Nonvanishing Functions Research Articles

A new version of circular symmetrization with applications to p-valent functions

A new version of circular symmetrization of sets, functions and condensers is proposed, which is different from classical symmetrization in the following respect: the symmetrized sets and condensers lie on the Riemann surface of the inverse function of a Chebyshev polynomial. As applications, Hayman's well-known results for nonvanishing p-valent holomorphic functions are supplemented as well as results for p-valent functions in a disc which have a zero of order p at the origin.Bibliography: 20 titles.

Banach–Stone theorems for vector valued functions on completely regular spaces

We obtain several Banach–Stone type theorems for vector-valued functions in this paper. Let X,Y be realcompact or metric spaces, E,F locally convex spaces, and ϕ a bijective linear map from C(X,E) onto C(Y,F). If ϕ preserves zero set containments, i.e., z(f)⊆z(g)⟺z(ϕ(f))⊆z(ϕ(g)),∀f,g∈C(X,E), then X is homeomorphic to Y, and ϕ is a weighted composition operator. The above conclusion also holds if we assume a seemingly weaker condition that ϕ preserves nonvanishing functions, i.e., z(f)=0̸⟺z(ϕf)=0̸,∀f∈C(X,E). These two results are special cases of the theorems in a very general setting in this paper, covering bounded continuous vector-valued functions on general completely regular spaces, and uniformly continuous vector-valued functions on metric spaces. Our results extend and generalize many recent ones.

Global injectivity and multiple equilibria in uni- and bi-molecular reaction networks

Dynamical system models of complex biochemical reaction networks are high-dimensional, nonlinear, and contain many unknown parameters.The capacity for multiple equilibria in such systems plays a key rolein important biochemical processes. Examples show that there is a verydelicate relationship between the structure of a reaction network andits capacity to give rise to several positive equilibria. In thispaper we focus on networks of reactions governed by mass-actionkinetics. As is almost always the case in practice, we assume that noreaction involves the collision of three or more molecules at the sameplace and time, which implies that the associated mass-actiondifferential equations contain only linear and quadratic terms. Wedescribe a general injectivity criterion for quadratic functions ofseveral variables, and relate this criterion to a network's capacityfor multiple equilibria. In order to take advantage of this criterionwe look for explicit general conditions that imply non-vanishing ofpolynomial functions on the positive orthant.In particular, we investigate in detail the case of polynomials withonly one negative monomial, and we fully characterize the case ofaffinely independent exponents.We describe several examples, including an example that shows howthese methods may be used for designing multistable chemical systemsin synthetic biology.

Solution to a Bergman space extremal problem for non-vanishing functions

For functions analytic and non-vanishing in {|z|<1} and of the form f(z)=1+az+a2z2+…, with a⩾0 given, we find the extremal function which minimizes the A2 norm and calculate the exact minimum value of this norm.

Parity violation in the Cosmic Microwave Background from a pseudoscalar inflaton

If the inflaton ϕ is a pseudoscalar, then it naturally interacts with gauge fields through the coupling ∝ϕ Fμν F̃μν. Through this coupling, the rolling inflaton produces quanta of the gauge field, that in their turn source the tensor components of the metric perturbations. Due to the parity-violating nature of the system, the right- and the left-handed tensor modes have different amplitudes. Such an asymmetry manifests itself in the form of non-vanishing TB and EB correlation functions in the Cosmic Microwave Background (CMB). We compute the amplitude of the parity-violating tensor modes and we discuss two scenarios, consistent with the current data, where parity-violating CMB correlation functions will be detectable in future experiments.

Area extremal problems for non-vanishing functions

We consider the problem of finding the extremal function f f\, which minimises the Bergman space A 2 A^2\, norm for the class of non-vanishing functions whose first n + 1 n+1 Taylor coefficients are given. \, We define an analytic function K K in terms of f f and show that the functions K K and f f satisfy a certain differential equation. This equation yields a set of relationships between the area moments and the circle moments of | f | 2 |f|^2 , which in particular shows that the outer part of f f\, is a polynomial of degree at most n n .

