On φ-Families of Probability Distributions

We generalize the exponential family of probability distributions. In our approach, the exponential function is replaced by a φ-function, resulting in a φ-family of probability distributions. We show how φ-families are constructed. In a φ-family, the analogue of the cumulant-generating function is a normalizing function. We define the φ-divergence as the Bregman divergence associated to the normalizing function, providing a generalization of the Kullback–Leibler divergence. A formula for the φ-divergence where the φ-function is the Kaniadakis κ-exponential function is derived.     

Convergent sequences of composition operators

Composition operators Cφ on the Hilbert Hardy space H² over the unit disk are considered. We investigate when convergence of sequences {φn} of symbols, (i.e., of analytic selfmaps of the unit disk) towards a given symbol φ, implies the convergence of the induced composition operators, Cφn→Cφ. If the composition operators Cφn are Hilbert-Schmidt operators, we prove that convergence in the Hilbert-Schmidt norm, ‖Cφn−CφHS_‖→0 takes place if and only if the following conditions are satisfied: ‖φn−φ_‖2→0, ∫1/(1−²|φ|)<∞, and ∫1/(1−²|φn|)→∫1/(1−²|φ|). The convergence of the sequence of powers of a composition operator is studied.     

On the standing wave for a class of nonlinear Schrödinger equations

This paper is concerned with the standing wave for a class of nonlinear Schrödinger equations

iφt+Δφ−²|x|φ+μ|φ|^p−1φ+γ|φ|^q−1φ=0,     

Groups generated by analytic Schrödinger operators

If the potential in a two-particle system is the boundary value of an analytic function, the physical Hamiltonian H(0) has an analytic continuation H(φ) which is not normal. In case the potential is local and belongs to suitable $\text{L}$^p-spaces, there is a bounded operator P(0, φ) projecting onto the continuous subspace of H(φ). This paper shows that P(0, φ) H(φ) e^?2iφ generates a strongly differentiable group. It is proved that P(0, φ) H(φ) is spectral, and details of the spectral projection operators are presented. The reasoning is based on the Paley-Wiener theorem for functions in a strip. It applies to larger systems provided the resolvent of the multiparticle operator H(φ) satisfies certain regularity conditions that come from the theory of smooth operators. There are no smallness conditions on the potential.     

Continued fractions and numeration in the Fibonacci base

Let φ be the golden ratio. We define and study a continued φ-fraction algorithm, inspired by Euclid's algorithm. We show that any non-negative element of Q(φ) has a finite continued φ-fraction.     

Bloch-to-Hardy composition operators

Let φ be a holomorphic mapping between complex unit balls. We characterize those regular φ for which the composition operators C _φ : f ? f ○ φ map the Bloch space into the Hardy space.     

On hyperbolic once-punctured-torus bundles III: Comparing two tessellations of the complex plane

To each once-punctured-torus bundle, Tφ, over the circle with pseudo-Anosov monodromy φ, there are associated two tessellations of the complex plane: one, Δ(φ), is (the projection from ∞ of) the triangulation of a horosphere at ∞ induced by the canonical decomposition into ideal tetrahedra, and the other, CW(φ), is a fractal tessellation given by the Cannon-Thurston map of the fiber group switching back and forth between gray and white each time it passes through ∞. In this paper, we fully describe the relation between Δ(φ) and CW(φ).     

On a class of C*-algebras generated by a countable family of partial isometries

The paper investigates the properties of an operator T _φ on the Hilbert space l ²(?), which are induced by the mapping φ of the set ? into itself. It is shown if the mapping φ is such that every preimage has finite, but not equipotentionally bounded cardinality, then the operator T _φ allows a closure and can be represented as a countable sum of partial isometries. The C*-algebras U _φ , P _φ and U _φ associated with given mappings and generated by the mentioned partial isometries are considered. Some properties of these algebras and some relations between them are given.     

Invertible weighted composition operators

A weighted composition operator Cψ,φ takes an analytic map f on the open unit disc of the complex plane to the analytic map ψ⋅f°φ where φ is an analytic map of the open unit disc into itself and ψ is an analytic map on the open unit disc. This paper studies the invertibility of such operators. The two maps ψ and φ are characterized when Cψ,φ acts on the Hardy-Hilbert space of the unit disc H²(D). Depending upon the nature of the fixed points of φ spectra are then investigated.     

