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

adjacency spectra of random and complete hypergraphs abstract we present progress on the problem of asymptotically describing the adjacency eigenvalues of random and complete uniform hypergraphs there is a natural conjecture arising from analogy with random matrix theory that connects these spectra to that of the all ones hypermatrix several of the ingredients along a possible path to this conjecture are established and may be of independent interest in spectral hypergraph hypermatrix theory in particular we provide a bound on the spectral radius of the symmetric bernoulli hyperensemble and show that the spectrum of the complete k uniform hypergraph for k 2 3 is close to that of an appropriately scaled all ones hypermatrix

the toda hierarchy and the kdv hierarchy this paper considers the relation between the periodic kdv hierarchy and the limit of the periodic toda hierarchies by choosing the initial data of the toda flows in a canonical way the behavior of a certain toda flow can mimic kdv flows conjecturally a method of deforming the kdv hierarchy is given

continuous limits for an integrable coupling of the kac van moerbeke hierarchy in this paper we present an integrable coupling of lattice hierarchy and its continuous limits by using lie algebra sl 4 by introducing a complex discrete spectral problem the integrable coupling of kac van moerbeke lattice hierarchy is derived it is shown that a new complex lattice spectral problem converges to the integrable coupling of discrete soliton equation hierarchy which has the integrable coupling system of mkdv hierarchy as a new kind of continuous limit

a bi hamiltonian formulation for triangular systems by perturbations a bi hamiltonian formulation is proposed for triangular systems resulting from perturbations around solutions from which infinitely many symmetries and conserved functionals of triangular systems can be explicitly constructed provided that one operator of the hamiltonian pair is invertible through our formulation four examples of triangular systems are exhibited which also show that bi hamiltonian systems in both lower dimensions and higher dimensions are many and varied two of four examples give local 2 1 dimensional bi hamiltonian systems and illustrate that multiscale perturbations can lead to higher dimensional bi hamiltonian systems

meyer vietoris type formula for determinants of elliptic differential operators abstract for a closed codimension one submanifold of a compact manifold m let m be the manifold with boundary obtained by cutting m along let a be an elliptic differential operator on m and b and c be two complementary boundary conditions on if a b is an elliptic boundary valued problem on m then one defines an elliptic pseudodifferential operator r of neumann type on and prove the following factorization formula for the regularized determinants det a det a b k det r with k a local quantity depending only on the jets of the symbols of a b and c along the particular case when m has dimension 2 a is the laplace beltrami operator and b resp c is the dirichlet resp neumann boundary condition is considered

analytic fields on riemann surfaces ii the properties of analytic fields on a riemann surface represented by a branch covering of 1 are investigated in detail branch points are shown to correspond to the vertex operators with simple conformal properties as applications we compute determinants of bar partial j operators forzn symmetric surfaces and obtain various representations for the two loop measure in the bosonic string theory together with various identities for theta functions of hyperelliptic surfaces we also present an integral representation for the quantum part of the twist field correlation functions which describe propagation of the string on the orbifold background we also calculate the quantum part of the structure constants of the twist field operator algebra generalizing the results of dixon friedan martinec and shenker

precise constants in bosonization formulas on riemann surfaces i a computation of the constant appearing in the spin 1 bosonization formula is given this constant relates faltings delta invariant to the zeta regularized determinant of the laplace operator with respect to the arakelov metric

improvements to platt s smo algorithm for svm classifier design this article points out an important source of inefficiency in platt s sequential minimal optimization smo algorithm that is caused by the use of a single threshold value using clues from the kkt conditions for the dual problem two threshold parameters are employed to derive modifications of smo these modified algorithms perform significantly faster than the original smo on all benchmark data sets tried

the multi component coupled burgers hierarchy of soliton equations and its multi component integrable couplings system with two arbitrary functions abstract a new loop algebra g m is constructed which is devoted to establishing an isospectral problem by making use of tu scheme the multi component coupled burgers hierarchy is obtained furthermore an expanding loop algebra f m of the loop algebra g m is presented it follows that the multi component integrable coupling system of the multi component coupled burgers hierarchy is worked out in general the method can be applied to other evolution equations hierarchies

training a support vector machine in the primal most literature on support vector machines svms concentrates on the dual optimization problem in this letter we point out that the primal problem can also be solved efficiently for both linear and nonlinear svms and that there is no reason for ignoring this possibility on the contrary from the primal point of view new families of algorithms for large scale svm training can be investigated

