Numbers, Functions, Equations 2013 (NFE 13)

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Dedicated to Professors

Zolt´

an Dar´

oczy

Imre K´

atai

on the occasion of their 75th birthday

Hotel Visegr´

ad, Visegr´

ad, Hungary

June 28–30, 2013

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List of Participants

1. Baron, Karol

University of Silesia, Katowice, Poland [email protected]

2. Bencz´ur, Andr´as

Institute of Computer Science and Control

Hungarian Academy of Sciences (MTA SZTAKI), Budapest, Hungary [email protected]

3. Bessenyei, Mih´aly

University of Debrecen, Debrecen, Hungary [email protected]

4. Boros, Zolt´an

5. Burai, P´al

6. Burcsi, P´eter

Eötvös Loránd University, Budapest, Hungary [email protected]

7. Chakraborty, Kalyan

Harish-Chandra Research Institute, Allahabad, India [email protected]

8. Czirbusz, S´andor

9. Dar´oczy, Zolt´an

10. Dar´oczy, B´alint

11. De Koninck, Jean-Marie Universit´e Laval, Qu´ebec, Canada [email protected]

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12. Demetrovics, J´anos

13. Farkas, G´abor

14. Fridli, S´andor

15. Fülöp, Ágnes

16. Ger, Roman

17. G lazowska, Dorota

University of Zielona G´ora, Zielona G´ora, Poland [email protected]

18. Gombos, Gerg˝o

19. Gonda, J´anos

20. Gselmann, Eszter

21. H´azy, Attila

University of Debrecen, Miskolc, Hungary [email protected]

22. Indlekofer, Karl-Heinz

University of Paderborn, Paderborn, Germany [email protected]

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23. Iv´anyi, Antal

24. Ivi´c, Aleksandar

Universiteta u Beogradu, Beograd, Serbia aivic [email protected]

25. J´arai, Antal

26. Jarczyk, Justyna

27. Jarczyk, Witold

28. Kaˇcinskait´e, Roma ˇ

Siauliai University, ˇSiauliai, Lithuania [email protected]

29. Kall´os, G´abor

Sz´echenyi Istv´an University, Gy˝or, Hungary [email protected]

30. K´atai, Imre

31. K´atain´e, Frank Edit

32. Kaya, Erdener

University of Mersin, Mersin, Turkey [email protected]

33. Kiss, Attila

34. Kotora-Dar´oczy, Erzs´ebet

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35. Kov´acs, Attila

36. Kozma, L´aszl´o

37. Lajk´o, K´aroly

38. Lakatos, Piroska

39. Laurinˇcikas, Antanas

Vilnius University, Vilnius, Lithuania [email protected] 40. L´en´ard, Margit

College of Ny´ıregyh´aza, Ny´ıregyh´aza, Hungary [email protected]

41. Losonczi, L´aszl´o

42. Maksa, Gyula

43. Manstaviˇcius, Eugenijus

Vilnius University, Vilnius, Lithuania [email protected] 44. Matkowski, Janusz

45. Matuszka, Tam´as

46. M´esz´aros, Fruzsina

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47. Miseviˇcius, Gintautas

Vilnius University, Vilnius, Lithuania [email protected] 48. Moln´ar, Lajos

49. Nagy, G´abor

50. N´emeth, Zsolt

51. P´ales, Zsolt

52. Phong, Bui Minh

53. Reich, Ludwig

University of Graz, Graz, Austria [email protected] 54. R´onyai, Lajos

55. Sablik, Maciej

56. Schipp, Ferenc

57. Simon, P´eter

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58. Stepanauskas, Gediminas

Vilnius University, Vilnius, Lithuania [email protected] 59. Székelyhidi, László

University of Botswana, Gaborone, Botswana University of Debrecen, Debrecen, Hungary [email protected]

60. Szili, L´aszl´o

61. Ton Quoc Binh

University of Hanoi, Hanoi, Vietnam [email protected]

62. T´oth, Vikt´oria

63. Totik, Vilmos

University of Szeged, Szeged, Hungary [email protected]

64. Varbanets, Pavel Dmitrievich

Odessa National University, Odessa, Ukraine [email protected]

65. Varbanets, Sergey Pavlovich

Odessa National University, Odessa, Ukraine [email protected]

66. Vatai, Emil

67. V´arter´esz, Magdolna

68. Weisz, Ferenc

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Program

June 28, Friday 12:30–14:00 Lunch

1ST _{Afternoon Session} _{Chairman: Zsolt P´}_ales 14:00–14:10 Conference opening

