numbers, functions, equations 2013 (nfe’13)

Numbers, Functions, Equations 2013

(NFE’13)

Dedicated to Professors

Zoltan Daroczy Imre Katai

on the occasion of their 75th birthday

Hotel Visegrad, Visegrad, Hungary

June 28–30, 2013http://compalg.inf.elte.hu/~nfe13/

Numbers, Functions, Equations 2013 (NFE’13)Hotel Visegrad, Visegrad (Hungary), June 28–30, 2013

List of Participants

1. Baron, KarolUniversity of Silesia, Katowice, [email protected]

2. Benczur, AndrasInstitute of Computer Science and ControlHungarian Academy of Sciences (MTA SZTAKI), Budapest, [email protected]

3. Bessenyei, MihalyUniversity of Debrecen, Debrecen, [email protected]

4. Boros, ZoltanUniversity of Debrecen, Debrecen, [email protected]

5. Burai, PalUniversity of Debrecen, Debrecen, [email protected]

6. Burcsi, PeterEotvos Lorand University, Budapest, [email protected]

7. Chakraborty, KalyanHarish-Chandra Research Institute, Allahabad, [email protected]

8. Czirbusz, SandorEotvos Lorand University, Budapest, [email protected]

9. Daroczy, ZoltanUniversity of Debrecen, Debrecen, [email protected]

10. Daroczy, BalintInstitute of Computer Science and ControlHungarian Academy of Sciences (MTA SZTAKI), Budapest, [email protected]

11. De Koninck, Jean-MarieUniversite Laval, Quebec, [email protected]


12. Demetrovics, JanosInstitute of Computer Science and ControlHungarian Academy of Sciences (MTA SZTAKI), Budapest, [email protected]

13. Farkas, GaborEotvos Lorand University, Budapest, [email protected]

14. Fridli, SandorEotvos Lorand University, Budapest, [email protected]

15. Fulop, AgnesEotvos Lorand University, Budapest, [email protected]

16. Ger, RomanUniversity of Silesia, Katowice, [email protected]

17. G lazowska, DorotaUniversity of Zielona Gora, Zielona Gora, [email protected]

18. Gombos, GergoEotvos Lorand University, Budapest, [email protected]

19. Gonda, JanosEotvos Lorand University, Budapest, [email protected]

20. Gselmann, EszterUniversity of Debrecen, Debrecen, [email protected]

21. Hazy, AttilaUniversity of Debrecen, Miskolc, [email protected]

22. Indlekofer, Karl-HeinzUniversity of Paderborn, Paderborn, [email protected]


23. Ivanyi, AntalEotvos Lorand University, Budapest, [email protected]

24. Ivic, AleksandarUniversiteta u Beogradu, Beograd, Serbiaaivic [email protected]

25. Jarai, AntalEotvos Lorand University, Budapest, [email protected]

26. Jarczyk, JustynaUniversity of Zielona Gora, Zielona Gora, [email protected]

27. Jarczyk, WitoldUniversity of Zielona Gora, Zielona Gora, [email protected]

28. Kacinskaite, RomaSiauliai University, Siauliai, [email protected]

29. Kallos, GaborSzechenyi Istvan University, Gyor, [email protected]

30. Katai, ImreEotvos Lorand University, Budapest, [email protected]

31. Kataine, Frank EditEotvos Lorand University, Budapest, [email protected]

32. Kaya, ErdenerUniversity of Mersin, Mersin, [email protected]

33. Kiss, AttilaEotvos Lorand University, Budapest, [email protected]

34. Kotora-Daroczy, ErzsebetUniversity of Debrecen, Debrecen, [email protected]


35. Kovacs, AttilaEotvos Lorand University, Budapest, [email protected]

36. Kozma, LaszloEotvos Lorand University, Budapest, [email protected]

37. Lajko, KarolyUniversity of Debrecen, Debrecen, [email protected]

38. Lakatos, PiroskaUniversity of Debrecen, Debrecen, [email protected]

39. Laurincikas, AntanasVilnius University, Vilnius, [email protected]

40. Lenard, MargitCollege of Nyıregyhaza, Nyıregyhaza, [email protected]

41. Losonczi, LaszloUniversity of Debrecen, Debrecen, [email protected]

42. Maksa, GyulaUniversity of Debrecen, Debrecen, [email protected]

43. Manstavicius, EugenijusVilnius University, Vilnius, [email protected]

44. Matkowski, JanuszUniversity of Zielona Gora, Zielona Gora, [email protected]

45. Matuszka, TamasEotvos Lorand University, Budapest, [email protected]

46. Meszaros, FruzsinaUniversity of Debrecen, Debrecen, [email protected]


47. Misevicius, GintautasVilnius University, Vilnius, [email protected]

48. Molnar, LajosUniversity of Debrecen, Debrecen, [email protected]

49. Nagy, GaborEotvos Lorand University, Budapest, [email protected]

50. Nemeth, ZsoltEotvos Lorand University, Budapest, [email protected]

51. Pales, ZsoltUniversity of Debrecen, Debrecen, [email protected]

52. Phong, Bui MinhEotvos Lorand University, Budapest, [email protected]

53. Reich, LudwigUniversity of Graz, Graz, [email protected]

54. Ronyai, LajosInstitute of Computer Science and ControlHungarian Academy of Sciences (MTA SZTAKI), Budapest, [email protected]

55. Sablik, MaciejUniversity of Silesia, Katowice, [email protected]

56. Schipp, FerencEotvos Lorand University, Budapest, [email protected]

57. Simon, PeterEotvos Lorand University, Budapest, [email protected]


58. Stepanauskas, GediminasVilnius University, Vilnius, [email protected]

59. Szekelyhidi, LaszloUniversity of Botswana, Gaborone, BotswanaUniversity of Debrecen, Debrecen, [email protected]

60. Szili, LaszloEotvos Lorand University, Budapest, [email protected]

61. Ton Quoc BinhUniversity of Hanoi, Hanoi, [email protected]

62. Toth, ViktoriaEotvos Lorand University, Budapest, [email protected]

63. Totik, VilmosUniversity of Szeged, Szeged, [email protected]

64. Varbanets, Pavel DmitrievichOdessa National University, Odessa, [email protected]

65. Varbanets, Sergey PavlovichOdessa National University, Odessa, [email protected]

66. Vatai, EmilEotvos Lorand University, Budapest, [email protected]

67. Varteresz, MagdolnaUniversity of Debrecen, Debrecen, [email protected]

68. Weisz, FerencEotvos Lorand University, Budapest, [email protected]


Program

June 28, Friday

12:30–14:00 Lunch

1ST Afternoon Session Chairman: Zsolt Pales

14:00–14:10 Conference opening

14:10–14:35 Jarai, Antal: On regularity properties of solutions definedon positive definite matrices of a Cauchy-type functional equation

14:35–15:00 Ger, Roman: Alienation of additive and logarithmic equations

15:00–15:25 Losonczi, Laszlo: Solved and unsolved problems for Pales means

15:25–15:50 Gselmann, Eszter: On some classes of partial differenceequations

15:50–16:15 Coffee Break

2ND Afternoon Session Chairman: Karl-Heinz Indlekofer

16:15–16:40 Manstavicius, Eugenijus: Additive function theory onpermutations

16:40–17:05 Matkowski, Janusz: Invariance formula for generalized quasi-arithmetic means

