Continuous Everywhere Research Articles

Maximum mutational robustness in genotype-phenotype maps follows a self-similar blancmange-like curve.

Phenotype robustness, defined as the average mutational robustness of all the genotypes that map to a given phenotype, plays a key role in facilitating neutral exploration of novel phenotypic variation by an evolving population. By applying results from coding theory, we prove that the maximum phenotype robustness occurs when genotypes are organized as bricklayer's graphs, so-called because they resemble the way in which a bricklayer would fill in a Hamming graph. The value of the maximal robustness is given by a fractal continuous everywhere but differentiable nowhere sums-of-digits function from number theory. Interestingly, genotype-phenotype maps for RNA secondary structure and the hydrophobic-polar (HP) model for protein folding can exhibit phenotype robustness that exactly attains this upper bound. By exploiting properties of the sums-of-digits function, we prove a lower bound on the deviation of the maximum robustness of phenotypes with multiple neutral components from the bricklayer's graph bound, and show that RNA secondary structure phenotypes obey this bound. Finally, we show how robustness changes when phenotypes are coarse-grained and derive a formula and associated bounds for the transition probabilities between such phenotypes.

Complex-tensor theory of simple smectics

Matter self-assembling into layers generates unique properties, including structures of stacked surfaces, directed transport, and compact area maximization that can be highly functionalized in biology and technology. Smectics represent the paradigm of such lamellar materials — they are a state between fluids and solids, characterized by both orientational and partial positional ordering in one layering direction, making them notoriously difficult to model, particularly in confining geometries. We propose a complex tensor order parameter to describe the local degree of lamellar ordering, layer displacement and orientation of the layers for simple, lamellar smectics. The theory accounts for both dislocations and disclinations, by regularizing singularities within defect cores and so remaining continuous everywhere. The ability to describe disclinations and dislocation allows this theory to simulate arrested configurations and inclusion-induced local ordering. This tensorial theory for simple smectics considerably simplifies numerics, facilitating studies on the mesoscopic structure of topologically complex systems.

Fractional isospectral and non-isospectral AKNS hierarchies and their analytic methods for N-fractal solutions with Mittag-Leffler functions

Ablowitz–Kaup–Newell–Segur (AKNS) linear spectral problem gives birth to many important nonlinear mathematical physics equations including nonlocal ones. This paper derives two fractional order AKNS hierarchies which have not been reported in the literature by equipping the AKNS spectral problem and its adjoint equations with local fractional order partial derivative for the first time. One is the space-time fractional order isospectral AKNS (stfisAKNS) hierarchy, three reductions of which generate the fractional order local and nonlocal nonlinear Schrödinger (flnNLS) and modified Kortweg–de Vries (fmKdV) hierarchies as well as reverse-t NLS (frtNLS) hierarchy, and the other is the time-fractional order non-isospectral AKNS (tfnisAKNS) hierarchy. By transforming the stfisAKNS hierarchy into two fractional bilinear forms and reconstructing the potentials from fractional scattering data corresponding to the tfnisAKNS hierarchy, three pairs of uniform formulas of novel N-fractal solutions with Mittag-Leffler functions are obtained through the Hirota bilinear method (HBM) and the inverse scattering transform (IST). Restricted to the Cantor set, some obtained continuous everywhere but nondifferentiable one- and two-fractal solutions are shown by figures directly. More meaningfully, the problems worth exploring of constructing N-fractal solutions of soliton equation hierarchies by HBM and IST are solved, taking stfisAKNS and tfnisAKNS hierarchies as examples, from the point of view of local fractional order derivatives. Furthermore, this paper shows that HBM and IST can be used to construct some N-fractal solutions of other soliton equation hierarchies.

