Sufficient Conditions For Invertibility Research Articles

On the invertibility of matrices with a double saddle-point structure

We establish necessary and sufficient conditions for invertibility of symmetric three-by-three block matrices having a double saddle-point structure that guarantee the unique solvability of double saddle-point systems. We consider various scenarios, including the case where all diagonal blocks are allowed to be rank deficient. Under certain conditions related to the nullity of the blocks and intersections of their kernels, an explicit formula for the inverse is derived.

Identification of unobserved distribution factors and preferences in the collective household model

This paper develops a non-parametric collective random utility model with a continuous choice of consumption and sharing. We allow for a heterogeneous population by introducing unobserved distribution factors and preferences that are non-separable in the stochastic household choice system. We provide necessary and sufficient conditions for invertibility with respect to the vector of unobserved heterogeneity. Further, we show non-parametric identification of each individual’s idiosyncratic preferences and Pareto weights from the distribution of observed choices. Based on this result we develop a non-parametric two-step estimation procedure and estimate individual counterfactuals in a collective labour supply model for households with children using the LISS panel.

Inversion and Symmetries of the Star Transform

The star transform is a generalized Radon transform mapping a function of two variables to its integrals along “star-shaped” trajectories, which consist of a finite number of rays emanating from a common vertex. Such operators appear in mathematical models of various imaging modalities based on scattering of elementary particles. The paper presents a comprehensive study of the inversion of the star transform. We describe the necessary and sufficient conditions for invertibility of the star transform, introduce a new inversion formula and discuss its stability properties. As an unexpected bonus of our approach, we prove a conjecture from algebraic geometry about the zero sets of elementary symmetric polynomials.

Closed Range Type Properties of Toeplitz Operators on the Bergman Space and the Berezin Transform

We characterize the multiplication operators with closed range on the Bergman space in terms of the Berezin transform, and apply this characterization to finite products of interpolating Blaschke products. We give some necessary and some sufficient conditions for invertibility of general Toeplitz operators on the Bergman space. We determine the Fredholm Toeplitz operators with $$BMO^1$$ symbols and the invertible Toeplitz operators with nonnegative symbols, when their Berezin transform is bounded and of vanishing oscillation.

A Multi-Term Basis Criterion for Families of Dilated Periodic Functions

In this paper we formulate a concrete method for determining whether a system of dilated periodic functions forms a Riesz basis in L^2(0,1) . This method relies on a general framework developed by Hedenmalm, Lindqvist and Seip about 20 years ago, which turns the basis question into one about the localisation of the zeros and poles of a corresponding analytic multiplier. Our results improve upon various criteria formulated previously, which give sufficient conditions for invertibility of the multiplier in terms of sharp estimates on the Fourier coefficients. Our focus is on the concrete verification of the hypotheses by means of analytical or accurate numerical approximations. We then examine the basis question for profiles in a neighbourhood of a non-basis family generated by periodic jump functions. For one of these profiles, the p -sine functions, we determine a threshold for positive answer to the basis question which improves upon those found recently.

Invertibility of Multipliers for Continuous G-frames

In this paper we study the concept of multipliers for continuous $g$-Bessel families in Hilbert spaces. We present necessary conditions for invertibility of multipliers for continuous $g$-Bessel families and sufficient conditions for invertibility of multipliers for continuous $g$-frames.

Necessary and Sufficient Conditions for the Invertibility of Nonlinear Differentiable Maps

We establish necessary and sufficient conditions of invertibility for nonlinear differentiable maps in the case of arbitrary Banach spaces. We also establish conditions for the existence and uniqueness of bounded and almost periodic solutions of nonlinear differential and difference equations.

A study on the inverse of pseudo-differential operators on $$\mathbb {S}^{1}$$ S 1

In this paper we give a necessary condition for a pseudo-differential operator on \(L^{p}(\mathbb {S}^{1}),\ 1\le p<\infty \) to be invertible. In case \(p=2\), we also find a sufficient condition for invertibility of these operators and a formula for the symbol of the inverse operator. This formula helps us to get a sufficient condition for those operators to be unitary. Finally as an application of our results we get some information about the spectrum of a pseudo-differential operator on \(\mathbb {S}^{1}\).

Sufficient conditions for invertibility of linear stationary systems

Sufficient conditions for the invertibility of linear stationary dynamical systems are formulated. It is shown that the a priori information that the input signal is bounded substantially expands the class of systems for which the inversion problem is solvable.

Spectral Analysis of Abstract Parabolic Operators in Homogeneous Function Spaces

We use methods of harmonic analysis and group representation theory to study the spectral properties of the abstract parabolic operator $\mathscr L = -d/dt+A$ in homogeneous function spaces. We provide sufficient conditions for invertibility of such operators in terms of the spectral properties of the operator $A$ and the semigroup generated by $A$. We introduce a homogeneous space of functions with absolutely summable spectrum and prove a generalization of the Gearhart-Pr\"uss Theorem for such spaces. We use the results to prove existence and uniqueness of solutions of a certain class of non-linear equations.

Fitting of self-exciting threshold autoregressive moving average nonlinear time-series model through genetic algorithm and development of out-of-sample forecasts

Self-exciting threshold autoregressive moving average (SETARMA) nonlinear time-series model is considered here. Sufficient conditions for invertibility and stationarity are derived. Parameter estimation algorithm is developed by employing real-coded genetic algorithm stochastic optimization procedure. A significant feature of the work done is that optimal out-of-sample forecasts up to three-step ahead and their forecast error variances are derived analytically. Relevant computer programs are written in statistical analysis system (SAS) and C. As an illustration, annual mackerel catch time-series data are considered. Forecast performance of the fitted model for hold-out data is evaluated by using Naive and Monte Carlo approaches. It is found that optimal out-of-sample forecast values are quite close to actual values and estimated variances are quite close to theoretical values. Superiority of the SETARMA model over the SETAR model for equal predictive ability through Diebold–Mariano test is also established.