14:10–14:35 Járai, Antal: On regularity properties of solutions defined on 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, László: Solved and unsolved problems for Páles means 15:25–15:50 Gselmann, Eszter: On some classes of partial difference

equations 15:50–16:15 Coffee Break

2N D _{Afternoon Session} _{Chairman: Karl-Heinz Indlekofer} 16:15–16:40 Manstaviˇcius, Eugenijus: Additive function theory on

permutations

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

17:05–17:30 Kov´acs, Attila: Some results on simultaneous number systems in the ring of integers of imaginary quadratic fields

17:30–17:55 T´oth, Vikt´oria: Collision and avalanche effect in pseudorandom sequences

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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 simultaneous

difference equations

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

9:50–10:15 G lazowska, Dorota: Invariance of the quasi-arithmetic mean in the class of generalized weighted quasi-arithmetic means 10:15–10:40 Jarczyk, Justyna and P´al Burai: Conditional homogenity

and translativity of Mak´o–P´ales means

1ST _{Morning Session (N)} _{Chairman: Aleksandar Ivi´}_c 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 prime factors

9:50–10:15 Kallós, Gábor: On Rényi–Parry expansions

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

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Program

June 29, Saturday, Morning Two sections: (FE) and (N)

2N D _{Morning Session (FE)} _{Chairman: Antal J´}_arai 11:05–11:30 Reich, Ludwig: Generalized Acz´el–Jabotinsky differential

equations, Briot–Bouquet equations and implicit functions 11:30–11:55 Sablik, Maciej and Agata Nowak: Chini’s equations

in actuarial mathematics

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

12:20–12:45 Lajkó, Károly; Fruzsina Mészáros; Gyula Pap: Characteri-zation of a bivariate distribution through a functional equation 2N D Morning Session (N) Chairman: Eugenijus Manstaviˇcius 11:05–11:30 Varbanets, Pavel and Sergey Varbanets: Twisted exponential

sums and the distribution of solutions of the congruence f(x, y)≡0 (mod pm) over Z[i]

11:30–11:55 Laurinˇcikas, Antanas; Kohji Matsumoto; J¨orn Steuding: Hybrid universality of certain composite functions involving DirichletL-functions

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

12:20–12:45 Nagy, G´abor: Number systems of the Gaussian integers with modified canonical digit sets

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Program

June 29, Saturday, Afternoon Two sections: (FE) and (N)

1ST _{Afternoon Session (FE)} _{Chairman: L´}_aszl´_{o Kozma}

14:00–14:25 Totik, Vilmos: A conjecture of Petruska and a conjecture of Widom 14:25–14:50 Moln´ar, Lajos: Sequential isomorphisms and endomorphisms

of Hilbert space effect algebras

14:50–15:15 Bessenyei, Mihály: Convexity structures induced by regular pairs 15:15–15:40 Boros, Zoltán and Noémi Nagy: Notes on approximately convex

functions

1ST _{Afternoon Session (N)} _{Chairman: Jean-Marie De Koninck} 14:00–14:25 Ivi´c, Aleksandar: The divisor problem in short intervals

14:25–14:50 Miseviˇcius, Gintautas and Audron´e Rimkeviˇcien´e: Joint limit theorems for periodic Hurwitz zeta-function II.

14:50–15:15 Kaˇcinskait´e, Roma: A note on functional independence of some zeta-functions

15:15–15:40 Fridli, S´andor and Ferenc Schipp: Construction of rational function systems with applications

15:40–16:00 Coffee Break

2N D Afternoon Session (Laudations) Chairman: Ferenc Schipp 16:00–16:30 De Koninck, Jean-Marie: Laudation to Imre K´atai

16:30–17:00 Páles, Zsolt: Laudation to Zoltán Daróczy 18:00–21:00 Banquet

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Program

June 30, Sunday Two sections: (FE) and (N) 8:00–9:00 Breakfast

1ST _{Morning Session (FE)} _{Chairman: Vilmos Totik} 9:00–9:25 Székelyhidi, László: Exponential polynomials on Abelian

groups

9:25–9:50 Burai, P´al: Monotone operators and local-global minimum property of nonlinear optimization problems

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

10:15–10:40 Daróczy, Bálint; András A. Benczúr; Lajos Rónyai: Fisher kernels for image descriptors: a theoretical overview and experimental results

1ST _{Morning Session (N)} _{Chairman: Gediminas Stepanauskas} 9:00–9:25 Kozma, László: On semantic descriptions of software systems 9:25–9:50 Iványi, Antal: Testing of random sequences

9:50–10:15 Fülöp, Ágnes: Estimation of the Kolmogorov entropy in the generalized number system

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

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Karol Baron

On some orthogonally additive functions

on inner product spaces

LetE 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

and

f(f(x)) =x for x∈E, then f is additive.

