Nonvanishing Functions Research Articles

Новый метод оценки модулей начальных тейлоровских коэффициентов на классе ограниченных не обращающихся в нуль функций

The task of obtaining the sharp estimate of the modulus of the n-th Taylor coefficient on the class B of bounded non-vanishing functions has been reduced to the problem of estimating a functional over the class of normalized bounded functions, which in turn has been reduced to the problem of finding the constrained maximum of a non-negative objective function of 2n-3 real arguments with constraints of the inequality type, that allows us to apply the standard numerical methods of finding constrained extrema. Analytical expressions of the first six objective functions have been obtained and their Lipschitz continuity has been proved. Based on the Lipschitz continuity of the objective function with number n, a method for the sharp estimating of the modulus of the n-th Taylor coefficient on the class B is rigorously proven. An algorithm of finding the global constrained maximum of the objective function is being discussed. The first step of this algorithm involves a brute-force search with a relatively large step. The second step of the algorithm uses a method for finding a local maximum with the initial points obtained at the previous step. The results of the numerical calculations are presented graphically and confirm the Krzyz conjecture for n=1,…,6. Based on these calculations, as well as on so-called asymptotic estimates, a sharp estimate of the moduli of the first six Taylor coefficients on the class B is derived. The obtained results are compared with previously known estimates of the moduli of initial Taylor coefficients on the class B and its subclasses B_t, t≥0. The extremals for B_t subclasses are discussed and the Krzyz hypothesis is updated for B_t subclasses. A brief historical overview of research of the estimations of moduli of initial Taylor coefficients on the class B is provided.

Gram Points in the Universality of the Dirichlet Series with Periodic Coefficients

Let a={am:m∈N} be a periodic multiplicative sequence of complex numbers and L(s;a), s=σ+it a Dirichlet series with coefficients am. In the paper, we obtain a theorem on the approximation of non-vanishing analytic functions defined in the strip 1/2<σ<1 via discrete shifts L(s+ihtk;a), h>0, k∈N, where {tk:k∈N} is the sequence of Gram points. We prove that the set of such shifts approximating a given analytic function is infinite. This result extends and covers that of [Korolev, M.; Laurinčikas, A. A new application of the Gram points. Aequat. Math. 2019, 93, 859–873]. For the proof, a limit theorem on weakly convergent probability measures in the space of analytic functions is applied.

ON JOINT DISCRETE UNIVERSALITY OF THE RIEMANN ZETA-FUNCTION IN SHORT INTERVALS

In the paper, we prove that the set of discrete shifts of the Riemann zeta-function approximating analytic nonvanishing functions f1(s),...,fr(s) defined on has a positive density in the interval [N,N + M] with with real algebraic numbers a1,...,ar linearly independent over Q. A similar result is obtained for shifts of certain absolutely convergent Dirichlet series.

Simple blow-up solutions of singular Liouville equations

In a recent series of important works Wei-Zhang [Adv. Math. 380 (2021), Paper No. 107606, 45; Proc. Lond. Math. Soc. (3) 124 (2022), pp. 106–131; Laplacian vanishing theorem for quantized singular Liouville equation, Preprint, arXiv:2202.10825, 2022] proved several vanishing theorems for non-simple blow-up solutions of singular Liouville equations. It is well known that a non-simple blow-up situation happens when the spherical Harnack inequality is violated near a quantized singular source. In this article, we further strengthen the conclusions of Wei-Zhang by proving that if the spherical Harnack inequality does hold, there exist blow-up solutions with non-vanishing coefficient functions.

Zalcman Functional and Majorization Results for Certain Subfamilies of Holomorphic Functions

In this paper, we investigate sharp coefficient functionals, like initial four sharp coefficient bounds, sharp Fekete–Szegö functionals, and, for n=1 and 2, sharp Zalcman functionals are evaluated for class of functions associated with tangent functions. Furthermore, we provide some majorization results for some non-vanishing holomorphic functions, whose ratios are related to various domains in the open unit disk.

