Math-for-industry Education&Research Hug

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Keywords. degenerate parabolic equations, blow-up, asymptotic behavior, type 2, eventual monotonicity

File：JMI2011C-1

Abstract. We propose a new numerical scheme for simulation of flow of two immiscible and incompressible phases in porous media. The method is based on a combination of the mixedhybrid finite element (MHFE) and discontinuous Galerkin (DG) methods. The combined approach allows for accurate approximation of the flux at the boundary between neighboring finite elements, especially in heterogeneous media. We extend the method proposed in [12] to simulate the nonwetting phase pooling at material interfaces. In order to show its applicability, the MHFE-DG method is tested against benchmark solutions and using laboratory data from literature.

Keywords. MHFE-DG method, Two-phase flow, Heterogeneous porous media

File：JMI2011C-2

Abstract. In this paper we consider the motion of non-closed planar polygonal curves governed by generalized crystalline curvature flow with a driving force. In the context of ”crystalline motion”, we usually restrict the curves in the special class of polygonal curves, so-called ”admissible class.” We here extend the previous results to wider class which is called ”essentially admissible class.” In such a class, there are no order-preserving structure, thus, controlling the movement of the solution curves becomes more difficult. In this paper we investigate the estimate of the movement of each facet of the solution curves in the essentially admissible class and show the global existence of V-shaped solutions.

Keywords. Motion by crystalline curvature, non-closed polygonal curves, essentially admissible
curves, V-shaped interface.

File：JMI2011C-3

Abstract. In this paper the numerical results for steady and unsteady flows of viscous and viscoelastic fluids are presented. The governing system of equations is based on the system of balance laws for mass and momentum for incompressible fluid. Two rheological models for the stress tensor are tested. The models used in this study are generalized Newtonian model with power-law viscosity model and Oldroyd-B model with constant viscosity. Numerical results for these models are presented.

Keywords. Navier-Stokes equations, generalized Newtonian fluids, Oldroyd-B fluids

File：JMI2011C-4

Abstract. This paper deals with segmentation of image data using a partial differential equation of level-set type. The first part of this paper describes the level-set formulation and modification of the level-set equation. The evolution process are controlled by the segmented image data in such a way that the edges of objects can be found. The semi-implicit complementary-volume numerical scheme is used for solving the level-set equation. The final part of the paper describes algorithm parameters and their setting used for segmentation of the left heart ventricle in the cardiac MRI images.

Keywords. Cardiac MRI, co-volume method, image segmentation, level set method, PDE

File：JMI2011C-5

Abstract. The article summarizes application of the nonlinear Galerkin method to the numerical solution of reaction-diffusion systems - the Gray-Scott model and simplified equation. The space discretization is performed by the finite-element method. For time integration we use the Runge- Kutta method with time step adaptivity. The computational results demonstrate properties of the method and the comparison with the finite difference method.

Keywords. Nonlinear Galerkin method, Gray-Scott model, reaction-diffusion equation, method of lines, finite difference method, finite element method, Runge-Kutta-Merson method.

File：JMI2011C-6

Abstract. The paper deals with the numerical modeling of compressible single-phase flow of a mixture composed of several components in a porous medium. The mathematical model is formulated by means of Darcy’s law, components continuity equations, constitutive relations, and appropriate initial and boundary conditions. The problem is solved numerically using a combination of the mixed-hybrid finite element method for Darcy’s law discretization and the finite volume method for the discretization of the transport equations. This approach provides exact local mass balance.
The time discretization is carried out by the Euler method. The resulting large system of nonlinear algebraic equations is solved by the Newton-Raphson iterative method. The dimensions of obtained system of linear algebraic equations are significantly reduced so that they do not depend on the number of mixture components. The convergence of the numerical scheme is verified on two problems of methane injection into a homogeneous 2D reservoir filled with propane which is horizontally or vertically oriented.

