List of publications
Preprints:
- Rataj J., Winter S.: On volume and surface area of parallel sets. II. Surface measures and (non-)differentiability of the volume.
arXiv 2301.07429. To appear in Bull. London Math. Soc.
Research papers:
- Rataj J., Winter S., Zähle M.: Mean Lipschitz-Killing curvatures for homogeneous random fractals.
arXiv 2107.14431. J. Fractal Geom. 10 (2023), 1-42.
- Rataj J.: Mean Euler characteristic of stationary random closed sets. <\i> Stoch. Proc. Appl. 137 (2021), 252-271.
- Rataj J., Zajíček L.: Smallness of the set of critical values of distance functions in two-dimensional Euclidean and Riemannian spaces.
arXiv 1903.09492. Mathematika 66 (2020), 297-324.
- Pokorný D., Rataj J., Zajíček L.: On the structure of WDC sets.
arXiv 1703.09520. Math. Nachr. 292 (2019), 1595-1626.
- Kiderlen M., Rataj J.: Dilation volumes of sets of bounded perimeter.
arXiv 1708.09191.
Adv. in Appl. Probab. 50 (2018), 1095–1118, DOI 10.1017/apr.2018.52.
- Hug D., Rataj J.: Mixed curvature measures of translative integral geometry.
arXiv 1606.04224. Geom. Dedicata 195 (2018), 101–120, DOI 10.1007/s10711-017-0278-1.
- Hug D., Rataj J., Weil W.: Flag representations of mixed volumes and mixed functionals of convex bodies.
arXiv 1705.04816. J. Math. Anal. Appl. 460 (2018), 745–776,
DOI 10.1016/j.jmaa.2017.12.039.
- Rataj J., Zajíček L.: On the structure of sets with positive reach.
arXiv 1604.08841. Math. Nachr. 290 (2017), 1806-1829, DOI 10.1002/mana.201600237.
- Goodey P., Hinderer W., Hug D., Rataj J., Weil W.: A flag representation of projection functions.
arXiv 1502.06747 Adv. Geom. 17 (2017), 303-322, DOI 10.1515/advgeom-2017-0022
- Fu J.H.G., Pokorný D., Rataj J.: Kinematic formulas for sets defined by differences of convex functions.
arXiv 1505.03388. Adv. Math. 311 (2017), 796-832
- Rataj J., Zähle M.: Legendrian cycles and curvatures.
arXiv 1402.2249 J. Geom. Anal. 25 (2015), 2133-2147,
DOI 10.1007/s12220-014-9506-1
- Rataj J.: Random sets of finite perimeter. arXiv 1401.1634
Math. Nachr. 288 (2015), 1047–1056, DOI 10.1002/mana.201300341
- Hug D., Rataj J., Weil W.: A product integral representation of mixed volumes of two convex bodies.
arXiv 1112.0955. Adv. Geom. 13 (2013), 633–662
- Pokorný D., Rataj J.: Normal cycles and curvature measures of sets with d.c. boundary,
arXiv 1209.2305. Adv. Math. 248 (2013), 963-985
- Rataj J., Zähle M.: Curvature densities of self-similar sets, arXiv 1009.6162. Indiana Univ. Math. J. 61 (2012) 4, 1425-1449
- Rataj J., Winter S.: Characterization of Minkowski measurability in terms of surface area, arXiv 1111.1825. J. Math. Anal. Appl. 400 (2013), 120-132
- Rataj J., Zajíček L.: Critical values and level sets of distance
functions in Riemannian, Alexandrov and Minkowski spaces, Houston J. Math. 38 (2012) 2, 445-467
- Honzl O., Rataj J.: Almost sure asymptotic behaviour of the $r$-neighbourhood surface area of Brownian paths. Czechoslovak Math. J. 62 (2012) 1, 67-75
- Auneau J., Rataj J., Jensen E.B.V.: Closed form of
the rotational Crofton formula. Math. Nachr. 285 (2012) 2-3, 164-180
- Rataj J., Winter S.: On volume and surface area of parallel sets,
arXiv 0905.3279. Indiana Univ. Math. J. 59 (2010) 5, 1661-1686
- Rataj J., Zajíček L.: Properties of distance functions on convex surfaces and Alexandrov spaces, arXiv 0902.4020. Czechoslovak Math. J. 61 (2011), 247–269
- Mrkvička T., Rataj J.: On estimation of intrinsic volume densities of stationary random closed sets via parallel sets in the plane. Kybernetika 45 (2009), 931-945
- Rataj J., Spodarev E., Meschenmoser D.: Approximations of
the Wiener sausage and its curvature measures. Ann. Appl. Probab. 19 (2009), 1840-1859
- Rataj J., Schmidt V., Spodarev E.: On the expected surface area of the Wiener sausage. Math. Nachr. 282 (2009) 4, 591-603
- Vedel Jensen E.B., Rataj J.: A rotational formula for intrinsic volumes. Adv. Appl. Math. 41 (2008), 530-560
- Rataj J.: Convergence of total variation of curvature measures. Monatsh. Math. 153 (2008), 153–164
- Mrkvička T., Rataj J.: On estimation of intrinsic volume densities of stationary random
closed sets. Stoch. Proc. Appl. 118 (2008) 2, 213-231
- Kiderlen M., Rataj J.: On infinitesimal increase of volumes of morphological transforms. Mathematika 53 (2006), 103-127