17:05–17:30 Kovacs, Attila: Some results on simultaneous number systemsin the ring of integers of imaginary quadratic fields

17:30–17:55 Toth, Viktoria: Collision and avalanche effect in pseudorandomsequences

18:00–20:00 Dinner


Program

June 29, Saturday, Morning

Two sections: (FE) and (N)

8:00-9:00 Breakfast

1ST Morning Session (FE) Chairman: Ludwig Reich

9:00–9:25 Jarczyk, Witold: Uniqueness of solutions of simultaneousdifference equations

9:25–9:50 Maksa, Gyula: Some remarks and problems related to realderivations

9:50–10:15 G lazowska, Dorota: Invariance of the quasi-arithmetic meanin the class of generalized weighted quasi-arithmetic means

10:15–10:40 Jarczyk, Justyna and Pal Burai: Conditional homogenityand translativity of Mako–Pales means

1ST Morning Session (N) Chairman: Aleksandar Ivic

9:00–9:25 Indlekofer, Karl-Heinz: On spaces of arithmetical functions

9:25–9:50 De Koninck, Jean-Marie; Nicolas Doyon; Florian Luca:Consecutive integers divisible by the square of their largest primefactors

9:50–10:15 Kallos, Gabor: On Renyi–Parry expansions

10:15–10:40 Farkas, Gabor: Sieving for large Cunningham chains of length 3of the first kind



Program

June 29, Saturday, Morning


2ND Morning Session (FE) Chairman: Antal Jarai

11:05–11:30 Reich, Ludwig: Generalized Aczel–Jabotinsky differentialequations, Briot–Bouquet equations and implicit functions

11:30–11:55 Sablik, Maciej and Agata Nowak: Chini’s equationsin actuarial mathematics

11:55–12:20 Baron, Karol: On some orthogonally additive functionson inner product spaces

12:20–12:45 Lajko, Karoly; Fruzsina Meszaros; Gyula Pap: Characteri-zation of a bivariate distribution through a functional equation

2ND Morning Session (N) Chairman: Eugenijus Manstavicius

11:05–11:30 Varbanets, Pavel and Sergey Varbanets: Twisted exponentialsums and the distribution of solutions of the congruencef(x, y) ≡ 0 (mod pm) over Z[i]

11:30–11:55 Laurincikas, Antanas; Kohji Matsumoto; Jorn Steuding:Hybrid universality of certain composite functions involvingDirichlet L-functions

11:55–12:20 Kaya, Erdener: Sums of independent random variables onadditive arithmetical semigroups

12:20–12:45 Nagy, Gabor: Number systems of the Gaussian integers withmodified canonical digit sets

12:45–14:00 Lunch


Program

June 29, Saturday, Afternoon


1ST Afternoon Session (FE) Chairman: Laszlo Kozma

14:00–14:25 Totik, Vilmos: A conjecture of Petruska and a conjecture of Widom

14:25–14:50 Molnar, Lajos: Sequential isomorphisms and endomorphismsof Hilbert space effect algebras

14:50–15:15 Bessenyei, Mihaly: Convexity structures induced by regular pairs

15:15–15:40 Boros, Zoltan and Noemi Nagy: Notes on approximately convexfunctions

1ST Afternoon Session (N) Chairman: Jean-Marie De Koninck

14:00–14:25 Ivic, Aleksandar: The divisor problem in short intervals

14:25–14:50 Misevicius, Gintautas and Audrone Rimkeviciene: Joint limittheorems for periodic Hurwitz zeta-function II.

14:50–15:15 Kacinskaite, Roma: A note on functional independence of somezeta-functions

15:15–15:40 Fridli, Sandor and Ferenc Schipp: Construction of rationalfunction systems with applications


2ND Afternoon Session (Laudations) Chairman: Ferenc Schipp

16:00–16:30 De Koninck, Jean-Marie: Laudation to Imre Katai

16:30–17:00 Pales, Zsolt: Laudation to Zoltan Daroczy

18:00–21:00 Banquet


Program

June 30, Sunday


8:00–9:00 Breakfast

1ST Morning Session (FE) Chairman: Vilmos Totik

9:00–9:25 Szekelyhidi, Laszlo: Exponential polynomials on Abeliangroups

9:25–9:50 Burai, Pal: Monotone operators and local-global minimum propertyof nonlinear optimization problems

9:50–10:15 Lenard, Margit: A (0, 2)-type Pal interpolation problem

10:15–10:40 Daroczy, Balint; Andras A. Benczur; Lajos Ronyai: Fisherkernels for image descriptors: a theoretical overview and experimentalresults

1ST Morning Session (N) Chairman: Gediminas Stepanauskas

9:00–9:25 Kozma, Laszlo: On semantic descriptions of software systems

9:25–9:50 Ivanyi, Antal: Testing of random sequences

9:50–10:15 Fulop, Agnes: Estimation of the Kolmogorov entropy in thegeneralized number system

10:15–10:40 Vatai, Emil: Speeding up distributed sieving with the inverse sieve

12:00–14:00 Lunch


Abstracts


Karol BaronUniversity of Silesia, Katowice, Poland

[email protected]

On some orthogonally additive functionson inner product spaces

Let E be a real inner product space of dimension at least 2. If f : E → E satisfies

f(x + y) = f(x) + f(y) for all orthogonal x, y ∈ E

andf(f(x)) = x for x ∈ E,

then f is additive.


Mihaly BessenyeiUniversity of Debrecen, Debrecen, Hungary

[email protected]

Convexity structures induced by regular pairs

The notion of (planar) convexity can be generalized using regular pairs. A regularpair is a pair of continuous functions with the property that each pair of points ofthe Cartesian plane (with distinct first coordinates) can be interpolated by a uniquemember of the linear hull of the components. Replacing classical lines by members ofthe linear hull, one can define a convexity structure on the plane. The aim of the talkis to determine the Radon-, Helly-, and Caratheodory-numbers of such structures andgive some applications in separation problems.


Zoltan BorosUniversity of Debrecen, Debrecen, Hungary

[email protected]

(joint work with Noemi Nagy)

Notes on approximately convex functions

The purpose of this note is to show that approximately convex (respectively, ap-proximately Wright-convex) functions with locally small error terms are, in fact, convex(respectively, Wright-convex).

Let I ⊂ R be an open interval, c ≥ 0 , p > 1 , and assume that the functionf : I → R fulfils the inequality

f(λx+ (1− λ)y) ≤ λf(x) + (1− λ)f(y) + c (λ(1− λ)|x− y|)p (1)

for every x, y ∈ I and λ ∈ [0, 1]. Then f is convex (i.e., f satisfies inequality (1) withc = 0 as well).

The proof is strongly related to the usual characterization of convexity with the aidof one sided derivatives. Establishing and applying results concerning the behavior ofcertain generalized derivatives, it is possible to prove an analogous result for Wright-convexity. Namely, under the same assumptions on c and p , if the function f : I → Rfulfils the inequality

f(λx+ (1− λ)y) + f(λy + (1− λ)x) ≤ f(x) + f(y) + c (λ(1− λ)|x− y|)p (2)

for every x, y ∈ I and λ ∈ [0, 1], then f is Wright-convex (i.e., f satisfies inequality (2)with c = 0 as well).


