Left Inverse Research Articles

A matrix pencil approach to the existence of compactly supported reconstruction functions in average sampling

The aim of this work is to solve a question raised for average sampling in shift-invariant spaces by using the well-known matrix pencil theory. In many common situations in sampling theory, the available data are samples of some convolution operator acting on the function itself: this leads to the problem of average sampling, also known as generalized sampling. In this paper we deal with the existence of a sampling formula involving these samples and having reconstruction functions with compact support. Thus, low computational complexity is involved and truncation errors are avoided. In practice, it is accomplished by means of a FIR filter bank. An answer is given in the light of the generalized sampling theory by using the oversampling technique: more samples than strictly necessary are used. The original problem reduces to finding a polynomial left inverse of a polynomial matrix intimately related to the sampling problem which, for a suitable choice of the sampling period, becomes a matrix pencil. This matrix pencil approach allows us to obtain a practical method for computing the compactly supported reconstruction functions for the important case where the oversampling rate is minimum. Moreover, the optimality of the obtained solution is established.

Inverses and quotients of mappings between ordered sets

In this paper, we study inverses and quotients of mappings between ordered sets, in particular between complete lattices, which are analogous to inverses and quotients of positive numbers. We investigate to what extent a generalized inverse can serve as a left inverse and as a right inverse, and how an inverse of an inverse relates to the identity mapping. The generalized inverses and quotients are then used to create a convenient formalism for dilations and erosions as well as for cleistomorphisms (closure operators) and anoiktomorphisms (kernel operators).

Refunctionalization at work

We present the left inverse of Reynolds’ defunctionalization and we show its relevance to programming and to programming languages. We propose two methods to transform a program that is almost in defunctionalized form into one that is actually in defunctionalized form, and we illustrate them with a recognizer for Dyck words and with Dijkstra’s shunting-yard algorithm.

Stability of localized operators

Let ℓ p , 1 ⩽ p ⩽ ∞ , be the space of all p-summable sequences and C a be the convolution operator associated with a summable sequence a. It is known that the ℓ p -stability of the convolution operator C a for different 1 ⩽ p ⩽ ∞ are equivalent to each other, i.e., if C a has ℓ p -stability for some 1 ⩽ p ⩽ ∞ then C a has ℓ q -stability for all 1 ⩽ q ⩽ ∞ . In the study of spline approximation, wavelet analysis, time–frequency analysis, and sampling, there are many localized operators of non-convolution type whose stability is one of the basic assumptions. In this paper, we consider the stability of those localized operators including infinite matrices in the Sjöstrand class, synthesis operators with generating functions enveloped by shifts of a function in the Wiener amalgam space, and integral operators with kernels having certain regularity and decay at infinity. We show that the ℓ p -stability (or L p -stability) of those three classes of localized operators are equivalent to each other, and we also prove that the left inverse of those localized operators are well localized.

Analytic projections, Corona problem and geometry of holomorphic vector bundles

The main result of the paper is a theorem giving a sufficient condition for the existence of a bounded analytic projection onto a holomorphic family of generally infinite dimensional subspaces (a holomorphic sub-bundle of a trivial bundle). This sufficient condition is also necessary in the case of finite dimension or codimension of the bundle. A simple lemma of N. Nikolski connects the existence of a bounded analytic projection with the Operator Corona Problem (existence of a bounded analytic left inverse for an operator-valued function), so as corollaries of the main result we obtain new results about the Operator Corona Problem. In particular, we find a new sufficient condition, a complete solution in the case of finite codimension, and a solution of the generalized Corona Problem.

Refunctionalization at Work

We present the left inverse of Reynolds's defunctionalization and we show its relevance to programming and to programming languages. We propose two methods to transform a program that is almost in defunctionalized form into one that is actually in defunctionalized form, and we illustrate them with a recognizer for Dyck words and with Dijkstra's shunting-yard algorithm.

