An algorithm for numerically computing an interval matrix containing the geometric mean of two Hermitian positive definite (HPD) matrices is proposed. We consider a special continuous-time algebraic Riccati equation (CARE) where the geometric mean is the unique HPD solution, and compute an interval matrix containing a solution to the equation. We invent a change of variables designed specifically for

We introduce two inertial extragradient algorithms for solving a bilevel pseudomonotone variational inequality problem in real Hilbert spaces. The advantages of the proposed algorithms are that they can work without the prior knowledge of the Lipschitz constant of the involving operator and only one projection onto the feasible set is required. Strong convergence theorems of the suggested algorithms

The Lyapunov exponent is used to quantify the chaos of a dynamical system, by characterizing the exponential sensitivity of an initial point on the dynamical system. However, we cannot directly compute the Lyapunov exponent for a dynamical system without its dynamical equation, although some estimation methods do exist. Information dynamics introduces the entropic chaos degree to measure the strength

For the Stokes equation over 2D and 3D domains, explicit a posteriori and a priori error estimation are novelly developed for the finite element solution. The difficulty in handling the divergence-free condition of the Stokes equation is solved by utilizing the extended hypercircle method along with the Scott-Vogelius finite element scheme. Since all terms in the error estimation have explicit values

We propose a forecasting method based on a logistic curve model with missing data, which are a ubiquitous problem in social science forecasting, especially in marketing. The method completely recovers parameters of the difference equation when data are on an exact solution curve because it uses an unequal step difference equation that has an exact solution. It makes full use of data without wasting

In numerical integration, cubature methods are effective, especially when the integrands can be well-approximated by known test functions, such as polynomials. However, the construction of cubature formulas has not generally been known, and existing examples only represent the particular domains of integrands, such as hypercubes and spheres. In this study, we show that cubature formulas can be constructed

In this paper, we introduce a preconditioned Euler-extrapolated single-step Hermitian and skew-Hermitian splitting (PE-SHSS) iteration method for solving a class of complex symmetric system of linear equations. The convergence properties of the PE-SHSS iteration method are investigated under suitable restrictions. In addition, the spectral properties of the corresponding preconditioned matrix are discussed

In this paper, we present a respectively scaled splitting (RSS) iteration method for the block 4-by-4 linear system from eddy current electromagnetic problems. Unconditional convergence properties of the RSS iteration method are established. Theoretical results show that the quasi-optimal iterative parameter that minimizes the spectral radius is \(\alpha _{opt}=1\) and the corresponding convergence

The original article has been corrected.

In this paper, we study the invading phenomenon of an alien predator to the habitat of two aborigine preys by traveling waves connecting the predator-free state to the co-existence state. We characterize the minimal wave speed of this invading process based on an application of Schauder’s fixed point theorem with the help of (generalized) upper-lower-solutions. New form of upper-lower-solutions are

Recently, stochastic differential equation (SDE) has been used for many applications in option pricing models which satisfy the non-negativity. So, constructing new numerical method preserves non-negativity for solving SDE is very important. This paper investigates the numerical analyses; convergence, non-negativity and stability of the multi-step Milstein method for SDE. We derive the new general

We consider Lipschitz-type backward stochastic differential equations (BSDEs) driven by cylindrical martingales on the space of continuous functions. We show the existence and uniqueness of the solution of such infinite-dimensional BSDEs and prove that the sequence of solutions of corresponding finite-dimensional BSDEs approximates the original solution. We also consider the hedging problem in bond

This article points out how a driver might misperceive the curvature of a road curve if the curve is combined with a slope change that the driver does not recognize. We investigated the relationship between the apparent curvature of a road and the change of the slope, and clarified the difference between the perceived curvature and the actual curvature when the driver mistakenly believes that the slope

We consider the radial symmetric stationary solutions of \(u_{t}=\varDelta u-|x|^{q}u^{-p}\). We first give a result on the existence of the negative value functions that satisfy the radial symmetric stationary problem on a finite interval for \(p \in 2{\mathbb{N}}\), \(q\in{\mathbb{R}}\). Moreover, we give the asymptotic behavior of u(r) and \(u'(r)\) at both ends of the finite interval. Second, we

We propose a two-dimensional fast Fourier transform (FFT) network to retrieve the prices of options that depend on two Lévy processes. Applications include, but are not limited to, the valuation of options on two stocks under the Lévy processes, and options on a single stock under a random time-change Lévy process. The proposed numerical scheme can be applied to different multivariate Lévy constructions

In this paper we consider a general optimal consumption and portfolio selection problem of a finitely-lived agent whose consumption rate process is subject to time-varying subsistence consumption constraints. That is, her consumption rate should be greater than or equal to some convex, non-decreasing and continuous function of time t. Using martingale duality approach and Feynman–Kac formula, we derive