Quasi Extended Chebyshev spaces and weight functions

We recently showed that the class of Quasi Extended Chebyshev spaces is the largest class of sufficiently differentiable functions permitting design. In previous articles we mentioned a simple procedure to build such spaces by means of both generalised derivatives associated with non-vanishing weight functions and two-dimensional Chebyshev spaces. In the present one we prove that, conversely, on a closed bounded interval, any Quasi Extended Chebyshev space can be obtained via the latter procedure. We then draw a few interesting consequences from the latter result.

The Heyde theorem for locally compact Abelian groups

We prove a group analogue of the well-known Heyde theorem where a Gaussian measure is characterized by the symmetry of the conditional distribution of one linear form given another. Let X be a locally compact second countable Abelian group containing no subgroup topologically isomorphic to the circle group T , G be the subgroup of X generated by all elements of order 2, and Aut ( X ) be the set of all topological automorphisms of X. Let α j , β j ∈ Aut ( X ) , j = 1 , 2 , … , n , n ⩾ 2 , such that β i α i − 1 ± β j α j − 1 ∈ Aut ( X ) for all i ≠ j . Let ξ j be independent random variables with values in X and distributions μ j with non-vanishing characteristic functions. If the conditional distribution of L 2 = β 1 ξ 1 + ⋯ + β n ξ n given L 1 = α 1 ξ 1 + ⋯ + α n ξ n is symmetric, then each μ j = γ j ∗ ρ j , where γ j are Gaussian measures, and ρ j are distributions supported in G.

The sheaf of nonvanishing meromorphic functions in the projective algebraic case is not acyclic

Let X / C be a projective algebraic manifold, and M X ∗ be the sheaf of nonvanishing meromorphic functions on X in the analytic topology. We prove a number of nonvanishing results for H • ( X , M X ∗ ) . In particular, M X ∗ is acyclic iff dim X = 1 .

Extremal Problems in the Fock Space

This paper is devoted to examining some extremal problems in the Fock space. We discuss the order and type of Fock space functions and pose an extremal problem for a zero-based subspace corresponding to a finite zero set. We examine the zeros of the extremal function and solve an extremal problem for non-vanishing functions.

Non-vanishing functions and Toeplitz operators on tube-type domains

We prove an index theorem for Toeplitz operators on irreducible tube-type domains and we extend our results to Toeplitz operators with matrix symbols. In order to prove our index theorem, we proved a result asserting that a non-vanishing function on the Shilov boundary of a tube-type bounded symmetric domain, not necessarily irreducible, is equal to a unimodular function defined as the product of powers of generic norms times an exponential function.

A nonhomogenizable linear transport equation in R2

In this paper I investigate the homogenizability of linear transport equations with periodic data. Some results on homogenizability and on the form of the limit are known in literature. In particular, in [9], I proved the homogenizability in the two-dimensional case for nonvanishing functions, and, on the other hand I gave an example of a nonhomogenizable equation in the three-dimensional case. In this paper, I describe an example of a nonhomogenizable equation in two dimensions. As in [9], I study the problem using an equivalent formulation in terms of dynamical system properties of the associated ODEs.

Region of variability for spiral-like functions with respect to a boundary point

For $\mu\in\mathbb C$ such that ${\rm Re\,}\mu>0$ let ${\mathcal F}_{\mu} $ denote the class of all non-vanishing analytic functions $f$ in the unit disk $\mathbb{D}$ with $f(0)=1$ and $$ {\rm Re} \bigg(\frac{2\pi}{\mu}\, \frac{zf'(z)}{f(z)}+ \frac{1+z}

Independent Linear Statistics on the Two-Dimensional Torus

Let $X={\bf T}^2$ be the two-dimensional torus, ${\rm Aut}({\bf T}^2)$ be the group of topological automorphisms of ${\bf T}^2$, $\Gamma({\bf T}^2)$ be the set of Gaussian distributions on ${\bf T}^2$, and $\xi_1,\xi_2$ be independent random variables taking on values in ${\bf T}^2$ and having distributions $\mu_j$ with the nonvanishing characteristic functions. Consider $\delta\in{\rm Aut}({\bf T}^2)$ and assume that the linear forms $L_1=\xi_1+\xi_2$ and $L_2=\xi_1+\delta\xi_2$ are independent. We give the description of possible distributions $\mu_j$ depending on $\delta$. In particular we give the complete description of $\delta$ such that the independence of $L_1$ and $L_2$ implies that $\mu_1,\mu_2\in\Gamma({\bf T}^2)$. It turned out that the corresponding Gaussian distributions are either degenerate or concentrated on shifts of the same dense in ${\bf T}^2$ one-parameter subgroup.