Semigroups associated with analytic Schrödinger operators

If the potential in a two-particle system is the boundary value of an analytic function, the physical Hamiltonian H(0) has an analytic continuation H(φ). The continuous spectrum of H(φ) consists of the half-line Y(0, φ) which runs from 0 to ∞e^2iφ. Integrating along lines parallel to Y(0, φ), this paper examines the Fourier transform of the resolvent R(λ, φ). The integration path passing through ±iεe^2iφ yields semigroups {U(t, ±iεe^2iφ, φ)} (t > 0 and t < 0). Under the assumption that the potential is local and belongs to suitable $\text{L}$^p-spaces, it is shown that the semigroups tend to norm limits as ε tends to 0. The proof is based on the Paley-Wiener theorem for functions in a strip. It generalizes to multiparticle systems under conditions on R(λ, φ) that are to be verified with the help of the theory of smooth operators.     

Weighted Composition Operators Between Zygmund Type Spaces and Their Essential Norms

Let ${u \in \mathcal{H}(\mathbb{D})}$ and φ be an analytic self-map of ${\mathbb{D}}$ . We estimate the essential norms of weighted composition operators uC _φ acting on Zygmund type spaces in terms of u, φ, their derivatives and the n-th power φ ⁿ of φ. Moreover, we give similar characterizations for boundedness of uC _φ between Zygmund type spaces.     

Precise and fast computation of a general incomplete elliptic integral of third kind by half and double argument transformations

This is a continuation of our works to compute the incomplete elliptic integrals of the first and second kind (Fukushima (2010, 2011) [21] and [23]). We developed a method to compute an associate incomplete elliptic integral of the third kind, J(φ,n|m)≡[Π(φ,n|m)−F(φ|m)]/n, by the half argument formulas of the sine and cosine amplitude functions and the double argument transformation of the integral. The relative errors of J(φ,n|m) computed by the new method are sufficiently small as less than 20 machine epsilons. Meanwhile, the simplicity of the adopted algorithm makes the new method run 1.5 to 3.7 times faster than Carlson’s duplication method. The combination of the new method and that to compute simultaneously two other associate incomplete elliptic integrals of the second kind, B(φ|m)≡[E(φ|m)−(1−m)F(φ|m)]/m and D(φ|m)≡[F(φ|m)−E(φ|m)]/m, which we established recently [23], enables a precise and fast computation of arbitrary linear combination of Legendre’s incomplete elliptic integrals of all three kinds, F(φ|m), E(φ|m), and Π(φ,n|m). These new procedures share the same device, the half argument transformations, while the double argument transformation of J(φ,n|m) includes those of B(φ|m) and D(φ|m) as its sub component. As a result, the simultaneous computation of the three associate integrals is significantly faster than computing them separately. In fact, our combined procedure is 2.7 to 5.9 times faster than the combination of Carlson’s duplication method to compute R_D and R_J.     

Subcentrality of restrictions of boundary measures on state spaces of C1-algebras

Let F be a closed face of the weak¹ compact convex state space of a unital C¹-algebra A. The author has already shown that F is a Choquet simplex if and only if p_φ^Fπ_φ(A)″p_φ^F is abelian for any φ in F with associated cyclic representation ($\text{H}$_φ,π_φ,ξ_φ), where p_φ^F is the orthogonal projection of $\text{H}$_φ onto the subspace spanned by vectors η defining vector states a → 〈π_φ(a)η, η)〉 lying in F. It is shown here that if B is a C¹-subalgebra of A containing the unit and such that ξ_φ is cyclic in $\text{H}$_φ for π_φ(B) for any φ in F, then the boundary measures on F are subcentral as measures on the state space of B if and only if p_φ^F(π_φ(A), π_φ(B)′)″p_φ^F is abelian for all φ in F. If A is separable, this is equivalent to the condition that any state in F with (π_φ(A)′ ∩ π_φ(B)″) one-dimensional is pure. Taking A to be the crossed product of a discrete C¹-dynamical system (B, G, α), these results generalise known criteria for the system to be G-central.     

