Inverse Theorem Research Articles

Canonical systems whose Weyl coefficients have dominating real part

For a two-dimensional canonical system y′ (t) = zJH(t)y(t) on the half-line (0, ∞) whose Hamiltonian H is a.e. positive semi-definite, denote by qH its Weyl coefficient. De Branges’ inverse spectral theorem states that the assignment H ↦ qH is a bijection between Hamiltonians (suitably normalised) and Nevanlinna functions.The main result of the paper is a criterion when the singular integral of the spectral measure, i.e. Re qH(iy), dominates its Poisson integral Im qH(iy) for y → +∞. Two equivalent conditions characterising this situation are provided. The first one is analytic in nature, very simple, and explicit in terms of the primitive M of H. It merely depends on the relative size of the off-diagonal entries of M compared with the diagonal entries. The second condition is of geometric nature and technically more complicated. It involves the relative size of the off-diagonal entries of H, a measurement for oscillations of the diagonal of H, and a condition on the speed and smoothness of the rotation of H.

A modulus of smoothness for some Banach function spaces

UDC 517.5 Based on the Steklov operator, we consider a modulus of smoothness of functions in some Banach function spaces, which can be not translation invariant and establish its main properties. A constructive characterization of the Lipschitz class is obtained by using the Jackson-type direct theorem and inverse theorem of trigonometric approximation. As application, we present several examples of related (weighted) function spaces.

Metric regularity, pseudo-Jacobians and global inversion theorems on Finsler manifolds

Our aim in this paper is to study the global invertibility of a locally Lipschitz map f:X→Y between (possibly infinite-dimensional) Finsler manifolds, stressing the connections with covering properties and metric regularity of f. To this end, we introduce a natural notion of pseudo-Jacobian Jf in this setting, as is a kind of set-valued differential object associated to f. By means of a suitable index, we study the relations between properties of pseudo-Jacobian Jf and local metric properties of the map f, which lead to conditions for f to be a covering map, and for f to be globally invertible. In particular, we obtain a version of Hadamard integral condition in this context.

A general framework for tropical differential equations

We construct a general framework for tropical differential equations based on idempotent semirings and an idempotent version of differential algebra. Over a differential ring equipped with a non-archimedean norm enhanced with additional differential information, we define tropicalization of differential equations and tropicalization of their solution sets. This framework includes rings of interest in the theory of p-adic differential equations: rings of convergent power series over a non-archimedean normed field. The tropicalization records the norms of the coefficients. This gives a significant refinement of Grigoriev’s framework for tropical differential equations. We then prove a differential analogue of Payne’s inverse limit theorem: the limit of all tropicalizations of a system of differential equations is isomorphic to a differential variant of the Berkovich analytification.

Inverse Maximum Theorems and Their Relations with Equilibrium and Fixed Point Theorems

Inverse Maximum Theorems and Their Relations with Equilibrium and Fixed Point Theorems

On the constants in abstract inverse theorems of approximation theory

In the classical inverse theorems of constructive function theory, structural characteristics of an approximated function are estimated in terms of its best approximations. Most of the known proofs of the inverse theorems utilize Bernstein’s idea to expand the function in polynomials of its best approximation. In the present paper, Bernstein’s proof is modified by using integrals instead of sums. With this modification, it turns out that desired inequalities are based on identities similar to Frullani integrals. The considerations here are quite general, which allows one to obtain analogs of the inverse theorems for functionals in abstract Banach or even seminormed spaces. Then these abstract results are specified and inverse theorems in concrete spaces of functions are deduced, including weighted spaces, with explicit constants.

Hölder classes in the 𝐿^{𝑝} norm on a chord-arc curve in ℝ³

The Hölder classes L p α ( L ) L_p^{\alpha } (L) in the L p ( L ) L_p(L) norm on a chord-arc curve L L in R 3 \mathbb {R}^3 are defined and direct and inverse approximation theorems are proved for functions from these classes by functions harmonic in a neighborhood of the curve. The approximation is estimated in the L p ( L ) L^p(L) norm, the direct theorem is proved for a certain subclass of L p α ( L ) L^\alpha _p(L) and the inverse theorem covers the entire Hölder class.

Revisiting Takahashi's inversion theorem in discrete symmetry-based dual frameworks

The so-called Takahashi's Inversion Theorem, the reconstruction of a given spinor based on its bilinear covariants [1], are re-examined, considering alternative dual structures. In contrast to the classical results, where the Dirac dual structure plays the central role, new duals are built using the discrete symmetries C,P,T. Their combinations are also taken into account. Furthermore, the imposition of a new adjoint structure led us also to re-examine the representation of the Clifford algebra basis elements, uncovering new bilinear structures and a new Fierz aggregate. Those results might help construct theories for new beyond standard model fields.

Local null controllability for the thermistor problem

In this paper, we deal with the local null controllability of an initial boundary value problem for a thermistor equation. The control is distributed, locally in space. The main ingredients of the proof are suitable Carleman estimates for an adjoint system and Liusternik’s Inverse Mapping Theorem in Hilbert spaces. As a complement to the theoretical results, we present formulations and numerical simulations for the controllability problem, the results were obtained through the finite element method and a quasi-Newton method, simulated in the software Freefem++. Finally, large time null controllability, stabilization and others possible extensions and open problems concerning other systems are presented.

