On finite pseudorandom sequences of k symbols

In earlier papers C. Mauduit and A. Sárközy have introduced and studied the measures of pseudorandomness for finite binary sequences. In [8] they extend this theory to sequences of k symbols: they give the definitions and also construct a “good” pseudorandom sequence of k symbols. In this paper these measures are studied for a “truely random” sequence.     

More constructions of pseudorandom sequences of k symbols

We present some new constructions of families of pseudorandom sequences of k symbols, which generalize several previous constructions for the binary case.     

The study of collision and avalanche effect in a family of pseudorandom binary sequences

In an earlier paper we studied collisions and avalanche effect in two of the most important constructions given for large families of binary sequences possessing strong pseudorandom properties. It turned out that one of the two constructions (which is based on the use of the Legendre symbol) is ideal from this point of view, while the other construction (which is based on the size of the modulo p residue of f(n) for some polynomial f(x) ∈ $ \mathbb{F}_p $ \mathbb{F}_p [x]) is not satisfactory since there are “many” collisions in it. Here 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.     

Collision and avalanche effect in families of pseudorandom binary 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. In the applications one usually needs large families of binary sequences of this type. In this paper we adapt the notions of collision and avalanche effect to study these pseudorandom properties of families of binary sequences. We test two of the most important constructions for these pseudorandom properties, and it turns out that one of the two constructions is ideal from this point of view as well, while the other construction does not possess these pseudorandom properties. Communicated by Attila Pethő     

Computation of k1 Via mennicke symbols

For any ring A, the group K₁A is filtered by the Whitehead determinants of invertible matrices over A of different sizes. We want to compute the corresponding graded group (especially the highest degree non-zero term) in terms of symbols which generalize Mennicke’s symbol. In particular, we generalize the Bass-Milnor-Serre result which presents SK₁A of a Dedekind ring A via the Mennicke symbol, to an arbitrary commutative ring A satisfying the Bass second stable range condition. As an application, SK₁ is computed for some rings of continuous functions. Some of our theorems are partially known, but we have often weakened hypotheses, using stable range conditions rather than Krull dimension (having in mind applications to rings of continuous functions).     

On finite pseudorandom lattices of k symbols

In earlier papers Mauduit and Sárközy have introduced and studied the measures of pseudorandomness for finite binary sequences and sequences of k symbols. Later they (with further coauthors) extended the notation of binary sequences to binary lattices. In this paper measures of pseudorandom lattices of k symbols are introduced and studied for “truly random” lattices.     

Extension theorems for Fredholm and invertibility symbols

This paper is a continuation of [GK3] where the theory of Invertibility Symbol in Banach algebras was developed. In the present paper we generalize these results for the case when the Invertibility Symbol is defined on a subalgebra of the Banach algebras. The difficulty which arises here in this more general case is connected with the fact that some elements of the subalgebra may have the inverses which do not belong to the subalgebra. This generalization of the theory allows us to study the Fredholm Symbols of linear operators. Applications to subalgebras generated by two idempotents and to algebras generated by singular integral operators are presented.     

Extension of Henrici's method to matrix sequences

In this paper we generalize the definition of linear convergence to matrix sequences. This new definition is used to establish some new results useful to study the new extension of Henrici's method. A convergence theorem, an algorithm for implementation of this method and some numerical examples are given.     

Stability of the notion of approximating class of sequences and applications

Given an approximating class of sequences {{B_n,m}_n}_m for {A_n}_n, we prove that (X⁺ being the pseudo-inverse of Moore–Penrose) is an approximating class of sequences for , where {A_n}_n is a sparsely vanishing sequence of matrices A_n of size d_n with d_k>d_q for k>q,k,qN. As a consequence, we extend distributional spectral results on the algebra generated by Toeplitz sequences, by including the (pseudo) inversion operation, in the case where the sequences that are (pseudo) inverted are distributed as sparsely vanishing symbols. Applications to preconditioning and a potential use in image/signal restoration problems are presented.     

On a notion of partially conditionally identically distributed sequences

A notion of conditionally identically distributed (c.i.d.) sequences has been studied as a form of stochastic dependence weaker than exchangeability, but equivalent to it in the presence of stationarity. We extend such notion to families of sequences. Paralleling the extension from exchangeability to partial exchangeability in the sense of de Finetti, we propose a notion of partially c.i.d. dependence, which is shown to be equivalent to partial exchangeability for stationary processes. Partially c.i.d. families of sequences preserve attractive limit properties of partial exchangeability, and are asymptotically partially exchangeable. Moreover, we provide strong laws of large numbers and two central limit theorems. Our focus is on the asymptotic agreement of predictions and empirical means, which lies at the foundations of Bayesian statistics. Natural examples of partially c.i.d. constructions are interacting randomly reinforced processes satisfying certain conditions on the reinforcement.     

Operator method for calculating Q symbols and their relation to weyl-wigner symbols and symplectic tomogram symbols

We propose a new method for calculating Husimi symbols of operators. In contrast to the standard method, it does not require using the anti-normal-ordering procedure. According to this method, the coordinate and momentum operators \(\hat q\) and \(\hat p\) are assigned other operators \(\hat X\) and \(\hat P\) satisfying the same commutation relations. We then find the result of acting with the \(\hat X\) and \(\hat P\) operators and also polynomials in these operators on the Husimi function. After the obtained expression is integrated over the phase space coordinates, the integrand becomes a Husimi function times the symbol of the operator chosen to act on that function. We explicitly evaluate the Husimi symbols for operators that are powers of \(\hat X\) or \(\hat P\) .     

Necklaces of beads in k colors and k-ary de Bruijn sequences

This paper constructs a decreasing lexicographic list of necklaces of length n in beads of k colors. This list, somewhat modified, produces a k-ary de Bruijn sequence of length kⁿ.     

Characterization of forcibly k-variegated degree sequences,k⩾3

We characterize forcibly k-variegated degree sequences for k?3.     

On the distribution of (k,r)-integers in Piatetski-Shapiro sequences

Czechoslovak Mathematical Journal - A natural number n is said to be a (k, r)-integer if n = akb, where k > r > 1 and b is not divisible by the rth power of any prime. We study the...     

On the notion of measurability

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I wish to record my thanks to Dr. Kummer for his constant encouragement and many helpful suggestions in preparing this paper.