lift of noninvariant solutions of heavenly equations from three to four dimensions and new ultra hyperbolic metrics we demonstrate that partner symmetries provide a lift of noninvariant solutions of the three dimensional boyer finley equation to noninvariant solutions of the four dimensional hyperbolic complex monge ampere equation the lift is applied to noninvariant solutions of the boyer finley equation obtained earlier by the method of group foliation to yield noninvariant solutions of the hyperbolic complex monge ampere equation using these solutions we construct new ricci flat ultra hyperbolic metrics with non zero curvature tensor that have no killing vectors

partner symmetries and non invariant solutions of four dimensional heavenly equations we extend our method of partner symmetries to the hyperbolic complex monge ampere equation and the second heavenly equation of plebanski we show the existence of partner symmetries and derive the relations between them for certain simple choices of partner symmetries the resulting differential constraints together with the original heavenly equations are transformed to systems of linear equations by an appropriate legendre transformation the solutions of these linear equations are generically non invariant as a consequence we obtain explicitly new classes of heavenly metrics without killing vectors

quantum classical correspondence via a deformed kinetic operator we propose an approach to quantum classical correspondence based on a deformation of the momentum and kinetic operators of quantum mechanics making use of the factorization method we construct classical versions of the momentum and kinetic operators which in addition to the standard quantum expressions contain terms that are functionals of the n particle density we show that this implementation of quantum classical correspondence is related to witten s deformation of the exterior derivative and laplacian introduced in the context of supersymmetric quantum mechanics the corresponding deformed action is also shown to be related to the fisher information finally we briefly consider the possible relevance of our approach to the construction of kinetic energy density functionals

partner symmetries of the complex monge ampere equation yield hyper kahler metrics without continuous symmetries we extend the mason newman lax pair for the elliptic complex monge ampere equation so that this equation itself emerges as an algebraic consequence we regard the function in the extended lax equations as a complex potential their differential compatibility condition coincides with the determining equation for the symmetries of the complex monge ampere equation we shall identify the real and imaginary parts of the potential which we call partner symmetries with the translational and dilatational symmetry characteristics respectively then we choose the dilatational symmetry characteristic as the new unknown replacing the kahler potential this directly leads to a legendre transformation studying the integrability conditions of the legendre transformed system we arrive at a set of linear equations satisfied by a single real potential this enables us to construct non invariant solutions of the legendre transform of the complex monge ampere equation using these solutions we obtained explicit legendre transformed hyper kahler metrics with a anti self dual riemann curvature 2 form that admit no killing vectors they satisfy the einstein field equations with euclidean signature we give the detailed derivation of the solution announced earlier and present a new solution with an added parameter we compare our method of partner symmetries for finding non invariant solutions to that of dunajski and mason who use hidden symmetries for the same purpose

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

factorization little or great algorithm the progress of the factorization method since the 1935 work of dirac is briefly reviewed though linked with older mathematical theories the factorization seems an autonomous driving force offering substantial support to the present day darboux and backlund approaches

on quasi harada rings abstract m harada ring theory proceedings of 1978 antwerp conference pp 669 690 dekker new york 1979 studied the following two conditions every non small leftr module contains a non zero injective submodule every non cosmall rightr module contains a non zero projective direct summand k oshiro hokkaido math j 13 1984 310 338 further studied the above conditions and called a left artinian ring with a left harada ring abbreviated left h ring and a ring satisfying the ascending chain condition for right annihilator ideals with a right co harada ring abbreviated right co h ring k oshiro math j okayama univ 31 1989 161 178 showed that left h rings and right co h rings are the same rings here we are particularly interested in the following characterization of a left h ring given in harada s paper above a ringris a left h ring if and only ifris a perfect ring and for any left non small primitive idempotenteofrthere exists a non negative integertesuch that a rre sk rre is injective for anyk 0 te and b rre ste 1 rre is a small module wheresk rre denotes thek th socle of the leftr modulere this characterization implies ste 1 rre is a uniserial leftr module for any left non small primitive idempotentein a left h ringr in this paper we generalize left h rings by removing concretely since a left h ringris also characterized by the statement thatris left artinian and for any primitive idempotentgofrthere exist a primitive idempotentegofrand a non negative integerkgsuch that the injective hull of the leftr modulerg jgis isomorphic tore skg rreg wherejis the jacobson radical ofrr we define a more general class of rings by the condition that for any primitive idempotentgofrthere exists a primitive idempotenteofrsuch that the injective hull of the leftr modulerg jgis isomorphic tore x re grx 0 and call it a left quasi harada ring abbreviated left qh ring in section 1 we characterize a left qh ring by generalizing the characterizations of a left h ring given by m harada and k oshiro we also consider the weaker rings left qf 2 rings and right qf 2 rings k oshiro math j okayama univ 32 1990 111 118 described the connection between left h rings and qf rings in section 2 we describe the connection between two sided qh rings and qf rings

