1st Conference of PhD Students in Mathematics

Transcription

1 1st Conference of PhD Students in Mathematics Volume of extended abstracts Organized by Bolyai Institute, University of Szeged June 29 July 2, 2010 Szeged, Hungary

2 Scientific Committee: Gábor Czédli László Hatvani László Kérchy András Krámli Tibor Krisztin Péter Major Gyula Pap Ágnes Szendrei Mária B. Szendrei Vilmos Totik Organizing Committee: Gábor Nagy Róbert Vajda Judit Nagy-György Attila Máder Béla Nagy Ábel Garab H-6720 Szeged, Aradi Vertanúk tere 1, Bolyai Institute, Univ. of Szeged, Hungary fax: phone: Sponsors: TÁMOP / Programme for Dissemination of Scientific Results at the Univ. of Szeged. Bolyai Institute, University of Szeged University of Szeged, TTIK HÖK Layout editors: Attila Máder Judit Nagy-György 2

3 Preface 3

4 Contents Preface 3 Contents 4 Program 5 Invited talks 6 Buchberger, Bruno, Can Invention in Mathematics and Computer Science be Automated? 7 Kovács, Mihály, An introduction to fractional calculus Accepted abstracts 9 Adamcsek, Edit, Application of ensemble transform Kalman filter in numerical weather prediction at the Hungarian Meteorological Service Backhausz, Ágnes, Local degree distributions in scale free random graph models Bárány, Balázs, On the dimension theory of iterated function systems with fixed point correspondence Bartha, Ferenc, Rigorous numerics for dissipative PDEs Csóka, Endre, Maximum flow is approximable by deterministic constant-time algorithm in sparse networks Dénes, Attila, Stability properties of a population dynamical model for two fish species in Lake Tanganyika Garab, Ábel, Unique periodic solutions of a delay differential equation with piecewise linear feedback function Horváth, Illés, Diffusive limit for the myopic (or true ) self-avoiding random walk in d Koi, Tamás, Capacity region of the discrete asynchronous multiple acces channel Kórus, Péter, Single and double sine series with general monotonic coefficients Máder, Attila, The maximum number of rectangular islands - Experiments in the classroom Maróti, Attila, Average dimension of fixed point spaces with applications Mészáros, Viola, Separated matchings in colored convex sets Nándori, Péter, Billiards and random walks Nagy, Béla, Potential theory on curves Pusztai, B. Gábor, Trigonometric BC n Sutherland models and Hermann actions Rakić, Mirjana, Doubly biased connectivity game Röst, Gergely, From simple dynamics to chaos through nonmonotone delayed feedback. 29 Sáfár, Orsolya, Time-delayed patchy environment model of the capillary migration assay 30 Skublics, Benedek, Matrices in modular lattices Szimjanovszki, Irma, Computer-aided study of the competition of species for territory. 32 Şuteu Szöllősi, István, On the Hall product of preinjective Kronecker modules Timár, Ádám, Self-similar groups and graphs Varga, Tamás, Uniform spacing of zeros of orthogonal polynomials on intervals with doubling property Vető, Bálint, Limit theorems for the one-dimensional myopic self-avoiding walk Vas, Gabriella, Periodic solutions for an equation with delay Invited speakers 39 List of Participants 39 4

5 Program 5

6 Invited talks 6

7 Can Invention in Mathematics and Computer Science be Automated? We first present some general ideas about Bruno Buchberger the practical role of logical formality for creativity in mathematics (including computer science) the creativity spiral in mathematics by which, in open-ended rounds, areas which need individual creativity for handling individual problem instances, after deeper mathematical research, can be trivialized in the sense that one single algorithm handles all problem instances in this area. In this way, mathematics (including computer science) can be conceived as the permanent effort to trivialize itself on higher and higher levels. Thus, mathematics is the prototype of automation in the center of the current age of automation in all areas of science, technology, economy, and society. We then present our recent formal method, called lazy thinking, by which it was possible to synthesize automatically algorithms for some non-trivial problems. 7

8 An introduction to fractional calculus Mihály Kovács Fractional differential equations have recently made a renaissance, mainly driven by scientists in Physics, Finance, and Hydrology, as they can be derived via stochastic limit theorems and hence provide robust and parsimonious models predicting power-law tails. This is because fractional derivatives derive from sums of random movements with power law probability tails, for which the usual central limit theorem is replaced by its heavy tail analogue. In this talk we give an introduction to fractional differential equations involving mathematical topics ranging from scaling limits of random walks to fractional powers of linear operators on a Banach space. We also give several motivational examples to the theory, including contaminant transport in subsurface hydrology, human travel patterns and spreading of invasive species. 8

