the quaternionic evolution operator in this chapter we study the theory of semigroups and groups of quaternionic linear operators the development of which started in 79 we show that the theory of hille phillips and yosida can be restored in the quaternionic setting

the cauchy formula with s monogenic kernel and a functional calculus for noncommuting operators abstract the new notion of slice monogenic functions introduced in the paper f colombo i sabadini d c struppa slice monogenic functions israel j math 171 2009 385 403 led us to define a new functional calculus for an n tuple of not necessarily commuting operators see f colombo i sabadini d c struppa a new functional calculus for noncommuting operators j funct anal 254 2008 2255 2274 in this paper we prove a cauchy formula with slice monogenic kernel for the slice monogenic functions this new cauchy formula is the fundamental tool to prove that our functional calculus apply to a more general setting moreover we deduce some fundamental properties of the functional calculus for example some algebraic properties the spectral mapping theorem and the spectral radius theorem

h measures a new approach for studying homogenisation oscillations and concentration effects in partial differential equations new mathematical objects called h measures are introduced for studying oscillations and concentration effects in partial differential equations applications to transport properties and to homogenisation are given as an example of the new results which can be obtained by this approach

a new method for roots of monic quaternionic quadratic polynomial the purpose of this paper is to show how the problem of finding roots or zeros of the monic quaternionic quadratic polynomials mqqp can be solved by its equivalent real quadratic form the real quadratic form matrices firstly defined in this paper are used to form a simple equivalent real quadratic form of mqqp some necessary and sufficient conditions for the existence of roots of mqqp are also presented the main idea of the practical method proposed in this work can be summarized in two steps translating mqqp into its equivalent real quadratic form and giving directly the quaternionic roots of mqqp by solving its equivalent real quadratic form

regular subspaces of a quaternionic hilbert space from quaternionic hermite polynomials and associated coherent states we define quaternionic hermite polynomials by analogy with two families of complex hermite polynomials as in the complex case these polynomials constitute orthogonal families of vectors in ambient quaternionic l2 spaces using these polynomials we then define regular and anti regular subspaces of these l2 spaces the associated reproducing kernels and the ensuing quaternionic coherent states

factoring skew polynomials over hamilton s quaternion algebra and the complex numbers abstract we show that all non constant polynomials in a skew polynomial ring h t over hamilton s quaternions decompose into a product of linear factors and that all non constant polynomials in the skew polynomial ring c t decompose into a product of linear and quadratic irreducible factors our proofs use nonassociative algebras constructed out of skew polynomial rings as introduced by petit applying results by petit we also characterize the real division algebras which are two dimensional vector spaces over c with c in their left and middle nuclei

slice segal bargmann transform the segal bargmann transform is a unitary map between the schrodinger and fock space which is used for example to show the integrability of quantum rabi models slice monogenic functions provide the framework in which functional calculus for quaternionic quantum mechanics can be developed in this paper a generalisation of the segal bargmann transform to the context of slice monogenic functions is constructed and studied in detail it is shown to interact appropriately with the recently constructed slice fourier transform this leads furthermore to a construction of a slice fock space which is shown to be a reproducing kernel space

finite bandwidth kramers kronig relations for acoustic group velocity and attenuation derivative applied to encapsulated microbubble suspensions kramers kronig kk analyses of experimental data are complicated by the conflict between the inherently bandlimited data and the requirement of kk integrals for a complete infinite spectrum of input information for data exhibiting localized extrema kk relations can provide accurate transforms over finite bandwidths due to the local weighting properties of the kk kernel recently acoustic kk relations have been derived for the determination of the group velocity cg and the derivative of the attenuation coefficient components of the derivative of the acoustic complex wave number these relations are applicable to bandlimited data exhibiting resonant features without extrapolation or unmeasured parameters in contrast to twice subtracted finite bandwidth kk predictions for phase velocity and attenuation coefficient components of the undifferentiated wave number these more recently derived relations for cg and provide stricter tests of causal consistency because the resulting shapes are inv