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Mih´

aly Bessenyei

Convexity structures induced by regular pairs

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

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Zolt´

an Boros

(joint work with

No´

emi 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 function f :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 everyx, y ∈I and λ ∈[0,1]. Then f is convex (i.e., f satisfies inequality (1) with c= 0 as well).

The proof is strongly related to the usual characterization of convexity with the aid of one sided derivatives. Establishing and applying results concerning the behavior of certain generalized derivatives, it is possible to prove an analogous result for Wright-convexity. Namely, under the same assumptions onc and p, if the functionf :I →_R fulfils 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 everyx, y ∈I and λ∈[0,1], thenf is Wright-convex (i.e., f satisfies inequality (2) with c= 0 as well).

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P´

al Burai

TU Berlin and University of Debrecen, Germany and Hungary [email protected] and [email protected]

Monotone operators and local-global minimum

property 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 monotone operators 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, Duke Math. 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 series the Institute for Fluid Dynamics and Applied Mathematics, vol. 48, University of Maryland, College Park, Mayland, 1967.

[5] Drábek, Pavel and Jaroslav Milota, Methods of Nonlinear Analysis, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2007, Applications to differential equations.

[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 for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000, Reprint of the 1970 original.

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B´

alint Dar´

oczy

(joint work with

Andr´

as A. Bencz´

ur

and

Lajos R´

onyai)

Fisher kernels for image descriptors:

a theoretical overview and experimental results

Visual words have recently proved to be a key tool in image classification. Best performing Pascal VOC and ImageCLEF systems use Gaussian mixtures or k-means clustering 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 information based distance over the mixtures yields the most accurate classification results. In this presentation we overview the theoretical foundations of the Fisher kernel method. We indicate that it yields a natural metric over images characterized by low level content descriptors generated from a Gaussian mixture. We justify the theoretical observations by reproducing standard measurements over the Pascal VOC 2007 data. Our accuracy is comparable to the most recent best performing image classification systems.

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Jean-Marie De Koninck

Universit´e Laval, Qu´ebec, 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 given power 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.

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G´

abor Farkas

Sieving for large Cunningham chains

of length 3 of the first kind

Our prospective purpose to reach new world record, namely find the largest known Cunningham chains of the first kind of length 3. This means the proof the primality of natural numbersp, 2p+ 1, 4p+ 3, each in the magnitude of 234944 _{(more than 10 500} decimal digits). This would be impossible without sieving which reduces the estimated time of our project. I would like to talk about the details of the theoretical background of the sieving process.

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S´

andor Fridli

Eötvös Loránd 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 biorthogonal systems that are inspired by applications including ECG processing and surface repre-sentation. We show several examples that demonstrate the benefit of such systems in various situations. Constructions in the one and also in the two dimensional cases are shown. Also the problem of discretization is addressed.

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´

Agnes F¨

ul¨

op

[email protected]

Estimation of the Kolmogorov entropy

in 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 an estimation of the Kolmogorov entropy by the Grassberger–Procaccia algorithm for a finite approximation ofBγ.

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Roman Ger

Silesian University, Katowice, Poland

[email protected]

Alienation of additive and

logarithmic equations

Let (R,+,·) stand for an Archimedean totally ordered unitary ring and let (H,+) be an Abelian group. Denote byC the cone of all strictly positive elements in R. We study 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. We show that modulo an additive constant equation (E) forces f and g to split back to the 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).

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Dorota G lazowska

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 generalized

weighted quasi-arithmetic means. We solve this equation under four times continuous differentiability of the unknown functionsϕ, 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 unknown functionsϕ, f, g.

References

[1] Baj´ak, Sz. and Zs. P´ales, Invariance equation for generalized quasi-arithmetic means,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 Seminar on Functional Equations and Inequailities Hajd´uszoboszl´o)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}, Seminar LV, No. 30, 6 pp., 28.04.2008.

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Eszter Gselmann

On some classes of partial difference equations

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

f(x1, . . . , xn) = m

X

i=1

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

wherem ∈_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 hyperboparabo-lic subclasses, respectively. After that, we show solution methods in each of these classes. Finally, some examples will follow.