Keywords. Mixed-hybrid finite element method, finite volume method, Newton-Raphson method, single-phase compressible multicomponent flow, miscible displacement

File：JMI2011C-7

Abstract. Nonlinear diffusion equations exhibit a wide variety of phenomena in the several fields of fluid dynamics, plasma physics and population dynamics. Among these the interaction between diffusion and absorption suggests a remarkable property in the behavior of the support of solution; that is, after support splitting phenomena appear, the support merges, and thereafter the support splits again. Moreover, numerically repeated support splitting and merging phenomena are observed.
In this paper, making use of the properties of the particular solutions, we construct an initial function for which such phenomena appear.

Keywords. nonlinear diffusion, support splitting, support merging, finite difference scheme

File：JMI2011C-8

Abstract. In the catalytic oxidation of carbon monoxide molecules (CO) on platinum surface (Pt), various pattern formations of densities of CO molecules have attracted many chemists and mathematicians since the great contributions by Ertl (e.g., [15]). Hildebrand [2] has proposed a reaction-diffusion-advection system to give mathematical understand for such pattern formations from macroscopic point of view. In a previous paper [6], we obtain sufficient conditions of the existence (or nonexistence) of stationary patterns of the system. However, the L^∞-boundedness for all stationary patterns have not yet been obtained. In this paper, we show that all stationary patterns of the system possess a universal L^∞ bound. This result yields a validity of the system from the modelling point of view.

Keywords. reaction-diffusion-advection system, stationary pattern, universal bound

File：JMI2011C-9

Abstract. Modern GPUs are well suited for performing image processing tasks. We utilize their high computational performance and memory bandwidth for image segmentation purposes.
We segment cardiac MRI data by means of numerical solution of an anisotropic partial differential equation of the Allen-Cahn type. We implement two different algorithms for solving the equation on the CUDA architecture. One of them is based on the Runge-Kutta-Merson method for the approximation of solutions of ordinary differential equations, the other uses the GMRES method for the numerical solution of systems of linear equations. In our experiments, the CUDA implementations of both algorithms are about 3–9 times faster than corresponding 12-threaded OpenMP implementations.

Keywords. CUDA, image segmentation, Allen-Cahn equation, Runge-Kutta-Merson, GMRES

File：JMI2011C-10

Abstract. Regeneration phenomena in cricket legs and Planarians have recently been studied at the single cellular level. Within a cell, Dachsous and Fat molecules, and between cells, Dachsous-Fat heterodimers, are considered related to regeneration phenomena. Inspired by recent studies on Dachsous and Fat, we modeled a cell chain with heterodimers and analyzed it. We parameterized redistribution of heterodimers during cell division, which is poorly understood. We then derived equations in parameters to regenerate the heterodimeric pattern even if part of the cell chain is excised. This excision model contained eight parameters, and hence we used a few algebraic methods to suit models that are described by a set of polynomials. A number of biological phenomena have recently been analyzed through algebraic methods; thereby, we can directly derive equations in parameters. The derived equations show that some specific relation between the redistribution ratio of heterodimers allows a cell chain to regenerate its heterodimeric pattern.

Keywords. Dachsous-Fat system, regeneration, algebraic methods, developmental biology

File：JMI2011B-1

Abstract. One way of improving efficiency of Gentry’s fully homomorphic encryption from ideal lattices is controlling the number of operations, but our recollection is that any scheme which controls the bound has not proposed. In this paper, we propose a key generation algorithm for Gentry’s scheme that controls the bound of the circuit depth by using the relation between the circuit depth and the eigenvalues of a basis of a lattice. We present experimental results that show that the proposed algorithm is practical. We discuss security of the basis of the lattices generated by the algorithm for practical use.

Keywords. homomorphic encryption, ideal lattice, circulant matrix

File：JMI2011B-2

Abstract. Pollard’s rho method is the asymptotically fastest known attack for the elliptic curve discrete logarithm problem (ECDLP) except special cases. It works by giving a pseudo-random sequence defined by an iteration function and then detecting a collision in the sequence. We note that the number of iterations before obtaining a collision is significant for the running time of the rho method and depends on the choice of an iteration function. For many iteration functions suitable for the ECDLP on elliptic curves except Koblitz curves, the number of iterations before obtaining a collision had been investigated. In this paper, we propose a new iteration function on Koblitz curves which is an extension of the iteration function proposed by Gallant et al. and analyze the performance on our iteration function experimentally.

