Hiroyuki Ochiai's research while affiliated with Kyushu University and other places
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Publications (87)
Fuchsian differential equations of order 3 with three singular points and with an accessory parameter are studied. When local exponents are generic, no shift operator is found, for codimension-1 subfamilies, neither. We found shift operators for several codimension-2 subfamilies of which accessory parameter is assigned as a cubic polynomial in the...
Fuchsian differential equations $H_j$ of order $j=3,\dots,6$ with three singular points and one accessory parameter are presented. The shift operators for $H_6$ are studied, which lead to assign the accessory parameter of $H_6$ a cubic polynomial of local exponents so that the equations have nice properties: shift operators and several symmetries....
We consider the convergence rate of the alternating projection method for the nontransversal intersection of a semialgebraic set and a linear subspace. For such an intersection, the convergence rate is known as sublinear in the worst case. We study the exact convergence rate for a given semialgebraic set and an initial point, and investigate when t...
Batch verification for digital signature scheme is a method to verify multiple signatures simultaneously. The complex exponent test (CE test for short) proposed by Cheon and Lee is one of the most efficient batch verification tests for several digital signature schemes on certain types of elliptic curves (including Koblitz curves). The security of...
A Fuchsian differential equation of order six with nine free exponents as parameters and with three singular points is presented. This equation has various symmetries, which specify the accessory parameter as a polynomial of the local exponents. For some shifts of exponents, the shift operators are found, which lead to reducibility conditions of th...
We study the exact convergence rate of the alternating projection method for the nontransversal intersection of a semialgebraic set and a linear subspace. If the linear subspace is a line, the exact rates are expressed by multiplicities of the defining polynomials of the semialgebraic set, or related power series. Our methods are also applied to gi...
The anomalous nanoscale electromagnetic field arising from light–matter interactions in a nanometric space is called a dressed photon. While the generic technology realized by utilizing dressed photons has demolished the conventional wisdom of optics, for example, the unexpectedly high-power light emission from indirect-transition type semiconducto...
A Fuchsian system of rank 8 in 3 variables with 4 parameters is presented. The singular locus consists of six planes and a cubic surface. The restriction of the system onto the intersection of two singular planes is an ordinary differential equation of order four with three singular points. A middle convolution of this equation turns out to be the...
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-...
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 a...
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-Leffle...
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...
We study a class of nonlinear evolution systems of time fractional partial differential equations depending on an arbitrary function using Lie symmetry analysis. We obtain not only infinitesimal symmetries but also a complete group classification and a classification of group invariant solutions of the systems. The class of systems is divided into...
This synthesis lecture presents an intuitive introduction to the mathematics of motion and deformation in computer graphics. Starting with familiar concepts in graphics, such as Euler angles, quaternions, and affine transformations, we illustrate that a mathematical theory behind these concepts enables us to develop the techniques for efficient/eff...
In physics, a rigid body means as an object which preserves the distances between any two points of it with or without external forces over time. So describing rigid transformation (or rigid motion) means finding the non-flip congruence transformations parametrized over time. For a rigid body X, an animation (or a motion) X(t) indexed by a time par...
Next we present our 3D application based on the concepts and techniques in Chapters 3 and 4. So let us consider how to parametrize rigid or non-rigid transformations. As we’ve learned, quater– nion or Euler angle parametrizes rotations, and dual quaternion with axis-angle presentation parametrizes the rigid transformation [Kavan2008]. These paramet...
In this chapter, we discuss affine transformations, i.e., matrices for deformations.
We have presented several mathematical basics that give the theoretical foundation for the deformation/animation techniques in computer graphics. As for your more visual understanding of this book, we would also recommend you to refer to our SIGGRAPH 2016 course notes [Ochiai2016a] and the video assocated with it [Ochiai2016b].
Geometric transformations that we have described give a basic mathematical framework for ge– ometric operations in computer graphics, such as rotation, shear, translation, and their composi– tions. Each affine transformation is then represented by a 4 x 4-homogeneous matrix (3.2) with usual operations: addition, scalar product, and product. While t...
As a simple case, this chapter deals with interpolating two triangles via affine transformations. We note that interpolating affine transformations itself may also have other interesting applications; see, for example, [Shoemake1994b] and [Alexa2002].