9 Accepted abstracts 9

10 Application of ensemble transform Kalman filter in numerical weather prediction at the Hungarian Meteorological Service Edit Adamcsek The main objective of numerical weather prediction (NWP) is to give a precise estimation of the future atmospheric state, i.e. a forecast. The changes in the state of the atmosphere can be described by a system of non-linear partial differential equations. These hydro-thermodynamic equations represent physical laws, such as conservation of energy (First Law of Thermodynamics), momentum (Newton s Second Law of Motion) and mass (continuity equation). This system with initial conditions is the NWP problem, which cannot be solved analytically. Limited area models, where the horizontal domain of the model is regional, need boundary condition as well. NWP models solve this problem numerically, and these models are extremely sensitive to initial conditions, so it is crucial to precisely specify the state of the atmosphere at the initial time in the model. The so-called analysis scheme creates the best possible initial conditions for the forecast model. Measurements from many different sources are available, like traditional weather surface observations, radiosonde and aircraft measurements, weather satellites and radars for instance. For creating the initial conditions, that is, the analysis, basically two sources of information are accessible: forecasts from the NWP model and the observations, both including their error characteristics. Within the scheme at every analysis time (typically at every 6 hours) an estimation of the state of the atmosphere is provided by the forecast model, and this information has to be optimally combined with the measured data to create the initial conditions. For linear models, the widely applied Kalman Filter would give solution for the problem: an initial state for the forecast model (the analysis), given that measured data are available, and the forecast made from this analysis would be the estimate of the future atmospheric states. Additionally, it also provides the estimation error and estimation error covariance matrix, which is the main interest in weather prediction, i.e. the forecast error. For non-linear models, such as NWP models, Kalman Filter is not applicable. In addition, due to computational reasons, an approximation is also necessary. NWP models, such as ALADIN/HU, which is the operational weather prediction model at the Hungarian Meteorological Service, are of dimension n 10 7, and Kalman Filter would need the cost of model integration, i.e. it would need to run the forecast model 2n times, which is not applicable in such a high dimensional system. Ensemble Kalman Filter (EnKF) is a Monte-Carlo implementation of the classical Kalman Filter, which is applicable in NWP and can be used for non-linear models. The main idea is to use a set of initial conditions, that is, an ensemble of analysis, for the forecast model, which would be run k times (the size of the ensemble) to provide the ensemble of the estimates. The provided forecasts would represent a statistical sample of the state of the atmosphere. The ensemble mean would represent the best estimate for the atmospheric state, the deviations from the mean would represent estimation error, i.e. forecast error, and the ensemble (empirical) covariance matrix would represent the estimation error covariance matrix. Ensemble Transform Kalman Filter (ETKF) is a version of EnKF, where a transformation simplifies the analysis scheme. Instead of running the analysis scheme k times, for creating the ensemble of initial conditions, it transforms the ensemble of forecasts into the ensemble of analysis. This realization of Kalman Filter is to be implemented into the operational ALADIN/HU model at the Hungarian Meteorological Service. At the current, early stage of the research, the frame of the ETKF procedure is working within the ALADIN/HU model with the ensemble size of 11. As a primary validation of the scheme, the transformation part of the procedure was tested, by using an estimation error covariance matrix constant in time, since the timedependent computation of the covariance matrix is still under construction. Results show that the created ensemble has too small spread, i.e. it does not represent the estimation error well. Thus further improvement of the transformation technique is needed, and the computation of the time-dependent covariance matrix has to be developed. 10

11 Local degree distributions in scale free random graph models Ágnes Backhausz Since the end of the nineties several complex real world networks and their random graph models were investigated. Many of them possess the scale free property: the tail of the degree distribution decreases polynomially fast, that is, if c d denotes the proportion of vertices of degree d, then for large values of d c d C d γ holds [1]. γ is called the characteristic exponent. If the whole network is completely known, the empirical estimator of the characteristic exponent may have nice properties. However, real world networks usually are too large and complex, hence our knowledge of the graph is partial. For several models of evolving random graphs the degree distribution and the characteristic exponent change when attention is restricted to a set of selected vertices that are close to the initial configuration ([2, 3]). Starting from these phenomena, the degree distribution constrained on a set of selected vertices is investigated, assuming that the graph model possesses the scale free property with characteristic exponent γ > 1, and the number of the selected vertices grows regularly with exponent 0 < α 1. Sufficient conditions for the almost sure existence of asymptotic degree distribution are given. Loosely speaking, these conditions ensure that the degree of the new vertex is not too large and that the neighbors of the new vertex determine whether it belongs to the set of selected vertices or not. Moreover, it is shown that if these conditions hold, then the characteristic exponent of the constrained degree distribution is equal to α (γ 1) + 1. The proofs are based on the methods of martingale theory [4]. We present several graph models that satisfy the sufficient conditions, e. g. generalizations of the Barabási tree, random multitrees. All of these models possess the preferential attachment property, and in each model, the characteristic exponent of the constrained degree distribution is less then the original one. One reason for that is the following: the selected vertices are closer to the initial configuration in some sense. There are more old vertices among them and their degree is larger than that of the typical ones. We also present a few random graph models showing the neccessity of some of the conditions. Keywords: scale free graphs, degree distribution, martingales, AMS: 60G42, 05C80 References [1] Barabási, A-L. and Albert, R. (1999), Emergence of scaling in random networks, Science 286, [2] Móri, T. F. (2006), A surprising property of the Barabási Albert random tree, Studia Sci. Math. Hungar. 43, [3] Móri, T. F. (2007), Degree distribution nearby the origin of a preferential attachment graph, Electron. Comm. Probab. 12, [4] Neveu, J. (1975), Discrete-Parameter Martingales, North-Holland, Amsterdam. 11