Keywords. Pollard’s rho method, ECDLP, Koblitz curves, Frobenius map

File：JMI2011B-3

Abstract. We prove the non-existence of elliptic curves having good reduction everywhere over some real quadratic fields. These results of computations give best-possible data including structures of Mordell-Weil groups over real quadratic fields \mathbb{Q}(\sqrt{m}) up to 100 via two-descent.

Keywords. Elliptic curves having everywhere good reduction, Mordell-Weil groups, Two-descent.

File：JMI2011B-4

Abstract. The differential-geometric structures of ideal magnetohydrodynamics are studied by calculating sectional curvatures of a semidirect product of the volume-preserving diffeomorphism group and the vector field on it. Some propositions on the negativeness and positiveness of the sectional curvatures are derived. In particular, the curvature of the section corresponding to the sausage instability of plasma is shown to be negative.

Keywords. ideal magnetohydrodynamics, differential geometry, sectional curvature, instability

File：JMI2011B-5

Abstract. A complete description of the geodesic curves on the Riemann manifold of multivariate normal distributions equipped with the Fisher information metric has been accomplished by Eriksen in 1987, and later by Calvo and Oller in 1991 but in a different manner. The former describes geodesic curves in terms of an exponential map in somewhat mysterious way and the latter obtains a solution of the differential equation of a geodesic curve explicitly by solving much general system of differential equations. The method what Erikson had taken seems to have a group theoretinature while it is still unclear. The purposes of this short note are to derive the explicit formula of the geodesic curve from the result obtained by Eriksen and to clarify why such exponential map may give geodesic curves for the one dimensional normal distribution case.

Keywords. geodesics, multivariate normal distribution, statistical manifold, Fisher’s information
metric, Riemann symmetric spaces, Lorentz group.

File：JMI2011B-6

Abstract. A fast analysis method is proposed to obtain time-periodic nonlinear fields in the presence of extremely slow decay fields. First, the analysis variables are time-averaged to reduce effects of the harmonic wave components, and next, second order time-derivatives of the timeaveraged values are used to correct the variables toward the time-periodic steady-state field. The time width in time-averaging operation can be set much shorter than one half period, and so the variables can be corrected in early stage of time evolution. The presented method was validated in two-variable simultaneous equations as a simple problem and a magnetic field simulation by the finite element method as a multivariable problem. Furthermore, harmonic TDC and a serial usage of (harmonic) TDC and TP-EEC are proposed for the case that higher order time-harmonic waves are included in the corrected objectives. In addition, the conventional simplified three-phase AC TP-EEC method is expanded to a general form for the three-phase AC system.

Keywords. time-periodic, steady-state, correction, transient field

File：JMI2011B-7

Abstract. In this paper we determine the complete coset weight distributions of the second order Reed-Muller code RM(2, 6) of length 64. Our method fully uses the interaction between the Jacobi polynomials for the code RM(2, 6) and those of the dual code RM(3, 6). The method also employs the group theoretic reduction processes to diminish the runtimes of computing the Jacobi polynomials for the code RM(2, 6) in great effect.


Keywords. second order Reed-Muller code of length 64, coset weight distributions, Jacobi polynomials

File：JMI2011A-1

Abstract. We study a general mathematical framework for variation of potential energy with respect to domain deformation. It enables rigorous derivation of the integral formulas for the energy release rate in crack problems. Applying a technique of shape sensitivity analysis, we formulate the shape derivative of potential energy as a variational problem with a parameter. Key tools of our abstract theory are a new parameter variational principle and the classical implicit function theorem in Banach spaces.


Keywords. shape derivative, variational principle, energy release rate, fracture mechanics

File：JMI2011A-2

Abstract. Ring Signcryption is used to provide a graceful way to leak trustworthy secrets in an anonymous, authenticated and confidential way. To the best of our knowledge, seven identity based ring signcryption schemes were reported in the literature. In this paper, we show that five of them are insecure in one or the other way. We then propose a new scheme and formally prove its security properties. A comparison of our scheme with the only existing secure scheme by Huang et al. shows that our scheme is more efficient than the scheme by Huang et al.