Pal BuraiTU Berlin and University of Debrecen, Germany and Hungary

[email protected] and [email protected]

Monotone operators and local-global minimumproperty of nonlinear optimization problems

Our main goal is to prove that every local minimizer of a certain nonlinear opti-mization problem is global. For this we use some results from the theory of monotoneoperators and connected functions.

At last we show applications of the main results in control theory.

References

[1] Avriel, M. and I. Zang, Generalized arcwise-connected functions and charac-terizations of local-global minimum properties, J. Optim. Theory Appl., 32 (1980),no. 4, 407–425.

[2] Browder, Felix E., The solvability of non-linear functional equations, DukeMath. J., 30 (1963), 557–566.

[3] Burai, P., Local-global minimum property in unconstrained minimization prob-lems, submitted (2012).

[4] de Figueiredo, D.G., Topics in Nonlinear Functional Analysis, Lecture seriesthe Institute for Fluid Dynamics and Applied Mathematics, vol. 48, University ofMaryland, College Park, Mayland, 1967.

[5] Drabek, Pavel and Jaroslav Milota, Methods of Nonlinear Analysis,Birkhauser Advanced Texts: Basler Lehrbucher. [Birkhauser Advanced Texts:Basel Textbooks], Birkhauser Verlag, Basel, 2007, Applications to differentialequations.

[6] Minty, G.J., Monotone networks, Proc. Roy. Soc. London. Ser. A, 257 (1960),194–212.

[7] Minty, George J., Monotone (nonlinear) operators in Hilbert space, Duke Math.J., 29 (1962), 341–346.

[8] Ortega, J.M. and W.C. Rheinboldt, Iterative Solution of Nonlinear Equa-tions in Several Variables, Classics in Applied Mathematics, vol. 30, Society forIndustrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000, Reprint ofthe 1970 original.


Balint DaroczyInstitute of Computer Science and Control

Hungarian Academy of Sciences (MTA SZTAKI), Budapest, Hungary

[email protected]

(joint work with Andras A. Benczur and Lajos Ronyai)

Fisher kernels for image descriptors:

a theoretical overview and experimental results

Visual words have recently proved to be a key tool in image classification. Bestperforming Pascal VOC and ImageCLEF systems use Gaussian mixtures or k-meansclustering to define visual words based on the content-based features of points of in-terest. In most cases, Gaussian Mixture Modeling (GMM) with a Fisher informationbased distance over the mixtures yields the most accurate classification results. In thispresentation we overview the theoretical foundations of the Fisher kernel method. Weindicate that it yields a natural metric over images characterized by low level contentdescriptors generated from a Gaussian mixture. We justify the theoretical observationsby reproducing standard measurements over the Pascal VOC 2007 data. Our accuracyis comparable to the most recent best performing image classification systems.


Jean-Marie De KoninckUniversite Laval, Quebec, Canada

[email protected]

(joint work with Nicolas Doyon and Florian Luca)

Consecutive integers divisible by the square

of their largest prime factors

We examine the occurrence of strings of consecutive integers divisible by a givenpower of their largest prime factor. In particular, we count how many strings of con-secutive integers, each of which is divisible by the square of its largest prime factor,are located below a given number.


Gabor FarkasEotvos Lorand University, Budapest, Hungary

[email protected]

Sieving for large Cunningham chainsof length 3 of the first kind

Our prospective purpose to reach new world record, namely find the largest knownCunningham chains of the first kind of length 3. This means the proof the primality ofnatural numbers p, 2p + 1, 4p + 3, each in the magnitude of 234944 (more than 10 500decimal digits). This would be impossible without sieving which reduces the estimatedtime of our project. I would like to talk about the details of the theoretical backgroundof the sieving process.


Sandor FridliEotvos Lorand University, Budapest, Hungary

[email protected]

(joint work with Ferenc Schipp)

Construction of rational function systems

with applications

In our talk we present our latest results on rational orthogonal and biorthogonalsystems that are inspired by applications including ECG processing and surface repre-sentation. We show several examples that demonstrate the benefit of such systems invarious situations. Constructions in the one and also in the two dimensional cases areshown. Also the problem of discretization is addressed.


Agnes FulopEotvos Lorand University, Budapest, Hungary

[email protected]

Estimation of the Kolmogorov entropyin the generalized number system

I study the dynamical properties of the complex number systems. A map is in-troduced on the transition graph by analogy of statistical physics systems. I gain anestimation of the Kolmogorov entropy by the Grassberger–Procaccia algorithm for afinite approximation of Bγ.


Roman GerSilesian University, Katowice, Poland

[email protected]

Alienation of additive andlogarithmic equations

Let (R,+, ·) stand for an Archimedean totally ordered unitary ring and let (H,+)be an Abelian group. Denote by C the cone of all strictly positive elements in R. Westudy the solutions f, g : C −→ H of a Pexider type functional equation

(E) f(x+ y) + g(xy) = f(x) + f(y) + g(x) + g(y)

resulting from summing up the additive and logarithmic equations side by side. Weshow that modulo an additive constant equation (E) forces f and g to split back tothe system of two Cauchy equations

f(x+ y) = f(x) + f(y)g(xy) = g(x) + g(y)

for every x, y ∈ C (alienation phenomenon).


Dorota G lazowskaUniversity of Zielona Gora, Zielona Gora, Poland

[email protected]

Invariance of the quasi-arithmetic mean

in the class of generalized weighted

quasi-arithmetic means

We consider the following invariance equation

Aϕ (Mf1,g1 ,Mf2,g2) = Aϕ, (1)

where Aϕ stands for the quasi-arithmetic mean and Mf1,g1 ,Mf2,g2 are two generalizedweighted quasi-arithmetic means. We solve this equation under four times continuousdifferentiability of the unknown functions ϕ, f1, f2, g1, g2.

We examine also a special case of the equation (1), i.e.

Aϕ (Mf,g,Mg,f ) = Aϕ.

We give the solution of this equation under natural assumptions on the unknownfunctions ϕ, f, g.

References

[1] Bajak, Sz. and Zs. Pales, Invariance equation for generalized quasi-arithmeticmeans, Aequationes Math., 77 (2009), 133–145.

[2] Matkowski, J., Invariant and complementary quasi-arithmetic means, Aequa-tiones Math., 57 (1999), 87–107.

[3] Matkowski, J., Remark 1, (at the Second Debrecen-Katowice Winter Seminaron Functional Equations and Inequailities Hajduszoboszlo) Ann. Math. Silesianae,16 (2002), p. 93.

[4] Matkowski, J. and P. Volkmann, A functional equation with two un-known functions, http://www.mathematik.uni-karlsruhe.de/~semlv, SeminarLV, No. 30, 6 pp., 28.04.2008.


Eszter GselmannUniversity of Debrecen, Debrecen, Hungary

[email protected]

On some classes of partial difference equations

In [1], J. A. Baker initiated the systematic investigation of some partial differenceequations. The main purpose of my talk is the investigation of partial difference equa-tions of the form

f(x1, . . . , xn) =m∑i=1

γi(x1, . . . , xn)f (x1 + ρ1,ih, . . . , xn + ρi,nh) ,

where m ∈ N, ρj,i ∈ R, i = 1, . . . ,m, j = 1, . . . , n are fixed as well as the functions γi,i = 1, . . . ,m.