12 On the dimension theory of iterated function systems with fixed point correspondence Balázs Bárány Let us denote the Hausdorff dimension of a compact subset Λ of R by dim H Λ and respectively the Box dimension by dim B Λ. Let {f 0,..., f m 1 } be a family of contracting similarity map such that f i (x) f i (y) = λ i x y for all x, y and for some 1 < λ i < 1. Then there exists a unique, nonempty compact subset Λ of R which satisfies Λ = m 1 i=0 f i(λ). We call this set Λ the attractor of the iterated function system (IFS) {f 0 (x),..., f m 1 (x)}. In this case we say that the attractor Λ (or the IFS itself) is self-similar. It is well known that the Hausdorff dimension and the Box dimension of the attractor is the unique solution of m 1 i=0 λ i s = 1, (1) if the open set condition (OSC) holds, i.e. there exists a nonempty open set U such that f i (U) U for every i = 0,..., m 1 and f i (U) f j (U) = for every i j, see [2]. Even if the OSC does not hold, the solution of equation (1) is called similarity dimension of the IFS. The similarity dimension is always an upper bound for the Hausdorff dimension of the attractor. In the case when the IFS has overlapping structure, i.e. the open set condition does not hold, the Hausdorff dimension of the attractor Λ of IFS {f i (x) = λ i x + d i } m 1 i=0 is dim B Λ = dim H Λ = min {s, 1} for a.e. (d 0,..., d m 1 ) R m (2) where s is the unique solution of (1), see [3]. In [1] we considered the IFS {γx, λx, λx + 1}, γ < λ on the real line. The problem of the computation of the dimension of the attractor of this IFS was raised by Pablo Shmerkin at the conference in Greifswald in The novelty of the result obtained in [1] about the dimension of Λ was to tackle the difficulty which comes from the fact that the first two maps have the same fixed point. It is clear that the IFS does not satisfy the OSC, and trivially we are not able to apply the formula (2). However, we are able to calculate the Hausdorff dimension for almost every contraction ratios. The talk is based on the paper [1]. Theorem. Let Λ be the attractor of {γx, λx, λx + 1} such that 0 < γ < λ then dim H Λ = min {1, s} for Lebesgue-a.e. γ, where s is the unique solution of 2λ s + γ s λ s γ s = 1. References [1] B. Bárány, On The Hausdorff Dimension of a Family of Self-Similar Sets with Complicated Overlaps, Fundamenta Mathematicae, 206, (2009), [2] J. E. Hutchinson, Fractals and Self-Similarity, Indiana Univ. Math. Journal, Vol. 30, No. 5, [3] K. Simon, B. Solomyak On the dimension of self-similar sets Fractals, 10, No. 1, (2002),

13 Rigorous numerics for dissipative PDEs Ferenc Bartha I would like to introduce the basics of a numerical method by Piotr Zgliczynski (Krakow), that can produce rigorous results and proofs for a certain class of dissipative partial differential equations. It starts with a spectral-type decomposition in the Fourier domain, then we separate the important coefficients and track their dynamics, while we use some uniform argument for the rest. In my work we are trying to generalize this method and weaken some of the conditions. The algorithm is realized as a computer program, thus during the implementation one must take extra care with accepting the results from the pc. For this reason, I would like to introduce the basic concepts of interval arithmetics and automatic differentiation as well; these tools make possible to produce real mathematical proofs with the aid of the computer. 13

14 Maximum flow is approximable by deterministic constant-time algorithm in sparse networks Endre Csóka In the last decade it became apparent that a large number of the most interesting structures and phenomena of the world can be described by networks which are so large that the data about them can be collected only by indirect means like random local sampling. This yielded the motivation of property testing and parameter testing, which became an intensively studied field recently. A parameter tester of bounded-degree graphs means an algorithm which chooses a constant number of random nodes, and to these constant radius neighbourhoods, assigns an estimation (a number) which is at most ε far from the true parameter with at least 1 ε probability. We call a parameter testable if there exists a tester with arbitrary small errors. There is a strongly connected concept called constant-time algorithm, introduced by Nguyen and Onak. For example, consider the maximum matching problem. Here, a constant-time algorithm is a "local function" that decides about each edge that whether it chooses to the mathing or not, so that the chosen edges form a matching, and its size is at most εn less than the size of the maximum matching, with at least 1 ε probability. Locality means that it depends only on the constant-size neighbourhood of the edge, including some random numbers, as follows. We assign independent random numbers to the nodes uniformly from [0, 1], and the neighbourhood consists not only of the induced subgraph of nodes at most constant far from the chosen node, but also of the random numbers of these nodes. In the paper of Nguyen and Onak, they showed a constant-time algorithm for this and some other problems. About the connection of the two concepts, notice that if we have a constant-time algorithm producing the maximum matching then the ratio of the size of the maximum matching and the number of nodes is a testable parameter. Because we can make a tester which simply calculates the probability that the chosen random node would be covered by the matching produced by the constant-time algorithm, and the average of these probabilities is a good approximation. In this paper, we show a constant-time deterministic algorithm for a version of the maximum flow problem. This determinism means that we will get the analogous result using no random numbers. If we delete all edges from all sources or targets then the value of the maximum flow decreases to 0 while the distribution of the local neighbourhoods remains asymptotically the same. That is why the value of the maximum flow in a graph with 1, or even with o(n) sources or targets cannot be tested in any reasonable way. Similarly, one new edge with high capacity between a source and a target would increase the value of the maximum flow by this arbitrary large value. These are some reasons why we will deal only with multiple sources and targets and bounded capacities. The idea of our method is the following. We notice that if there are no short augmenting paths then the flow is almost maximal. That is why we use the Edmonds Karp algorithm until we augmented on all paths with bounded length. We notice that if the algorithm augments on the same-length paths in random order then the flow on each edge can be determined with high probability only by its constant radius neighbourhood. Then we modify the algorithm to be surely local without changing the result so much. Thus we get an expectedly well-approximating constant-time random algorithm. As the space of flows is convex and the value of flows is linear, if we average on all random orderings, we get an approximating deterministic constant-time algorithm. 14

