The absolute convergence of the Fourier-Haar series for two-dimensional functions

It is well-known that if an one-dimensional function is continuously differentiable on [0, 1], then its Fourier-Haar series converges absolutely. On the other hand, if a function of two variables has continuous partial derivatives f _x ^′ and f _y ^′ on T ², then its Fourier series does not necessarily absolutely converge with respect to a multiple Haar system (see [1]). In this paper we state sufficient conditions for the absolute convergence of the Fourier-Haar series for two-dimensional continuously differentiable functions.     

Certain summation and transformation formulas for generalized hypergeometric series

We derive summation formulas for generalized hypergeometric series of unit argument, one of which upon specialization reduces to Minton’s summation theorem. As an application we deduce a reduction formula for a certain Kampé de Fériet function that in turn provides a Kummer-type transformation formula for the generalized hypergeometric function _pF_p(x).     

Transformation formula for a double Clausenian hypergeometric series,its q-analogue,and its invariance group

A transformation formula for a double basic hypergeometric series of type Φ_0:2;2^1:2;2 is derived. This transformation yields a double series analogue of Sears’ transformation for a terminating ₃Φ₂ series. In the limit q→1, the formula reduces to a transformation for a terminating double Clausenian hypergeometric series of unit argument (one of the proper Kampé de Fériet series, F_0:2;2^1:2;2(1,1)). This formula is a double series analogue of Whipple's terminating ₃F₂ transformation. This transformation gives rise to a transformation group (the invariance group) acting on the parameters of the double series. The invariance group is examined and shown to be a subgroup of a double copy of the symmetries of the square.     

Ergodic Theory and Maximal Abelian Subalgebras of the Hyperfinite Factor

Let T be a free ergodic measure-preserving action of an abelian group G on (X,μ). The crossed product algebra R_T=L^∞(X,μ)? G has two distinguished masas, the image C_T of L^∞(X,μ) and the algebra S_T generated by the image of G. We conjecture that conjugacy of the singular masas S_T⁽¹⁾ and S_T⁽²⁾ for weakly mixing actions T⁽¹⁾ and T⁽²⁾ of different groups implies that the groups are isomorphic and the actions are conjugate with respect to this isomorphism. Our main result supporting this conjecture is that the conclusion is true under the additional assumption that the isomorphism γ : R_T⁽¹⁾→R_T⁽²⁾ such that γ(S_T⁽¹⁾)=S_T⁽²⁾ has the property that the Cartan subalgebras γ(C_T⁽¹⁾) and C_T⁽²⁾ of R_T⁽²⁾ are inner conjugate. We discuss a stronger conjecture about the structure of the automorphism group Aut(R_T,S_T), and a weaker one about entropy as a conjugacy invariant. We study also the Pukanszky and some related invariants of S_T, and show that they have a simple interpretation in terms of the spectral theory of the action T. It follows that essentially all values of the Pukanszky invariant are realized by the masas S_T, and there exist non-conjugate singular masas with the same Pukanszky invariant.     

Kirchhoff index of graphs and some graph operations

Let T be a rooted tree, G a connected graph, x, y ∈ V(G) be fixed and G _i ’s be |V(T)| disjoint copies of G with x _i and y _i denoting the corresponding copies of x and y in G _i , respectively. We define the T-repetition of G to be the graph obtained by joining y _i to x _j for each i ∈ V(T) and each child j of i. In this paper, we compute the Kirchhoff index of the T-repetition of G in terms of parameters of T and G. Also we study how Kf(G) behaves under some graph operations such as joining vertices or subdividing edges.     

Taylor series and divisibility

LetA be an augmentedK-algebra; defineT:A →A ?k _kA byT(a)=1?a ?a?1,a ∈A. We prove, under some conditions, thatg is in the subalgebraK[f] ofA generated byf if and only ifT(g) is in the principal ideal generated byT(f) inA?k _kA. WhenA=K[[X]],T(f) is a multiple ofT(X) if and only iff belongs to the ringL obtained by localizingK[X] at (X).     

On complete characterization of divergence sets of fourier-haar series

The paper proves that E ? [0, 1] is a set of divergence points of Fourier-Haar series of a function f ∈ L ^∞[0, 1] if and only if E is a G _δσ type set of zero measure.     