Embedding formalism for CFTs in general states on curved backgrounds

We present a generalization of the embedding space formalism to conformal field theories (CFTs) on nontrivial states and curved backgrounds, based on the ambient metric of Fefferman and Graham. The ambient metric is a Lorentzian Ricci-flat metric in $d+2$ dimensions and replaces the Minkowski metric of the embedding space. It is canonically associated with a $d$-dimensional conformal manifold, which is the physical spacetime where the ${\mathrm{CFT}}_{d}$ lives. We propose a construction of ${\mathrm{CFT}}_{d}$ $n$-point functions in nontrivial states and on curved backgrounds using appropriate geometric invariants of the ambient space as building blocks. This captures the contributions of nonvanishing one-point functions of multi-stress-energy tensors, at least in holographic CFTs. We apply the formalism to two-point functions of thermal CFT, finding exact agreement with a holographic computation and expectations based on thermal operator product expansions (OPEs), and to CFTs on squashed spheres where no prior results are known and existing methods are difficult to apply, demonstrating the utility of the method.

Two coefficient conjectures for nonvanishing Hardy functions, II

Recently the author proved that the Hummel-Scheinberg-Zalcman conjecture of 1977 on coefficients of nonvanishing $H^p$ functions is true for all $p = 2m, \ m \in \mathbb N$, i.e., for the Hilbertian Hardy spaces $H^{2m}$. As a consequence, this also implies the proof of the Krzyz conjecture for bounded nonvanishing functions, which originated this direction. In the present paper, we solve the problem for all spaces $H^p$ with $p \ge 2$.

Two Coefficient Conjectures for Nonvanishing Hardy Functions, I

There are two eminent still open conjectures for nonvanishing holomorphic Hardy functions f(z) on the unit disk. The Hummel–Scheinberg–Zalcman conjecture posed in 1977 extends Krzyz’s conjecture of 1968 to Hp spaces with finite p > 1 and states that Taylor’s coefficients of nonvanishing holomorphic functions f ∈ Hp with norm ‖f‖p ≤ 1 are sharply estimated by |cn| ≤ (2/e)1 − 1/p, with appropriate extremal functions. Both conjectures have been investigated by many authors; however still remain open. The desired Krzyz’s estimate |cn| ≤ 2/e for f ∈ H∞ with ‖f‖∞ ≤ 1 was established only for the initial coefficients cn with n ≤ 5 The only known results for the Hummel–Scheinberg–Zalcman conjecture are that it is true for n = 1 and n = 2 as well as some results for special subclasses of Hp. We prove here that the Hummel–Scheinberg–Zalcman conjecture is true for all spaces H2m with m ∈ N. In the limit as m → ∞, this also provides the proof of Krzyz’s conjecture. Our approach involves deep results from Teichmüller space theory, especially the Bers isomorphism theorem for Teichmüller spaces of punctured Riemann surfaces, and special quasiconformal deformations of H2m functions.

The full power of the half-power

We use the complex square root to define a very simple homotopic invariant over the non-vanishing functions defined on the circle. As a consequence we provide easy proofs of the plane Brouwer fixed point theorem and the Fundamental Theorem of Algebra. The relation of this new invariant with the winding number and the Brouwer degree will be fully unveiled.

Note on boundedness of the $L$-index in the direction of the composition of slice entire functions

We study a composition of two functions belonging to a class of slice holomorphic functions in the whole $n$-dimensional complex space. The slice holomorphy in the space means that for some fixed direction $\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}$ and for every point $z^0\in\mathbb{C}^n$ the function is holomorphic on its restriction on the slice $\{z^0+t\mathbf{b}: t\in\mathbb{C}\}.$ An additional assumption on joint continuity for these functions allows to construct an analog of theory of entire functions having bounded index. The analog is applicable to study properties of slice holomorphic solutions of directional differential equations, describe local behavior and value distribution.In particular, we found conditions providing boundedness of $L$-index in the direction $\mathbf{b}$ for a function $f(\underbrace{\Phi(z),\ldots,\Phi(z)}_{m\text{ times}}),$where $f: \mathbb{C}^n\to\mathbb{C}$ is a slice entire function, $\Phi: \mathbb{C}^n\to\mathbb{C}$ is a slice entire function,${L}: \mathbb{C}^n\to\mathbb{R}_+$ is a continuous function.The obtained results are also new in one-dimensional case, i.e. for $n=1,$ $m=1.$ They are deduced using new approach in this area analog of logarithmic criterion.For a class of nonvanishing outer functions in the composition the sufficient conditions obtained by logarithmic criterion are weaker than the conditions by the Hayman theorem.