bi hamiltonian structure of an asymmetric heavenly equation in the paper of sheftel and malykh 2009 j phys a math theor 42 395202 on the classification of second order pdes with four independent variables that possess partner symmetries an asymmetric heavenly equation appears as one of the canonical equations admitting partner symmetries here for the asymmetric heavenly equation formulated in a two component form we present the lax pair of olver ibragimov shabat type and obtain its multi hamiltonian structure therefore by magri s theorem it is a completely integrable bi hamiltonian system in four dimensions

self duality of quasi harada rings and locally distributive rings abstract recently y baba y baba on self duality of auslander rings of local serial rings comm algebra 30 6 2002 2583 2592 proved that certain quasi harada rings have a self duality in this paper we also investigate left quasi harada rings and obtain several results including an improvement of the result of y baba and applications to self duality of locally distributive rings particularly we shall obtain several results related to azumaya s conjecture every exact artinian ring has a self duality

the fisher information based uncertainty relation cramer rao inequality and kinetic energy for the d dimensional central problem the inequality with l being the grand orbital quantum number and its conjugate relation for r2 p 2 are shown to be fulfilled in the d dimensional central problem their use has allowed us to improve the fisher information based uncertainty relation i i const and the cramer rao inequalities r2i d2 p2i d2 in addition the kinetic energy and the radial expectation value r2 are shown to be bounded from below by the fisher information in position and momentum spaces denoted by i and i respectively

group foliation and non invariant solutions of the heavenly equation the main physical results of this paper are new exact analytical solutions of the heavenly equation of importance in the general theory of relativity these solutions are not invariant under any subgroup of the symmetry group of the equation the main mathematical result is a new method of obtaining non invariant solutions of partial differential equations with infinite dimensional symmetry groups the method involves the compatibility of the given equations with a differential constraint which is automorphic under a specific symmetry subgroup the latter acting transitively on the submanifold of the common solutions by studying the integrability of the resulting conditions one can provide an explicit foliation of the entire solution manifold of the considered equations

quantization from an exponential distribution of infinitesimal action abstract a statistical model of quantization based on an exponential distribution of infinitesimal action is proposed a trajectory which does not extremize the action along an infinitesimal short segment of path is allowed to occur with a very small probability following an exponential law planck s constant is argued to give the average deviation from the infinitesimal stationary action

variational approach to dequantization we present a dequantization procedure based on a variational approach whereby quantum fluctuations latent in the quantum momentum are suppressed this is done by adding generic local deformations to the quantum momentum operator which give rise to a deformed kinetic term quantifying the amount of fuzziness caused by such fluctuations considered as a functional of such deformations the deformed kinetic term is shown to possess a unique minimum which is seen to be the classical kinetic energy furthermore we show that extremization of the associated deformed action functional introduces an essential nonlinearity to the resulting field equations which are seen to be the classical hamilton jacobi and continuity equations thus a variational procedure determines the particular deformation that has the effect of suppressing the quantum fluctuations resulting in dequantization of the system

almost self duality and harada rings abstract the existence of self duality for left harada rings was investigated by j kado and k oshiro j algebra 211 1999 384 408 recently the author constructed examples of left harada rings without self duality j algebra 241 2001 731 744 in this paper we investigate almost self duality and rings of a certain class which contains right co harada rings equivalently left harada rings here almost self duality is a generalization of self duality the main purpose of the paper is to show that every ring of the class particularly every right co harada ring has an almost self duality

