We provide some examples and counterexamples regarding hyperfinite representations of real distributions. We examine some hyperfinite representatives of the null distribution and of the Dirac ...distribution, we discuss hyperfinite representations of an infinite Lp norm, and we study the failure of an energy inequality for the discrete Laplacian in dimension 1. These examples are relevant for the study of partial differential equations with hyperfinite techniques.
•Eigenproperties of a multi-cracked Euler-Bernoulli inextensible circular arch.•Cracks are accounted for in governing equation by means of the distribution theory.•Governing equation defined over a ...unique integration domain regardless n° cracks.•Integration via Laplace transform is adopted to infer closed form mode shapes.•Both validations and parametric analyses are presented.
Despite the numerous explicit solutions of free vibration of arches with regular cross sections, in case of concentrated defects such as cracks, no procedure is available to analyse arch vibrations without sub-division of the integration domain. As a result, curved sub-elements comprised between crack and external constraints, or successive cracks, are considered.
In this paper a distributional approach is adopted to provide a formulation of the free vibration differential governing equations of circular inextensible arches over a unique integration domain in presence of multiple concentrated open (non-breathing) cracks. Discontinuities due to the presence of an arbitrary number of cracks are modelled by means of Dirac's deltas. An integration procedure is devised to offer closed form solutions of the relevant vibration modes together with the relevant frequency determinantal equation. Natural frequencies and mode shapes of damaged arches with different damage and restraint configurations have been evaluated and compared with experimental results available in the literature as well as finite element numerical simulations. The presented closed form solutions are also employed for two parametric studies to evaluate the influence of an increasing number of along axis concentrated cracks as well as of the location of cracks along the arch span.
In this paper, we discuss the implications of applying traditional diffuse-interface techniques to problems involving mass flux across the interface such as the phase change front. In a simplified ...setting of stationary radial flow and linear viscous fluid, we confirm by analytical tools in the framework of Colombeau algebra that the numerical solutions to such problems approximate in fact modified physical problems that involve additional surface tension-like stress localized in the interfacial zone. The arising dynamical surface tension depends on the viscosity and density profiles within the interface. Expanding the setting to models of power-law fluids, we show that the dynamic surface tension vanishes in the limit of interfacial width going to zero for shear-thinning fluids. In contrast, for the shear-thickening case, the diffuse interface numerical solutions to the considered class of problems cannot be assigned any straightforward physical meaning, as the dynamic surface tension becomes unbounded with decreasing interfacial width and the traction jump in the limiting case cannot be even represented by any classical distribution. Consequently, our findings raise questions regarding the broad applicability of diffuse interface techniques in scenarios involving non-material interfaces, underscoring the necessity for further investigation.
•Diffuse interface techniques face difficulties treating non-material interfaces.•An artificial dynamic surface tension emerges within the diffuse interface.•For non-linear rheologies, dynamic surface tension blows up or vanishes in the limit.•Colombeau algebra enables study of arising singularities outside distribution theory.
We present an extension of some results of higher order calculus of variations and optimal control to generalized functions. The framework is the category of generalized smooth functions, which ...includes Schwartz distributions, while sharing many nonlinear properties with ordinary smooth functions. We prove the higher order Euler–Lagrange equations, the D’Alembert principle in differential form, the du Bois–Reymond optimality condition and the Noether’s theorem. We start the theory of optimal control proving a weak form of the Pontryagin maximum principle and the Noether’s theorem for optimal control. We close with a study of a singularly variable length pendulum, oscillations damped by two media and the Pais–Uhlenbeck oscillator with singular frequencies.
