Harmonic Forms Research Articles

Devising an analytical method for solving the eighth-order Kolmogorov equations for an asymmetric Markov chain

The object of research is a complex system of three subsystems, which function independently of each other and are in a working or failed state. There is a need to analytically model and manage the Markov random process in the system, varying the intensity of their development-restoration and degradation-destruction flows. In the study, an analytical method for solving Kolmogorov equations of the eighth order for an asymmetric Markov chain was devised. The corresponding Kolmogorov equations of the eighth order have an ordered transition probability matrix. The distribution of the eight roots of this equation in the complex plane has central symmetry. The results are analytical solutions for the probabilities of the eight states of the Markov chain in time in the form of ordered determinants with respect to the indices of the eight roots and the indices of the eight states, including the column vector of the initial conditions. Symmetry has been established in the distribution on the complex plane of eight real, negative roots of the characteristic Kolmogorov equation centered at the point defined as Re ϑ = –a7/8, where a7 is the coefficient of the characteristic equation of the eighth degree at the seventh power. Formulas expressing eight roots of the characteristic Kolmogorov equation have been heuristically derived, one of which is zero, due to the intensities of failures and recovery of three subsystems, the eight states of which in general make up an asymmetric Markov chain. For structures consisting of three independently functioning processes, the random process of the transition of the structure through eight possible states with a known initial state is determined in time. An analytical solution to Kolmogorov differential equations of the eighth order for an asymmetric state graph is proposed in harmonic form for the purpose of analysis and synthesis of a random Markov process in a triple system.

High-Efficiency Forward Modeling of Gravitational Fields in Spherical Harmonic Domain with Application to Lunar Topography Correction

Gravity forward modeling as a basic tool has been widely used for topography correction and 3D density inversion. The source region is usually discretized into tesseroids (i.e., spherical prisms) to consider the influence of the curvature of planets in global or large-scale problems. Traditional gravity forward modeling methods in spherical coordinates, including the Taylor expansion and Gaussian–Legendre quadrature, are all based on spatial domains, which mostly have low computational efficiency. This study proposes a high-efficiency forward modeling method of gravitational fields in the spherical harmonic domain, in which the gravity anomalies and gradient tensors can be expressed as spherical harmonic synthesis forms of spherical harmonic coefficients of 3D density distribution. A homogeneous spherical shell model is used to test its effectiveness compared with traditional spatial domain methods. It demonstrates that the computational efficiency of the proposed spherical harmonic domain method is improved by four orders of magnitude with a similar level of computational accuracy compared with the optimized 3D GLQ method. The test also shows that the computational time of the proposed method is not affected by the observation height. Finally, the proposed forward method is applied to the topography correction of the Moon. The results show that the gravity response of the topography obtained with our method is close to that of the optimized 3D GLQ method and is also consistent with previous results.

Generalisation of the Asai-Kaneko-Ninomiya identity to higher level

Let j be the classical modular invariant, which is a modular function of weight zero for SL2(Z). Then for every z,τ in the upper half-plane, the Asai-Kaneko-Ninomiya (AKN) identity relates j(z)−j(τ) to the generating function of the Hecke system jm(τ) of modular functions for SL2(Z). In [7] the modular functions jm(z) were generalised to modular functions JN,m for higher level using harmonic weak Maass forms. In this paper, we show that under certain conditions on JN,m, the AKN identity can be generalised to higher level Γ0(N) using Poincaré series of weight 2 and level N. We also prove an infinite product formula for JN,1 which generalises the denominator formula for the Monster Lie algebra.