Cosmic birefrigence: cross-spectra and cross-bispectra with CMB anisotropies

Parity-violating extensions of Maxwell electromagnetism induce a rotation of the linear polarization plane of photons during propagation. This effect, known as cosmic birefringence, impacts on the Cosmic Microwave Background (CMB) observations producing a mixing of E and B polarization modes which is otherwise null in the standard scenario. Such an effect is naturally parametrized by a rotation angle which can be written as the sum of an isotropic component α 0 and an anisotropic one δα(n̂). In this paper we compute angular power spectra and bispectra involving δα and the CMB temperature and polarization maps. In particular, contrarily to what happens for the cross-spectra, we show that even in absence of primordial cross-correlations between the anisotropic birefringence angle and the CMB maps, there exist non-vanishing three-point correlation functions carrying signatures of parity-breaking physics. Furthermore, we find that such angular bispectra still survive in a regime of purely anisotropic cosmic birefringence, which corresponds to the conservative case of having α o = 0. These bispectra represent an additional observable aimed at studying cosmic birefringence and its parity-violating nature beyond power spectrum analyses. They provide also a way to perform consistency checks for specific models of cosmic birefringence. Moreover, we estimate that among all the possible birefringent bispectra,〈δαTB〉and〈δαEB〉are the ones which contain the largest signal-to-noise ratio. Once the cosmic birefringence signal is taken to be at the level of current constraints, we show that these bispectra are within reach of future CMB experiments, as LiteBIRD.

Two coefficient conjectures for nonvanishing Hardy functions, I

Two coefficient conjectures for nonvanishing Hardy functions, I

Cosmology from pregeometry

We discuss cosmological solutions for a diffeomorphism invariant gauge theory of the non-compact Lorentz group $SO(1,3)$. Besides the gauge bosons our model of pregeometry contains a vector field in the vector representation of $SO(1,3)$ and a scalar singlet. General relativity and variable gravity emerge as effective theories for large distances and times in Planck units. We propose an approximation to the effective action with up to two derivatives. For a suitable range of parameters the universe approaches for large times stable Minkowski space. For late cosmology the model predicts dynamical dark energy and provides for a candidate for dark matter. Early cosmology is characterized by an inflationary epoch. The beginning of the universe in the infinite past is great emptiness, corresponding to an ultraviolet fixed point with the associated quantum scale symmetry. The beginning universe is a vacuum state with vanishing expectation values and finite non-vanishing correlation functions for the fluctuations of all fields. There is no physical big bang singularity.

Khinchin Families and Hayman Class

We give criteria, following Hayman and Báez-Duarte, for non-vanishing functions with non-negative coefficients to be Gaussian and strongly Gaussian. We use these criteria to show in a simple and unified manner asymptotics for a number of combinatorial objects, and, particularly, for a variety of partition questions like Ingham’s theorem on partitions with parts in an arithmetic sequence, or Wright’s theorem on plane partitions and, of course, Hardy–Ramanujan’s partition theorem.

Many-body quantum chaos and dual-unitarity round-a-face.

We propose a new type of locally interacting quantum circuits-quantum cellular automata-that are generated by unitary interactions round-a-face (IRF). Specifically, we discuss a set (or manifold) of dual-unitary IRFs with local Hilbert space dimension d [DUIRF (d)], which generate unitary evolutions both in space and time directions of an extended 1+1 dimensional lattice. We show how arbitrary dynamical correlation functions of local observables can be evaluated in terms of finite-dimensional completely positive trace preserving unital maps in complete analogy to recently studied circuits made of dual-unitary brick gates (DUBGs). The simplest non-vanishing local correlation functions in dual-unitary IRF circuits are shown to involve observables non-trivially supported on two neighboring sites. We completely characterize the ten-dimensional manifold of DUIRF (2) for qubits ( d=2) and provide, for d=3,4,…,7, empirical estimates of its dimensionality based on numerically determined dimensions of tangent spaces at an ensemble of random instances of dual-unitary IRF gates. In parallel, we apply the same algorithm to determine dimDUBG(d) and show that they are of similar order though systematically larger than dimDUIRF(d) for d=2,3,…,7. It is remarkable that both sets have a rather complex topology for d≥3 in the sense that the dimension of the tangent space varies among different randomly generated points of the set. Finally, we provide additional data on dimensionality of the chiral extension of DUBG circuits with distinct local Hilbert spaces of dimensions d≠d' residing at even/odd lattice sites.

