Inverse Theorem Research Articles

A class of regularized adaptive multichannel equalization algorithms for speech dereverberation

The multiple input/output inverse theorem (MINT) is a typical speech dereverberation technique based on acoustic channel equalization. Although the adaptive MINT (A-MINT) method has been developed to meet the requirements of real-time application systems, its performance often deteriorates due to the effect of the identification errors of room impulse responses (RIRs). In this paper, we present a regularized A-MINT approach to reduce the sensitivity of the adaptive equalizer to the RIR identification errors. According to the regularized MINT criterion, Newton's and quasi-Newton methods are used to establish two adaptive speech dereverberation algorithms, respectively, which obtain faster convergent rate and better dereverberation performance. The effectiveness of the proposed algorithms is validated through numerical experiments with the RIRs measured in real acoustic environments.

Contramodules for algebraic groups: Induction and projective covers

In this paper we investigate contramodules for algebraic groups. Namely, we give contramodule analogs to two 20th century results about comodules. Firstly, we show that induction of contramodules over coordinate rings of algebraic groups is exact if and only if the associated quotient variety is affine. Secondly, we give an inverse limit theorem for constructing projective covers of simple G-modules using G-structures of projective covers of simple modules of the first Frobenius kernel, G1. We conclude by showing that the inverse limit theorem is a special case of a more general phenomenon between injective covers in k[G]-Comod and projective covers in k[G]-Contra.

Quantum data compression under localized features

In this paper, we first introduce the local quantum information processing task by compressing the density operator based on local quantum Bernoulli noises. Then, the local quantum compression theorem is given, that is, the local quantum entropy is the minimum achievable rate of local compression. Finally, we prove the theorem with the proof of the direct coding theorem and the inverse theorem. The direct coding theorem shows that a scheme with such a local compression rate exists, and that this local compression rate converges to the local quantum entropy. The inverse theorem shows that the compression scheme with the rate below the local entropy is unachievable. With the continuous development of quantum technology, local quantum data compression technology will have broad application prospects and development space.

The Dirac-Dolbeault Operator Approach to the Hodge Conjecture

The Dirac-Dolbeault operator for a compact Kähler manifold is a special case of Dirac operator. The Green function for the Dirac Laplacian over a Riemannian manifold with boundary allows the expression of the values of the sections of the Dirac bundle in terms of the values on the boundary, extending the mean value theorem of harmonic analysis. Utilizing this representation and the Nash–Moser generalized inverse function theorem, we prove the existence of complex submanifolds of a complex projective manifold satisfying globally a certain partial differential equation under a certain injectivity assumption. Thereby, internal symmetries of Dolbeault and rational Hodge cohomologies play a crucial role. Next, we show the existence of complex submanifolds whose fundamental classes span the rational Hodge classes, proving the Hodge conjecture for complex projective manifolds.

Inverse spectral Love problem via Weyl–Titchmarsh function

In this paper, we prove an inverse resonance theorem for the half-solid with vanishing stresses on the surface via Weyl–Titchmarsh function. Using a semi-classical approach, it is possible to simplify this three-dimensional problem of the elastic wave equation for the half-solid as a Schrödinger equation with Robin boundary conditions on the half-line. The goal of the paper is to establish a method to recover the potential from eigenvalues and resonances through the Weyl–Titchmarsh function for non-self-adjoint problems and to establish a one-to-one and onto map between suitable function spaces. Moreover, we produce an algorithm in order to retrieve the shear modulus from the eigenvalues and resonances, via the spectral data.

Inverse Theorem for Approximation on Subsets of a Domain with Cusps

Inverse Theorem for Approximation on Subsets of a Domain with Cusps

Direct and inverse approximation theorems for functions defined in Damek–Ricci spaces

UDC 517.5 We introduce the notion of k th modulus of smoothness and establish the direct and inverse theorems in terms of the quantities E s ( f ) and the moduli of smoothness generated by the spherical mean operator defined on the L 2 -space for the Damek–Ricci spaces. These theorems are analogous to the well-known theorems of Jackson and Bernstein. We also consider some problems related to the constructive characteristics of functional classes defined by the majorants of the moduli of smoothness of their elements.

