on resource bounded instance complexity we prove the instance complexity conjecture of ko orponen schoning and watanabe for all recursive tally sets and for all recursive sets which are np hard under honest reductions in particular it holds for all natural np hard problems on the other hand the conjecture is shown to be oracle dependent additional results concern the nonrecursiveness of the instance complexity measure and a comparison of time bounded c and cd complexity

weyl von neumann berg theorem for quaternionic operators we prove the weyl von neumann berg theorem for right linear operators not necessarily bounded in a quaternionic hilbert space let n be a right linear normal need not be bounded operator in a quaternionic separable infinite dimensional hilbert space h then for a given 0 there exists a compact operator k with k and a diagonal operator d on h such that n d k

indefinite sequences with a finite order singularity in the complex moment problem in this paper we consider indefinite sequences having singularity at the point 1 1 i p of order at most 2x we show that they are related to both the classical moment problem and the l6vy khinchin formula for negative definite sequences we establish an integral representation for these sequences and characterize the exponentially bounded ones by means of finitely many positive definiteness conditions even in the case x 1 our approach provides new l6vy khinchin type integral representations let no be the set of all non negative integers and fix the positive integer p we denote by l z i the algebra of polynomials in z and where z zl zp e r p this algebra acts on the sequences ct m n m n e ng n following the correspondence

algebraic methods for diagonalization of a quaternion matrix in quaternionic quantum theory by means of complex representation and real representation of a quaternion matrix this paper studies the problem of diagonalization of a quaternion matrix gives two algebraic methods for diagonalization of quaternion matrices in quaternionic quantum theory

spectral theory for unitary operators on a quaternionic hilbert space the spectral theorem for a unitary operator on a quaternionic hilbert space is established and a number of related results are proved

harmonic analysis on conelike bodies and holomorphic functions on tube domains in this paper we establish bochner weil type theorems and integral formulas of levy khinchin type in the setting of locally compact commutative semigroups with involution these results are used to prove some new holomorphic extension results for a conelike involution semigroup s in a finite dimensional real space v we establish integral characterizations for continuous positive definite and definitizable functions on s in particular we obtain holomorphic and cauchy riemann extensions of these functions by means of an integral representation we prove that ifz0is a point in the complexified vector space w v iv and if a function is defined onz0 s then either positive definiteness or definitizability offextends holomorphy from a domain containingz0to a tube domain in w containing the sum of 0 and the convex cone spanned by s we also obtain the holomorphic extension as a generalized fourier laplace transform of some measure or a levy khinchin type integral representation

information measures for infinite sequences we revisit the notion of computational depth and sophistication for infinite sequences and study the density of the sets of deep and sophisticated infinite sequences koppel defined the sophistication of an object as the length of the shortest total program that given some data as input produces the object and such that the sum of the size of the program with the size of the data is as concise as the smallest description of the object however the notion of sophistication is not properly defined for all sequences we propose a new definition of sophistication for infinite sequences as the limit of the ratio of the sophistication of the initial segments and its length we prove that the set of sequences with sophistication equal to zero has lebesgue measure one and that the set of sophisticated sequences is dense when the sophistication is respectively defined either using lim inf or lim sup antunes fortnow van melkebeek and vinodchandran captured the notion of useful information by computational depth the difference between time bounded and unbounded kolmogorov complexities we show that the set of deep infinite sequences is dense we also prove that sophistication and depth for infinite sequences are distinct information measures

searching for shortest and least programs abstract the kolmogorov complexity of a string x is defined as the length of a shortest program p of x for some appropriate universal machine u that is u p x and p is a shortest string with this property neither the plain nor the prefix free version of kolmogorov complexity are recursive but for both versions it is well known that there are recursive exact solovay functions that is recursive upper bounds for kolmogorov complexity that are infinitely often tight let a coding function for a machine m be a function f such that f x is always a program of x for m from the existence of exact solovay functions it follows easily that for every universal machine there is a recursive coding function that maps infinitely many strings to a shortest program extending a recent line of research in what follows it is investigated in which situations there is a coding function for some universal machine that maps infinitely many strings to the length lexicographically least program the main results which hold in the plain as well as in the prefix free setting are the following for every universal machine there is a recursive coding function that maps infinitely many strings to their least programs there is a partial recursive coding function defined in the natural way for some universal machine that for every set maps infinitely many prefixes of the set to their least programs exactly for every set that is bennett shallow not deep there is a recursive coding function for some universal machine that maps all prefixes of the set to their least programs differences between the plain and the prefix free frameworks are obtained by considering effective sequences i 1 i 2 of mutually disjoint finite sets and asking for a recursive coding function for some universal machine that maps at least one string in each set i n to its least code such coding functions do not exist in the prefix free setting but exist in the plain setting in case the sets i n are not too small