normal matrices over hermitian discrete valuation rings abstract a generalization of the classical spectral theorem for normal complex matrices is proved for matrices with entries from a class of discrete valuation rings which we have called hermitian the hermitian discrete valuation rings include the convergent complex power series in one variable and the complex formal power series in one variable thus one obtains an algebraic proof of rellich s theorem on diagnolizability of hermitian analytic matrices which is simultaneously valid for both the rings of convergent and formal power series

a new loop algebra and a corresponding integrable hierarchy as well as its integrable coupling a type of new interesting loop algebra gm m 1 2 with a simple commutation operation just like that in the loop algebra a1 is constructed with the help of the loop algebra gm a new multicomponent integrable system m akns kn hierarchy is worked out as reduction cases the m akns hierarchy and m kn hierarchy are engendered respectively in addition the system 1 akns kn which is a reduced case of the m akns kn hierarchy above is a unified expressing integrable model of the akns hierarchy and the kn hierarchy obviously the m akns kn hierarchy is again a united expressing integrable model of the multicomponent akns hierarchy m akns and the multicomponent kn hierarchy m kn this article provides a simple method for obtaining multicomponent integrable hierarchies of soliton equations finally we work out an integrable coupling of the m akns kn hierarchy

quasianalytic multiparameter perturbation of polynomials and normal matrices we study the regularity of the roots of multiparameter families of complex univariate monic polynomials p x z zn p n 1 1 jaj x znij with fixed degree n whose coefficients belong to a certain subringc of c 1 functions we require that c includes polynomial but excludes flat functions quasianalyticity and is closed under composition derivation division by a coordinate and taking the inverse examples are quasianalytic denjoy carleman classes in particular the class of real analytic functions c we show that there exists a locally finite covering k of the parameter space where each k is a composite of finitely many c mappings each of which is either a local blow up with smooth center or a local power substitution in coordinates given by x 7 x 1 1 x q q i n 0 such that for each k the family of polynomials p k admits a c parameterization of its roots if p is hyperbolic all roots real then local blow ups suffice using this desingularization result we prove that the roots of p can be parameterized by sbvloc functions whose classical gradients exist almost ev erywhere and belong to l1 in general the roots cannot have gradients in l p for any 1 p neither can the roots be in w 1 1 loc or v mo we obtain the same regularity properties for the eigenvalues and the eigen vectors of c families of normal matrices a further consequence is that every continuous subanalytic function belongs to sbvloc

perturbation theory for normal operators let e 3 x 7 a x be a c mapping with its values being un bounded normal operators with common domain of denition and compact resolvent here c stands for c1 c real analytic c m denjoy carleman of beurling or roumieu type c0 1 locally lipschitz or ck the param eter domain e is either r or rn or an innite dimensional convenient vector space we completely describe the c dependence on x of the eigenvalues and the eigenvectors of a x thereby we extend previously known results for self adjoint operators to normal operators partly improve them and show that they are best possible for normal matrices a x we obtain partly stronger results

connected components of the moduli spaces of abelian differentials with prescribed singularities consider the moduli space of pairs c where c is a smooth compact complex curve of a given genus and is a holomorphic 1 form on c with a given list of multiplicities of zeroes we describe connected components of this space this classification is important in the study of dynamics of interval exchange transformations and billiards in rational polygons and in the study of geometry of translation surfaces

spectral determinants on mandelstam diagrams we study the regularized determinant of the laplacian as a functional on the space of mandelstam diagrams noncompact translation surfaces glued from finite and semi infinite cylinders a mandelstam diagram can be considered as a compact riemann surface equipped with a conformal flat singular metric omega 2 where omega is a meromorphic one form with simple poles such that all its periods are pure imaginary and all its residues are real the main result is an explicit formula for the determinant of the laplacian in terms of the basic objects on the underlying riemann surface the prime form theta functions the canonical meromorphic bidifferential and the divisor of the meromorphic form omega as an important intermediate result we prove a decomposition formula of the type of burghelea friedlander kappeler for the determinant of the laplacian for flat surfaces with cylindrical ends and conical singularities

