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Uncertainty is ubiquitous in biological systems. For example, since gene expression is intrinsically governed by noise, nature shows a fascinating degree of variability. If we want to use a model to predict the behaviour of such an intrinsically stochastic system, we have to cope with the fact that the model parameters are never exactly known, but vary according to some distribution. A key question is then to determine how the uncertainties in the parameters affect the model outcome. Knowing the latter uncertainties is crucial when a model is used for, e.g., experimental design, optimisation, or decision-making. To establish how parameter and model prediction uncertainties are related, Monte Carlo approaches could be used. Then, the model is evaluated for a huge number of parameters sets, drawn from the multivariate parameter distribution. However, when model solutions are computationally expensive this approach is intractable. To overcome this problem, so-called spectral expansion (SE) methods have been developed to quantify prediction uncertainty within a probabilistic framework. Such SE methods have a basis in, e.g., computational mathematics, engineering, physics, and fluid dynamics, and, to a lesser extent, systems biology. The computational costs of SE schemes mainly stem from the calculation of the expansion coefficients. Furthermore, SE effectively leads to a surrogate model which captures the dependence of the model on the uncertainty parameters, but is much simpler to execute compared to the original model. In this paper, we present an innovative scheme for the calculation of the expansion coefficients. It guarantees that the model has to be evaluated only a restricted number of times. Especially for models of high complexity this may be a huge computational advantage. By applying the scheme to a variety of examples we show its power, especially in challenging situations where solutions slowly converge due to high computational costs, bifurcations, and discontinuities. [ABSTRACT FROM AUTHOR]

In this article, we discuss a conformable fractional Sturm-Liouville boundary-value problem on the whole line. We prove the existence of a spectral function for the singular conformable fractional Sturm-Lioville problem. Further, we establish a Parseval equality and spectral expansion formula by terms of the spectral function for conformable fractional Sturm-Liouville problem on the whole line. [ABSTRACT FROM AUTHOR]

This paper proposes a new highly effective method for determining the optimal control of the stress-strain state of spatially multi-connected composite bodies using a stationary temperature field. The proposed method is considered based on the example of a stationary axisymmetric thermoelastic problem for a space with a spherical inclusion and cavity. The proposed method is based on the generalized Fourier method and reduces the original problem to an equivalent problem of optimal control, in which the state of the object is determined by an infinite system of linear algebraic equations, the right-hand side of which parametrically depends on the control. At the same time, the functional of the cost of the initial problem is transformed into a quadratic functional, which depends on the state of the equivalent system and parametrically on the control. The limitation on the temperature distribution is replaced by the value of the control norm in the space of square summable sequences. In fact, this paper considers for the first time the problem of optimal control of an infinite system of linear algebraic equations and develops a method for its solution. The proposed method is based on presenting the solutions of infinite systems in a parametric form, which makes it possible to reduce equivalent problem to the problem of conditional extremum of a quadratic functional, which explicitly depends on the control. A further solution to this problem A further solution to this problem is found by the Lagrange method using the spectral decomposition of the quadratic functional matrix. found by the Lagrange method using the spectral decomposition of the quadratic functional matrix. The method developed in this paper is strictly justified. For all infinite systems, the Fredholm property of their operators is proved. As an important result necessary for substantiation, for the first time, an estimate from below of the module of the multi-parameter determinant of the resolving system of the boundary value problem of conjugation – space with a spherical inclusion – was obtained when solving it using the Fourier method. The theorem that establishes the conditions for the existence and uniqueness of the solution of equivalent problem or optimal control problem without restrictions in the space of square summable sequences is proved. The numerical algorithm is based on a reduction method for infinite systems of linear algebraic equations. Estimates of the practical accuracy of the numerical algorithm demonstrated the stability of the method and sufficiently high accuracy even with close location of the boundary surfaces. Graphs showing the optimal temperature distribution for various geometric parameters of the problem and their analysis are provided. The proposed method extends to boundary value problems with different geometries.

