nonlinear wave interactions in a complex hamiltonian formalism a hamiltonian formalism is used to study the set of equations governing the changes in the amplitudes of some coherent waves due to their nonlinear interactions the main point of the present description is that it uses these complex wave amplitudes as canonical variables for a real hamiltonian thus greatly simplifying the algebra three examples are then dealt with first the hamiltonian for the four wave interactions is derived in accordance with previous studies such systems are then known to be integrable as a second example one looks at a very special case of three interacting waves where a fundamental interacts simultaneously with its second and third harmonic even though this system can be hamiltonian it seems to be nonintegrable finally attention is focused on the elliptically polarised third harmonic generation both in anisotropic and in isotropic media this set of equations can be hamiltonian under certain restrictions on the coupling coefficients but it also appears to be nonintegrable because not enough independent invariants are found

frequency domain methods for the solutions of n order fuzzy differential equations abstract the solutions of the fuzzy differential equations in refs 8 10 12 14 16 have so far been obtained by integration or fuzzy integration in the time domain it is often more convenient particularly in the fuzzy random vibration problems 12 and the fuzzy stochastic dynamic systems 14 to obtain the solutions by integration or fuzzy integration in the frequency domain this is accomplished by the use to generalized fuzzy harmonic analysis 12 14 in the second paper of a series of reports on fuzzy differenrial equations we continue studying the n th order fuzzy differential equation x n t a n 1 t x n 1 t a 0 t x t f t where x n t x n 1 t x 1 t are n th n 1 th 1st same oorder or reverse order derived functions of an unknown fuzzy set valued function x t respectively x t is known fuzzy set valued function a i t i 0 1 n 1 are deterministic functions of parameter t the solving processes of frequency domain for n th order fuzzy differential equations are put forward one example is considered in order to demonstrate the rationality and validity of the methods the work provides an indispensable mathematical tool for setting up the theories of fuzzy stochastic differential equations 8 fuzzy dynamical systems 2 fuzzy random vibration 12 fuzzy stochastic dynamical systems 14 18 20 and fuzzy stochastic systems 21 23

koszul duality of affine kac moody algebras and cyclotomic rational double affine hecke algebras abstract we give a proof of the parabolic singular koszul duality for the category o of affine kac moody algebras the main new tool is a relation between moment graphs and finite codimensional affine schubert varieties we apply this duality to q schur algebras and to cyclotomic rational double affine hecke algebras this yields a proof of a conjecture of chuang miyachi relating the level rank duality with the ringel koszul duality of cyclotomic rational double affine hecke algebras

proof of integrability for five wave interactions in a case with unequal coupling constants complete integrability is proved for a system of ode describing the double three wave interaction between five waves when the coupling constant in one triplet is twice as large as in the other known results from the painleve analysis for this case are combined with the yoshida kovalevskaya approach 1983 to gain insight about the degree of the missing first integral this integral of degree six is then obtained via a direct search based on irreducible forms elementary building blocks for polynomial first integrals in involution with the manley rowe invariants

the theory of fuzzy stochastic processes abstract in this paper we deal with the general theory of fuzzy stochastic processes we give the suitable definitions of fuzzy random function fuzzy stochastic process and their fall shadow distribution functions the concepts of separability and measurability of a fuzzy random function are introduced and some properties of these new concepts are derived we investigate the continuity of fuzzy sample function of a fuzzy stochastic process

complex mathcal pt symmetric extensions of the nonlinear ultra short light pulse model the short pulse equation is symmetric which arises in nonlinear optics for the ultra short pulse case we present a family of new complex symmetric extensions of the short pulse equation based on the complex symmetric extension principle some properties of these equations with some chosen parameters are studied including the hamiltonian structures and exact solutions such as solitary wave solutions doubly periodic wave solutions and compacton solutions our results may be useful to understand complex symmetric nonlinear physical models this article is part of a special issue of journal of physics a mathematical and theoretical devoted to quantum physics with non hermitian operators