Firstly, we present how such type of equations can be classified into elliptic, parabo-lic and hyperbolic subclasses, respectively. After that, we show solution methods ineach of these classes. Finally, some examples will follow.

References

[1] Baker, John A., An analogue of the wave equation and certain related functionalequations, Canadian Mathematical Bulletin. Bulletin Canadien de Mathematiques,12 (1969), 837–846.


Karl-Heinz IndlekoferUniversity of Paderborn, Paderborn, Germany

[email protected]

On spaces of arithmetical functions

Here we consider

- almost even functions,

- limit periodic functions,

- almost multiplicative functions

and extend the results to general algebras of arithmetical functions.


Antal IvanyiEotvos Lorand University, Budapest, Hungary

[email protected]

Testing of random sequences

Let ξ be a random integer vector, having uniform distribution

Pξ = (i1, i2, . . . , in) = 1/nn, for 1 ≤ i1, . . . , in ≤ n.

A realization (i1, i2, . . . , in) of ξ is called good, if its elements are different. We analyzealgorithms Backward, Forward, Linear, Garbage, Random, Tree, Modu-lar which decide whether a given random realization is good.

References

[1] Behrens, W. U., Feldversuchsanordnungen mit verbessertem Ausgleich derBodenunterschiede, Zeitschrift fur Landwirtschaftliches Versuchs- und Unter-suchungswesen, 2 (1956), 176–193.

[2] Crook, J. F., A pencil-and-paper algorithm for solving Sudoku puzzles. NoticesAmer. Math. Soc. 56(4) (2009), 460–468.

[3] Egorychev, G.P., Ivanyi, A.M. and A. I. Makosij, Analysis of two combina-torial sums characterizing the speed of computers with block memory. (Russian)Ann. Univ. Sci. Budapest., Sect. Comp., 7 (1987), 19–32.

[4] Ivanyi, A. and I. Katai, Estimates for speed of computers with interleavedmemory systems. Ann. Univ. Sci. Budap. Rolando Eotvos, Sect. Math. 19 (1976),159–164.

[5] Ivanyi, A. and I. Katai, Modeling of priorityless processing in an interleavedmemory with a perfectly informed processor. Autom. Remote Control 46(4)(1985), 520–526 (1985); translation from Avtom. Telemekh. 1985, No.4, 129–136.

[6] Ivanyi, A. and I. Katai, Testing of random matrices. Acta Univ. Sapientiae,Inform. 3(1) (2011), 99–126.

[7] Ivanyi, A. and I. Katai, Quick testing of random sequences. Egri-Nagy, Attilaet al. (ed.), Proceedings 8th Int. Conf. Appl. Math. (ICAI 2010), (Eger, Hungary,January 27–30, 2010.) 2 Volumes. Eger: BVB Nyomda es Kiado Kft., 2012, 379–386.

[8] Ivanyi, A. and I. Katai, Quick testing of random matrices. (Hungarian) Sapien-tia Matinfo Konferencia. Kivonatok (Marosvasarhely, May 25–26, 2013), MITIS,Kolozsvar, 2013, 13–14.

[9] Ivanyi, A. and B. Novak, Testing of sequences by simulation. Acta Univ.Sapientiae, Inform. 2(2) (2010), 135–153.


Aleksandar IvicSerbian Academy of Sciences and Arts, Beograd, Serbia

ivicrgf.bg.ac.rs, aivic 2000yahoo.com

The divisor problem in short intervals

Some new results involving ∆(X + U) − ∆(X) are presented, where the interval[X + U, X] is “short” in the sense that one can have U = o(X) as X →∞. As usual,

∆(x) :=∑n6x

d(n)− x log x− (2γ − 1)x

is the error term in the classical Dirichlet divisor problem, d(n) =∑

δ|n 1 is the number

of divisors of n, and γ = −Γ′(1) = 0.5772157 . . . is Euler’s constant. The author (2009)proved, sharpening a result of M. Jutila (1984): For 1 U = U(T ) ≤ 1

2

√T we have

(c3 = 8π−2)

2T∫T

(∆(x+ U)−∆(x)

)2dx = TU

3∑j=0

cj logj(√TU

)+Oε(T

1/2+εU2) +Oε(T1+εU1/2),

so that for T ε ≤ U = U(T ) ≤ T 1/2−ε this is a true asymptotic formula. The speakerand W. Zhai (2012) proved: Suppose log2 T U 6 T 1/2/2, T 1/2 H 6 T, then wehave

T+H∫T

max06u6U

∣∣∣∆(x+ u)−∆(x)∣∣∣2 dx HUL5 + TL4 logL+

+H1/3T 2/3U2/3L10/3(logL)2/3),

where L := log T. They also proved: Suppose T, U,H are large parameters and C > 1is a large constant such that

T 131/416+ε U 6 C−1T 1/2L−5, CT 1/4UL5 logL 6 H ≤ T.

Then in the interval [T, T + H] there are HU−1 subintervals of length U suchthat on each subinterval one has ±∆(x) ≥ c±T

1/4 for some c± > 0.


Antal JaraiEotvos Lorand University, Budapest, Hungary

[email protected]

On regularity properties of solutions defined on

positive definite matrices of a Cauchy-type

functional equation

For the real-valued solution of the functional equation

f(x1/2yx1/2

)= f(x)f(y)

satisfied for all positive definite matrices x, y it is proved that all measureble solutionsor solutions having the property of Baire are infinitely many times differentiable. Moregeneral equations are also treated.


Justyna JarczykUniversity of Zielona Gora, Zielona Gora, Poland

[email protected]

(joint work with Pal Burai)

Conditional homogenity and translativity

of Mako–Pales means

We characterize the homogeneous and translative members of the class of Mako-Pales means (see [1]). This is a common generalization of the classes of weighted quasi-arithmetic means and Lagrangian means. So, as an application we get the descriptionof homogeneous and translative means also within these classes.

References

[1] Mako, Z. and Zs. Pales, On the equality of generalized quasi-arithmetic means,Publ. Math. Debrecen, 72 (2008), 407–440.


Witold JarczykUniversity of Zielona Gora, Zielona Gora, Poland

[email protected]

Uniqueness of solutions of simultaneousdifference equations

Given a set T of positive numbers, a function c : T → R and a real number p wedeal with the simultaneous linear difference equations

ϕ(tx) = ϕ(x) + c(t)xp, (Et)

where t runs through T . We present uniqueness result in the case when individualequations (Et) usually have a lot of solutions in the considered class of functions.


Roma KacinskaiteSiauliai University, Siauliai, Lithuania

[email protected]

A note on functional independenceof some zeta-functions

In 1887, O. Holder proved [1] the algebraic-differential independence for the gammafunction Γ(s), which means that there exists no polynomial P 6≡ 0 such that, forn ∈ N ∪ 0 and all s ∈ C,

P (s,Γ(s), ...,Γ(n)(s)) = 0.