building sparse twin support vector machine classifiers in primal space abstract twin support vector machines tsvm obtain faster training speeds than classical support vector machines svm however tsvm augmented vectors lose sparsity in this paper a rapid sparse twin support vector machine stsvm classifier in primal space is proposed to improve the sparsity and robustness of tsvm based on a simple back fitting strategy the stsvm iteratively builds each nonparallel hyperplanes by adding one support vector sv from the corresponding class at one time this process is terminated using an adaptive and stable stopping criterion stsvm learning is implemented by linear equation computing systems through introducing a quadratic function to approximate the empirical risk the computational results on several synthetic and benchmark datasets indicate that the stsvm obtains a sparse separating hyperplane at a low cost without sacrificing its generalization performance

the trace identity a powerful tool for constructing the hamiltonian structure of integrable systems a new approach to hamiltonian structures of integrable systems is proposed by making use of a trace identity for a variety of isospectral problems that can be unified to one model x u it is shown that both the related hierarchy of evolution equations and the hamiltonian structure can be obtained from the same solution of the equation vx u v

asymptotics of the spectrum and the selberg zeta function on the space of riemann surfaces letz s r be the selberg zeta function of a compact riemann surfacer we study the behavior ofz s r asr tends to infinity in the moduli space of stable curves the main result is an estimate forz s r valid fors in a neighborhood depending only on the genus ofs 1 our analysis gives an alternate proof of the belavin knizhnik double pole result 5

on the approximation property of excellent rings where d is a smooth finite type a algebra a theorem which would imply the existence of an embedding of the above type yields the positive solution of the bass quillen conjecture so far this theorem is only known in special cases there is the following result by artin 3 4

variational identities and applications to hamiltonian structures of soliton equations abstract this is an introductory report concerning our recent research on hamiltonian structures we will discuss variational identities associated with continuous and discrete spectral problems and their applications to hamiltonian structures of soliton equations our illustrative examples are the akns hierarchy and the volterra lattice hierarchy associated with semisimple lie algebras and two hierarchies of their integrable couplings associated with non semisimple lie algebras the resulting hamiltonian structures generate infinitely many commuting symmetries and conservation laws for the four soliton hierarchies the presented variational identities can be applied to hamiltonian structures of other soliton hierarchies

a discrete variational identity on semi direct sums of lie algebras the discrete variational identity under general bilinear forms on semi direct sums of lie algebras is established the constant involved in the variational identity is determined through the corresponding solution to the stationary discrete zero curvature equation an application of the resulting variational identity to a class of semi direct sums of lie algebras in the volterra lattice case furnishes hamiltonian structures for the associated integrable couplings of the volterra lattice hierarchy

differentiable perturbation of unbounded operators if a t is a c 1 curve of unbounded self adjoint operators with compact resolvents and common domain of definition then the eigenvalues can be parameterized c 1 in t if a is c then the eigenvalues can be parameterized twice differentiably

many parameter holder perturbation of unbounded operators if u mapsto a u is a c0 mapping for 0 1 having as values unbounded self adjoint operators with compact resolvents and common domain of definition parameterized by u in an even infinite dimensional space then any continuous in u arrangement of the eigenvalues of a u is indeed c0 in u

canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant this article contains an elementary constructive proof of resolution of singularities in characteristic zero our proof applies in particular to schemes of finite type and to analytic spaces so we recover the great theorems of hironaka we introduce a discrete local invariant rm inv x a whose maximum locus determines a smooth centre of blowing up leading to desingularization to define rm inv x we need only to work with a category of local ringed spaces x x cal o x satisfying certain natural conditions if a in x then rm inv x a depends only on widehat cal o x a more generally rm inv x is defined inductively after any sequence of blowings up whose centres have only normal crossings with respect to the exceptional divisors and lie in the constant loci of rm inv x cdot the paper is self contained and includes detailed examples one of our goals is that the reader understand the desingularization theorem rather than simply know it is true

the h functional calculus based on the s spectrum for quaternionic operators and for n tuples of noncommuting operators abstract in this paper we extend the h functional calculus to quaternionic operators and to n tuples of noncommuting operators using the theory of slice hyperholomorphic functions and the associated functional calculus called s functional calculus the s functional calculus has two versions one for quaternionic valued functions and one for clifford algebra valued functions and can be considered the riesz dunford functional calculus based on slice hyperholomorphicity because it shares with it the most important properties the s functional calculus is based on the notion of s spectrum which in the case of quaternionic normal operators on a hilbert space is also the notion of spectrum that appears in the quaternionic spectral theorem the main purpose of this paper is to construct the h functional calculus based on the notion of s spectrum for both quaternionic operators and for n tuples of noncommuting operators we remark that the h functional calculus for n 1 tuples of operators applies in particular to the dirac operator

