on additive transformations preserving a multiplicative matrix function abstract let m n k be the ring of all n n matrices over a division ring k and f be a multiplicative matrix function from m n k to a multiplicative abelian group with zero g 0 f ab f a f b a b m n k we call an additive transformation on m n k preserves a multiplicative matrix function f if f a f a a m n k in this paper we characterize all additive surjective transformations on m n k over any division ring k chk 2 that leave a non trivial multiplicative matrix function invariant applications to several related preservers are considered

a generalization of the craig sakamoto theorem to euclidean jordan algebras abstract letac and massam 10 extended the craig sakamoto theorem to euclidean jordan algebras in this paper we give another proof of this generalization by reformulating the result in terms of rank and determinant equalities and by proving the result in each of the simple euclidean jordan algebras

a novel constellation reshaping method for papr reduction of ofdm signals in this paper we propose a constellation reshaping method for peak to average power ratio papr reduction in orthogonal frequency division multiplexing ofdm systems the key idea of the proposed method is to generate a reshaped quadrature amplitude modulation r qam constellation in which the minimum distance of the r qam is the same as that of the traditional qam constellation when the novel r qam constellation is employed with the traditional partial transmit sequence pts and selected mapping slm methods in ofdm systems called as the r pts and r slm methods in this paper respectively the mean square errors between the constellations of the received data and each r qam rotated by different phase rotation factors are calculated at the receiver and the phase rotation factor with the minimum mean square error is selected to recover the original data without side information therefore the flexibility in phase rotation choice and the ability to avoid data rate loss make the r pts and r slm methods more suitable for good bit error rate performance in ofdm systems theoretical analysis and simulation results show that the r pts and r slm methods could provide the same papr reduction and bit error rate as those of the conventional pts and slm methods with perfect side information respectively

dynamic structure factor of vibrating fractals proteins as a case study we study the dynamic structure factor s k t of proteins at large wave numbers k k r g ensuremath gg 1 where r g is the gyration radius at this regime measurements are sensitive to internal dynamics and we focus on vibrational dynamics of folded proteins exploiting the analogy between proteins and fractals we perform a general analytic calculation of the displacement two point correlation functions ensuremath langle stackrel p vec u i t ensuremath stackrel p vec u j 0 2 ensuremath rangle we confront the derived expressions with numerical evaluations that are based on protein data bank pdb structures and the gaussian network model gnm for a few proteins and for the sierpinski gasket as a controlled check we use these calculations to evaluate s k t with arrested rotational and translational degrees of freedom and show that the decay of s k t is dominated by the spatially averaged mean square displacement of an amino acid the latter has been previously shown to evolve subdiffusively in time ensuremath langle stackrel p vec u i t ensuremath stackrel p vec u i 0 2 ensuremath rangle ensuremath sim t ensuremath nu where ensuremath nu is the anomalous diffusion exponent that depends on the spectral dimension d s and fractal dimension d f as a result for wave numbers obeying k 2 ensuremath langle stackrel p vec u 2 ensuremath rangle ensuremath gtrsim 1 s k t effectively decays as a stretched exponential s k t ensuremath simeq s k e ensuremath ensuremath gamma k t ensuremath beta with ensuremath beta ensuremath simeq ensuremath nu where the relaxation rate is ensuremath gamma k ensuremath sim k b t m ensuremath omega o 2 1 ensuremath beta k 2 ensuremath beta t is the temperature and m ensuremath omega o 2 the gnm effective spring constant describing the interaction between neighboring amino acids the static structure factor is dominated by the fractal character of the native fold s k ensuremath sim k ensuremath d f with negligible to marginal influence of vibrations the analytical expressions are first confronted with numerically based calculations on the sierpinski gasket and very good agreement is found between simulations and theory we then perform pdb gnm based numerical calculations for a few proteins and an effective stretched exponential decay of the dynamic structure factor is found albeit their relatively small size however when rotational and translational diffusion are added we find that their contribution is never negligible due to finite size effects while we can still attribute an effective stretching exponent ensuremath beta to the relaxation profile this exponent is significantly larger than the anomalous diffusion exponent ensuremath nu we compare our theory with recent neutron spin echo studies of myoglobin and hemoglobin and conclude that experiments in which the rotational and translational degrees of freedom are arrested e g by anchoring the proteins to a surface will improve the detection of internal vibrational dynamics

