Singular Functions Research Articles

Interpolating Meshless Methods for 3D Elastic Problems

Interpolating meshless methods can directly impose boundary conditions because of the interpolation property which shows advantages in dealing with problems with boundary conditions. The interpolating element-free Galerkin method (IEFGM), the improved interpolating element-free Galerkin method (IIEFGM), and the radial point interpolation method (RPIM) are applied in this paper to solve the two-dimensional and three-dimensional elastic problems. IEFGM and IIEFGM are two different ways to change the status that the traditional element-free Galerkin method (EFG) does not have the interpolation property. IEFGM uses an improved interpolating moving least-squares (IMLS) method that employed singular weight functions while IIEFGM takes the improved interpolating moving least-squares method based on non-singular weight function. RPIM, one of the most widely used interpolating meshless methods, is compared with IEFGM and IIEFGM in this paper. The numerical results of two-dimensional and three-dimensional elastic problems show that the three types of interpolating meshless methods obtain high precision displacement solutions and stress solutions.

Newly modified unified auxiliary equation method and its applications

This study aims to propose a new modified unified auxiliary equation method for the first time. To illustrate the validity of the new modified method, we consider the dimensionless time-dependent paraxial equation which is investigated previously in Tarla and Yilmazer (2022) using the unified auxiliary equation method. As a result, some different and more general solutions are obtained such as singular, periodic, hyperbolic, dark-bright, trigonometric, exponential, and Jacobi elliptic functions. The obtained results are compared with the solutions obtained by the unified auxiliary equation method in Tarla and Yilmazer (2022) in the result and discussion section. We conclude that our findings are novel and distinctive, and we give the comparison in the section on results and discussion. It is demonstrated that the modified unified auxiliary equation method provides a more efficient mathematical technique for solving nonlinear partial differential equations. In addition, we draw 3D profiles and counter plots for some obtained solutions with the assistance of a computer program by giving a specific value for the involved parameters.

A REVIEW OF FRACTAL FUNCTIONS AND APPLICATIONS

In this paper, we mainly investigate continuous functions with unbounded variation on closed intervals. Given the increasing number of proposals and definitions of different kinds of functions with fractal structure in the scope of fractal functions, it is important to introduce a systematic classification and discuss elementary definitions of fractal functions. Different examples of fractal functions with certain fractal dimensions have been given. Then, we have introduced different kind of fractal function such as singular fractal function, irregular fractal function, regular fractal function and complete regular fractal function. We also introduce fractal functions of integer dimension with one and two, respectively. Other continuous and discontinuous functions with fractal structure have been given. Applications of fractal functions in interpolation, approximation and relationship between fractional calculus have been investigated. Finally, we studied fractal dimension spaces with different fractal dimensions.

Finite-time flocking control of multiple nonholonomic mobile agents

This paper focuses on the problem of finite-time flocking in the multi-agent nonholonomic system with proximity graph. With the aid of singular communication function and results from graph theory, a novel distributed control protocol is proposed, where each agent merely obtains state information of its neighbors. Based on the theory of finite-time stability, the proposed protocol can achieve flocking within a finite-time and an upper bound on the setting time can be derived. Furthermore, we can deduce several sufficient conditions to the problem under the assumption that the positions and relative distances of agents are unknown. These sufficient conditions show that there are always suitable gains that enable our proposed protocol to perform finite-time flocking control and maintain the connectivity of the graph for any given initial connected graphs. Finally, the theoretical results are validated by numerical simulations.

Fractional Romanovski–Jacobi tau method for time-fractional partial differential equations with nonsmooth solutions

In this paper, we introduce a new finite class of non-polynomial basis functions for the spectral tau approximation of time-fractional partial differential equations on a semi-infinite interval. Different from many other spectral tau approaches, the singular basis functions are based on a finite class of Romanovski–Jacobi polynomials through a fractional coordinate transform. As such, the singularity of the new basis can be tailored to suit that of the singular solutions to a class of time-fractional partial differential equations, leading to spectrally accurate approximations. We introduce the fractional Romanovski-Jacobi-Gauss-type quadrature formulae along with basic approximation results for some weighted projection operators defined in suitable weighted Sobolev spaces. We discuss the relationship between such kind of finite orthogonal functions and other classes of infinite orthogonal functions such as fractional Laguerre functions and fractional modified generalized Laguerre functions. Moreover, we derive a new formula expressing explicitly the fractional integral of the fractional Romanovski-Jacobi functions in terms of the same functions themselves. This family of orthogonal systems offers great flexibility to match a wide range of time-fractional differential equations with Robin boundary conditions and with nonsmooth solutions.