In 1973, S. M. Voronin proved [2] the functional independence of the Riemann zeta-function ζ(s) showing that ζ(s) does not satisfy any differential equation of the form

smFm(y(s), y′(s), ..., y(n−1)(s)) + ...+ F0(y(s), y′(s), ..., y(n−1)(s)) = 0,

where F0, ..., Fm are continuous functions not all identically zero.

In the talk, we present the generalization of Voronin’s result for some zeta-functions.In 2011, A. Laurincikas and the author considered [3] the functional independence ofperiodic zeta-function ζ(s; a) and periodic Hurwitz zeta-function ζ(s, α; b). One moregeneralization of the Voronin theorem for Dirichlet L-function L(s, χ) and the functionζ(s, α; b) by the author is obtained in [4].

References

[1] Holder, O., Uber die Eigenschaft der Gammafunktion keiner algebraischen Diffe-rentialgleichung zu genugen, Math. Ann., 28 (1887), 1–13.

[2] Voronin, S. M., On differential independence of ζ(s) functions, Dokl. Akad.Nauk SSSR, 209 (1973), no. 6, 1264–1266, (in Russian) = Sov. Math. Dokl., 14(1973), 607–609.

[3] Kacinskaite and A. Laurincikas, The joint distribution of periodic zeta-functions, Stud. Sci. Math. Hung., 48 (2011), no. 2, 257–279.

[4] Kacinskaite, R., Functional independence of Dirichlet L-function and periodicHurwitz zeta-function, to appear.


Gabor KallosSzechenyi Istvan University, Gyor, Hungary

[email protected]

On Renyi–Parry expansions

The properties of the expansions of real numbers in noninteger bases have beensystematically investigated since the late 50s, when the seminal works by AlfredRenyi and William Parry were published. The original idea was to choose the digitsin ”greedy” way, but the analysis was extended soon to general β- or q-expansions. Inthis talk we give a brief review about the results in this field.

The central questions in the presentation are as follows:

• What kind of expansions can be defined?

• How many possible expansions can exist?

• Which parts can occur in general and special expansions?

• What type of regularities can be found in expansions?

• Especially, what are the 1-expansions like?

Jointly, pay a tribute to activities of Professors Zoltan Daroczy and ImreKatai, some beautiful parts of their results will be presented; namely the field ofthe univoque numbers and new normal number-family constructions.

References

[1] Daroczy, Z. and I. Katai, Univoque sequences, Publ. Math. Debrecen, 42(1993), 397–407.

[2] Komornik, V., Expansions in noniteger bases, Integers, 11B (2011), 1–30.

[3] De Koninck J.-M. and I. Katai, Normal numbers created from primes andpolynomials, Uniform Distribution Theory, 7/2 (2012), 1–20.

[4] Parry, W., On the β expansions of real numbers, Acta Math. Hung., 11 (1960),401–406.

[5] Renyi, A., Representations for real numbers and their ergodic properties, ActaMath. Hung., 8 (1957), 477–493.


Erdener KayaMersin University, Mersin, Turkey

[email protected]

Sums of independent random variableson additive arithmetical semigroups

Let (G, ∂) be an additive arithmetical semigroup if G, generated by a countable setP of prime elements, is a commutative semigroup with identity element 1G, and ∂ is aninteger valued degree mapping ∂ : G → N0 [5]. Assume that the generating functionZ of G is an element of the exp− log class F (the class of Indlekofer [3]).

In this paper, we embed the additive arithmetical semigroup in a probability spaceΩ := (βG, σ(A), δ) where βG denote the Stone-Cech compactification of G [2, 4]. Weshow that every additive function g on G, g(a) =

∑pk‖a

g(pk) (a ∈ G), can be identified

with a sum g =∑p

Xp of independent random variables on Ω. The main result will

be that the existence of the limit distribution of a real-valued additive function g isequivalent to the a.e. convergence of g. And finally, we characterize the essentiallydistributed additive functions in case that the generating function Z of G belongs to asubclass of the exp− log class F [1].

References

[1] Barat, A., K.-H. Indlekofer and E. Kaya, Two-series theorem in additivearithmetical semigroups, Annales Univ. Sci. Budapest., Sect. Comp., 38, 309–318(2012).

[2] Indlekofer, K.-H., New approach to probabilistic number theory-compactifications and integration, Adv. Stud. in Pure Math., 49, Probability andNumber Theory-Kanazawa, 133–170 (2005).

[3] Indlekofer, K.-H., Tauberian theorems with applications to arithmetical semi-groups and probabilistic combinatorics, Annales Univ. Sci. Budapest., Sect.Comp., 34, 135–177 (2011).

[4] Indlekofer, K.-H. and E. Kaya, The three-series theorem in additive arithmeti-cal semigroups, Annales Univ. Sci. Budapest., Sect. Comp., 38, 161–181 (2012).

[5] Knopfmacher J. and W.-B. Zhang, Number theory arising from finite fields,Analytic and probabilistic theory, 241 Pure and Appl. Math., Marcel Decker, NewYork, (2001).


Attila KovacsEotvos Lorand University, Budapest, Hungary

[email protected]

Some results on simultaneous number systems inthe ring of integers of imaginary quadratic fields

Let Λ be a lattice in Rn, M : Λ→ Λ be a linear operator such that det(M) 6= 0, andlet D be a finite subset of Λ containing 0. The triple (Λ,M,D) is called a generalizednumber system (GNS) if every element x of Λ has a unique, finite representation of theform x =

∑li=0M

idi, where di ∈ D, l ∈ N and dl 6= 0.

If (Λ,M,D) is a number system then (1) D must be a full residue system moduloM , (2) M must be expansive and (3) det(I − M) 6= ±1. If a system fulfils theseconditions then it is a radix system and the operator M is called a radix base.

In [2] block diagonal systems were introduced and investigated. Let the radix sys-tems (Λi,Mi, Di) be given (1 ≤ i ≤ k). Let Λ = ⊗Λi the direct product of thelattices, M = ⊕k

i=1Mi the direct sum of the bases and Dh = (dT1 ‖dT2 ‖ · · · ‖dTk )T : di ∈∈ Di the homomorphic digit set. Here dT is a row vector and ‖ means the concatena-tion operator. Katai et. al. [1] introduced the notion of simultaneous radix systemswhen the blocks N1, N2, . . . , Nk are mutual coprime integers (none of them is 0,±1)and D = δe where e = (1, 1, . . . , 1)T , δ = 1, 2, . . . , |N1N2 · · ·Nk| − 1. The computa-tion of the expansions are based on Chinese Remaindering. Katai et al. showed thatthe system (Z2, N1 ⊕N2, D) is GNS if and only if N1 < N2 ≤ −2 and N2 = N1 + 1.

The simultaneous radix expansions were also investigated in the lattice of Gaussianintegers [4, 2, 3]. In this talk we present the newest results achieved in the research ofsimultaneous number expansions in the ring of integers of imaginary quadratic fields.

References

[1] Indlekofer, K-H., I. Katai and P. Racsko, Number systems and fractalgeometry, Probability Theory and Applications, Kluwer Academic Press, (1993)319–334.

[2] Kovacs, A., Number System Constructions with Block Diagonal Bases, submit-ted (2012)

[3] Kovacs, A., Algorithmic construction of simultaneous number systems in thelattice of Gaussian integers, Annales Univ. Sci. Budapest., Sect. Comp., 39 (2013)279–290.