References

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

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Karl-Heinz Indlekofer

University of Paderborn, Paderborn, Germany [email protected]

On spaces of arithmetical functions

Here we consider

- almost even functions, - limit periodic functions,

- almost multiplicative functions

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Antal Iv´

anyi

Testing of random sequences

Letξ be a random integer vector, having uniform distribution

P{ξ= (i1, i2, . . . , in) = 1/nn}, for 1≤i1, . . . , in ≤n.

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

References

[1] Behrens, W. U., Feldversuchsanordnungen mit verbessertem Ausgleich der Bodenunterschiede, Zeitschrift f¨ur Landwirtschaftliches Versuchs- und Unter-suchungswesen, 2 (1956), 176–193.

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

[3] Egorychev, G.P., Iv´anyi, 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] Iványi, A. and I. Kátai, Estimates for speed of computers with interleaved memory systems.Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math.19 (1976), 159–164.

[5] Iványi, A. and I. Kátai, Modeling of priorityless processing in an interleaved memory with a perfectly informed processor. Autom. Remote Control 46(4) (1985), 520–526 (1985); translation fromAvtom. Telemekh. 1985, No.4, 129–136. [6] Iványi, A. and I. Kátai, Testing of random matrices. Acta Univ. Sapientiae,

Inform. 3(1)(2011), 99–126.

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

[8] Iványi, A. and I. Kátai,Quick testing of random matrices. (Hungarian) Sapien-tia Matinfo Konferencia. Kivonatok (Marosvásárhely, May 25–26, 2013), MITIS, Kolozsvár, 2013, 13–14.

[9] Iv´anyi, A. and B. Nov´ak, Testing of sequences by simulation. Acta Univ. Sapientiae, Inform. 2(2)(2010), 135–153.

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Aleksandar Ivi´

c

Serbian 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) asX → ∞. As usual,

∆(x) := X

n6x

d(n)−xlogx−(2γ−1)x

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

δ|n1 is the number

of divisors ofn, andγ =−Γ0_{(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 Z T ∆(x+U)−∆(x)2dx=T U 3 X j=0 cjlogj √ T U +Oε(T1/2+εU2) +Oε(T1+εU1/2),

so that for Tε ≤ U = U(T) ≤ T1/2−ε this is a true asymptotic formula. The speaker and W. Zhai (2012) proved: Suppose log2T U ₆ T1/2_{/2, T}1/2 _H ₆ _T, _{then we} have T+H Z T max 06u6U ∆(x+u)−∆(x) 2 dxHUL5 ₊_T_L4_log_L+ +H1/3T2/3U2/3L10/3_(log_L)2/3_),

whereL := logT. They also proved: SupposeT, U, H are large parameters andC >1 is a large constant such that

T131/416+εU ₆C−1T1/2L−5_{, CT}1/4_UL5_log_L

6H ≤T.

Then in the interval [T, T +H] there are HU−1 _{subintervals of length} _U _such that on each subinterval one has ±∆(x)≥c±T1/4 for somec±>0.

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Antal J´

arai

[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 matricesx, y it is proved that all measureble solutions or solutions having the property of Baire are infinitely many times differentiable. More general equations are also treated.

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Justyna Jarczyk

(joint work with

P´

al Burai

)

Conditional homogenity and translativity

of Mak´

o–P´

ales means

We characterize the homogeneous and translative members of the class of Mak´ o-P´ales 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 description of homogeneous and translative means also within these classes.

References

[1] Mak´o, Z. and Zs. P´ales, On the equality of generalized quasi-arithmetic means,

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Witold Jarczyk

Uniqueness of solutions of simultaneous

difference equations

Given a set T of positive numbers, a function c: T → _R and a real number p we deal 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 individual equations (Et) usually have a lot of solutions in the considered class of functions.

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Roma Kaˇ

cinskait ˙e

ˇ

Siauliai University, ˇSiauliai, Lithuania [email protected]

A note on functional independence

of some zeta-functions

In 1887, O. H¨older proved [1] the algebraic-differential independence for the gamma function Γ(s), which means that there exists no polynomial P 6≡ 0 such that, for n∈_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-functionζ(s) showing that ζ(s) does not satisfy any differential equation of the form

smFm(y(s), y0(s), ..., y(n−1)(s)) +...+F0(y(s), y0(s), ..., y(n−1)(s)) = 0,

whereF0, ..., 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. Laurinˇcikas and the author considered [3] the functional independence of periodic zeta-functionζ(s;a) and periodic Hurwitz zeta-function ζ(s, α;b). One more generalization of the Voronin theorem for DirichletL-functionL(s, χ) and the function ζ(s, α;b) by the author is obtained in [4].