Autoencoders for a manifold learning problem with a jacobian rank constraint

We formulate the manifold learning problem as the problem of finding an operator that maps any point to a close neighbor that lies on a “hidden” k-dimensional manifold. We call this operator the correcting function. Under this formulation, autoencoders can be viewed as a tool to approximate the correcting function. Given an autoencoder whose Jacobian has rank k, we deduce from the classical Constant Rank Theorem that its range has a structure of a k-dimensional manifold. A k-dimensionality of the range can be forced by the architecture of an autoencoder (by fixing the dimension of the code space), or alternatively, by an additional constraint that the rank of the autoencoder mapping is not greater than k. This constraint is included in the objective function as a new term, namely a squared Ky-Fan k-antinorm of the Jacobian function. We claim that this constraint is a factor that effectively reduces the dimension of the range of an autoencoder, additionally to the reduction defined by the architecture. We also add a new curvature term into the objective. To conclude, we experimentally compare our approach with the CAE+H method on synthetic and real-world datasets.

Understanding Representation Learnability of Nonlinear Self-Supervised Learning

Self-supervised learning (SSL) has empirically shown its data representation learnability in many downstream tasks. There are only a few theoretical works on data representation learnability, and many of those focus on final data representation, treating the nonlinear neural network as a ``black box". However, the accurate learning results of neural networks are crucial for describing the data distribution features learned by SSL models. Our paper is the first to analyze the learning results of the nonlinear SSL model accurately. We consider a toy data distribution that contains two features: the label-related feature and the hidden feature. Unlike previous linear setting work that depends on closed-form solutions, we use the gradient descent algorithm to train a 1-layer nonlinear SSL model with a certain initialization region and prove that the model converges to a local minimum. Furthermore, different from the complex iterative analysis, we propose a new analysis process which uses the exact version of Inverse Function Theorem to accurately describe the features learned by the local minimum. With this local minimum, we prove that the nonlinear SSL model can capture the label-related feature and hidden feature at the same time. In contrast, the nonlinear supervised learning (SL) model can only learn the label-related feature. We also present the learning processes and results of the nonlinear SSL and SL model via simulation experiments.

Approximation on hexagonal domains by Taylor-Abel-Poisson means

Approximative properties of the Taylor-Abel-Poisson linear summation method of Fourier series are considered for functions of several variables, periodic with respect to the hexagonal domain, in the integral metric. In particular, direct and inverse theorems are proved in terms of approximations of functions by the Taylor-Abel-Poisson means and K-functionals generated by radial derivatives. Bernstein type inequalities for L1-norm of radial derivatives of arbitrary order of the Poisson kernel are also obtained.

Quantitative inverse theorem for Gowers uniformity norms and in

AbstractWe prove quantitative bounds for the inverse theorem for Gowers uniformity norms $\mathsf {U}^5$ and $\mathsf {U}^6$ in $\mathbb {F}_2^n$ . The proof starts from an earlier partial result of Gowers and the author which reduces the inverse problem to a study of algebraic properties of certain multilinear forms. The bulk of the work in this paper is a study of the relationship between the natural actions of $\operatorname {Sym}_4$ and $\operatorname {Sym}_5$ on the space of multilinear forms and the partition rank, using an algebraic version of regularity method. Along the way, we give a positive answer to a conjecture of Tidor about approximately symmetric multilinear forms in five variables, which is known to be false in the case of four variables. Finally, we discuss the possible generalization of the argument for $\mathsf {U}^k$ norms.

Construction and approximation for a class of feedforward neural networks with sigmoidal function

As we know, feedforward neural networks (FNNs) with sigmoidal activation function are universal approximators. Theoretically, for any continuous function defined on a compact set, there exists an FNN such that the FNN can approximate the function with arbitrary accuracy, which constitutes a theoretical guarantee that FNN can be used as an efficient learning machine. This paper addresses the construction and approximation for FNNs. We construct an FNN with sigmoidal activation function and estimate its approximation error. In particular, an inverse theorem of the approximation is established, which implies the equivalence characterization theorem of the approximation and reveals the relationship between the topological structure of the FNN and its approximation ability. As keys in this study, the concepts of modulus of continuity of function, K-functional, and their relationship are utilized, and two Bernstein-type inequalities are established.

The K-functional and saturated linear approximation processes

It is shown that already the knowledge of the saturation class , {{mathfrak {F}}}(J_t, L^p) of a saturated convolution approximation process , {J_t} on , L^p({{mathbb {R}}}^n),, 1le p< infty , completely determines its norm approximation behavior. This is achieved by using the , K-functional , K(t,f;L^p,{{mathfrak {F}}}(J_t, L^p)) as a comparison scale, which relates on the one hand the approximation process and on the other appropriate moduli of smoothness. This implies that simultaneously one gets the so-called direct and inverse theorems. There are open problems if the saturation order is slightly perturbed, e.g., by a , |log t|^lambda -factor, , lambda >0. Proofs are mainly based on the Fourier transformation.