Null controllability of a volume-surface reaction-diffusion equation with dynamic boundary conditions

This article focuses on investigating the null controllability of a volume-surface reaction-diffusion equation with dynamic boundary conditions. Notably, the reaction and diffusion coefficients depend on the state variable in the bulk and on its surface. Our main result establishes the local null controllability of the quasi-linear equation under certain conditions on the regularity of diffusion and reaction coefficients as well as initial data. To this end, we approach the problem by first addressing the question of null controllability in the framework of a inhomogeneous linearized equation. Next, we derive new estimates of both control and state, allowing us to apply a local inversion theorem to obtain the local null controllability of the quasi-linear equation.

Hexadecapole at the heart of nonlinear electromagnetic fields

In classical Maxwell’s electromagnetism, the monopole term of the electric field is proportional to r −2, while higher order multipole terms, sourced by anisotropic sources, fall-off faster. However, in nonlinear electromagnetism even a spherically symmetric field has multipole-like contributions. We prove that the leading subdominant term of the electric field, defined by nonlinear electromagnetic Lagrangian obeying Maxwellian weak field limit, in a static, spherically symmetric, asymptotically flat spacetime, is of the order O(r−6) as r→∞ . Moreover, using Lagrange inversion theorem and Faà di Bruno’s formula, we derive the series expansion of the electric field from the Taylor series of an analytic electromagnetic Lagrangian.

A Lie-bracket-based notion of stabilizing feedback in optimal control

For a control system two major objectives can be considered: the stabilizability with respect to a given target, and the minimization of an integral functional (while the trajectories reach this target). Here we consider a problem where stabilizability or controllability are investigated together with the further aim of a ‘cost regulation’, namely a state-dependent upper bound of the functional. This paper is devoted to a crucial step in the programme of establishing a chain of equivalences among degree-k stabilizability with regulated cost, asymptotic controllability with regulated cost and the existence of a degree-k Minimum Restraint Function (which is a special kind of Control Lyapunov Function). Besides the presence of a cost we allow the stabilizing ‘feedback’ to give rise to directions that range in the union of original directions and the family of iterated Lie bracket of length ≤ k . In the main result asymptotic controllability [resp. with regulated cost] is proved to be necessary for degree-k stabilizability [resp. with regulated cost]. Further steps of the above-mentioned logical chain as, for instance, a Lyapunov-type inverse theorem – i.e. the possibility of deriving existence of a Minimum Restraint Function from stabilizability – are proved in companion papers.

Herglotz-type vakonomic dynamics and its Noether symmetry for nonholonomic constrained systems

In this paper, Herglotz-type vakonomic dynamics and Noether theory of nonholonomic systems are studied. Firstly, Herglotz-type vakonomic dynamical equations for nonholonomic systems are derived on the premise of Herglotz variational principle. Secondly, in terms of the Herglotz-type vakonomic dynamical equations, the Noether symmetry of Herglotz-type vakonomic dynamics is explored, and the Herglotz-type vakonomic dynamical Noether theorems and their inverse theorems are deduced. Finally, the conservation laws of Appell–Hamel case with non-conservative forces are analyzed to show the validity of our results.

On a Version of Dontchev and Hager’s Inverse Mapping Theorem

By revisiting an open question raised by Kirk and Shahzad, we are able to prove a generalized version of Nadler’s fixed-point theorem in the context of strong b-metric space. Such a result leads us to prove a new version of Dontchev and Hager’s inverse mapping theorem. Some examples are provided to illustrate the results.

