Inverse Theorem Research Articles

On Nash–Moser–Ekeland Inverse Mapping Theorem

We present a criterion for local surjectivity of mappings between graded Frechet spaces in the spirit of a well-known criterion in Banach spaces. As applications, we get “hard inverse mapping theorem” in the flavor of Nash–Moser. The technology of proofs was strongly influenced by a recent paper of Ekeland.

Noether’s Symmetries and Its Inverse for Fractional Logarithmic Lagrangian Systems

Abstract In this paper, Noether’s theorem and its inverse theorem are proved for the fractional variational problems based on logarithmic Lagrangian systems. The Hamilton principle of the systems is derived. And the definitions and the criterions of Noether’s symmetry and Noether’s quasi-symmetry of the systems based on logarithmic Lagrangians are given. The intrinsic relation between Noether’s symmetry and the conserved quantity is established. At last an example is given to illustrate the application of the results.

The valuative tree is the projective limit of Eggers-Wall trees

Consider a germ $C$ of reduced curve on a smooth germ $S$ of complex analytic surface. Assume that $C$ contains a smooth branch $L$. Using the Newton-Puiseux series of $C$ relative to any coordinate system $(x,y)$ on $S$ such that $L$ is the $y$-axis, one may define the {\em Eggers-Wall tree} $\Theta_L(C)$ of $C$ relative to $L$. Its ends are labeled by the branches of $C$ and it is endowed with three natural functions measuring the characteristic exponents of the previous Newton-Puiseux series, their denominators and contact orders. The main objective of this paper is to embed canonically $\Theta_L(C)$ into Favre and Jonsson's valuative tree $\mathbb{P}(\mathcal{V})$ of real-valued semivaluations of $S$ up to scalar multiplication, and to show that this embedding identifies the three natural functions on $\Theta_L(C)$ as pullbacks of other naturally defined functions on $\mathbb{P}(\mathcal{V})$. As a consequence, we prove an inversion theorem generalizing the well-known Abhyankar-Zariski inversion theorem concerning one branch: if $L'$ is a second smooth branch of $C$, then the valuative embeddings of the Eggers-Wall trees $\Theta_{L'}(C)$ and $\Theta_L(C)$ identify them canonically, their associated triples of functions being easily expressible in terms of each other. We prove also that the space $\mathbb{P}(\mathcal{V})$ is the projective limit of Eggers-Wall trees over all choices of curves $C$. As a supplementary result, we explain how to pass from $\Theta_L(C)$ to an associated splice diagram.

Inverse results of approximation and the saturation order for the sampling Kantorovich series

In the present paper, we study the saturation order for the sampling Kantorovich series in the space of uniformly continuous and bounded functions. In order to achieve the above result, we first need to establish a relation between the sampling Kantorovich operators and the classical generalized sampling series of P.L. Butzer. Further, for the latter operators, we also need to prove a Voronovskaja-type formula. Moreover, inverse theorems of approximation are also obtained, in order to give a characterization for the rate of convergence of the above operators. Finally, several examples of kernel functions for which the above theory can be applied have been presented and discussed in details.

Controlling the Electrical State via Uncertain Power Injections in Three-Phase Distribution Networks

We consider the problem of controlling the electrical state of a three-phase distribution network by enforcing the explicit power injection. More precisely, we assume that the power injection is constrained to reside in some uncertainty set, and the problem is to ensure that the electrical state remains in a set that satisfies feasibility constraints. To formalize this, we say that a set S of power injections is a “domain of V-control” if any continuous trajectory of the electrical state that starts in the set V must stay in V as long as the corresponding trajectory of the power injection stays in S. First, we show that the existence and uniqueness of load-flow solution is not enough to guarantee V-control, and give sufficient additional conditions for V-controllability to hold. Incidentally, the derivation of our conditions establishes that local uniqueness implies nonsingularity of the load-flow Jacobian (the converse is well known by the inverse function theorem). Then, we give a concrete algorithm to determine whether a given set of power injections is a domain of V-control for some feasible and nonsingular set V. The algorithm is evaluated on IEEE test feeders. Our method can be used to perform admissibility tests in the control of distribution networks by power injections.

Analysis of a time-dependent fluid-solid interaction problem above a local rough surface

This paper is concerned with the mathematical analysis of time-dependent fluid-solid interaction problem associated with a bounded elastic body immersed in a homogeneous air or fluid above a local rough surface. We reformulate the unbounded scattering problem into an equivalent initial-boundary value problem defined in a bounded domain by proposing a transparent boundary condition (TBC) on a hemisphere. Analyzing the reduced problem with Lax-Milgram lemma and abstract inversion theorem of Laplace transform, we prove the well-posedness and stability for the reduced problem. Moreover, an a priori estimate is established directly in the time domain for the acoustic wave and elastic displacement with using the energy method.