Proper condensates and long range order

Within the framework of the algebra of canonical commutation relations in position space, a long range order between particles in bounded regions is established in states with a sufficiently large particle number. It occurs whenever homogeneous proper (infinite) condensates form locally in the states in the limit of infinite densities. The condensates are described by eigenstates of the momentum operator, covering also those cases where they are streaming with a constant velocity. The arguments given are model independent and lead to a new criterion for the occurrence of condensates. It makes use of a novel approach to the identification of condensates, based on a characterization of regular and singular wave functions.

Singular elliptic equations with nonlinearities of subcritical and critical growth

AbstractIn this paper, we consider the problem $-\Delta u =-u^{-\beta }\chi _{\{u>0\}} + f(u)$ in $\Omega$ with $u=0$ on $\partial \Omega$, where $0<\beta <1$ and $\Omega$ is a smooth bounded domain in $\mathbb {R}^{N}$, $N\geq 2$. We are able to solve this problem provided $f$ has subcritical growth and satisfy certain hypothesis. We also consider this problem with $f(s)=\lambda s+s^{\frac {N+2}{N-2}}$ and $N\geq 3$. In this case, we are able to obtain a solution for large values of $\lambda$. We replace the singular function $u^{-\beta }$ by a function $g_\epsilon (u)$ which pointwisely converges to $u^{-\beta }$ as $\epsilon \rightarrow 0$. The corresponding energy functional to the perturbed equation $-\Delta u + g_\epsilon (u) = f(u)$ has a critical point $u_\epsilon$ in $H_0^{1}(\Omega )$, which converges to a non-trivial non-negative solution of the original problem as $\epsilon \rightarrow 0$.

A weighted fractional problem involving a singular nonlinearity and an L 1 datum

In this article, we show the existence of a unique entropy solution to the following problem: where the domain is bounded and contains the origin, , , , sp<N, , for q>1 and h is a general singular function with singularity at 0. Further, the fractional p-Laplacian with weight α is given by

Multifractal Features of Particle-Size Distribution and Their Relationships With Soil Erosion Resistance Under Different Vegetation Types in Debris Flow Basin

Understanding the influence of vegetation types on soil particle-size distribution (PSD) is essential to evaluate the effects of sediment control by vegetation restoration. In this work, we studied the effects of different vegetation types, including bare land, meadow, shrub and forest on soil PSD in Jiangjiagou gully, Yunnan province, China. A total of 60 soil samples were collected and analyzed for soil particle size distribution using the laser diffraction method. Fractal theory was used to calculate multifractal parameters. The volume fraction of silt particles in shrub and forest is significantly higher than that in bare land, meadow, whereas the total volume fraction of sand particles in bare land and meadow exceed that in shrub and forest. The soil particle size distribution along soil layers has no significant difference in each vegetation type. The volumetric fractal dimension is significantly higher in forest and shrub than in bare land and grassland, but there is no significant difference between forest and shrub. In addition, soil erosion resistance exhibits significant differences of forest &amp;gt; shrub &amp;gt; grassland &amp;gt; bare land. Multifractal parameters are highest in bare land except for multifractal spectrum values (f (αmax) and f (αmin)) and the maximum value of singularity index (αmin). All generalized dimensions spectra curves of the PSD are sigmoidal, whereas the singular spectrum function shows an asymmetric upward convex curve. Furthermore, soil erosion resistance has significant relationships with multifractal parameters. Our results suggest that multifractal parameters of the soil PSD can predict its anti-ability to erosion. This study also provides an important insight for the evaluation of soil structure improvement and the effects of erosion control by vegetation restoration in dry-hot valley areas.

