hilbert kunz density function and asymptotic hilbert kunz multiplicity for projective toric varieties abstract for a toric pair x d where x is a projective toric variety of dimension d 1 1 and d is a very ample t cartier divisor we show that the hilbert kunz density function hkd x d is the d 1 dimensional volume of p d z where p d r d is a compact d dimensional set which is a finite union of convex polytopes we also show that for k 1 the function hkd x k d can be replaced by another compactly supported continuous function k d which is linear in k this gives the formula for the associated coordinate ring r m lim k e h k r m k e 0 r m k d k d 1 e 0 r m d 1 0 d d where d see proposition 1 2 is solely determined by the shape of the polytope p d associated to the toric pair x d moreover d is a multiplicative function for segre products this yields explicit computation of d and hence the limit for smooth fano toric surfaces with respect to anticanonical divisor in general due to this formulation in terms of the polytope p d one can explicitly compute the limit for two dimensional toric pairs and their segre products we further show that theorem 6 3 the renormalized limit takes the minimum value if and only if the polytope p d tiles the space m r r d 1 with the lattice m z d 1 as a consequence one gets an algebraic formulation of the tiling property of any rational convex polytope

quaternion involutions and anti involutions an involution or anti involution is a self inverse linear mapping in this paper we study quaternion involutions and anti involutions we review formal axioms for such involutions and anti involutions we present two mappings one a quaternion involution and one an anti involution and a geometric interpretation of each as reflections we present results on the composition of these mappings and show that the quaternion conjugate may be expressed using three mutually perpendicular anti involutions finally we show that projection of a vector or quaternion can be expressed concisely using three mutually perpendicular anti involutions

a system of generalized sylvester quaternion matrix equations and its applications let h m n be the set of all m n matrices over the real quaternion algebra we call that a h n n is hermitian if a a i j k where i j k are the quaternion units denote a a in this paper we derive some necessary and sufficient conditions for the solvability to the system of generalized sylvester real quaternion matrix equations a i x i y i b i c i z d i e i i 1 2 and give an expression of the general solution to the above mentioned system as applications we give some solvability conditions and general solution for the generalized sylvester real quaternion matrix equation a 1 x y b 1 c 1 z d 1 e 1 where z is required to be hermitian we also present some solvability conditions and general solution for the system of real quaternion matrix equations involving hermicity a i x i a i x i b i y b i c i i 1 2 where y is required to be hermitian our results include some well known results as special cases

hilbert kunz functions of a hirzebruch surface abstract here we compute hilbert kunz functions of any nontrivial ruled surface over p k 1 with respect to all ample line bundles on it

notes on the hilbert kunz function abstract we note certain properties of the hilbert kunz function and hilbert kunz multiplicity including a strengthened inequality between hilbert kunz and hilbert samuel multiplicities and a characterization of a mixed hilbert function incorporating both ordinary and frobenius powers of ideals

preconditioning techniques for large linear systems a survey this article surveys preconditioning techniques for the iterative solution of large linear systems with a focus on algebraic methods suitable for general sparse matrices covered topics include progress in incomplete factorization methods sparse approximate inverses reorderings parallelization issues and block and multilevel extensions some of the challenges ahead are also discussed an extensive bibliography completes the paper

a new real structure preserving quaternion qr algorithm abstract new real structure preserving decompositions are introduced to develop fast and robust algorithms for the right eigenproblem of general quaternion matrices under the orthogonally j r s symplectic transformations the francis j r s qr step and the j r s qr algorithm are firstly proposed for j r s symmetric matrices and then applied to calculate the schur form of quaternion matrices a novel quaternion givens matrix is defined and utilized to compute the qr factorization of quaternion hessenberg matrices an implicit double shift quaternion qr algorithm is presented with a technique for automatically choosing shifts and within real operations numerical experiments are provided to demonstrate the efficiency and accuracy of newly proposed algorithms

lsqr algorithm with structured preconditioner for the least squares problem in quaternionic quantum theory abstract the solution of a linear quaternionic least squares qls problem can be transformed into that of a linear least squares ls problem with jrs symmetric real coefficient matrix which is suitable to be solved by developing structured iterative methods when the coefficient matrix is large and sparse the main aim of this work is to construct a structured preconditioner to accelerate the lsqr convergence the preconditioner is based on structure preserving tridiagonalization to the real counterpart of the coefficient matrix of the normal equation and the incomplete inverse upper lower factorization related to only one symmetric positive definite tridiagonal matrix rather than four so it is reliable and has low storage requirements the performances of the lsqr algorithm with structured preconditioner are demonstrated by numerical experiments