Sign patterns that require a positive or nonnegative left inverse

An m by n sign pattern A is an m by n matrix with entries in {+, −, 0}. The signpattern A requires a positive (resp. nonnegative) left inverse provided each real matrix with sign pattern A has a left inverse with all entries positive (resp. nonnegative). In this paper, necessary and sufficient conditions are given for a sign pattern to require a positive or nonnegative left inverse. It is also shown that for n ≥ 2, there are no square sign patterns of order n that require a positive (left) inverse, and that an n by n sign pattern requiring a nonnegative (left) inverse is permutationally equivalent to an upper triangular sign pattern with positive main diagonal entries and nonpositive off-diagonal entries.

Synthèse d'un filtre de détection de défauts robuste. Approche par inversion du système

In this paper, we present an unified approach for fault detection and isolation in discrete-time systems affected by noises and faults on measurement and state equations. The proposed robust fault detection filter will be designed under less restrictive conditions compared with classical fault detection filter. After having parameterized the minimum-time left inverse of the system, the degrees of freedom remaining available will be computed to generate an optimal faults estimation. The latter is minimally sensitive to state and measurement noise.

On one-pass CPS transformations

AbstractWe bridge two distinct approaches to one-pass CPS transformations, i.e, CPS transformations that reduce administrative redexes at transformation time instead of in a post-processing phase. One approach is compositional and higher-order, and is independently due to Appel, Danvy and Filinski, and Wand, building on Plotkin's seminal work. The other is non-compositional and based on a reduction semantics for the lambda-calculus, and is due to Sabry and Felleisen. To relate the two approaches, we use three tools: Reynolds's defunctionalization and its left inverse, refunctionalization; a special case of fold–unfold fusion due to Ohori and Sasano, fixed-point promotion; and an implementation technique for reduction semantics due to Danvy and Nielsen, refocusing. This work is directly applicable to transforming programs into monadic normal form.

Largest subsemigroups of the full transformation monoid

In this paper we are concerned with the following question: for a semigroup S, what is the largest size of a subsemigroup T ⩽ S where T has a given property? The semigroups S that we consider are the full transformation semigroups; all mappings from a finite set to itself under composition of mappings. The subsemigroups T that we consider are of one of the following types: left zero, right zero, completely simple, or inverse. Furthermore, we find the largest size of such subsemigroups U where the least rank of an element in U is specified. Numerous examples are given.

A New Error Correction Intuitive Approach in Quantum Communications Protocols

Quantum Error Correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions. We obtain necessary and sufficient conditions for the perfect recovery of an encoded state after its degradation by an interaction. The conditions depend only on the behavior of the logical states. We use them to give a recovery operator independent definition of error-correcting codes. We relate this definition to four others: The existence of a left inverse of the interaction, an explicit representation of the error syndrome using tensor products, perfect recovery of the completely entangled state and an information theoretic identity. Two notions of fidelity and error for imperfect recovery are introduced, one for pure and the other for entangled states. The latter is more appropriate when using codes in a quantum memory or in applications of quantum teleportation to communication. We show that the error for entangled states is bounded linearly by the error for pure states. A formal definition of independent interactions for qubits is given. This leads to lower bounds on the number of qubits required to correct errors and a formal proof that the classical bounds on the probability of error- correcting codes applies to error correcting quantum codes, provided that the interaction is dominated by an identity component.

On One-Pass CPS Transformations

We bridge two distinct approaches to one-pass CPS transformations, i.e., CPS transformations that reduce administrative redexes at transformation time instead of in a post-processing phase. One approach is compositional and higher-order, and is independently due to Appel, Danvy and Filinski, and Wand, building on Plotkin's seminal work. The other is non-compositional and based on a reduction semantics for the lambda-calculus, and is due to Sabry and Felleisen. To relate the two approaches, we use three tools: Reynolds's defunctionalization and its left inverse, refunctionalization; a special case of fold-unfold fusion due to Ohori and Sasano, fixed-point promotion; and an implementation technique for reduction semantics due to Danvy and Nielsen, refocusing.<br /> <br />This work is directly applicable to transforming programs into monadic normal form.

Refunctionalization at Work

We present the left inverse of Reynolds's defunctionalization and we show its relevance to programming and to programming languages. We present two methods to put a program that is almost in defunctionalized form into one that is actually in defunctionalized form, and we illustrate them with a recognizer for Dyck words and with Dijkstra's shunting-yard algorithm.