the radon transform between monogenic and generalized slice monogenic functions in bures et al elements of quaternionic analysis and radon transform 2009 the authors describe a link between holomorphic functions depending on a parameter and monogenic functions defined on mathbb r n 1 using the radon and dual radon transforms the main aim of this paper is to further develop this approach in fact the radon transform for functions with values in the clifford algebra mathbb r n is mapping solutions of the generalized cauchy riemann equation i e monogenic functions to a parametric family of holomorphic functions with values in mathbb r n and analogously the dual radon transform is mapping parametric families of holomorphic functions as above to monogenic functions the parametric families of holomorphic functions considered in the paper can be viewed as a generalization of the so called slice monogenic functions an important part of the problem solved in the paper is to find a suitable definition of the function spaces serving as the domain and the target of both integral transforms

scattering theory in the energy space for a class of non linear wave equations we study the asymptotic behaviour in time of the solutions and the theory of scattering in the energy space for the non linear wave equation square varphi f varphi 0 in n n 3 we prove the existence of the wave operators asymptotic completeness for small initial data and forn 4 asymptotic completeness for arbitrarily large data the assumptions onf cover the case wheref behaves slightly better than a single powerp 1 4 n 2 both near zero and at infinity see 1 5 1 6 and 1 8

a note on the computation of all zeros of simple quaternionic polynomials polynomials with quaternionic coefficients located on only one side of the powers we call them simple polynomials may have two different types of zeros isolated and spherical zeros we will give a new characterization of the types of the zeros and based on this characterization we will present an algorithm for producing all zeros including their types without using an iteration process which requires convergence the main tool is the representation of the powers of a quaternion as a real linear combination of the quaternion and the number one as introduced by pogorui and shapiro complex var and elliptic funct 49 2004 pp 379 389 and the use of a real companion polynomial which already was introduced for the first time by niven amer math monthly 48 1941 pp 654 661 there are several examples

semigroups over real alternative algebras generation theorems and spherical sectorial operators the aim of this paper is twofold on one hand generalizing some recent results obtained in the quaternionic setting but using simpler techniques we prove the generation theorems for semigroups in banach spaces whose set of scalars belongs to the class of real alternative algebras which includes besides real and complex numbers quaternions octonions and clifford algebras on the other hand in this new general framework we introduce the notion of spherical sectorial operator and we prove that a spherical sectorial operator generates a semigroup that can be represented by a cauchy integral formula it follows that such a semigroup is analytic in time

generalized strichartz inequalities for the wave equation strichartz inequalities for the wave equation u f are estimates of the solution u of the cauchy problem for that equation in the form of space time integral norms in terms of similar norms of the inhomogeneity f and of suitable norms of the initial data

finite bandwidth effects on the causal prediction of ultrasonic attenuation of the power law form kramers kronig k k relations exist as a consequence of causality placing nonlocal constraints on the relationship between dispersion and absorption the finite bandwidth method of applying these relations is examined where the k k integrals are restricted to the spectrum of the experimental data these finite bandwidth k k relations are known to work with resonant type data and here are applied to dispersion data consistent with a power law attenuation coefficient exponent from 1 to 2 bandwidth restricted forms of the zero and once subtracted k k relations are used to determine the attenuation coefficient from phase velocity analytically it is shown that these transforms produce the proper power law form of the attenuation coefficient as a stand alone term summed with artifacts that are dependent on the integration limits calculations are performed to demonstrate how these finite bandwidth artifacts affect the k k predictions under a variety of conditions the predictions are studied in a local context as a function of subtraction frequency bandwidth and power law exponent the k k predictions of the power law exponent within various decades of the spectrum are also examined in general the agreement between finite bandwidth k k predictions and exact values grows as the power law exponent approaches 1 and with increasing bandwidth

on existence and scattering with minimal regularity for semilinear wave equations abstract we prove existence and scattering results for semilinear wave equations with low regularity data we also determine the minimal regularity that is needed to ensure local existence and well posedness and we give counterexamples to well posedness more specifically we show that equations of the type u u p with initial data u u t in r n 1 r n have a local solution if p n and we construct counterexamples if p n the existence results rely on mixed norm space time estimates of strichartz type

