Abstract: This paper proposes a unified framework for the investigation of constrained learning theory in reflexive Banach spaces of features via regularized empirical risk minimization. The focus is placed on Tikhonov-like regularization with totally convex functions. This broad class of regularizers provides a flexible model for various priors on the features, including, in particular, hard constraints and powers of Banach norms. In such context, the main results establish a new general form of the representer theorem and the consistency of the corresponding learning schemes under general conditions on the loss function, the geometry of the feature space, and the modulus of total convexity of the regularizer. In addition, the proposed analysis gives new insight into basic tools such as reproducing Banach spaces, feature maps, and universality. Even when specialized to Hilbert spaces, this framework yields new results that extend the state of the art.

Abstract: In this paper we will develop a family of non-conforming " Crouzeix-Raviart " type finite elements in three dimensions. They consist of local polynomials of maximal degree p ∈ N on simplicial finite element meshes while certain jump conditions are imposed across adjacent simplices. We will prove optimal a priori estimates for these finite elements. The characterization of this space via jump conditions is implicit and the derivation of a local basis requires some deeper theoretical tools from orthogonal polynomials on triangles and their representation. We will derive these tools for this purpose. These results allow us to give explicit representations of the local basis functions. Finally we will analyze the linear independence of these sets of functions and discuss the question whether they span the whole non-conforming space.

Abstract: We study a new mathematical model which describes the equilibrium of a locking material in contact with a foundation. The contact is frictionless and is modeled with a nonsmooth multivalued interfa...

Abstract: This paper studies the planar deformations of a beam composed of a linearly elastic material. Starting from the field equations for the plane-stress problem and adopting a series expansion for the displacement vector about the bottom surface, we deduce the beam equations with two unknowns in a consistent manner. The success relies on using the field equations together with the bottom traction conditions to establish the exact recursion relations, such that all quantities can be represented in terms of the two leading expansion coefficients of the displacements. Another feature is that the remainders of the series can be carried over to the beam equations. Then, based on the general solutions and the error terms of the beam equations, pointwise error estimates for displacement and stress fields are rigorously established. Three benchmark problems are considered, for which the two-dimensional exact solutions are available. It is shown that this new beam theory recovers the exact solutions for these problems. Two cases with boundary layer effects are also discussed in the appendix.

Abstract: In this paper, we investigate in a Hilbert space setting a second-order dynamical system of the form ẍ(t) + γ(t)ẋ(t) + x(t) − Jλ(t)A(x(t) − λ(t)D(x(t)) − λ(t)β(t)B(x(t))) = 0, where A : ℋimageℋ is ...

Abstract: We provide the full asymptotic description of the quasi-normal modes (resonances) in any strip of fixed width for Dirac fields in slowly rotating Kerr–Newman–de Sitter black holes. The resonances s...

Abstract: General radiation magnetic hydrodynamics models include two main parts that are coupled: one part is the macroscopic magnetic fluid part, which is governed by the ideal compressible magnetohydrodyn...

Abstract: We study an initial boundary value problem for the nonhomogeneous heat conducting fluids with non-negative density First of all, we show that for the initial density allowing vacuum, the strong so

Abstract: The problem of minimizing a separable convex function under linearly coupled constraints arises from various application domains such as economic systems, distributed control, and network flow. The...

Abstract: The paper studies optimal strategies for a borrower who needs to repay his debt, in an infinite time horizon. An instantaneous bankruptcy risk is present, which increases with the size of the debt. This induces a pool of risk-neutral lenders to charge a higher interest rate, to compensate for the possible loss of part of their investment. Solutions are interpreted as Stackelberg equilibria, where the borrower announces his repayment strategy u(t) at all future times, and lenders adjust the interest rate accordingly. This yields a highly non-standard problem of optimal control, where the instantaneous dynamics depend on the entire future evolution of the system. Our analysis shows the existence of optimal open-loop controls, deriving necessary conditions for optimality and characterizing possible asymptotic limits as t → +∞.

Abstract: We combine techniques in meshfree methods and Gaussian process regressions to construct kernel-based estimators for numerical derivatives from noisy data. Specially, we construct meshfree estimator...

Abstract: We study continuity properties in Lebesgue spaces for a class of Fourier integral operators arising in the study of the Boltzmann equation. The phase has a Holder-type singularity at the origin. We...

Abstract: We present a hydrodynamic model for the ensemble of thermodynamic Cucker–Smale (TCS) particles in the presence of a temperature field, and study its global-in-time well-posedness in Sobolev space. ...

Abstract: The standard Sobolev space W2s(ℝd), with arbitrary positive integers s and d for which s > d/2, has the reproducing kernel Kd,s(x,t) =∫ℝd ∏j=1dcos(2π(x j − tj)uj) 1 +∑0<|α|1≤s∏j=1d(2πuj)2αjdu for a...

Abstract: In this paper, we investigate the global well-posedness of classical solutions to three-dimensional Cauchy problem of the compressible Navier–Stokes type system with a Korteweg stess tensor under the condition that the initial energy is small. This result improves previous results obtained by Hattori–Li in [H. Hattori and D. Li, Solutions for two dimensional system for materials of Korteweg type, SIAM J. Math. Anal. 25 (1994) 85–98; H. Hattori and D. Li. Global solutions of a high-dimensional system for Korteweg materials. J. Math. Anal. Appl. 198 (1996) 84–97.], where the existence of the classical solution is established for initial data close to an equilibrium in some Sobolev space Hs.

Abstract: Many problems in applications are defined on a bounded interval Therefore, wavelets and framelets on a bounded interval are of importance in both theory and application There is a great deal of e