Approximation of Certain Non-vanishing Analytic Functions in a Parabolic Region

Approximation of Certain Non-vanishing Analytic Functions in a Parabolic Region

Discrete Approximation by a Dirichlet Series Connected to the Riemann Zeta-Function

In the paper, a Dirichlet series ζuN(s) whose shifts ζuN(s+ikh), k=0,1,⋯, h>0, approximate analytic non-vanishing functions defined on the right-hand side of the critical strip is considered. This series is closely connected to the Riemann zeta-function. The sequence uN→∞ and uN≪N2 as N→∞.

Majorization Results for Certain Subfamilies of Analytic Functions

Let h 1 z and h 2 z be two nonvanishing holomorphic functions in the open unit disc with h 1 0 = h 2 0 = 1 . For some holomorphic function q z , we consider the class consisting of normalized holomorphic functions f whose ratios f z / z q z and q z are subordinate to h 1 z and h 2 z , respectively. The majorization results are obtained for this class when h 1 z is chosen either h 1 z = cos z or h 1 z = 1 + sin z or h 1 z = 1 + z and h 2 z = 1 + sin z .

Density deconvolution under general assumptions on the distribution of measurement errors

In this paper, we study the problem of density deconvolution under general assumptions on the measurement error distribution. Typically, deconvolution estimators are constructed using Fourier transform techniques, and it is assumed that the characteristic function of the measurement errors does not have zeros on the real line. This assumption is rather strong and is not fulfilled in many cases of interest. In this paper, we develop a methodology for constructing optimal density deconvolution estimators in the general setting that covers vanishing and nonvanishing characteristic functions of the measurement errors. We derive upper bounds on the risk of the proposed estimators and provide sufficient conditions under which zeros of the corresponding characteristic function have no effect on estimation accuracy. Moreover, we show that the derived conditions are also necessary in some specific problem instances.

Inhomogeneous interval fields based on scaled inverse distance weighting interpolation

This paper introduces a novel method to model non-deterministic quantities based on experimental measurement data. The focus of this work is on quantities that vary over a continuous domain, e.g., material properties, time-dependent strain rate effects, or stress–strain curves. These quantities are modelled by means of the recently introduced concept of interval fields. An interval field defines intervals that are defined throughout the continuous domain and have dependence in this domain by expanding them over a set of basis functions, describing the spatial nature of the non-determinism of the modelled quantities. One of the more intuitive concepts of defining basis functions in an interval field is through inverse distance weighting interpolation (IDW), which starts from known intervals at specific control points within the domain. For each of these control points, a corresponding basis function is defined, the relative weight of which is decreasing inversely with the distance. Through this definition, all intervals have non-vanishing basis functions throughout the model domain. This makes the application of standard IDW extremely challenging when the interval uncertainty varies inhomogeneously over the domain, i.e., when local effects are present in the model.Therefore, in this paper standard IDW is adapted by changing the distance measure. More specifically, the weight of intervals is increased locally, while diminishing the weight in other regions. For this purpose, a function is introduced that maps the domain to a higher dimension feature space, in which the distances that determine the weight are measured. This mapping function is based on either the size of the intervals at the control points or experimental data, which both yield additional control resulting in increased agreement with experimental data.This paper demonstrates that this method outperforms standard IDW in controllability, while limiting the number of control points. This is illustrated in three case studies: a first case concerning modelling local non-determinism; a second case where a mix of global and local effects is modelled; and the third case, where the interval field is based on experimental stress strain curves. In all these cases, multiple configurations demonstrate the effects of the parameters, and how the new technique is applied. The proposed technique outperforms standard IDW in all three case studies, with an increased coefficient of determination, R2, between 22% and 56%, in comparison to standard IDW.