Authors:
Damir Ferizović Katholieke Universiteit Leuven, Celestijnenlaan 200B, Leuven, 3001, Belgium

Search for other papers by Damir Ferizović in
Current site
Google Scholar
PubMed
Close
,
Julian Hofstadler Universität Passau, Innstraße 33, Passau, 94032, Germany

Search for other papers by Julian Hofstadler in
Current site
Google Scholar
PubMed
Close
, and
Michelle Mastrianni University of Minnesota, 206 Church St. SE, Minneapolis, 55455, Minnesota, USA

Search for other papers by Michelle Mastrianni in
Current site
Google Scholar
PubMed
Close
Restricted access

In this paper we show that the spherical cap discrepancy of the point set given by centers of pixels in the HEALPix tessellation (short for Hierarchical, Equal Area and iso-Latitude Pixelation) of the unit 2-sphere is lower and upper bounded by order square root of the number of points, and compute explicit constants. This adds to the currently known (short) collection of explicitly constructed sets whose discrepancy converges with order 𝑁−1/2, matching the asymptotic order for i.i.d. random point sets. We describe the HEALPix framework in more detail and give explicit formulas for the boundaries and pixel centers. We then introduce the notion of an 𝑛-convex curve and prove an upper bound on how many fundamental domains are intersected by such curves, and in particular we show that boundaries of spherical caps have this property. Lastly, we mention briefly that a jittered sampling technique works in the HEALPix framework as well.

  • [1]

    C. Aistleitner, J. S. Brauchart, and J. Dick. Point Sets on the Sphere 𝕊2 with Small Spherical Cap Discrepancy. Discrete Comput. Geom., 48(4):9901024, 2012.

    • Search Google Scholar
    • Export Citation
  • [2]

    K. Alishahi and M. Zamani. The spherical ensemble and uniform distribution of points on the sphere. Electron. J. Probab., 20:2327, 2015.

    • Search Google Scholar
    • Export Citation
  • [3]

    J. Beck. Sums of distances between points on a sphere—an application of the theory of irregularities of distribution to discrete geometry. Mathematica, 31(1):3341, 1984.

    • Search Google Scholar
    • Export Citation
  • [4]

    J. Beck. Some upper bounds in the theory of irregularities of distribution. Acta Arithmetica, 43(2):115130, 1984.

  • [5]

    J. Beck and W. L. Chen. Irregularities of Distribution. Cambridge University Press, Cambridge, 1987.

  • [6]

    Beltrán, C. and U. Etayo. The Diamond ensemble: A constructive set of spherical points with small logarithmic energy. J. Complex., 59:101471, 2020.

    • Search Google Scholar
    • Export Citation
  • [7]

    C. Beltrán, J. Marzo, and J. Ortega-Cerdà. Energy and discrepancy of rotationally invariant determinantal point processes in high dimensional spheres. J. Complex, 37:76109, 2016.

    • Search Google Scholar
    • Export Citation
  • [8]

    J. Brauchart, P. Grabner. Distributing many points on spheres: Minimal energy and designs. J. Complex., 31(3):293326, 2015.

  • [9]

    S. Borodachov, D. Hardin, and E. Saff. Discrete Energy on Rectifiable Sets. Springer (2019).

  • [10]

    A. Bondarenko, D. Radchenko, and M. Viasovska. Well separated spherical designs. Constr. Approx., 41(1):93112, 2014.

  • [11]

    W. L. Chen, A. Srivastav, and G. Travaglini. A Panorama of Discrepancy Theory. Lecture Notes in Mathematics 2107 (2014).

  • [12]

    U. Etayo. Spherical Cap Discrepancy of the Diamond Ensemble. Discrete Comput. Geom., 66:12181238, 2021.

  • [13]

    D. Ferizović. Spherical cap discrepancy of perturbed lattices under the Lambert projection. Discrete Comput Geom (to appear 2023). (arXiv:2202.13894)

    • Search Google Scholar
    • Export Citation
  • [14]

    Górski, K. M., E. Hivon, A. J. Banday, B. D. Wandelt, F. K. Hansen, M. Reinecke, and M. Bartelmann. HEALPix: A Framework for High-Resolution Discretization and Fast Analysis of Data Distributed on the Sphere. Astrophys. J., 622:759771, 2005.