A homogenous random fractal model for time series produced by constant energy molecular dynamics simulations

In the present work we present an analysis of time series of instantaneous temperature and pressure produced during microcanonical (constant energy) molecular dynamics (MD). Simulations were applied to a nickel oxide grain boundary for a temperature range from about 0.15T_m up to about 0.80T_m, T_m being the melting point of the system. We performed a series of analysis for these time series including test for randomness, power spectrum, Hurst exponent, structure function test and test for multifractality. The obtained results show evidence of an homogenous random fractal model. Pressure time series presents 1/f^α noise over the whole range of frequencies of the system while temperature time series presents a white noise behavior. The origins of this observed behavior are discussed. A comparison also is made with results already obtained from constant temperature MD where the temperature time series present a two-regime behavior: white noise at low frequencies and 1/f^α at high frequencies with α increasing as a function of temperature. The origins of this difference in the behavior are discussed.     

Near-best multivariate approximation by Fourier series, Chebyshev series and Chebyshev interpolation

A set of results concerning goodness of approximation and convergence in norm is given for L_∞ and L₁ approximation of multivariate functions on hypercubes. Firstly the trigonometric polynomial formed by taking a partial sum of a multivariate Fourier series and the algebraic polynomials formed either by taking a partial sum of a multivariate Chebyshev series of the first kind or by interpolating at a tensor product of Chebyshev polynomial zeros are all shown to be near-best L_∞ approximations. Secondly the trigonometric and algebraic polynomials formed by taking, respectively, a partial sum of a multivariate Fourier series and a partial sum of a multivariate Chebyshev series of the second kind are both shown to be hear-best L₁ approximations. In all the cases considered, the relative distance of a near-best approximation from a corresponding best approximation is shown to be at most of the order of Π log n_j, where n_j (j = 1, 2,…, N) are the respective degrees of approximation in the N individual variables. Moreover, convergence in the relevant norm is established for all the sequences of near-best approximations under consideration, subject to appropriate restrictions on the function space.     

Algebraic properties of codimension series of PI-algebras

Let c _n (R), n = 0, 1, 2, …, be the codimension sequence of the PI-algebra R over a field of characteristic 0 with T-ideal T(R) and let c(R, t) = c ₀(R) + c ₁(R)t + c ₂(R)t ² + … be the codimension series of R (i.e., the generating function of the codimension sequence of R). Let R ₁,R ₂ and R be PI-algebras such that T(R) = T(R1)T(R ₂). We show that if c(R ₁, t) and c(R ₂, t) are rational functions, then c(R, t) is also rational. If c(R ₁, t) is rational and c(R ₂, t) is algebraic, then c(R, t) is also algebraic. The proof is based on the fact that the product of two exponential generating functions behaves as the exponential generating function of the sequence of the degrees of the outer tensor products of two sequences of representations of the symmetric groups S _n .     

A note on the time series which is the product of two stationary time series

The time series […,x_-1y_-1,x₀y₀,x₁y₁,…]> which is the product of two stationary time series x_t and y_t is studied. Such sequences arise in the study of nonlinear time series, censored time series, amplitude modulated time series, time series with random parameters, and time series with missing observations. The mean and autocovariance function of the product sequence are derived.     

Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation

In this paper, the initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation over an open bounded domain G×(0,T), G∈Rn are considered. Based on an appropriate maximum principle that is formulated and proved in the paper, too, some a priory estimates for the solution and then its uniqueness are established. To show the existence of the solution, first a formal solution is constructed using the Fourier method of the separation of the variables. The time-dependent components of the solution are given in terms of the multinomial Mittag-Leffler function. Under certain conditions, the formal solution is shown to be a generalized solution of the initial-boundary-value problem for the generalized time-fractional multi-term diffusion equation that turns out to be a classical solution under some additional conditions. Another important consequence from the maximum principle is a continuously dependence of the solution on the problem data (initial and boundary conditions and a source function) that - together with the uniqueness and existence results - makes the problem under consideration to a well-posed problem in the Hadamard sense.     

On the supremum of the pseudocompact group topologies

P is the class of pseudocompact Hausdorff topological groups, and P^′ is the class of groups which admit a topology T such that (G,T)∈P. It is known that every G=(G,T)∈P is totally bounded, so for G∈P^′ the supremum T^∨(G) of all pseudocompact group topologies on G and the supremum T#(G) of all totally bounded group topologies on G satisfy T^∨⊆T#.The authors conjecture for abelian G∈P^′ that T^∨=T#. That equality is established here for abelian G∈P^′ with any of these (overlapping) properties. (a) G is a torsion group; (b) |G|?c²; (c) r₀(G)=|G|=ω|G|; (d) |G| is a strong limit cardinal, and r₀(G)=|G|; (e) some topology T with (G,T)∈P satisfies w(G,T)?c; (f) some pseudocompact group topology on G is metrizable; (g) G admits a compact group topology, and r₀(G)=|G|. Furthermore, the product of finitely many abelian G∈P^′, each with the property T^∨(G)=T#(G), has the same property.     