alternative descriptions in quaternionic quantum mechanics we characterize the quasianti hermitian quaternionic operators in quaternionic quantum mechanics by means of their spectra moreover we state a necessary and sufficient condition for a set of quasianti hermitian quaternionic operators to be anti hermitian with respect to a uniquely defined positive scalar product in a infinite dimensional right quaternionic hilbert space according to such results we obtain two alternative descriptions of a quantum optical physical system in the realm of quaternionic quantum mechanics while no alternative can exist in complex quantum mechanics and we discuss some differences between them

an algebraic method for quaternion and complex least squares coneigen problem in quantum mechanics in the study of theory and numerical computations of quantum theory in order to well understand the perturbation theory experimental proposals and theoretical discussions underlying the quaternion and complex formulations one meets problems of approximate solutions of quaternion and complex problems such as approximate solution of quaternion and complex linear equations a or a that is appropriate when there are errors in the vector and i e quaternion and complex least squares coneigen problem in quantum mechanics this paper by means of representation of quaternion matrices and complex matrices studies the problems of quaternion and complex least squares coneigen problem and give practical algebraic methods of computing approximate coneigenvalues and coneigenvectors for quaternion and complex matrices in quantum mechanics

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 a basic theorem of quaternion modules a complete proof of the equivalence of the star operation in the operator algebra isomorphic to quaternions and the adjoint operation in a quaternion hilbert module is given

the complexity of malign measures this paper analyzes the concept of malignness which is the property of probability ensembles making the average case running time equal to the worst case running time for a class of algorithms the author derives lower and upper bounds on the complexity of malign ensembles which are tight for exponential time algorithms and which show that no polynomial time computable malign ensemble exists for the class of polynomial time algorithms furthermore it is shown that for no class of superlinear algorithms a polynomial time samplable malign ensemble exists unless every language in p has an expected polynomial time constructor

lowness and logical depth abstract bennett s concept of logic depth 3 seeks to capture the idea that a language has a lot of useful information thus we would expect that neither sufficiently random nor sufficiently computationally trivial sequences are deep a question of moser and stephan 11 explores the boundary of this assertion asking if there is a low computably enumerable bennett deep language we answer this question affirmatively by constructing a superlow computably enumerable bennett deep language

quaternionic quantum field theory we show that a quaternionic quantum field theory can be formulated when the numbers of bosonic and fermionic degrees of freedom are equal and the fermions as well as the bosons obey a second order wave equation the theory takes the form of either a functional integral with quaternion imaginary lagrangian or a schrodinger equation and transformation theory for quaternion valued wave functions with a quaternion imaginary hamiltonian the connection between the two formulations is developed in detail and many related issues including the breakdown of the correspondence principle and the hilbert space structure are discussed

complex structure on a real hilbert space and symplectic structure on a complex hilbert space alternative ways of complexifying a real hilbert space and quaternionizing a complex hilbert space are described the work gives some insight into why even though in the finite dimensional case a complex hilbert space when viewed as a real hilbert space and a quaternionic hilbert space when viewed as a complex hilbert space have twice their original dimensions the degrees of freedom of the linear operators remain unchanged many ramifications are discussed among them the reconciliation of the linearity of the adjoint of a semilinear antilinear map from one complex hilbert space to another with the semilinearity antilinearity of the adjoint of a semilinear antilinear map from one complex hilbert space to itself groundwork is prepared for the study of the noncommutative algebra of additive operators on a quaternionic hilbert space

analytic plane wave solutions for the quaternionic potential step by using the recent mathematical tools developed in quaternionic differential operator theory we solve the schrodinger equation in the presence of a quaternionic step potential the analytic solution for the stationary states allows one to explicitly show the qualitative and quantitative differences between this quaternionic quantum dynamical system and its complex counterpart a brief discussion on reflected and transmitted times performed by using the stationary phase method and its implication on the experimental evidence for deviations of standard quantum mechanics is also presented the analytic solution given in this paper represents a fundamental mathematical tool to find an analytic approximation to the quaternionic barrier problem up to now solved by numerical method

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

relative to a random oracle a bf p a ne bf np a ne text co bf np a with probability 1 let a be a language chosen randomly by tossing a fair coin for each string x to determine whether x belongs to a with probability 1 each of the relativized classes textbf logspace a bf p a bf np a bf pp a and textbf pspace a is properly contained in the next also bf np a ne text co bf np a with probability 1 by contrast with probability 1 the class bf p a coincides with the class bf bpp a of languages recognized by probabilistic oracle machines with error probability uniformly bounded below tfrac 1 2 bf np a is shown with probability 1 to contain a bf p a immune set i e a set having no infinite subset in bf p a the relationship of bf p a immunity to p sparseness and bf np a completeness is briefly discussed bf p a immune sets in bf np a can be sparse or moderately dense but not co sparse relativization with respect to a random length preserving permutation pi instead of a rand

