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Inspired by the classical Cauchy transform in L 2 (∂ B (R)) , we first derive the Laurent expansion for Hermitian monogenic functions in Hermitian Clifford analysis, and we obtain direct applications of this expansion. Then we use the Laurent expansion to study the L 2 -boundary values of Hermitian monogenic functions, we prove that every f ∈ L 2 (S 2 m - 1 ; V) can be decomposed as a sum of boundary values of functions, which are h-monogenic inside and outside the unit ball respectively. [ABSTRACT FROM AUTHOR]

We study Jacobi processes (X t) t ≥ 0 on [ - 1 , 1 ] N and [ 1 , ∞ [ N which are motivated by the Heckman–Opdam theory and associated integrable particle systems. These processes depend on three positive parameters and degenerate in the freezing limit to solutions of deterministic dynamical systems. In the compact case, these models tend for t → ∞ to the distributions of the β -Jacobi ensembles and, in the freezing case, to vectors consisting of ordered zeros of one-dimensional Jacobi polynomials. We derive almost sure analogues of Wigner's semicircle and Marchenko–Pastur limit laws for N → ∞ for the empirical distributions of the N particles on some local scale. We there allow for arbitrary initial conditions, which enter the limiting distributions via free convolutions. These results generalize corresponding stationary limit results in the compact case for β -Jacobi ensembles and, in the deterministic case, for the empirical distributions of the ordered zeros of Jacobi polynomials. The results are also related to free limit theorems for multivariate Bessel processes, β -Hermite and β -Laguerre ensembles, and the asymptotic empirical distributions of the zeros of Hermite and Laguerre polynomials for N → ∞ . [ABSTRACT FROM AUTHOR]

Let μ be a self-similar measure with compact support K. The Hausdorff dimension of K is α. The Cauchy transform of μ is denoted by F (z) . For 0 < β < 1 , we define the function F [ β ] , which compares with the fractional derivative of F of order β. Let Φ (z) = F (1 / z) , | z | < 1 . In this paper, we prove that Φ [ β ] belongs to A p for 0 < p < 1 / (β + 1) , and (Φ ′) [ β ] belongs to A p for 1 ≤ p < 1 / β ≤ 1 / (2 − α) , where A p is the Bergman space. At the same time, we give a value distribution property of F, which is similar to the big Picard theorem. [ABSTRACT FROM AUTHOR]

We give estimates of the Cauchy transform in Lebesgue integral norms in Clifford analysis framework which are the generalizations of Cauchy transform in complex plane, and mainly establish the (L p , L q) -boundedness of the Clifford Cauchy transform in Euclidean space R n + 1 using the Clifford algebra and the Hardy–Littlewood maximal function. Furthermore, we prove Hedberg estimate and Kolmogorov's inequality related to Clifford Cauchy transform. As applications, some respective results in complex plane are directly obtained. Based on the properties of the Clifford Cauchy transform and the principle of uniform boundedness, we solve existence of solutions to integral equations with Cauchy kernel in quaternionic analysis. [ABSTRACT FROM AUTHOR]

Finite rank perturbations T=N+K of a bounded normal operator N acting on a separable Hilbert space are studied thanks to a natural functional model of T; in its turn the functional model solely relies on a perturbation matrix/characteristic function previously defined by the second author. Function theoretic features of this perturbation matrix encode in a closed-form the spectral behavior of T. Under mild geometric conditions on the spectral measure of N and some smoothness constraints on K we show that the operator T admits invariant subspaces, or even it is decomposable.

Let μ be a self-similar measure with compact support K. The Hausdorff dimension of K is α. The Cauchy transform of μ is denoted by F(z). For 0<β<1, we define the function F[β], which compares with the fractional derivative of F of order β. Let Φ(z)=F(1/z),|z|<1. In this paper, we prove that Φ[β] belongs to Ap for 0

In this work we describe the process of construction of solution to Initial Value Problem for some quasi-linear non-strictly hyperbolic equation. Using a General integral, the non-linear version of Cauchy problem is investigated. We also depict the structure of domain where the solution is defined. [ABSTRACT FROM AUTHOR]

Let $ S_j(z) = \varepsilon_j +(z-\varepsilon_j)/2 $ be an iterated function system, where $ \varepsilon_j = e^{2j\pi i/3} $ for $ j = 0, 1, 2 $. Then, there exists a uniform self-similar measure $ \mu $ supported on a compact set $ K $, which is the attractor of $ \{S_j\}_{j = 0}^2 $. The Hausdorff dimension of the attractor $ K $ is $ \alpha = \log 3/\log 2 $. Let $ F(z) = \int_{K}(z-\omega)^{-1}d\mu(\omega) $ be the Cauchy transform of $ \mu $. In this paper, we consider the Hardy space and the multiplier property of $ F $. We prove that $ F' $ belongs to $ H^p $ for $ 0 < p < 1/(2-\alpha) $ and that $ F $ is a multiplier of some class of function space.