symmetric versus hermitian formulations of quantum mechanics a non hermitian hamiltonian that has an unbroken symmetry can be converted by means of a similarity transformation to a physically equivalent hermitian hamiltonian this raises the following question in which form of the quantum theory the non hermitian or the hermitian one is it easier to perform calculations this paper compares both forms of a non hermitian ix3 quantum mechanical hamiltonian and demonstrates that it is much harder to perform calculations in the hermitian theory because the perturbation series for the hermitian hamiltonian is constructed from divergent feynman graphs for the hermitian version of the theory dimensional continuation is used to regulate the divergent graphs that contribute to the ground state energy and the one point green s function the results that are obtained are identical to those found much more simply and without divergences in the non hermitian symmetric hamiltonian the contribution to the ground state energy of the hermitian version of the theory involves graphs with overlapping divergences and these graphs are extremely difficult to regulate in contrast the graphs for the non hermitian version of the theory are finite to all orders and they are very easy to evaluate

quasi exactly solvable quartic potential a new two parameter family of quasi exactly solvable quartic polynomial potentials is introduced heretofore it was believed that the lowest degree one dimensional quasi exactly solvable polynomial potential is sextic this belief is based on the assumption that the hamiltonian must be hermitian however it has recently been discovered that there are huge classes of non hermitian symmetric hamiltonians whose spectra are real discrete and bounded below replacing hermiticity by the weaker condition of symmetry allows for new kinds of quasi exactly solvable theories the spectra of this family of quartic potentials discussed here are also real discrete and bounded below and the quasi exact portion of the spectra consists of the lowest j eigenvalues these eigenvalues are the roots of a jth degree polynomial

integrability of restricted multiple three wave interactions using a hamiltonian framework with complex canonical variables allows for the determination of irreducible forms which serve as building blocks for polynomial invariants all the independent invariants in involution are thus obtained for the restricted multiple three wave interactions where all triads are coupled through a common pump or daughter wave in the case of equal coupling strengths in all triads the mixed common pump daughter wave case is not integrable

a new pt symmetric complex hamiltonian with a real spectrum we construct an isospectrum system in terms of a real and a complex potential to show that the underlying pt symmetric complex hamiltonian possesses a real spectrum which is shared by its real partner

complex square well a new exactly solvable quantum mechanical model recently a class ofpt invariant quantum mechanical models described by the non hermitian hamiltonianhdp2 x2 ix was studied it was found that the energy levels for this theory are real for all 0 here the limit as 1is examined it is shown that in this limit the theory becomes exactly solvable a generalization of this hamiltonian h d p 2 x 2m ix md 1 2 3 is also studied and thispt symmetric hamiltonian becomes exactly solvable in the large limit as well in effect what is obtained in each case is a complex analogue of the hamiltonian for the square well potential expansions about the large limit are obtained

new solvable singular potentials we obtain three new solvable real shape invariant potentials starting from the harmonic oscillator poschl teller i and poschl teller ii potentials on the half axis and extending their domain to the full line while taking special care to regularize the inverse square singularity at the origin the regularization procedure gives rise to a delta function behaviour at the origin our new systems possess underlying nonlinear potential algebras which can also be used to determine their spectra analytically

the general theory for response analysis of fuzzy stochastic dynamical systems abstract most real life structural mechanical systems have complex geometrical and material properties and operate under complex fuzzy environmental conditions these systems are certainly subjected to fuzzy random excitations induced by the environment for an analytical treatment of such a system subjected to fuzzy random excitations it becomes necessary to establish the general theory of fuzzy stochastic dynamical systems in the first part of this paper on the dynamic response of fuzzy stochastic systems we construct the mathematical models of fuzzy random algebraic equation dynamic systems nth order fuzzy random differential equation dynamic systems and a system of first order fuzzy random differential equations the theory of the response fuzzy mean response and fuzzy covariance response of fuzzy random algebraic equation dynamic systems in time domain and frequency domain is brought forward in the second part the theory of the response fuzzy mean response and fuzzy covariance response of the nth order fuzzy random differential equation dynamic systems in time domain and frequency domain is put forward the theory of the response analysis of a system to both stationary and nonstationary fuzzy random excitations is established the method of the response analysis of a system to nonstationary fuzzy random excitation with evolutionary spectra is brought forward and two examples are considered in the third part the theory of the response fuzzy mean response and fuzzy covariance response of a system of the first order fuzzy random differential equations with constant coefficients and variable coefficients in time domain and frequency domain is put forward an example is given to show the rationality and validity of the theory

