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Abstract We study groups of surjective linear isometries on Banach algebras of Lipschitz maps with values in some unital C ⁎ -algebras. In this paper, these spaces are endowed with the sum norm. For the case where the C ⁎ -algebras are commutative whose groups of all surjective isometries are algebraically reflexive, we prove that the group of all surjective isometries on the corresponding Banach algebra of Lipschitz maps are algebraically reflexive. We also prove that the group of unital surjective isometries between matrix-valued Lipschitz algebras are reflexive. [ABSTRACT FROM AUTHOR]

We consider a class of (ill-posed) optimal control problems in which a distributed vector-valued control is enforced to take values pointwise in a finite set M ⊂ Rm. After convex relaxation, one obtains a well-posed optimization problem, which still promotes control values in M. We state the corresponding well-posedness and stability analysis and exemplify the results for two specific cases of quite general interest, optimal control of the Bloch equation and optimal control of an elastic deformation. We finally formulate a semismooth Newton method to numerically solve a regularized version of the optimal control problem and illustrate the behavior of the approach for our example cases. [ABSTRACT FROM AUTHOR]

Learning partial differential equations (LPDEs) from training data for particular tasks has been successfully applied to many image processing problems. In this paper, we propose a more effective LPDEs model for vector-valued image tasks. The PDEs are also formulated as a linear combination of fundamental differential invariants, but have several distinctions. First, we simplify the current LPDEs system by omitting a PDE which works as an indicate function in current ones. Second, instead of using L2-norm, we use the L1-norm to regularize the coefficients with respect to the fundamental differential invariants. Third, as the objective function is not smooth, we resort to the alternating direction method to optimize it. We illustrate the properties of our LPDEs system by several examples in denoising and demosaicking of RGB color images. The experiments demonstrate the advantage of the proposed method over other PDE-based methods. [ABSTRACT FROM AUTHOR]

We consider the multiplicative complexity of Boolean functions with multiple bits of output, studying how large a multiplicative complexity is necessary and sufficient to provide a desired nonlinearity. For so-called ΣΠΣ circuits, we show that there is a tight connection between error correcting codes and circuits computing functions with high nonlinearity. Combining this with known coding theory results, we show that functions with n inputs and n outputs with the highest possible nonlinearity must have at least 2.32 n AND gates. We further show that one cannot prove stronger lower bounds by only appealing to the nonlinearity of a function; we show a bilinear circuit computing a function with almost optimal nonlinearity with the number of AND gates being exactly the length of such a shortest code. Additionally we provide a function which, for general circuits, has multiplicative complexity at least 2 n − 3 . Finally we study the multiplicative complexity of “almost all” functions. We show that every function with n bits of input and m bits of output can be computed using at most 2.5 ( 1 + o ( 1 ) ) m 2 n AND gates. [ABSTRACT FROM AUTHOR]

We present a unified approach to study the problem of the commons for agents with other-regarding preferences. This situation is modeled as a game with vector-valued utilities. Several types of agents are characterized depending on the importance assigned to the components of their utility functions. We obtain the set of equilibria of the game with two types of agents, pro-social and pro-self, and some refinements of this set for conservative agents. The most relevant result is that only a pro-social agent is required to avoid the tragedy of the commons, regardless of the behavior of the rest of the agents. [ABSTRACT FROM AUTHOR]

An extension of the well-posedness analysis of the scalar and the vector advection–reaction problem in Banach graph spaces of power p ∈ ( 1 , ∞ ) is proposed. This analysis is based on the sign of the associated Friedrichs tensor, taking positive, null or reasonably negative values. [ABSTRACT FROM AUTHOR]

In this paper, we establish UMD lattice-valued variational inequalities for ergodic averages of contractively regular operators and analytic semigroups. These results generalize, on the one hand some scalar-valued variational inequalities in ergodic theory, on the other hand Xu's very recent result on UMD lattice-valued maximal inequality. As an application, we deduce the jump estimates and obtain quantitative information on the rate of the pointwise convergence for semigroups acting on UMD lattice-valued functions. [ABSTRACT FROM AUTHOR]

We first obtain that sub differentials of set-valued mapping from finite-dimensional spaces to finite-dimensional possess certain relaxed compactness. Then using this weak compactness, we establish gap functions for generalized Stampacchia vector variational-like inequalities which are defined by means of subdifferentials. Finally, an existence result of generalized weakly efficient solutions for vector optimization problem involving a subdifferentiable and preinvex set-valued mapping is established by exploiting the existence of a solution for the weak formulation of the generalized Stampacchia vector variational-like inequality via a Fan-KKM lemma. [ABSTRACT FROM AUTHOR]

In this article we present a generalization of the concept of function of bounded variation, in the sense of Riesz, for functions defined on a rectangle in R2, which take values in a Banach space. As applications, we obtain generalizations of some results due to Chistyakov and a counterpart of the classical Riesz's Lemma. [ABSTRACT FROM AUTHOR]

Abstract: Wavelet analysis has been applied to many aspects in science and technology. The notion of vector-valued wavelets with four-scale dilation factor associated with an orthogonal vector-valued scaling function is introduced. The existence of orthogonal vector-valued wavelets with quarternary-scale is discussed. A necessary and sufficient condition is presented by means of vector-valued multiresolution analysis and paraunitary vector filter bank theory. An algorithm for constructing a sort of orthogonal vector-valued wavelets with compact support is proposed, and their orthogonal properties are discussed. [Copyright &y& Elsevier]

Several ways to generalize scalar total variation to vector-valued functions have been proposed in the past. In this paper, we give a detailed analysis of a variant we denote by TVJ, which has not been previously explored as a regularizer. The contributions of the manuscript are twofold: on the theoretical side, we show that TVJ can be derived from the generalized Jacobians from geometric measure theory. Thus, within the context of this theory, TVJ is the most natural form of a vectorial total variation. As an important feature, we derive how TVJ can be written as the support functional of a convex set in 픏². This property allows us to employ fast and stable minimization algorithms to solve inverse problems. The analysis also shows that in contrast to other total variation regularizers for color images, the proposed one penalizes across a common edge direction for all channels, which is a major theoretical advantage. Our practical contribution consist of an extensive experimental section, where we compare the performance of a number of provable convergent algorithms for inverse problems with our proposed regularizer. In particular, we show in experiments for denoising, deblurring, superresolution, and inpainting that its use leads to a significantly better restoration of color images, both visually and quantitatively. Source code for all algorithms employed in the experiments is provided online. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the well-known transitive algebra problem and reductive algebra problem on vector valued reproducing analytic Hilbert spaces. For an analytic Hilbert space with complete Nevanlinna–Pick kernel k, it is shown that both transitive algebra problem and reductive algebra problem on multiplier invariant subspaces of have positive answer if the algebras contain all analytic multiplication operators. This extends several known results on the problems. [Copyright &y& Elsevier]

Abstract: In this paper, we give existence and uniqueness results for solutions of nonlocal boundaryvector value problems of the form,, where n2, ƒ : [0, 1] x (R N 1 ) n → R N 1 is a Carathéodory function, g : [0, l] → R N 1 × R N 1 is a Lebesgue measurable N 1 × N 1-matrix function and it satisfies g(0) = 0, the integral is in sense of Riemann-Stieltjes. The existence of a solutions is proven by the coincidence degree theory. As an application, we also give one example to demonstrate our results. [Copyright &y& Elsevier]