We give a negative answer to the Newman-Shapiro problem on weighted approximation for entire functions formulated in 1966 and motivated by the theory of operators on the Fock space. There exists a function in the Fock space such that its exponential multiples do not approximate some entire multiples in the space. Furthermore, we establish several positive results under different restrictions on the function in question. [ABSTRACT FROM AUTHOR]

In this paper we analyse the integrability of Cauchy transforms of functions and measures. As an application of general Paley–Wiener theorems, we show that the integrability of the Cauchy transform of a function or measure is deeply related to the integrability of the Fourier transform of the corresponding function (measure). Our main results are representation theorems for the Cauchy transform in weighted spaces of Bergman–Dirichlet type. [ABSTRACT FROM AUTHOR]

In resonance to a recent geometric framework proposed by Douglas and Yang, afunctional model for certain linear bounded operators with rank-oneself-commutator acting on a Hilbert space is developed. By taking advantage ofthe refined existing theory of the principal function of a hyponormal operatorwe transfer the whole action outside the spectrum, on the resolvent of theunderlying operator, localized at a distinguished vector. The wholeconstruction turns out to rely on an elementary algebra body involving analyticmultipliers and Cauchy transforms. A natural field theory interpretation of theresulting resolvent functional model is proposed.

We establish the mapping properties of the Bergman projections, the harmonic Bergman projections, the Cesáro operator and the Hilbert matrix on exponential Orlicz spaces and Zygmund spaces. We also have the mapping properties of the Berezin transform on Zygmund spaces. A sharpened result for the mapping properties of the Cauchy transform from the Lebesgue spaces to space of bounded function is also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence of a solution interpreted as the Cauchy transform of a signed measure of a particular algebraic quadratic equation of the form zC~(z) -- P (z) C{z) + Q (z) =0 for some polynomials P (z) and Q (z). This issue requires the description of the critical graph of a related quadratic differential in the Riemann sphere C. In particular, we discuss the existence of finite critical trajectories of this quadratic differential. [ABSTRACT FROM AUTHOR]

Rotational smoothing is a phenomenon consisting in a gain of regularity by means of averaging over rotations. This phenomenon is present in operators that regularize only in certain directions, in contrast to operators regularizing in all directions. The gain of regularity is the result of rotating the directions where the corresponding operator performs the smoothing effect. In this paper we carry out a systematic study of the rotational smoothing for a class of operators that includes k -vector-space Riesz potentials in R n with k < n , and the convolution with fundamental solutions of elliptic constant-coefficient differential operators acting on k -dimensional linear subspaces. Examples of the latter type of operators are the planar Cauchy transform in R n , or a solution operator for the transport equation in R n. The analysis of rotational smoothing is motivated by the resolution of some inverse problems under low-regularity assumptions. [ABSTRACT FROM AUTHOR]

A conjecture of T. Richards is proven which yields a quantitative version of the classical Gauss-Lucas theorem: if K is a convex set, then for every ε>0 there is an αε<1 such that if a polynomial Pn of degree at most n has k≥αεn zeros in K, then P'n has at least k−1 zeros in the ε-neighborhood of K. Estimates are given for the dependence of αε on ε. [ABSTRACT FROM AUTHOR]

Near 100% of diffractive efficiency for diffractive optical elements (DOEs) is one of the most required optical performances in broadband imaging applications. Of all flat DOEs, none seems to interest researchers as much as Two-Materials Composed Diffractive Fresnel Lens (TM-DFL) among the most promising flat DOEs. An approach of the near 100% of diffractive efficiency for TM-DFL once developed to determine the design rules mainly takes the advantage of numerical computation by methods of mapping and fitting. Despite a curved line of near 100% of diffractive efficiency can be generated in the Abbe and partial dispersion diagram, it is not able to analytically elaborate the relationship between two optical materials that compose the TM-DFL. Here, we present a theoretical framework, based on the fundaments of Cauchy's equation, Abbe number, partial dispersion, and the diffraction theory of Fresnel lens, for obtaining a general design formalism, so to perform the perfect material matching between two different optical materials for achieving the near 100% of diffractive efficiency for TM-DFL in the broadband imaging applications. [ABSTRACT FROM AUTHOR]

