Bijective Operator Research Articles

Factorization of abstract operators into two second degree operators and its applications to integro-differential equations

Boundary value problem B1x = f with an abstract linear operator B1, corresponding to an Fredholm integro-differential equation with ordinary or partial differential operator is researched. An exact solution to B1x = f in the case when a bijective operator B1 has a factorization of the form B1 =BB0 where B and B0 are two linear more simple than B1 second degree abstract operators, received. Conditions for factorization of the operator B1 and a criterion for bijectivity of B1 are found.

Rowmotion Markov chains

Rowmotion is a certain well-studied bijective operator on the distributive lattice J(P) of order ideals of a finite poset P. We introduce the rowmotion Markov chainMJ(P) by assigning a probability px to each x∈P and using these probabilities to insert randomness into the original definition of rowmotion. More generally, we introduce a very broad family of toggle Markov chains inspired by Striker's notion of generalized toggling. We characterize when toggle Markov chains are irreducible, and we show that each toggle Markov chain has a remarkably simple stationary distribution.We also provide a second generalization of rowmotion Markov chains to the context of semidistrim lattices. Given a semidistrim lattice L, we assign a probability pj to each join-irreducible element j of L and use these probabilities to construct a rowmotion Markov chain ML. Under the assumption that each probability pj is strictly between 0 and 1, we prove that ML is irreducible. We also compute the stationary distribution of the rowmotion Markov chain of a lattice obtained by adding a minimal element and a maximal element to a disjoint union of two chains.We bound the mixing time of ML for an arbitrary semidistrim lattice L. In the special case when L is a Boolean lattice, we use spectral methods to obtain much stronger estimates on the mixing time, showing that rowmotion Markov chains of Boolean lattices exhibit the cutoff phenomenon.

Gram matrix associated to controlled frames

Controlled frames have been recently introduced in Hilbert spaces to improve the numerical efficiency of interactive algorithms for inverting the frame operator. In this paper, unlike the cross-Gram matrix of two different sequences which is not always a diagnostic tool, we define the controlled-Gram matrix of a sequence as a practical implement to diagnose that a given sequence is a controlled Bessel, frame or Riesz basis. Also, we discuss the cases that the operator associated to controlled Gram matrix will be bounded, invertible, Hilbert–Schmidt or a trace-class operator. Similar to standard frames, we present an explicit structure for controlled Riesz bases and show that every [Formula: see text]-controlled Riesz basis [Formula: see text] is in the form [Formula: see text], where [Formula: see text] is a bijective operator on [Formula: see text]. Furthermore, we propose an equivalent accessible condition to the sequence [Formula: see text] being a [Formula: see text]-controlled Riesz basis.

Bridging the Gap Between Imitation Learning and Inverse Reinforcement Learning.

Learning from demonstrations is a paradigm by which an apprentice agent learns a control policy for a dynamic environment by observing demonstrations delivered by an expert agent. It is usually implemented as either imitation learning (IL) or inverse reinforcement learning (IRL) in the literature. On the one hand, IRL is a paradigm relying on the Markov decision processes, where the goal of the apprentice agent is to find a reward function from the expert demonstrations that could explain the expert behavior. On the other hand, IL consists in directly generalizing the expert strategy, observed in the demonstrations, to unvisited states (and it is therefore close to classification, when there is a finite set of possible decisions). While these two visions are often considered as opposite to each other, the purpose of this paper is to exhibit a formal link between these approaches from which new algorithms can be derived. We show that IL and IRL can be redefined in a way that they are equivalent, in the sense that there exists an explicit bijective operator (namely, the inverse optimal Bellman operator) between their respective spaces of solutions. To do so, we introduce the set-policy framework that creates a clear link between the IL and the IRL. As a result, the IL and IRL solutions making the best of both worlds are obtained. In addition, it is a unifying framework from which existing IL and IRL algorithms can be derived and which opens the way for the IL methods able to deal with the environment's dynamics. Finally, the IRL algorithms derived from the set-policy framework are compared with the algorithms belonging to the more common trajectory-matching family. Experiments demonstrate that the set-policy-based algorithms outperform both the standard IRL and IL ones and result in more robust solutions.

The Bijectivity of the Tight Frame Operators in Lebesgue Spaces

 Abstract—The motivation of this dissertation mainly is which affine frame wavelet systems span Lebesgue spaces have been investigated less. The technique we used is analogous to technique of Calderon-Zygmund operators, but we rely on Calderon-Zygmund decomposition theorem. We prove our main results without smoothness assumption on frame wavelets. We prove that affine tight frame wavelets span Lebesgue spaces under the condition, we also show that the affine tight frame operator extends from 2 () L  to a bounded, linear and bijective operator on () p L  , for 1. p    Under such condition, the

Higher-order finite volume methods II: Inf–sup condition and uniform local ellipticity

The main purpose of this paper is to study the construction of higher-order finite volume methods (FVMs) of triangle meshes. We investigate the relationship of the three theoretical notions crucial in the construction of FVMs: the uniform ellipticity of the family of its discrete bilinear forms, its inf–sup condition and its uniform local ellipticity. Both the uniform ellipticity of the family of the discrete bilinear forms and its inf–sup condition guarantee the unique solvability of the FVM equations and the optimal error estimate of the approximate solution. We characterize the uniform ellipticity in terms of the inf–sup condition and a geometric condition on the bijective operator mapping from the trial space onto the test space involved in the construction of FVMs. We present a geometric interpretation of the inf–sup condition. Moreover, since the uniform local ellipticity is a convenient sufficient condition for the uniform ellipticity, we further provide sufficient conditions and necessary conditions of the uniform local ellipticity.