- Rataj J.: Estimation of intrinsic volumes from parallel neighbourhoods. Suppl. Rend. Circ. Mat. Palermo, Ser. II, 77 (2006), 553-563
- Rataj J.: On boundaries of unions of sets with positive reach. Beiträge zur
Algebra und Geometrie 46 (2005), 397-404
- Rataj J., Zähle M.: General normal cycles and Lipschitz manifolds of bounded curvature. Ann. Global Anal. Geom. 27 (2005), 135-156
- Rataj J.: On estimation of the Euler number by projections of thin slabs. Adv. in Appl. Probab. 36 (2004), b715-724
- Rataj J.: Absolute curvature measures for unions of sets with
positive reach. Mathematika 49 (2002), 33-44
- Rataj J.: On set covariance and three-point test sets.
Czechoslovak Math. J. 54 (2004), 205-214
- Rataj J., Zahle M.: Normal cycles of Lipschitz manifolds by approximation with parallel sets. Diff. Geom. Appl. 19 (2003), 113-126
- Rataj J., Zähle M.: A remark on mixed curvature measures for sets
with positive reach. Beiträge zur Algebra und Geometrie 43
(2002), 171-179
- Rataj J.: Determination of spherical area measures by means of
dilation volumes. Math. Nachr. 235 (2002), 143-162
- Rataj J., Zähle M.: Curvatures and Currents for Unions of Sets
with Positive Reach, II. Ann. Global Anal. Geom. 20 (2001), 1-21
- Rataj J.: A translative integral formula for absolute curvature
measures. Geom. Dedicata 84 (2001), 245-252
- Beneš V., Rataj J., Krejčíř P., Ohser J.: Projection measures and estimation variances of intensities. Statistics 32 (1999) 4, 369-393
- Rataj J.: Translative and kinematic formulae for curvature measures of flat sections. Math. Nachr. 197 (1999), 89-101
- Rataj J.: The iterated version of a translative integral formula
for sets of positive reach. Rend. Circ. Mat. Palermo Ser. II,
Suppl. 46 (1997), 129-138
- Rataj J., Saxl I.: Boolean cluster models: Mean cluster dilations
and spherical contact distances. Math. Bohemica 122
(1997), 21-36
- Saxl I., Rataj J.: Spherical contact and nearest neighbour
distances in Boolean cluster fields. Acta Stereol. 15
(1996), 91-96
- Rataj J.: Remarks to a translative formula for sets of positive
reach. Geom. Dedicata 65 (1997), 59-62
- Rataj J.: Estimation of oriented normal direction distribution
via mixed areas. Acta Stereol. 15 (1996), 77-82
- Rataj J.: Estimation of oriented direction distribution of a
planar body. Adv. Appl. Probab. 28 (1996),
394-404
- Rataj J., Zähle M.: Mixed curvature measures for sets of positive
reach and a translative integral formula. Geom. Dedicata 57
(1995), 259-283
- Rataj J.: Characterization of compact sets by their dilation
volume. Math. Nachr. 173 (1995), 287-295
- Rataj J., Saxl I., Pelikán K.: Convergence of oscillating point
patterns to the Poisson point process. Applications of Mathematics
38 (1993), 221-235
- Rataj J.: Random distances and edge correction. Statistics
24 (1993), 377-385
- Rataj J.: An intersection formula for Hausdorff rectifiable sets
using mixed volumes of associated zonoids. Geometriae Dedicata
43 (1992), 65-69
- Rataj J., Saxl I.: Estimation of direction distribution of a
planar fibre system. Acta Stereol. 11 Suppl.I (1992), 631-636
- Lešanovský A., Rataj J., Hojek S.: 0-1 sequences having the same numbers of (1-1) couples of given distances Math. Bohemica 117(1992), 271-282
- Lešanovský A., Rataj J.: Solution to Problem 89-5. IMS
Bulletin 20 (1991), 564
- Saxl I., Rataj J.: Sférická kontaktní vzdálenost - charakteristika prostorového rozmístění částic (Spherical contact distance - characterization of spatial arrangement of particles). Pokroky práškové metalurgie 1-2 (1990), 88-160
- Rataj J., Pelikán K.: Korekce okrajových jevů při odhadu charakteristik geometrických struktur (Edge effect correction in estimation of characteristics of geometrical structures). Pokroky práškové metalurgie 1-2 (1990), 75-87
- Rataj J.: A characterization of weak convergence. Časopis
pěst. mat. 115 (1990), 48-60
- Kroupa F., Nejezchleb K., Rataj J., Saxl I.: Non-homogeneous
distribution of internal stresses and cracks in poled PZT ceramics. Ferroelectrics
100 (1989), 281-290
- Rataj J., Saxl I.: Analysis of planar anisotropy by means of the
Steiner compact set. J. Appl. Probab. 26
(1989), 490-502
- Saxl I., Rataj J.: Random paths and interparticle distances. Acta
Stereol. 7 (1988), 101-106
- Rataj J., Saxl I.: Analysis of planar anisotropy by means of the
Steiner compact set: a simple graphical method. Acta Stereol.