Real‐time power dispatch can coordinate wind farms, automatic generation control units and non‐automatic generation control units. In real‐time power dispatch, the probable wind power forecast errors should be appropriately formulated to ensure system security with high probability and minimize operational cost. Previous studies and the authors' onsite tests show that Cauchy distribution effectively fits the "leptokurtic" feature of small‐timescale wind power forecast errors distributions. In this paper, a chance‐constrained real‐time dispatch model with the wind power forecast errors represented by multivariate Cauchy distribution is proposed. Since the Cauchy distribution is stable and has promising mathematical characteristics, the proposed chance‐constrained real‐time dispatch model can be analytically transformed to a convex optimization problem considering the dependence among wind farms' outputs. Moreover, the proposed model incorporates an affine control strategy compatible with automatic generation control systems. This strategy makes the chance‐constrained real‐time dispatch adaptively take into account both the potential power ramping requirement and power variation on transmission lines caused by the generation adjustment to offset the wind power forecast errors in real‐time power dispatch stage. Numerical test results show that the proposed method is reliable and effective. Meanwhile it is very efficient and suitable for real‐time application. [ABSTRACT FROM AUTHOR]

We investigate some spectral properties of the weighted quaternionic Cauchy transform when acting on the right quaternionic Hilbert space of Gaussian integrable functions. We study its boundedness, compactness, and memberships to the k -Schatten class, and we identify its range. This is done by means of its restriction to the n -th S-polyregular Bargmann space of the second kind, for which we provide an explicit closed expression for its action on the quaternionic Itô–Hermite polynomials constituting an orthogonal basis. We also exhibit an orthogonal basis of eigenfunctions of its n -Bergman projection leading to the explicit determination of its singular values. The provided results extend to the quaternionic setting those obtained by A. Intissar and A. Intissar in (2006) [32] for the weighted Cauchy transform on the complex plane. [ABSTRACT FROM AUTHOR]

For n ≥ 3, let Qn ⊂ C be an arbitrary regular n-sided polygon. We prove that the Cauchy transform FQn of the normalised two-dimensional Lebesgue measure on Qn is univalent and starlike but not convex in Ĉ \ Qn. [ABSTRACT FROM AUTHOR]

We develop the complex-analytic viewpoint on the tree convolutions studied by the second author and Weihua Liu in Jekel and Liu (2020), which generalize the free, boolean, monotone, and orthogonal convolutions. In particular, for each rooted subtree T of the N-regular tree (with vertices labeled by alternating strings), we define the convolution ... for arbitrary probability measures µ1 ⋯ µN on ℝ using a certain fixed-point equation for the Cauchy transforms. The convolution operations respect the operad structure of the tree operad from Jekel and Liu (2020). We prove a general limit theorem for iterated T -free convolution similar to Bercovici and Pata's results in the free case Bercovici and Pata (1999), and we deduce limit theorems for measures in the domain of attraction of each of the classical stable laws. [ABSTRACT FROM AUTHOR]

Let Δ be an equilateral triangle, we define the Cauchy transform of the normalized two-dimensional Lebesgue measure μ on Δ by F (z) = ∫ Δ d μ (w) z − w . We prove that F is univalent and starlike, but not convex in C ˆ ∖ Δ. In particular, for the regular triangle P 3 with vertexes { e π i / 3 , − 1 , e 5 π i / 3 } , the Cauchy transform is convex of order α = − 1 − 3 3 2 (3 log ⁡ 2 − π) (≈ 0.339) in | z | > 1. [ABSTRACT FROM AUTHOR]

Free regular convolution semigroups describe the distribution of free subordinators, while Bondesson class convolution semigroups correspond to classical subordinators with completely monotone Lévy density. We show that these two classes of convolution semigroups are in bijection with the class of complete Bernstein functions, and we establish an integral identity linking the two semigroups. We provide several explicit examples that illustrate this result. [ABSTRACT FROM AUTHOR]