structure of some categories of representations of infinite dimensional lie algebras the highest weight representations of a finite dimensional complex semisimple lie algebra 9 have been studied extensively and a number of interesting results have been obtained in the last few years an extensive bibliography is given in 7 in 2 3 a category cy of representations of g was introduced which turns out to be the proper setting for considering various questions regarding highest weight representations the aim of this paper is to extend the study of this notion to a kac moody algebra or more generally to a contragredient lie algebra g a corresponding to a square matrix a over c the study of such algebras was begun independently in 8 14 and subsequently a number of interesting results were proved these results show that many important properties from the finite dimensional setup i e from the context of 9 can be proved in the general setup involving g a in particular the notion of category was extended in 9 to the case of kac moody algebras and used in the proof of combinatorial identities

double three wave interaction of four waves lax representations and exact solutions the nonlinear resonant interaction of coherent waves is a fundamental process in the study of wave phenomena which has received a great deal of attention in its many aspects in the present article a system of four interacting waves which constitute two resonant triplets is considered the system is described in a simplified model by a hamiltonian system of eight autonomous ordinary differential equations with time as the independent variable both positive and negative energy waves are allowed in the interaction two distinct lax representations for this system two new classes of exact solutions in terms of elliptic functions a solution in the form of a convergent generic laurent series expansion around a movable pole in the independent variable and a stabilization criterion for the explosive instability that may occur when waves of different energy sign interact are obtained herein

integrability of restricted multiple three wave interactions ii coupling constants with ratios 1 and 2 the system of ordinary differential equations describing the multiple three wave interactions with a common pump daughter wave is proved to be completely integrable by obtaining the necessary 2n 1 independent first integrals in involution for the case when the coupling constants and the frequency mismatches have ratio 1 and or 2 this case was deemed integrable on the basis of a painleve analysis but a direct proof has been lacking for some time the first integrals are the n 1 quadratic manley rowe relations the cubic hamiltonian n 1 quartic integrals analogous to the ones needed for complete integrability in the case of equal coupling constants and detunings in all wave triads and a new sixth order integral involving all wave quantities the form of this last invariant was deduced from the recent result for the analogous interaction between five waves n 2 and essentially made possible by the proper use of irreducible forms elementary building blocks for polynomial first integrals in involution with the manley rowe invariants

exact solutions of the schrodinger equation for nonseparable anharmonic oscillator potentials in two dimensions an explicit procedure is given to construct exact closed form solutions to the time independent schrodinger equation in two dimensions 2 w x y 0 where w x y is a polynomial potential of degree greater than two not separable in cartesian coordinates several examples are discussed for which w x y is a sextic polynomial as has already been seen in studies of the corresponding one dimensional problem a complete set of eigenvalues and wave functions is not found however these closed form solutions can be used to check the accuracy and efficiency of numerical algorithms

non hermitian matrix description of the calpcalt symmetric anharmonic oscillators the symmetric differential schrodinger equation h e with the operator h h x p2 ax4 i x3 cx2 i x h x on l2 is studied at a 0 it is rearranged as a linear algebraic diagonalization with rigorous proof our non variational construction of bound states offers an infinite dimensional analogue to the recent finite dimensional quasi exact solution available at the less common a 0

generalized pt symmetry and real spectra the fact that eigenvalues of pt symmetric hamiltonians h can be real for some values of ap arameter and complex for others is explained by showing that the matrix elements ofh a ndhence the secular equation are real not only for pt but also for an ya ntiunitary operator a satisfying a 2k 1w ith k odd the argument is illustrated by a 2 2m atrix hamiltonian and two examples of the generalization are given

pt symmetric quantum mechanics this paper proposes to broaden the canonical formulation of quantum mechanics ordinarily one imposes the condition h h on the hamiltonian where represents the mathematical operation of complex conjugation and matrix transposition this conventional hermiticity condition is sufficient to ensure that the hamiltonian h has a real spectrum however replacing this mathematical condition by the weaker and more physical requirement h h where represents combined parity reflection and time reversal pt one obtains new classes of complex hamiltonians whose spectra are still real and positive this generalization of hermiticity is investigated using a complex deformation h p2 x2 ix e of the harmonic oscillator hamiltonian where e is a real parameter the system exhibits two phases when e 0 the energy spectrum of h is real and positive as a consequence of pt symmetry however when 1 e 0 the spectrum contains an infinite number of complex eigenvalues and a finite number of real positive eigenvalues b