We have presented several mathematical basics that give the theoretical foundation for the de– formation/animation techniques in computer graphics. As for your more visual understanding of this book, we would also recommend you to refer to our SIGGRAPH 2016 course notes [Ochiai2016a] and the video assocated with it [Ochiai2016b].
Next we present our 3D application based on the concepts and techniques in Chapters 3 and 4. So let us consider how to parametrize rigid or non-rigid transformations. As we’ve learned, quaternion or Euler angle parametrizes rotations, and dual quaternion with axis-angle presentation parametrizes the rigid transformation [Kavan2008]. These parametri...
Geometric transformations that we have described give a basic mathematical framework for geometric operations in computer graphics, such as rotation, shear, translation, and their compositions. Each affine transformation is then represented by a 4 × 4-homogeneous matrix (3.2) with usual operations: addition, scalar product, and product. While the p...
In physics, a rigid body means as an object which preserves the distances between any two points of it with or without external forces over time. So describing rigid transformation (or rigid motion) means finding the non-flip congruence transformations parametrized over time. For a rigid body X, an animation (or a motion) X(t) indexed by a time par...
In this chapter, we discuss affine transformations, i.e., matrices for deformations.
As a simple case, this chapter deals with interpolating two triangles via affine transformations. We note that interpolating affine transformations itself may also have other interesting applications; see, for example, [Shoemake1994b] and [Alexa2002].
This course presents an elementary, intuitive, and visual introduction to several mathematical basics for beginners in computer graphics. The mathematical concepts covered in this course include 2D/3D translation, rotation, affine transformation, quaternion, dual quaternion, exponential and logarithm. These are quite useful for various aspects of c...
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 solve the problem, and discuss its applications in texturing.
“Progress in Expressive Image Synthesis” (MEIS2015), was held in Fukuoka, Japan, September 25–27, 2015. The aim of the symposium was to provide a unique venue where various issues in computer graphics (CG) application fields could be discussed by mathematicians, CG researchers, and practitioners. Through the previous symposiums MEIS2013 and MEIS201...
Good parametrisation of affine transformations is essential to interpolation,
deformation and analysis of shapes and animation. It has been one of the
central research topics in computer graphics. However, there is no single
perfect method and each one has both advantages and disadvantages. In this
paper, we propose a novel parametrisation of affin...
This synthesis lecture presents an intuitive introduction to the mathematics of motion and deformation in computer graphics. Starting with familiar concepts in graphics, such as Euler angles, quaternions, and affine transformations, we illustrate that a mathematical theory behind these concepts enables us to develop the techniques for efficient/eff...
One of the roles of mathematics is to serve as a language to describe science and technology. The terminology is often common over several branches of science and technology. In this chapter, we describe several basic notions with the emphasis on what is the point of a definition and what are key properties. The objects are taken from set theory, g...
This chapter is intended to give a summary of Lie groups and Lie algebras for computer graphics, including an example from interpolations and blending of motions and deformation. In animation and filmmaking procedure, we want to get a smooth transition of the drawing of given starting and ending point (the drawings at these points are called key fr...
We introduce a new presentation of the two dimensional rigid transformation
which is more concise and efficient than the standard matrix presentation.
By modifying the ordinary dual number construction for the complex numbers,
we define the ring of anti-commutative dual complex numbers,
which parametrizes two dimensional rotation and translation al...
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 sm...
While many technical terms, such as Euler angle, quaternion, and affine transformation, now become quite popular in computer graphics, their graphical meanings are sometimes a bit far from the original mathematical entities, which might cause misunderstanding or misuse of the mathematical techniques. This course presents an intuitive introduction t...
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 sm...
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 th...
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.
This is a survey on the non-commutative harmonic oscillator, which is a generalization of usual (scalar) harmonic oscillators to the system introduced by Parmeggiani and Wakayama.With the definitions and the basic properties, we summarize the positivity of several related operators with sl2 interpretations. We also mention some unsolved questions,...
We give a combinatorial formula of the dimension of global solutions to a generalization of Gauss-Aomoto-Gelfand hypergeometric system, where the quadratic differential operators are replaced by higher order operators. We also derive a polynomial estimate of the dimension of global solutions for the case in 3×3 variables.
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 $...
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...
Differential operators on Siegel modular forms which behave well under the restriction of the domain are essentially intertwining operators of the tensor product of holomorphic discrete series to its irreducible components. These are characterized by polynomials in the tensor of pluriharmonic polynomials with some invariance properties. We give a c...