Purely leptonic decays of heavy-flavored charged mesons

We study the purely leptonic decays of heavy-flavored charged pseudoscalar (P) and vector (V) mesons (D(s)(*)+, B(c)(*)+) in the relativistic independent quark (RIQ) model based on an average flavor-independent confining potential in an equally mixed scalar-vector harmonic form. We first compute the mass spectra of the ground-state-mesons and fix the model parameters necessary for the present analysis. Using the meson wave functions derivable in the RIQ model, and model parameters so fixed from hadron spectroscopy, we predict the decay constants: fP(V), ratios of decay constants: fV/fP, fP1/fP2, fV1/fV2, and the branching fractions (BFs): B(P(V)→l+νl), l=e, μ, τ, which agree with the available experimental data and other Standard Model (SM) predictions. For the unmeasured decay constants, especially in the purely leptonic decays of the charged vector mesons, our predictions could be tested in the upcoming Belle-II, SCTF, CEPC, FCC-ee, and LHCb experiments in the near future. Published by the American Physical Society 2024

3D electro-elastic static analysis of advanced plates and shells

The proposed three-dimensional (3D) coupled electro-elastic shell model allows the static analysis of smart structures embedding classical piezoelectric and functionally graded piezoelectric layers. Plates, cylindrical and spherical panels are investigated in both sensor and actuator configurations. The primary variables of the coupled model are displacement components and the electric potential. Therefore, displacements, stresses, strains, electric potential and electric displacements are calculated for smart structures used as sensors (applied mechanical load) and smart structures used as actuators (applied electric potential). In the case of spherical shells, 3D equilibrium equations are coupled with the 3D divergence electric displacement equation. The obtained coupled system is also valid for cylindrical shells and plates thanks to the use of a mixed curvilinear orthogonal reference system. The partial differential governing equations are solved using the Navier harmonic form and the exponential matrix method for simply supported structures. Preliminary assessments are proposed to validate the model and further benchmarks are analyzed to discuss the effects connected with the thickness ratio, the lamination scheme, the load conditions, the geometry of the structures and the employed materials. The main innovation point of the proposed model is the possibility to analyze several geometries including different materials and applied loads by means of a unique and general formulation where partial differential equations are solved in closed form. The use of functionally graded piezoelectric materials allows to eliminate all stress component discontinuities at each layer interface.

Three-dimensional vibration analysis of multilayered composite and functionally graded piezoelectric plates and shells

In the present paper, a coupled 3D exact electro-elastic shell model for Functionally Graded (FG) and composite piezoelectric structures is proposed. Primary variables of the electro-elastic model are the electric potential and the three displacement components. The model allows to evaluate the piezoelectric effect in terms of frequencies and vibration modes. Both closed and open circuit configurations are analyzed and compared. The 3D equilibrium equations and the 3D divergence electric displacement equation for spherical shells give the set of partial differential equations for the electro-elastic problem. The proposed model for spherical shells automatically degenerates into simpler models for plates and cylindrical shells via properly considerations on radii of curvature along in-plane directions. The orthogonal mixed curvilinear coordinates α, β and z are employed. The partial differential governing equations have constant coefficients considering fictitious layers and they are solved using the Navier harmonic form and the exponential matrix method. These features lead to an exact solution for simply-supported boundary conditions. Free vibration analyses are conducted and circular frequencies for the first three thickness vibration modes are computed. After a global assessment phase to verify the correctness of the developed model, new benchmarks are proposed: different thickness ratios and material configurations are investigated. The present work can be intended as a reference general solution for those scientists interested in the study of piezoelectric structures via 2D analytical and numerical formulations.