This paper proposes a new highly effective method for determining the optimal control of the stress-strain state of spatially multi-connected composite bodies using a stationary temperature field. The proposed method is considered based on the example of a stationary axisymmetric thermoelastic problem for a space with a spherical inclusion and cavity. The proposed method is based on the generalized Fourier method and reduces the original problem to an equivalent problem of optimal control, in which the state of the object is determined by an infinite system of linear algebraic equations, the right-hand side of which parametrically depends on the control. At the same time, the functional of the cost of the initial problem is transformed into a quadratic functional, which depends on the state of the equivalent system and parametrically on the control. The limitation on the temperature distribution is replaced by the value of the control norm in the space of square summable sequenc es. In fact, this paper considers for the first time the problem of optimal control of an infinite system of linear algebraic equations and develops a method for its solution. The proposed method is based on presenting the solutions of infinite systems in a parametric form, which makes it possible to reduce equivalent problem to the problem of conditional extremum of a quadratic functional, which explicitly depends on the control. A further solution to this problem A further solution to this problem is found by the Lagrange method using the spectral decomposition of the quadratic functional matrix. found by the Lagrange method using the spectral decomposition of the quadratic functional matrix. The method developed in this paper is strictly justified. For all infinite systems, the Fredholm property of their operators is proved. As an important result necessary for substantiation, for the first time, an estimate from below of the module of the multi-parameter determinant of the resolving system of the boundary value problem of conjugation – space with a spherical inclusion – was obtained when solving it using the Fourier method. The theorem that establishes the conditions for the existence and uniqueness of the solution of equivalent problem or optimal control problem without restrictions in the space of square summable sequences is proved. The numerical algorithm is based on a reduction method for infinite systems of linear algebraic equations. Estimates of the practical accuracy of the numerical algorithm demo nstrated the stability of the method and sufficiently high accuracy even with close location of the boundary surfaces. Graphs showing the optimal temperature distribution for various geometric parameters of the problem and their analysis are provided. The proposed method extends to boundary value problems with different geometries. [ABSTRACT FROM AUTHOR]

Deterministic complex networks that use iterative generation algorithms have been found to more closely mirror properties found in real world networks than the traditional uniform random graph models. In this paper we introduce a new, Iterative Independent Model (IIM), generalizing previously defined models in Bonato et al. (2020, 2009, 2017). These models use ideas from Structural Balance Theory to generate edges through a notion of cloning where "the friend of my friend is my friend" and anticloning where "the enemy of my enemy is my friend" (Bonato et al., 2020; Easley et al., 2010). In this paper, we vastly generalize these notions by allowing each vertex added at a given time step to choose independently of the other vertices if it will be cloned or anticloned. While it may seem natural to focus on a randomized model, where we randomly determine whether or not to clone any given vertex, we found the general deterministic model exhibited certain structural properties regardless of the probabilities. This allows applications to then explore the particulars, while having the theoretical model explain the structural phenomenons that occur in all possible scenarios. Throughout the paper we demonstrate that all IIM graphs have spectral gap bounded away from zero, which indicates the clustering properties also found in social networks. Furthermore, we show bounds on the diameter, domination number, and clique number further indicating the well clustered behavior of IIM graphs. Finally, for any fixed graph F all IIM graphs will eventually contain an induced copy of F. [ABSTRACT FROM AUTHOR]

As one myth of polynomial interpolation and quadrature, Trefethen [Math. Today (Southend-on-Sea) 47 (2011), pp. 184–188] revealed that the Chebyshev interpolation of |x-a| (with |a|<1) at the Clenshaw-Curtis points exhibited a much smaller error than the best polynomial approximation (in the maximum norm) in about 95% range of [-1,1] except for a small neighbourhood near the singular point x=a. In this paper, we rigorously show that the Jacobi expansion for a more general class of \Phi-functions also enjoys such a local convergence behaviour. Our assertion draws on the pointwise error estimate using the reproducing kernel of Jacobi polynomials and the Hilb-type formula on the asymptotic of the Bessel transforms. We also study the local superconvergence and show the gain in order and the subregions it occurs. As a by-product of this new argument, the undesired \log n-factor in the pointwise error estimate for the Legendre expansion recently stated in Babus̆ka and Hakula [Comput. Methods Appl. Mech Engrg. 345 (2019), pp. 748–773] can be removed. Finally, all these estimates are extended to the functions with boundary singularities. We provide ample numerical evidences to demonstrate the optimality and sharpness of the estimates. [ABSTRACT FROM AUTHOR]