integrable couplings of vector akns soliton equations by enlarging the associated matrix spectral problems two specific classes of multicomponent integrable couplings of the physically important vector akns soliton equations are constructed which can be linked to each other by a backlund transformation the resulting two hierarchies of integrable couplings possess the enlarged zero curvature representations and recursion structures and thus each system in the two hierarchies has infinitely many commuting symmetries

perturbation of spectral subspaces and solution of linear operator equations abstract let a and b be normal operators on a hilbert space let k a and k b be subsets of the complex plane at distance at least from each other let e be the spectral projector for a belonging to k a and let f be the spectral projector for b belonging to k b our main results are estimates of the form ef c e a b f in some special situations the constant c is as low as 1 as an application we prove for an absolute constant d that if the space is finite dimensional and if a and b are normal with a b d then the spectrum of b can be obtained from that of a multiplicities counted by moving each eigenvalue by at most our main results have equivalent formulations as statements about the operator equation aq qb s let a and b be normal operators on perhaps different hilbert spaces assume a k a and b k b where k a k b and are as before then we give estimates of the forms q c aq qb

the convenient setting for non quasianalytic denjoy carleman differentiable mappings abstract for quasianalytic denjoy carleman differentiable function classes c q where the weight sequence q q k is log convex stable under derivations of moderate growth and also an l intersection see 1 6 we prove the following the category of c q mappings is cartesian closed in the sense that c q e c q f g c q e f g for convenient vector spaces applications to manifolds of mappings are given the group of c q diffeomorphisms is a regular c q lie group but not better

le saut en zero de la fonction de decalage spectral abstract we give a formula for the jump at zero of the spectral shift function associated with schrodinger operator on manifolds with conical ends we show that according to its decay a zero energy resonant state has a non integer contribution

nonlinear regularization path for quadratic loss support vector machines regularization path algorithms have been proposed to deal with model selection problem in several machine learning approaches these algorithms allow computation of the entire path of solutions for every value of regularization parameter using the fact that their solution paths have piecewise linear form in this paper we extend the applicability of regularization path algorithm to a class of learning machines that have quadratic loss and quadratic penalty term this class contains several important learning machines such as squared hinge loss support vector machine svm and modified huber loss svm we first show that the solution paths of this class of learning machines have piecewise nonlinear form and piecewise segments between two breakpoints are characterized by a class of rational functions then we develop an algorithm that can efficiently follow the piecewise nonlinear path by solving these rational equations to solve these rational equations we use rational approximation technique with quadratic convergence rate and thus our algorithm can follow the nonlinear path much more precisely than existing approaches such as predictor corrector type nonlinear path approximation we show the algorithm performance on some artificial and real data sets

nonnegative functions as squares or sums of squares abstract we prove that for n 4 there are c nonnegative functions f of n variables and even flat ones for n 5 which are not a finite sum of squares of c 2 functions for n 1 where a decomposition in a sum of two squares is always possible we investigate the possibility of writing f g 2 we prove that in general one cannot require a better regularity than g c 1 assuming that f vanishes at all its local minima we prove that it is possible to get g c 2 but that one cannot require any additional regularity

bosonization on higher genus riemann surfaces we prove the equivalence between certain fermionic and bosonic theories in two spacetime dimensions the theories have fields of arbitrary spin on compact surfaces with any number of handles global considerations require that we add new topological terms to the bosonic action the proof that our prescription is correct relies on methods of complex algebraic geometry

greedy function approximation a gradient boosting machine function estimation approximation is viewed from the perspective of numerical optimization in function space rather than parameter space a connection is made between stagewise additive expansions and steepest descent minimization a general gradient descent boosting paradigm is developed for additive expansions based on any fitting criterion specific algorithms are presented for least squares least absolute deviation and huber m loss functions for regression and multiclass logistic likelihood for classification special enhancements are derived for the particular case where the individual additive components are regression trees and tools for interpreting such treeboost models are presented gradient boosting of regression trees produces competitive highly robust interpretable procedures for both regression and classification especially appropriate for mining less than clean data connections between this approach and the boosting methods of freund and shapire and friedman hastie and tibshirani are discussed

