Tube Domains Research Articles

The holomorphic discrete series contribution to the generalized Whittaker Plancherel formula

For a Hermitian Lie group G of tube type we find the contribution of the holomorphic discrete series to the Plancherel decomposition of the Whittaker space L2(G/N,ψ), where N is the unipotent radical of the Siegel parabolic subgroup and ψ is a certain non-degenerate unitary character on N. The holomorphic discrete series embeddings are constructed in terms of generalized Whittaker vectors for which we find explicit formulas in the bounded domain realization, the tube domain realization and the L2-model of the holomorphic discrete series. Although L2(G/N,ψ) does not have finite multiplicities in general, the holomorphic discrete series contribution does.Moreover, we obtain an explicit formula for the formal dimensions of the holomorphic discrete series embeddings, and we interpret the holomorphic discrete series contribution to L2(G/N,ψ) as boundary values of holomorphic functions on a domain Ξ in a complexification GC of G forming a Hardy type space H2(Ξ,ψ).

Faster than FFT: Conformal accelerations method

Fourier-Laplace transform technique allows one to represent several classes of important probability distributions and solutions of basic boundary problems for wide classes of fractional differential equations as integrals of functions enjoying two key properties: analytic continuation to a cone or the union of a cone and tube domain, and regular decay at infinity. Integral representations for Wiener-Hopf factors, fractional moments and special functions enjoy these properties as well. In the paper, we present the general methodology which allows one to evaluate the integrals enjoying these properties very fast and accurately. Among applications, we derive new efficient realizations of the Fourier, Laplace and Z-transforms, representations for probability distributions in Levy models, stable ones including, and algorithms for pricing contingent claims, Monte-Carlo simulations and evaluation of special functions.

A novel holographic framework preserving reflection positivity in dSd spacetime

This manuscript introduces a novel holographic correspondence in d-dimensional de Sitter (dSd) spacetime, connecting bulk dSd scalar unitary irreducible representations (UIRs) with their counterparts at the dSd boundary I±, all while preserving reflection positivity. The proposed approach, with potential applicability to diverse dSd UIRs, is rooted in the geometry of the complex dSd spacetime and leverages the inherent properties of the (global) dSd plane waves, as defined within their designated tube domains.

Geometry of Hermitian symmetric spaces under the action of a maximal unipotent group

Let [Formula: see text] be a non-compact irreducible Hermitian symmetric space of rank [Formula: see text] and let [Formula: see text] be an Iwasawa decomposition of [Formula: see text]. The group [Formula: see text] acts on [Formula: see text] by biholomorphisms and the real [Formula: see text]-dimensional submanifold [Formula: see text] intersects every [Formula: see text]-orbit transversally in a single point. Moreover [Formula: see text] is contained in a complex [Formula: see text]-dimensional submanifold of [Formula: see text] biholomorphic to [Formula: see text], the product of [Formula: see text] copies of the upper half-plane in [Formula: see text]. This fact leads to a one-to-one correspondence between [Formula: see text]-invariant domains in [Formula: see text] and tube domains in [Formula: see text]. In this setting we prove an analogue of Bochner’s tube theorem. Namely, an [Formula: see text]-invariant domain [Formula: see text] in [Formula: see text] is Stein if and only if the base [Formula: see text] of the associated tube domain is convex and “cone invariant”. We also prove the univalence of [Formula: see text]-invariant holomorphically separable Riemann domains over [Formula: see text]. This yields a precise description of the envelope of holomorphy of an arbitrary [Formula: see text]-invariant domain in [Formula: see text]. Finally, we obtain a characterization of several classes of [Formula: see text]-invariant plurisubharmonic functions on [Formula: see text] in terms of the corresponding classes of convex functions on [Formula: see text]. As an application we present an explicit Lie group theoretical description of all [Formula: see text]-invariant potentials of the Killing metric on [Formula: see text] and of the associated moment maps.