Sign Patterns That Allow a Positive or Nonnegative Left Inverse

An $m$ by $n$ sign pattern $\mathcal{S}$ is an $m$ by $n$ matrix with entries in $\{+, -, 0 \}$. Such a sign pattern allows a positive (resp., nonnegative) left inverse, provided that there exist an $m$ by $n$ matrix $A$ with the sign pattern $\mathcal{S}$ and an $n$ by $m$ matrix $B$ with only positive (resp., nonnegative) entries satisfying $BA=I_{n}$, where $I_{n}$ is the $n$ by $n$ identity matrix. For $m>n \geq 2$, a characterization of $m$ by $n$ sign patterns with no rows of zeros that allow a positive left inverse is given. This leads to a characterization of all $m$ by $n$ sign patterns with $m \geq n \geq 2$ that allow a positive left inverse, giving a generalization of the known result for the square case, which involves a related bipartite digraph. For $m \geq n$, $m$ by $n$ sign patterns with all entries in $\{+,0\}$ and $m$ by 2 sign patterns with $m \geq 2$ that allow a nonnegative left inverse are characterized, and some necessary or sufficient conditions for a general $m$ by $n$ sign pattern to allow a nonnegative left inverse are presented.

A FOLIATION FOR UNKNOWN INPUTS RECONSTRUCTION

This paper deals with a new method to compute the left inverse for a class of dynamical systems. This method is based on a projection which decompose the state space into two transverse foliations: one is tangent to the unknown inputs and the other is transverse. Contrary to the traditional methods, our method enables us to write the inverse dynamic system under consideration. Within this work we highlight our propositions by examples.

Multidimensional Multichannel FIR Deconvolution Using GrÖbner Bases

We present a new method for general multidimensional multichannel deconvolution with finite impulse response (FIR) convolution and deconvolution filters using Gröbner bases. Previous work formulates the problem of multichannel FIR deconvolution as the construction of a left inverse of the convolution matrix, which is solved by numerical linear algebra. However, this approach requires the prior information of the support of deconvolution filters. Using algebraic geometry and Gröbner bases, we find necessary and sufficient conditions for the existence of exact deconvolution FIR filters and propose simple algorithms to find these deconvolution filters. The main contribution of our work is to extend the previous Gröbner basis results on multidimensional multichannel deconvolution for polynomial or causal filters to general FIR filters. The proposed algorithms obtain a set of FIR deconvolution filters with a small number of nonzero coefficients (a desirable feature in the impulsive noise environment) and do not require the prior information of the support. Moreover, we provide a complete characterization of all exact deconvolution FIR filters, from which good FIR deconvolution filters under the additive white noise environment are found. Simulation results show that our approaches achieve good results under different noise settings.

A DIVISION ALGORITHM

A divisibility test of Arend Heyting, for polynomials over a field in an intuitionistic setting, may be thought of as a kind of division algorithm. We show that such a division algorithm holds for divisibility by polynomials of content 1 over any commutative ring in which nilpotent elements are zero. In addition, for an arbitrary commutative ring R, we characterize those polynomials g such that the R-module endomorphism of R[X] given by multiplication by g has a left inverse.

The Volterra Operator is not Supercyclic

It is shown that the classical Volterra operator, which is cyclic, is not supercyclic on any of the spaces L p [0, 1], 1 ≤ p < ∞. This solves a question posed by Hector Salas. This contrasts with the fact that the derivative operator, the left inverse of the Volterra operator, although unbounded, is hypercyclic on L p [0, 1].

Wave diffraction by a strip grating: the two‐straight line approach

AbstractBoundary‐value problems of wave diffraction by a periodic strip grating are associated with a Toeplitz operator acting on a space of functions defined on a two‐straight line contour. Simple formulas are given for the left inverse of the operator associated with the Neumann boundary‐value problem and for the right inverse of the operator associated with the Dirichlet boundary‐value problem when the period of the grating is equal to double the width of the strips. (© 2004 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Classroom note: On postulates of a group

A semigroup G is a group if it has a left identity and every element has a left inverse. The purpose of this note is to weaken this condition further in two different ways.