causal theories and data for acoustic attenuation obeying a frequency power law this study compares causal theories based on kramers kronig relations fractional calculus and on those derived from new time domain causal relationships to diverse data all these theories are based on the assumptions that the functional form of the attenuation persists beyond the measurement range and that attenuation is much smaller than the wave number the data for lossy media with attenuation having a power law frequency dependence with an exponent y include cases for both liquids and solids ranging from acoustic to ultrasound frequencies data are in closer correspondence with the new theory which predicts decreasing dispersion as the power exponent y approaches zero or an even integer experimental results and supporting evidence show that the classical case of frequency squared attenuation is dispersionless an approximate nearly local kramers kronig theory is in agreement with the time causal theory when the exponent is close to one but deviates for other values the comprehensive time causal theory is shown to be equivalent to two other theories derived from exact kramers kronig relations and from fractional calculus and it covers the y odd integer cases which are missing or incomplete in these approaches attenuation dispersion relations are presented in two forms one for a general frequency range and another for a finite range it is demonstrated that complete velocity dispersion within a signal bandwidth can be predicted from knowledge of the attenuation data and velocity at a single frequency including the velocity at either zero frequency y 1 and or at a high frequency limit 0 y 1 in addition attenuation data can be predicted from velocity data

a geometric method for detecting chaotic dynamics a new method of detection of chaos in dynamical systems generated by time periodic nonautonomous differential equations is presented it is based on the existence of some sets called periodic isolating segments in the extended phase space satisfying some topological conditions by chaos we mean the existence of a compact invariant set such that the poincare map is semiconjugated to the shift on two symbols and the counterimage by the semiconjugacy of any periodic point in the shift contains a periodic point of the poincare map as an application we prove that the planar equationz 1 ei t z 2 zgenerates chaotic dynamics provided 0 1 288

julia theory for slice regular functions the theory of slice regular functions is nowadays widely studied and has found elegant applications to functional calculus for quaternionic linear operators and schur analysis however much less is known about their boundary behaviors in this paper we initiate the study of julia theory for slice regular functions more precisely we establish the quaternionic versions of the julia lemma the julia caratheodory theorem the boundary schwarz lemma the hopf lemma and the burns krantz rigidity theorem for slice regular self mappings of the open unit ball b h and of the right half space h we provide an explicit example to show that in quaternionic hopf lemma the slice derivative of a slice regular function f at the boundary fixed point may not be real in contrast to the complex version our result implies that the commonly believed version of the hopf lemma turns out to be totally wrong this new quaternionic version of the hopf lemma also improves osserman estimate even in the complex setting and is essentially the first significant theorem belonging to the theory of quaternions itself other than the theory of complex analysis since the result involves the lie brackets which reflect the non commutative feature of quaternions

global existence for maxwell bloch systems abstract maxwell bloch equations describe the propagation of an electromagnetic wave through a quantum medium for any number of quantum levels in space dimension 3 we show the global existence of weak l 2 solutions to the initial value problem in the case of smoother electromagnetic fields with curl in l 2 the solution is unique for smooth data h s s 2 the solutions remain smooth for all times

slice polynomial functions and twistor geometry of ruled surfaces in mathbb cp 3 in the present paper we introduce the class of slice polynomial functions slice regular functions defined over the quaternions outside the real axis whose restriction to any complex half plane is a polynomial these functions naturally emerge in the twistor interpretation of slice regularity introduced in gentili et al j eur math soc 16 11 2323 2353 2014 and developed in altavilla j geom phys 123 184 208 2018 to any slice polynomial function p we associate its companion p vee and its extension to the real axis p mathbb r that are quaternionic functions naturally related to p then using the theory of twistor spaces we are able to show that for any quaternion q the cardinality of simultaneous pre images of q via p p vee and p mathbb r is generically constant giving a notion of degree with the brand new tool of slice polynomial functions we compute the twistor discriminant locus of a cubic scroll mathcal c in mathbb cp 3 and we conclude by giving some qualitative results on the complex structures induced by mathcal c via the twistor projection

