Continuity Properties Research Articles

New insights in the oxidation behavior of (FeCoCrNi)94Al4Ti2Six high entropy alloys at 1100 °C

A new high-entropy alloy (FeCoNiCr)94Al4Ti2Six with excellent high-temperature oxidation resistance at 1100°C has been developed. The inner oxide behavior with Si added is the key factor of its oxidation resistance. The effect of Si content on the types, continuity, spalling resistance, and diffusion inhibition property of the inner oxide layer were revealed. An internal nitridation of Ti phenomenon was observed in this HEA and the preferential formation of TiN was confirmed by DFT calculation. A novel model for the oxidation resistance of high-entropy alloys affected by silicon is proposed.

Mappings of generalized finite distortion and continuity

AbstractWe study continuity properties of Sobolev mappings , , that satisfy the following generalized finite distortion inequality for almost every . Here and are measurable functions. Note that when , we recover the class of mappings of finite distortion, which are always continuous. The continuity of arbitrary solutions, however, turns out to be an intricate question. We fully solve the continuity problem in the case of bounded distortion , where a sharp condition for continuity is that is in the Zygmund space for some . We also show that one can slightly relax the boundedness assumption on to an exponential class with , and still obtain continuous solutions when with . On the other hand, for all with , we construct a discontinuous solution with and , including an example with and .

Approximation Properties of Extended Beta-Type Szász–Mirakjan Operators

In this article, we introduce generalized beta extension of Szacute{a}sz-integral type operators and study their approximation properties. First, we calculate the some estimates for these operators. Further, we study the uniform convergence and order of approximation in terms of Korovkin-type theorem and modulus of continuity for the space of univariate continuous functions and bivariate continuous functions in their sections.. Moreover, numerical estimates and graphical representations for convergence of one- and two-dimensional sequences of operators are studied. In continuation, local and global approximation properties are studied in terms of the first- and second-order modulus of smoothness, Peetre’s K-functional and weight functions in various functional spaces.

Learning continuous models for continuous physics

Dynamical systems that evolve continuously over time are ubiquitous throughout science and engineering. Machine learning (ML) provides data-driven approaches to model and predict the dynamics of such systems. A core issue with this approach is that ML models are typically trained on discrete data, using ML methodologies that are not aware of underlying continuity properties. This results in models that often do not capture any underlying continuous dynamics—either of the system of interest, or indeed of any related system. To address this challenge, we develop a convergence test based on numerical analysis theory. Our test verifies whether a model has learned a function that accurately approximates an underlying continuous dynamics. Models that fail this test fail to capture relevant dynamics, rendering them of limited utility for many scientific prediction tasks; while models that pass this test enable both better interpolation and better extrapolation in multiple ways. Our results illustrate how principled numerical analysis methods can be coupled with existing ML training/testing methodologies to validate models for science and engineering applications.

Strong unique continuation for variable coefficient parabolic operators with Hardy type potential

In this paper, we prove the strong unique continuation property at the origin for solutions of the following scaling critical parabolic differential inequality|div(A(x,t)∇u)−ut|≤M|x|2|u|, where the coefficient matrix A is Lipschitz continuous in x and t. Our main result sharpens a previous one of Vessella concerned with the subcritical case as well as extends an earlier result of one of us with Garofalo and Manna for the heat operator.

Pseudo-differential calculi and entropy estimates with Orlicz modulation spaces

We deduce continuity properties for pseudo-differential operators with symbols in Orlicz modulation spaces when acting on other Orlicz modulation spaces. In particular we extend well-known results in the literature. For example we generalize the classical result by Cordero and Nicola that if1p′+1q′⩽1p1+1p2,1p′+1q′⩽1q1+1q2,pj,qj⩽q′,q⩽p and a∈Mp,q, then the pseudo-differential operator Op(a) is continuous from Mp1,q1 to Mp2′,q2′.We also show that the entropy functional Eϕ possess suitable continuity properties on a suitable Orlicz modulation space MΦ satisfying Mp⊆MΦ⊆M2, though Eϕ is discontinuous on M2=L2.

Mild Solutions of Fractional Integrodifferential Diffusion Equations with Nonlocal Initial Conditions via the Resolvent Family

Fractional integrodifferential diffusion equations play a significant role in describing anomalous diffusion phenomena. In this paper, we study the existence and uniqueness of mild solutions to these equations. Firstly, we construct an appropriate resolvent family, through which the related equicontinuity, strong continuity, and compactness properties are studied using the convolution theorem of Laplace transform, the probability density function, the Cauchy integral formula, and the Fubini theorem. Then, we construct a reasonable mild solution for the considered equations. Finally, we obtain some sufficient conditions for the existence and uniqueness of mild solutions to the considered equations by some fixed point theorems.