recursive finite newton algorithm for support vector regression in the primal some algorithms in the primal have been recently proposed for training support vector machines this letter follows those studies and develops a recursive finite newton algorithm ihlf svr rfn for training nonlinear support vector regression the insensitive huber loss function and the computation of the newton step are discussed in detail comparisons with libsvm 2 82 show that the proposed algorithm gives promising results

nonlinearization of spectral problems for the perturbation kdv systems abstract the theory of nonlinearization of spectral problems is developed for investigating the perturbation kdv systems and a kind of finite dimensional integrable hamiltonian systems is generated from spectral problems of the perturbation kdv systems to establish the liouville integrability of the obtained hamiltonian systems we identify them with the perturbation systems of the garnier system which can be directly proved to be integrable a useful formula to compute the functional gradient of the spectral parameter with respect to the potential is also presented for arbitrary order matrix spectral problems

on 2 step corank 2 nilpotent sub riemannian metrics in this paper we study the nilpotent 2 step corank 2 sub riemannian metrics that are nilpotent approximations of general sub riemannian metrics we exhibit optimal syntheses for these problems it turns out that in general the cut time is not equal to the first conjugate time but has a simple explicit expression as a by product of this study we get some smoothness properties of the spherical hausdorff measure in the case of a generic 6 dimensional 2 step corank 2 sub riemannian metric

isomonodromic tau function on the space of admissible covers abstract the isomonodromic tau function of the fuchsian differential equations associated to frobenius structures on hurwitz spaces can be viewed as a section of a line bundle on the space of admissible covers we study the asymptotic behavior of the tau function near the boundary of this space and compute its divisor this yields an explicit formula for the pullback of the hodge class to the space of admissible covers in terms of the classes of compactification divisors

functional determinants and geometry in this paper we study the zeta function determinant in the context of elliptic boundary value problems our main technique is to relate the determinant of an operator or a ratio of determinants to the boundary values of the solutions of the operator this has the advantage of restricting attention to the solutions of the operator which do not depend on the boundary conditions and can often be written down explicitly rather than the eigenvalues which are usually difficult to work with in addition the problem is reduced to a calculation over the boundary of the manifold which is a closed manifold of dimension one less than the original manifold this has special significance in the case that the manifold is a finite interval in this case the boundary is a pair of points and the determinant of an ordinary differential operator is expressed in terms of the determinant of a finite matrix

continuous limits for the kac van moerbeke hierarchy and for their restricted flows the continuous limit for the kac van moerbeke kvm hierarchy for their bi hamiltonian formulation recursion relation and square eigenfunction relation is studied a new family of integrable symplectic maps ism are reduced from the kvm hierarchy via constraint for a higher flow of the hierarchy in terms of square eigenfunctions their integrability is deduced from the discrete zero curvature representation of the kvm hierarchy it is shown that these isms provide maps which approximate many well known integrable mechanical systems e g neumann garnier embedded into the kdv hierarchy as their restricted flows

building support vector machines with reduced classifier complexity support vector machines svms though accurate are not preferred in applications requiring great classification speed due to the number of support vectors being large to overcome this problem we devise a primal method with the following properties 1 it decouples the idea of basis functions from the concept of support vectors 2 it greedily finds a set of kernel basis functions of a specified maximum size dmax to approximate the svm primal cost function well 3 it is efficient and roughly scales as o ndmax2 where n is the number of training examples and 4 the number of basis functions it requires to achieve an accuracy close to the svm accuracy is usually far less than the number of svm support vectors

on the continuous limit of integrable lattices i the kac moerbeke system and kdv theory kdv theory is constructed systematically through the continuous limit of the kac moerbeke system the infinitely many commuting vector fields the conserved functionals the lax pairs and the bihamiltonian structure are recovered as the limits of suitably defined linear combinations of homologous objects for the kac moerbeke system the combinatorial aspects of this recombination method are treated in detail

a direct method for integrable couplings of td hierarchy a direct method for establishing integrable couplings is proposed in this paper as an example illustration integrable couplings of td hierarchy are obtained by constructing a suitable transformation of lax pairs and a new lie algebra

