A statissztikus fizika es kvantummechanika matematikai problemaival foglalkozunk. A hidrodinamika mikroszkopikus modelljeinek tag osztalyait vizsgaltuk, kulonos tekintettel hiperbolikus skalazasu rendszerek makroszkopikus viselkedesere a lokeshulllamok tartomanyaban. Levezettuk a rugalmassagtan nemlinearis egyenleteit. Attorest jelentő eredmenyek szulettek aszimmetrikus modellek 1/3 kitevős skalazasaval kapcsolatban (szubdiffuziv viselkedes). Erdős-Renyi-Barabasi tipusu omszervező strukturak kritikus viselkedeset, a graf komponenseinek aszimptokus meretet tisztaztuk. Nem-Markov bolyongasok hatareloszlasat meghatarozva meglepő kulonbseget tapasztaltunk az iranyitott es nem iranyitott elek eseteinel. Leirtuk veletlen matrixok sajatertekeinek pomtfolyamatat. Sinai biliardok es altalanosabb kaotikus dinamikai rendszerek ergodikus viselkedeset tisztaztuk, eredmenyesen targyaltuk a sikbeli Lorentz folyamat rekurrenciajanak es Brown-kozelitesenek problemajat. Meghataroztuk iteralt fuggvenyrendszerek attraktorainak Hausdorff dimnenziojat, veletlen Cantor halmazok kulonbseget. Tisztaztuk a kvantummechanikai allapotter geometriajanak kerdeseit. A kvantumrendszerek allapotbecsleseit vizsgalva azok megbizhatosagat jellemeztuk. Algoritmust adtunk n-szintű rendszer becsleseire, a becslesi strategia szamitogepes vizsgalatara is sor kerult. A folytonos optimalizacio kulonfele algoritmusit konstrualtuk meg, tisztaztuk azok hatekonysagat. | Mathematical problems of statistical physics and quantum mechanics are investigated. Various microscopic models of hydrodynamics are introduced, existence of hyperbolic scaling limits is prroven including the derivation of the equations of nonlinear elastodynamics. Fairly deep results have been obtained on the 1/3 exponent scaling of asymmetric systems (subdiffusive behaviour of the tagged particle). In case of Erdős-Renyi-Barabasi type self-organized systems the size of the graph components has been determined. Investigating the limit distribution of non-Markovian random walks, a considerable difference in the behavior of models with directed and non-directed bonds has been observed. Characterization of the point process of eigenvalues of random matrices was presented. Ergodic behavior of Sinai billiards and of more general dynamical systems has been described, discussion of recurrence and Brownian approximation of the planar Lorentz gas has been completed. Hausdorff dimension of attractors of iterated maps has been calculated, difference of random Cantor sets was also studied. Geometry of quantum mechanical and reliability of estimators for the state of quantum mechanical systems has been investigated. Algorithms for constrained estimation for n-level quantum systems have been introduced and investigated by computer simulations. Various methods of continuous optimalization and their effectiveness was also studied.

Applications in quantum information theory and quantum tomography have raised current interest in complex Hadamard matrices. In this note we investigate the connection between tiling of Abelian groups and constructions of complex Hadamard matrices. First, we recover a recent, very general construction of complex Hadamard matrices due to Dita [2] via a natural tiling construction. Then we find some necessary conditions for any given complex Hadamard matrix to be equivalent to a Dita-type matrix. Finally, using another tiling construction, due to Szabó [8], we arrive at new parametric families of complex Hadamard matrices of order 8, 12 and 16, and we use our necessary conditions to prove that these families do not arise with Dita's construction. These new families complement the recent catalogue [10] of complex Hadamard matrices of small order.