Absolutely convergent Fourier series and function classes. II

We study the smoothness property of a function f with absolutely convergent Fourier series, and give best possible sufficient conditions in terms of its Fourier coefficients to ensure that f belongs to one of the Zygmund classes Λ_∗(α) and λ_∗(α) for some 0<α?2. This paper is a natural supplement to our earlier one [F. Móricz, Absolutely convergent Fourier series and function classes, J. Math. Anal. Appl. 324 (2) (2006) 1168-1177] under the same title, and we keep its notations.     

Extending the factorization principle to hypergeometric series of general form

It is shown that the formulas of operator factorization of hypergeometric functions obtained in the author’s previous works can be extended to hypergeometric series of the most general form. This generalization does not make the technical apparatus of the factorization method more complicated. As an example illustrating the practical effectiveness of the formulas obtained in the paper, we analyze transformation properties of the Horn seriesG ₃, whose structure is typical for general hypergeometric functions. It is shown that Erdélyi’s transformation formula relating the seriesG ₃ to the Appell functionF ₂, contains erroneous expressions in the arguments ofG ₃. The correct analog of Erdélyi’s formula is found, and some new transformations of the seriesG ₃ are presented. Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 573–581, April, 2000.     

Measure preserving composition operators

We consider composition operators T induced on functional Hilbert spaces H = L₂(S, ∑, μ) byTf(·) = f(h(·)) where h: S → S is a nonsingular transformation. For these mappings T: H → H we give conditions under which they accept invariant Borel probability measures, and we relate the two structures of T, i.e., that of a bounded linear operator to that of a measure preserving transformation.     

On the role of effective representations of Lie groupoids

In this paper, we undertake the study of the Tannaka duality construction for the ordinary representations of a proper Lie groupoid on vector bundles. We show that for each proper Lie groupoid G, the canonical homomorphism of G into the reconstructed groupoid T(G) is surjective, although — contrary to what happens in the case of groups — it may fail to be an isomorphism. We obtain necessary and sufficient conditions in order that G may be isomorphic to T(G) and, more generally, in order that T(G) may be a Lie groupoid. We show that if T(G) is a Lie groupoid, the canonical homomorphism G→T(G) is a submersion and the two groupoids have isomorphic categories of representations.     

A new symmetry for Biedenharn's G-functions and classical hypergeometric series

We show that the general bisymmetric polynomials ^m_μG⁽ⁿ⁾_q(γ₁,…, γ_n; δ₁,…, δ_m) are a limiting case of the bisymmetric, invariant polynomials ⁿ_μG⁽ⁿ⁾_q(γ₁,…, γ_n; δ₁,…, δ_n) which characterize U(n) tensor operators 〈p, q,…, q, 0, …, 0〉. By taking suitable limits of a pair of difference equations for ⁿ_μG⁽ⁿ⁾_q(γ; δ) we then deduce “transposition symmetry” for ^m_μG⁽ⁿ⁾_q(γ; δ) from the same symmetry for ⁿ_μG⁽ⁿ⁾_q(γ; δ). As an application of transposition symmetry for ^m_μG⁽ⁿ⁾_q(γ; δ) we derive an elegant, new contiguous relation for classical, well-poised hypergeometric series, and also prove an identity between these series and multiple hypergeometric series well-poised in SU(n).     

Characterization of double domination subdivision number of trees

In a graph G, a vertex dominates itself and its neighbors. A subset S⊆V(G) is a double dominating set of G if S dominates every vertex of G at least twice. The double domination numberdd(G) is the minimum cardinality of a double dominating set of G. The double domination subdivision numbersd_dd(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the double domination number. In this paper first we establish upper bounds on the double domination subdivision number for arbitrary graphs in terms of vertex degree. Then we present several different conditions on G which are sufficient to imply that sd_dd(G)?3. We also prove that 1?sd_dd(T)?2 for every tree T, and characterize the trees T for which sd_dd(T)=2.