Let G be a reductive algebraic group over the complex number filed, and K = G^{\theta} be the fixed points of an involutive automorphism \theta of G so that (G, K) is a symmetric pair. We take parabolic subgroups P and Q of G and K respectively and consider a product of partial flag varieties G/P and K/Q with diagonal K-action. The double flag vari...
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 fu...
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 I...
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.
We find a symmetry for the reflection groups in the double shuffle space of depth three. The space was introduced by Ihara, Kaneko and Zagier and consists of polynomials in three variables satisfying certain identities which are connected with the double shuffle relations for multiple zeta values. Goncharov has defined a space essentially equivalen...
Spectra of real alternating operators seem to be quite interesting from the view point of explaining the Riemann Hypothesis for various zeta functions. Unfortunately we have not sufficient experiments concerning this theme. Necessary works would be to supply new examples of spectra related to zeros and poles of zeta functions. A century ago Hilbert...
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 differen...
We define the Casimir energies of permutations of the natural numbers by using the analytic continuation of the zeta functions associated with such permutations. We discuss the analytic properties of such zeta functions and compute the explicit values of Casimir energies for several examples.
We show that the fake projective planes that are constructed from dyadic discrete subgroups discovered by Cartwright, Mantero, Steger, and Zappa are realized as connected components of certain unitary Shimura surfaces. As a corollary we show that these fake projective planes have models defined over the number field .
We prove existence of scaling limits of sequences of functions defined by the recursion relation w(n+1)(1) (x) = -w(n)(x)(2). which is a successive approximation to w(1) (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 -> infinity...
We prove existence of scaling limits of sequences of functions defined by the recursion relation w� n+1(x )= −wn(x) 2. Namely, wn approach the exact solution as n →∞ in asymptotically conformal ways, wn(x) � qn ¯ w(qnx), for a sequence of numbers {qn} .W e also discuss the implications of the results in terms of random sequential bisections of a ro...
We study basic properties of Milnor’s multiple gamma functions. Moreover, we introduce new multiple sine functions from these Milnor’s functions and show the difficult special values of the Dirichlet L-functions can be expressed by these functions.
We investigate the structure of invariant distributions on a non-isotropic non-Riemannian symmetric space of rank one. Especially, the J-criterion related to the generalized Gelfand pair is shown for this space without imposing the condition on the eigenfuction of the Laplace-Bertrami operator.
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, w...
We consider a reductive dual pair (G, G') in the stable range with G' the smaller member and of Hermitian symmetric type. We study the theta lifting of nilpotent K'_C-orbits, where K' is a maximal compact subgroup of G' and we describe the precise K_C-module structure of the regular function ring of the closure of the lifted nilpotent orbit of the...
We show the structure theorem of the monoidal absolute derivations of the matrix algebra over a commutative ring. In particular, those over the ring of rational integers are inner.
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.
We give a formula for the coefficients of the Yablonskii-Vorob'ev polynomial. Also the reduction modulo a prime number of the polynomial is studied.
Just as the function ring case we expect the existence of the coe卤cient field for the integer ring. Using the notion of one element field in place of such a coe卤cient field, we calculate abso-lute derivations of arithmetic rings. Notable examples are the matrix rings over the integer ring, where we obtain some absolute rigidity. Knitting up prime n...
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 som...
We give a formula for the coefficients of the Yablonskii-Vorob'ev polynomial. Also the reduction modulo a prime number of the polynomial is studied.
For a prime p and for the number a(n) of solutions of x
p
= 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 p
2
. 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...
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 prove that each counting function of the m-simple branched covers with a fixed genus of an elliptic curve is expressed as a polynomial of the Eisenstein series E_2, E_4 and E_6 . The special case m=2 is considered by Dijkgraaf.
The p-adic norms of the Taylor coefficients of the function exp(x + xp/p) are expressed in terms of a p-adic analytic function for p ≤ 23.
Just as the function ring case we expect the existence of the coefficient field for the integer ring. Using the notion of one element field in place of such a coefficient field, we calculate abso- lute derivations of arithmetic rings. Notable examples are the matrix rings over the integer ring, where we obtain some absolute rigidity. Knitting up pr...