Affine deformations of quasi‐divisible convex cones

AbstractWe study subgroups of obtained by adding a translation part to the holonomy of a finite‐volume convex projective surface. Under a natural condition on the translations added to the peripheral parabolic elements, we show that the affine action of the group on has convex domains of discontinuity which are regular, generalizing a result of Mess for globally hyperbolic flat spacetimes. We then classify all such domains arising from a fixed group and show that the quotient of each of them is an affine manifold foliated by convex surfaces with constant affine Gaussian curvature (CAGC). The proof is based on the analysis of CAGC surfaces developed in a previous work, along with a correspondence between the geometry of an affine space endowed with a convex cone and the geometry of a convex tube domain. We also show that the moduli space of such groups is a vector bundle over the moduli space of finite‐volume convex projective structures, with rank equal to the dimension of the Teichmüller space.

Integral representations in weighted Bergman spaces on tube domains

ABSTRACT Herein, the Laplace transform representations for functions of weighted holomorphic Bergman spaces on the tube domains are developed. Then a weighted version of the edge-of-the-wedge theorem is derived as a byproduct of the main results.

Three Related Problems of Bergman Spaces of Tube Domains over Symmetric Cones

Three Related Problems of Bergman Spaces of Tube Domains over Symmetric Cones

Boundedness of Bergman projectors on homogeneous Siegel domains

In this paper we study the boundedness of Bergman projectors on weighted Bergman spaces on homogeneous Siegel domains of Type II. As it appeared to be a natural approach in the special case of tube domains over irreducible symmetric cones, we study such boundedness on the scale of mixed-norm weighted Lebesgue spaces. The sharp range for the boundedness of such operators is essentially known only in the case of tube domains over Lorentz cones. In this paper we prove that the boundedness of such Bergman projectors is equivalent to variuos notions of atomic decomposition, duality, and characterization of boundary values of the mixed-norm weighted Bergman spaces, extending results mostly known only in the case of tube domains over irreducible symmetric cones. Some of our results are new even in the latter simpler context. We also study the simpler, but still quite interesting, case of the “positive” Bergman projectors, the integral operator in which the Bergman kernel is replaced by its modulus. We provide a useful characterization which was previously known for tube domains.

A geometrical point of view for branching problems for holomorphic discrete series of conformal Lie groups

This paper is devoted to branching problems for holomorphic discrete series representations of a conformal group [Formula: see text] of a tube domain [Formula: see text] over a symmetric cone [Formula: see text] restricted to the conformal group [Formula: see text] of a tube domain [Formula: see text] holomorphically embedded in [Formula: see text]. The goal of this work is the explicit construction of the symmetry breaking and holographic operators in this geometrical setting. We answer this program with the introduction of another functional model for holomorphic discrete series representations. This model leads to a geometrical interpretation and a close relation to the theory of orthogonal polynomials for such branching problems. This program is illustrated by three cases. First, we consider the [Formula: see text]-fold tensor product of holomorphic discrete series of the universal covering of [Formula: see text]. Then, it is tested on the restrictions of a member of the scalar-valued holomorphic discrete series of the conformal group [Formula: see text] to the subgroup [Formula: see text], and finally to the subgroup [Formula: see text].

Three-dimensional simulations of reshocked inclined Richtmyer-Meshkov instability: Effects of initial perturbations

A Mach 1.55 shock and subsequent reshock interacting with predominantly single-mode and multi-mode interfaces between nitrogen and carbon dioxide is studied through three-dimensional adaptive mesh refinement simulations using the FLASH code. The simulations are directly compared with the PLIF/PIV experimental measurements to validate the computational code. Development of instability after first shock and chaotic turbulent mixing after reshock are discussed for the entire shock tube domain.

Numerical Analysis of Flow-Induced Vibration of Heat Exchanger Tube Bundles Based on Fluid-Structure Coupling Dynamics

According to the needs generated by the industry, there is an urgent need to investigate the flow vibration response law of the elastic tube bundle of heat exchangers subjected to the coupling effect of shell and tube domain at different inlet flow velocities. In this paper, based on the continuity equation, momentum equation, turbulence model, and dynamic grid technology, the vibration response of the elastic tube bundle under the mutual induction of shell and tube fluids is studied by using pressure-velocity solver and two-way fluid-structure coupling (FSC) method. The results show that the monitoring points on the same connection block have the same vibration frequency, while the vibration amplitude is different. Monitoring point A is the most deformed by the impact of fluid in the shell and tube domain. When the inlet velocity( v shell = v tube = 0.15 , 0.4, 0.5 m/s) of shell and tube is low, the amplitude of tube bundle vibration in the Y direction is greater than that in X and Z direction, and the tube bundle produces periodic vibration in the vertical direction. The vibration equilibrium position of the tube bundle along the shell flow direction gradually moves up with the increase of the inlet velocity. The amplitude in the Y -direction of the elastic bundle decreases with the increase of shell-side and tube-side velocity. The relationship between the vibration amplitude in the Y direction and the entrance velocity is a linear function.