A thick-restart Lanczos type algorithm is proposed for Hermitian J-symmetric matrices. Since Hermitian J-symmetric matrices possess doubly degenerate spectra or doubly multiple eigenvalues with a simple relation between the degenerate eigenvectors, we can improve the convergence of the Lanczos algorithm by restricting the search space of the Krylov subspace to that spanned by one of each pair of the

We investigate the optimal consumption, portfolio, and life insurance decisions problem of a liquidity constrained household whose preference is given by the CES (constant elasticity of substitution) utility function. By applying the martingale and duality method, we obtain the closed-form solution for the household’s value function and optimal strategies. We provide a rigorous proof for the optimality

The reflection principle for Brownian motion gives a way to calculate the joint distribution of a hitting time and a one dimensional marginal. The Carr–Nadtochiy transform is a formulation that generalizes the reflection principle in this respect. The transform originated from a way to hedge so-called barrier options in the literature of financial mathematics. The existence of the transform has been

In this article, we consider a coefficient stability problem for one-dimensional stochastic differential equations driven by an \(\alpha\)-stable process with \(\alpha \in (1,2)\). More precisely, we find an upper bound for the \(L^{\alpha -1}(\varOmega ,{\mathbb {P}})\) distance between two solutions in terms of the \(L^{\alpha }\left( {\mathbb {R}},\mu^{\alpha }_{x_0}\right)\) distance of the coefficients

We propose a general theory of estimating interpolation error for smooth functions in two and three dimensions. In our theory, the error of interpolation is bound in terms of the diameter of a simplex and a geometric parameter. In the two-dimensional case, our geometric parameter is equivalent to the circumradius of a triangle. In the three-dimensional case, our geometric parameter also represents

Stationary waves of constant shape and constant propagation speed on rotational flows of two layers are computed numerically. Two layers are assumed to be of distinct constant vorticity distributions. Three different kinds of waves of finite depth are considered: pure capillary, capillary-gravity, and gravity waves. The problem is formulated as a bifurcation problem, which involves many parameters

The shortest bibranching problem is a common generalization of the minimum-weight edge cover problem in bipartite graphs and the minimum-weight arborescence problem in directed graphs. For the shortest bibranching problem, an efficient primal-dual algorithm is given by Keijsper and Pendavingh (J Comb Theory Ser B 73:130–145, 1998), and the tractability of the problem is ascribed to total dual integrality

In simulating physical systems, conservation of the total energy is often essential, especially when energy conversion between different forms of energy occurs frequently. Recently, a new fourth order energy-preserving integrator named MB4 was proposed based on the so-called continuous stage Runge–Kutta methods (Miyatake and Butcher in SIAM J Numer Anal 54(3):1993–2013, 2016). A salient feature of

NIST SP800-22 is one of the most widely used statistical testing tools for pseudorandom number generators (PRNGs). This tool consists of 15 tests (one-level tests) and two additional tests (two-level tests). Each one-level test provides one or more p-values. The two-level tests measure the uniformity of the obtained p-values for a fixed one-level test. One of the two-level tests categorizes the p-values

We present a numerical approach for determination of the spectral stability of solitary waves by computing eigenvalues and eigenfunctions of the corresponding eigenvalue problems, along with their continuation, for nonlinear wave equations in one space dimension. We illustrate the approach for the nonlinear Schrödinger (NLS) equation with a small potential, and numerically determine the spectral stability

Mathematical models for self-propelled motions are often utilized for understanding the mechanism of collective motions observed in biological systems. Indeed, several patterns of collective motions of camphor disks have been reported in experimental systems. In this paper, we show the existence of asymmetrically rotating solutions of a two-camphor model and give necessary conditions for their existence

The purpose of the paper is to establish higher-order Karush–Kuhn–Tucker higher-order necessary optimality conditions for set-valued optimization where the derivatives of objective and constraint functions are separated. We first introduce concepts of higher-order weak lower inner epiderivatives for set-valued maps and discuss some useful properties about new epiderivatives, for instance, convexity

In this paper, a new preconditioner for a class of \(2\times 2\) block linear systems is proposed. The proposed new preconditioner is a better approximation to the original coefficient matrix than the previous ones. The eigenvalue distribution and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are discussed. Finally, two numerical examples are provided to show the

This paper presents a new projected Barzilai–Borwein method for the complementarity problem over symmetric cone by applying the Barzilai–Borwein-like steplengths to the projected method. A new descent direction is employed and a non-monotone line search is used in the method in order to guarantee the global convergence. The projected Barzilai–Borwein method is proved to be globally convergent under