differential invariants and group foliation for the complex monge ampere equation we apply the method of group foliation to the complex monge ampere equation cma 2 with the goal of establishing a regular framework for finding its non invariant solutions we employ the infinite symmetry subgroup of the equation the group of unimodular biholomorphisms to produce a foliation of the solution space into leaves which are orbits of solutions with respect to the symmetry group accordingly cma 2 is split into an automorphic system and a resolvent system which we derive in this paper this is an intricate system and here we make no attempt to solve it in order to obtain non invariant solutions we obtain all differential invariants up to third order for the group of unimodular biholomorphisms and in particular all the basis differential invariants we construct the operators of invariant differentiation from which all higher differential invariants can be obtained consequently we are able to write down all independent partial differential equations with one real unknown and two complex independent variables which keep the same infinite symmetry subgroup as cma 2 we prove explicitly that applying operators of invariant differentiation to third order invariants we obtain all fourth order invariants at this level we have all the information which is necessary and sufficient for group foliation we propose a new approach in the method of group foliation which is based on the commutator algebra of operators of invariant differentiation the resolving equations are obtained by applying this algebra to differential invariants with the status of independent variables furthermore this algebra together with jacobi identities provides the commutator representation of the resolvent system this proves to be the simplest and most natural way of arriving at the resolving equations

an informational characterization of schrodinger s uncertainty relations heisenberg s uncertainty relations employ commutators of observables to set fundamental limits on quantum measurement the information concerning incompatibility non commutativity of observables is well included but that concerning correlation is missing schrodinger s uncertainty relations remedy this defect by supplementing the correlation in terms of anti commutators however both heisenberg s uncertainty relations and schrodinger s uncertainty relations are expressed in terms of variances which are not good measures of uncertainty in general situations e g when mixed states are involved by virtue of the wigner yanase skew information we will establish an uncertainty relation along the spirit of schrodinger from a statistical inference perspective and propose a conjecture the result may be interpreted as a quantification of certain aspect of the celebrated wigner araki yanase theorem for quantum measurement which states that observables not commuting with a conserved quantity cannot be measured exactly

incompleteness of trajectory based interpretations of quantum mechanics trajectory based approaches to quantum mechanics include the de broglie bohm interpretation and nelson s stochastic interpretation it is shown that the usual route to establishing the validity of such interpretations via a decomposition of the schr dinger equation into a continuity equation and a modified hamilton jacobi equation fails for some quantum states a very simple example is provided by a quantum particle in a box described by a wavefunction that is initially uniform over the interior of the box for this example there is no corresponding continuity or modified hamilton jacobi equation and the space time dependence of the wavefunction has a known fractal structure examples with finite average energies are also constructed

an analytic algorithm for decoupling and integrating systems of nonlinear partial differential equations abstract an analytic method is presented which by taking advantage of integrability conditions decides if a system of equations in which the functions and their partial derivatives have only positive integer exponents allows solutions and leads to new equations of lower order the method is especially striking if the equations are overdetermined and can be implemented on a computer an application to the full vacuum field equations of general relativity in the presence of a killing vector leads to the known formulation with potentials

killing vectors in self dual euclidean einstein spaces using the formalism of complex h spaces we show that all real euclidean self dual spaces that admit at least one killing vector may be gauged so that only two distinct types of killing vectors appear in kahler coordinates these are the generators of a translational or a rotational symmetry we give explicit forms both for the killing vectors and for the constraint on the kahler potential function which allows for such a killing vector in the translational case we show how all such spaces are determined by the general solution of the three dimensional flat laplace s equation and how these are related to the multi taub nut metrics of gibbons and hawking in the rotational case we simplify the equation determining but this is not sufficient to obtain the general solution

injectivity of quasi projective modules projectivity of quasi injective modules and projective cover of injective modules abstract in k r fuller on indecomposable injectives over artinian rings pacific j math 29 1969 115 135 theorem 3 1 k r fuller gave necessary and sufficient conditions for projective left modules to be injective over a left artinian ring in y baba and k oshiro on a theorem of fuller preprint we studied this theorem in the present paper first we generalize the theorem by giving necessary and sufficient conditions for quasi projective modules to be injective and ones for quasi injective modules to be projective in k r fuller on indecomposable injectives over artinian rings pacific j math 29 1969 115 135 theorem 2 4 fuller also gave a natural one to one correspondence between the homogeneous components of the k th upper resp lower loewy factor of an injective right r module e and the k th lower resp upper loewy factor of fr whenever rf is the projective cover of the socle of e where f is a primitive idempotent in a right artinian ring r second we shall give a complete correspondence between simple submodules of the 1st upper loewy factor of e and the 1st lower loewy factor of rf