equivalence of slice semi regular functions via sylvester operators abstract the aim of this paper is to study some features of slice semi regular functions sem on a circular domain contained in the skew symmetric algebra of quaternions h via the analysis of a family of linear operators built from left and right multiplication on sem this class of operators includes the family of sylvester type operators s f g our goal is achieved by a strategy based on a matrix interpretation of these operators as we show that sem can be seen as a 4 dimensional vector space on the field sem r we then study the rank of s f g and describe its kernel and image when it is not invertible finding meaningful differences in the cases when the rank is either 2 or 3 by using these results we are able to characterize when the functions f and g are either equivalent under conjugation or intertwined by means of a zero divisor thus proving a number of statements on the behaviour of slice semi regular functions in this way information about the operator obtained by linear algebra techniques give as a significant application the solution of a problem in an area of function theory which had a remarkable development in the last decade see 16 we also provide a complete classification of idempotents and zero divisors on product domains of h

an extension problem for the cr fractional laplacian we show that the conformally invariant fractional powers of the sub laplacian on the heisenberg group are given in terms of the scattering operator for an extension problem to the siegel upper halfspace remarkably this extension problem is different from the one studied among others by caffarelli and silvestre we also prove an energy identity that yields a sharp trace sobolev embedding

extension results for slice regular functions of a quaternionic variable in this paper we prove a new representation formula for slice regular functions which shows that the value of a slice regular function f at a point q x yi can be recovered by the values of f at the points q yj and q yk for any choice of imaginary units i j k this result allows us to extend the known properties of slice regular functions defined on balls centered on the real axis to a much larger class of domains called axially symmetric domains we show in particular that axially symmetric domains play for slice regular functions the role played by domains of holomorphy for holomorphic functions

verification of kramers kronig relationship in porous materials having a rigid frame abstract the propagation of acoustic waves in porous materials having a rigid frame is well described by several models a doubt about the causality of these models has been raised recently in the literature a verification of the causality of these models is studied in this paper using the kramers kronig dispersion relations adapted to the frequency power law dependence of the attenuation it is shown that these models are causal in the high and low frequency range a time domain wave equation and time causal theory have been treated

addendum to factoring skew polynomials over hamilton s quaternion algebra and the complex numbers j algebra 427 2015 20 29 abstract let d be the quaternion division algebra over a real closed field f then every non constant polynomial in a skew polynomial ring d t decomposes into a product of linear factors and thus has a zero in d this improves 8 theorem 2

counterexamples to local existence for semi linear wave equations in this paper we study how much regularity of initial data is needed to ensure existence of a local solution to a semi linear wave equation we give counterexamples to local existence for the typical model equations the counterexamples we construct are sharp i e one does have a local solution if the data has slightly more regularity

generalized 2d laguerre polynomials and their quaternionic extensions a class of orthogonal polynomials in two quaternionic variables is introduced this class serves as an analogous to the classical zernike polynomials z m n z z arxiv 1502 07256 2014 a number of interesting properties such as the orthogonality condition recurrence relations raising and lowering operators are discussed in detail particularly the ladder operators realized as differential operators in terms of the so called cullen derivatives for these quaternionic polynomials are also studied some physically interesting summation and integral formulas are also proved and their physical relevance briefly discussed

computing the zeros of quaternion polynomials abstract a method is developed to compute the zeros of a quaternion polynomial with all terms of the form q k x k this method is based essentially in niven s algorithm 1 which consists of dividing the polynomial by a characteristic polynomial associated to a zero the information about the trace and the norm of the zero is obtained by an original idea which requires the companion matrix associated to the polynomial the companion matrix is represented by a matrix with complex entries three numerical examples using mathematica 2 2 version are given

one dimensional quaternion homogeneous polynomial differential equations in this paper we study the dynamics of real four dimensional polynomial differential equations of the forms q aqkq m or q qkaqm where q a are quaternions q is the conjugate of q and k and m are non negative integers in many parts of the work the quaternion structure is used along the proofs