A kutatasok harom temakorre csoportosithatok: 1. Veletlen matrixok A nagy elteres tetelek tipikusan egy matrix sajaterteksűrűsegere vonatkoztak. Ket veletlen es fuggetlen projekcio esete teljesen ujfajta eredmenyhez vezetett. Vojculescu szabad valoszinűseg elmeleti szallitasi koltseg egyenlőtlenseget tobb iranyba is tovabb fejlesztettuk, a korvonalon es a szamegyenesen adott mertekekre. A bizonyitasban veletlen matrixok sajaterteksűrűsegevel valo megkozelitest es nagy elteres teteleket alkalmaztunk. 2. Matrixok nyomegyenlőtlensegeinek alkalmazasai Ket matrixra a kozepeknek jol kidolgozott elmelete van, de harom matrixra ilyen nem volt Tobb matrixra is ertelmeztuk a kozepeket egy algoritmus segitsegevel. Matrix egyenlőtlensegek kerultek alkalmazasra a Neumann-entropia erős szubadditivitasanak bizonyitasaban. 3. A kvantummechanikai rendszerek allapottere Az allapotbecsles az ismeretlen allapot parametereit meresekkel kivanja megbecsulni. Bizonyos becslesi eljarasok statisztikus osszehasonlitasa es a negyzetes hibamatrix alapjan valo optimalizalas a kutatott tema. Peldaul, a hibmatrix determinansa minimalis, ha a meresek komplementarisak. Bevezettuk a komplementaris reszrendszerek fogalmat es szamos tobb problemat meoldottunk a ket kvantumbitből allo rendszer esetere. Statisztikailag elegseges reszalgebrak tobb uj jellemzese kerult bizonyitasra, kozottuk egy faktorizacios tulajdonsag. | The research can be grouped in three fields: 1. Random matrices A new kind of large deviation result was obtained about the eigenvalue densities of two independent random projections. Voiculescu's free transportation cost inequality was extended to different directions, measures on the circle and on the real line. In the proof approximation by eigenvalue densities of random matrices and large deviation results were used. 2. Matrix inequalities The matrix means for two matrices have a well-known theory. We defined the mean for several matrices by an algorithm. Matrix inequalities were used in a new proof of the strong subadditivity of the von Neumann entropy. 3. State space of quantum systems The state determination means giving an estimate of the unknown system on the basis of measurement. Statistical arguments are used here. Estimation schemes can be compared and optimized. It is proven that the determinant of the quadratic error matrix is minimal, if the complementary measurements are used. The concept of complementarity has been extended to subalgebras and a few relevant questions are answered for a system of two quantum bits.

Let U m be an m×m Haar unitary matrix and U[ m,n ] be its n×n truncation. In this paper the large deviation is proven for the empirical eigenvalue density of U[ m,n ] as m/n→λ and n→∞. The rate function and the limit distribution are given explicitly. U[ m,n ] is the random matrix model of quq, where u is a Haar unitary in a finite von Neumann algebra, q is a certain projection and they are free. The limit distribution coincides with the Brown measure of the operator quq.

Let U m be an m × m Haar unitary matrix and U [m,n] be its n × n truncation. In this paper the large deviation is proven for the empirical eigenvalue density of U [m,n] as m/n → λ and n → ∞. The rate function and the limit distribution are given explicitly. U [m,n] is the random matrix model of quq, where u is a Haar unitary in a finite von Neumann algebra, q is a certain projection and they are free. The limit distribution coincides with the Brown measure of the operator quq.

Let U n be an n × n Haar unitary matrix. In this paper, the asymptotic normality and independence of Tr U n , Tr U n 2 ,..., Tr U n k are shown by using elementarymethods. More generally, it is shown that the renormalized truncated Haar unitariesconverge to a Gaussian random matrix in distribution.

A summary is given about the subjects lectured by the department for chemical engineering students. The most important task of the department is to lecture general and inorganic chemistry for first year students and to run the related seminars on chemical calculations, and laboratory practice. In the second level of tuition the department is responsible for quantum chemistry subject. Besides, the department offers some criterion and elective subjects for the students and participates in English language course at the Chemical Engineering Faculty.