The resolvent of a dilation-analytic three-particle system

If the potential in a three-particle system is the boundary value of an analytic function, the physical Hamiltonian H(0) has a dilation-analytic continuation H(φ). The continuous spectrum of H(φ) consists of half-lines Y(λ_p, φ) starting at the thresholds λ_p of scattering channels and making angles 2φ with the positive real axis. If the interaction is the sum of local two-body potentials in suitable $\text{L}$^p-spaces, each half-line Y(λ_p, φ) is associated with an operator P(λ_p, φ) that projects onto an invariant subspace of H(φ). Suppose Y(λ_p, φ) does not pass through any two- or three-particle eigenvalues λ ≠ λ_p when φ runs through some interval $\text{0 < α ? φ ? β <}\text{π}\text{2}$. For φ in [α, β], this paper shows that the resolvent R(λ, φ) has smoothness properties near Y(λ_p, φ) that are sufficient for P(λ_p, φ)[H(φ) ? λ_p] e^?2iφ to be spectral and to generate a strongly differentiable group. The projection, the group, and the spectral resolution operators are norm continuous in φ. These results are not affected by any spurious poles of the resolvent equation. At a spurious pole λ = λ_p + ze^2iφ, the resolvent R(λ_p + ze^2iφ,φ) is examined by a method that uses two resolvent equations in succession and shows that there is norm continuity in z, φ. The case of spurious poles on Y(λ_p, φ) is included.     

Estimates of the smoothness of dyadic orthogonal wavelets of Daubechies type

Suppose that ω(φ, ·) is the dyadic modulus of continuity of a compactly supported function φ in L ²(?⁺) satisfying a scaling equation with 2 ⁿ coefficients. Denote by α _φ the supremum for values of α > 0 such that the inequality ω(φ, 2^?j ) ≤ C2 ^?αj holds for all j ∈ ?. For the cases n = 3 and n = 4, we study the scaling functions φ generating multiresolution analyses in L ²(?₊) and the exact values of α _φ are calculated for these functions. It is noted that the smoothness of the dyadic orthogonal wavelet in L ²(?₊) corresponding to the scaling function φ coincides with α _φ .     

The Exact Value of Normal Structure Coefficients and WCS coefficients in a Class of Orlicz Function Spaces

Let φ be an N-function. Then the normal structure coefficients N and the weakly convergent sequence coefficients WCS of the Orlicz function spaces L ^φ[0, 1] generated by φ and equipped with the Luxemburg and Orlicz norms have the following exact values. (i) If F _φ(t) = t _?(t)/φ(t) is decreasing and 1 < C _φ < 2 (where \(C_\Phi = \lim _{t \to + \infty } t\varphi (t)/\Phi (t)\)), then N(L ^(φ)[0, 1]) = N(L ^φ[0, 1]) = WCS(L ^(φ)[0, 1]) = WCS(L ^φ[0, 1]) = 2^1?1/Cφ. (ii) If F _φ(t) is increasing and C _φ > 2, then N(L ^(φ)[0, 1]) = N(L ^φ[0, 1]) = WCS(L ^(φ)[0, 1]) = WCS(L ^φ[0, 1]) = 2^1/Cφ.     

On character amenability of Banach algebras

We continue our work [E. Kaniuth, A.T. Lau, J. Pym, On φ-amenability of Banach algebras, Math. Proc. Cambridge Philos. Soc. 144 (2008) 85-96] in the study of amenability of a Banach algebra A defined with respect to a character φ of A. Various necessary and sufficient conditions of a global and a pointwise nature are found for a Banach algebra to possess a φ-mean of norm 1. We also completely determine the size of the set of φ-means for a separable weakly sequentially complete Banach algebra A with no φ-mean in A itself. A number of illustrative examples are discussed.     

On the residual finiteness of descending HNN-extensions of groups

Let G be a group of finite generic rank, φ an injective endomorphism of the group G, and G(φ) the descending HNN-extension of G corresponding to the endomorphism φ. Let the index of the subgroup Gφ in G be finite and equal to n. It is proved that, if the group G is almost residually π-finite for some set π of primes coprime to n, then the group G(φ) is residually finite. This generalizes a series of known results, including the Wise-Hsu theorem on the residual finiteness of an arbitrary descending HNN-extension of any almost polycyclic group.     

Sharp inequalities for the dyadic square function in the BMO setting

We introduce a method for the simultaneous study of a BMO function φ and its dyadic square function S(φ) that can yield sharp norm inequalities between the two. One of the applications is the sharp bound for the p-th moment of S(φ), 0<p<∞, which in turn implies the square-exponential integrability of the square function. We also present sharp refinements of these inequalities in the more restrictive case when φ is assumed to be bounded.     

Dilating covariant representations of the non-commutative disc algebras

Let φ be an isometric automorphism of the non-commutative disc algebra An for n?2. We show that every contractive covariant representation of (An,φ) dilates to a unitary covariant representation of (On,φ). Hence the C^∗-envelope of the semicrossed product Anφ_×Z⁺ is Onφ_×Z.