This study is based upon investigations of the existence of a singular Hahn difference equation of q,ω$$ q,\omega $$‐Sturm–Liouville problem with transmission conditions. Moreover, the Parseval identity and expansion formula in eigenfunctions are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we construct the spectral expansion for the nonself-adjoint differential operators generated in the space of vector functions by an ordinary differential expression of arbitrary order with periodic matrix coefficients, by using the essential spectral singularities, singular quasimomenta, and series with parenthesis. [ABSTRACT FROM AUTHOR]

In the work, we consider issues of well-posed and unique solvability of all possible boundary value problems for one class of operatordifferential equations of the third order with a multiple characteristic in Sobolev type spaces of the third order on the semi-axis. In addition, we find exact values of the norms of intermediate derivatives operators in Sobolev-type spaces and establish their connection with solvability conditions. Note that the solvability conditions obtained are expressed only by the operator coefficients of the equation, which makes it easy to check them in the application. [ABSTRACT FROM AUTHOR]

Spectral problems for the Dirac operator with regular but not strongly regular boundary conditions and complex‐valued potential summable over a finite interval are considered. The purpose of this paper is to find conditions under which the root function system forms a usual Riesz basis rather than a Riesz basis with parentheses. [ABSTRACT FROM AUTHOR]

We discuss multiple change-point estimation as edge detection in piecewise smooth functions with finitely many jump discontinuities. In this paper we propose change-point estimators using concentration kernels with Fourier coefficients. The change-points can be located via the signal based on Fourier transformation system. This method yields location and amplitude of the change-points with refinement via concentration kernels. We prove that, in an appropriate asymptotic framework, this method provides consistent estimators of change-points with an almost optimal rate. In a simulation study the proposed change-point estimators are compared and discussed. Applications of the proposed methods are provided with Nile flow data and daily won-dollar exchange rate data. [ABSTRACT FROM AUTHOR]

In this paper, we study doubly skewed CIR processes, which are extensions of skew Brownian motion. We use modified spectral expansion to obtain some properties, including the transition densities and first hitting time distributions, of doubly skewed CIR processes. [ABSTRACT FROM AUTHOR]

In NMR or MRI, the measured signal is a function of the accumulated magnetization phase inside the measurement voxel, which itself depends on microstructural tissue parameters. Usually the phase distribution is assumed to be Gaussian and higher-order moments are neglected. Under this assumption, only the x-component of the total magnetization can be described correctly, and information about the local magnetization and the y-component of the total magnetization is lost. The Gaussian Local Phase (GLP) approximation overcomes these limitations by considering the distribution of the local phase in terms of a cumulant expansion. We derive the cumulants for a cylindrical muscle tissue model and show that an efficient numerical implementation of these terms is possible by writing their definitions as matrix differential equations. We demonstrate that the GLP approximation with two cumulants included has a better fit to the true magnetization than all the other options considered. It is able to capture both oscillatory and dampening behavior for different diffusion strengths. In addition, the introduced method can possibly be extended for models for which no explicit analytical solution for the magnetization behavior exists, such as spherical magnetic perturbers.

