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The method of the fundamental solutions (MFS) is used to construct an approximate solution for a partial differential equation in a bounded domain. It is demonstrated by combining the fundamental solutions shifted to the points outside the domain and determining the coefficients of the linear sum to satisfy the boundary condition on the finite points of the boundary. In this paper, the existence of the approximate solution by the MFS for the Neumann problems of the modified Helmholtz equation in disk domains is rigorously demonstrated. We reveal the sufficient condition of the existence of the approximate solution. Applying the Green formula to the Neumann problem of the modified Helmholtz equation, we bound the error between the approximate solution and exact solution into the difference of the function of the boundary condition and the normal derivative of the approximate solution by boundary integrations. Using this estimate of the error, we show the convergence of the approximate solution by the MFS to the exact solution with exponential order, that is, N 2 a N order, where a is a positive constant less than one and N is the number of collocation points. Furthermore, it is demonstrated that the error tends to 0 in exponential order in the numerical simulations with increasing number of collocation points N .

We study a class of nonlinear evolution systems of time fractional partial differential equations using Lie symmetry analysis. We obtain not only infinitesimal symmetries but also a complete group classification and a classification of group invariant solutions of this class of systems. We find that the class of systems of differential equations studied is naturally divided into two cases on the basis of the type of a function that they contain. In each case, the dimension of the Lie algebra generated by the infinitesimal symmetries is greater than 2, and for this reason we present the structures and one-dimensional optimal systems of these Lie algebras. The reduced systems corresponding to the optimal systems are also obtained. Explicit group invariant solutions are found for particular cases., Comment: 14 pages. To appear in Applied Mathematics and Computation

Zeta functions of periodic cubical lattices are explicitly derived by computing all the eigenvalues of the adjacency operators and their characteristic polynomials. We introduce cyclotomic-like polynomials to give factorization of the zeta function in terms of them and count the number of orbits of the Galois action associated with each cyclotomic-like polynomial to obtain its further factorization. We also give a necessary and sufficient condition for such a polynomial to be irreducible and discuss its irreducibility from this point of view., Comment: Advanced Studies in Pure Mathematics 84, 2020, Various Aspects of Multiple Zeta Functions, in honor of Professor Kohji Matsumoto's 60th birthday, pp. 93-121

We consider a certain tiling problem of a planar region in which there are no long horizontal or vertical strips consisting of copies of the same tile. Intuitively speaking, we would like to create a dappled pattern with two or more kinds of tiles. We give an efficient algorithm to turn any tiling into one satisfying the condition, and discuss its applications in texturing.

We implemented a Mathematica module of CGA which includes functions to denote CGA elements and compute several operations in CGA. We can draw the figure in 3D space which is corresponding to a CGA element. Our draw function is using Gr\"{o}bner Basis for simplifying equations of figures. It can be used for any dimensional figures. One of our motivations is to realize 3D origami system using our own CGA Library. We follow the 2D computational origami system E-Origami-System developed by Ida et.al. and formulated simple fold operations in 3D by using CGA points and motions. Then, we proved some geometric theorems about origami properties by computing CGA equation formulas.

It is well-known that one-dimensional time fractional diffusion-wave equations with variable coefficients can be reduced to ordinary fractional differential equations and systems of linear fractional differential equations via scaling transformations. We then derive exact solutions to classes of linear fractional differential equations and systems thereof expressed in terms of Mittag-Leffler functions, generalized Wright functions and Fox H-functions. These solutions are invariant solutions of diffusion-wave equations obtained through certain transformations, which are briefly discussed. We show that the solutions given in this work contain previously known results as particular cases., Comment: 25 pages

We prove, for graphical models for nearest neighbour interactions, a conjecture stated by Letac and Massam in 2007. Our result is important in the analysis of Wishart distributions on cones related to graphical models and in its statistical applications.

Let K be a complex quadratic extension of and let denote the adeles of K. We find special values at all of the critical points of twisted tensor L-functions attached to cohomological cuspforms on GL2() and establish Galois equivariance of the values. To investigate the values, we determine the archimedean factors of a class of integral representations of these L–functions, thus proving a conjecture due to Ghate. We also investigate analytic properties of these L–functions, such as their functional equations.