    • Search Google Scholar
    • Export Citation
  • [15]

    D. P. Hardin, T. Michaels, and E. B. Saff. A Comparison of Popular Point Configurations on 𝕊2. Dolomites Research Notes on Approximation, 9:1649, 2016.

    • Search Google Scholar
    • Export Citation
  • [16]

    A. B. J. Kuijlaars and E. B. Saff. Distributing many points on a sphere. Math. Intell., 19:511, 1997.

  • [17]

    J. Matousek. Geometric Discrepancy: An Illustrated Guide. Algorithms and Comb., 18, 2010.

  • [18]

    A. Lubotzky, R. Phillips, and P. Sarnak. Hecke Operators and Distributing Points on the Sphere I. Commun. Pure Appl. Math., 39, 1968.

  • [19]

    E. A. Rakhmanov, E. B. Saff, and Y. M. Zhou. Minimal discrete energy on the sphere. Math. Res. Lett., 1:647662, 1994.

  • [20]

    N. Sauer. On the density of families of sets. J. Comb. Theory, Series A, 13(1):145147, 1972.

  • [21]

    S. Shelah. A combinatorial problem; stability and order for models and theories in infinitary languages. Pac. J. Math., 41(1):247261, 1972.

    • Search Google Scholar
    • Export Citation
  • Collapse
  • Expand

The LaTeX template package can be downloaded from HERE.

Editors in Chief

Gábor SIMONYI (Rényi Institute of Mathematics)
András STIPSICZ (Rényi Institute of Mathematics)
Géza TÓTH (Rényi Institute of Mathematics) 

Managing Editor

Gábor SÁGI (Rényi Institute of Mathematics)

Editorial Board

  • Imre BÁRÁNY (Rényi Institute of Mathematics)
  • Károly BÖRÖCZKY (Rényi Institute of Mathematics and Central European University)
  • Péter CSIKVÁRI (ELTE, Budapest) 
  • Joshua GREENE (Boston College)
  • Penny HAXELL (University of Waterloo)
  • Andreas HOLMSEN (Korea Advanced Institute of Science and Technology)
  • Ron HOLZMAN (Technion, Haifa)
  • Satoru IWATA (University of Tokyo)
  • Tibor JORDÁN (ELTE, Budapest)
  • Roy MESHULAM (Technion, Haifa)
  • Frédéric MEUNIER (École des Ponts ParisTech)
  • Márton NASZÓDI (ELTE, Budapest)
  • Eran NEVO (Hebrew University of Jerusalem)
  • János PACH (Rényi Institute of Mathematics)
  • Péter Pál PACH (BME, Budapest)
  • Andrew SUK (University of California, San Diego)
  • Zoltán SZABÓ (Princeton University)
  • Martin TANCER (Charles University, Prague)
  • Gábor TARDOS (Rényi Institute of Mathematics)
  • Paul WOLLAN (University of Rome "La Sapienza")

STUDIA SCIENTIARUM MATHEMATICARUM HUNGARICA
Gábor Sági
Address: P.O. Box 127, H–1364 Budapest, Hungary
Phone: (36 1) 483 8344 ---- Fax: (36 1) 483 8333
E-mail: smh.studia@renyi.mta.hu

Indexing and Abstracting Services:

  • CABELLS Journalytics
  • CompuMath Citation Index
  • Essential Science Indicators
  • Mathematical Reviews
  • Science Citation Index Expanded (SciSearch)
  • SCOPUS
  • Zentralblatt MATH

2022  
Web of Science  
Total Cites
WoS
570
Journal Impact Factor 0.7
Rank by Impact Factor

Mathematics (Q3)

Impact Factor
without
Journal Self Cites
0.7
5 Year
Impact Factor
0.8
Journal Citation Indicator 0.65
Rank by Journal Citation Indicator

Mathematics (Q2)