A robust and accurate interpolant for tabulated sound speed profiles

An new interpolation algorithm that is a hybrid of the cubic spline and monotone piecewise cubic Hermite polynomial interpolants is introduced. The resulting algorithm has several properties that make it an attractive choice for the interpolation of sound speed profiles in acoustic propagation models. It has been shown that when interpolants do not have a continuous first and/or second derivative, the resulting acoustic model predictions can exhibit a variety of artifacts such as false or omitted caustics. The first derivative of our interpolant is continuous everywhere. Our interpolant also automatically respects the monotonicity of the underlying data in the sense that it will be monotone (increasing or decreasing) on any interval where the associated data is also monotone. This property avoids the oscillations between data points that can be a source of frustration when using some interpolants such as cubic splines. Finally, the interpolant will also possess a continuous second derivative everywhere, except at those data points where monotonicity considerations take priority. In some cases, the interpolant will coincide exactly with the cubic spline.An new interpolation algorithm that is a hybrid of the cubic spline and monotone piecewise cubic Hermite polynomial interpolants is introduced. The resulting algorithm has several properties that make it an attractive choice for the interpolation of sound speed profiles in acoustic propagation models. It has been shown that when interpolants do not have a continuous first and/or second derivative, the resulting acoustic model predictions can exhibit a variety of artifacts such as false or omitted caustics. The first derivative of our interpolant is continuous everywhere. Our interpolant also automatically respects the monotonicity of the underlying data in the sense that it will be monotone (increasing or decreasing) on any interval where the associated data is also monotone. This property avoids the oscillations between data points that can be a source of frustration when using some interpolants such as cubic splines. Finally, the interpolant will also possess a continuous second derivative everywhere, e...

Zero-Error Consensus Tracking With Preassignable Convergence for Nonaffine Multiagent Systems

In this paper, we investigate the consensus tracking control problem for networked multiagent systems (MASs) with unknown nonaffine dynamics. Our goal is to achieve asymptotic (rather than ultimately uniformly bounded) consensus tracking, which is quite challenging especially if nonvanishing/nonparametric uncertainties are involved and at the same time the control protocol is required to be fully distributed and continuous everywhere. Here, we present a conceptually new and structurally simple solution with distributed and continuous control action. The developed method is capable of ensuring zero-error tracking with a unique converging feature in that the consensus tracking error first converges to a small adjustable residual set around zero within a prescribed finite time, and then further shrinks to zero exponentially. The key technique lies in the utilization of a state transformation based on certain scaling function. Our method also prevents the restrictive requirement that all subsystems have access to the linearly parameterized information as imposed in most existing consensus tracking results for nonlinear MAS.

Time and Space Fractional Schrödinger Equation with Fractional Factor**Supported by National Natural Science Foundation of China under Grant Nos. 11472247 and 11872335

In this paper, we introduce a new definition of fractional derivative which contains a fractional factor, and its physical meanings are given. When studying the fractional Schrödinger equation (FSE) with this form of fractional derivative, the result shows that under the description of time FSE with fractional factor, the probability of finding a particle in the whole space is still conserved. By using this new definition to construct space FSE, we achieve a continuous transition from standard Schrödinger equation to the fractional one. When applying this form of Schrödinger equation to a particle in an infinite symmetrical square potential well, we find that the probability density distribution loses spatial symmetry and shows a kind of attenuation property. For the situation of a one-dimensional infinite δ potential well, the first derivative of time-independent wave function Φ to space coordinate x can be continuous everywhere when the particle is at some special discrete energy levels, which is much different from the standard Schrödinger equation.

Fractional Calculus on Fractal Interpolation for a Sequence of Data with Countable Iterated Function System

In recent years, the concept of fractal analysis is the best nonlinear tool towards understanding the complexities in nature. Especially, fractal interpolation has flexibility for approximation of nonlinear data obtained from the engineering and scientific experiments. Random fractals and attractors of some iterated function systems are more appropriate examples of the continuous everywhere and nowhere differentiable (highly irregular) functions, hence fractional calculus is a mathematical operator which best suits for analyzing such a function. The present study deals the existence of fractal interpolation function (FIF) for a sequence of data \({\{(x_n,y_n):n\geq 2\}}\) with countable iterated function system, where \({x_n}\) is a monotone and bounded sequence, \({y_n}\) is a bounded sequence. The integer order integral of FIF for sequence of data is revealed if the value of the integral is known at the initial endpoint or final endpoint. Besides, Riemann–Liouville fractional calculus of fractal interpolation function had been investigated with numerical examples for analyzing the results.

Smooth Neuroadaptive PI Tracking Control of Nonlinear Systems With Unknown and Nonsmooth Actuation Characteristics

This paper considers the tracking control problem for a class of multi-input multi-output nonlinear systems subject to unknown actuation characteristics and external disturbances. Neuroadaptive proportional-integral (PI) control with self-tuning gains is proposed, which is structurally simple and computationally inexpensive. Different from traditional PI control, the proposed one is able to online adjust its PI gains using stability-guaranteed analytic algorithms without involving manual tuning or trial and error process. It is shown that the proposed neuroadaptive PI control is continuous and smooth everywhere and ensures the uniformly ultimately boundedness of all the signals of the closed-loop system. Furthermore, the crucial compact set precondition for a neural network (NN) to function properly is guaranteed with the barrier Lyapunov function, allowing the NN unit to play its learning/approximating role during the entire system operation. The salient feature also lies in its low complexity in computation and effectiveness in dealing with modeling uncertainties and nonlinearities. Both square and nonsquare nonlinear systems are addressed. The benefits and the feasibility of the developed control are also confirmed by simulations.