References

[1] H¨older, O.,Uber die Eigenschaft der Gammafunktion keiner algebraischen Diffe-¨ rentialgleichung zu gen¨ugen, 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] Kaˇcinskait ˙e and A. Laurinˇcikas, The joint distribution of periodic zeta-functions,Stud. Sci. Math. Hung., 48 (2011), no. 2, 257–279.

[4] Kaˇcinskait ˙e, R., Functional independence of Dirichlet L-function and periodic Hurwitz zeta-function, to appear.

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G´

abor Kall´

os

Sz´echenyi Istv´an University, Gy˝or, Hungary

[email protected]

On R´

enyi–Parry expansions

The properties of the expansions of real numbers in noninteger bases have been systematically investigated since the late 50s, when the seminal works by Alfr´ed R´enyiandWilliam Parrywere published. The original idea was to choose the digits

in ”greedy” way, but the analysis was extended soon to generalβ- or q-expansions. In this 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 Zoltán Daróczy and Imre Kátai, some beautiful parts of their results will be presented; namely the field of

the univoque numbers and new normal number-family constructions.

References

[1] Dar´oczy, Z. and I. K´atai, 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. K´atai, Normal numbers created from primes and

polynomials,Uniform Distribution Theory,7/2 (2012), 1–20.

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

[5] R´enyi, A., Representations for real numbers and their ergodic properties, Acta Math. Hung., 8 (1957), 477–493.

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Erdener Kaya

Mersin University, Mersin, Turkey [email protected]

Sums of independent random variables

on additive arithmetical semigroups

Let (G, ∂) be an additive arithmetical semigroup ifG, generated by a countable set

P of prime elements, is a commutative semigroup with identity element 1G, and∂ is an

integer valued degree mapping ∂ : G → _N0 [5]. Assume that the generating function

Z of Gis an element of the exp−log classF (the class of Indlekofer [3]).

In this paper, we embed the additive arithmetical semigroup in a probability space Ω := (βG, σ( ¯A),δ¯) where βG denote the Stone- ˇCech compactification of G[2, 4]. We show that every additive function g on G,g(a) = P

pk_k_a

g(pk) (a∈G), can be identified

with a sum ¯g = P

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 is equivalent to the a.e. convergence of ¯g. And finally, we characterize the essentially distributed additive functions in case that the generating functionZ of Gbelongs to a subclass of the exp−log classF [1].

References

[1] Bar´at, A., K.-H. Indlekofer and E. Kaya, Two-series theorem in additive arithmetical 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 and Number 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,241Pure and Appl. Math., Marcel Decker, New York, (2001).

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Attila Kov´

acs

[email protected]

Some results on simultaneous number systems in

the 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, and}

letD be a finite subset of Λ containing 0. The triple (Λ, M, D) is called a generalized number system (GNS) if every elementx of Λ has a unique, finite representation of the formx=Pl

i=0Midi, wheredi ∈D,l ∈N and dl 6= 0.

If (Λ, M, D) is a number system then (1) D must be a full residue system modulo M, (2) M must be expansive and (3) det(I − M) 6= ±1. If a system fulfils these conditions then it is aradix 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 the

lattices,M =⊕k

i=1Mi the direct sum of the bases and Dh ={(dT1kdT2k · · · kdTk)T : di ∈

∈Di}the homomorphic digit set. Here dT is a row vector andk means the

concatena-tion operator. K´atai et. al. [1] introduced the notion of simultaneous radix systems when 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, . . . ,}_|N

1N2· · ·Nk| −1. The

computa-tion of the expansions are based on Chinese Remaindering. K´atai et al. showed that the 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 Gaussian integers [4, 2, 3]. In this talk we present the newest results achieved in the research of simultaneous number expansions in the ring of integers of imaginary quadratic fields.

References

[1] Indlekofer, K-H., I. K´atai and P. Racsk´o, Number systems and fractal geometry, Probability Theory and Applications, Kluwer Academic Press, (1993) 319–334.

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

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

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

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L´

aszl´

o Kozma

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 action semantics. As a result we suggest to use action semantics for describing semantic properties of software systems including programming and MDE languages.

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K´

aroly Lajk´

o

University of Debrecen, Debrecen, Hungary

[email protected]

(joint work with

Fruzsina M´

esz´

aros

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 satisfied almost everywhere on its domain for the unknown density functions. We consider the case of conditionals in location-scale families with specified moments (we have linear regressions and conditional standard deviations).

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Antanas Laurinˇ

cikas

Vilnius University, Lithuania

[email protected]

(joint work with

Kohji Matsumoto

and

J¨

orn 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].