Harmonic weak Maass forms and periods II

Harmonic weak Maass forms and periods II

Dark patterns: EU's regulatory efforts

AbstractIn a world where technology is rapidly advancing, regulation of dark pattern practices has become a topic of increasing importance. Society has become somewhat desensitized to these deceptive online practices that manipulate users into taking actions, which are not in their best interests, such as difficulty unsubscribing from a service, prominence of consent buttons, and countless other advanced tactics to obscure transparency. However, these ongoing practices harm both the individual user, and society in general, by impeding informed decision‐making. This Article addresses the European Union's leading efforts to tackle dark pattern practices, and in particular, addresses the numerous legislative acts which have been enacted to regulate and eliminate them. The acts explored in this Article include the General Data Protection Regulation, the Uniform Commercial Practices Directive, the Data Act, the Digital Markets Act, the Digital Services Act, the Amendment to the Directive on Financial Services Contracts Concluded at a Distance, and the Artificial Intelligence Act. This Article then discusses the interplay between the numerous acts, and the resulting ambiguities and overlap which have led to a level of regulatory redundancy. This Article examines not only the difficulty in interpretation of the various acts, but additionally, explores the issues which arise in implementation from a jurisdictional perspective. Further, this Article suggests potential solutions to address the fragmented legislation, including a hybrid form of harmonization, as well as methods for consolidation and centralization.

Rotorcraft Flight Control Design for Rotor Noise Abatement

This article presents the development and implementation of active noise abatement flight control laws utilizing redundant control allocation in rotorcraft. The study starts by finding a periodic equilibrium of coupled rotorcraft flight dynamics and acoustics using a modified harmonic balance trim solution method. Next, the rotorcraft nonlinear time-periodic dynamics are linearized about its periodic equilibrium and transformed into an equivalent higher-order linear time-invariant system in harmonic decomposition form. Composite aeroacoustic measures are formulated as outputs of this system. To enable faster runtime and tractable control design, a reduced 8‐state model representative of the coupled rigid‐body dynamics and acoustics is obtained through residualization. An explicit model following control law with pseudoinverse allocation to redundant control surfaces–an active horizontal stabilizer in this study–is synthesized using the 8‐state model. The article further assesses the impact of noise alleviation control on handling qualities of conventional and compound rotorcraft in forward flight. The results demonstrate that redundant control allocation is an effective method for active reduction of unsteady rotor noise, with increasing effectiveness during aggressive maneuvers. Future-generation rotorcraft, particularly future vertical lift configurations featuring high control redundancy and aggressive maneuvering capabilities, can significantly benefit from noise abatement through redundant control allocation. Furthermore, the article highlights that while implementing noise abatement control in conventional helicopters can degrade handling qualities, the use of redundant control allocation in compound configurations offers noise reduction without compromising handling qualities.

The book as a genre phenomenon in modern prose of the Russian abroad (“Venice. Quarantine Chronicles” by E. Margolis)

The article studies the book as a special type of artistic unity, representing an original model for expressing artistic consciousness in modern Russian literature abroad. The article proves the productivity of this genre phenomenon, identifies the main trends in book creation associated with an appeal to the sacral text, with the actualization of the autobiographical, diary and memoir prose traditions. The representative sample is analyzed on the basis of the modern literary scholarship provisions, which conceptualize the book as an original genre phenomenon. Using the example of E. Margolis’ book “Venice. Quarantine Chronicles” (2020), we analyze the mechanisms of constructing the artistic world, organically embodied in book format. The article reveals the nature of the artistic integrity of the book, conditioned by the interaction of verbal and visual components, the motif complex associated with the Christian tradition (which includes motifs of insight, internal movement, silence and healing of the soul), and reveals the features of the palimpsest, which serves as the principle of genre modeling. The conceptual and formal anchor in the book is the image of Venice, which connects everyday and metaphysical, visible and invisible aspects. The article proves that the book represents a form of the world harmonization and a space for a dialogue between different cultural codes at whose junction the author’s hybrid identity is formed.

Left-invariant almost complex structures on the higher dimensional Kodaira–Thurston manifolds

We develop computational techniques which allow us to calculate the Kodaira dimension as well as the dimension of spaces of Dolbeault harmonic forms for left-invariant almost complex structures on the generalised Kodaira–Thurston manifolds.