weakly non local fluid mechanics the schrodinger equation a weakly non local extension of ideal fluid dynamics is derived from the second law of thermodynamics it is proved that in the reversible limit the additional pressure term can be derived from a potential the requirement of the additivity of the specific entropy function determines the quantum potential uniquely the relation to other known derivations of the schrodinger equation stochastic fisher information exact uncertainty is clarified

unique additive information measures boltzmann gibbs shannon fisher and beyond abstract it is proved that the only additive and isotropic information measure that can depend on the probability distribution and also on its first derivative is a linear combination of the boltzmann gibbs shannon and fisher information measures power law equilibrium distributions are found as a result of the interaction of the two terms the case of second order derivative dependence is investigated and a corresponding additive information measure is given

on classification of second order pdes possessing partner symmetries recently we have demonstrated how to use partner symmetries for obtaining noninvariant solutions of the heavenly equations of plebanski that govern heavenly gravitational metrics in this paper we present a classification of scalar second order pdes with four variables that possess partner symmetries and contain only second derivatives of the unknown we present a general form of such a pde together with recursion relations between partner symmetries this general pde is transformed to several simplest canonical forms containing two heavenly equations of plebanski among them and two other nonlinear equations which we call the mixed heavenly equation and asymmetric heavenly equation we have calculated all the point and contact symmetries of all the canonical equations which can be used as an input in our recursion relations on an example of the mixed heavenly equation we show how to use partner symmetries for obtaining noninvariant solutions of pdes by a lift from invariant solutions finally we present ricci flat self dual metrics governed by solutions of the mixed heavenly equation and its legendre transform

spreading lengths of hermite polynomials the renyi shannon and fisher spreading lengths of the classical or hypergeometric orthogonal polynomials which are quantifiers of their distribution all over the orthogonality interval are defined and investigated these information theoretic measures of the associated rakhmanov probability density which are direct measures of the polynomial spreading in the sense of having the same units as the variable share interesting properties invariance under translations and reflections linear scaling and vanishing in the limit that the variable tends towards a given definite value the expressions of the renyi and fisher lengths for the hermite polynomials are computed in terms of the polynomial degree the combinatorial multivariable bell polynomials which are shown to characterize the finite power of an arbitrary polynomial play a relevant role for the computation of these information theoretic lengths indeed these polynomials allow us to design an error free computing approach for the entropic moments weighted l q norms of hermite polynomials and subsequently for the renyi and tsallis entropies as well as for the renyi spreading lengths sharp bounds for the shannon length of these polynomials are also given by means of an information theoretic based optimization procedure moreover the existence of a linear correlation between the shannon length as well as the second order renyi length and the standard deviation is computationally proved finally the application to the most popular quantum mechanical prototype system the harmonic oscillator is discussed and some relevant asymptotical open issues related to the entropic moments mentioned previously are posed

fisher information and kinetic energy functionals a dequantization approach we strengthen the connection between information theory and quantum mechanical systems using a recently developed dequantization procedure whereby quantum fluctuations latent in the quantum momentum are suppressed the dequantization procedure results in a decomposition of the quantum kinetic energy as the sum of a classical term and a purely quantum term the purely quantum term which results from the quantum fluctuations is essentially identical to the fisher information the classical term is complementary to the fisher information and in this sense it plays a role analogous to that of the shannon entropy we demonstrate the kinetic energy decomposition for both stationary and nonstationary states and employ it to shed light on the nature of kinetic energy functionals

schrodinger equation from an exact uncertainty principle an exact uncertainty principle formulated as the assumption that a classical ensemble is subject to random momentum fluctuations of a strength which is determined by and scales inversely with uncertainty in position leads from the classical equations of motion to the schrodinger equation

geometry of the schrodinger equation and stochastic mass transportation the schrodinger equation is deduced from a geometric principle lagrange hamilton jacobi and hamilton formalisms are defined almost analogously to the deterministic case which can be identified as geometric optic the form of these formalisms is identical to the deterministic formalisms furthermore it will be shown how the deterministic case is superficial to the stochastic one if planck s constant is very small an elementary proof of heisenberg s uncertainty relation finishes the paper