Design and analysis of the Extended Hybrid High-Order method for the Poisson problem

We propose an Extended Hybrid High-Order scheme for the Poisson problem with solution possessing weak singularities. Some general assumptions are stated on the nature of this singularity and the remaining part of the solution. The method is formulated by enriching the local polynomial spaces with appropriate singular functions. Via a detailed error analysis, the method is shown to converge optimally in both discrete and continuous energy norms. Some tests are conducted in two dimensions for singularities arising from irregular geometries in the domain. The numerical simulations illustrate the established error estimates, and show the method to be a significant improvement over a standard Hybrid High-Order method.

Metamorphic dynamical quantum phase transition in double-quench processes at finite temperatures

By deriving a general framework and analyzing concrete examples, we demonstrate a class of dynamical quantum phase transitions (DQPTs) in one-dimensional two-band systems going through double-quench processes. When this type of DQPT occurs, the Loschmidt amplitude vanishes and the rate function remains singular after the second quench, meaning the final state continually has no overlap with the initial state. This type of DQPT is named metamorphic DQPT to differentiate it from ordinary DQPTs that only exhibit zero Loschmidt amplitude and singular rate function at discrete time points. The metamorphic DQPTs occur at zero as well as finite temperatures. Our examples of the Su-Schrieffer-Heeger (SSH) model and Kitaev chain illustrate the conditions and behavior of the metamorphic DQPT. Since ordinary DQPTs have been experimentally realized in many systems, similar setups with double quenches will demonstrate the metamorphic DQPT. Our findings thus provide additional controls of dynamical evolution of quantum systems.

Alternative fuels’ blending model to facilitate the implementation of carbon offsetting and reduction Scheme for International Aviation

The aviation sector contributes 4.1% to the global economy, although it was responsible for 2.8% of the net global CO2 emissions in 2019, and it contributed around 12% of the CO2 emissions produced from the transportation sector alone in 2018. Hence, the UN’s Carbon Offsetting and Reduction Scheme for International Aviation (CORSIA) can serve as a cost-effective mechanism to promote low carbon fuels and mitigate CO2 emissions, where integrating alternative jet fuels may reduce the cost associated with CORSIA's offsetting requirements. Therefore, this study considers the following alternative production pathways; Gas to Liquid, Oil to Jet, Gas to Jet, Sugar to Jet, Catalytic Hydro-thermolysis Jet and Alcohol to Jet; for the development of an optimal fuel blending model. A multi-objective model is developed and transformed into a singular objective function for a convenient minimisation of the total fuel-related costs for operators under CORSIA. The model considered the reclaiming possibility of offsetting costs when a CORSIA's eligible fuel is utilised. The results demonstrated that an increase in carbon price increases the total net cost of fuel purchasing and carbon taxing, therefore, enhancing the opportunity for airlines to integrate larger shares of alternative fuels. In addition, the presented model managed to select optimal fuels' blending ratios considering carbon credit approach, which achieved a total cost reduction by 0.13% to 8.08%, assuming a carbon tax range of 3–180 $/tonne CO2.

Cellular automata that generate symmetrical patterns give singular functions

In this paper, we mainly study linear one-dimensional and two-dimensional elementary cellular automata that generate symmetrical spatio-temporal patterns. For spatio-temporal patterns of cellular automata from the single site seed, we normalize the number of nonzero states of the patterns, take the limits, and give one-variable functions for the limit sets. We can obtain a one-variable function for each limit set and show that the resulting functions are singular functions, which are non-constant, are continuous everywhere, and have a zero derivative almost everywhere. We show that for Rule 90, a one-dimensional elementary cellular automaton (CA), and a two-dimensional elementary CA, the resulting functions are Salem’s singular functions. We also discuss two nonlinear elementary CAs, Rule 22 and Rule 126. Although their spatio-temporal patterns are different from that of Rule 90, their resulting functions from the number of nonzero states equal the function of Rule 90.

Different human papillomavirus types share early natural history transitions in immunocompetent women.