Applications of Differential Geometry Linking Topological Bifurcations to Chaotic Flow Fields

At every point p on a smooth n-manifold M there exist n+1 skew-symmetric tensor spaces spanning differential r-forms ω with r=0,1,⋯,n. Because d∘d is always zero where d is the exterior differential, it follows that every exact r-form (i.e., ω=dλ where λ is an r−1-form) is closed (i.e., dω=0) but not every closed r-form is exact. This implies the existence of a third type of differential r-form that is closed but not exact. Such forms are called harmonic forms. Every smooth n-manifold has an underlying topological structure. Many different possible topological structures exist. What distinguishes one topological structure from another is the number of holes of various dimensions it possesses. De Rham’s theory of differential forms relates the presence of r-dimensional holes in the underlying topology of a smooth n-manifold M to the presence of harmonic r-form fields on the smooth manifold. A large amount of theory is required to understand de Rham’s theorem. In this paper we summarize the differential geometry that links holes in the underlying topology of a smooth manifold with harmonic fields on the manifold. We explore the application of de Rham’s theory to (i) visual, (ii) mechanical, (iii) electrical and (iv) fluid flow systems. In particular, we consider harmonic flow fields in the intracellular aqueous solution of biological cells and we propose, on mathematical grounds, a possible role of harmonic flow fields in the folding of protein polypeptide chains.

Harmonic Maass forms associated with CM newforms

Abstract In this paper, we use a regularized theta lifting to construct harmonic Maass forms corresponding to binary theta functions of weight k ≥ 2 k\geq 2 under the 𝜉-operator. As a result, we show that their holomorphic parts have algebraic Fourier coefficients, with compatible Galois action. As an application, we prove rationality properties of coefficients of harmonic Maass forms corresponding to CM newforms, answering a question of Bruinier, Ono and Rhoades.

Weighted p-basic harmonic forms and its applications

In this paper, we study the weighted p-basic harmonic forms on a weighted Riemannian foliation (M,g,F,e−fν) for some basic function f. At the same time, we prove that there is no non-trivial Lfp-weighted p-basic harmonic form under some assumptions about the generalized weighted curvature. Finally, we consider the Liouville type theorem for (F,F′)(p,f)-harmonic map between (M,g,F,e−fν) and (M′,g′,F′) as applications.

Primitive decompositions of Dolbeault harmonic forms on compact almost-Kähler manifolds

Primitive decompositions of Dolbeault harmonic forms on compact almost-Kähler manifolds

On the spaces of $(d + d^{c})$-harmonic forms and $(d + d^{\Lambda})$-harmonic forms on almost Hermitian manifolds and complex surfaces

In this paper, we study the spaces of (d+d^{c}) -harmonic forms and of (d+d^{\Lambda}) -harmonic forms, a natural generalization of the spaces of Bott–Chern harmonic forms (respectively, symplectic harmonic forms) from complex (respectively, symplectic) manifolds to almost Hermitian manifolds. We apply the same techniques to compact complex surfaces, computing their Bott–Chern and Aeppli numbers and their spaces of (d+d^{\Lambda}) -harmonic forms. We give several applications to compact quotients of Lie groups by a lattice.

JURISDICTION OF TRANSNATIONAL ADMINISTRATIVE-LEGAL ACTS IN THE PROCESS OF EUROPEAN INTEGRATION

Transnational administrative-legal acts exhibit the feature of exerting influence not only within the jurisdiction of the originating state but also across the borders, encompassing the territories of other states. Such acts are one form of harmonization and "integration" of activities within the European Union. The European integration process entails progressively aligning and synchronizing laws, policies, and regulations across European Union member states to establish a unified market and foster deeper political and economic collaboration. This progression involved the delegation of specific governmental authorities from national administrations to EU institutions. Consequently, transnational administrative-legal acts play a crucial role in promoting and regulating this integration. European Union law exerts a varied impact on the collaboration and organization of administrative entities, along with its influence on the judiciary. In part, this includes the obligation to recognize administrative bodies or judicial decisions of other states.However, there are few areas in which the enforcement of EU law is fully regulated by national law. There are frequent cases where the enforcement standards of the member states are applied to the law of the European Union, which, in some cases, damages the idea of a unified application of the law of the European Union. Transnational administrative-legal acts often require cooperation and coordination among member states in order to effectively address existing challenges. The European integration process promotes this cooperation through various mechanisms. In this way, it is important that when applying the national legal norms, the practical realization of the objective of the European Union norm is not hindered and, at the same time, the national interests of the candidate country are not harmed. This concerns the application of both substantive and procedural law norms. The present article discusses the issue of transnational administrative-legal acts in the process of European integration; The typology of transnational administrative-legal acts is presented, the precedents of the European Court of Justice and their influence on the principle of territoriality and the right to effective justice are analyzed. Keywords: Transnational administrative-legal act, Jurisdiction