The continuity of 𝑝-rationality and a lower bound for 𝑝’-degree characters of finite groups

Let p p be a prime and G G a finite group. We propose a strong bound for the number of p ′ p’ -degree irreducible characters of G G in terms of the commutator factor group of a Sylow p p -subgroup of G G . The bound arises from a recent conjecture of Navarro and Tiep [Forum Math. Pi 9 (2021), pp. 1–28] on fields of character values and a phenomenon called the continuity of p p -rationality level of p ′ p’ -degree characters. This continuity property in turn is predicted by the celebrated McKay-Navarro conjecture (see G. Navarro [Ann. of Math. (2) 160 (2004), pp. 1129–1140]). We achieve both the bound and the continuity property for p = 2 p=2 .

Wave breaking in the unidirectional non-local wave model

In this paper we study wave breaking in the unidirectional non-local wave model describing the motion of a collision-free plasma in a magnetic field. By analyzing the monotonicity and continuity properties of a system of the Riccati-type differential inequalities involving the extremal slopes of flows, we show a new sufficient condition on the initial data to exhibit wave breaking. Moreover, the estimates of life span and wave breaking rate are derived.

Asymptotic behavior of the Zhang–Kawazumi's φ-invariants

The φ-invariant was introduced by Shou-Wu Zhang and Nariya Kawazumi. We study the continuity property of the φ-invariants for degenerating graphs, show that this continuity property induces an adelic divisor on the moduli space of Riemann surfaces, and then give an asymptotic expression of the Zhang–Kawazumi's invariants for Riemann surfaces near the boundary of the moduli space. We mainly use Yuan–Zhang's adelic divisors, and follow Yuan's idea of globalization of φ-invariants.

Robustness of Stochastic Optimal Control to Approximate Diffusion Models Under Several Cost Evaluation Criteria

In control theory, typically a nominal model is assumed based on which an optimal control is designed and then applied to an actual (true) system. This gives rise to the problem of performance loss because of the mismatch between the true and assumed models. A robustness problem in this context is to show that the error because of the mismatch between a true and an assumed model decreases to zero as the assumed model approaches the true model. We study this problem when the state dynamics of the system are governed by controlled diffusion processes. In particular, we discuss continuity and robustness properties of finite and infinite horizon α-discounted/ergodic optimal control problems for a general class of nondegenerate controlled diffusion processes as well as for optimal control up to an exit time. Under a general set of assumptions and a convergence criterion on the models, we first establish that the optimal value of the approximate model converges to the optimal value of the true model. We then establish that the error because of the mismatch that occurs by application of a control policy, designed for an incorrectly estimated model, to a true model decreases to zero as the incorrect model approaches the true model. We see that, compared with related results in the discrete-time setup, the continuous-time theory lets us utilize the strong regularity properties of solutions to optimality (Hamilton–Jacobi–Bellman) equations, via the theory of uniformly elliptic partial differential equations, to arrive at strong continuity and robustness properties. Funding: The research of S. Yüksel was partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).

On the Unique Continuation Property for a Coupled System of Third-order Nonlinear Schrodinger Equations

In this paper, we study the unique continuation properties for a coupled system of third-order nonlinear Schrodinger equations and show the Carleman estimates of L2 and Lp (p > 2) types, as well as exponential decay properties of the solutions. As a consequence we obtain that if (\(\mathit{u}\), \(\mathit{w}\)) = (\(\mathit{u}\)(\(\mathit{x}\), \(\mathit{t}\)), \(\mathit{w}\)(\(\mathit{x}\), \(\mathit{t}\))) is a suffciently smooth solution of the system such that there exists \(\mathit{l}\) \(\in\) \(\mathbb{R}\) with supp \(\mathit{u}\)(.,tj) \(\subseteq\)(\(\mathit{l}\), \(\infty\)) (\(\mathit{or}\)(-\(\infty\), \(\mathit{l}\))) and supp \(\mathit{w}\)(.,tj) \(\subseteq\)(\(\mathit{l}\), \(\infty\)) (\(\mathit{or}\)(-\(\infty\), \(\mathit{l}\))), for \(\mathit{j}\) = 1,2 (t1 \(\neq\) t2), then \(\mathit{u}\) \(\equiv\) 0 and \(\mathit{w}\) \(\equiv\) 0.

Multiple-order singularity expansion method

Physical systems and signals are characterized by complex functions of the frequency in the harmonic domain. The extension of such functions to the complex frequency plane, and in particular expansions and factorized forms of the harmonic-domain functions in terms of their poles and zeros, is of high interest to describe the physical properties of a system, and study its response dynamics in the temporal and harmonic domains. In this work, we start from a general property of continuity and differentiability of the complex functions to derive the multiple-order singularity expansion method. We rigorously derive the common singularity and zero expansion and factorization expressions, and generalize them to the case of singularities of arbitrary order, while deducing the behavior of these complex frequencies from the simple hypothesis that we are dealing with physically realistic signals.