on the distribution of the peak to average power ratio in ofdm signals the distribution of the peak to average power ratio papr in strictly band limited orthogonal frequency division multiplexing ofdm signals is studied assuming that the base band ofdm signal is characterized as a band limited complex gaussian process we first attempt to derive the exact distribution of the papr in the band limited ofdm signals since this distribution cannot be expressed in a closed form we further develop a simple closed form approximation based on the level crossing rate analysis comparisons of the proposed distributions with those obtained by computer simulations show good agreement and convergence with an increase in the number of subcarriers

pairs of hermitian and skew hermitian quaternionic matrices canonical forms and their applications canonical forms are described for pairs of quaternionic matrices or equivalently matrix pencils where each one of the matrices is either hermitian or skew hermitian under strict equivalence and simultaneous congruence several applications are developed including structure of selfadjoint and skew adjoint matrices with respect to a regular sesquilinear quaternion valued form or with respect to a regular skew sesquilinear form another application gives canonical forms for quaternionic matrices under congruence

quaternion singular value decomposition based on bidiagonalization to a real or complex matrix using quaternion householder transformations we present a practical and efficient means to compute the singular value decomposition svd of a real or complex quaternion matrix a based on bidiagonalization of a to a real or complex bidiagonal matrix b using quaternionic householder transformations computation of the svd of b using an existing subroutine library such as lapack provides the singular values of a the singular vectors of a are obtained trivially from the product of the householder transformations and the real or complex singular vectors of b we show in the paper that left and right quaternionic householder transformations are different because of the non commutative multiplication of quaternions and we present formulae for computing the householder vector and matrix in each case

on exponential of split quaternionic matrices abstract the exponential of a matrix plays an important role in the theory of lie groups the main purpose of this paper is to examine matrix groups over the split quaternions and the exponential map from their lie algebras into the groups since the set of split quaternions is a noncommutative algebra the way of computing the exponential of a matrix over the split quaternions is more difficult than calculating the exponential of a matrix over the real or complex numbers therefore we give a method of finding exponential of a split quaternion matrix by its complex adjoint matrix

general mapping between random walks and thermal vibrations in elastic networks fractal networks as a case study we present an approach to mapping between random walks and vibrational dynamics on general networks random walk occupation probabilities first passage time distributions and passage probabilities between nodes are expressed in terms of thermal vibrational correlation functions recurrence is demonstrated equivalent to the landau peierls instability fractal networks are analyzed as a case study in particular we show that the spectral dimension governs whether or not the first passage time distribution is well represented by its mean we discuss relevance to universal features arising in protein vibrational dynamics

color sparse representations for image processing review models and prospects sparse representations have been extended to deal with color images composed of three channels a review of dictionary learning based sparse representations for color images is made here detailing the differences between the models and comparing their results on the real and simulated data these models are considered in a unifying framework that is based on the degrees of freedom of the linear filtering transformation of the color channels moreover this allows it to be shown that the scalar quaternionic linear model is equivalent to constrained matrix based color filtering which highlights the filtering implicitly applied through this model based on this reformulation the new color filtering model is introduced using unconstrained filters in this model spatial morphologies of color images are encoded by atoms and colors are encoded by color filters color variability is no longer captured in increasing the dictionary size but with color filters this gives an efficient color representation

commuting graphs of some subsets in simple rings abstract let d be a division ring with center f and n 1 a natural number for s m n d the commuting graph of s denoted by s is the graph with vertex set s z s such that distinct vertices a and b are adjacent if and only if ab ba in this paper we prove that if n 2 and a n i t are the sets of all non invertible nilpotent idempotent matrices and involutions respectively then for any division ring d a n i and t are connected graphs we show that if n 2 and u is the set of all upper triangular matrices then for every algebraic division ring d u is a connected graph also it is shown that if r is the set of all reducible matrices and m n d is algebraic over f then for n 2 r is a connected graph finally we prove that for n 2 m n h is a connected graph where h is the ring of real quaternions

on j conjugate product of quaternion polynomial matrices in this paper the concept of j conjugate product over quaternion polynomial matrices is proposed further some interesting properties of the proposed j conjugate products are derived by means of j conjugate product necessary and sufficient conditions for consimilarity over quaternion matrices are finally established