regular articlegeneralized strichartz inequalities for the wave equation we make a synthetic exposition of the generalized strichartz inequalities for the wave equation obtained in 6 together with the limiting cases recently obtained in 13 with as simple proofs as possible the proofs combine stationary phase estimates dyadic decompositions the hardy littlewood inequality or the holder inequality in time and abstract duality and interpolation arguments

on the applicability of kramers kronig relations for ultrasonic attenuation obeying a frequency power law in the recent literature concern has been raised regarding the validity of kramers kronig relations for media with ultrasonic attenuation obeying a frequency power law it is demonstrated however that the kramers kronig dispersion relations for application to these types of media are available the developed dispersion relations are compared with measurements on several liquids and agreement is found to better than 1 m s over the experimentally available bandwidth a discussion regarding the validity of these dispersion relations in particular how the dispersion relations relate to the so called paley wiener conditions forms the conclusion

quaternionic valued ordinary differential equations ii coinciding sectors abstract we give some sufficient conditions for the existence of at least one periodic solution of the quaternionic polynomial equations in some cases we are able to prove uniqueness of periodic solutions inside some sectors of h and detect solutions heteroclinic to the periodic ones

on periodic solutions of planar polynomial differential equations with periodic coefficients abstract we consider the planar equation z ak l t zk z l where ak l is a t periodic complex valued continuous function equal to 0 for almost all k l n we present sufficient conditions imposed on ak l which guarantee the existence of its t periodic solutions and in the case a0 0 0 the conditions for the existence of nonzero ones we use a method which computes the fixed point index of the poincare andronov operator in isolated sets of fixed points generated by so called periodic blocks the method is based on the lefschetz fixed point theorem and the topological principle of wazewski

simplified expressions of the subtracted kramers kronig relations using the expanded forms applied to ultrasonic power law systems the kramers kronig kk relations are a large class of integral transformations that exploit the broad principle of simple causality in order to link the physical properties of matter and materials in applications to the complex valued wavenumber for acoustic propagation the method of subtractions is used to form convergent integral relations between the phase velocity and the attenuation coefficient when the method of subtractions is applied in the usual manner the integrands in the relations become unnecessarily complicated in this work an expanded form of the subtracted relations is presented which is essentially a truncated taylor series expansion of the hilbert transforms the implementation of the relations only requires the explicit evaluation of two simply expressed integrals involving the hilbert transform kernel these two integrals determine the values of the other terms in the subtracted relations demonstrating the computational efficiency of the technique the method is illustrated analytically through its application to power law attenuation coefficients and its associated dispersion which are observed in a wide variety of materials this approach explicitly shows the central role of the hilbert transform kernel in the kk relations which can become obscured in other formulations

the global cauchy problem for the critical non linear wave equation abstract we study the global cauchy problem for the non linear wave equation p 1 0 for the critical value p n 2 n 2 in space dimension n 3 we identify a weak space time integrability property stip of the solutions and prove that it is sufficient to ensure the uniqueness of weak solutions the global existence of finite energy solutions with the naturally associated stip and the global existence of regular solutions with some n dependent restrictions on the regularity for spherically symmetric solutions we prove that the previous crucial stip follows from the morawetz inequality actually in a much stronger form than necessary thereby proving that all the previous results hold in the spherically symmetric case

smoothing properties and retarded estimates for some dispersive evolution equations smoothing properties in the form of space time integrability properties play an important role in the study of dispersive evolution equations a number of them follow from a combination of general arguments and specific estimates we present a general formulation which makes the separation between the two types of ingredients as clear as possible and we illustrate it with the examples of the schrodinger equation of the wave equation and of a class of 1 1 dimensional equations related to the benjamin ono equation of special interest for the cauchy problem are retarded estimates expressed in terms of those properties we derive a number of such estimates associated with the last example and we mention briefly an application of those estimates to the cauchy problem for the generalized benjamin ono equation

slice regular functions on real alternative algebras in this paper we develop a theory of slice regular functions on a real alternative algebra a our approach is based on a well known fueter s construction two recent function theories can be included in our general theory the one of slice regular functions of a quaternionic or octonionic variable and the theory of slice monogenic functions of a clifford variable our approach permits to extend the range of these function theories and to obtain new results in particular we get a strong form of the fundamental theorem of algebra for an ample class of polynomials with coefficients in a and we prove a cauchy integral formula for slice functions of class c 1