Arithmetic-analytical expression of the Koch-type curves and their generalizations (I)

In this paper, by constructing the one-to-one correspondence between its IFS and the quaternary fractional expansion, the n-th iteration analytical expression and the limit representation of the family of the Koch-type curves with arbitrary angles are obtained. The distinction between our method and that of H. Sagan is that we provide the generation process analytically and represent it as a graph of a series function which looks like the Weierstrass function. With these arithmetic expressions, we further analyze and prove some of the fractal properties of the Koch-type curves such as the self-similarity, the Holder exponent and with the property of continuous everywhere but differentiable nowhere. Then, we will show that the Koch-type curves can be approximated by different constructed generators. Based on the analytic transformation of the Koch-type curves, we also constructed more continuous but nowhere-differentiable curves represented by arithmetic expressions. This result implies that the analytical expression of a fractal has theoretical and practical significance.

Smoothed particle hydrodynamics with smoothed pseudo-density

Abstract In this paper, we present a new formulation of smoothed particle hydrodynamics (SPH), which, unlike the standard SPH (SSPH), is well behaved at the contact discontinuity. The SSPH scheme cannot handle discontinuities in density (e.g., the contact discontinuity and the free surface), because it requires that the density of fluid is positive and continuous everywhere. Thus there is inconsistency in the formulation of the SSPH scheme at discontinuities of the fluid density. To solve this problem, we introduce a new quantity associated with particles and the “density” of that quantity. This “density” evolves through the usual continuity equation with an additional artificial diffusion term, in order to guarantee the continuity of the “density.” We use this “density,” or pseudo-density, instead of the mass density, to formulate our SPH scheme. We call our new method SPH with smoothed pseudo-density, and we show that it is physically consistent and can handle discontinuities quite well.

Truncated hierarchical Catmull–Clark subdivision with local refinement

Abstract In this paper we present a new method termed Truncated Hierarchical Catmull–Clark Subdivision (THCCS), which generalizes truncated hierarchical B-splines to control grids of arbitrary topology. THCCS basis functions satisfy partition of unity, are linearly independent, and are locally refinable. THCCS also preserves geometry during adaptive h -refinement and thus inherits the surface continuity of Catmull–Clark subdivision, namely C 2 -continuous everywhere except at the local region surrounding extraordinary nodes, where the surface continuity is C 1 . Adaptive isogeometric analysis is performed with THCCS basis functions on a benchmark problem with extraordinary nodes. Local refinement on complex surfaces is also studied to show potential wide application of the proposed method.

Generalized Kiesswetter's Functions

In 1966, Kiesswetter found an interesting example of continuous everywhere but differentiable nowhere functions using base-4 expansion of real numbers. In this paper we show how Kiesswetter’s function can be extended to general cases. We also provide an equivalent form for such functions via a recurrence relation.

Analytic behavior of the QED polarizability function at finite temperature

We revisit the analytical properties of the static quasi-photon polarizability function for an electron gas at finite temperature, in connection with the existence of Friedel oscillations in the potential created by an impurity. In contrast with the zero temperature case, where the polarizability is an analytical function, except for the two branch cuts which are responsible for Friedel oscillations, at finite temperature the corresponding function is non analytical, in spite of becoming continuous everywhere on the complex plane. This effect produces, as a result, the survival of the oscillatory behavior of the potential. We calculate the potential at large distances, and relate the calculation to the non-analytical properties of the polarizability.

Applications of Matlab in Mathematical Analysis

In international academia, Matlab has already been accepted as an accurate and reliable standard computational soft ware. It is widely applied in applied algebra, statistics processing, automation and digital communication. Matlab has also been incorporated into the study of calculus. In this paper, we demonstrate Weierstrass function which has the property of being continuous everywhere but differentiable at no where, the graph of Weierstrass function is given by the use of Matlab. Besides, we study the numerical approximation of π by using Matlab and Romberg method.

