Representation of Cyclotomic Fields and Their Subfields

Let \K be a finite extension of a characteristic zero field \F. We say that the pair of $n\times n$ matrices $(A,B)$ over \F represents \K if \K \cong \F[A]/ where \F[A] denotes the smallest subalgebra of M_n(\F) containing $A$ and $$ is an ideal in \F[A] generated by $B$. In particular, $A$ is said to represent the field \K if there exists an irreducible polynomial q(x)\in \F[x] which divides the minimal polynomial of $A$ and \K \cong \F[A]/. In this paper, we identify the smallest circulant-matrix representation for any subfield of a cyclotomic field. Furthermore, if $p$ is any prime and \K is a subfield of the $p$-th cyclotomic field, then we obtain a zero-one circulant matrix $A$ of size $p\times p$ such that (A,\J) represents \K, where \J is the matrix with all entries 1. In case, the integer $n$ has at most two distinct prime factors, we find the smallest 0-1 companion-matrix that represents the $n$-th cyclotomic field. We also find bounds on the size of such companion matrices when $n$ has more than two prime factors.Comment: 17 page

Crossover of magnetoconductance autocorrelation for a ballistic chaotic quantum dot

The autocorrelation function C_{\varphi,\eps}(\Delta\varphi,\,\Delta \eps)= \langle \delta g(\varphi,\,\eps)\, \delta g(\varphi+\Delta\varphi,\,\eps+\Delta \eps)\rangle ($\varphi$ and \eps are rescaled magnetic flux and energy) for the magnetoconductance of a ballistic chaotic quantum dot is calculated in the framework of the supersymmetric non-linear $\sigma$-model. The Hamiltonian of the quantum dot is modelled by a Gaussian random matrix. The particular form of the symmetry breaking matrix is found to be relevant for the autocorrelation function but not for the average conductance. Our results are valid for the complete crossover from orthogonal to unitary symmetry and their relation with semiclassical theory and an $S$-matrix Brownian motion ensemble is discussed.Comment: 7 pages, no figures, accepted for publication in Europhysics Letter

Probability density of determinants of random matrices

In this brief paper the probability density of a random real, complex and quaternion determinant is rederived using singular values. The behaviour of suitably rescaled random determinants is studied in the limit of infinite order of the matrices

The effects of grain shape and frustration in a granular column near jamming

We investigate the full phase diagram of a column of grains near jamming, as a function of varying levels of frustration. Frustration is modelled by the effect of two opposing fields on a grain, due respectively to grains above and below it. The resulting four dynamical regimes (ballistic, logarithmic, activated and glassy) are characterised by means of the jamming time of zero-temperature dynamics, and of the statistics of attractors reached by the latter. Shape effects are most pronounced in the cases of strong and weak frustration, and essentially disappear around a mean-field point.Comment: 17 pages, 19 figure

On pressure rise in rockets

The phenomenon of pressure rise in rockets has been discussed on the basis of r=CP/Sup/n as the law of burning. An explicit expression for the chamber pressure at any instant has been derived and illustrated by a table and a graph

Level number variance and spectral compressibility in a critical two-dimensional random matrix model

We study level number variance in a two-dimensional random matrix model characterized by a power-law decay of the matrix elements. The amplitude of the decay is controlled by the parameter b. We find analytically that at small values of b the level number variance behaves linearly, with the compressibility chi between 0 and 1, which is typical for critical systems. For large values of b, we derive that chi=0, as one would normally expect in the metallic phase. Using numerical simulations we determine the critical value of b at which the transition between these two phases occurs.Comment: 6 page

Generalization of the Poisson kernel to the superconducting random-matrix ensembles

We calculate the distribution of the scattering matrix at the Fermi level for chaotic normal-superconducting systems for the case of arbitrary coupling of the scattering region to the scattering channels. The derivation is based on the assumption of uniformly distributed scattering matrices at ideal coupling, which holds in the absence of a gap in the quasiparticle excitation spectrum. The resulting distribution generalizes the Poisson kernel to the nonstandard symmetry classes introduced by Altland and Zirnbauer. We show that unlike the Poisson kernel, our result cannot be obtained by combining the maximum entropy principle with the analyticity-ergodicity constraint. As a simple application, we calculate the distribution of the conductance for a single-channel chaotic Andreev quantum dot in a magnetic field.Comment: 7 pages, 2 figure

Calculation of some determinants using the s-shifted factorial

Several determinants with gamma functions as elements are evaluated. This kind of determinants are encountered in the computation of the probability density of the determinant of random matrices. The s-shifted factorial is defined as a generalization for non-negative integers of the power function, the rising factorial (or Pochammer's symbol) and the falling factorial. It is a special case of polynomial sequence of the binomial type studied in combinatorics theory. In terms of the gamma function, an extension is defined for negative integers and even complex values. Properties, mainly composition laws and binomial formulae, are given. They are used to evaluate families of generalized Vandermonde determinants with s-shifted factorials as elements, instead of power functions.Comment: 25 pages; added section 5 for some examples of application

Smoothing of sandpile surfaces after intermittent and continuous avalanches: three models in search of an experiment

We present and analyse in this paper three models of coupled continuum equations all united by a common theme: the intuitive notion that sandpile surfaces are left smoother by the propagation of avalanches across them. Two of these concern smoothing at the `bare' interface, appropriate to intermittent avalanche flow, while one of them models smoothing at the effective surface defined by a cloud of flowing grains across the `bare' interface, which is appropriate to the regime where avalanches flow continuously across the sandpile.Comment: 17 pages and 26 figures. Submitted to Physical Review

Performance Improvement of GNSS Receiver by Mitigation of Multipath Effects

The Rake Receiver is excessively used in modern CDMA communication but for navigation application it is not self-sufficient to provide satisfactory performance. To overcome the shortcomings of Rake Receiver for navigation purpose one needs to introduce some differential rake architecture which will mitigate the multipath corresponding to the incoming data after it has been processed and demodulated. In this manner a particular will be able to cancel out the multipath and at the end will have only the strongest multipath from which decision can be made to recover the data back. The other very critical module is Delay Locked Loop (DLL) which will align the code at the receiver end so as to minimize the pseudo range error. The DLL will try to lock the incoming signal with the local code and in order to do so it will consider 3 different locally generated codes Early, Prompt and Late. According to the parameter defined, it may accuracy up to one tenth of a chip. The DLL will use the code-phase provided by the previous blocks and try to find the local code which will give us the minimum pseudo range error. If the multipath signals are delayed by more than 1.5 chips then matched filter algorithm will detect all three signals by processing on auto-correlation function. But when delay is less than 1.5 chips then NLMS algorithm is used for multipath detection. This two algorithm is incorporated in this design. Whenever incoming signal is received it will first try to find out multipath components within 1.5 chips. After that it will go to the next step in order to find multipath components outside 1.5 chips. In this project the above mentioned approaches are combined so as to get a system which will give the optimum performance in terms of the SNR and the pseudo range