a new theory of regular functions of a quaternionic variable abstract in this paper we develop the fundamental elements and results of a new theory of regular functions of one quaternionic variable the theory we describe follows a classical idea of cullen but we use a more geometric formulation to show that it is possible to build a rather complete theory our theory allows us to extend some important results for polynomials in the quaternionic variable to the case of power series

time domain wave equations for lossy media obeying a frequency power law for attenuation described by a slowly varying power law function of frequency 0 y classical lossy time domain wave equations exist only for the restricted cases where y 0 or y 2 for the frequently occurring practical situation in which attenuation is much smaller than the wave number a lossy dispersion characteristic is derived that has the desired attenuation general power law dependence in order to obtain the corresponding time domain lossy wave equation time domain loss operators similar in function to existing derivative operators are developed through the use of generalized functions three forms of lossy wave equations are found depending on whether y is an even or odd integer or a noninteger a time domain expression of causality analogous in function to the kramers kronig relations in the frequency domain is used to derive the causal wave equations final causal versions of the time domain wave equations are obtained even for the cases where y 1 which according to the paley wiener th

power series and analyticity over the quaternions we study power series and analyticity in the quaternionic setting we first consider a function f defined as the sum of a power series sum nolimits n in mathbb n q n a n in its domain of convergence which is a ball b 0 r centered at 0 at each p in b 0 r f admits expansions in terms of appropriately defined power series centered at p namely sum nolimits n in mathbb n q p n b n the expansion holds in a ball p r p defined with respect to a non euclidean distance we thus say that f is analytic in b 0 r furthermore we remark that p r p is not always a euclidean neighborhood of p when it is we say that f is strongly analytic at p it turns out that f is strongly analytic in a neighborhood of b 0 r cap mathbb r that can be strictly contained in b 0 r we then relate these notions of analyticity to the class of quaternionic functions introduced in gentili and struppa adv math 216 1 279 301 2007 and recently extended in colombo et al adv math 222 5 1793 1808 2009 under the name of slice regular functions indeed analyticity proves equivalent to slice regularity in the same way as complex analyticity is equivalent to holomorphy hence the theory of slice regular quaternionic functions which is quickly developing reveals a new feature that reminds the nice properties of holomorphic complex functions

kramers kronig relations applied to finite bandwidth data from suspensions of encapsulated microbubbles in this work the kramers kronig k k relations are applied to experimental data of resonant nature by limiting the interval of integration to the measurement spectrum the data are from suspensions of encapsulated microbubbles albunex and have the characteristics of an ultrasonic notch filter the goal is to test the consistency of this dispersion and attenuation data with the kramers kronig relations in a strict manner without any parameters from outside the experimental bandwidth entering in to the calculations in the course of reaching the goal the artifacts associated with the truncation of the integrals are identified and it is shown how their impacts on the results can be minimized the problem is first approached analytically by performing the kramers kronig calculations over a restricted spectral band on a specific hilbert transform pair lorentzian curves the resulting closed form solutions illustrate the type of artifacts that can occur due to truncation and also show that accurate res

the theory of quaternion matrix derivatives a systematic framework for the calculation of the derivatives of quaternion matrix functions with respect to quaternion matrix variables is introduced the proposed approach is equipped with the matrix product and chain rules and applies to both analytic and nonanalytic functions of quaternion variables this rectifies a mathematical shortcut in the existing methods which incorrectly use the traditional product rule we also show that within the proposed framework the derivatives of quaternion matrix functions can be calculated directly without using quaternion differentials or resorting to the isomorphism with real vectors illustrative examples show how the proposed quaternion matrix derivatives can be used as an important tool for solving optimization problems in signal processing applications

the fundamental theorem of algebra for hamilton and cayley numbers in this paper we prove the fundamental theorem of algebra for polynomials with coefficients in the skew field of hamilton numbers quaternions and in the division algebra of cayley numbers octonions the proof inspired by recent definitions and results on regular functions of a quaternionic and of a octonionic variable follows the guidelines of the classical topological argument due to gauss