Curvature positivity of invariant direct images of Hermitian vector bundles

We prove that the invariant part, with respect to a compact group action satisfying certain condition, of the direct image of a Nakano positive Hermitian holomorphic vector bundle over a bounded pseudoconvex domain is Nakano positive. We also consider the action of the noncompact group \({\mathbb {R}}^m\) and get the same result for a family of tube domains, which leads to a new method to the matrix-valued Prekopa’s theorem originally proved by Raufi.

A Novel Guide Vane System Design to Mitigate Rotating Vortex Rope in High Head Francis Model Turbine

Guide vanes are a mechanical system used to direct flow in the desired direction. At lower operating conditions, implementing a guide vane system in the draft tube of a hydro-turbine can decrease the excess swirl in the flow and, thus, reduce pressure fluctuations. The present study discusses a numerical methodology to design an effective guide vane system in the draft tube of a high head Francis model turbine. The numerical method is computationally efficient and thus, saves excess computational time and data storage required for parametric analysis of the guide vane system. The aim is to mitigate the ‘rotating’ vortex rope with minimum additional losses in the turbine. The factors considered for the guide vane system design are a) number of guide vanes, b) chord length, c) span, d) inlet-outlet angles of the guide vanes, and e) their position in the draft tube. The parametric study comprises a) ideal guide vane design and b) realistic guide vane design study. The ideal guide vane design study was with the standalone draft tube domain. The realistic guide vane design study used the passage domains of the distributor, runner, and complete draft tube. From the ideal guide vane design study, a guide vane system with two or three guide vanes of chord 86% of runner radius and leading-edge span of 30% of runner radius effectively mitigates the rotating vortex rope. The system with three guide vanes is the most efficient when placed at some distance below the runner exit with mitigation above 95%

Complex Geodesics in Tube Domains and Their Role in the Study of Harmonic Mappings in the Disc

We continue the research on the structure of complex geodesics in tube domains over (bounded) convex bases. In some special cases a more explicit form of the geodesics than the existing ones are provided. As one of the consequences of our study an effective formula for the Kobayashi—Royden metric in the tube domain \({T_{{\mathbb{B}_n}}}\) at the origin is given. The results on the Kobayashi—Royden metric in a natural way provide versions of the Schwarz Lemma for harmonic mappings. We also present a result on harmonic mappings defined on the disc that may be seen as a generalisation of the Radó—Kneser—Choquet Theorem for a class of harmonic bivalent mappings that lets understand better the geometry of complex geodesics in tube domains.

Asymptotic total geodesy of local holomorphic curves exiting a bounded symmetric domain and applications to a uniformization problem for algebraic subsets

The current article stems from our study on the asymptotic behavior of holomorphic isometric embeddings of the Poincaré disk into bounded symmetric domains. As a first result we prove that any holomorphic curve exiting the boundary of a bounded symmetric domain $\Omega$ must necessarily be asymptotically totally geodesic. Assuming otherwise we derive by the method of rescaling a hypothetical holomorphic isometric embedding of the Poincaré disk with $\operatorname{Aut}(\Omega^\prime)$-equivalent tangent spaces into a tube domain $\Omega^\prime \subset \Omega$ and derive a contradiction by means of the Poincaré–Lelong equation. We deduce that equivariant holomorphic embeddings between bounded symmetric domains must be totally geodesic. Furthermore, we solve a uniformization problem on algebraic subsets $Z \subset \Omega$. More precisely, if $\check{\Gamma} \subset \operatorname{Aut} (\Omega)$ is a torsion-free discrete subgroup leaving $Z$ invariant such that $Z / \check{\Gamma}$ is compact, we prove that $Z \subset \Omega$ is totally geodesic. In particular, letting $\Gamma \subset \operatorname{Aut} (\Omega)$ be a torsion-free cocompact lattice, and $\pi : \Omega \to \Omega / \Gamma =: X_\Gamma$ be the uniformization map, a subvariety $Y \subset X_\Gamma$ must be totally geodesic whenever some (and hence any) irreducible component $Z$ of $\pi-(Y)$ is an algebraic subset of $\Omega$. For cocompact lattices this yields a characterization of totally geodesic subsets of $X_\Gamma$ by means of bi-algebraicity without recourse to the celebrated monodromy result of André–Deligne on subvarieties of Shimura varieties, and as such our proof applies to not necessarily arithmetic cocompact lattices. In place of the monodromy result of André–Deligne we exploit the existence theorem of Aubin and Yau on Kähler–Einstein metrics for projective manifolds $Y$ satisfying $c_1 (Y) \lt 0$ and make use of Nadel’s semisimplicity theorem on automorphism groups of noncompact Galois covers of such manifolds, together with the total geodesy of equivariant holomorphic isometric embeddings between bounded symmetric domains.

