Search

Start Over Creator Rataj, Jan

1. Almost sure asymptotic behaviour of the $r$-neighbourhood surface area of Brownian paths

Creator:
Honzl, Ondřej and Rataj, Jan
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
Minkowski content, Kneser function, Brownian motion, and Wiener sausage
Language:
English
Description:
We show that whenever the $q$-dimensional Minkowski content of a subset $A\subset \mathbb R^d$ exists and is finite and positive, then the “S-content” defined analogously as the Minkowski content, but with volume replaced by surface area, exists as well and equals the Minkowski content. As a corollary, we obtain the almost sure asymptotic behaviour of the surface area of the Wiener sausage in $\mathbb R^d$, $d\geq 3$.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

4. Konečná diagnóza druhé republiky?

Creator:
Rataj, Jan
Type:
article and TEXT
Language:
Czech
Description:
Stať je diskusní recenzí knihy Jana Gebharta a Jana Kuklíka Druhá republika 1938 -1939: Svár demokracie a totality v politickém, společenském a kulturním životě (Praha, Paseka 20 a zároveň úvahou nad charakterem politického režimu druhé republiky. V první části autor seznamuje s hlavními tezemi i obsahem jednotlivých kapitol dotyčné knihy, vytýká některé její konkrétní nedostatky, přičemž ji celkově hodnotí jako faktograficky bohatou a zasvěcenou syntézu dějin druhé republiky, opřenou o precizně shromážděné prameny. S jejími autory ovšem vede spor o interpretaci druhé republiky. Nesouhlasí s jejich výkladem, že v ní šlo primárně o zápas demokratických sil, snažících se uchovat co nejvíce z první republiky, a extrémních sil směřujících k totalitě, který se odehrával pod umírněně restriktivní kontrolou vlády Rudolfa Berana, jež musela reagovat na hrubý nátlak nacistického Německa. Proti tomu staví tezi, že po Mnichovu 1938 byl nastolen autoritářský režim, který se vlastní vývojovou logikou posouval k totalitnímu modelu a napájel se z myšlenkových zdrojů autoritářské konzervativní pravice za první republiky a českého fašismu., This is both a review of Jan Gebhart and Jan Kuklík’s Druhá republika 1938–1939: Svár demokracie a totality v politickém, společenském a kulturním životě (The Second Republic, 1938–39: A Conflict between Democracy and Totalitarianism in Political, Social, and Cultural Life), published by Paseka, Prague, in 2004, and an essay about the character of the political régime of the Second Republic. In the first part of the review-essay, the author introduces the main theses and contents of the individual chapters of Gebhart and Kuklík’s book, criticizing some specific shortcomings, though on the whole he considers it a factually rich, well-informed synthesis of the history of the Second Republic, based on precisely marshalled sources. He disagrees, however, with the authors’ interpretation of the Second Republic. In particular, he takes issue with their interpretation that the Second Republic represented primarily a struggle between democratic forces on the one hand, which were trying to preserve as much as possible of the First Republic, and extremist forces on the other, which were heading towards totalitarianism, a struggle played out under the moderately restrictive control of the Beran government, which had to react to the crude pressure applied by Nazi Germany. In contrast he presents the thesis that after the Munich Agreement of 1938 an authoritarian régime was established which, by the logic of developments moved to the totalitarian model and drew on the intellectual sources of the authoritarian conservative rightwing during the First Republic and Czech Fascism., and Diskuse
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

5. On estimation of intrinsic volume densities of stationary random closed sets via parallel sets in the plane

Creator:
Mrkvička, Tomáš and Rataj, Jan
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
random closed set, convex ring, curvature measure, and intrinsic volume
Language:
English
Description:
A method of estimation of intrinsic volume densities for stationary random closed sets in Rd based on estimating volumes of tiny collars has been introduced in T. Mrkvička and J. Rataj, On estimation of intrinsic volume densities of stationary random closed sets, Stoch. Proc. Appl. 118 (2008), 2, 213-231. In this note, a stronger asymptotic consistency is proved in dimension 2. The implementation of the method is discussed in detail. An important step is the determination of dilation radii in the discrete approximation, which differs from the standard techniques used for measuring parallel sets in image analysis. A method of reducing the bias is proposed and tested on simulated data.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

6. On set covariance and three-point test sets

Creator:
Rataj, Jan
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
convex body, set with positive reach, normal measure, and set covariance
Language:
English
Description:
The information contained in the measure of all shifts of two or three given points contained in an observed compact subset of $\mathbb{R}^d $ is studied. In particular, the connection of the first order directional derivatives of the described characteristic with the oriented and the unoriented normal measure of a set representable as a finite union of sets with positive reach is established. For smooth convex bodies with positive curvatures, the second and the third order directional derivatives of the characteristic is computed.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

7. Properties of distance functions on convex surfaces and applications

Creator:
Rataj, Jan and Zajíček, Luděk
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
distance function, convex surface, Alexandrov space, DC manifold, ambiguous locus, skeleton, and $r$-boundary
Language:
English
Description:
If $X$ is a convex surface in a Euclidean space, then the squared intrinsic distance function $\mathop {{\rm dist}}^2(x,y)$ is DC (d.c., delta-convex) on $X\times X$ in the only natural extrinsic sense. An analogous result holds for the squared distance function $\mathop {{\rm dist}}^2(x,F)$ from a closed set $F \subset X$. Applications concerning $r$-boundaries (distance spheres) and ambiguous loci (exoskeletons) of closed subsets of a convex surface are given.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public