handbook of differential equations chapter headings definitions and concepts transformations exact analytical methods exact methods for odes exact methods for pdes approximate analytical methods numerical methods concepts numerical methods for odes numerical methods for pdes list of tables list of programs list of figures subject index

borel functional calculus for quaternionic normal operators in this article we give an approach to borel functional calculus for quaternionic normal operators which are not necessarily bounded first we establish the definition of functional calculus for a subclass of quaternion valued borel functions and then we extend the same to the class of quaternion valued borel functions as well as l functions we also prove spectral mapping theorem as a consequence

planar nonautonomous polynomial equations the riccati equation abstract we give a few sufficient conditions for the existence of two periodic solutions of the riccati ordinary differential equation in the plane we give also examples of the equation without periodic solutions

the classification and the computation of the zeros of quaternionic two sided polynomials already for a long time it is known that quaternionic polynomials whose coefficients are located only at one side of the powers may have two classes of zeros isolated zeros and spherical zeros only recently a classification of the two types of zeros and a means to compute all zeros of such polynomials have been developed in this investigation we consider quaternionic polynomials whose coefficients are located at both sides of the powers and we show that there are three more classes of zeros defined by the rank of a certain real 4 4 matrix this information can be used to find all zeros in the same class if only one zero in that class is known the essential tool is the description of the polynomial p by a matrix equation p z a z z b z where a z is a real 4 4 matrix determined by the coefficients of the given polynomial p and p z b are real column vectors with four rows this representation allows also to include two sided polynomials which contain several terms of the same degree we applied newton s method to p z 0 this method turned out to be a very effective tool in finding the zeros this method allowed also to prove that the essential number of zeros of a quaternionic two sided polynomial p of degree n is in general not bounded by n we conjecture that the bound is 2n there are various examples

singularities of slice regular functions over real alternative algebras abstract the main goal of this work is classifying the singularities of slice regular functions over a real alternative algebra a this function theory has been introduced in 2011 as a higher dimensional generalization of the classical theory of holomorphic complex functions of the theory of slice regular quaternionic functions launched by gentili and struppa in 2006 and of the theory of slice monogenic functions constructed by colombo sabadini and struppa since 2009 along with this generalization step the larger class of slice functions over a has been defined we introduce here a new type of series expansion near each singularity of a slice regular function this instrument which is new even in the quaternionic case leads to a complete classification of singularities this classification also relies on some recent developments of the theory concerning the algebraic structure and the zero sets of slice functions peculiar phenomena arise which were not present in the complex or quaternionic case and they are studied by means of new results on the topology of the zero sets of slice functions the analogs of meromorphic functions called slice semiregular functions are introduced and studied

roots of quaternion polynomials theory and computation abstract a quaternion polynomial f t in the single variable t is one whose coefficients are in the skew field h of quaternions in this manuscript an elementary proof is given of the fact that such an f has a root in h moreover an algorithm is proposed for finding all roots of f t along with their multiplicities the algorithm is based on computing the real part of first and then using the multiplication rule in h the imaginary part of is computed via a linear quaternion equation several numerical examples are also presented to illustrate the performance of the method

quaternionic valued ordinary differential equations the riccati equation abstract we give some sufficient conditions for the existence of at least two periodic solutions of the quaternionic riccati equation in some cases we are able to give a full description of dynamics and detect solutions heteroclinic to the periodic ones we also provide an example of the riccati equation without periodic solutions which appears in the euler vorticity dynamics

polynomial interpolation over quaternions abstract interpolation theory for complex polynomials is well understood in the non commutative quaternionic setting the polynomials can be evaluated on the left and on the right if the interpolation problem involves interpolation conditions of the same left or right type the results are very much similar to the complex case a consistent problem has a unique solution of a low degree less than the number of interpolation conditions imposed and the solution set of the homogeneous problem is an ideal in the ring h z the problem containing both left and right interpolation conditions is quite different there may exist infinitely many low degree solutions and the solution set of the homogeneous problem is a quasi ideal in h z