mass fractal dimension and the compactness of proteins vibrational dynamics and energy flow in a protein are related by alexander orbach theory to the protein s mass fractal dimension d and spectral dimension d proteins struct funct bioinf 55 529 2004 recently proposed a relation between d and protein size based on their computational analysis of a set of proteins ranging from about 100 to several thousand amino acids we report here values for d computed for 200 proteins from the protein data bank pdb ranging from about 100 to over 10 000 amino acids and examine variation of d with protein size the average d is found to be 2 5 significantly smaller than a completely compact three dimensional collapsed polymer indeed we find that on average a protein in its pdb configuration fills about three quarters of the volume within the protein surface protein mass is also found to scale with radius of gyration with an exponent of 2 5 for this set of proteins

consimilarity of quaternion matrices and complex matrices this paper discusses consimilarity of quaternion matrices obtains the jordan canonical form of a quaternion matrix under consimilarity and some necessary and sufficient conditions for consimilarity of quaternion matrices the consimilarity of quaternion matrices is a natural extension of the complex consimilarity of complex matrices has many good properties including some that are essentially different from those of complex consimilarity moreover using the consimilarity of quaternion matrices we obtain a new way to discuss complex consimilarity of complex matrices

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

a system of matrix equations with five variables in this paper we give some necessary and sufficient conditions for the consistence of the system of quaternion matrix equations a 1 x c 1 y b 1 d 1 a 2 w c 2 z b 2 d 2 a 3 v c 3 v b 3 c 4 a 4 v b 4 c 5 a 5 x y b 5 c 6 w z d 6 e 6 v f 6 g 6 and constitute an expression of the general solution to the system when it is solvable the outcomes of this paper encompass some recognized results in the collected works in addition we establish an algorithm and a numerical example to illustrate the theory constructed in the paper

quaternionic differential operators motivated by a quaternionic formulation of quantum mechanics we discuss quaternionic and complex linear differential equations we touch only a few aspects of the mathematical theory namely the resolution of the second order differential equations with constant coefficients we overcome the problems coming out from the loss of the fundamental theorem of the algebra for quaternions and propose a practical method to solve quaternionic and complex linear second order differential equations with constant coefficients the resolution of the complex linear schrodinger equation in the presence of quaternionic potentials represents an interesting application of the mathematical material discussed in this paper

optimization in quaternion dynamic systems gradient hessian and learning algorithms the optimization of real scalar functions of quaternion variables such as the mean square error or array output power underpins many practical applications solutions typically require the calculation of the gradient and hessian however real functions of quaternion variables are essentially nonanalytic which are prohibitive to the development of quaternion valued learning systems to address this issue we propose new definitions of quaternion gradient and hessian based on the novel generalized hamilton real ghr calculus thus making a possible efficient derivation of general optimization algorithms directly in the quaternion field rather than using the isomorphism with the real domain as is current practice in addition unlike the existing quaternion gradients the ghr calculus allows for the product and chain rule and for a one to one correspondence of the novel quaternion gradient and hessian with their real counterparts properties of the quaternion gradient and hessian relevant to numerical applications are also introduced opening a new avenue of research in quaternion optimization and greatly simplified the derivations of learning algorithms the proposed ghr calculus is shown to yield the same generic algorithm forms as the corresponding real and complex valued algorithms advantages of the proposed framework are illuminated over illustrative simulations in quaternion signal processing and neural networks

on the finiteness of the cone spectrum of certain linear transformations on euclidean jordan algebras abstract let l be a linear transformation on a finite dimensional real hilbert space h and k be a closed convex cone with dual k in h the cone spectrum of l relative to k is the set of all real for which the linear complementarity problem x k y l x x k and x y 0 admits a nonzero solution x in the setting of a euclidean jordan algebra h and the corresponding symmetric cone k we discuss the finiteness of the cone spectrum for z transformations and quadratic representations on h

vibrational shortcut to the mean first passage time problem what is the average time a random walker takes to get from a to b on a fractal structure and how does this mean time scale with the size of the system and the distance between source and target we take a nonprobabilistic approach toward this problem and show how the solution is readily obtained using an analysis of thermal vibrations on fractals invariance under scaling and continuity with respect to the spectral dimension are shown to be emergent properties of the solution obtained via vibrational analysis our result emphasizes the duality between diffusion and vibrations on fractal structures applications to biological systems are discussed