Let $ G $ be a connected reductive algebraic group over $ \C $. We denote by $ K = (G^{\theta})_{0} $ the identity component of the fixed points of an involutive automorphism $ \theta $ of $ G $. The pair $ (G, K) $ is called a symmetric pair. Let $Q$ be a parabolic subgroup of $K$. We want to find a pair of parabolic subgroups $P_{1}$, $P_{2}$ of $G$ such that (i) $P_{1} \cap P_{2} = Q$ and (ii) $P_{1} P_{2}$ is dense in $G$. The main result of this article states that, for a simple group $G$, we can find such a pair if and only if $(G, K)$ is a Hermitian symmetric pair. The conditions (i) and (ii) yield to conclude that the $K$-orbit through the origin $(e P_{1}, e P_{2})$ of $G/P_{1} \times G/P_{2}$ is closed and it generates an open dense $G$-orbit on the product of partial flag variety. From this point of view, we also give a complete classification of closed $K$-orbits on $G/P_{1} \times G/P_{2}$., Comment: 7 pages

Let $ G $ be a connected, simply connected semisimple algebraic group over the complex number field, and let $ K $ be the fixed point subgroup of an involutive automorphism of $ G $ so that $ (G, K) $ is a symmetric pair. We take parabolic subgroups $ P $ of $ G $ and $ Q $ of $ K $ respectively and consider the product of partial flag varieties $ G/P $ and $ K/Q $ with diagonal $ K $-action, which we call a \emph{double flag variety for symmetric pair}. It is said to be \emph{of finite type} if there are only finitely many $ K $-orbits on it. In this paper, we give a parametrization of $ K $-orbits on $ G/P \times K/Q $ in terms of quotient spaces of unipotent groups without assuming the finiteness of orbits. If one of $ P \subset G $ or $ Q \subset K $ is a Borel subgroup, the finiteness of orbits is closely related to spherical actions. In such cases, we give a complete classification of double flag varieties of finite type, namely, we obtain classifications of $ K $-spherical flag varieties $ G/P $ and $ G $-spherical homogeneous spaces $ G/Q $., 47 pages, 3 tables; add all the details of the classification

We explicitly give new group invariant solutions to a class of Riemann-Liouville time fractional evolution systems with variable coefficients. These solutions are derived from every element in an optimal system of Lie algebras generated by infinitesimal symmetries of evolution systems in the class. We express the solutions in terms of Mittag-Leffler functions, generalized Wright functions, and Fox H-functions and show that these solutions solve diffusion-wave equations with variable coefficients. These solutions contain previously known solutions as particular cases. Some plots of solutions subject to the order of the fractional derivative are illustrated.

The non-commutative harmonic oscillator is a 2×2-system of harmonic oscillators with a non-trivial correlation. We write down explicitly the special value at s=2 of the spectral zeta function of the non-commutative harmonic oscillator in terms of the complete elliptic integral of the first kind, which is a special case of a hypergeometric function.

Let V be a neighborhood of a singular locus of a hyperbolic 3-cone-manifold, which is a quotient space of the 3-dimensional hyperbolic space. In this paper we give an explicit expression of a harmonic vector field v on the hyperbolic manifold V in terms of hypergeometric functions. The expression is obtained by solving a system of ordinary differential equations which is induced by separation of the variables in the vector-valued partial differential equation (∆ + 4)τ = 0, where ∆ is the Laplacian of V and τ is the dual 1-form of v. We transform this system of ordinary differential equations to single-component differential equations by elimination of unknown functions and solve these equations. The most important step in solving them consists of two parts, decomposing their differential operators into differential operators of the type appearing in Riemann’s P -equation in the ring of differential operators and then describing the projections to the components of this decomposition in terms of differential operators that are also of the type appearing in Riemann’s P -equation.

We prove existence of scaling limits of sequences of functions defined by the recursion relation w′n+1(x) = –wn(x)2. which is a successive approximation to w′(x) = –w(x)2, a simplest non-linear ordinary differential equation whose solutions have moving singularities. Namely, the sequence approaches the exact solution as n → ∞ in an asymptotically conformal way, wn(x) $\\asymp$ qn $\\bar w$(qnx), for a sequence of numbers {qn} and a function $\\bar w$. We also discuss implication of the results in terms of random sequential bisections of a rod.

We consider the connection problem for the Heun differential equation, which is a Fuchsian differential equation that has four regular singular points. We consider the case in which the parameters in this equation satisfy a certain set of conditions coming from the eigenvalue problem of the non-commutative harmonic oscillators. As an application, we describe eigenvalues with multiplicities greater than 1 and the corresponding odd eigenfunctions of the non-commutative harmonic oscillators. The existence of a rational or a certain algebraic solution of the Heun equation implies that the corresponding eigenvalues has multiplicities greater than 1.

The hypergeometric equations with polyhedral monodromy groups derive 3-integral-parameter families of polynomials.

We propose a method for representing the solutions of a certain type of ordinary differential operator L in terms of those of more fundamental differential operators. This method consists of two steps, decomposing L in the ring of differential operators and then describing the projections to the components of this decomposition also in terms of some differential operators. We provide a concrete algorithm for the application of this method and show that this algorithm is successful for a specific example of a 6-th order homogeneous linear ordinary differential operator of Fuchsian type with three singular points.