We consider a new approach for the numerical approximation to some classes of stochastic differential equations driven by white noise. The proposed method shares some features with the stochastic collocation techniques, and in particular, it takes advantage of the assumption of smoothness of the functional to be approximated, to achieve fast convergence. The solution to the stochastic differential equation (SDE) is represented by means of Lagrange polynomials. The coefficients of the polynomial basis are functions of time, and they can be computed by solving a system of deterministic ordinary differential equations. Another motivation of this work lies in the novelty of the numerical scheme that does not belong to classical techniques to solve SDEs. Numerical examples are presented to illustrate the accuracy and the efficiency of the proposed method. In particular, we observe a spectral convergence for the mean and the variance of the solution. [ABSTRACT FROM AUTHOR]

The current investigation of our paper is concentrated on an infinite-buffer single-server queueing model with discrete-time batch Markovian arrival process and Markovian service process. The steady-state behaviour of this discrete-time queueing system at various significant time epochs are presented. We first analyze the system-length distribution at outside observer's epoch using the R G -factorization technique combined with the spectral expansion method to compute the stochastic matrix G . For the purpose of comparative study, a comprehensive analysis of the system-length distribution at outside observer's epoch is also carried out using the roots method. Next, we obtain the explicit closed-form expressions of the system-length distributions at pre-arrival, intermediate, post-departure and random epochs by developing the relations among them in equilibrium state. The analysis of waiting-time distribution in the queue measured in slots of an arbitrary customer in an arriving batch is also discussed. Computational experiences with adequate numerical results are conferred in the structure of tables and graphs. Finally, we suggest that our queueing model seems to be fitted for outpatient service of any specialized hospital in a smart city. [ABSTRACT FROM AUTHOR]

In this paper, we consider the eigenvalue problem for ordinary differential equations of fourth order with a spectral parameter contained in two of boundary conditions. We obtain refined asymptotic formulas for eigenvalues and eigenfunctions, and study uniform convergence of Fourier series expansions of continuous functions in the system of eigenfunctions of this problem. [ABSTRACT FROM AUTHOR]

In this work, we prove the existence of a spectral function for singular conformable fractional Dirac system. Further, we establish a Parseval equality and spectral expansion formula by terms of the spectral function. [ABSTRACT FROM AUTHOR]

On-line social networks, such as in Facebook and Twitter, are often studied from the perspective of friendship ties between agents in the network. Adversarial ties, however, also play an important role in the structure and function of social networks, but are often hidden. Underlying generative mechanisms of social networks are predicted by structural balance theory, which postulates that triads of agents, prefer to be transitive, where friends of friends are more likely friends, or anti-transitive, where adversaries of adversaries become friends. The previously proposed Iterated Local Transitivity (ILT) and Iterated Local Anti-Transitivity (ILAT) models incorporated transitivity and anti-transitivity, respectively, as evolutionary mechanisms. These models resulted in graphs with many observable properties of social networks, such as low diameter, high clustering, and densification. We propose a new, generative model, referred to as the Iterated Local Model (ILM) for social networks synthesizing both transitive and anti-transitive triads over time. In ILM, we are given a countably infinite binary sequence as input, and that sequence determines whether we apply a transitive or an anti-transitive step. The resulting model exhibits many properties of complex networks observed in the ILT and ILAT models. In particular, for any input binary sequence, we show that asymptotically the model generates finite graphs that densify, have clustering coefficient bounded away from 0, have diameter at most 3, and exhibit bad spectral expansion. We also give a thorough analysis of the chromatic number, domination number, Hamiltonicity, and isomorphism types of induced subgraphs of ILM graphs. [ABSTRACT FROM AUTHOR]

Spectral problems for the Dirac operator specified on a finite interval with regular, but not strongly regular boundary conditions and a complex-valued integrable potential are studied. This work is aimed at finding the conditions under which the root function system forms a common Riesz basis rather than a Riesz basis with parentheses. [ABSTRACT FROM AUTHOR]

This paper focuses on pointwise error estimates for piecewise functions expanded in the form of Laguerre polynomial series. By utilizing the reproducing kernel of Laguerre polynomials, we derive an explicit pointwise error formula, which enables the estimation of convergence orders of pointwise values for piecewise functions by using the Hilb-type formula and generalized van der Corput-type Lemmas. The presented convergence analyses illustrate that the piecewise functions exhibit local superconvergence away from the singular points. Plenty of numerical experiments are carried out to validate the accuracy of these theoretical results. [ABSTRACT FROM AUTHOR]