jacobi method for quaternion matrix singular value decomposition the study of quaternion matrices has gained interest in many areas in recent years and the problem of diagonalizing such matrices has also attracted attention in this article we present an algorithm for computing the svd of a matrix with quaternion coefficients directly in quaternion arithmetic using a generalization of classical complex jacobi methods the extension of the jacobi transformation to the quaternion case is introduced for the diagonalization of a hermitian quaternion valued matrix based on this an implicit jacobi algorithm is proposed for computing the svd of a quaternion matrix the performance of the proposed algorithm is presented and compared with an already known algorithm using a complex equivalent of the quaternion matrix and shown to be superior in execution time and accuracy

basic classes of matrices with respect to quaternionic indefinite inner product spaces basic classes of matrices or linear transformations in finite dimensional quaternionic vector spaces with a regular indefinite inner product are introduced and studied the classes include plus matrices selfadjoint skew adjoint and unitary matrices in particular results are proved concerning extensions of invariant semidefinite subspaces canonical form for unitary matrices is developed and subsequently applied to stability of periodic hamiltonian systems

diagonalization of bounded linear operators on separable quaternionic hilbert space by using the representation of its complex conjugate pairs we have investigated the diagonalization of a bounded linear operator on separable infinite dimensional right quaternionic hilbert space the sufficient condition for diagonalizability of quaternionic operators is derived the result is applied to anti hermitian operators which is essential for solving schrodinger equation in quaternionic quantum mechanics

on the solutions of tridiagonal linear systems tridiagonal linear systems are of special importance being appeared in space researches and also in several well known algorithms there are many methods for solving a tridiagonal linear system most of them are suffering from the accumulation of the round off errors in this paper a new method will be given to overcome the defects of the other methods also two direct methods for solving a system of linear equations having matrices of quaternion entries as coefficients and independent terms are proposed second kind chebyshev polynomials in a matrix argument are used as a tool to do certain calculations

analysis and design of ofdm oqam systems based on filterbank theory a discrete time analysis of the orthogonal frequency division multiplex offset qam ofdm oqam multicarrier modulation technique leading to a modulated transmultiplexer is presented the conditions of discrete orthogonality are established with respect to the polyphase components of the ofdm oqam prototype filter which is assumed to be symmetrical and with arbitrary length fast implementation schemes of the ofdm oqam modulator and demodulator are provided which are based on the inverse fast fourier transform non orthogonal prototypes create intersymbol and interchannel interferences isi and ici that in the case of a distortion free transmission are expressed by a closed form expression a large set of design examples is presented for ofdm oqam systems with the number of subcarriers going from four up to 2048 which also allows a comparison between different approaches to get well localized prototypes

strict double diagonal dominance in euclidean jordan algebras abstract in the first part of the paper we deal with euclidean jordan algebraic generalizations of some results of brualdi on inclusion regions for the eigenvalues of complex matrices using directed graphs as a consequence the theorems of brauer ostrowski and brauer on the location of eigenvalues are extended to the setting of euclidean jordan algebras in the second part motivated by the work of li and tsatsomeros on the class of doubly diagonally dominant matrices with complex entries and its subclasses we present some inter relations between the h property generalized strict diagonal dominance invertibility and strict double diagonal dominance in euclidean jordan algebras in addition we show that in a euclidean jordan algebra the schur complements of a strictly doubly diagonally dominant element inherit this property

a structure preserving jacobi algorithm for quaternion hermitian eigenvalue problems abstract a new real structure preserving jacobi algorithm is proposed for solving the eigenvalue problem of quaternion hermitian matrix by employing the generalized jrs symplectic jacobi rotations the new quaternion jacobi algorithm can preserve the symmetry and jrs symmetry of the real counterpart of quaternion hermitian matrix moreover the proposed algorithm only includes real operations without dimension expanding and is generally superior to the state of the art algorithm numerical experiments are reported to indicate its efficiency and accuracy