15 Stability properties of a population dynamical model for two fish species in Lake Tanganyika Attila Dénes This is a joint work with Professors László Hatvani and László L. Stachó. We investigate a population dynamical model initiated by László Stachó. This model describes the change of the amount of two fish species a carnivore and a herbivore living in Lake Tanganyika. The model consists of two parts: the development of the population during a year is described by a system of differential equations while the reproduction at the end of each year is described by a discrete dynamical system. The system of differential equations after a series of simplifications has the form: L = C LG Ġ = (L λ(t))g, (1) where λ : [0, ) (0, ) is a given continuous function and lim t λ(t) = λ 1 > 0 exists. This equation does not have an equilibrium, but its limit equation has the equilibrium L = C LG Ġ = (L λ 1 )G, (λ 1, C λ1 ). We showed that (λ 1, C λ1 ) (2) is a globally eventually uniformasymptotically stable point of the non-autonomous system (1). In the proof we use linearization, the method of limit equations and Lyapunov s direct method. We also show some results about the behaviour of the whole system with the discrete part of the reproduction. We have also studied the slightly modified system where the growth of the vegetation left on its own would be exponential: L = (C G)L Ġ = (L λ(t))g. (3) 15

16 Unique periodic solutions of a delay differential equation with piecewise linear feedback function Ábel Garab We consider the delayed differential equation: ẋ(t) = a x(t) + b f(c x(t 1)), (1) where a, b, c R are positive parameters such that a < bc and f(ξ) = 1 2 ( ξ + 1 ξ 1 ), for all ξ R. This equation is often applied in models of neural networks and time delay appears due to finite conduction velocities or synaptic transmission. In neural systems, periodic solutions are of great importance. Our purpose is to give sufficient conditions for nonexistence and uniqueness of periodic solutions of (1) with prescribed oscillation frequencies. Such conditions have been provided by Krisztin and Walther [2] in the case when f is an odd C 1 function satisfying some convexity properties. To formulate our main result, we need some preparation. The natural phase space for (1) is C := C([ 1, 0], R). If x( ) is a real function on some interval I, then we let x t C be defined by x t (θ) = x(t + θ) for all θ [ 1, 0], at least where it makes sense. Now, define the following functional: V : C\{0} {0, 2, 4,..., }, V (ϕ) = { sc(ϕ) if sc(ϕ) is even or infinite, sc(ϕ) + 1 if sc(ϕ) is odd, where sc(ϕ) denotes the number of sign changes of ϕ on the interval [ 1, 0]. According to [3], V is a discrete Lyapunov functional and consequently for any periodic solution x : R R of (1), there exists an appropriate k = k(x) {0, 1, 2,... } such that V (x t ) = 2k, for all t R. Thus, we can write the last formula in a shorter way: V (x) = 2k. Now we are ready to formulate our theorem. The proof is based on the technique applied in [2]. Theorem. Let a, b, c > 0 and k N be fixed such that a < bc and let ν = ν(a, b, c) = (bc) 2 a 2 + arccos a bc. Then the following statements hold. 1. If ν < 2kπ then there exists no periodic solution of (1) in V 1 (2l) where l k, l N. 2. If 2kπ < ν < (2k + 2)π, then equation (1) has at most one periodic solution in V 1 (2l) for all l k, l N. 3. Otherwise, if ν = 2kπ is the case, then equation (1) has infinitely many periodic solutions in V 1 (2k) (namely the periodic solutions of the linearized equation of (1)), but all their ranges are contained in the interval [ 1/c, 1/c]. Győri and Hartung [1] proved that in case of a > b and c = 1, all solutions of equation (1) tends to an equilibrium point. After this, Vas [4] showed that if b 0 = b 0 (a) is defined by the equation b 2 0 a 2 + arccos a b 0 = 2π and b > b 0, then there exists a periodic solution of (1). It ramained an open problem whether there exists a periodic function in the case of a b b 0. As a corollary of our theorem we obtain that there exists no periodic solution in this case. This is a joint work with Dr. Tibor Krisztin. 16

17 References [1] I. Győri, F. Hartung, Stability analysis of a single neuron model with delay, J. Comput. Appl. Math. 157 (2003) [2] T. Krisztin and H.-O. Walther, Unique periodic orbits for delayed positive feedback and the global attractor, J. Dynam. Differential Equations 13 (2001), [3] J. Mallet-Paret and G. Sell, Systems of differential delay equations: Floquet multipliers and discrete Lyapunov functions, J. Diff. Eq. 125 (1996), [4] G. Vas, Asymptotic constancy and periodicity for a single neuron model with delay, Nonlinear Anal. 71 (2009),