EXOTIC LAPLACIANS AND DERIVATIVES OF WHITE NOISE

In this paper, we give a relationship between the exotic Laplacians and the Lévy Laplacians in terms of the higher-order derivatives of white noise by introducing a bijective and continuous linear operator acting on white noise functionals. Moreover, we study a relationship between exotic Laplacians, acting on higher-order singular functionals, each other in terms of the constructed operator.

A characterization of product preserving maps with applications to a characterization of the Fourier transform

It is shown that a product preserving bijective (not necessarily real linear or continuous) operator on an appropriate class of complex valued functions must have either the form $[\phi\mapsto \phi\circ u]$ or $[\phi\mapsto\overline{\phi\circ u}]$ where $u$ is a fixed diffeomorphism of the base.

On New Bijective Convolution Operator Acting for Analytic Functions

Problem statement: We introduced a new bijective convolution linear operator defined on the class of normalized analytic functions. This operator was motivated by many researchers namely Srivastava, Owa, Ruscheweyh and many others. The operator was essential to obtain new classes of analytic functions. Approach: Simple technique of Ruscheweyh was used in our preliminary approach to create new bijective convolution linear operator. The preliminary concept of Hadamard products was mentioned and the concept of subordination was given to give sharp proofs for certain sufficient conditions of the linear operator aforementioned. In fact, the subordinating factor sequence was used to derive different types of subordination results. Results: Having the linear operator, subordination theorems were established by using standard concept of subordination. The results reduced to well-known results studied by various researchers. Coefficient bounds and inclusion properties, growth and closure theorems for some subclasses were also obtained. Conclusion: Therefore, many interesting results could be obtained and some applications could be gathered.

On New Bijective Convolution Operator Act for Analytic Functions

On New Bijective Convolution Operator Act for Analytic Functions

A computable version of Banach’s Inverse Mapping Theorem

Given a program of a linear bounded and bijective operator T , does there exist a program for the inverse operator T − 1 ? And if this is the case, does there exist a general algorithm to transfer a program of T into a program of T − 1 ? This is the inversion problem for computable linear operators on Banach spaces in its non-uniform and uniform formulation, respectively. We study this problem from the point of view of computable analysis which is the Turing machine based theory of computability on Euclidean space and other topological spaces. Using a computable version of Banach’s Inverse Mapping Theorem we can answer the first question positively. Hence, the non-uniform version of the inversion problem is solvable, while a topological argument shows that the uniform version is not. Thus, we are in the striking situation that any computable linear operator has a computable inverse while there exists no general algorithmic procedure to transfer a program of the operator into a program of its inverse. As a consequence, the computable version of Banach’s Inverse Mapping Theorem is a powerful tool which can be used to produce highly non-constructive existence proofs of algorithms. We apply this method to prove that a certain initial value problem admits a computable solution. As a preparation of Banach’s Inverse Mapping Theorem we also study the Open Mapping Theorem and we show that the uniform versions of both theorems are limit computable, which means that they are effectively Σ 2 0 -measurable with respect to the effective Borel hierarchy.

Commutativity Preserving Maps of Factors

By a von Neumann algebra M we mean a weakly closed, self-adjoint algebra of operators on a Hilbert space which contains I, the identity operator. A factor is a von Neumann algebra whose centre consists of scalar multiples of I.In all that follows ϕ:M → N will be a one to one, *-linear map from the von Neumann factor M onto the von Neumann algebra N such that both ϕ and ϕ−1 preserve commutativity. Our main result states that if M is not of type I2 then where is an isomorphism or an antiisomorphism, c is a non-zero scalar, and λ is a *-linear map from M into ZN, the centre of N.Our interest in this problem was aroused by several recent results. In [1], Choi, Jafarian, and Radjavi proved that if S is the real linear space of n × n matrices over any algebraically closed field, n ≧ 3, and ψ a linear operator on S which preserves commuting pairs of matrices, then either ψ(S) is commutative or there exists a unitary matrix U such thatfor all A in S. They proved an analogous result for the collection of all bounded self-adjoint operators on an infinite dimensional Hilbert space when ψ is one to one. Subsequently, Omladic [7] proved that if ψ:L(X) → L(X) is a bijective linear operator preserving commuting pairs of operators where X is a non-trivial Banach space, thenwhere U is a bounded invertible operator on X and A′ is the adjoint of A.

On operators preserving commutativity

Let L( X) be the algebra of all bounded operators on a non-trivial complex Banach space X and F: L( X) → L( X) a bijective linear operator such that F and F −1 both send commuting pairs of operators into commuting pairs. Then, either F( A) = σUAU −1 + p( A) I, or F( A) = σUA′ U −1 + p( A) I, where p is a linear functional on L( X), U is a bounded linear bijective operator between the appropriate two spaces, σ is a complex constant, and A′ is the adjoint of A. The form of an operator F for which F and F −1 both send projections of rank one into projections of rank one is also determined.

On the boundary operator in electromagnetic scattering

SynopsisFor radiating solutions to the time-harmonic Maxwell equations, it is shown that the boundary operator mapping the tangential components of the electric field into the tangential components of the magnetic field is a bounded bijective operator from the space of Holder continuous tangential fields with Hölder continuous surface divergence onto itself.