Invariant measures for certain linear fractional transformations mod 1

Explicit invariant measures are derived for a family of finite-to-one, ergodic transformations of the unit interval having indifferent periodic orbits.

     

Reflexive linear transformations

A linear operator A is called reflexive if the only operators that leave invariant the invariant subspaces of A are the operators in the weak closure of the algebra of polynomials in A. In this note we completely characterize reflexive operators on finite-dimensional spaces.     

Capacities and geometric transformations of subsets inn-space

Generalized capacity of a set and condenser is introduced as a lower bound of the functionalI(v), provided with some symmetrical features where functionv satisfies certain normalizing conditions. It is established that, depending on the type of symmetry of the functionalI(v), the generalized capacity does not increase under such geometric transformations of sets and condensers as polarization, Gonchar standardization, Steinerk-dimensional symmetrization,k-dimensional spherical symmetrization and dissymmetrization. These results strengthen many known statements of this kind for the particular cases of capacities.     

A certain class of q-series transformations

A large number of summation and transformation formulas for a certain class of double hypergeometric series are observed here to follow fairly readily from a single known result which, in turn, is a very special case of one of six general expansion formulas given in the literature. Generalizations and unifications of these expansion formulas involving series with essentially arbitrary terms are presented. It is also shown how the various series transformations considered in this paper admit themselves of $\text{q}$-extensions which are capable of unifying numerous scattered results in the theory of basic double hypergeometric functions.     

Reversible linear differential equations

Let ? be a meromorphic connection on a vector bundle over a compact Riemann surface Γ. An automorphism $\sigma :\Gamma \to \Gamma$ is called a symmetry of ? if the pullback bundle and the pullback connection can be identified with ?. We study the symmetries by means of what we call the Fano group; and, under the hypothesis that ? has a unimodular reductive Galois group, we relate the differential Galois group, the Fano group and the symmetries by means of an exact sequence.     

Generating all linear transformations

For an n×n Boolean matrix R, let A_R={n×n matrices A over a field F such that if r_ij=0 then a_ij=0}. We show that a collection A_R〈1〉,…,A_R〈k〉 generates all n×n matrices over F if and only if the matrix J all of whose entries are 1 can be expressed as a Boolean product of Hall matrices from the set {R〈1〉,…,R〈k〉}. We show that J can be expressed as a product of Hall matrices R〈i〉 if and only if ΣR〈i〉?R〈i〉 is primitive.     

Harmonious orbits of linear transformations

We describe necessary and sufficient conditions for orbits of linear transformations onR ⁿ ,n1, and sets arising as sums of elements from orbits, to be harmonious subsets. This is done via a generalization of the notion of Pisot-Vijayaraghavan and Salem numbers.     

Extension of linear operator transformations

Acta Mathematica Hungarica -     

Balanced subsets of the semigroup of partial transformations

For the elements of subsets of the semigroup of partial transformations we study when the sets of commuting and noncommuting elements are of the same cardinality.     

Algebraic reflexivity of linear transformations

Let be the set of all linear transformations from to , where and are vector spaces over a field . We show that every -dimensional subspace of is algebraically -reflexive, where denotes the largest integer not exceeding , provided is less than the cardinality of .

     

Hessenberg pairs of linear transformations

Let denote a field and V denote a nonzero finite-dimensional vector space over . We consider an ordered pair of linear transformations A:V→V and A^*:V→V that satisfy (i)–(iii) below.

1. [(i)]Each of A,A^* is diagonalizable on V.

2. [(ii)]There exists an ordering of the eigenspaces of A such that

where V_-1=0, V_d+1=0.

3. [(iii)]There exists an ordering of the eigenspaces of A^* such that

where , .

We call such a pair a Hessenberg pair on V. In this paper we obtain some characterizations of Hessenberg pairs. We also explain how Hessenberg pairs are related to tridiagonal pairs.

A certain family of fractional wavelet transformations

In the present paper, a fractional wavelet transform of real order α is introduced, and various useful properties and results are derived for it. These include (for example) Perseval's formula and inversion formula for the fractional wavelet transform. Multiresolution analysis and orthonormal fractional wavelets associated with the fractional wavelet transform are studied systematically. Fractional Fourier transforms of the Mexican hat wavelet for different values of the order α are compared with the classical Fourier transform graphically, and various remarkable observations are presented. A comparative study of the various results, which we have presented in this paper, is also represented graphically.