HKT Manifolds: Hodge Theory, Formality and Balanced Metrics

ABSTRACT Let $(M,I,J,K,\Omega)$ be a compact HKT manifold, and let us denote with $\partial$ the conjugate Dolbeault operator with respect to I, $\partial_J:=J^{-1}\overline\partial J$, $\partial^\Lambda:=[\partial,\Lambda]$, where Λ is the adjoint of $L:=\Omega\wedge-$. Under suitable assumptions, we study Hodge theory for the complexes $(A^{\bullet,0},\partial,\partial_J)$ and $(A^{\bullet,0},\partial,\partial^\Lambda)$ showing a similar behavior to Kähler manifolds. In particular, several relations among the Laplacians, the spaces of harmonic forms and the associated cohomology groups, together with Hard Lefschetz properties, are proved. Moreover, we show that for a compact HKT $\mathrm{SL}(n,\mathbb{H})$-manifold, the differential graded algebra $(A^{\bullet,0},\partial)$ is formal and this will lead to an obstruction for the existence of an HKT $\mathrm{SL}(n,\mathbb{H})$ structure $(I,J,K,\Omega)$ on a compact complex manifold (M, I). Finally, balanced HKT structures on solvmanifolds are studied.

𝐿-values of harmonic Maass forms

Bruinier, Funke, and Imamoglu have proved a formula for what can philosophically be called the “central L L -value” of the modular j j -invariant. Previously, this had been heuristically suggested by Zagier. Here, we interpret this “ L L -value” as the value of an actual L L -series, and extend it to all integral arguments and to a large class of harmonic Maass forms which includes all weakly holomorphic cusp forms. The context and relation to previously defined L L -series for weakly holomorphic and harmonic Maass forms are discussed. These formulas suggest possible arithmetic or geometric meaning of L L -values in these situations. The key ingredient of the proof is to apply a recent theory of Lee, Raji, and the authors to describe harmonic Maass L L -functions using test functions.

Linearized Models of the Coupled Rotorcraft Flight Dynamics and Acoustics for Real-Time Noise Prediction

This article demonstrates the linearization of the coupled rotorcraft flight dynamics and aeroacoustics to provide realtime acoustic predictions in generalized maneuvering flight. To demonstrate the methodology, the study makes use of a nonlinear simulation model of a generic utility helicopter (PSUHeloSim) that is coupled with an aeroacoustic solver based on a marching cubes algorithm. A periodic equilibrium of the coupled rotorcraft flight dynamics and acoustics is first found at a desired flight condition using a modified harmonic balance trim solution method. Next, the nonlinear time-periodic dynamics are linearized about that periodic equilibrium and transformed into an equivalent higher order linear time-invariant system in harmonic decomposition form. Composite aeroacoustic measures are included as an output of this system. To speed up runtime and make control design tractable, the order of these harmonic decomposition models is reduced via residualization to an 8-state model where the states are representative of the rigid-body dynamics of the aircraft. This 8-state model is shown to provide accurate acoustic response predictions for small-amplitude pilot inputs and to abate runtime by a factor of approximately 104, thus enabling acoustic predictions in generalized maneuvering flight that are significantly faster than real time. The 8-state model is subsequently used to demonstrate the use of linear system tools for the dynamic analysis of the coupled rotorcraft flight dynamics and acoustics.