We study a method of successive approximation to dφ dx (x )= −φ(x)2, a simplest first order non- linear ordinary differential equation whose solutions have moving singularities. We give a sufficient condition for a series of approximate solutions φn, n =0 , 1, 2, ··· , to have a scaling limit, namely, φ(x) = lim n→∞ q −1 n φn(q −1 n x) exists, wher...
Citations
... Although such knowledge might be new or old, it is often not widespread in the relevant community. In our case, we have three pieces of such important knowledge: two of which, the Greenberg-Robinson (GR) theorem [12,13] and the micro-macro duality (MMD) theory of Ojima [14,15], are considered old information, whereas the Clebsch dual space-like electromagnetic (CDSE) theory [16,17,18] built on the two preceding pieces of knowledge is considered new knowledge. As demonstrated by the Schrödinger's cat, the prevailing understanding of the invisible microscopic quantum world and its connection to the visible macroscopic world is unsatisfactory. ...
... However, there are only few methods to find the solutions of nonlinear systems of fractional PDEs with Riemann-Liouville fractional derivatives. Some authors have solved a few fractional nonlinear systems of PDEs with the help of Lie symmetry approach [3][4][5]10,13,[18][19][20]32,33]. It motivated us to solve some physically important time fractional nonlinear systems and analyse their behaviour by graphical representation. ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/595952821870592-1519097633300_Q64/Khongorzul-Dorjgotov.jpg)
... Therefore, nowadays, symmetry properties of multi-dimensional FDEs and systems of FDEs are much less investigated. There is a relatively small number of papers devoted to these topics (we mention here only some recent papers [42][43][44][45][46][47][48][49][50][51][52][53], see also brief overview in the last section of [24]). However, there is a practically significant class of multi-dimensional FDEs that consists of equations with a single fractional derivative (usually, time-fractional derivatives are used in such equations). ...
... Multiple trigonometric functions have periodicity and multiplication formulas, and also satisfy algebraic differential equations similar to the usual trigonometric functions. (See [K1,KOW,KKo]. We also refer to Manin [M] for an excellent survey from the viewpoint of absolute mathematics.) ...
... Let Ω be the free Zmodule generated by the variables ξ p . Let d : Z → Ω be the map defined by d0 = 0 and by dn = n p |n v p (n) p · ξ p for n 0, where p varies over the different prime divisors of n. (A version of d : Z → Ω and generalizations can be found in [6].) Note that n · v p (n)/p ∈ Z when p|n, so dn ∈ Ω for all n ∈ Z. ...
... Affine registration stage is then just one more anti-symmetric update before the other updates. We use affine parametrization by Kaji and Ochiai (2016). ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/279327938760721-1443608378515_Q64/Shizuo-Kaji.jpg)
... In addition to generating physically plausible animations, we also need techniques to control the smoke. There is considerable work on controlling fluid animations [4][5][6][12][13][14][15][16][17][18]. These works control the fluid body using a set of spatiallycomplete keyframe shapes. ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/279514883084305-1443652949374_Q64/Syuhei-Sato.jpg)
![[object Object]](https://i1.rgstatic.net/ii/profile.image/277069188616194-1443069850049_Q64/Yoshinori-Dobashi.jpg)
![[object Object]](https://i1.rgstatic.net/ii/profile.image/277091141603328-1443075084076_Q64/Kei-Iwasaki.jpg)
![[object Object]](https://i1.rgstatic.net/ii/profile.image/277128999391234-1443084110460_Q64/Tomoyuki-Nishita.jpg)
... In the 19 th century, Cli¤ord described the dual numbers with in the form A = a+"a , where a; a 2 R, " 2 = 0 and " 6 = 0 [4]. Up to this time, there are number of studies in the literature that concern about the dual numbers and dual complex numbers [1,3,5,[8][9][10][17][18][19][20]. For instance, Fjelstad and Gal examined the extensions of the hyperbolic complex numbers to n dimensions and they presented n dimensional dual complex numbers [9]. ...
Reference: Dual-Complex Generalized k-Horadam Numbers
... He showed that After his discovery, many people studied their generalizations and analogues: (a) integrals on a different interval, (b) indefinite integrals, (c) integrals involving products of a polynomial of higher degree and logarithms of trigonometric functions, (d) q-analogues, and (e) integrals of a logarithm of generalized multiple sine function. These works can be, for instance, found in [19] for (a)-(c), in [14,17] for (d), and in [13,15,16,18] for (e). It is attractive to build a theory that enables us to handle some of them together. ...