For numerical simulation of elastic structures, data-driven computational approaches attempt to use a data set of material responses, without resorting to conventional modeling of the material constitutive equation. In a material data set in the stress–strain space, the data points are considered to lie on or near a low-dimensional manifold, rather distribute ubiquitously in the space. This paper presents

We propose a mathematical model to understand the transmission dynamics of HIV/AIDS in an environment. In addition to previous approaches, we incorporate two classes of isolated. By isolated we do not mean physical separation, but commitment to keep its status. We establish the well-posedness of our model and fully analyze the asymptotic behavior of the solutions which depends on the basic reproduction

The nonlinear matrix equation \(X^p=R+M^T(X^{-1}+B)^{-1}M\), where p is a positive integer, M is an arbitrary \(n\times n\) real matrix, R and B are symmetric positive semidefinite matrices, is considered. When \(p=1\), this matrix equation is the well-known discrete-time algebraic Riccati equation (DARE), we study the convergence rate of an iterative method which was proposed in Meng and Kim (J Comput

In this paper we will establish nonlinear a priori lower and upper bounds for the solutions to a large class of equations which arise from the study of traveling wave solutions of reaction–diffusion equations, and we will apply our nonlinear bounds to the Lotka–Volterra system of two and four competing species as examples. The idea used in a series of papers by the first author et al. for the establishment

This paper considers a bidimensional continuous-time renewal risk model, in which the two components of each pair of claim sizes are linked via the strongly asymptotic independence structure and the two claim-number processes from different lines of business are (almost) arbitrarily dependent. Precise asymptotic formulas for three kinds of finite-time ruin probabilities are established when the claim

Inner product functional encryption (IPFE) is one class of functional encryption supporting only inner product functionality. All previous IPFE schemes are bounded schemes, meaning that the vector length that can be handled in the scheme is fixed in the setup phase. In this paper, we propose the first unbounded IPFE schemes, in which we do not have to fix the lengths of vectors in the setup phase and

In this paper, a new iterative refinement for ill-conditioned linear systems is derived based on discrete gradient methods for gradient systems. It is proved that the new method is convergent for any initial values irrespective of the choice of the stepsize h. Some properties of the new iterative refinement are presented. It is shown that the condition number of the coefficient matrix in the linear

For continuous functions, midpoint convexity characterizes convex functions. By considering discrete versions of midpoint convexity, several types of discrete convexities of functions, including integral convexity, L\(^\natural\)-convexity and global/local discrete midpoint convexity, have been studied. We propose a new type of discrete midpoint convexity that lies between L\(^\natural\)-convexity

The original article has been corrected.

In this paper we focus on qualitative properties of solutions to a nonlocal nonlinear partial integro-differential equation (PIDE). Using the theory of abstract semilinear parabolic equations we prove existence and uniqueness of a solution in the scale of Bessel potential spaces. Our aim is to generalize known existence results for a wide class of Lévy measures including with a strong singular kernel

In this paper, we deduce the asymptotic error distribution of the Euler method for the nonlinear filtering problem with continuous-time observations. As studied in previous works by several authors, the error structure of the method is characterized by conditional expectations of some functionals of multiple stochastic integrals. Our main result is to prove the stable convergence of a sequence of such

The existence of a solution to an important singular coagulation equation with a multiple fragmentation kernel has been recently proved in Jpn J Ind Appl Math 35(3):1283–1302, 2018. This paper proves the uniqueness of the solution to the same problem in the function space \(\varOmega _{.,r_2} (T) = \bigcup _{\lambda >0 }\varOmega _{\lambda , r_2} (T)\), where \(\varOmega _{\lambda , r_2} (T)\) is the

In this paper, we address the weighted linear matroid intersection problem from computation of the degree of the determinant of a symbolic matrix. We show that a generic algorithm computing the degree of noncommutative determinants, proposed by the second author, becomes an \(O(mn^3 \log n)\) time algorithm for the weighted linear matroid intersection problem, where two matroids are given by column

We propose some conjectures on the asymptotic distribution of the probabilistic Burgers cellular automaton (PBCA), which is defined by a simple rule of particle motion with a probabilistic parameter. Asymptotic distribution of configurations converges to a unique steady state for PBCA. We propose a new and widely-applicable approach to analyze probabilistic particle systems and apply it concretely

In this study, we discuss the sensitivity of a quantum PageRank. By utilizing the finite-dimensional perturbation theory, we estimate the change of the quantum PageRank under a small analytical perturbation on the Google matrix. In addition, we will show the way to estimate the lower bound of the convergence radius and the error bound of the finite sum in the expansion of the perturbed PageRank.