extremal ranks of a quaternion matrix expression subject to consistent systems of quaternion matrix equations with applications abstract assume that the linear quaternion matrix expression f x 1 x 2 a a 3 x 1 b 3 a 4 x 2 b 4 where x 1 x 2 are variant quaternion matrices in this paper we derive the maximal and minimal ranks of f x 1 x 2 subject to consistent systems of quaternion matrix equations a 1 x 1 c 1 x 1 b 1 c 2 and a 2 x 2 c 3 x 2 b 2 c 4 moreover corresponding results on some special cases are presented as applications we give necessary and sufficient conditions for the existence of solutions to some systems of quaternion matrix equations some previous known results can be regarded as the special cases of this paper

performance of wavelet packet division multiplexing in impulsive and gaussian noise wavelet packet division multiplexing wpdm is a high capacity flexible and robust multiple signal transmission technique in which the message signals are waveform coded onto wavelet packet basis functions for transmission we derive an expression for the probability of error for a wpdm scheme in the presence of both impulsive and gaussian noise sources and demonstrate that wpdm can provide greater immunity to impulsive noise than both a time division multiplexing scheme and an orthogonal frequency division multiplexing scheme

right eigenvalues for quaternionic matrices a topological approach abstract we apply the lefschetz fixed point theorem to show that every square matrix over the quaternions has right eigenvalues we classify them and discuss some of their properties such as an analogue of jordan canonical form and diagonalization of elements of the compact symplectic group sp n

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

residue network in protein native structure belongs to the universality class of a three dimensional critical percolation cluster single protein molecules are regarded as contact networks of amino acid residues relationships between the shortest path lengths and the numbers of residues within single molecules in the native structures are examined for various sized proteins a universal scaling among proteins is obtained which shows that the residue networks are fractal networks this universal fractal network is characterized with three kinds of dimensions the network topological dimension d c ensuremath approx 1 9 the fractal dimension d f ensuremath approx 2 5 and the spectral dimension d s ensuremath approx 1 3 these values are in surprisingly good coincidence with those of the three dimensional critical percolation cluster hence the residue contact networks in the protein native structures belong to the universality class of the three dimensional percolation cluster the criticality is relevant to the ambivalence in the protein native structures the coexistence of stability and instability both of which are necessary for protein functions

quaternionic eigenvalue problem we discuss the right eigenvalue equation for h c and r linear quaternionic operators the possibility to introduce an isomorphism between these operators and real complex matrices allows us to translate the quaternionic problem into an equivalent real or complex counterpart interesting applications are found in solving differential equations within quaternionic formulations of quantum mechanics

quaternions and matrices of quaternions abstract we give a brief survey on quaternions and matrices of quaternions present new proofs for certain known results and discuss the quaternionic analogues of complex matrices the methods of converting a quaternion matrix to a pair of complex matrices and homotopy theory are emphasized

a structure preserving algorithm for the quaternion cholesky decomposition in this paper we discuss the cholesky decomposition of the hermitian positive definite quaternion matrix for the first time the structure preserving gauss transformation is defined and then a novel structure preserving algorithm which is applied to its real representation matrix is proposed our algorithm needs only real number operations does not depend on the quaternion toolbox for matlab qtfm and has more portability although the flops of our algorithm are theoretically about the same as those based on quaternion arithmetic operations or qtfm numerical experiments show that our algorithm runs faster

conjugacy invariants of sl 2 h abstract in this paper we use the quaternionic formalism of mobius transformations on r 4 to derive conjugacy invariants on sl 2 h and hence on psl 2 h we then use these invariants to distinguish between the various conjugacy classes of psl 2 h

a comparison of dct based ofdm and dft based ofdm in frequency offset and fading channels a precise method for calculating the bit error probability bep of a discrete cosine transform dct based orthogonal frequency division multiplexing ofdm system on additive white gaussian noise awgn channels in the presence of frequency offset is derived these accurate results are used to examine and compare the bep performance of a dct ofdm system and the conventional discrete fourier transform dft based ofdm system in an awgn environment several signaling formats such as binary phase shift keying quaternary phase shift keying and 16 ary quadrature amplitude modulation are considered the performance of a dct ofdm with a zero padding guard interval scheme is then compared with a zero padded dft ofdm with the employment of minimum mean square error mmse detection and mmse decision feedback detection with ordering scheme over frequency selective fast rayleigh fading channels analysis and simulation results show that the dct ofdm system outperforms the dft ofdm system in the presence of frequency offset and in frequency selective fast fading environments