Competitive equilibria in matching markets with budgets

Consider a market with n unit demand buyers and m sellers, each selling one unit of an indivisible good. The buyers specify their preferences over items via utility functions u ij ( p j ), which is the utility of buyer i for item j when its price is p j . So far, this is the classic Shapley-Shubik assignment model [Shapley and Shubik 1971] which captures a variety of matching markets including housing markets and ad auctions [Edelman et al. 2007], except for the extension to general utility functions instead of the quasi-linear utilities in the original model. Shapley and Shubik show that a competitive equilibrium always exists in their model, and later work [Crawford and Knoer 1981, Quinnzi 1984, Gale 1984] shows that a competitive equilibrium must also exist for the model with general utility functions u ij (·), provided these u ij (·) are strictly decreasing and continuous everywhere.

About “black holes” and dark matter

It is shown that the complete system of classical gravitational equations for an isolated centrally symmetric body yields that: (1) in terms of Galilean coordinates all metric coefficients of the Riemannian space induced by the body cannot be equal to zero or infinity anywhere; (2) they, together with the first-order derivatives, should be continuous everywhere. The equations do not contain solutions corresponding to “black holes,” but admit solutions corresponding to objects for which the surface radius (in terms of standard coordinates) is equal to the double mass of matter under this surface. These objects can make the main contribution to the dark matter of the Universe and explain observed effects, such as gravitational microlensing and other effects. Under certain conditions they can become powerful X-ray sources.

Evaluation of deformable registration of patient lung 4DCT with subanatomical region segmentations

Deformable registration is needed for a variety of tasks in establishing the voxel correspondence between respiratory phases. Most registration algorithms assume or imply that the deformation field is smooth and continuous everywhere. However, the lungs are contained within closed invaginated sacs called pleurae and are allowed to slide almost independently along the chest wall. This sliding motion is characterized by a discontinuous vector field, which cannot be generated using standard deformable registration methods. The authors have developed a registration method that can create discontinuous vector fields at the boundaries of anatomical subregions. Registration is performed independently on each subregion, with a boundary-matching penalty used to prevent gaps. This method was implemented and tested using both the B-spline and Demons registration algorithms in the Insight Segmentation and Registration Toolkit. The authors have validated this method on four patient 4DCT data sets for registration of the end-inhalation and end-exhalation volumes. Multiple experts identified homologous points in the lungs and along the ribs in the two respiratory phases. Statistical analyses of the mismatch of the homologous points before and after registration demonstrated improved overall accuracy for both algorithms.

Determination of periodic orbits of nonlinear oscillators by means of piecewise-linearization methods

An approximate method based on piecewise linearization is developed for the determination of periodic orbits of nonlinear oscillators. The method is based on Taylor series expansions, provides piecewise analytical solutions in three-point intervals which are continuous everywhere and explicit three-point difference equations which are P-stable and have an infinite interval of periodicity. It is shown that the method presented here reduces to the well-known Stor- mer technique, is second-order accurate, and yields, upon applying Taylor series expansion and a Padeapproximation, another P-stable technique whenever the Jacobian is different from zero. The method is generalized for single degree- of-freedom problems that contain the velocity, and (approximate) analytical solutions are presented. Finally, by intro- ducing the inverse of a vector and the vector product and quotient, and using Taylor series expansions and a Pade ´ approximation, the method has been generalized to multiple degree-of-freedom problems and results in explicit three-point finite difference equations which only involve vector multiplications. � 2005 Elsevier Ltd. All rights reserved.

Mean value coordinates for closed triangular meshes

Constructing a function that interpolates a set of values defined at vertices of a mesh is a fundamental operation in computer graphics. Such an interpolant has many uses in applications such as shading, parameterization and deformation. For closed polygons, mean value coordinates have been proven to be an excellent method for constructing such an interpolant. In this paper, we generalize mean value coordinates from closed 2D polygons to closed triangular meshes. Given such a mesh P , we show that these coordinates are continuous everywhere and smooth on the interior of P . The coordinates are linear on the triangles of P and can reproduce linear functions on the interior of P . To illustrate their usefulness, we conclude by considering several interesting applications including constructing volumetric textures and surface deformation.

Semi-analytical and numerical solutions of multi-degree-of-freedom nonlinear oscillation systems with linear coupling

This research focuses on the development of an approach for solving multi-degree-of-freedom (MDOF) nonlinear oscillation problems with linear coupling. The original physical information included in the governing equations is mostly transferred into semi-analytical and numerical solutions. The semi-analytical solutions generated by the present approach are continuous everywhere and reflect more accurately the characteristics of the motion of the nonlinear dynamic systems. General procedures for three types of nonlinear oscillation problems are formulated in detail for allocation in nonlinear dynamic analysis. Two nonlinear oscillation systems with quadratic and cubic nonlinearities are solved to demonstrate the applications of the present approach.