the abhyankar jung theorem abstract we show that every quasi ordinary weierstrass polynomial p z z d a 1 x z d 1 a d x k x z x x 1 x n over an algebraically closed field of characteristic zero k such that a 1 0 is quasi ordinary that means that if the discriminant p k x is equal to a monomial times a unit then the ideal a i d i x i 2 d is monomial and generated by one of a i d i x we use this result to give a constructive proof of the abhyankar jung theorem that works for any henselian local subring of k x and the function germs of quasi analytic families

relative zeta functions relative determinants and scattering theory we use the method of zeta function regularization to regularize the ratio det a det a0 of the determinants of two elliptic self adjoint operators a a0 satisfying certain natural assumptions this is of interest especially if the regularized determinants of the individual operators don t exist as for example in the case of elliptic operators on a noncompact manifold

semidirect sums of lie algebras and discrete integrable couplings a relation between semidirect sums of lie algebras and integrable couplings of lattice equations is established and a practicable way to construct integrable couplings is further proposed an application of the resulting general theory to the generalized toda spectral problem yields two classes of integrable couplings for the generalized toda hierarchy of lattice equations the construction of integrable couplings using semidirect sums of lie algebras provides a good source of information on complete classification of integrable lattice equations

moduli spaces of meromorphic functions and determinant of laplacian the hurwitz space is the moduli space of pairs x f where x is a compact riemann surface and f is a meromorphic function on x we study the laplace operator delta df 2 of the flat singular riemannian manifold x df 2 we define a regularized determinant for delta df 2 and study it as a functional on the hurwitz space we prove that this functional is related to a system of pde which admits explicit integration this leads to an explicit expression for the determinant of the laplace operator in terms of the basic objects on the underlying riemann surface the prime form theta functions the canonical meromorphic bidifferential and the divisor of the meromorphic differential df the proof has several parts that can be of independent interest as an important intermediate result we prove a decomposition formula of the type of burghelea friedlander kappeler for the determinant of the laplace operator on flat surfaces with conical singularities and euclidean or conical ends we introduce and study the s matrix s lambda of a surface with conical singularities as a function of the spectral parameter lambda and relate its behavior at lambda 0 with the schiffer projective connection on the riemann surface x we also prove variational formulas for eigenvalues of the laplace operator of a compact surface with conical singularities when the latter move

on determinants of laplacians on riemann surfaces determinants of laplacians on tensors and spinors of arbitrary weights on compact hyperbolic riemann surfaces are computed in terms of values of selberg zeta functions at half integer points

extraction and prediction of coherent patterns in incompressible flows through space time koopman analysis abstract we develop methods for detecting and predicting the evolution of coherent spatiotemporal patterns in incompressible time dependent fluid flows driven by ergodic dynamical systems our approach is based on representations of the generators of the koopman and perron frobenius groups of operators governing the evolution of observables and probability measures on lagrangian tracers respectively in a smooth orthonormal basis learned from velocity field snapshots through the diffusion maps algorithm these operators are defined on the product space between the state space of the fluid flow and the spatial domain in which the flow takes place and as a result their eigenfunctions correspond to global space time coherent patterns under a skew product dynamical system moreover using this data driven representation of the generators in conjunction with leja interpolation for matrix exponentiation we construct model free prediction schemes for the evolution of observables and probability densities defined on the tracers we present applications to periodic gaussian vortex flows and aperiodic flows generated by lorenz 96 systems

real analytic desingularization and subanalytic sets an elementary approach we give a proof of a theorem on desingularization of real analytic functions which is a weaker version of h hironaka s result but has the advantage of being completely self contained and elementary and not involving any machinery from algebraic geometry we show that the basic facts about subanalytic sets can be proved from this result

the spectral theorem for quaternionic unbounded normal operators based on the s spectrum in this paper we prove the spectral theorem for quaternionic unbounded normal operators using the notion of s spectrum the proof technique consists of first establishing a spectral theorem for quaternionic bounded normal operators and then using a transformation which maps a quaternionic unbounded normal operator to a quaternionic bounded normal operator with this paper we complete the foundation of spectral analysis of quaternionic operators the s spectrum has been introduced to define the quaternionic functional calculus but it turns out to be the correct object also for the spectral theorem for quaternionic normal operators the lack of a suitable notion of spectrum was a major obstruction to fully understand the spectral theorem for quaternionic normal operators a prime motivation for studying the spectral theorem for quaternionic unbounded normal operators is given by the subclass of unbounded anti self adjoint quaternionic operators which play a crucial role in the quaternionic quantum mechanics