Many applications arising from machine learning, statistics and image processing can be formulated as a convex minimization model with separable structures both in objective function and constraints. The Peaceman–Rachford splitting method is very efficient for solving these problems, but it is not convergent in the absence of some restrictive assumptions. In this paper, we propose a linearized Peaceman–Rachford

This paper considers a dependent risk model with stochastic return and Brownian perturbation, where there exists a dependence structure between the claim sizes and the inter-arrival times and the price process of the investment portfolio is a geometric Lévy process. When the claim sizes have heavy-tailed distributions, the asymptotic lower and upper bounds of the finite-time ruin probability have been

Let \((B_t)_{t\in [0,\infty )}\) be a real Brownian motion on a probability space \((\varOmega ,{\mathcal {F}},P)\). Our concern is whether and how a noncausal type stochastic differential \(dX_t=a(t,\omega )\,dB_t+b(t,\omega )\,dt\) is determined from its stochastic Fourier coefficients (SFCs for short) \((e_n,dX):\)\(=\int _{0}^L\overline{e_n(t)}\,dX_t\) with respect to a CONS \((e_n)_{n\in {\mathbb

To facilitate the numerical analysis of particle methods, we derive truncation error estimates for the approximate operators in a generalized particle method. Here, a generalized particle method is defined as a meshfree numerical method that typically includes other conventional particle methods, such as smoothed particle hydrodynamics or moving particle semi-implicit methods. A new regularity of discrete

In this paper, for the implicit complementarity problem, it is shown that the solution’s sign patterns can be calculated via solving a linear system under some assumptions. Next, Newton iteration is applied to a equivalent nonlinear equation with quadratic convergence and the non-singularity of the Jacobian is discussed. Moreover, a superlinearly convergent hybrid method is established by combining

Quadratic integer programming problems with cardinality constraint have many applications in real life. Portfolio selection is an important application in financial optimization. In this paper we develop an exact and efficient algorithm for quadratic integer programming problems with cardinality constraint. This iterative algorithm is actually a branch and bound method, which adopts a domain cut and

Empty houses, Akiya, are of substantial concern to the ageing society of Japan; the ratio of Akiya is projected to worsen to 30.5% in 2033. Such a high ratio of Akiya will influence the social system and the Japanese economy overall in the near future. Nonetheless, Akiya continues to increase unabated because there are no specific government measures to address the problem. In this study, we develop

We consider numerical methods for computing eigenvalues located in the interior part of the spectrum of a large symmetric matrix. For such difficult eigenvalue problems, an effective solution is to use the Harmonic Ritz pairs in projection methods because the error bounds on the Harmonic Ritz pairs are well studied. In this paper, we prove global convergence of the iterative projection methods with

In this paper, we consider the blow-up curve of semilinear wave equations. Merle and Zaag (Am J Math 134:581–648, 2012) considered the blow-up curve for \(\partial _t^2 u- \partial _x^2 u = |u|^{p-1}u\) and showed that there is the case that the blow-up curve is not differentiable at some points when the initial value changes its sign. Their analysis depends on the variational structure of the problem

We study the partial differential equation (PDE) approach for efficient and accurate valuation of a variable annuity (VA) contract with a surrender option. Specifically, using the Laplace–Carson Transform (LCT), we derive an analytic pricing formula for a VA contract with a surrender option which is formulated as a PDE with an optimal surrender boundary. To demonstrate the efficiency and accuracy of

We consider a generalized version of the Takagi function, which is one of the most famous example of nowhere differentiable continuous functions. We investigate a set of conditions to describe the rate of convergence of Takagi class functions from the probabilistic point of view: The law of large numbers, the central limit theorem, and the law of the iterated logarithm. On the other hand, we show that

In this article, we study a discounted stochastic game to model resource optimal intrusion detection in wireless sensor networks. To address the problem of uncertainties in various network parameters, we propose a globalized robust game-theoretic framework for discounted robust stochastic games. A robust solution to the considered problem is an optimal point that is feasible for all realizations of

We consider an optimal investment, consumption, and life insurance purchase problem for a wage earner. We treat a stochastic factor model that the mean returns of risky assets depend linearly on underlying economic factors formulated as the solutions of linear stochastic differential equations. We discuss the partial information case that the wage earner can not observe the factor process and use only

We investigate upper and lower hedging prices of multivariate contingent claims from the viewpoint of game-theoretic probability and submodularity. By considering a game between “Market” and “Investor” in discrete time, the pricing problem is reduced to a backward induction of an optimization over simplexes. For European options with payoff functions satisfying a combinatorial property called submodularity

We consider unbiased simulation methods for one-dimensional stochastic differential equations with reflection at zero. In particular, we propose improvements of the forward unbiased simulation method provided by Alfonsi et al. (Parametrix methods for one-dimensional reflected SDEs. Modern problems of stochastic analysis and statistics: selected contributions in honor of Valentin Konakov. Springer,