[4] Nagy, G., On the simultaneous number systems of Gaussian integers, AnnalesUniv. Sci. Budapest., Sect. Comp., 35 (2011) 223–238.


Laszlo KozmaEotvos Lorand University, Budapest, Hungary

[email protected]

On semantic descriptions of software systems

Formal methods are essential for giving precise descriptions of software systems.In our talk we analysed some approaches to conventional semantics and to actionsemantics. As a result we suggest to use action semantics for describing semanticproperties of software systems including programming and MDE languages.


Karoly LajkoUniversity of Debrecen, Debrecen, Hungary

[email protected]

(joint work with Fruzsina Meszaros and Gyula Pap)

Characterization of a bivariate distribution

through a functional equation

We deal with a special characterization problem of a conditionally specified abso-lutely continuous bivariate distribution by the help of a functional equation satisfiedalmost everywhere on its domain for the unknown density functions. We consider thecase of conditionals in location-scale families with specified moments (we have linearregressions and conditional standard deviations).


Antanas LaurincikasVilnius University, Lithuania

[email protected]

(joint work with Kohji Matsumoto and Jorn Steuding)

Hybrid universality of certain composite functions

involving Dirichlet L-functions

We generalize a hybrid joint universality theorem for Dirichlet L-functions L(s, χ)obtained in [1].

Let D =s ∈ C : 1

2< σ < 1

. Denote by H(D) the space of analytic functions on

D equipped with the topology of uniform convergence on compacta. Moreover, denoteby K the class of compact subsets of D with connected complements, and let H(K),K ∈ K, stand for the class of continuous functions on K which are analytic in theinterior of K.

Let β1 > 0, . . . , βr > 0, and β = min1≤j≤r

βj. We say that the operator F : Hr(D) →H(D) belongs to the Lipschitz class Lip(β1, . . . , βr) if the following hypotheses aresatisfied:

1 For each polynomial p = p(s), and any K ∈ K, there exists an element (g1, . . . , gr)∈ F−1p ⊂ Hr(D) such that gj(s) 6= 0 on K, j = 1, . . . , r;

2 For any K ∈ K, there exist a constant c > 0, and a set K ∈ K such that

sups∈K|F (g11(s), . . . , g1r(s))− F (g21(s), . . . , g2r(s))| ≤ c sup

1≤j≤rsups∈K|g1j(s)− g2j(s)|βj

for all (gk1, . . . gkr) ∈ Hr(D), k = 1, 2.

In the report, we will present the following theorem.

Theorem 1. Suppose that χ1, . . . , χr are pairwise non-equivalent Dirichlet characters,and that F ∈ Lip(β1, . . . , βr). Let K ∈ K and f(s) ∈ H(K). Moreover, let αj : j =1, . . . ,m be any sequence of real numbers linearly independent over Q and θj : j =1, . . . ,m be any sequence of real numbers. Then, for every ε > 0,

lim infT→∞

1

Tmeas

τ ∈ [0, T ] : sup

s∈K|F (L(s+ iτ, χ1), . . . , L(s+ iτ, χr))− f(s)| < ε,

max1≤j≤m

‖ταj − θj‖ <(ε

2

) 1β

> 0.

References

[1] Pankowski, L., Hybrid joint universality theorem for Dirichlet L-functions, ActaArith., 141(1) (2010), 59–79.


Margit LenardCollege of Nyıregyhaza, Nyıregyhaza, Hungary

[email protected]

A (0,2)-type Pal interpolation problem

In 1975 L. Pal introduced a modification of Hermite-Fejer interpolation in whichthe function values and the first derivatives were prescribed on two sets of nodes xiand x∗i , respectively. In [2] he investigated the case when pn(xi) = 0 for i = 1, . . . , nand p′n(x∗i ) = 0 for i = 1, . . . , n− 1, where pn is a polynomial of degree n with distinctzeros. In 1983 L. Szili [3] studied the inverse problem when the roles of pn and p′n areinterchanged, namely, the first derivatives are interpolated at the zeros of pn and thefunction values are interpolated at the zeros of p′n. Several authors studied these twoproblems, the Pal-type (0;1)- and (1;0)-interpolation using different choices of nodes.

We consider the Pal interpolation with other kind of interpolation conditions. In-stead of the Lagrange condition (only the function values are given) we have Hermiteconditions (consecutive derivatives up to a higher order are prescribed). The lacunarycondition we substitute with the weighted (0,1,. . . ,r-2,r)-interpolation (r ≥ 2), thegeneralization of P. Turan’s problem [4], introduced by J. Balazs [1]. We study theregularity of the problem and the explicit representation of the fundamental polyno-mials of interpolation. We give some examples on the zeros of the classical orthogonalpolynomials.

References

[1] Balazs, J., Weighted (0, 2)-interpolation on the roots of theultrasphericalpolynomials, (in Hungarian: Sulyozott (0, 2)-interpolacio ultraszferikus polinomgyokein), MTA III.oszt. Kozl., 11 (1961), 305–338.

[2] Pal, L.G., A new modification of Hermite-Fejer interpolation, Analysis Math., 1(1975), 197–205.

[3] Szili, L., An interpolation process on the roots of integrated Legendre polynomi-als, Analysis Math., 9 (1983), 235–245.

[4] Turan, P., On some open problems of approximation theory, J. Approx. Theory,29 (1980), 23–85.


Laszlo LosoncziUniversity of Debrecen, Debrecen, Hungary

[email protected]

Solved and unsolved problems for Pales means

Given two continuous functions f, g : I → R on the nonempty open interval I suchthat g is positive and f/g is strictly monotone and a probability measure µ on theBorel subsets of [0, 1], the two variable mean Mf,g;µ : I2 → I, called Pales mean, isdefined by

Mf,g;µ(x, y) :=

(f

g

)−1

1∫0

f(tx+ (1− t)y

)dµ(t)

1∫0

g(tx+ (1− t)y

)dµ(t)

(x, y ∈ I).

The aim of our talk is to report on solved and unsolved problems (among othersequality, homogeneity, comparison problems) of these means.

References

[1] Losonczi, L., On homogeneous Pales means, Annales Univ. Sci. Budapest., SectioComputatorica (submitted).

[2] Losonczi, L., Homogeneous symmetric means of two variables, AequationesMath., 74 (2007), 262–281.

[3] Losonczi, L. and Zs. Pales, Equality of two-variable functional means generatedby different measures, Aequationes Math., 81 (2011), 31–53.

[4] Losonczi, L. and Zs. Pales, Comparison of means generated by two functionsand a measure, J. Math. Anal. Appl., 345 (2008), 135–146.


Gyula MaksaUniversity of Debrecen, Debrecen, Hungary

[email protected]

Some remarks and problemsrelated to real derivations

A real derivation is a function d : R → R (the set of all real numbers) for which thefunctional equations

d(x + y) = d(x) + d(y) (x, y ∈ R) (1)

andd(xy) = xd(y) + yd(x) (x, y ∈ R) (2)

hold simultaneously for all x, y ∈ R. The solutions of equation (1) are called additivefunctions and it is also known (see [2]) that there is non-zero additive function d whichfulfils (2), too. This makes possible to construct surprising counter-examples by usingreal derivations.