18 Diffusive limit for the myopic (or true ) self-avoiding random walk in d 3 Illés Horváth, Bálint Tóth, Bálint Vető The myopic (or true ) self-avoiding walk model (MSAW) was introduced in the physics literature by Amit, Parisi and Peliti in It is a random motion in Z d pushed towards domains less visited in the past by a kind of negative gradient of the occupation time measure: let w be a fixed smooth rate function and t X(t) Z d a continuous time nearest neighbor jump process on the integer lattice Z d whose law is given as follows: where P ( X(t + dt) = y { ) w(l(t, x) l(t, y)) dt + o(dt) if x y = 1 past, X(t) = x = o(dt) otherwise l(t, z) := l(0, z) + {0 s t : X(s) = z}, z Z d is the occupation time measure of the walk X(t) with some initial values l(0, z), z Z d. This is a continuous time version of the original true self-avoiding random walk. We investigate the asymptotic behaviour of MSAW in the non-recurrent dimensions. For a wide class of self-interaction functions, we identify a natural stationary (in time) and ergodic distribution of the environment (the local time profile) as seen from the moving particle and we establish diffusive lower and upper bounds for the displacement of the random walk. For a particular, more restricted class of interactions, we prove full CLT for the finite dimensional distributions of the displacement. This result settles part of the conjectures (based on non-rigorous renormalization group arguments) posed by Amit, Parisi and Peliti. Keywords: self-repelling random motion, local time, central limit theorem 18

19 Modelling the strategies for age specific vaccination scheduling during influenza pandemic outbreaks Diána Knipl The A(H1N1)v is a subtype of influenza A virus which appeared in March Finding optimal policies to reduce the morbidity and mortality for influenza pandemics is a top public health priority. Using a compartmental model with age structure and vaccination status, we examined the effect of age specific scheduling of vaccination during a pandemic infuenza outbreak, when there is a race between the vaccination campaign and the dynamics of the pandemic. Our results agree with some recent studies on that age specifcity is paramount to vaccination planning. However, little is known about the effectiveness of such control measures when they are applied during the outbreak. We found that without reallocating any vaccines between age groups, the best scheduling scheme can decrease the overall attack rate by up to 10%. We demonstrate the importance of early start of the vaccination campaign, since ten days delay may increase the attack rate by up to 6%. Taking into account the delay between developing immunity and vaccination is a key factor in evaluating the impact of vaccination campaigns. We provide a general framework which will be useful for the next pandemic waves as well. The applicability of our population dynamic model is demonstrated for the first wave of A(H1N1)v in Hungary. Keywords: pandemic influenza, vaccination strategies, compartmental models, system of differential equations References [1] D. Knipl & G. Röst, Modelling the strategies for age specific vaccination scheduling during influenza pandemic outbreaks, submitted 19

20 Capacity region of the discrete asynchronous multiple acces channel Tamás Koi Briefly multiple access channel means that many senders send messages to one receiver simultaneously. For the sake of simplicity let us assume that there are two senders. The capacity region consists of the rate pairs (R 1, R 2 ), such that the first and second sender can send information simultaneously with rate R 1, R 2, respectively, with arbitrary small average probability. If the users are synchronous then the capacity region is the convex closure of the union of some pentagons. The method of time sharing makes possible to achieve the convex closure. The asynchronous multiple acces channel (AMAC) arises when the senders do not synchronize the starting times of their codewords, rather there is an unkown delay between the senders. Cover [2] showed that if a bound b n is present for the delay; depending on the codeword length n such that b n n 0 then the convex closure is still achievable by a generalised time sharing method. However, if the delay is not bounded then time sharing seems impossible, and it is widely accepted that in this case the union of pentagons is the capacity region. This view is supported by the papers [3, 4, 5]. However the assertion appears not have been proved completely. Since the AMAC seems to be an important mathematical model our main purpose was to give a mathematically complete formulation of model and a full derivation of the capacity region. It is shown by the technique of Gray from [1] and the techniques of Grant, Rimoldi and Urbanke [6] that the union is achievable regardless of the distributions of the delay. Our proof differs from [6] in that the delay is not known to the decoder. This setting is the same as in [4] by Hui and Humblet, but their proof contains errors. Note that [3, 5] assumed that the delay is known to the decoder. More importantly a complete derivation of the converse is provided. Our new approach shows how the capacity region depends on the distribution of the delay. The familiar regions are recovered in the two classical cases when the delay is bounded, and when it is uniformly distributed. An interesting new case where the delay is uniformly distributed over the even numbers is presented. In this case the capacity region obtained is the convex combinations with weights of the pentagons. References [1] R. M. Gray, Sliding-Block Joint Source/Noisy-Channel Coding Theorems, IEEE Trans. Inform. Theory, vol. IT-22, NO. 6, Nov [2] T. M. Cover, R. J. McEliece, E. C. Posner, Asynchronous Multiple-Acces Channel Capacity, IEEE Trans. Inform. Theory, vol. IT-27, NO. 4, July [3] G. Sh. Poltyrev, Coding in an Asynchronous Multiple-Access Channel, Probl. Peredachi Inf., vol. 19, no. 3, pp , [4] J. Y. N. Hui, P. A. Humblet, The Capacity Region of the Totally Asynchronous Multiple-Access Channel, IEEE Trans. Information Theory, vol. IT-31, NO. 2, Mar [5] Sergio Verdu Multiple-Access Channels with Memory with and without Frame Synchronism, IEEE Trans. Inform. Theory, vol. 35, NO. 3, May [6] A. J. Grant, B. Rimoldi, R. L. Urbanke, P. A. Whiting, Rate-Splitting Multiple Acces For Discrete Memoryless Channels, IEEE Trans. Inform. Theory, vol. 47, NO. 3, Mar

