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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 the convergence rate is linear or sublinear. As a consequence, we show that the exact rates are expressed by multiplicities of the defining polynomials of the semialgebraic set, or related power series in the case that the linear subspace is a line, and we also decide the convergence rate for given data by using elimination theory. Our methods are also applied to give upper bounds for the case that the linear subspace has the dimension more than one. The upper bounds are shown to be tight by obtaining exact convergence rates for a specific semialgebraic set, which depend on the initial points., 22 pages, 1 figure

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 the CE test relies essentially on the cardinality of a subset of a residue ring of the endomorphism ring of an elliptic curve over an ideal. They have evaluated the cardinality of the above subset, and have illustrated the effectiveness of the CE test by using the evaluation. The aim of this paper is to point out that their evaluation contains a flaw. The flaw is generally related to two roots of a quadratic equation which is used in their argument. We mend the flaw of their evaluation. Our correct evaluation shows that the CE test can achieve the same security as the underlying signature scheme on Koblitz curves. As a result, the CE test is a secure batch verification when the underlying signature scheme uses Koblitz curves.

The anomalous nanoscale electromagnetic field arising from light&ndash, 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 semiconductors, dressed photons are still considered to be too elusive to justify because conventional optical theory has never explained the mechanism causing them. The situation seems to be quite similar to that of the dark energy/matter issue in cosmology. Regarding these riddles in different disciplines, we find a common important clue for their resolution in the form of the relevance of space-like momentum support, without which quantum fields cannot interact with each other according to a mathematical result of axiomatic quantum field theory. Here, we show that a dressed photon, as well as dark energy, can be explained in terms of newly identified space-like momenta of the electromagnetic field and dark matter can be explained as the off-shell energy of the Weyl tensor field.

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

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 .

The claudin family comprises four-pass transmembrane proteins involved in the formation of tight junctions (TJs). Relatively recently, claudin domain containing (CLDND) 1, also known as claudin-25, was identified as a novel member of the claudin family. In the present study, we revealed that in the adult murine brain, CLDND1 is abundant in the cerebellum among common sites of intracerebral hemorrhage. Thus, the dynamics of CLDND1 after cerebellar hemorrhage were examined. Both CLDND1 mRNA and protein levels transiently decreased at 24 hr after hemorrhagic insult. For immunostaining, an anti-CLDND1 antibody that recognizes the specific epitope in the extracellular first loop was prepared. Dual immunohistochemical staining with CD31 using coronal cryosections of intact murine cerebellum tissue revealed that CLDND1 is expressed on endothelial cells. We therefore performed an in vitro permeability test using a human brain endothelial cell (HBEC) line to reveal whether CLDND1 contributes to cell adhesion like other claudins. CLDND1 was expressed on HBECs as well as in murine cerebellum tissue, and a strong signal was observed at TJs. RNA interference against CLDND1 decreased both the mRNA and protein levels without cytotoxicity. The permeability to small molecules, but not to large ones, across confluent HBECs increased on CLDND1 knockdown compared with mock-treated cells. These results suggest that the transient decrease of CLDND1 after cerebellar hemorrhage is responsible for low-molecular-weight selective vascular hyperpermeability. © 2017 Wiley Periodicals, Inc.

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.

Good parametrizations of affine transformations are essential to interpolation, deformation, and analysis of shape, motion, 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 parametrization of affine transformations, which is a generalization to or an improvement of existing methods. Our method adds yet another choice to the existing toolbox and shows better performance in some applications. A C++ implementation is available to make our framework ready to use in various applications.

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

The wnt protein family has important members involved in cell differentiation, proliferation and plasticity expression; however, little is known about its biosynthesis processes. On the other hand, an increase in the intracerebral cyclic adenosine 3', 5'-monophosphate (cAMP) level leads to synaptic plasticity via the de novo synthesis of any protein. Here, the effect of dibutyryl cAMP (dbcAMP), a membrane permeability cAMP analog, on the wnt family was investigated in rat primary-cultured glial cells containing astrocytes and microglia. Among wnt3a, 4, 5a, 7a and 11 mRNA, only wnt4 expression was increased by longer treatment (24 h), compared with short treatment (2 h), with dbcAMP in a concentration-dependent manner, and its effect reached statistical significance at 1 mM. In cultures of isolated astrocytes or microglia, wnt4 expression was not affected by 1 mM dbcAMP for 24 h, and microglial wnt4 protein was undetectable even when cells were treated with the drug. Mixed glial cells treated for 24 h with 1 mM dbcAMP showed significantly increased wnt4 protein, as well as mRNA. Immunofluorescence manifested that cells that expressed wnt4 protein were astrocytes, but not microglia. Intraperitoneal injection of 1.25 mg/kg rolipram, a phosphodiesterase (PDE) IV inhibitor that can pass through the blood brain barrier and inhibits cAMP degradation specifically, showed a tendency to increase wnt4 expression in the adult rat brain after 24 h, and the increases in wnt4 mRNA and protein levels reached statistical significance in the hippocampus and striatum, respectively. This is the first finding to help elucidate the selective biosynthesis of central wnt4 through cAMP-stimulated microglia and astrocytes interaction.

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.

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/effective creation of computer animation. This book, therefore, serves as a good guidepost to mathematics (differential geometry and Lie theory) for students of geometric modeling and animation in computer graphics. Experienced developers and researchers will also benefit from this book, since it gives a comprehensive overview of mathematical approaches that are particularly useful in character modeling, deformation, and animation. Table of Contents: Preface / Symbols and Notations / Introduction / Rigid Transformation / Affine Transformation / Exponential and Logarithm of Matrices / 2D Affine Transformation between Two Triangles / Global 2D Shape Interpolation / Parametrizing 3D Positive Affine Transformations / Further Readings / Bibliography / Authors' Biographies

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