We study the behaviour of singular integral operators T k t of convolution type on C associated with the parametric kernels k t (z) : = (Re z) 3 | z | 4 + t · Re z | z | 2 , t ∈ R , k ∞ (z) : = Re z | z | 2 ≡ Re 1 z , z ∈ C \ { 0 }. It is shown that for any positive locally finite Borel measure with linear growth the corresponding L 2 -norm of T k 0 controls the L 2 -norm of T k ∞ and thus of the Cauchy transform. As a corollary, we prove that the L 2 ( H 1 ⌊ E) -boundedness of T k t with a fixed t ∈ (- t 0 , 0) , where t 0 > 0 is an absolute constant, implies that E is rectifiable. This is so in spite of the fact that the usual curvature method fails to be applicable in this case. Moreover, as a corollary of our techniques, we provide an alternative and simpler proof of the bi-Lipschitz invariance of the L 2 -boundedness of the Cauchy transform, which is the key ingredient for the bilipschitz invariance of analytic capacity. [ABSTRACT FROM AUTHOR]

We show that a Toeplitz operator on the space of real analytic functions on the real line is left invertible if and only if it is an injective Fredholm operator, it is right invertible if and only if it is a surjective Fredholm operator. The characterizations are given in terms of the winding number of the symbol of the operator. Our results imply that the range of a Toeplitz operator (and also its adjoint) is complemented if and only if it is of finite codimension. Similarly, the kernel of a Toeplitz operator (and also its adjoint) is complemented if and only if it is of finite dimension. [ABSTRACT FROM AUTHOR]

We introduce and characterize the class of Toeplitz operators on the Fréchet space of all entire functions. We completely describe Fredholm, semi-Fredholm, invertible and one-sided invertible operators in this class. [ABSTRACT FROM AUTHOR]

In this paper, we study the weak asymptotic in the C -plane of some wave functions resulting from the WKB-techniques applied to a Schrödinger equation with quartic oscillator and having some boundary condition. As a first step, we make transformations of our problem to obtain a Heun equation satisfied by the polynomial part of the WKB wave functions. Especially, we investigate the properties of the Cauchy transform of the root counting measure of re-scaled solutions of the Schrodinger equation, to obtain a quadratic algebraic equation of the form C 2 (z) + r (z) C (z) + s (z) = 0 , where r, s are also polynomials. As a second step, we discuss the existence of solutions (in the form of Cauchy transform of a signed measure) of this algebraic equation. It remains to describe the critical graph of a related quadratic differential - p (z) d z 2 where p(z) is a quartic polynomial. In particular, we discuss the existence (and their number) of finite critical trajectories of this quadratic differential. [ABSTRACT FROM AUTHOR]

Refractive indices and optical transmittance spectra of (1-x)Pb(Zn1/3Nb2/3)O3-xPbTiO3 (PZN-xPT) single crystals poled along [001] direction have been investigated. Cauchy dispersion equations of the refractive indices were obtained by least square fitting, which can be used to calculate the refractive indices in the low absorption wavelength range. After poled along [001] direction, the transmittance of PZN-12%PT single crystal is more than 65% from 0.5 to 5.8 μm, which is much higher than that of PZN-5%PT and PZN-8%PT single crystals. Optical energy bandgap was obtained from absorption coefficient spectra. Our results show that as the PT content increases, the refractive indices of PZN-xPT single crystals increase, while the optical bandgap decreases. [ABSTRACT FROM AUTHOR]

The Cauchy-like relation M∞=A+BG∞ has recently been found to hold for the high frequency limit values of the longitudinal modulus M∞ and transverse modulus G∞ of viscoelastic liquids, with B≃3 in all the investigated systems. The Brillouin scattering results here reported for curing epoxy systems and thermal glass formers give evidence for the validity of a Cauchy-like relation M′=A+BG′ for the real part of the elastic moduli measured at finite frequencies. Our results suggest as well the validity of a pure Cauchy relation ΔM=3 ΔG for the relaxation strengths of longitudinal and shear moduli in relaxing liquids. [ABSTRACT FROM AUTHOR]

The authors report the results of temperature-dependent Brillouin scattering from both transverse and longitudinal acoustic waves, heat capacity studies as well as room temperature Raman scattering studies on M2O–MgO–WO3–P2O5 glasses (M=K,Rb). These results were used to obtain information about structure and various properties of the studied glasses such as fragility, elastic moduli, ratio of photoelastic constants, and elastic anharmonicity. They have found that both glasses have similar properties but replacement of K+ ions by Rb+ ions in the glass network leads to decrease of elastic parameters and P44 photoelastic constant due to increase of fragility. Based on Brillouin spectroscopy they show that a linear correlation between longitudinal and shear elastic moduli holds over a large temperature range. This result supports the literature data that the Cauchy-type relation represents a general rule for amorphous solids. An analysis of the Boson peak revealed that the form of the frequency distribution of the excess density of states is in agreement with the Euclidean random matrix theory. The reason of the observed shift of the maximum frequency of the Boson peak when K+ ions are substituted for Rb+ ions is also briefly discussed. [ABSTRACT FROM AUTHOR]