a necessary condition for waveforms with better papr than ofdm this paper establishes a necessary condition that must be satisfied by the modulation waveforms of any generalized waveforms for multicarrier gwmc system with better peak to average power ratio papr than conventional orthogonal frequency division multiplexing ofdm gwmc systems include in particular all classical multicarrier modulation systems as a consequence we show that ofdm has the best papr performance over all gwmc systems that do not satisfy this necessary condition we also identify an infinite family of gwmc systems with the same papr performance as ofdm to illustrate our results we present the simulations of the papr behavior for different gwmc systems including some with better papr performance than ofdm

analysis of heat kernel highlights the strongly modular and heat preserving structure of proteins abstract in this paper we study the structure and dynamical properties of protein contact networks with respect to other biological networks together with simulated archetypal models acting as probes we consider both classical topological descriptors such as modularity and statistics of the shortest paths and different interpretations in terms of diffusion provided by the discrete heat kernel which is elaborated from the normalized graph laplacians a principal component analysis shows high discrimination among the network types by considering both the topological and heat kernel based vector characterizations furthermore a canonical correlation analysis demonstrates the strong agreement among those two characterizations providing thus an important justification in terms of interpretability for the heat kernel finally and most importantly the focused analysis of the heat kernel provides a way to yield insights on the fact that proteins have to satisfy specific structural design constraints that the other considered networks do not need to obey notably the heat trace decay of an ensemble of varying size proteins denotes subdiffusion a peculiar property of proteins

optimal stochastic restart renders fluctuations in first passage times universal stochastic restart may drastically reduce the expected run time of a computer algorithm expedite the completion of a complex search process or increase the turnover rate of an enzymatic reaction these diverse first passage time fpt processes seem to have very little in common but it is actually quite the other way around here we show that the relative standard deviation associated with the fpt of an optimally restarted process i e one that is restarted at a constant nonzero rate which brings the mean fpt to a minimum is always unity we interpret further generalize and discuss this finding and the implications arising from it

music algorithm for vector sensors array using biquaternions in this paper we use a biquaternion formalism to model vector sensor signals carrying polarization information this allows a concise and elegant way of handling signals with eight dimensional 8 d vector valued samples using this model we derive a biquaternionic version of the well known array processing music algorithm and we show its superiority to classically used long vector approach new results on biquaternion valued matrix spectral analysis are presented of particular interest for the biquaternion music bq music algorithm is the decomposition of the spectral matrix of the data into orthogonal subspaces we propose an effective algorithm to compute such an orthogonal decomposition of the observation space via the eigenvalue decomposition evd of a hermitian biquaternionic matrix by means of a newly defined quantity the quaternion adjoint matrix the bq music estimator is derived and simulation results illustrate its performances compared with two other approaches in polarized antenna processing lv music and psa music the proposed algorithm is shown to be superior in several aspects to the existing approaches compared with lv music the bq music algorithm is more robust to modelization errors and coherent noise while it can detect less sources in comparaison with psa music our approach exhibits more accurate estimation of direction of arrival doa for a small number of sources while keeping the polarization information accessible

on solutions of the quaternion matrix equation ax b and their applications in color image restoration by using the complex representation of quaternion matrices and the moore penrose generalized inverse we derive the expressions of the least squares solution with the least norm the least squares pure imaginary solution with the least norm and the least squares real solution with the least norm for the quaternion matrix equation ax b respectively finally we discuss their applications in color image restoration

gleason s problem and schur multipliers in the multivariable quaternionic setting abstract we define and study the counterparts of gleason s problem of the drury arveson s space and of schur multipliers when the unit ball of c n is replaced by the unit ball of h n where h denotes the algebra of real quaternions schur multipliers are characterized in terms of coisometric operator matrices in quaternionic spaces finally we define the counterpart of blaschke factors in this setting and study a one point interpolation problem