closed form solutions of the schrodinger equation for a coupled harmonic potential in three dimensions using the ansatz method we obtain the exact closed form solutions of a time independent schrodinger equation for a coupled harmonic potential and its variant in three dimensions some authors r s kaushal quantum mechanics of noncentral harmonic and anharmonic potentials in two dimensions ann phys n y 206 1991 90 105 have raised the difficulty to solve the schrodinger equation for coupled potentials but we found that by imposing some restrictions on ansatz parameters and at the cost of certain constraints on the potential parameters ground state as well as excited state solutions of the schrodinger equation are obtained the number of constraints increases with increase in anharmonicity of the potential

generalized grassmannian coherent states for pseudo hermitian n level systems the purpose of this paper is to generalize fermionic coherent states for two level systems described by pseudo hermitian hamiltonian cherbal et al 2007 j phys a math theor 40 1835 to n level systems central to this task is the expression of the coherent states in terms of generalized grassmann variables these kinds of grassmann coherent states satisfy the bi overcompleteness condition instead of the overcompleteness one as it is reasonably expected because of the biorthonormality of the system choosing an appropriate grassmann weight function the resolution of identity is examined moreover grassmannian coherent and squeezed states of the deformed group suq 2 for the three level pseudo hermitian system are presented

exactly solvable models with symmetry and with an asymmetric coupling of channels bound states generated by the k coupled symmetric square wells are studied in a series of models where the hamiltonians are assumed pseudo hermitian and symmetric specific rotation like generalized parities are considered such that at some integers n we show how our assumptions make the models exactly solvable and quasi hermitian this means that they possess the real spectra as well as the standard probabilistic interpretation

comments on frequency time domain methods for solutions of n order fuzzy differential equations by zhang wang and liu abstract we argue that the method of solving fuzzy differential equations in 1 and 2 is not useful since the solution may exist for only an arbitrarily short time

the semi infinite intersection cohomology sheaf abstract we introduce the semi infinite category of sheaves on the affine grassmannian and construct a particular object in it which we call the semi infinite intersection cohomology sheaf we relate it to several other entities naturally appearing in the geometric langlands theory

fuzzy initial value problem for n th order linear differential equations we present two methods of solving nth order linear differential equation that have fuzzy initial conditions for the second order constant coefficient case we show that 1 the first method always gives a solution and 2 the second solution quite often does not exist and when it does exist it may be equal to the solution determined using the first method we apply the first method to three example problems including an electrical circuit and a vibrating mass

quantum mechanics of complex hamiltonian systems in one dimension with a view to obtaining further insight into the nature of eigenvalues and eigenfunctions of a stationary state one dimensional schrodinger equation corresponding to a non hermitian hamiltonian h x p we investigate the ground state solutions for a variety of potentials within the framework of an extended complex phase space characterized by x x1 ip2 p p1 ix2 where x1 p1 and x2 p2 are real and considered as canonical pairs the analyticity property of the eigenfunction alone is found sufficient to throw light on the nature of eigenvalues and eigenfunctions for different systems it is noted that the imaginary part of the eigenvalue ei turns out to be zero for all potentials v x with real couplings whereas it turns out to be nonzero for the case when the couplings are complex the prescription is also extended to study the excited states the problems related to the normalization of the eigenfunction and the boundary conditions to be used within this framework are also discussed

localization of g modules on the affine grassmannian we consider the category of modules over the affine kac moody algebra of critical level with regular central character in our previous paper we conjectured that this category is equivalent to the category of hecke eigen d modules on the affine grassmannian g t g t this conjecture was motivated by our proposal for a local geometric langlands correspondence in this paper we prove this conjecture for the corresponding i 0 equivariant categories where i 0 is the radical of the iwahori subgroup of g t our result may be viewed as an affine analogue of the equivalence of categories of g modules and d modules on the flag variety g b due to beilinson bernstein and brylinski kashiwara

non hermitian localization and population biology the time evolution of spatial fluctuations in inhomogeneous d dimensional biological systems is analyzed a single species continuous growth model in which the population disperses via diffusion and convection is considered time independent environmental heterogeneities such as a random distribution of nutrients or sunlight are modeled by quenched disorder in the growth rate linearization of this model of population dynamics shows that the fastest growing localized state dominates in a time proportional to a power of the logarithm of the system size using an analogy with a schr odinger equation subject to a constant imaginary vector potential we propose a delocalization transition for the steady state of the nonlinear problem at a critical convection threshold separating localized and extended states in the limit of high convection velocity the linearized growth problem in d dimensions exhibits singular scaling behavior described by a d ensuremath 1 dimensional generalization of the noisy burgers equation with universal singularities in the density of states associated with disorder averaged eigenvalues near the band edge in the complex plane the burgers mapping leads to unusual transverse spreading of convecting delocalized populations