21 Single and double sine series with general monotonic coefficients Péter Kórus One of the basic theorems in the theory of uniform convergence of sine series is due to Chaundry and Jolliffe (1916). They proved that if {a k, k = 1, 2,...} is a nonnegative sequence converging monotonically to zero, then the series a k sin kx converges uniformly in x if and only if ka k 0 (k ). Recently a number of papers were published to extend this theorem by enlarging the class of monotonic coefficients, while keeping the condition ka k 0 still necessary and sufficient. Some of the class conditions are the following: RBV : NBV : a k Ca n ; GBV : k=n 2n k=n a k C( a n + a 2n ); MVBV : 2n k=n 2n k=n a k C a k C n max a k ; n k n+n 0 [λn] k=[λ 1 n] a k. In the theorems involving classes NBVS and MVBVS, the sufficiency part is proved for sine series with complex coefficients as well, while the necessity part stands for sine series with nonnegative coefficients. The following two classes of appropriate sequences of coefficients were introduced by me quite recently: SBVS : 2n 1 k=n a k C n sup 2m m [n/λ] k=m a k ; SBVS 2 : 2n 1 k=n a k C n sup 2m m b(n) k=m a k where {b(k) : k = 1, 2,...} [0, ) tends monotonically to infinity. In the above enumeration, each class contains the previous classes as a subclass. In the second part of my talk I present my new results on the uniform convergence of double sine series of the form j=1 k=1 a jk sin jx sin ky. In this case, the first known result was proved by Žak and Šneider (1966) for double sine series with monotonic double sequences of nonnegative numbers, which are defined by the conditions a jk 0, 10 a jk 0, 01 a jk 0, 11 a jk 0. It was proved by them that such a double sine series is uniformly regularly convergent if and only if jka jk 0 (j + k ). To extend this result, larger classes than the monotonic ones similar to the one variable case have been defined in 2009, named MVBVDS and NBVDS. This time I shall introduce the notions SBVDS and SBVDS 2 and characterize the uniform convergence of the above double sine series with coefficients from these classes. 21

22 The maximum number of rectangular islands - Experiments in the classroom Attila Máder We consider the combinatorial problem of rectangular islands by elementary means. The topic of islands and the methods for its investagation is suitable also for high school students, although some of the corresponding results are quite new. We believe that it is worth to show the topic of islands to teachers and young students because not only new research questions, but also many easy and novel exercises can be created in this topic. Moreover, the motivated students can be involved in the process of constructing definitions for a suitable mathematical model, which is not typical in the standard math curriculum. Because most of the problems are of finitary type, experimental mathematics with computer support proves to be useful for the formulation of general conjectures related to the bounds of the number of islands in particular configurations. The use of calculators and computer algebra systems is gaining greater and greater importance in education today. Their functionality is twofold here: we construct several graphical representations and interactive games which facilitate detecting finite patterns which might lead to general conjectures. To make interactive computer games and demonstrations we used the computer algebra system Mathematica 6. The aim of the fisrt game is to create as many islands as we can on a rectangular board. In addition, by increasing or decreasing a cell height, the new configuration is replotted and the sum of the number of islands will be immediately recomputed by the computer. Moreover, the user gets a checkmark if the maximum number is reached. In the second game the user starts with a 3D cuboid representation of a given island system. By a slider, he can increase or decrease the water level. The tool brings the idea of a particular enumeration of the islands closer to the learner. The use of interactive games can support the shift from frontal teaching to project based, self-paced or active small group learning. The learning process is an active one and it can only be successful and effective if students actively participate. Therefore, the joint interpretation of the observations and results gained by using the interactive computer games will probably encourage and motivate students to formulate conjectures or simply ask reasonble questions. The role of the teacher her is to organize, facilitate, control and the guide the empirical activities of the students. For the integration of computers into math education, we refer to Buchberger s White Box/Black Box principle. References [1] J. Barát, P. Hajnal and E. K. Horváth, Elementary proof techniques for the maximum number of islands, European Journal of Combinatorics, submitted. [2] G. Czédli : The number of rectangular islands by means of distributive lattices, European Journal of Combinatorics 30 (2009), [3] E. K. Horváth, A. Máder, A. Tepavčević, Introducing Czédli-type islands, The College Mathematical Journal, submitted. [4] A. Máder, R. Vajda, Elementary Approaches to the Teaching of the Combinatorial Problem of Rectangular Islands, International Journal of Computers for Mathematical Learning, submitted. [5] A. Máder, G. Makay, The maximum number of rectangular islands, Novi Sad Journal of Mathematics, submitted 22

23 Average dimension of fixed point spaces with applications Attila Maróti Let G be a finite group, F a field, and V a finite dimensional F G-module with no trivial composition factor. Then the arithmetic average dimension of the fixed point spaces of elements of G acting on V is at most (1/p) dim V where p is the smallest prime divisor of the order of G. This answers an old question of Peter M. Neumann and Michael R. Vaughan-Lee. Several applications of this result is given. This is joint work with Robert M. Guralnick. 23

24 Separated matchings in colored convex sets Viola Mészáros There is a two-colored point set of 2n points, n points red and n points blue, in the plane in convex position. Without loss of generality we may assume that the points are on the circle C. Erdős asked to estimate the number of vertices on the longest non-crossing, alternating path. Kynčl, Pach and Tóth proved the upper bound 4 3 n + c n for certain coloring and the lower bound n + c n logn for arbitrary colored point set where c and c are positive constants. The upper bound is conjectured to be tight. In this talk we concentrate on separated matchings, a notion closely related to the original problem. A separated matching is a matching where no two edges cross geometrically and all edges can be crossed by a line. It is believed that the size (number of matched points) of the maximum separated matching must be at least 4 3n. We support this conjecture with a broad class of examples. Our examples also exhibit the currently known best upper bound for the Erdős problem. Our construction contains two types of arcs that we repeat along the circle in arbitrary order an equal number of times. We show that this class of configurations allows at most 4 3 n + O( n) vertices in the maximum separated matching. In the previous papers only one example was given with a coloring of high discrepancy. Our example suggests that the discrepancy doesn t play an important role. We investigate the case of coloring with small discrepancy. If the discrepancy is two or three, we show that the number of vertices in the maximum separated matching is at least 4 3 n. Acknowledgement Work on this paper was partially supported by OTKA Grant K76099 and by the grant no. MSM of the Ministry of Education of the Czech Republic. 24