Keywords. Ring Signcryption, Cryptanalysis, Provable Security, Confidentiality, Chosen Plaintext Attack, Adaptive Chosen Ciphertext Attack, Bilinear Pairing, Random Oracle Model.

File：JMI2011A-3

Abstract. The target matrix of the dhLV algorithm is already shown to be a class of nonsymmetric band matrix with complex eigenvalues. In the case where the band width M = 1 in the dhLV algorithm, it is applicable to a tridiagonal matrix, with real eigenvalues, whose upper and lower subdiagonal entries are restricted to be positive and 1, respectively. In this paper, we first clarify that the dhLV algorithm is also applicable to the eigenvalue computation of nonsymmetric tridiagonal matrix with relaxing the restrictions for subdiagonal entries. We also demonstrate that the wellknown packages are not always desirable for computing nonsymmetric eigenvalues with respect to numerical accuracy. Through some numerical examples, it is shown that the tridiagonal eigenvalues computed by the dhLV algorithm are to high relative accuracy.


Keywords. matrix eigenvalues, tridiagonal matrix, discrete hungry Lotka-Volterra system

File：JMI2011A-4

Abstract. Various solutions to the discrete Schwarzian KdV equation are discussed. We first derive the bilinear difference equations of Hirota type of the discrete Schwarzian KP equation, which is decomposed into three discrete two-dimensional Toda lattice equations. We then construct two kinds of solutions in terms of the Casorati determinant. We derive the discrete Schwarzian KdV equation on an inhomogeneous lattice and its solutions by a reduction process. We finally discuss the solutions in terms of the τ functions of some Painlev´e systems.


Keywords. discrete integrable systems, discrete differential geometry, bilinear equation, τ function

File：JMI2011A-5

Abstract. Radial Basis Function (RBF) interpolation is a common approach to scattered data interpolation. Gaussian Process regression is also a common approach to estimating statistical data. Both techniques play a central role, for example, in statistical or machine learning, and recently they have been increasingly applied in other fields such as computer graphics. In this survey we describe the formulation of both techniques as instances of functional regression in a Reproducing Kernel Hilbert Space. We then show that the RBF and Gaussian Process techniques can in some cases be reduced to an identical formulation, differing primarily in their assumptions on when the data locations and values are known, as well as in their (respectively) deterministic and stochastic perspectives. The scope and effectiveness of the RBF and Gaussian process techniques are illustrated through several applications in computer graphics.


Keywords. Reproducing Kernel Hilbert Space, Radial Basis Function, scattered data interpolation,
Gaussian process regression.

File：JMI2011A-6

Abstract. The quadratic Wiener functional coming from the Anderson-Darling test statistic is investigated in the framework of the Malliavin calculus. The functional gives a completely new and important example of a quadratic Wiener functional.


Keywords. Anderson-Darling test, the Malliavin calculus, quadratic Wiener functional, Legendre polynomial

File：JMI2011A-7

Abstract. Within the last 20 years, new biological structures called fractones, named in honor ofthe late Dr. Benoit Mandelbrot due to their fractal-like appearance, have been discovered by cellbiologists. Their primary purposes are theorized to pertain to the major processes of the life cycleof cells, namely cell division, migration, and differentiation. Building on the back of the discretizeddiffusion equations, we built a mathematical model of how these fractones interact with the cellsand the associated growth factors produced in order to gain insight into the growth process as a whole. As it is shown in this paper, the complexity of this biological process opens the door to entirely new questions in the field of control theory.


Keywords. Morphogenesis, Fractones, Control Theory, Computational Modeling.

File：JMI2011A-8

Abstract. There are two decision methods for the decomposition of multiple points on an elliptic curve, one based on Semaev’s summation polynomials and the other based on Stange’s elliptic nets.
This paper presents some relations between these two methods. Using these relations, we show that an index calculus attack for the elliptic curve discrete logarithm problem (ECDLP) over extension fields via an elliptic net is equivalent to such an attack via Semaev’s summation polynomials.


Keywords. Index calculus attack, Semaev’s summation polynomials, elliptic nets.

File：JMI2011A-9