Optimal $$L^2$$-Extensions on Tube Domains and a Simple Proof of Prékopa’s Theorem

We prove the optimal $$L^2$$ -extension theorem of Ohsawa–Takegoshi type on a tube domain. As an application, we give a simple proof of Prékopa’s theorem.

Standard Subspaces of Hilbert Spaces of Holomorphic Functions on Tube Domains

In this article we study standard subspaces of Hilbert spaces of vector-valued holomorphic functions on tube domains E + i C^0, where C \subeq E is a pointed generating cone invariant under e^{R h} for some endomorphism h \in \End(E), diagonalizable with the eigenvalues 1,0,-1 (generalizing a Lorentz boost). This data specifies a wedge domain W(E,C,h) \subeq E and one of our main results exhibits corresponding standard subspaces as being generated using test functions on these domains. We also investigate aspects of reflection positivity for the triple (E,C,e^{\pi i h}) and the support properties of distributions on E, arising as Fourier transforms of operator-valued measures defining the Hilbert spaces H. For the imaginary part of these distributions, we find similarities to the well known Huygens' principle, relating to wedge duality in the Minkowski context. Interesting examples are the Riesz distributions associated to euclidean Jordan algebras.

The envelope of holomorphy of a classical truncated tube domain

We present the envelope of holomorphy of a classical truncated tube domain.

Reduced Numerical Modeling of a High Head Francis Turbine Draft Tube at Part Load

In the present study, a reduced model of the Francis-99 model turbine was investigated numerically at part load operating condition. The reduced model consists of a standalone draft tube domain of the Francis-99 model turbine. Numerical studies performed in the past on nearly complete hydro-turbine models (inclusive of the spiral casing, distributor domains, runner, and draft tube) reportedly consist of a large number of computational grids. This may increase the computational costs and data storage required to perform numerical analysis, which could be a setback for future research on new design concepts and optimization study of the draft tube domain. The reduced model was developed by mapping the phase averaged axial, radial, and tangential velocity profiles from the runner exit to the inlet of the standalone draft tube domain. Additionally, turbulent kinetic energy (k) and turbulent eddy dissipation (ε) variables were also considered for better flow prediction inside the draft tube domain. Two methods for mapping inlet boundary conditions were considered in the present study. In the first method, the entire planar profile of the runner-draft tube interface was considered. In the second method, the variables along a radial profile at the runner exit were considered with an axis-symmetric flow assumption over the entire draft tube inlet plane. The numerical results obtained from the Francis-99 reduced model turbine were validated against the numerical model of the NVKS Francis-99 model turbine (with available structured mesh) that was also analysed using the passage flow numerical technique and available experimental results. The results were found to be in reasonable agreement, with each other. The present study could be useful for the future mitigation study of rotating vortex rope by modifying the draft tube domain.

Hilbert-type inequalities in homogeneous cones

We prove L^p-L^q bounds for the class of Hilbert-type operators associated with generalized powers Q^a in a homogeneous cone \Omega . Our results extend and slightly improve earlier work from [16], where the problem was considered for scalar powers \mathbf a = (\alpha, \dots, \alpha) and symmetric cones. We give a more transparent proof, provide new examples, and briefly discuss the open question regarding characterization of L^p boundedness for the case of vector indices \mathbf a . Some applications are given to boundedness of Bergman projections in the tube domain over \Omega .