Inverse Theorem Research Articles

A refined version of the inverse decomposition theorem for modular multiplicative inverse operators

Recently, we started in the IOP Conference Series [1, 2] and in [3] the study of modular multiplicative inverse operators in an algorithmic functional context on (ℤ/ϱℤ)* with ϱ > 3, where it is possible to distinguish different formulas that break up these operators over (ℤ/ϱℤ)*, all this thanks to the so-called “inverse decomposition theorem (IDT)”. In this paper, we will study these issues a little deeper. Though we mainly focus the paper on derived for these operators a refined version of the (IDT), also appear on the scene some additional contributions. Finally, in a brief overview, we compare the differences that exist among the new result here established and the old ones. However, along with this, there are still plenty of questions to be answered in this research field.

Direct and inverse approximation theorems of functions in the Musielak-Orlicz type spaces

In Musilak-Orlicz type spaces ${\mathcal S}_{\bf M}$, direct and inverse approximation theorems are obtained in terms of the best approximations of functions and generalized moduli of smoothness. The question of the exact constants in Jackson-type inequalities is studied.

Blind Signal Dereverberation Based on Mixture of Weighted Prediction Error Models

We extend the linear prediction-based dereverberation method called weighted prediction error (WPE). WPE optimizes a causal finite impulse response (FIR) filter that predicts the late reverberation components of an observed signal. However, by the multi-input/output inverse (MINT) theorem, in general, such FIR filters exist only when the number of sources is fewer than that of the microphones and no ambient noise exists. To mitigate the model error of WPE in adverse environments, we propose a mixture model of multiple WPEs in which the time frames are divided into clusters in each frequency bin and the WPE's causal FIR filter is optimized in each cluster. Experimental results show that our proposed method significantly improves the dereverberation performance of WPE when the noise level is high or the number of microphones is small.

Second order necessary conditions for optimal control problems of evolution equations involving final point equality constraints

We establish some second order necessary conditions for optimal control problems of evolution equations involving final point equality and inequality constraints. Compared with the existing works, the main difference is due to the presence of end-point equality constraints. With such constraints, we cannot simply use the variational techniques since perturbations of a given control may be no longer admissible. We also cannot use the Ekeland’s variational principle, which is a first order variational principle, to obtain second order necessary conditions. Instead, we combine some inverse mapping theorems on metric spaces and second order linearization of data to obtain our results.

Approximation theorems in weighted Lebesgue spaces with variable exponent

In this work, approximation properties of de la Vall?e-Poussin means are investigated in weighted Lebesgue spaces with variable exponent where weight function belongs to Muckenhoupt class. For this purpose direct, inverse and simultaneous theorems of approximation theory are proved and constructive characterizations of functions are obtained in weighted Lebesgue spaces with variable exponent.

Online Speech Dereverberation Using Mixture of Multichannel Linear Prediction Models

We extend the state-of-the-art online dereverberation method, online weighted prediction error (WPE), which predicts late reverberation components using a multichannel linear prediction (MCLP) filter. The multi-input/output inverse theorem states that in general such an MCLP filter for WPE exists only if the number of sources is less than that of microphones M and there is no additive noise; otherwise, the WPE model has error, degrading its performance especially when M is small. To mitigate this WPE drawback, we recently developed an offline dereverberation method called switching WPE (SwWPE), which has multiple MCLP filters and switches them for each time-frequency bin. We here propose a recursive least squares (RLS) algorithm for the online optimization of SwWPE. Our proposed RLS has forgetting weights for each MCLP filter, and their decay rates are controlled based on a switching mechanism of SwWPE to stably improve the dereverberation performance of the online WPE. Experimental results show that our online SwWPE clearly outperforms online WPE in speech dereverberation tasks.

Multiplicator type operators and approximation of periodic functions of one variable by trigonometric polynomials

The norms of the images of multiplier type operators generated by an arbitrary generator are estimated in terms of the best approximations of univariate periodic functions by trigonometric polynomials in the -spaces, . As corollaries, estimates for the quality of approximation by Fourier means, an inverse theorem of approximation theory, comparison theorems, an analogue of the Marchaud inequality for generalized moduli of smoothness defined by a periodic generator, as well as some constructive sufficient conditions for generalized smoothness and Bernstein type inequalities for generalized derivatives of trigonometric polynomials are obtained. Bibliography: 49 titles.

Nonlocal Generalized Implicit Function Theorems in Hilbert Spaces

A nonlocal generalized implicit function theorem is obtained for mappings acting in Hilbert spaces. Its proof is based on the theory of ordinary differential equations. Under natural assumptions, we use this theorem to derive a global generalized implicit function theorem as well as a global implicit function theorem and a global inverse function theorem as particular cases of the former. Using the estimates produced, we prove an assertion on the continuation of an implicit function from a given closed set to the entire parameter space.