Inversion theorem and quantitative uncertainty principles for the Dunkl Gabor transform on $${\mathbb {R}}^{d}$$ R d

We prove a new inversion formula for the Dunkl Gabor transform. We also prove several versions of Heisenberg-type inequalities, Donoho–Stark’s uncertainty principles, local Cowling–Price’s type inequalities and finally Faris–Price’s uncertainty principle for the previous transform.

Arc spaces, motivic measure and Lipschitz geometry of real algebraic sets

We investigate connections between Lipschitz geometry of real algebraic varieties and properties of their arc spaces. For this purpose we develop motivic integration in the real algebraic set-up. We construct a motivic measure on the space of real analytic arcs. We use this measure to define a real motivic integral which admits a change of variables formula not only for the birational but also for generically one-to-one Nash maps. As a consequence we obtain an inverse mapping theorem which holds for continuous rational maps and, more generally, for generically arc-analytic maps. These maps appeared recently in the classification of singularities of real analytic function germs. Finally, as an application, we characterize in terms of the motivic measure, germs of arc-analytic homeomorphism between real algebraic varieties which are bi-Lipschitz for the inner metric.

On Furstenberg's intersection conjecture, self-similar measures, and the $L^q$ norms of convolutions

We study a class of measures on the real line with a kind of self-similar structure, which we call dynamically driven self-similar measures, and contain proper self-similar measures such as Bernoulli convolutions as special cases. Our main result gives an expression for the $L^q$-dimensions of such dynamically driven self-similar measures, under certain conditions. As an application, we settle Furstenberg's long-standing conjecture on the dimension of the intersections of $\times p$ and $\times q$-invariant sets. Among several other applications, we also show that Bernoulli convolutions have an $L^q$ density for all finite $q$, outside of a zero-dimensional set of exceptions. The proof of the main result is inspired by M. Hochman's approach to the dimensions of self-similar measures and his inverse theorem for entropy. Our method can be seen as an extension of Hochman's theory from entropy to $L^q$ norms, and likewise relies on an inverse theorem for the decay of $L^q$ norms of discrete measures under convolution. This central piece of our approach may be of independent interest, and is an application of well-known methods and results in additive combinatorics: the asymmetric version of the Balog-Szemeredi-Gowers Theorem due to Tao-Vu, and some constructions of Bourgain.

Inverse and Saturation Theorems for Some Summation Integral Linear Positive Operators

Inverse and saturation theorems have been studied for linear combination of some summation integral hybrid linear positive operators. Linear approximating technique and Steklov means have been used to obtain better approximation results.

О задаче продолжения функции внутрь круга в пространствах с весом, имеющим особенность на границе

It has been proven that any quadratically summable function with a power weight in the unit circle is uniquely represented as an orthogonal sum of its analytic and coanalytic components; therefore, it is natural to consider the coanalytic component of a function as a certain characteristic of its nonanalyticity. The article considers the problem of finding the extension of a function with the unit circle inside the circle that it would have the minimal deviation from the Sobolev weight subspace of analytic functions (the problem of minimizing the coanalytic deviation). Similar coanalytic problems were considered by other researchers in the weightless case for a unit circle, half-band, and an arbitrary bounded simply connected domain with a smooth boundary. The problem is formulated in spaces with a weight having a power singularity at the unit circle entire boundary, and the boundary values of the functions are taken from the corresponding Besov space. The known properties of weight spaces, including the direct and inverse theorems about the traces of functions from the considered classes are formulated. By using these properties, a theorem about the existence and uniqueness of the problem solution is proved in the framework of the ideas of the monotonic operators theory.

On the Distribution of Zero Sets of Holomorphic Functions. III. Inversion Theorems

Пусть $M$ - субгармоническая функции в области $D\subset \mathbb C^n$ с мерой Рисса $\nu_M$, ${\mathsf Z}\subset D$. Как было показано в первой из предшествующих статей, если существует голоморфная функция $f\neq 0$ в $D$, $f({\mathsf Z})=0$, $|f|\leq \exp M$ на $D$, то имеет место некоторая шкала интегральных равномерных оценок сверху распределения множества $\mathsf Z$ через $\nu_M$. В настоящей статье показано, что при $n=1$ этот результат «почти обратим». Из такой шкалы оценок распределения точек последовательности ${\mathsf Z}:=\{{\mathsf z}_k \}_{k=1,2,…\subset D\subset \mathbb C$ через $\nu_M$ следует, что существует ненулевая голоморфной функции $f$ в $D$, $f(\mathsf Z)=0$, $|f|\leq \exp M^{\uparrow}$ на $D$, где функция $M^{\uparrow}\geq M$ на $D$ строится через усреднения функции $M $ по быстро сужающимся кругам при приближении к границе области $D$ с некоторой возможной аддитивной логарифмической добавкой, связанной со скоростью сужения этих кругов.

Exact constants in direct and inverse approximation theorems for functions of several variables in the spaces Sp

In the paper, exact constants in direct and inverse approximation theorems for functions of several variables are found in the spaces Sp. The equivalence between moduli of smoothness and some K-functionals is also shown in the spaces Sp.