An extension of Bernstein inequality Bang, Ha Huy; Huy, Vu Nhat
Journal of mathematical analysis and applications,
11/2021, Volume:
503, Issue:
1
Journal Article
Peer reviewed
In this paper, we obtain the following extension of Bernstein inequality for polynomial differential operators: if 1≤p≤∞, K is an arbitrary compact set in R and P(x) is a polynomial, then there ...exists a constant C such that‖Pm(D)f‖p≤Cmsupx∈K|Pm(x)|‖f‖p for all m∈N,p∈1,∞ and all f∈Vp,K, where Vp,K={f∈Lp(R):suppfˆ⊂K} and fˆ is the Fourier transform of f. Further, we use Nikolskii's idea to get Bernstein inequality for polynomial differential operators with different metrics. The corresponding results for polynomial integral operators are given. An application is also given.
Regularity theory in generalized function algebras of Colombeau type is largely based on the notion of G∞-regularity, which reduces to C∞-regularity when restricted to Schwartz distributions. ...Surprisingly, in the nonstandard version of the Colombeau algebras, this basic property of G∞-regularity does not hold.
Detrended Fluctuation Analysis (DFA) and its generalization for multifractal signals, Multifractal Detrend Fluctuation Analysis (MFDFA), are widely used techniques to investigate the fractal ...properties of time series by estimating their Hurst exponent (H). These methods involve calculating the fluctuation functions, which represent the square root of the mean square deviation from the detrended cumulative curve of a given time series. However, in the multifractal variant of the method, a particular case arises when the multifractal index vanishes. Consequently, it becomes necessary to define the fluctuation function differently for this specific case. In this paper, we propose an approach that eliminates the need for a piecewise definition of the fluctuation function, thereby enabling a unified formulation and interpretation in both DFA and MFDFA methodologies. Our formulation provides a more compact algorithm applicable to mono and multifractal time series. To achieve this, we express the fluctuation functions as generalized means, using the generalized logarithm and exponential functions from the context of the non-extensive statistical mechanics. We identified that the generalized formulation is the Box–Cox transformation of the dataset; hence we established a relationship between statistics parametrization and multifractality. Furthermore, this equivalence is related to the entropic index of the generalized functions and the multifractal index of the MFDFA method. To validate our formulation, we assess the efficacy of our method in estimating the (generalized) Hurst exponents H using commonly used signals such as the fractional Ornstein–Uhlenbeck (fOU) process, the symmetric Lévy distribution, pink, white, and Brownian noises. In addition, we apply our proposed method to a real-world dataset, further demonstrating its effectiveness in estimating the exponents H and uncovering the fractal nature of the data.
•A unified formulation and a new interpretation for DFA and MFDFA is proposed.•The fluctuation is written as a generalized mean, using generalized functions.•A relationship between the entropic and the multifractal indices is established.•Algorithm is validated with synthetic and real-world time series.
Since its invention in 1979 the Feichtinger algebra has become a useful Banach space of functions with applications in time-frequency analysis, the theory of pseudo-differential operators and several ...other topics. It is easily defined on locally compact Abelian groups and, in comparison with the Schwartz(-Bruhat) space, the Feichtinger algebra allows for more general results with easier proofs. This review paper develops the theory of Feichtinger’s algebra in a linear and comprehensive way. The material gives an entry point into the subject and it will also bring new insight to the expert. A further goal of this paper is to show the equivalence of the many different characterizations of the Feichtinger algebra known in the literature. This task naturally guides the paper through basic properties of functions that belong to this space, over operators on it, and to aspects of its dual space. Additional results include a seemingly forgotten theorem by Reiter on Banach space isomorphisms of the Feichtinger algebra, a new identification of Feichtinger’s algebra as the unique Banach space in
L
1
with certain properties, and the kernel theorem for the Feichtinger algebra. A historical description of the development of the theory, its applications, and a list of related function space constructions is included.
We construct Gelfand–Shilov spaces of type and discuss the continuity and boundedness of the ridgelet transform. We then extend the analysis to generalized functions and explore its application to ...the space of tempered ultradistributions. Finally, we extend our findings to Gevrey functions of compact support. We also discuss an application of the ridgelet transform in image processing.
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