LetD=s ∈_C: 1₂ < σ <1 . Denote by H(D) the space of analytic functions on

Dequipped with the topology of uniform convergence on compacta. Moreover, denote

by 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 the

interior ofK.

Let β1 > 0, . . . , βr >0, and β = min

1≤j≤rβj. We say that the operator F : H

r₍_D₎ _→

H(D) belongs to the Lipschitz class Lip(β1, . . . , βr) if the following hypotheses are

satisfied:

1◦For each polynomialp=p(s), and anyK ∈ K, there exists an element (g1, . . . , gr) ∈F−1{p} ⊂Hr₍_D_{) such that} _g

j(s)6= 0 onK,j = 1, . . . , r;

2◦ For anyK ∈ K, there exist a constant c >0, and a set ˆK ∈ K such that sup s∈K |F(g11(s), . . . , g1r(s))−F(g21(s), . . . , g2r(s))| ≤c sup 1≤j≤r sup s∈Kˆ |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 inf T→∞ 1 Tmeas τ ∈[0, T] : sup s∈K |F(L(s+iτ, χ1), . . . , L(s+iτ, χr))−f(s)|< ε, max 1≤j≤mkτ αj −θjk< ε 2 1_β >0.

References

[1] Pa´nkowski, L.,Hybrid joint universality theorem for DirichletL-functions,Acta Arith., 141(1) (2010), 59–79.

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Margit L´

en´

ard

College of Ny´ıregyh´aza, Ny´ıregyh´aza, Hungary [email protected]

A (0,2)-type P´

al interpolation problem

In 1975 L. Pál introduced a modification of Hermite-Fejér interpolation in which the function values and the first derivatives were prescribed on two sets of nodes {xi} and {x∗_i}, respectively. In [2] he investigated the case when pn(xi) = 0 fori= 1, . . . , n andp0_n(x∗_i) = 0 fori= 1, . . . , n−1, where pn is a polynomial of degree n with distinct zeros. In 1983 L. Szili [3] studied the inverse problem when the roles ofpn and p0n are interchanged, namely, the first derivatives are interpolated at the zeros of pn and the function values are interpolated at the zeros of p0_n. Several authors studied these two problems, the Pál-type (0;1)- and (1;0)-interpolation using different choices of nodes.

We consider the Pál interpolation with other kind of interpolation conditions. In-stead of the Lagrange condition (only the function values are given) we have Hermite conditions (consecutive derivatives up to a higher order are prescribed). The lacunary condition we substitute with the weighted (0,1,. . . ,r-2,r)-interpolation (r ≥ 2), the generalization of P. Turán’s problem [4], introduced by J. Balázs [1]. We study the regularity of the problem and the explicit representation of the fundamental polyno-mials of interpolation. We give some examples on the zeros of the classical orthogonal polynomials.

References

[1] Balázs, J., Weighted (0,2)-interpolation on the roots of theultraspherical polynomials, (in Hungarian: Súlyozott (0,2)-interpoláció ultraszférikus polinom gyökein),MTA III.oszt. Közl., 11 (1961), 305–338.

[2] P´al, L.G., A new modification of Hermite-Fej´er 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] Tur´an, P., On some open problems of approximation theory, J. Approx. Theory, 29 (1980), 23–85.

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L´

aszl´

o Losonczi

[email protected]

Solved and unsolved problems for P´

ales means

Given two continuous functions f, g:I →_R on the nonempty open intervalI such that g is positive and f /g is strictly monotone and a probability measure µ on the Borel subsets of [0,1], the two variable mean Mf,g;µ : I2 → I, called P´ales mean, is

defined by Mf,g;µ(x, y) := f g −1      1 R 0 f tx+ (1−t)ydµ(t) 1 R 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 others equality, homogeneity, comparison problems) of these means.

References

[1] Losonczi, L.,On homogeneous P´ales means,Annales Univ. Sci. Budapest., Sectio Computatorica (submitted).

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

[3] Losonczi, L. and Zs. P´ales,Equality of two-variable functional means generated by different measures,Aequationes Math.,81 (2011), 31–53.

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

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Gyula Maksa

Some remarks and problems

related to real derivations

A real derivation is a functiond:_R→_R(the set of all real numbers) for which the functional equations

d(x+y) =d(x) +d(y) (x, y ∈_R) (1)

and

d(xy) = xd(y) +yd(x) (x, y ∈_R) (2) hold simultaneously for all x, y ∈_R. The solutions of equation (1) are called additive functions and it is also known (see [2]) that there is non-zero additive functiondwhich fulfils (2), too. This makes possible to construct surprising counter-examples by using real derivations.