hilbert kunz theory for nodal cubics via sheaves abstract suppose b f x y z h is the homogeneous coordinate ring of a characteristic p degree 3 irreducible plane curve c with a node let j be a homogeneous x y z primary ideal and n e n be the hilbert kunz function of b with respect to j let q p n when j x y z it is known that e n 7 3 q 2 1 3 q r where r 5 3 if q 2 3 and is 1 otherwise we generalize this showing that e n q 2 q r where r only depends on q mod 3 we describe and r in terms of classification data for a vector bundle on c

expanded bethe peierls approximation for the blume capel model we have studied the blume capel model by using the expanded bethe peierls approximation in this approximation the system is taken as a group of chains composed of a central chain and its nearest neighbor chains the nearest neighbor chains are in an effective field produced by the other spins which can be determined by the condition that the magnetization of the central chain is equal to that of its nearest neighbor chains in the present approximation the lattice dimensionality can be distinguished in the formulations with the use of the transfer matrix method we calculate the free energy magnetization specific heat and transition temperature of the systems on simple cubic z 6 triangular z 6 and square z 4 lattices it is found that the behavior of the specific heat with temperature and the single ion anisotropy parameter is similar to that obtained by the effective field theory the temperatures at the tricritical point and the phase transition point on simple cubic lattice are higher than those on the triangular lattice the transition temperatures at zero single ion anisotropy on simple cubic and square lattices are much closer to those obtained by series expansion method than those obtained by effective field theory and bethe lattice approximation and the value of the single ion anisotropy at the tricritical point obtained on simple cubic lattice is in agreement with that obtained by the series expansion method

spin 1 ising model on tetrahedron recursive lattices exact results abstract we investigate the ferromagnetic spin 1 ising model on the tetrahedron recursive lattices an exact solution of the model is found in the framework of which it is shown that the critical temperatures of the second order phase transitions of the model are driven by a single equation simultaneously on all such lattices it is also shown that this general equation for the critical temperatures is equivalent to the corresponding polynomial equation for the model on the tetrahedron recursive lattice with arbitrary given value of the coordination number the explicit form of these polynomial equations is shown for the lattices with the coordination numbers z 6 9 and 12 in addition it is shown that the thermodynamic properties of all possible physical phases of the model are also completely driven by the corresponding single equations simultaneously on all tetrahedron recursive lattices in this respect the spontaneous magnetization the free energy the entropy and the specific heat of the model are studied in detail

decoupling approximation in spin s s 1 2 ising systems a decoupling approximation is introduced for studying the magnetic properties of a spin s s 1 2 ising system within the framework of the differential operator technique the formulation yields a statistical accuracy better than that of bethe peierls approximation it is also applied to the study of phase diagram in a spin s transverse ising system

exact results for the spin 1 ising model on pure square husimi lattices critical temperatures and spontaneous magnetization abstract we investigate the second order phase transitions of the ferromagnetic spin 1 ising model on pure husimi lattices built up from elementary squares with arbitrary values of the coordination number it is shown that the critical temperatures of the second order phase transitions are driven by a single equation simultaneously on all such lattices it is also shown that for arbitrary given value of the coordination number this equation is equivalent to the corresponding polynomial equation the explicit form of these polynomial equations is present for the lattices with the coordination numbers z 4 6 and 8 it is proven that at least for the small values of the coordination number the positions of the critical temperatures are uniquely determined in addition it is shown that the properties of all phases of the model are also driven by the corresponding single equations simultaneously on all pure husimi lattices built up from elementary squares the spontaneous magnetization of the model is investigated in detail

color image processing by using binary quaternion moment preserving thresholding technique this paper presents a new moment preserving thresholding technique called the binary quaternion moment preserving bqmp thresholding for color image data based on representing color data by the quaternions the statistical parameters of color data can be expressed through the definition of quaternion moments analytical formulas of the bqmp thresholding can thus be determined by using the algebra of the quaternions the computation time for the bqmp thresholding is of order of the data size by using the bqmp thresholding quaternion moment based operators are designed for the application of color image processing such as color image compression multiclass clustering of color data and subpixel color edge detection the experimental results show that the proposed operator for color image compression can have output picture quality acceptable to human eyes in addition the proposed edge operator can detect the color edge at the subpixel level therefore the proposed bqmp thresholding can be used as a tool for color image processing

a fast structure preserving method for computing the singular value decomposition of quaternion matrices abstract in this paper we propose a fast structure preserving algorithm for computing the singular value decomposition of quaternion matrices the algorithm is based on the structure preserving bidiagonalization of the real counterpart for quaternion matrices by applying orthogonal jrs symplectic matrices the algorithm is efficient and numerically stable

asymptotic hilbert kunz multiplicity abstract for a pair m i where m is finitely generated graded module over a standard graded ring r of dimension d 2 and i is a graded ideal with l r i and generated by elements of the same degree we prove that lim q e 1 m i q q d exists where e 1 m i q denotes the first coefficient of the hilbert samuel polynomial of m i q we use this to get an expression for lim k e h k m i k e 0 m i k d k d 1 where e h k denotes the hilbert kunz multiplicity in particular if dim m d then we deduce that the difference e h k m i k e m i k d grows at least as a fixed positive multiple of k d 1 as k this is proved using renormalized hk density functions