quaternionic matrices unitary similarity simultaneous triangularization and some trace identities abstract we construct six unitary trace invariants for 2 2 quaternionic matrices which separate the unitary similarity classes of such matrices and show that this set is minimal we have discovered a curious trace identity for two unit speed one parameter subgroups of sp 1 a modification gives an infinite family of trace identities for quaternions as well as for 2 2 complex matrices we were not able to locate these identities in the literature we prove two quaternionic versions of a well known characterization of triangularizable subalgebras of matrix algebras over an algebraically closed field finally we consider the problem of describing the semi algebraic set of pairs x y of quaternionic n n matrices which are simultaneously triangularizable even the case n 2 which we analyze in more detail remains unsolved

proteins coexistence of stability and flexibility we introduce an equation for protein native topology based on recent analysis of data from the protein data bank and on a generalization of the landau peierls instability criterion for fractals the equation relates the protein fractal dimension df the spectral dimension ds and the number of amino acids n deviations from the equation may render a protein unfolded the fractal nature of proteins is shown to bridge their seemingly conflicting properties of stability and flexibility over 500 proteins have been analyzed df ds and n and found to obey this equation of state

a 3d ginibre point field we introduce a family of three dimensional random point fields using the concept of the quaternion determinant the kernel of each field is an n dimensional orthogonal projection on a linear space of quaternionic polynomials we find explicit formulas for the basis of the orthogonal quaternion polynomials and for the kernel of the projection for number of particles n rightarrow infty we calculate the scaling limits of the point field in the bulk and at the center of coordinates we compare our construction with the previously introduced fermi sphere point field process

a quaternion qr algorithm this paper extends the francis qr algorithm to quaternion and antiquaternion matrices it calculates a quaternion version of the schur decomposition using quaternion unitary similarity transformations following a finite step reduction to a hessenberg like condensed form a sequence of implicit qr steps reduces the matrix to triangular form eigenvalues may be read off the diagonal eigenvectors may be obtained from simple back substitutions for serial computation the algorithm uses only half the work and storage of the unstructured francis qr iteration by preserving quaternion structure the algorithm calculates the eigenvalues of a nearby quaternion matrix despite rounding errors

quaternion var modelling and estimation quaternion vector autoregression var modeling is a natural extension of real and complex var we demonstrate how a quaternion var can be treated as a special case of structured real var we show that generalized least squares and under gaussianity maximum likelihood estimation of the parameters reduces to simple least squares estimation if the innovations are quaternion proper

quasistationary quaternionic hamiltonians and complex stochastic maps we show that the complex projections of time dependent quasianti hermitian quaternionic hamiltonian dynamics are complex stochastic dynamics in the space of complex quasi hermitian density matrices if and only if a quasistationarity condition is fulfilled i e if and only if is an hermitian positive time independent complex operator an example is also discussed

michaelis menten reaction scheme as a unified approach towards the optimal restart problem we study the effect of restart and retry on the mean completion time of a generic process the need to do so arises in various branches of the sciences and we show that it can naturally be addressed by taking advantage of the classical reaction scheme of michaelis and menten stopping a process in its midst only to start it all over again may prolong leave unchanged or even shorten the time taken for its completion here we are interested in the optimal restart problem i e in finding a restart rate which brings the mean completion time of a process to a minimum we derive the governing equation for this problem and show that it is exactly solvable in cases of particular interest we then continue to discover regimes at which solutions to the problem take on universal details independent forms which further give rise to optimal scaling laws the formalism we develop and the results obtained can be utilized when optimizing stochastic search processes and randomized computer algorithms an immediate connection with kinetic proofreading is also noted and discussed

quaternions further contributions to a matrix oriented approach abstract by representing quaternions as four dimensional vectors and the multiplication of quaternions as matrix by vector product we employ a matrix oriented approach to the topic we investigate properties of the fundamental real matrix associated with a quaternion and examine a frequently considered quaternion equation from which the n th power of a quaternion can be determined

burnside s theorem for matrix rings over division rings abstract a version of burnside s theorem states that if f is an arbitrary field and a m n f is an irreducible or equivalently transitive subalgebra containing a rank one matrix then a m n f the present paper shows that if f is replaced by a division ring d then every transitive left subalgebra of mn d containing a rank one matrix is equal to mn d here by a left algebra we mean a ring which is also a left d module counterexamples are given in case a is irreducible but not transitive moreover it is shown that irreducible left algebras of quaternionic matrices contain rank one idempotents and their structures are classified