Best Approximation and Inverse Results for Neural Network Operators

In the present paper we considered the problems of studying the best approximation order and inverse approximation theorems for families of neural network (NN) operators. Both the cases of classical and Kantorovich type NN operators have been considered. As a remarkable achievement, we provide a characterization of the well-known Lipschitz classes in terms of the order of approximation of the considered NN operators. The latter result has inspired a conjecture concerning the saturation order of the considered families of approximation operators. Finally, several noteworthy examples have been discussed in detail

Direct, Inverse Theorems of Approximation by Linear Combinations of Weighted Baskakov–Durrmeyer Operators in Orlicz Spaces

Direct, Inverse Theorems of Approximation by Linear Combinations of Weighted Baskakov–Durrmeyer Operators in Orlicz Spaces

Direct and Inverse Embedding Theorems for Different Dimensions for a Class of Multianisotropic Sobolev Spaces

Direct and Inverse Embedding Theorems for Different Dimensions for a Class of Multianisotropic Sobolev Spaces

Insensitizing controls for the micropolar fluids

In this paper, we investigate the existence of insensitizing controls for the micropolar fluids in a bounded domain with homogeneous Dirichlet boundary conditions and arbitrarily located internal controller. The study of insensitizing controls is essential to solve a stability problem, which means that we look for controls such that some functionals of the velocity fields (the so‐called sentinels) are insensitive to the small perturbations of initial data. The problem of insensitizing controls is transformed into a suitable controllability problem for a cascade system. Our proof relies on a new global Carleman inequality and the inverse mapping theorem.

Finding regulatory modules of chemical reaction systems

Within a cell, numerous chemical reactions form chemical reaction networks (CRNs), which are the origins of cellular functions. We previously developed a theoretical method called structural sensitivity analysis (SSA), which enables us to determine, solely from the network structure, the qualitative changes in the steady-state concentrations of chemicals resulting from the perturbations to a parameter. Notably, if a subnetwork satisfies specific topological conditions, it is referred to as a buffering structure, and the effects of perturbations to the parameter within the subnetwork are localized to the subnetwork (the law of localization). A buffering structure can be the origin of modularity in the regulation of cellular functions generated from CRNs. However, an efficient method to search for buffering structures in a large CRN has not yet been established. In this study, we prove the “inverse theorem” of the law of localization, which states that a certain subnetwork exhibiting a confined response range is always a buffering structure. In other words, we are able to identify buffering structures in terms of confined responses rather than the topological conditions. By leveraging this property, we develop an algorithm to enumerate all buffering structures for a given network by calculating responses. Additionally, we show that the hierarchy of perturbed response patterns corresponds to that of buffering structures, and present a method to illustrate the hierarchy. Our method will be a powerful tool for understanding the regulation of cellular functions generated from CRNs. Published by the American Physical Society 2024

Herglotz-type vakonomic dynamics and Noether theory of nonholonomic systems with delayed arguments

The Herglotz-type vakonomic dynamics of nonholonomic constrained systems with delayed arguments and its Noether theory are studied in this paper. First of all, the Herglotz-type equations of time-delayed vakonomic dynamics for nonholonomic systems are established, and the Herglotz-type local extremal equations are given. Secondly, on the basis of derivation of the variational formulas of Hamilton–Herglotz action with time delay, the Herglotz-type Noether symmetry criteria for time-delayed vakonomic dynamics are investigated. Thirdly, the Herglotz-type Noether’s theorems and inverse theorems for time-delayed vakonomic dynamics of nonholonomic systems are deduced. Finally, an example is presented to demonstrate the application of the results.

Approximate Noether theorem and its inverse for nonlinear dynamical systems with approximate nonstandard Lagrangian

The approximate Noether theorem and its inverse theorem for the nonlinear dynamical systems with approximate exponential Lagrangian and approximate power-law Lagrangian are investigated. For each case, the approximate differential equations of motion for the nonlinear dynamical systems with approximate nonstandard Lagrangian are established, the generalized Noether identities are given. The relationship between the approximate Noether symmetries and approximate conserved quantities for the system with approximate nonstandard Lagrangian are established, and the approximate Noether theorems and their inverse theorems are obtained. Two examples are given to illustrate the application of the results.

Simple proof of the global inverse function theorem via the Hopf–Rinow theorem

Simple proof of the global inverse function theorem via the Hopf–Rinow theorem