Refined mass-critical Strichartz estimates for Schrödinger operators

We develop refined Strichartz estimates at $L^2$ regularity for a class of time-dependent Schr\"{o}dinger operators. Such refinements begin to characterize the near-optimizers of the Strichartz estimate, and play a pivotal part in the global theory of mass-critical NLS. On one hand, the harmonic analysis is quite subtle in the $L^2$-critical setting due to an enormous group of symmetries, while on the other hand, the spacetime Fourier analysis employed by the existing approaches to the constant-coefficient equation are not adapted to nontranslation-invariant situations, especially with potentials as large as those considered in this article. Using phase space techniques, we reduce to proving certain analogues of (adjoint) bilinear Fourier restriction estimates. Then we extend Tao's bilinear restriction estimate for paraboloids to more general Schr\"{o}dinger operators. As a particular application, the resulting inverse Strichartz theorem and profile decompositions constitute a key harmonic analysis input for studying large data solutions to the $L^2$-critical NLS with a harmonic oscillator potential in dimensions $\ge 2$. This article builds on recent work of Killip, Visan, and the author in one space dimension.

Radon Transform on a Harmonic Manifold

We extend to a large class of noncompact harmonic manifolds the inversion formulas for the Radon transform on horospheres in hyperbolic spaces or Damek–Ricci spaces. Horospheres are defined here as level hypersurfaces of Busemann functions. The proof uses harmonic analysis on the manifolds considered, developed in a recent paper by Biswas, Knieper and Peyerimhoff; we also give a concise proof of their Fourier inversion theorem for harmonic manifolds.

On Lipschitz implicit function theorems in Banach spaces and applications

This paper presents several Lipschitz implicit function theorems for maps between Banach spaces. The results use semi-inner products as a means for understanding the geometry of normed spaces. The arguments rely on corresponding geometric properties of operators on normed spaces, and on Ekeland's variational principle, in contrast with classic results such as the inverse function theorems of Nash and Moser and their use of Newton iteration procedures. The results are illustrated with an application to a central problem in mathematical economics, characterizing Lipschitz behavior of Pareto optimal allocations in a model of trade over an infinite horizon.

Examples of compact Einstein four-manifolds with negative curvature

We give new examples of compact, negatively curved Einstein manifolds of dimension 4 4 . These are seemingly the first such examples which are not locally homogeneous. Our metrics are carried by a sequence of four-manifolds ( X k ) (X_k) previously considered by Gromov and Thurston (Pinching constants for hyperbolic manifolds, Invent. Math. 89 (1987), no. 1, 1–12). The construction begins with a certain sequence ( M k ) (M_k) of hyperbolic four-manifolds, each containing a totally geodesic surface Σ k \Sigma _k which is nullhomologous and whose normal injectivity radius tends to infinity with k k . For a fixed choice of natural number l l , we consider the l l -fold cover X k → M k X_k \to M_k branched along Σ k \Sigma _k . We prove that for any choice of l l and all large enough k k (depending on l l ), X k X_k carries an Einstein metric of negative sectional curvature. The first step in the proof is to find an approximate Einstein metric on X k X_k , which is done by interpolating between a model Einstein metric near the branch locus and the pull-back of the hyperbolic metric from M k M_k . The second step in the proof is to perturb this to a genuine solution to Einstein’s equations, by a parameter dependent version of the inverse function theorem. The analysis relies on a delicate bootstrap procedure based on L 2 L^2 coercivity estimates.

A method for inverting the Laplace transforms of two classes of rational transfer functions in control engineering

A dependable approach for inverting two classes of rational Laplace transforms, involving regular polynomials and partial sums with non-integer exponents is developed. Such types of Laplace transforms frequently emerge as the system transfer functions in the analysis of feedback control loops or process dynamics. The proposed method systematically translates the Laplace inversion problem into an integer or fractional order differential equation and yields the analytical inverse Laplace transform function utilizing the Adomian decomposition method. The method is especially useful in dealing with high-order rational transfer functions; where approaches based on the partial fraction expansion and the residue inversion theorem lose their practicality. The ready-to-use inversion formulas are presented and their usability and reliability is demonstrated through a number of case studies.

Lagrange Inversion Theorem for Dirichlet series

We prove an analogue of the Lagrange Inversion Theorem for Dirichlet series. The proof is based on studying properties of Dirichlet convolution polynomials, which are analogues of convolution polynomials introduced by Knuth in [5].

Distance estimates to feasible controls for systems with final point constraints and second order necessary optimality conditions

Abstract We prove an inverse mapping theorem on a metric space of controls that allows to “control” final points of trajectories of a nonlinear system. More precisely, our result provides local distance estimates of arbitrary controls from feasible ones. As an application we derive second-order necessary optimality conditions for L 1 -local minima for the Mayer optimal control problem with a general control constraint U ⊂ R m , state constraints described by inequalities and final point constraints, possibly having empty interior. Thanks to this inverse mapping theorem we first get a second-order variational inequality as a necessary optimality condition. Then the separation theorem leads in a straightforward way to second-order necessary conditions.