25 Billiards and random walks Péter Nándori Mathematical theory of billiards is a fascinating subject with an enormous progression in the last decades. With the recently developed technics, lots of the arguments used in stochastics (mainly in the theory of random walks) can be adapted to the case of such deterministic systems. For instance, the so-called Lorentz process is a deterministic model of the motion of an electron in metals: a freely moving point in the two dimensional space collides with fix, periodically situated scatterers and bounces off according to the classical law of mechanics (the angle of incidence is equal to the angle of reflection). The movement could be modeled with Simple Symmetric Random Walk (SSRW) - in this case the normality of the diffusively scaled limit is not difficult to prove. The same limes (i.e. Brownian motion) is by now proven for the Lorentz process. However, owing to the complexity of the methods, lots of interesting questions remain open. For example, consider a locally perturbed Lorentz process, that is finitely many scatterers are replaced by somehow different ones. In this case one could expect the same limit process to appear. This question for SSRW was treated by D. Szász and A. Telcs in 1981, and for Lorentz process with finite horizon by D. Dolgopyat, D. Szász an T. Varjú in Here, the assumption on finite horizon (i.e. when the free flight vector of the electron in bounded) is essential. In the case of infinite horizon, other interesting phenomena appear. For example, the scaling is super-diffusive (that is n log n instead of n). In this talk, we are going to focus on the treatment of the local perturbation in the case of infinite horizon on the random walk level, that is, we need to consider a distribution in the non normal domain of attraction of the Gaussian law. Among others, we need to estimate the probability of flying over a point in Z 2, as the perturbation can modify the trajectory in such a case, as well. 25

26 Potential theory on curves Béla Nagy The purpose of this talk is to show some old and new results in potential theory. Recently, polynomial inequalities have been proved in various settings and various classes of sets. See, for example, Sarantopoulos1991, Totik2000, NagyTotik2005. Roughly speaking, first they prove the polynomial inequalities on nice sets (ellipses, real lemniscates or complex lemniscates), then using an approximation argument, transfer the results to a wider class of sets. During this transfer, by a simple containing relation, supremum norms are preserved. This transfer is not possible in more general setting, since this containing relation is too restrictive. This is why I investigate potential theory on curves. By curves, I mean a finite system of Jordan curves. Further assumptions are sometimes needed, for example, each of these curves is simple, they do not intersect each other and they are smooth enough. In any case, this is a compact set K. A real lemniscate L is the inverse image of the [ 1, 1] interval via a polynomial r, that is, L = r 1 [ 1, 1]. We use (logarithmic) potential theory and Green s functions. We measure the approximation in terms of d(l, K) = sup g K (z) : z L where g K (z) = g C \K(z, ) is the Green s function of the unbounded component of the complement of K. Using the smoothness of K, we estimate the asymptotic behaviour d(l, K) with the degree of r. 26

27 Trigonometric BC n Sutherland models and Hermann actions B. Gábor Pusztai The Calogero Sutherland models are among the most actively studied interacting many particle systems. Their popularity mainly stems from the fact that they are exactly solvable due to their geometric origin. In a joint work with L. Fehér we have derived the trigonometric quantum BC n models from certain Hermann actions of U(N). The applied technique is based on our recently developed general method of quantum Hamiltonian reduction under polar group actions. Our approach provides a conceptual understanding of the problem and finds the place of the BC n models in the unifying framework of polar actions. 27