In this talk, firstly we show such a counter-example (see [1]) and then we presentsome characterizations of real derivations, finally we give sufficient conditions for ad-ditive functions to obtain that they are real derivations. Open problems will also beformulated.

References

[1] Daroczy, Z. and Gy. Maksa, Nonnegative information functions, Analytic func-tion methods in probability theory (Proc. Colloq. Methods of Complex Anal. in theTheory of Probab. and Statist., Kossuth L. Univ. Debrecen, 1977, 21, Colloq.Math. Soc. Janos Bolyai, 21, North-Holland, Amsterdam-New York, 1979, 67–78.

[2] Kuczma, M., An Introduction to the Theory of Functional Equations and In-equalities, Panstwowe Wydawnictwo Naukowe — Uniwersytet Slaski, Warszawa–Krakow–Katowice, 1985.


Eugenijus ManstaviciusVilnius University, Vilnius, Lithuania

[email protected]

Additive function theory on permutations

Influenced by probabilistic number theory (in particular, by the works of the hero ofthe day Professor Imre Katai and other hungarian mathematicians), we are dealing withthe value distribution of additive functions defined on permutations. By definition, areal two-dimensional array hj(s), where j ≥ 1, s ≥ 0, and hj(0) ≡ 0, via expression

h(σ) :=∑j≤n

hj(kj(σ)

),

determines an additive function h : Sn → R. Here Sn is the symmetric group of ordern ∈ N and kj(σ) ∈ Z+ denotes the number of cycles of length j in the permutationσ ∈ Sn. Assume that σ is taken at random with a probability

ν(θ)n

(σ

):= Λ−1

n

n∏j=1

θkj(σ)j , σ ∈ Sn, 00 := 1,

where θj ≥ 0 if j ≤ n and Λn is an appropriate normalizing sequence. The mainobjective is to find general conditions under which there exist sequences α(n) ∈ R and

β(n) > 0 such that ν(θ)n

(h(σ) − α(n) < xβ(n)

)weakly converges to a non-degenerate

distribution function as n → ∞. The case θj ≡ θ > 0 corresponds to the Ewensprobability measure on Sn. The interest to deal with it and more general weightedmeasures has been raised by some recent papers in physics.

By now, we have advanced in applying a probabilistic approach (see [1]) and amethod of factorial moments (see [2]). This will be discussed in the talk.

References

[1] Manstavicius, E., On total variation distance for random assemblies, DiscreteMath. and Theor. Computer Sci., AofA’12 Proc., 97–108 (2012).

[2] Baksajeva, T. and E. Manstavicius, On statistics of permutations chosenfrom the Ewens distribution (submitted; arXiv :0677678, 2013).


Janusz MatkowskiUniversity of Zielona Gora, Zielona Gora, Poland

[email protected]

Invariance formula for generalizedquasi-arithmetic means

A invariance formula in the class of generalized p-variable quasiarithmetic meanand an effective form of the limit of sequence of iterates of the mean-type mappings ofthis type will be given. Some examples and application in solving a functional equationwill be presented.


Gintautas MiseviciusVilnius Gediminas Technical University, Lithuania

[email protected]

(joint work with Audrone Rimkeviciene)

Joint limit theorems for periodic

Hurwitz zeta-function II.

Let s = σ + it be a complex variable, α, 0 < α 6 1, be a fixed parameter, anda = am : m ∈ N = N ∪ 0 be a periodic sequence of complex numbers with minimalperiod k ∈ N. The periodic Hurwitz zeta-function ζ(s, α; a) is defined, for σ > 1, bythe series

ζ(s, α, a) =∞∑

m=0

am(m+ α)s

,

and continues analytically to the whole complex plane, except, maybe, for a simplepole at the point s = 1.

In [1], two joint limit theorems on the weak convergence of probability measures onthe complex plane for periodic Hurwitz zeta-functions were proved. For j = 1, . . . , r,let ζ(s, αj, aj) be a periodic Hurwitz zeta-function with parameter αj, 0 < αj 6 1,and periodic sequence of complex numbers aj = amj : m ∈ N0 with minimal periodkj ∈ N. For brevity, we use the notation σ = (σ1, . . . , σr), σ+ it = (σ1 + it, . . . , σr + it),α = (α1, . . . , αr), a = (a1, . . . , ar) and ζ(s, α; a) =

(ζ(s, α1, ; a1), . . . ζ(s, αr; ar)

). Denote

by B(S) the class of Borel sets of the space S, and by measA the Lebesgue measure of ameasurable set A ⊂ R. Then in [5], the weak convergence as T →∞ of the probabilitymeasure

PT (A)def=

1

Tmeas

t ∈ [0, T ] : ζ(σ + it, α; a) ∈ A

, A ∈ B(Cr)),

was discussed. The cases of algebraically independent and rational parameters α1, . . . , αr

were considered.

The aim of this note is to consider the weak convergence of the probability measure

PT (A) =1

Tmeas

t ∈ [0, T ] : ζ

=(σ=

+it, α=, a=

) ∈ A, A ∈ B(Cr+r1).

References

[1] Rimkeviciene, A., Joint limit theorems for periodic Hurwitz zeta-functions,Siauliai Math. Seminar, 6(14) (2011), 53–68.


Lajos MolnarUniversity of Debrecen, Debrecen, Hungary

[email protected]

Sequential isomorphisms and endomorphismsof Hilbert space effect algebras

Let A be a C∗-algebra. The effect algebra E(A) of A consists of the self-adjointelements a for which 0 ≤ a ≤ 1. Effects play important role in the mathematicaldescriptions of the theory of quantum measurements. The sequential product on E(A)is defined by a b =

√ab√a.

In this talk we first recall a former result of ours on the structure of sequentialisomorphisms between effect algebras in the setting of von Neumann algebras. Wenext present a recent result that gives the general form of all (continuous) sequentialendomorphisms of finite dimensional Hilbert space effect algebras.

References

[1] Molnar, L., Sequential isomorphisms between the sets of von Neumann algebraeffects. Acta Sci. Math. (Szeged), 69 (2003), 755-772.

[2] Dolinar, G. and L. Molnar, Sequential endomorphisms of finite dimensionalHilbert space effect algebras, J. Phys. A: Math. Theor., 45 (2012), 065207.


Gabor NagyEotvos Lorand University, Budapest, Hungary

[email protected]

Number systems of the Gaussian integerswith modified canonical digit sets

I. Katai and J. Szabo determined in [1] all possible bases of canonical numbersystems of the Gaussan integers. Namely they proved that α ∈ Z[i] is the base of acanonical number system if and only if Re(α) < 0 and Im(α) = ±1. The aim of thetalk is to show how we can get number systems with the modification of the canonicaldigit set.

References

[1] Katai, I. and J. Szabo, Canonical number systems for complex integers, ActaSci. Math. (Szeged), 37 (1975), 255–260.