newton method in the context of quaternion analysis abstract in this paper we propose a version of newton method for finding zeros of a quaternion function of a quaternion variable based on the concept of quaternion radial derivative several numerical examples involving elementary functions are presented

existence of solutions for schrodinger evolution equations we study the existence uniqueness and regularity of the solution of the initial value problem for the time dependent schrodinger equationi u t 1 2 u v t x u u 0 u0 we provide sufficient conditions onv t x such that the equation generates a unique unitary propagatoru t s and such thatu t s u0ec1 l2 c0 h2 n foru0eh2 n the conditions are general enough to accommodate moving singularities of type x 2 n 4 or x n 2 n 3

a new series expansion for slice regular functions abstract a promising theory of quaternion valued functions of one quaternionic variable now called slice regular functions has been introduced by gentili and struppa in 2006 the basic examples of slice regular functions are the power series of type n n q n a n on their balls of convergence b 0 r q h q r conversely if f is a slice regular function on a domain h then it admits at each point q 0 an expansion of type f q n n q q 0 n a n where q q 0 n denotes the n th power of q q 0 with respect to an appropriately defined multiplication however the information provided by such an expansion is somewhat limited by a fact if q 0 does not lie on the real axis then the set of convergence of the series in the previous equation needs not be a euclidean neighborhood of q 0 we are now able to construct a new type of expansion that is not affected by this phenomenon an expansion into series of polynomials valid in open subsets of the domain along with this construction we present applications to the computation of the multiplicities of zeros and of partial derivatives

experimental validation of the use of kramers kronig relations to eliminate the phase sheet ambiguity in broadband phase spectroscopy the technique of broadband phase spectroscopy proposed in 1978 by sachse and pao j appl phys 49 4320 4327 1978 determines the phase velocity as a function of frequency from the fourier transforms of a received reference and through sample signal although quite successful this approach can be influenced by an ambiguity in the phase velocity calculation which stems from the boundedness of the inverse tangent operation used to calculate phase several empirical approaches to resolve the phase ambiguity have been reported an alternative approach that has not previously been considered appeals to the causal nature of the measurements this article experimentally validates a method which uses the causally consistent kramers kronig relations to eliminate the ambiguity in phase spectroscopy derived phase velocity calculations broadband pulse and narrow band tone burst measurements were performed on three gelatin based phantoms containing different concentrations of graphite particles 0 10 and 20 by volume the phantoms were constructed to have attenuation coefficients which vary approximately linear with frequency a dependence exhibited by many soft tissues the narrow band phase velocity measurements do not suffer from a phase ambiguity and thus they serve as a gold standard against which the broadband phase velocity measurements are compared the experimental results illustrate that using the kramers kronig dispersion relations in conjunction with phase spectroscopy derived phase velocity measurements is an effective means by which to resolve the phase sheet ambiguity in broadband phase spectroscopy

lie symmetries for lie systems a lie system is a nonautonomous system of first order differential equations admitting a superposition rule i e a map expressing its general solution in terms of a generic family of particular solutions and some constants using that a lie system can be considered as a curve in a finite dimensional lie algebra of vector fields a so called vessiot guldberg lie algebra we associate every lie system with a lie algebra of lie point symmetries induced by the vessiot guldberg lie algebra this enables us to derive lie symmetries of relevant physical systems described by first and higher order systems of differential equations by means of lie systems in an easier way than by standard methods a generalization of our results to partial differential equations is introduced among other applications lie symmetries for several new and known generalizations of the real riccati equation are studied

geometry of riccati equations over normed division algebras abstract this work introduces and studies riccati equations over finite dimensional normed division algebras we prove that a riccati equation over a finite dimensional normed division algebra a is a particular case of conformal riccati equation on a euclidean space and it can be considered as a curve in a lie algebra of vector fields v so dim a 1 1 previous results on known types of riccati equations are recovered from a new viewpoint a new type of riccati equations the octonionic riccati equations are extended to the octonionic projective line o p 1 as a new physical application quaternionic riccati equations are applied to study quaternionic schrodinger equations on 1 1 dimensions