Necessary stages of cervical carcinogenesis include acquisition of a carcinogenic human papillomavirus (HPV) type, persistence associated with the development of precancerous lesions, and invasion. Using prospective data from immunocompetent women in the Guanacaste HPV Natural History Study (NHS), the ASCUS-LSIL Triage Study (ALTS) and the Costa Rica HPV Vaccine Trial (CVT), we compared the early natural history of HPV types to inform transition probabilities for health decision models. We excluded women with evidence of high-grade cervical abnormalities at any point during follow-up and restricted the analysis to incident infections in all women and prevalent infections in young women (aged <30 years). We used survival approaches accounting for interval-censoring to estimate the time to clearance distribution for 20 529 HPV infections (64% were incident and 51% were carcinogenic). Time to clearance was similar across HPV types and risk classes (HPV16, HPV18/45, HPV31/33/35/52/58, HPV 39/51/56/59 and noncarcinogenic HPV types); and by age group (18-29, 30-44 and 45-54 years), among carcinogenic and noncarcinogenic infections. Similar time to clearance across HPV types suggests that relative prevalence can predict relative incidence. We confirmed that there was a uniform linear association between incident and prevalent infections for all HPV types within each study cohort. In the absence of progression to precancer, we observed similar time to clearance for incident infections across HPV types and risk classes. A singular clearance function for incident HPV infections has important implications for the refinement of microsimulation models used to evaluate the cost-effectiveness of novel prevention technologies.

Singular Euler–Maclaurin expansion on multidimensional lattices

We extend the classical Euler–Maclaurin (EM) expansion to sums over multidimensional lattices that involve functions with algebraic singularities. This offers a tool for a precise and fast evaluation of singular sums that appear in multidimensional long-range interacting systems. We find that the approximation error decays exponentially with the expansion order for band-limited functions and that the runtime is independent of the number of particles. First, the EM summation formula is generalised to lattices in higher dimensions, assuming a sufficiently regular summand function. We then develop this new expansion further and construct the singular Euler–Maclaurin expansion in higher dimensions, an extension of our previous work in one dimension, which remains applicable and useful even if the summand function includes a singular function factor. We connect our method to analytical number theory and show that all operator coefficients can be efficiently computed from derivatives of the Epstein zeta function. Finally we demonstrate the numerical performance of the expansion and efficiently compute singular lattice sums in infinite two-dimensional lattices, which are of relevance in condensed matter, statistical, and quantum physics. An implementation in mathematica is provided online along with this article.

Integration of the modified double layer potential of the vector boundary element method for eddy current problems

The boundary element method for the eddy current problem (BEM-ECP) was proposed in a number of papers and is applicable to important tasks such as the problem of inductive heating and transmission of electromagnetic energy. BEM-ECP requires the construction of a system of linear algebraic equations in which the matrix is inherently dense and is constructed out of element matrices. For the process of the element matrix computation, two cases are normally considered: far-field interaction and near-field interaction, because the construction of element matrices requires integration of a singular function. In this article, we suggest a transform that allows computing the matrix components of the near-singular interaction part while implementing only the single and double layer potentials. The previously suggested modified double layer potential (MDLP) can be integrated by means of this transform, which simplifies the program implementation of BEM-ECP significantly. Solving model problems, we analyse the drawbacks of the previously suggested approach. This analysis includes the proof of the MDLP singularity that makes the integration of this potential a rather difficult task without the help of our transform. The previously suggested approach does not work well with surfaces that are not smooth. Our approach does consider such cases, which is its main advantage. We demonstrate this on the model problems with known analytical solutions.

A Note on Sampling Recovery of Multivariate Functions in the Uniform Norm

We study the recovery of multivariate functions from reproducing kernel Hilbert spaces in the uniform norm. Our main interest is to obtain preasymptotic estimates for the corresponding sampling numbers. We obtain results in terms of the decay of related singular numbers of the compact embedding into $L_2(D,\varrho_D)$ multiplied with the supremum of the Christoffel function of the subspace spanned by the first $m$ singular functions. Here the measure $\varrho_D$ is at our disposal. As an application we obtain near optimal upper bounds for the sampling numbers for periodic Sobolev type spaces with general smoothness weight. Those can be bounded in terms of the corresponding benchmark approximation number in the uniform norm, which allows for preasymptotic bounds. By applying a recently introduced sub-sampling technique related to Weaver's conjecture we mostly lose a $\sqrt{\log n}$ and sometimes even less. Finally we point out a relation to the corresponding Kolmogorov numbers.