28 Doubly biased connectivity game Mirjana Rakić, Dan Hefetz, Miloš Stojaković Positional game is a pair (V, F), where V is a finite set and F 2 V. We refer to V as the board and F as the winning sets. We look at the so-called Maker-Breaker games, played by Maker and Breaker who take turns in occupying the previously unoccupied elements of the board. Maker s goal is to occupy all the elements of a winning set by the end of game, and Breaker s goal is to prevent him from doing that, i.e., his goal is to occupy at least one element in every winning set. In many games one of the players can win quite easily. To even out the players chances to win biased games are introduced. For two positive integers a and b, in biased (a : b) game Maker claims a elements in each move, and Breaker claims b elements in each move. Now, if Maker can win a (1 : 1) game easily, a standard approach is to look at the same game with (1 : b) bias, increasing b until the game becomes more balanced. Chvátal and Erdős [1] were first to consider biased Maker-Breaker positional games played on the edge set of the complete graph E(K n ), n. In Connectivity Game winning sets are the edge sets of spanning trees of K n, and Maker wins if he claims all the edges of one spanning tree. In [1] it is shown that Breaker wins the (1 : b) Connectivity Game, if b > n(1+ɛ) log n, ɛ > 0. Gebauer and Szabó proved in [2] that Maker can win the same game, if b (ln n ln ln n 6) n ln 2 n. We study doubly biased (a : b) Connectivity Game on E(K n ), where both a and b can be greater than one. For each a = a(n), our hope is to determine b 0 (a, n) = b 0 (a), so that for b < b 0 (a) the game is Maker s win, and for b > b 0 (a) the game is Breaker s win. We refer to b 0 (a) as the threshold bias for a. For a = o(ln n) we prove that an an(ln ln n + a 1) ln n ln 2 < b 0 (a) < an ln a (1 o(1))an n ln n ln 2 n. When a = c ln n, 0 < c 1, we obtain cn c+1 o(n) < b 0(a) < cn(1 o(1)). When a = c ln n, c > 1, we get cn c+1 o(n) < b 0(a) < 2cn 2c+1 +o(n). When a = ω(ln n) and a = o ( ) n ln n, then n n(ln 2 n+a) a ln n < b 0 (a) < n (1 o(1)) n ln n cn 2 ln n ln ln n 2a. When a = ln n, c > 0, we have n c (1 + o(1)) < b 0 (a) < ln n ln ln n n (1 o(1)) 2c. For a = ω ( n ln n), we obtain n References 2n(ln n ln a + 2) a < b 0 (a) < n (1 o(1)) (n 2a)(ln n ln 2a) + 4a. 2a [1] V. Chvátal and P. Erdős, Biased Positional Games, Annals of Discrete Math. 2(1978), [2] H. Gebauer and T. Szabó, Asymptotic random graph intuition for the biased connectivity game, Random Structures & Algorithms 35(2010),

29 From simple dynamics to chaos through nonmonotone delayed feedback Gergely Röst Some seemingly simple nonlinear delay differential equations still pose massive problems to the understanding of their global dynamics, even after many decades of intensive research. In the talk an overview of some recent results will be presented for two celebrated model equations with unimodal feedback: the Nicholson blowflies equation arisen in population dynamics, and the Mackey-Glass equation which has been proposed to model blood cell production and haematological diseases, and well known for its chaotic behavior. In particular, we give conditions that ensure that all solutions eventually enter the domain where the feedback is monotone, thus chaotic behavior can be excluded. We give sharp (in certain sense the sharpest) bounds for the global attractor and construct heteroclinic orbits from the trivial equilibrium to a slowly oscillating periodic orbit around the positive equilibrium. We discuss the coexistence of rapidly oscillating periodic solutions, and provide many numerical examples for different scenarios. Keywords: delay differential equation, nonlinear dynamics, chaotic behavior, global attractor References [1] G. Röst & J. Wu, Domain-decomposition method for the global dynamics of delay differential equations with unimodal feedback, Proc. R. Soc. Lond. Ser. A, Vol 463(2086), pp (2007) [2] E. Liz & G. Röst, On the global attractor of delay differential equations with unimodal feedback, Disc. Contin. Dyn. Sys. 24(4) (2009) [3] E. Liz & G. Röst Dichotomy results for delay differential equations with negative Schwarzian derivative, Nonlinear Anal RWA 11(3) (2010) 29

30 Time-delayed patchy environment model of the capillary migration assay Orsolya Sáfár The chemotaxis is the movement of the cells induced by chemical compounds. In our joint work the two chamber capillary assay was investigated, which is a frequently used method to measure chemotactic response of the eukaryotic protozoa. The differential equation system (Keller-Segel model) which was developed and successfully applied to describe the chemotactic response of bacteria, was not suited to forecast the behavior of our model cell the eukaryote ciliated protozoa Tetrahymena pyriformis. In the present study the patchy space interpolation was used to simplify the model. We used the time-delayed version of Fick s first law to model the density of the cells during the measurement: du 1 (t) dt du 2 (t) dt t = d = d t ae a(τ t) (u 2 (τ) u 1 (τ))dτ (1) ae a(τ t) (u 1 (τ) u 2 (τ))dτ, (2) where u 1 (t), u 2 (t) denotes the cell density in the two chamber, d > 0 is the diffusion coefficient which is proportional to the area of the interface between the two tanks, and the activity of the cells. The parameter a > 0 is the degree of delay. With an appropriate substitution, we reduced this time-delayed system to a system of ordinary differential equation with one relevant parameter. It was proved that the solutions have one asymptotic stable equilibrium irrespectively of the parameter value. This equilibrium undergoes a node-focus bifurcation. For feasible values of the parameter, the solutions are positive. This modified system describes the observed phenomena during measurement correctly, and gives us the opportunity to optimise the capillary assay. 30

31 Matrices in modular lattices Benedek Skublics Let (a 1,..., a m, c 12,..., c 1m ) be a spanning von Neumann m-frame of a modular lattice L, and let (u 1,..., u n, v 12,..., v 1n ) be a spanning von Neumann n-frame of the interval [0; a 1 ]. Assume that either m 4, or L is Arguesian and m 3. Let R denote the coordinate ring of (a 1,..., a m, c 12,..., c 1m ). If n 2, then there is a ring S such that R is isomorphic to the ring of all n n matrices over S. If n 4 or L is Arguesian and n 3, then we can choose S as the coordinate ring of (u 1,..., u n, v 12,..., v 1n ). The proof uses product frames which were defined by Czedli [1]. The talk is based on [2]. References [1] G. Czédli: The product of von Neumann n-frames, its characteristic, and modular fractal lattices, Algebra Universalis 60 (2009), [2] G. Czédli and B. Skublics: The ring of an outer von Neumann frame in modular lattices, Algebra Universalis, to appear. 31