ≤ (2s)^{w²/4}γₙ(K)$ for any s ∈ [0,1], where γₙ is the standard Gaussian probability measure. Some natural corollaries are deduced. Another conjecture of Vershynin is proved to be false.
5
Content available remote

A note on sums of independent uniformly distributed random variables

64%
Latała R., Oleszkiewicz K.
Colloquium Mathematicum
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1995
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tom 68
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nr 2
197-206
6
Content available remote

On the best constant in the Khinchin-Kahane inequality

64%
Latała R., Oleszkiewicz K.
Studia Mathematica
|
1994
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tom 109
|
nr 1
101-104
EN
We prove that if $r_i$ is the Rademacher system of functions then $(ʃ ∥∑_{i=1}^{n} x_{i}r_{i}(t)∥^2 dt)^{1/2} ≤ √2 ʃ ∥∑_{i=1}^{n}x_{i}r_{i}(t)∥dt$ for any sequence of vectors $x_i$ in any normed linear space F.
7
Content available remote

Polydisc slicing in $ℂ^n$

64%
Oleszkiewicz K., Pełczyński A.
Studia Mathematica
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2000
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tom 142
|
nr 3
281-294
EN
Let D be the unit disc in the complex plane ℂ. Then for every complex linear subspace H in $ℂ^n$ of codimension 1, $vol_{2n-2}(D^{n-1}) ≤ vol_{2n-2}(H ∩ D^{n}) ≤ 2vol_{2n-2}(D^{n-1})$. The lower bound is attained if and only if H is orthogonal to the versor $e_{j}$ of the jth coordinate axis for some j = 1,...,n; the upper bound is attained if and only if H is orthogonal to a vector $e_{j} + σe_{k}$ for some 1 ≤ j < k ≤ n and some σ ∈ ℂ with |σ| = 1. We identify $ℂ^n$ with $ℝ^{2n}$; by $vol_{k}(·)$ we denote the usual k-dimensional volume in $ℝ^{2n}$. The result is a complex counterpart of Ball's [B1] result for cube slicing.
8
Content available remote

Classes of measures closed under mixing and convolution. Weak stability

51%
Misiewicz J. K., Oleszkiewicz K., Urbanik K.
Studia Mathematica
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2005
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tom 167
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nr 3
195-213
EN
For a random vector X with a fixed distribution μ we construct a class of distributions ℳ(μ) = {μ∘λ: λ ∈ 𝓟}, which is the class of all distributions of random vectors XΘ, where Θ is independent of X and has distribution λ. The problem is to characterize the distributions μ for which ℳ(μ) is closed under convolution. This is equivalent to the characterization of the random vectors X such that for all random variables Θ₁, Θ₂ independent of X, X' there exists a random variable Θ independent of X such that $XΘ₁ + X'Θ₂ \stackrel{d}{=} XΘ$. We show that for every X this property is equivalent to the following condition: ∀ a,b ∈ ℝ ∃ Θ independent of X, $aX + bX' \stackrel{d}{=} XΘ$. This condition reminds the characterizing condition for symmetric stable random vectors, except that Θ is here a random variable, instead of a constant. The above problem has a direct connection with the concept of generalized convolutions and with the characterization of the extreme points for the set of pseudo-isotropic distributions.
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