Continuous selection of Lagrangian subspaces

ABSTRACT We study continuous selections of the set-valued map that takes every skew-symmetric bilinear form on a vector space to its corresponding set of maximal isotropic subspaces. Applications are made to establishing continuity properties of the Vergne polarizing subalgebras of completely solvable Lie algebras in terms of Schubert cells of suitable Grassmann manifolds.

Hölder regularity for collapses of point-vortices

The first part of this article studies the collapses of point-vortices for the Euler equation in the plane and for surface quasi-geostrophic equations in the general setting of α models. In these models the kernel of the Biot–Savart law is a power function of exponent . It is proved that, under a standard non-degeneracy hypothesis, the trajectories of the point-vortices have a Hölder regularity up to the time of collapse. The Hölder exponent obtained is and this exponent is proved to be optimal for all α by exhibiting an example of a three-vortex collapse. The same question is then addressed for the Euler point-vortex system in smooth bounded connected domains. It is proved that if a given point-vortex has an accumulation point in the interior of the domain as t → T, then it converges towards this point and displays the same Hölder continuity property. A partial result for point-vortices that collapse with the boundary is also established: we prove that their distance to the boundary is Hölder regular.

Simultaneous inversion for a fractional order and a time source term in a time-fractional diffusion-wave equation

Abstract In this article, we consider an inverse problem for determining simultaneously a fractional order and a time-dependent source term in a multi-dimensional time-fractional diffusion-wave equation by a nonlocal condition. Based on a uniformly bounded estimate of the Mittag-Leffler function given in this paper, we prove the uniqueness of the inverse problem and the Lipschitz continuity properties for the direct problem. Then we employ the Levenberg–Marquardt method to recover simultaneously the fractional order and the time source term, and establish a finite-dimensional approximation algorithm to find a regularized numerical solution. Moreover, a fast tensor method for solving the direct problem in the three-dimensional case is provided. Some numerical results in one and multidimensional spaces are presented for showing the robustness of the proposed algorithm.

Subordinacy theory on star-like graphs

We study Jacobi matrices on star-like graphs, which are graphs that are given by the pasting of a finite number of half-lines to a compact graph. Specifically, we extend subordinacy theory to this type of graphs, that is, we find a connection between asymptotic properties of solutions to the eigenvalue equations and continuity properties of the spectral measure with respect to the Lebesgue measure. We also use this theory in order to derive results regarding the multiplicity of the singular spectrum.

Characterisation of weyl spectrum in Hilbert spaces

In this paper, the study give some continuity properties of the Weyl spectrum of a continuous linear operator on a Banach space and show that the Weyl’s theorem holds for a spectral operator of finite type although the theorem fails for a spectral operator in general.

Classes of Kernels and Continuity Properties of the Double Layer Potential in Hölder Spaces

We prove the validity of regularizing properties of the boundary integral operator corresponding to the double layer potential associated to the fundamental solution of a nonhomogeneous second order elliptic differential operator with constant coefficients in Hölder spaces by exploiting an estimate on the maximal function of the tangential gradient with respect to the first variable of the kernel of the double layer potential and by exploiting specific imbedding and multiplication properties in certain classes of kernels of integral operators and a generalization of a result for integral operators on differentiable manifolds.

Inverse scattering with a parametrized spatial spectral volume integral equation for finite scatterers.

In wafer metrology, the knowledge of the photomask together with the deposition process only reveals the approximate geometry and material properties of the structures on a wafer as a priori information. With this prior information and a parametrized description of the scatterers, we demonstrate the performance of the Gauss-Newton method for the precise and noise-robust reconstruction of the actual structures, without further regularization of the inverse problem. The structures are modeled as 3D finite dielectric scatterers with a uniform polygonal cross-section along their height, embedded in a planarly layered medium. A continuous parametrization in terms of the homogeneous permittivity and the vertex coordinates of the polygons is employed. By combining the global Gabor frame in the spatial spectral Maxwell solver with the consistent parametrization of the structures, the underlying linear system of the Maxwell solver inherits all the continuity properties of the parametrization. Two synthetically generated test cases demonstrate the noise-robust reconstruction of the parameters by surpassing the reconstruction capabilities of traditional imaging methods at signal-to-noise ratios up to -3d B with geometrical errors below λ/7, where λ is the illumination wavelength. For signal-to-noise ratios of 10dB, the geometrical parameters are reconstructed with errors of approximately λ/60, and the material properties are reconstructed with errors of around 0.03%. The continuity properties of the Maxwell solver and the use of prior information are key contributors to these results.