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

References

[1] Daróczy, Z. and Gy. Maksa,Nonnegative information functions,Analytic func-tion methods in probability theory (Proc. Colloq. Methods of Complex Anal. in the Theory of Probab. and Statist., Kossuth L. Univ. Debrecen, 1977, 21, Colloq. Math. Soc. János Bolyai, 21, North-Holland, Amsterdam-New York, 1979, 67–78. [2] Kuczma, M., An Introduction to the Theory of Functional Equations and In-equalities, Państwowe Wydawnictwo Naukowe — Uniwersytet ´Sl¸aski, Warszawa– Kraków–Katowice, 1985.

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Eugenijus Manstaviˇ

cius

Vilnius University, Vilnius, Lithuania

[email protected]

Additive function theory on permutations

Influenced by probabilistic number theory (in particular, by the works of the hero of the day Professor Imre K´atai and other hungarian mathematicians), we are dealing with the value distribution of additive functions defined on permutations. By definition, a real two-dimensional array{hj(s)}, wherej ≥1,s≥0, and hj(0)≡0, via expression

h(σ) :=X

j≤n

hj kj(σ)

,

determines an additive function h:_Sn →R. Here Sn is the symmetric group of order

n ∈ _N and kj(σ) ∈ Z+ denotes the number of cycles of length j in the permutation

σ∈_Sn. Assume that σ is taken at random with a probability

ν_n(θ) {σ} := Λ−1_n n Y j=1 θkj(σ) j , σ∈Sn, 00 := 1,

where θj ≥ 0 if j ≤ n and Λn is an appropriate normalizing sequence. The main

objective is to find general conditions under which there exist sequencesα(n)∈_Rand

β(n) >0 such that νn(θ) h(σ)−α(n) < xβ(n)

weakly converges to a non-degenerate distribution function as n → ∞. The case θj ≡ θ > 0 corresponds to the Ewens

probability measure on _Sn. The interest to deal with it and more general weighted

measures has been raised by some recent papers in physics.

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

References

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

[2] Bakˇsajeva, T. and E. Manstaviˇcius, On statistics of permutations chosen from the Ewens distribution (submitted;arXiv:0677678, 2013).

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Janusz Matkowski

Invariance formula for generalized

quasi-arithmetic means

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

(44)

Gintautas Miseviˇ

cius

Vilnius Gediminas Technical University, Lithuania

[email protected]

(joint work with

Audron ˙e Rimkeviˇ

cien ˙e)

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, and

a={am :m∈N=N∪ {0}be a periodic sequence of complex numbers with minimal

period k ∈ _N. The periodic Hurwitz zeta-function ζ(s, α;a) is defined, for σ > 1, by the series ζ(s, α,a) = ∞ X m=0 am (m+α)s,

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

In [1], two joint limit theorems on the weak convergence of probability measures on the 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 numbersaj ={amj :m ∈ N0} with minimal period

kj ∈N. For brevity, we use the notationσ = (σ1, . . . , σr),σ+it= (σ1+it, . . . , σr+it),

α= (α1, . . . , αr),a= (a1, . . . ,ar) andζ(s, α;a) = ζ(s, α1,;a1), . . . ζ(s, αr;ar)

.Denote byB(S) the class of Borel sets of the spaceS, and by measAthe Lebesgue measure of a measurable setA⊂_R. Then in [5], the weak convergence asT → ∞of the probability measure b 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 parametersα1, . . . , 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(C r+r1 ).

References

[1] Rimkeviˇcien ˙e, A., Joint limit theorems for periodic Hurwitz zeta-functions,

ˇ

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Lajos Moln´

ar

[email protected]

Sequential isomorphisms and endomorphisms

of Hilbert space effect algebras

Let A be a C∗-algebra. The effect algebra E(A) of A consists of the self-adjoint elements a for which 0 ≤ a ≤ 1. Effects play important role in the mathematical descriptions of the theory of quantum measurements. The sequential product onE(A) is defined bya◦b =√ab√a.

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

References

[1] Moln´ar, L., Sequential isomorphisms between the sets of von Neumann algebra effects. Acta Sci. Math. (Szeged), 69 (2003), 755-772.

[2] Dolinar, G. and L. Moln´ar, Sequential endomorphisms of finite dimensional Hilbert space effect algebras, J. Phys. A: Math. Theor., 45 (2012), 065207.