For K ⊂ C a compact subset and μ a positive finite Borel measure supported on K , let R ∞ (K , μ) be the weak-star closure in L ∞ (μ) of rational functions with poles off K. We show that if R ∞ (K , μ) has no non-trivial L ∞ summands and f ∈ R ∞ (K , μ) , then f is invertible in R ∞ (K , μ) if and only if Chaumat's map for K and μ applied to f is bounded away from zero on the envelope with respect to K and μ. The result proves the conjecture ⋄ posed by J. Dudziak [6] in 1984. [ABSTRACT FROM AUTHOR]

Exterior power operations on the higher K -groups of a quasi-compact scheme have recently been constructed by Taelman and the authors by purely algebraic means. In this paper, we prove two formulae that help to compute these operations. The first is a formula for exterior powers of external products. The second is a formula for exterior powers of n -cubes, i.e., of acyclic binary multi-complexes supported on [ 0 , 1 ] n. These formulae provide evidence for the expectation that our exterior power operations agree with those defined by Hiller. [ABSTRACT FROM AUTHOR]

We extend some results of M. G. Krein to the class of entire functions which can be represented as ratios of discrete Cauchy transforms in the plane. As an application we obtain new versions of de Branges' Ordering Theorem for nearly invariant subspaces in a class of Hilbert spaces of entire functions. Examples illustrating sharpness of the obtained results are given. We also discuss applications to spectral theory of rank one perturbations of normal operators. [ABSTRACT FROM AUTHOR]

Abstract In this paper we study the additive splitting associated to the quaternionic Cauchy transform defined by the Cauchy formula of slice hyperholomorphic functions. Moreover, we introduce and study the analogue of the fundamental solution of the global operator of slice hyperholomorphic functions. We state our results in the quaternionic setting but several results hold for Clifford algebra-valued function with minor changes in the proofs. [ABSTRACT FROM AUTHOR]

Abstract We prove local in time well-posedness for a large class of quasilinear Hamiltonian, or parity preserving, Schrödinger equations on the circle. After a paralinearization of the equation, we perform several paradifferential changes of coordinates in order to transform the system into a paradifferential one with symbols which, at the positive order, are constant and purely imaginary. This allows to obtain a priori energy estimates on the Sobolev norms of the solutions. [ABSTRACT FROM AUTHOR]

Carbon nanotubes are modeled as point configurations and investigated by minimizing configurational energies including two- and three-body interactions. Optimal configurations are identified with local minima and their fine geometry is fully characterized in terms of lower-dimensional problems. Under moderate tension, we prove the existence of periodic local minimizers, which indeed validates the so-called Cauchy-Born rule in this setting. [ABSTRACT FROM AUTHOR]

In this paper, the notion of L-contractions is introduced and a new fixed point theorem for such contractions is established. [ABSTRACT FROM AUTHOR]

An example is constructed of a purely unrectifiable measure |$\mu$| for which the singular integral associated to the kernel |$K(z) = \frac{\bar{z}}{z^2}$| is bounded in |$L^2(\mu)$|⁠. The singular integral fails to exist in the sense of principal value |$\mu$| -almost everywhere. This is in sharp contrast with the results known for the kernel |$\tfrac{1}{z}$| (the Cauchy transform). [ABSTRACT FROM AUTHOR]

Highlights • We consider inverse problems for the poloidal flux equation. • We present two fast computational algorithms. • A new method referred to as Proper Solution Space , is applied to the inverse evaluation. Abstract This note is concerned with two methods for the evaluation of the boundary of tore supra in a Tokamak. One method is based on optimization which seeks to minimize a cost functional. The other method is based on obtaining the poloidal flux within two concentric circles and interpolating the location of the boundary. Both methods are iterative, and consider Cauchy problems for the poloidal flux. The second method uses the method of proper solution space to generate the value of the poloidal flux within two concentric circles. It then interpolates for the location of the boundary. Both algorithms show good robustness to noise. A number of numerical examples are used to show the applicability of the methods. [ABSTRACT FROM AUTHOR]

In this paper, we calculated the impedance characteristics of a two-element antenna array with a chiral metamaterial substrate. A system of singular integral equations with Cauchy-type singularity with respect to the unknown longitudinal distributions of the surface current density is obtained. The input impedance dependences on the normalized arm length for various substrate types, as well as various values of the chirality parameter of the substrate, are presented. [ABSTRACT FROM AUTHOR]