7 (1988), 107-112
- Rataj J., Zítek F.: Sur la norme de Fourier, III. Časopis
pěst. mat. 112 (1987), 312-319
- Rataj J.: Some remarks on Fourier seminorms. Časopis pěst.
mat. 112 (1987), 368-372
- Preiss D., Rataj J.: Maximal sets of orthogonal measures are not
analytic. Proc. Amer. Math. Soc. 93 (1985), 471-476
Monographs:
- Rataj J., Zähle M.: Curvature Measures of Singular Sets. Springer, Cham, 2019.
- Beneš V., Rataj J.: Stochastic
Geometry: Selected Topics. Kluwer Academic Publishers, Boston
2004
- Saxl I., Pelikán K., Rataj J., Besterci M.: Quantification
and Modelling of Heterogeneous Systems. Cambridge
Interscience Publishing, Cambridge 1995
Text book:
Rataj J: Bodové procesy [Point Processes]. Karolinum, Praha 2000,
second corrected edition 2006
Published conference contributions:
- Saxl I., Rataj J.: Random paths in non-convex bodies. In: Geobild
'89 (A.Hübler, W.Nagel, B.D.Ripley & G.Werner eds.),
Akademie-Verlag, Berlin (1989), 93-98
- Rataj J., Saxl I., Nejezchleb K., Kroupa F.: Nehomogenita
povrchových jevů (Non-homogeneity of superficial efects. In: Fraktografia
'89, Dom techniky ČSVTS, Košice (1989), 141-145
- Saxl I., Rataj J.: Anizotropie povrchových trhlin v
PZT-keramikách (Anisotropy of superficial cracks in PZT-ceramics). In: Fraktografia
'89, Dom techniky ČSVTS, Košice (1989), 451-455
- Lešanovský A., Rataj J.: Determination of compact sets in
Euclidean spaces by the volume of their dilation. In: DIANA III,
Mathematical
Institute,
Czechoslovak Academy of Sciences, Prague
(1990), 165-177
- Saxl I., Rataj J.: Distances of spherical contact in lattices of
figures and lattices of figures with faults. In: Geometrical
problems of image processing (U. Eckhardt, A.Hübler, W.Nagel &
G.Werner eds.), Research in Informatics 4, Akademie-Verlag, Berlin
(1991), 179-184
- Rataj J.: Flat section formulae in integral geometry. In: Proc.
S4G (International Conference on Stereology, Spatial Statistics and
Stochastic Geometry), Union of Czech Mathematicians and
Physicists, Prague 1999, 31-38
- Černý R., Rataj J.: How big is the Wiener sausage? In: Proc. S4G (International
Conference on Stereology, Spatial Statistics and
Stochastic Geometry), Union of Czech Mathematicians and
Physicists,
Prague 2006, 75-80
- Rataj J.: On estimation of boundary length and curvature
densities for planar stationary random sets. In: Stereology and Image Analysis, ECS
10 - Proceedings (V. Capasso et al. Eds.), The Miriam Project Series,
Esculapio Pub. Co., Bologna, Italy 2009 (electronic)
Theses:
- Rataj J.: Ortogonální pravděpodobnostní jádra [Orthogonal
Transition Kernels]. Diploma Thesis, Faculty of Mathematics and
Physics, Charles University, Prague (1985)
- Rataj J.: Charakterizace geometrických struktur v
euklidovském prostoru [Characterization of Geometrical structures in
Euclidean Spaces]. Ph.D. Thesis, Matematical Institute, Academy of
Sciences of the Czech Republic, Prague (1991)
- Rataj J.: Translative and kinematic integral geometry for
non-smooth bodies. Habilitation Thesis, Faculty of Mathematics and
Physics, Charles University, Prague 1999