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G´

abor Nagy

Number systems of the Gaussian integers

with modified canonical digit sets

I. K´atai and J. Szab´o determined in [1] all possible bases of canonical number systems of the Gaussan integers. Namely they proved that α ∈ _Z[i] is the base of a canonical number system if and only if Re(α) < 0 and Im(α) = ±1. The aim of the talk is to show how we can get number systems with the modification of the canonical digit set.

References

[1] K´atai, I. and J. Szab´o, Canonical number systems for complex integers, Acta Sci. Math. (Szeged), 37 (1975), 255–260.

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Ludwig Reich

Karl-Franzens-Universit¨at Graz, Graz, Austria [email protected]

Generalized

Acz´

el–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)) whereH(X) andG(X) are given formal power series of the formH(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 iteration theory, 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-lutionsφ6= 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 the non-generic case. The cases k = 1 and k ≥ 2 require different approaches. From now on we assume k = 1. Then we solve (AJ,(H,G)) also by transforming it to an implicit function problem. From this we easily deduce the dependence of the solutions on the internal parameter ρ and show that the solutions are local analytic functions of (ρ, X), if H and G are both convergent. If H = G we obtain the so called stan-dard formφ(ρ, X) =S−1(ρS(X)) of the solution of the translation equation, the case G(X) = hkX yields the transformation of (AJ,(H,H)) to its normal form.

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Maciej Sablik

(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 where sets of zeros of the unknown functions are nonempty, under very mild assumptions. In the 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, Actuarial Mathematics. JMAA,253 (2001), 16–34.

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L´

aszl´

o Sz´

ekelyhidi

University of Botswana, Gaborone, Botswana University 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 the theory of functional equations and spectral synthesis. In this talk we characterize these functions by the properties of the annihilator ideal of the variety generated by them. In particular, we obtain new characterization of polynomial functions on Abelian groups.

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Vikt´

oria T´

oth

Collision and avalanche effect in

pseudorandom sequences

Recently a constructive theory of pseudorandomness of binary sequences has been developed and many constructions for binary sequences with strong pseudorandom properties have been given. Motivated by applications, Mauduit and S´ark¨ozy in [1] generalized and extended this theory from the binary case to k-ary sequences, i.e., to

k symbols. They constructed a large family ofk-ary sequences with strong pseudoran-dom properties. I adapted the notions of collision and avalanche effect to study these pseudorandom properties of families of binary and k-ary sequences. Two of the most important 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 the other 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 can take 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-ary

sequences in [4]. The aim of this paper is twofold. First the definitions of collision and 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. S´ark¨ozy,On finite pseudorandom sequences ofk symbols,

Indag. Math.,13 (2002), 89–101.

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

[3] T´oth, V.,The study of collision and avalanche effect in a family of pseudorandom binary sequences,Periodica Math. Hungar., 59 (2009), 1–8.

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

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Vilmos Totik

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

[email protected]

A conjecture of Petruska and

a conjecture of Widom

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

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Pavel Varbanets

I.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

p

m

)

over

Z

[i]

Let G = _Z[i] be the ring of Gaussian integer numbers. For γ ∈ G we state Gγ(respectively, G∗γ) for residue (reduced residue) system modulo γ. Let χ be an

arbitrary multiplicative character moduloγ. We will obtain estimations of the twisted exponential sums Sχ(f, γ) = X x∈G∗ γ χ(x)e2πi<(f(γx)),

wheref is a rational function over G.

In particular, we give estimations for the twisted Kloosterman sum Kχ(α, β;γm) :=

X

x,y∈G xy≡1 (modγ)

χ(x)e2πi<(αx+γβy).

Also we study the norm Kloosterman sum K(α1, α2, α3;q) := X x1,x2,x3∈G∗q N(x1x2x3)≡1 (modq) e2πi<( α₁x₁₊α₂x₂₊α₃x₃ q )

and the character sum over norm group

X

u∈Gpn

N(u)≡1 (modpn₎

χ(Φ(u))epn(Ψ(u)),

wherep 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 overGare given. In particular, we have obtained an asymptotic formula for the number of solutions of the congruencey` _≡_f_{(x) (mod} _pm₎

in the incomplete residue system, where `, m ∈ _N, p is a Gaussian prime number, f(x)∈G[x].

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Emil Vatai

[email protected]

Speeding up distributed sieving

with the inverse sieve

The efficiency of sieving algorithms depends on memory management, especially when 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 gain anything by distributing the workload among processing units, the memory bus is always 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 be thought of as a kind of inverse sieve, because it removes the larger sieving primes instead of the candidates in the sieve table.

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