Scimago  
Scimago
H-index
26
Scimago
Journal Rank
0.351
Scimago Quartile Score

Mathematics (Q3)

Scopus  
Scopus
Cite Score
1.8
Scopus
CIte Score Rank
General Mathematics 128/387 (67th PCTL)
Scopus
SNIP
1.276

2021  
Web of Science  
Total Cites
WoS
589
Journal Impact Factor 0,739
Rank by Impact Factor Mathematics 229/332
Impact Factor
without
Journal Self Cites
0,710
5 Year
Impact Factor
0,654
Journal Citation Indicator 0,57
Rank by Journal Citation Indicator Mathematics 287/474
Scimago  
Scimago
H-index
26
Scimago
Journal Rank
0,265
Scimago Quartile Score Mathematics (miscellaneous) (Q3)
Scopus  
Scopus
Cite Score
1,3
Scopus
CIte Score Rank
General Mathematics 193/391 (Q2)
Scopus
SNIP
0,746

2020  
Total Cites 536
WoS
Journal
Impact Factor
0,855
Rank by Mathematics 189/330 (Q3)
Impact Factor  
Impact Factor 0,826
without
Journal Self Cites
5 Year 1,703
Impact Factor
Journal  0,68
Citation Indicator  
Rank by Journal  Mathematics 230/470 (Q2)
Citation Indicator   
Citable 32
Items
Total 32
Articles
Total 0
Reviews
Scimago 24
H-index
Scimago 0,307
Journal Rank
Scimago Mathematics (miscellaneous) Q3
Quartile Score  
Scopus 139/130=1,1
Scite Score  
Scopus General Mathematics 204/378 (Q3)
Scite Score Rank  
Scopus 1,069
SNIP  
Days from  85
submission  
to acceptance  
Days from  123
acceptance  
to publication  
Acceptance 16%
Rate

2019  
Total Cites
WoS
463
Impact Factor 0,468
Impact Factor
without
Journal Self Cites
0,468
5 Year
Impact Factor
0,413
Immediacy
Index
0,135
Citable
Items
37
Total
Articles
37
Total
Reviews
0
Cited
Half-Life
21,4
Citing
Half-Life
15,5
Eigenfactor
Score
0,00039
Article Influence
Score
0,196
% Articles
in
Citable Items
100,00
Normalized
Eigenfactor
0,04841
Average
IF
Percentile
13,117
Scimago
H-index
23
Scimago
Journal Rank
0,234
Scopus
Scite Score
76/104=0,7
Scopus
Scite Score Rank
General Mathematics 247/368 (Q3)
Scopus
SNIP
0,671
Acceptance
Rate
14%

 

Studia Scientiarum Mathematicarum Hungarica
Publication Model Hybrid
Submission Fee none
Article Processing Charge 900 EUR/article
Printed Color Illustrations 40 EUR (or 10 000 HUF) + VAT / piece
Regional discounts on country of the funding agency World Bank Lower-middle-income economies: 50%
World Bank Low-income economies: 100%
Further Discounts Editorial Board / Advisory Board members: 50%
Corresponding authors, affiliated to an EISZ member institution subscribing to the journal package of Akadémiai Kiadó: 100%
Subscription fee 2023 Online subsscription: 708 EUR / 860 USD
Print + online subscription: 796 EUR / 970 USD
Subscription Information Online subscribers are entitled access to all back issues published by Akadémiai Kiadó for each title for the duration of the subscription, as well as Online First content for the subscribed content.
Purchase per Title Individual articles are sold on the displayed price.

Studia Scientiarum Mathematicarum Hungarica
Language English
French
German
Size B5
Year of
Foundation
1966
Volumes
per Year
1
Issues
per Year
4
Founder Magyar Tudományos Akadémia  
Founder's
Address
H-1051 Budapest, Hungary, Széchenyi István tér 9.
Publisher Akadémiai Kiadó
Publisher's
Address
H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
Responsible
Publisher
Chief Executive Officer, Akadémiai Kiadó
ISSN 0081-6906 (Print)
ISSN 1588-2896 (Online)