matrix lsqr algorithm for structured solutions to quaternionic least squares problem abstract in this paper we employ matrix lsqr algorithm to deal with quaternionic least squares problem in order to find the minimum norm solutions with kinds of special structures and propose a strategy to accelerate convergence rate of the algorithm via right left preconditioning of the coefficient matrices we mainly focus on analyzing the minimum norm hermitian solution and the minimum norm bihermitian solution to the quaternionic least squares problem i j k other structured solutions also can be obtained using the proposed technique a number of numerical experiments are performed to show the efficiency of the preconditioned matrix lsqr algorithm

hypercomplex fourier transforms of color images fourier transforms are a fundamental tool in signal and image processing yet until recently there was no definition of a fourier transform applicable to color images in a holistic manner in this paper hypercomplex numbers specifically quaternions are used to define a fourier transform applicable to color images the properties of the transform are developed and it is shown that the transform may be computed using two standard complex fast fourier transforms the resulting spectrum is explained in terms of familiar phase and modulus concepts and a new concept of hypercomplex axis a method for visualizing the spectrum using color graphics is also presented finally a convolution operational formula in the spectral domain is discussed

real structure preserving algorithms of householder based transformations for quaternion matrices in this paper we survey three different forms of householder based transformations for quaternion matrices in the literature and propose a new form of quaternion householder based transformation we propose real structure preserving algorithms of these householder based transformations which make the procedure computationally more flexible and efficient we compare the computation counts and assignment numbers of these algorithms we also compare the effectiveness of these real structure preserving algorithms applying to the quaternion qrd and the quaternion svd all these four real structure preserving algorithms are more efficient comparing to the algorithms which apply quaternion toolbox using quaternion arithmetics or algorithms which directly performs real householder transformations on the real representation of a quaternion matrix among these four real structure preserving algorithms the most efficient ones are based on quaternion householder reflection and new proposed householder based transformation

solution to a system of real quaternion matrix equations encompassing η hermicity let h m n be the set of all m n matrices over the real quaternion algebra h c 0 c 1 i c 2 j c 3 k i 2 j 2 k 2 i j k 1 c 0 c 1 c 2 c 3 r a h n n is known to be hermitian if a a a i j k and a means the conjugate transpose of a we mention some necessary and sufficient conditions for the existence of the solution to the system of real quaternion matrix equations including hermicity a 1 x c 1 a 2 y c 2 y b 2 d 2 y y a 3 z c 3 z b 3 d 3 z z a 4 x a 4 x b 4 y b 4 c 4 z c 4 d 4 and also construct the general solution to the system when it is consistent the outcome of this paper diversifies some particular results in the literature furthermore we constitute an algorithm and a numerical example to comprehend the approach established in this treatise

a new structure preserving method for quaternion hermitian eigenvalue problems in this paper we propose a novel structure preserving algorithm for solving the right eigenvalue problem of quaternion hermitian matrices the algorithm is based on the structure preserving tridiagonalization of the real counterpart for quaternion hermitian matrices by applying orthogonal jrs symplectic matrices the algorithm is numerically stable because we use orthogonal transformations the algorithm is very efficient it costs about a quarter arithmetical operations and a quarter to one eighth cpu times comparing with standard general purpose algorithms numerical experiments are provided to demonstrate the efficiency of the structure preserving algorithm

lsqr iterative common symmetric solutions to matrix equations axb e and cxd f abstract a new matrix based iterative method is presented to compute common symmetric solution or common symmetric least squares solution of the pair of matrix equations axb e and cxd f by this iterative method for any initial matrix x0 a solution x can be obtained within finite iteration steps if exact arithmetic was used and the solution x with the minimum frobenius norm can be obtained by choosing a special kind of initial matrix in addition the unique nearest common symmetric solution or common symmetric least squares solution x to given matrix x in frobenius norm can be obtained by first finding the minimum frobenius norm common symmetric solution or common symmetric least squares solution of the new pair of matrix equations the given numerical examples show that the matrix based iterative method proposed in this paper has faster convergence than the iterative methods proposed in 1 2 to solve the same problems

global least squares method gl lsqr for solving general linear systems with several right hand sides abstract in this paper we propose a new method for solving general linear systems with several right hand sides this method is based on global least squares method and reduces the original matrix to the lower bidiagonal form we derive a simple recurrence formula for generating the sequence of approximate solutions x k some theoretical properties of the new method are discussed and we also show that how this method can be implemented for the sylvester equation finally some numerical experiments on test matrices are presented to show the efficiency of the new method