For a prime p and for the number a(n) of solutions of xp = 1 in the symmetric group on n letters, ordp\( (a(n)) \geq [n/p] - [n/p^2] \), and especially, ordp\( (a(n)) = [n/p] - [n/p^2] \) provided \( n \equiv 0 \) mod p2. Let r be an integer with \( 1 \leq r \leq p^2 - 1 \). If ordp\( (a(r)) \leq [r/p] + 1 \), then, for each positive integer m, ordp\( (a(mp^2 + r)) = m(p-1)+ ord_p (a(r)) \). Assume that ordp\( (a(r)) = [r/p] + 2 \). If \( a(p^2 + r) \equiv -p^{p-1} a(r) mod \quad p^{p+[r/p]+2} \), then ordp\( (a(mp^2 + r)) = m(p-1) + [r/p] + 2 \); otherwise, there exists a p-adic integer b such tha ordp\( (a(mp^2 + r)) = m(p-1) + [r/p] + 2 + ord_p(m-b) \).

The spectral problem for non-commutative harmonic oscillators is shown to be equivalent to solve Fuchsian ordinary differential equations with four regular singular points in a complex domain.

We give quotients V // G = Spec ( C [ V ] G ) for some prehomogeneous vector spaces ( G × H , V ).

The visual simulation of fluids has become an important element in many applications, such as movies and computer games. In these applications, large-scale fluid scenes, such as fire in a village, are often simulated by repeatedly rendering multiple small-scale fluid flows. In these cases, animators are requested to generate many variations of a small-scale fluid flow. This paper presents a method to help animators meet such requirements. Our method enables the user to generate flow field variations from a single simulated dataset obtained by fluid simulation. The variations are generated in both the frequency and spatial domains. Fluid velocity fields are represented using Laplacian eigenfunctions which ensure that the flow field is always incompressible. In generating the variations in the frequency domain, we modulate the coefficients (amplitudes) of the basis functions. To generate variations in the spatial domain, our system expands or contracts the simulation space, then the flow is calculated by solving a minimization problem subject to the resized velocity field. Using our method, the user can easily create various animations from a single dataset calculated by fluid simulation.

We study absolute zeta functions from the view point of a canonical normalization. We introduce the absolute Hurwitz zeta function for the normalization. In particular, we show that the theory of multiple gamma and sine functions gives good normalizations in cases related to the Kurokawa tensor product. In these cases, the functional equation of the absolute zeta function turns out to be equivalent to the simplicity of the associated non-classical multiple sine function of negative degree.

We introduce multiple versions of L-functions for Witten zeta functions. We study their algebraic and analytic properties. Especially we investigate the existence of zeros at negative integers. These results strongly suggest the universal zero at -2. We look at their absolute limits also.

We consider a generalization of the Mahler measure of a multivariable polynomial $P$ as the integral of $\log^k|P|$ in the unit torus, as opposed to the classical definition with the integral of $\log|P|$. A zeta Mahler measure, involving the integral of $|P|^s$, is also considered. Specific examples are computed, yielding special values of zeta functions, Dirichlet $L$-functions, and polylogarithms.

We study the structure of the twisted homology groups attached to weighted lines in the plane with resonant singular points, and find possible intersection pairings. As a typical example, we treat homology groups attached to 2-dimensional Selberg-type integrals.

Knowing the number of solutions for a Diophantine equation is an important step to study various arithmetic problems. Igusa originated the study of Igusa zeta functions associated to local Diophantine problems. Multiplying all these local Igusa zeta functions we obtain the global version in the natural way. Unfortunately, investigations on global Igusa zeta functions are rare up to now. Reformulating the global Igusa zeta function via the number of morphisms between algebraic systems we discover a new aspect: the multivariable Euler product of Igusa type and its applications. A purpose of this paper is to encourage experts for further studies on global Igusa zeta functions by treating a simple interesting example.

Commuting differential operators with symbols ∂ 1 2 + ∂ 2 2 , ∂ 1 2 ∂ 2 2 are related to the functional equation V 1 ( x 1 ) U + ( x 1 + x 2 ) + V 1 ( x 1 ) U − ( x 1 − x 2 ) + V 2 ( x 2 ) U + ( x 1 + x 2 ) − V 2 ( x 2 ) U − ( x 1 − x 2 ) = F + ( x 1 + x 2 ) + F − ( x 1 − x 2 ) + G 1 ( x 1 ) + G 2 ( x 2 ). We discuss several properties, such as a meromorphic extension of solutions and periodicity, of this functional equation and commuting differential operators.