Gamma convergence and renormalization group: Two sides of a coin?

We discuss, both from the point of view of Gamma convergence and from the point of view of the renormalization Group, the zero range strong contact interaction of three non-relativistic massive particles. Formally, the potential term is g (delta (x_3-x_1) + delta (x_3 -x_2)), ;, g < 0 and is the limit epsilon rightarrow 0 of approximating potentials V_epsilon (|x_i -x_3|) = epsilon ^{-3} V ( frac{|x_i - x_3|}{epsilon }) , V( x) in L^1(R^3) cap L^2 (R^3) . The presence of a delta function in the limit does not allow the use of standard tools of functional analysis. In the first approach (European Phys. J. Plus 136-363, 2021), (European Phys. J. Plus 1136-1161, 2021), we introduced a map mathcal{K}, called Krein Map , from L^2 (R^9) to a space (Minlos space) mathcal{M}) of more singular functions. In { mathcal M}, the system is represented by a one parameter family of self-adjoint operators. In the topology of L^2 (R^9), the system is an ordered family of weakly closed quadratic forms. By Gamma convergence, the infimum is a self-adjoint operator, the Hamiltonian H of the system. Gamma convergence implies resolvent convergence (An Introduction to Gamma Convergence Springer 1993) but not operator convergence!. This approach is variational and non-perturbative. In the second approach, perturbation theory is used. At each order of perturbation theory, divergences occur when epsilon rightarrow 0. A finite renormalized Hamiltonian H_R is obtained by redefining mass and coupling constant at each order of perturbation theory. In this approach, no distinction is made between self-adjoint operators and quadratic forms. One expects that H = H_R , i.e., that “renormalization” amounts to the difference between the Hamiltonian obtained by quadratic form convergence and the one obtained by Gamma convergence. We give some hints, but a formal proof is missing. For completeness, we discuss briefly other types of zero-range interactions.

Ionic diode-based self-powered ionic skins with multiple sensory capabilities

Ionic diode consisting of a pair of polycation and polyanion conductors can serve as self-powered ionic skin capable of transforming external stimuli into the potential signals through ions like human skin. The major barriers to their applications include unclear mechanoelectric conversion mechanism, limited transparency and stretchability, and singular function of pressure perception. Herein, we present a versatile strategy for constructing transparent and stretchable ionic diode-based self-powered multifunctional skins composed of a hydrogel-based ionic diode sandwiched between two ionic hydrogel electrodes. Benefiting from such sandwich structure, the self-powered skins can detect small variations in not only pressure/strain, but temperature, salinity and pH with high sensitivity and reproducibility based on thermodiffusion or salinity gradient-induced change of potential. In particular, we demonstrate for the first time that the ionic diode-based devices constitute two mechanoelectric conversion mechanisms, namely the thickness-dependent self-induced potentials of ionic diodes and varying potential losses at the diode/electrode interfaces. This work provides a route to the construction of high-performance bionic ionic sensory systems.

Accelerated inexact composite gradient methods for nonconvex spectral optimization problems

This paper presents two inexact composite gradient methods, one inner accelerated and another doubly accelerated, for solving a class of nonconvex spectral composite optimization problems. More specifically, the objective function for these problems is of the form $f_1 + f_2 + h$ where $f_1$ and $f_2$ are differentiable nonconvex matrix functions with Lipschitz continuous gradients, $h$ is a proper closed convex matrix function, and both $f_2$ and $h$ can be expressed as functions that operate on the singular values of their inputs. The methods essentially use an accelerated composite gradient method to solve a sequence of proximal subproblems involving the linear approximation of $f_1$ and the singular value functions underlying $f_2$ and $h$. Unlike other composite gradient-based methods, the proposed methods take advantage of both the composite and spectral structure underlying the objective function in order to efficiently generate their solutions. Numerical experiments are presented to demonstrate the practicality of these methods on a set of real-world and randomly generated spectral optimization problems.