Существование обратной функции в окрестности нерегулярного значения

The classical inverse function theorems guarantee the existence of an inverse function in a neighborhood of the value of a given point if the regularity condition is satisfied at this point, that is, the first derivative at a given point is nondegenerate. A more general condition for the existence of an implicit function is the 2-regularity condition. It holds, for example, for many quadratic mappings at zero. It is known that under natural smoothness assumptions, the existence of a continuous inverse function follows from a 2-regularity of a map at a point in a certain direction. In this paper, it is shown that, in the known statements guaranteeing the existence of an inverse function when the 2-regularity condition is satisfied, we can weaken the smoothness assumptions. However, the inverse function may not be continuous.

VALUE PATTERNS OF MULTIPLICATIVE FUNCTIONS AND RELATED SEQUENCES

We study the existence of various sign and value patterns in sequences defined by multiplicative functions or related objects. For any set$A$whose indicator function is ‘approximately multiplicative’ and uniformly distributed on short intervals in a suitable sense, we show that the density of the pattern$n+1\in A$,$n+2\in A$,$n+3\in A$is positive as long as$A$has density greater than$\frac{1}{3}$. Using an inverse theorem for sumsets and some tools from ergodic theory, we also provide a theorem that deals with the critical case of$A$having density exactly$\frac{1}{3}$, below which one would need nontrivial information on the local distribution of$A$in Bohr sets to proceed. We apply our results first to answer in a stronger form a question of Erdős and Pomerance on the relative orderings of the largest prime factors$P^{+}(n)$,$P^{+}(n+1),P^{+}(n+2)$of three consecutive integers. Second, we show that the tuple$(\unicode[STIX]{x1D714}(n+1),\unicode[STIX]{x1D714}(n+2),\unicode[STIX]{x1D714}(n+3))~(\text{mod}~3)$takes all the$27$possible patterns in$(\mathbb{Z}/3\mathbb{Z})^{3}$with positive lower density, with$\unicode[STIX]{x1D714}(n)$being the number of distinct prime divisors. We also prove a theorem concerning longer patterns$n+i\in A_{i}$,$i=1,\ldots ,k$in approximately multiplicative sets$A_{i}$having large enough densities, generalizing some results of Hildebrand on his ‘stable sets conjecture’. Finally, we consider the sign patterns of the Liouville function$\unicode[STIX]{x1D706}$and show that there are at least$24$patterns of length$5$that occur with positive upper density. In all the proofs, we make extensive use of recent ideas concerning correlations of multiplicative functions.

Notes on explicit and inversion formulas for the Chebyshev polynomials of the first two kinds

In the paper, starting from the Rodrigues formulas for the Chebyshev polynomials of the first and second kinds, by virtue of the Fa\`a di Bruno formula, with the help of two identities for the Bell polynomials of the second kind, and making use of a new inversion theorem for combinatorial coefficients, the authors derive two nice explicit formulas and their corresponding inversion formulas for the Chebyshev polynomials of the first and second kinds.

On Approximation by Linear Combinations of Modified Summation Operators of Integral Type in Orlicz Spaces

In this paper, the authors introduce the Orlicz spaces corresponding to the Young function and, by virtue of the equivalent theorem between the modified K-functional and modulus of smoothness, establish the direct, inverse, and equivalent theorems for linear combinations of modified summation operators of integral type in the Orlicz spaces.

A general inversion theorem for cointegration

A generalization of the Granger and the Johansen Representation Theorems valid for any (possibly fractional) order of integration is presented. This Representation Theorem is based on inversion results that characterize the order of the pole and the coefficients of the Laurent series representation of the inverse of a matrix function around a singular point. Explicit expressions of the matrix coefficients of the (polynomial) cointegrating relations, of the Common Trends and of the Triangular representations are provided, either starting from the Moving Average or the Auto Regressive form. This contribution unifies different approaches in the literature and extends them to an arbitrary order of integration. The role of deterministic terms is discussed in detail.

Dimension growth for iterated sumsets

We study dimensions of sumsets and iterated sumsets and provide natural conditions which guarantee that a set F subseteq mathbb {R} satisfies overline{dim }_{mathrm {B}}F+F > overline{dim }_{mathrm {B}}F or even dim _{mathrm {H}}n F rightarrow 1. Our results apply to, for example, all uniformly perfect sets, which include Ahlfors–David regular sets. Our proofs rely on Hochman’s inverse theorem for entropy and the Assouad and lower dimensions play a critical role. We give several applications of our results including an Erdős–Volkmann type theorem for semigroups and new lower bounds for the box dimensions of distance sets for sets with small dimension.

A Bernstein type inequality for the Askey–Wilson operator

We establish a Riesz-type interpolation formula on the interval [−1,1] for the Askey–Wilson operator. As consequences, sharp Bernstein inequality and Markov inequality are obtained when differentiation is replaced by the Askey–Wilson operator. Moreover, an inverse approximation theorem is proved using a Bernstein type inequality in L2-space. We conclude our paper with an overconvergence result which is applied to characterize all q-differentiable functions of Brown and Ismail.