On the Virtualization of Audio Transducers.

In audio transduction applications, virtualization can be defined as the task of digitally altering the acoustic behavior of an audio sensor or actuator with the aim of mimicking that of a target transducer. Recently, a digital signal preprocessing method for the virtualization of loudspeakers based on inverse equivalent circuit modeling has been proposed. The method applies Leuciuc's inversion theorem to obtain the inverse circuital model of the physical actuator, which is then exploited to impose a target behavior through the so called Direct-Inverse-Direct Chain. The inverse model is designed by properly augmenting the direct model with a theoretical two-port circuit element called nullor. Drawing on this promising results, in this manuscript, we aim at describing the virtualization task in a broader sense, including both actuator and sensor virtualizations. We provide ready-to-use schemes and block diagrams which apply to all the possible combinations of input and output variables. We then analyze and formalize different versions of the Direct-Inverse-Direct Chain describing how the method changes when applied to sensors and actuators. Finally, we provide examples of applications considering the virtualization of a capacitive microphone and a nonlinear compression driver.

Connectedness in asymmetric spaces

Inverse theorems for geometric approximation theory related to the structure of approximating sets are obtained. Various types of connectedness of sets in asymmetric spaces (generally nonmetrizable) are studied. In particular, conditions are established for an asymmetric space and a subset of this space that guarantee that the intersection of an open ball with this set is path-connected.

Significance of skewness and kurtosis on the solute dispersion in pulsatile Carreau–Yasuda fluid flow in a tube with wall absorption

Solute dispersion in Carreau–Yasuda fluid flow in a tube presented in Rana &amp; Murthy (Proc. R. Soc. Lond.A, vol. 472, 2016, p. 20160294) was limited to a steady-state velocity profile due to the nonlinearity associated with the Yasuda parameter$a$with power-law exponent$n$. This limitation is overcome and the velocity profile is obtained for all values of the Yasuda parameter by using the Lagrange inversion theorem, which admits power series solution for the flow field. An analytical solution for the concentration distribution in the circular tube is obtained for the unsteady and pulsatile flow with$n\leq 1$and$\alpha &lt;&lt;1$and the numerical solution is presented for all values of$\alpha$and$n$. The solute dispersion is analysed analytically using the Sankarasubramanian–Gill generalized dispersion method and also using the Aris–Barton method of moments considering up to fourth-order moments. The solute dispersion is also simulated numerically by using a new class of computationally explicit Runge–Kutta method. The axial mean concentration of the solute is estimated by the exchange, convective and dispersion coefficients. The third- and fourth-order moments give rise to skewness and kurtosis revealing the deviation from the Gaussianity and reduction in the peak of the mean concentration profile at a small time of the solute injection. All time variations of these five moments against flow governing parameters are thoroughly investigated. The flow and dispersion regimes that are derived here for moments provide a good understanding of the solute dispersion in the tube. The increase in the Womersley frequency parameter led to a phase lag at each period. This work is the initiation of estimating the skewness and kurtosis in a non-yield stress fluid flow in a tube.

Estimates for the Weyl coefficient of a two-dimensional canonical system

For a two-dimensional canonical system $y'(t)=zJH(t)y(t)$ on some interval $(a,b)$ whose Hamiltonian $H$ is a.e. positive semi-definite and which is regular at $a$ and in the limit point case at $b$, denote by $q_H$ its Weyl coefficient. De~Branges' inverse spectral theorem states that the assignment $H\mapsto q_H$ is a bijection between Hamiltonians (suitably normalised) and Nevanlinna functions. We give upper and lower bounds for $|q_H(z)|$ and $\Im q_H(z)$ when $z$ tends to $i\infty$ non-tangentially. These bounds depend on the Hamiltonian $H$ near the left endpoint $a$ and determine $|q_H(z)|$ up to universal multiplicative constants. We obtain that the growth of $|q_H(z)|$ is independent of the off-diagonal entries of $H$ and depends monotonically on the diagonal entries in a natural way. The imaginary part is, in general, not fully determined by our bounds (in forthcoming work we shall prove that for most Hamiltonians also $\Im q_H(z)$ is fully determined). We translate the asymptotic behaviour of $q_H$ to the behaviour of the spectral measure $\mu_H$ of $H$ by means of Abelian--Tauberian results and obtain conditions for membership of growth classes defined by weighted integrability condition (Kac classes) or by boundedness of tails at $\pm\infty$ w.r.t. a weight function. Moreover, we apply our results to Krein strings and Sturm--Liouville equations.

Direct and inverse results for popular differences in trees of positive dimension

AbstractWe establish analogues for trees of results relating the density of a set ${E \subset \mathbb {N}}$ , the density of its set of popular differences and the structure of E. To obtain our results, we formalize a correspondence principle of Furstenberg and Weiss which relates combinatorial data on a tree to the dynamics of a Markov process. Our main tools are Kneser-type inverse theorems for sets of return times in measure-preserving systems. In the ergodic setting, we use a recent result of the first author with Björklund and Shkredov and a stability-type extension (proved jointly with Shkredov); we also prove a new result for non-ergodic systems.