Sobolev Mappings and Moduli Inequalities on Carnot Groups

Abstract. We study the mappings that satisfy moduli inequalities on Carnot groups. We prove that the homeomorphisms satisfying the moduli inequalities (Q-homeomorphisms) with a locally integrable function Q are Sobolev mappings. On this base in the frameworks of the weak inverse mapping theorem, we prove that, on the Carnot groups 𝔾; the mappings inverse to Sobolev homeomorphisms of finite distortion of the class $$ {W}_{v,\mathrm{loc}}^1\left(\Omega; {\Omega}^{\prime}\right) $$ belong to the Sobolev class $$ {W}_{1,\mathrm{loc}}^1\left({\Omega}^{\prime };\Omega \right) $$ .

A bilinear version of Bogolyubov’s theorem

A theorem of Bogolyubov states that for every dense set A A in Z N \mathbb {Z}_N we may find a large Bohr set inside A + A − A − A A+A-A-A . In this note, motivated by work on a quantitative inverse theorem for the Gowers U 4 U^4 norm, we prove a bilinear variant of this result for vector spaces over finite fields. Given a subset A ⊂ F p n × F p n A \subset \mathbb {F}^n_p \times \mathbb {F}^n_p , we consider two operations: one of them replaces each row of A A by the set difference of it with itself, and the other does the same for columns. We prove that if A A has positive density and these operations are repeated several times, then the resulting set contains a bilinear analogue of a Bohr set, namely the zero set of a biaffine map from F p n × F p n \mathbb {F}^n_p \times \mathbb {F}^n_p to an F p \mathbb {F}_p -vector space of bounded dimension. An almost identical result was proved independently by Bienvenu and Lê.

Convexity of sets in metric Abelian groups

Abstract In the present paper, we introduce a new concept of convexity which is generated by a family of endomorphisms of an Abelian group. In Abelian groups, equipped with a translation invariant metric, we define the boundedness, the norm, the modulus of injectivity and the spectral radius of endomorphisms. Beyond the investigation of their properties, our first main goal is an extension of the celebrated Rådström cancellation theorem. Another result generalizes the Neumann invertibility theorem. Next we define the convexity of sets with respect to a family of endomorphisms, and we describe the set-theoretical and algebraic structure of the class of such sets. Given a subset, we also consider the family of endomorphisms that make this subset convex, and we establish the basic properties of this family. Our first main result establishes conditions which imply midpoint convexity. The next main result, using our extension of the Rådström cancellation theorem, presents further structural properties of the family of endomorphisms that make a subset convex.

An efficient algorithm for time-dependent failure credibility by combining adaptive single-loop Kriging model with fuzzy simulation

The time-dependent failure credibility (TDFC) can reasonably measure the safety level of the time-dependent structure under the fuzzy uncertainty, but the direct optimization algorithm to estimate the TDFC requires large computational cost and even results in locally optimal solutions. Therefore, an efficient method is proposed for estimating the TDFC by combining the fuzzy simulation and the single-loop Kriging model. In the proposed method, fuzzy inverse transformation theorem is firstly used to transform the estimation of the TDFC into a sample classification problem, in which the candidate sample pool generated by fuzzy simulation (FS) is classified into the failure group and the safety one. For improving the efficiency of the classification, a Kriging model is adaptively trained by an elaborate U-learning function in the candidate sample pool. After the candidate sample is divided into the failure group and the safety one by the convergent Kriging model, the TDFC can be estimated as a byproduct easily. The innovation of the proposed method includes two aspects: establishing the idea of the fuzzy simulation combined with the single-loop Kriging model to estimate TDFC efficiently and robustly, and designing an elaborate U-learning function to improve the efficiency of training the single-loop Kriging model. The presented examples validate the efficiency of the proposed method under the acceptable precision.

On Erdős-Ginzburg-ZIV inverse theorems for dihedral and dicyclic groups

Let G be a finite group and exp(G) = lcm{ord(g) ∣ g ∈ G}. A finite unordered sequence of terms from G, where repetition is allowed, is a product-one sequence if its terms can be ordered such that their product equals the identity element of G. We denote by s(G)(or E(G) respectively) the smallest integer l such that every sequence of length at least l has a product-one subsequence of length exp(G)(or ∣G∣ respectively). In this paper, we provide the exact values of s(G) and E(G) for Dihedral and Dicyclic groups and we provide explicit characterizations of all sequences of length s(G) — 1 (or E(G) — 1 respectively) having no product-one subsequence of length exp(G)(or ∣G∣ respectively).