This work links the conditional probability structure of Lancaster probabilities to a construction of reversible continuous-time Markov processes. Such a task is achieved by using the spectral expansion of the corresponding transition probabilities in order to introduce a continuous time dependence in the orthogonal representation inherent to Lancaster probabilities. This relationship provides a novel methodology to build continuous-time Markov processes via Lancaster probabilities. Particular cases of well-known models are seen to fall within this approach. As a byproduct, it also unveils new identities associated to well known orthogonal polynomials. [ABSTRACT FROM AUTHOR]

To handle the ubiquitous problem of "dependence learning," copulas are quickly becoming a pervasive tool across a wide range of data‐driven disciplines encompassing neuroscience, finance, econometrics, genomics, social science, machine learning, healthcare, and many more. At the same time, despite their practical value, the empirical methods of "learning copula from data" have been unsystematic with full of case‐specific recipes. Taking inspiration from modern LP‐nonparametrics, this paper presents a modest contribution to the need for a more unified and structured approach of copula modeling that is simultaneously valid for arbitrary combinations of continuous and discrete variables. [ABSTRACT FROM AUTHOR]

We consider one-dimensional Dirac operator with a summable potential on the interval G = (0; 2π). Componentwise uniform equiconvergence rate of spectral expansion for absolutely continuous function and trigonometric Fourier series is estimated on a compact. [ABSTRACT FROM AUTHOR]

We define a generalized hyperbolic Eisenstein series for a pair of a hyperbolic manifold of finite volume and its submanifold. We prove the convergence, the differential equation and the precise spectral expansion associated to the Laplace–Beltrami operator. We also derive the analytic continuation with the location of the possible poles and their residues from the spectral expansion. [ABSTRACT FROM AUTHOR]

In this paper, an even order ordinary differential operator on the interval G = (0, 1) is considered. Uniform equiconvergence of spectral expansion in eigenfunctions of the given operator with a trigonometric series is studied. The uniform equiconvergence rate on any compact K ⊂ G is established for the functions from the classes Wpµ (G), p ≥ 1. [ABSTRACT FROM AUTHOR]

In this paper, we consider the operator L generated in L2(R+) by the differential expression l(y) = -y′′+q(x)y, x∈R+ := [0,∞) and the boundary condition y′0/y0 =α0 + α1λ+α2λ², where q is a complex valued function and αi ∈ C, i = 0,1,2 with α2≠0. We have proved that spectral expansion of L in terms of the principal functions under the condition q ∈ AC(R+), limx→∞ q(x)=0, supx∈R+ [eε√x|q′x|] < ∞, ε > 0 taking into account the spectral singularities. We have also proved the convergence of the spectral expansion. [ABSTRACT FROM AUTHOR]

We consider the singular Hahn-Dirac system defined by... where λ is a complex spectral parameter and p and r are real-valued functions defined on (∞,∞) and continuous at ω0. We prove the existence of a spectral function for such a system. We also prove the Parseval equality and the spectral expansion formula in terms of the spectral function for this system on the whole line. [ABSTRACT FROM AUTHOR]

We study the convergence of means of spectral expansions corresponding to positive self-adjoint elliptic pseudo-differential operators for distributions from the Sobolev-Liouville class. [ABSTRACT FROM AUTHOR]

In this paper we construct the spectral expansion for the differential operator generated in L2 (-∞, ∞) by ordinary differential expression of arbitrary order with periodic complex-valued coefficients by introducing new concepts as essential spectral singularities and singular quasimomenta and using the series with parenthesis. Moreover, we find a criteria for which the spectral expansion coincides with the Gelfand expansion for the self-adjoint case. [ABSTRACT FROM AUTHOR]

In this work, we study the one-dimensional Dirac system on a whole line with impulsive conditions. We construct a spectral function of this system. Using the spectral function, we establish a Parseval equality and spectral expansion formula for such a system. [ABSTRACT FROM AUTHOR]