right eigenvalue equation in quaternionic quantum mechanics we study the right eigenvalue equation for quaternionic and complex linear matrix operators defined in n dimensional quaternionic vector spaces for quaternionic linear operators the eigenvalue spectrum consists of n complex values for these operators we give a necessary and sufficient condition for the diagonalization of their quaternionic matrix representations our discussion is also extended to complex linear operators whose spectrum is characterized by 2n complex eigenvalues we show that a consistent analysis of the eigenvalue problem for complex linear operators requires the choice of a complex geometry in defining inner products finally we introduce some examples of the left eigenvalue equations and highlight the main difficulties in their solution

polylog depth highness and lowness for e abstract we study the relations between the notions of highness lowness and logical depth in the setting of complexity theory we introduce a new notion of polylog depth based on time bounded kolmogorov complexity we show polylog depth satisfies all basic logical depth properties namely sets in p are not polylog deep sets with time bounded kolmogorov complexity greater than polylog are not polylog deep and only polylog deep sets can polynomially turing compute a polylog deep set we prove that if np does not have p measure zero then np contains polylog deep sets we show that every high set for e contains a polylog deep set in its polynomial turing degree and that there exist low e exp polylog deep sets keywords algorithmic information theory kolmogorov complexity bennett logical depth

on the polynomial depth of various sets of random strings we introduce a general framework for defining the depth of an infinite binary sequence with respect to a class of observers we show that our general framework captures all depth notions introduced in computability complexity theory so far we review most such notions show how they are particular cases of our general depth framework and review some classical results about the different depth notions we use our framework to define new notions of polynomial depth called monotone poly depth based on a polynomial version of monotone kolmogorov complexity we show that monotone poly depth satisfies all desirable properties of depth notions we give two natural examples of deep sets by showing that both the set of levin random strings and the set of kolmogorov random strings are monotone poly deep

average case complete problems many interesting combinatorial problems were found to be np complete since there is little hope to solve them fast in the worst case researchers look for algorithms which are fast just on average this matter is sensitive to the choice of a particular np complete problem and a probability distribution of its instances some of these tasks are easy and some not but one needs a way to distinguish the difficult on average problems such negative results could not only save positive efforts but may also be used in areas like cryptography where hardness of some problems is a frequent assumption it is shown in 1 that the tiling problem with uniform distribution of instances has no polynominal on average algorithm unless every np problem with every simple probability distribution has it

a pseudorandom generator from any one way function pseudorandom generators are fundamental to many theoretical and applied aspects of computing we show how to construct a pseudorandom generator from any one way function since it is easy to construct a one way function from a pseudorandom generator this result shows that there is a pseudorandom generator if and only if there is a one way function

additive functionals and operators on a quaternionic hilbert space it is shown that the structure of functionals and operators on a quaternionic hilbert space is much richer than is generally appreciated but one has to work simultaneously with the usual definition and an unusual one of multiplication by scalar for functionals the different definitions of course give rise to different vector spaces a generalized version of the riesz representation theorem for quaternionic hilbert spaces is proved along with the basic theorem on the algebra of additive operators on such a space

computational depth concept and applications we introduce computational depth a measure for the amount of nonrandom or useful information in a string by considering the difference of various kolmogorov complexity measures we investigate three instantiations of computational depth basic computational depth a clean notion capturing the spirit of bennett s logical depth we show that a turing machine m runs in time polynomial on average over the time bounded universal distribution if and only if for all inputs x m uses time exponential in the basic computational depth of x sublinear time computational depth and the resulting concept of shallow sets a generalization of sparse and random sets based on low depth properties of their characteristic sequences we show that every computable set that is reducible to a shallow set has polynomial size circuits distinguishing computational depth measuring when strings are easier to recognize than to produce we show that if a boolean formula has a nonnegligible fraction of its satisfying assignments with low depth then we can find a satisfying assignment efficiently

an algorithm for quaternionic linear equations in quaternionic quantum theory by means of complex representation and companion vector in this paper we introduce a definition of rank of a quaternion matrix study the problems of quaternionic linear equations and obtain an algorithm for quaternionic linear equations in quaternionic quantum theory

additive isometries on a quaternionic hilbert space a systematic study of additive isometries on a quaternionic hilbert space is presented a number of new results describing the properties of such operators are proved the work culminates in the first mathematical proof of wigner s theorem for quaternionic hilbert spaces of dimension other than 2 which asserts that any operator which preserves the absolute value of the inner product on a quaternionic hilbert space is equivalent in the sense of differing pointwise by a mere phase factor to a linear isometry a complete and concise description of the exceptional situation in a two dimensional quaternionic hilbert space is given

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

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