Extreme and Support Points of the Class of Non-Vanishing Univalent Functions

Let S0 be the standard class of non-vanishing univalent functions in the unit disc. In this article we present a generalization of the classical ellipse theorem for this class.

Measurement-based quantum computation beyond the one-way model

We introduce schemes for quantum computing based on local measurements on entangled resource states. This work elaborates on the framework established in Gross and Eisert [Phys. Rev. Lett. 98, 220503 (2007); quant-ph/0609149]. Our method makes use of tools from many-body physics-matrix product states, finitely correlated states, or projected entangled pairs states-to show how measurements on entangled states can be viewed as processing quantum information. This work hence constitutes an instance where a quantum information problem-how to realize quantum computation-was approached using tools from many-body theory and not vice versa. We give a more detailed description of the setting and present a large number of examples. We find computational schemes, which differ from the original one-way computer, for example, in the way the randomness of measurement outcomes is handled. Also, schemes are presented where the logical qubits are no longer strictly localized on the resource state. Notably, we find a great flexibility in the properties of the universal resource states: They may, for example, exhibit nonvanishing long-range correlation functions or be locally arbitrarily close to a pure state. We discuss variants of Kitaev's toric code states as universal resources, and contrast this with situations where they can be efficiently classically simulated. This framework opens up a way of thinking of tailoring resource states to specific physical systems, such as cold atoms in optical lattices or linear optical systems.

Generalized Spectral Coherences for Complex-Valued Harmonizable Processes

Complex-valued nonstationary random processes have nonvanishing complementary second-order moment functions. In this paper, we propose generalized dual-frequency and time-frequency coherence functions for harmonizable processes. The proposed generalized spectral coherences are based on widely linear estimators, and they result in coherence measures that combine Hermitian and complementary moment functions. We show that for analytic processes, and more surprisingly also for real-valued processes, additional second-order information becomes available through the generalized coherences. We offer illuminating geometrical interpretations of the proposed coherences through Hilbert space inner product formulations. Finally, we extend the theory to generalized cross-coherences between pairs of harmonizable processes

A minimal area problem for nonvanishing functions

The minimal area covered by the image of the unit disk is found for non- vanishing univalent functions normalized by the conditions f (0) = 1, f(0) = α .T wo different approaches are discussed, each of which contributes to the complete solution of the problem. The first approach reduces the problem, via symmetrization, to the class of typically real functions, where the well-known integral representation can be employed to obtain the solution upon ap rioriknowledge of the extremal function. The second approach, requiring smoothness assumptions, leads, via some variational formulas, to a boundary value problem for analytic functions, which admits an ex- plicit solution.

A conformal invariant for non-vanishing analytic functions and its applications

A conformal invariant for non-vanishing analytic functions and its applications

Exploratory study of three-point Green’s functions in Landau-gauge Yang-Mills theory

Green's functions are a central element in the attempt to understand nonperturbative phenomena in Yang-Mills theory. Besides the propagators, 3-point Green's functions play a significant role, since they permit access to the running coupling constant and are an important input in functional methods. Here we present numerical results for the two nonvanishing 3-point Green's functions in 3d pure SU(2) Yang-Mills theory in (minimal) Landau gauge, i.e. the three-gluon vertex and the ghost-gluon vertex, considering various kinematical regimes. In this exploratory investigation the lattice volumes are limited to ${20}^{3}$ and ${30}^{3}$ at $\ensuremath{\beta}=4.2$ and $\ensuremath{\beta}=6.0$. We also present results for the gluon and the ghost propagators, as well as for the eigenvalue spectrum of the Faddeev-Popov operator. Finally, we compare two different numerical methods for the evaluation of the inverse of the Faddeev-Popov matrix, the point-source and the plane-wave-source methods.