Ludwig ReichKarl-Franzens-Universitat Graz, Graz, Austria

[email protected]

Generalized

Aczel–Jabotinsky differential equations,

Briot–Bouquet equations and

implicit functions

We consider the differential equation

H(X) · dφ

dX(X) = G(φ(X)) (AJ,(H,G))

where H(X) and G(X) are given formal power series of the form H(X) = hkXk + . . .,

G(X) = gkXk + . . ., with hk, gk 6= 0, k ≥ 1, and the solution φ is an invertible power

series φ(X) = ρX + . . ..We discuss the origin of (AJ,(H,G)) in functional equation problems from iterationtheory, the theory of families of commuting power series, and from the theory of re-versibility. Then we present necessary and sufficient conditions for the existence of so-lutions φ 6= 0 and give a theorem an the structure of the set of solutions of (AJ,(H,G))by transforming (AJ,(H,G)) to a so called Briot-Bouquet differential equation in thenon-generic case. The cases k = 1 and k ≥ 2 require different approaches. Fromnow on we assume k = 1. Then we solve (AJ,(H,G)) also by transforming it to animplicit function problem. From this we easily deduce the dependence of the solutionson the internal parameter ρ and show that the solutions are local analytic functionsof (ρ,X), if H and G are both convergent. If H = G we obtain the so called stan-dard form φ(ρ,X) = S−1(ρS(X)) of the solution of the translation equation, the caseG(X) = hkX yields the transformation of (AJ,(H,H)) to its normal form.Our research is motivated by joint work with H. Fripertinger and W. Jablonski.


Maciej SablikUniversity of Silesia, Katowice, Poland

[email protected]

(joint work with Agata Nowak )

Chini’s equations in actuarial mathematics

We deal with an equation mentioned by M. Chini in [1] and recalled by A. Guer-raggio in [2]. This is a multiplicative form of the equation previously studied by T.Riedel, P. K. Sahoo and M. Sablik (cf. [3]). We solve it completely in the case wheresets of zeros of the unknown functions are nonempty, under very mild assumptions. Inthe remaining case, we reduce the problem to the one considered in [3].

References

[1] Chini, M., Sopra un’equazione da cui discendo due notevoli formule di Matem-atica attuariale. Period. Mat., 4 (1907), 264–270.

[2] Guerraggio, A., Le equazioni funzionali nei fondamenti della matematica fi-nanziera, Riv. Mat. Sci. Econom. Social., 9 (1986), 33–52.

[3] Riedel, T., M. Sablik and P.K. Sahoo, On a functional equation, ActuarialMathematics. JMAA, 253 (2001), 16–34.


Laszlo SzekelyhidiUniversity of Botswana, Gaborone, BotswanaUniversity of Debrecen, Debrecen, Hungary

[email protected]

[email protected]

Exponential polynomials on Abelian groups

Exponential monomials and polynomials on Abelian groups play a basic role in thetheory of functional equations and spectral synthesis. In this talk we characterize thesefunctions by the properties of the annihilator ideal of the variety generated by them. Inparticular, we obtain new characterization of polynomial functions on Abelian groups.


Viktoria TothEotvos Lorand University, Budapest, Hungary

[email protected]

Collision and avalanche effect inpseudorandom sequences

Recently a constructive theory of pseudorandomness of binary sequences has beendeveloped and many constructions for binary sequences with strong pseudorandomproperties have been given. Motivated by applications, Mauduit and Sarkozy in [1]generalized and extended this theory from the binary case to k-ary sequences, i.e., tok symbols. They constructed a large family of k-ary sequences with strong pseudoran-dom properties. I adapted the notions of collision and avalanche effect to study thesepseudorandom properties of families of binary and k-ary sequences. Two of the mostimportant binary constructions were tested in [2] for these pseudorandom properties,and it turned out that one of them is ideal from this point of view as well, while theother construction does not possess these pseudorandom properties. In another paper[3] it is shown that this weakness of the second construction can be corrected: one cantake a subfamily of the given family which is just slightly smaller and collision free.The study of the pseudorandom properties mentioned above was extended to k-arysequences in [4]. The aim of this paper is twofold. First the definitions of collisionand avalanche effect were extended to k-ary sequences, and then these related prop-erties were studied in a large family of pseudorandom k-ary sequences with ”small”pseudorandom measures.

References

[1] Mauduit, C. and A. Sarkozy, On finite pseudorandom sequences of k symbols,Indag. Math., 13 (2002), 89–101.

[2] Toth, V., Collision and avalanche effect in families of pseudorandom binary se-quences, Periodica Math. Hungar., 55 (2007), 185–196.

[3] Toth, V., The study of collision and avalanche effect in a family of pseudorandombinary sequences, Periodica Math. Hungar., 59 (2009), 1–8.

[4] Toth, V., Extension of the notion of collision and avalanche effect to sequencesof k symbols, Periodica Math. Hungar., 65 (2012), 229–238.


Vilmos TotikUniversity of Szeged, Hungary and University of South Florida, USA

[email protected]

A conjecture of Petruska anda conjecture of Widom

The talk will discuss with some historical background a conjecture of Petruska aboutnonconstant solutions of integral equations and a conjecture of Widom on the behaviorof Chebyshev numbers. Both conjectures turn out to be false.


Pavel VarbanetsI.I. Mechnikov Odessa National University, Odessa, Ukraine

[email protected]

(joint work with Sergey Varbanets)

Twisted exponential sums and the distribution

of solutions of the congruence

f (x, y) ≡ 0 (mod pm) over Z[i]

Let G = Z[i] be the ring of Gaussian integer numbers. For γ ∈ G we stateGγ(respectively, G∗γ) for residue (reduced residue) system modulo γ. Let χ be anarbitrary multiplicative character modulo γ. We will obtain estimations of the twistedexponential sums

Sχ(f, γ) =∑x∈G∗

γ

χ(x)e2πi<(f(x)γ ),

where f is a rational function over G.In particular, we give estimations for the twisted Kloosterman sum

Kχ(α, β; γm) :=∑x,y∈G

xy≡1 (mod γ)

χ(x)e2πi<(αx+βyγ ).

Also we study the norm Kloosterman sum

K(α1, α2, α3; q) :=∑

x1,x2,x3∈G∗q

N(x1x2x3)≡1 (mod q)

e2πi<(α1x1+α2x2+α3x3

q )

and the character sum over norm group∑u∈Gpn

N(u)≡1 (mod pn)

χ(Φ(u))epn(Ψ(u)),

where p is a rational prime number, Φ(u),Ψ(u) ∈ Gpn [u].There is an applications of the estimations of such sums in problem on distribution of

solutions of the special congruences overG are given. In particular, we have obtained anasymptotic formula for the number of solutions of the congruence y` ≡ f(x) (mod pm)in the incomplete residue system, where `,m ∈ N, p is a Gaussian prime number,f(x) ∈ G[x].


Emil VataiEotvos Lorand University, Budapest, Hungary

[email protected]

Speeding up distributed sievingwith the inverse sieve

The efficiency of sieving algorithms depends on memory management, especiallywhen the sieve table is very large. Furthermore, a naive parallel or distributed ap-proach does not help, because the operations are pretty simple, and one does not gainanything by distributing the workload among processing units, the memory bus isalways congested.

To reduce inter processor communication, the sieve is compressed and an appro-priate sieving method is used to reduce the bulk of the work. This approach can bethought of as a kind of inverse sieve, because it removes the larger sieving primesinstead of the candidates in the sieve table.


Notes

numbers, functions, equations 2013 (nfe’13)

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