A Sharp Sufficient Condition for B-Spline Vector Field Invertibility. Application to Diffeomorphic Registration and Interslice Interpolation

In this paper, a new sufficient condition for the invertibility of a B-spline vector field as well as its extension to spatio-temporal vector fields are presented. The new condition, while guaranteeing that the vector field is diffeomorphic, is sharp enough to allow large deformations. The complexity of the constraint is also discussed. A new diffeomorphic registration algorithm using this condition and its application to interslice interpolation are presented and evaluated on synthetic images and in vivo magnetic resonance $T_2$ images of the brain. The registration method has also been compared to some state-of-the-art registration methods.

Some properties of Toeplitz operators on Dirichlet space with respect to Berezin type transform

In this paper, we consider the Toeplitz operators and little Hankel operators on the Dirichletspace D of the unit disc D. By Berezin type transform, the invariant subspaces of Toeplitz operators, asymptoticmultiplicative property of the Berezin type symbols of Toeplitz operators and solvability of the Riccati equationsof Toeplitz operators are discussed. We also obtained a sufficient condition for invertibility of Toeplitz operatorsand little Hankel operators in terms of Berezin transform. Moreover, the Hankel operators and Berezin transformare used to characterize compactness of operators 2 Tuv - TuTv - TvTu for functions u and v in L 2,1.

Necessary and sufficient conditions for invertibility of operators in spaces of real interpolation

Let A be a bounded linear operator from a couple (X0,X1) to a couple (Y0,Y1) such that the restrictions of A on the end spaces X0 and X1 have bounded inverses defined on Y0 and Y1, respectively. We are interested in the problem of how to determine if the restriction of A on the space (X0,X1)θ,q has a bounded inverse defined on the space (Y0,Y1)θ,q. In this paper, we show that a solution to this problem can be given in terms of indices of two subspaces of the kernel of the operator A on the space X0+X1.

Inverse of a fuzzy matrix of fuzzy numbers

The aim of this paper is to extend the concept of inverse of a matrix with fuzzy numbers as its elements, which may be used to model uncertain and imprecise aspects of real-world problems. We pursue two main ideas based on employing real scenarios and arithmetic operators. In each case, exact and inexact strategies are provided. In the first idea, we give some necessary and sufficient conditions for invertibility of fuzzy matrices based on regularity of their scenarios. And then Zadeh's extension principle and interpolation on Rohn's approach for inverting interval matrices are followed to compute fuzzy inverse. In the second idea, Dubois and Prade's arithmetic operators will be employed for the same purpose. But with respect to the inherent difficulties which are derived from the positivity restriction on spreads of fuzzy numbers, the concept of ϵ-inverse of a fuzzy matrix and its relaxation are generalized and some useful theorems will be revealed. Finally fuzzifying the defuzzified version of the original problem for introducing fuzzy inverse, which can be followed by each idea, will be presented.

A Simple Regularizer for B-spline Nonrigid Image Registration That Encourages Local Invertibility

Nonrigid image registration is an important task for many medical imaging applications. In particular, for radiation oncology it is desirable to track respiratory motion for thoracic cancer treatment. B-splines are convenient for modeling nonrigid deformations, but ensuring invertibility can be a challenge. This paper describes sufficient conditions for local invertibility of deformations based on B-spline bases. These sufficient conditions can be used with constrained optimization to enforce local invertibility. We also incorporate these conditions into nonrigid image registration methods based on a simple penalty approach that encourages diffeomorphic deformations. Traditional Jacobian penalty methods penalize negative Jacobian determinant values only at grid points. In contrast, our new method enforces a sufficient condition for invertibility directly on the deformation coefficients to encourage invertibility globally over a 3D continuous domain. The proposed penalty approach requires substantially less compute time than Jacobian penalties per iteration.

A necessary and sufficient condition for invertibility of adapted perturbations of identity on Wiener space

Let ( W , H , μ ) be the classical Wiener space, assume that U = I W + u is an adapted perturbation of identity satisfying the Girsanov identity. Then, U is invertible if and only if the kinetic energy of u is equal to the relative entropy of the measure induced with the action of U on the Wiener measure μ, in other words U is invertible if and only if 1 2 ∫ W | u | H 2 d μ = ∫ W d U μ d μ log d U μ d μ d μ . To cite this article: A.S. Üstünel, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Multivariate MIMO FIR inverses

The problem of computing exact finite impulse response (FIR) inverses for multivariate multiple-input multiple-output (MIMO) FIR systems is considered. Necessary and sufficient conditions for invertibility are given, along with computation techniques. Random systems and structured systems are defined. Necessary and sufficient conditions for the almost sure invertibility of structured random systems are derived. Bounds on the orders of the inverse filters are computed.

The invertibility of convolution type operators on a union of intervals and the corona theorem

The invertibility of convolution type operators on unions of intervals is studied. Sufficient conditions of invertibility for some classes of these operators are established. Solvability results forn-term corona problems are obtained using two different approaches: one involving reduction ton−1 Riemann-Hilbert problems in two variables and another involving reduction to two-term corona problems. The invertibility of the convolution operators on a union of intervals is also related to the invertibility of associated convolution operators on single intervals. Formulas for the inverse operators are given.

Invertibility of irreducible matrices

We obtain new sufficient conditions for invertibility of an irreducible complex matrix. Remarks are also given on eigenvalues (and the associated eigenvectors) that lie on the boundary of various spectrum inclusion regions of an irreducible matrix. Our results extend, strengthen, or clarify the recent work of Brualdi, Brualdi and Mellendorff, Farid, Solov'ev, and Zhang and Gu.