preservation of depth in the local geometric langlands correspondence it is expected that under mild conditions the local langlands correspondence preserves depths of representations in this article we formulate a conjectural geometrisation of this expectation we prove half of this conjecture by showing that the depth of a categorical representation of the loop group is greater than or equal to the depth of its underlying geometric langlands parameter a key ingredient of our proof is a new definition of the slope of a meromorphic connection a definition which uses opers

wakimoto modules opers and the center at the critical level abstract wakimoto modules are representations of affine kac moody algebras in fock modules over infinite dimensional heisenberg algebras in this paper we present the construction of the wakimoto modules from the point of view of the vertex algebra theory we then use wakimoto modules to identify the center of the completed universal enveloping algebra of an affine kac moody algebra at the critical level with the algebra of functions on the space of opers for the langlands dual group on the punctured disc giving another proof of the theorem of b feigin and the author

geometric langlands duality and representations of algebraic groups over commutative rings as such it can be viewed as a first step in the geometric langlands program the connected complex reductive groups have a combinatorial classification by their root data in the root datum the roots and the co roots appear in a symmetric manner and so the connected reductive algebraic groups come in pairs if g is a reductive group we write g for its companion and call it the dual group g the notion of the dual group itself does not appear in satake s paper but was introduced by langlands together with its various elaborations in li l2 and is a cornerstone of the langlands program it also appeared later in physics mo gno in this paper we discuss the basic relationship between g and g we begin with a reductive g and consider the affine grassmannian qx the grassmannian for the loop group of g for technical reasons we work with formal algebraic loops the affine grassmannian is an infinite dimen sional complex space we consider a certain category of sheaves the spherical perverse sheaves on r these sheaves can be multiplied using a convolution product and this leads to a rather explicit construction of a hopf algebra by what has come to be known as tannakian formalism the resulting hopf algebra turns out to be the ring of functions on g in this interpretation the spherical perverse sheaves on the affine grassman nian correspond to finite dimensional complex representations of g thus instead of defining g in terms of the classification of reductive groups we pro vide a canonical construction of g starting from g we can carry out our construction over the integers the spherical perverse sheaves are then those with integral coefficients but the grassmannian remains a complex algebraic object

some spectral equivalences between schrodinger operators spectral equivalences of the quasi exactly solvable sectors of two classes of schrodinger operators are established using gaudin type bethe ansatz equations in some instances the results can be extended leading to full isospectrality in this manner we obtain equivalences between pt symmetric problems and hermitian problems we also find equivalences between some classes of hermitian operators

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

granular differentiability of fuzzy number valued functions in this paper using the concept of horizontal membership functions a new definition of fuzzy derivative called granular derivative is proposed based on granular difference moreover a new definition of fuzzy integral called granular integral is defined and its relation with the granular derivative is given a new definition of a metric granular metric on the space of type 1 fuzzy numbers and a concept of continuous fuzzy functions are also presented restrictions associated to previous approaches hukuhara differentiability strongly generalized hukuhara differentiability generalized hukuhara differentiability generalized differentiability zadeh s extension principle and fuzzy differential inclusions dealing with fuzzy differential equations fdes are expressed it is shown that the proposed approach does not have the drawbacks of the previous approaches it is also demonstrated how this approach enables researchers to solve fdes more conveniently than ever before moreover we showed that this approach does not necessitate that the diameter of the fuzzy function be monotonic it is also proved that the result of each of the four basic operations on fuzzy numbers introduced based on the proposed approach leads to a fuzzy number moreover the condition for the existence of the granular derivative of a fuzzy function is provided by a theorem additionally by two examples it is shown that the existence of the granular derivative of a fuzzy function does not imply the existence of the generalized hukuhara differentiability of the fuzzy function and vice versa the terms doubling property and unnatural behavior in modeling phenomenon are also introduced furthermore using some examples the paper proceeds to elaborate on the efficiency and effectiveness of the proposed approach moreover as an application of the proposed approach the response of boeing 747 to impulsive elevator input is obtained in the presence of uncertain initial conditions and parameters

the solution of the schrodinger equation for complex hamiltonian systems in two dimensions we investigate the ground state solutions of the schrodinger equation for complex non hermitian hamiltonian systems in two dimensions within the framework of an extended complex phase space approach the eigenvalues and eigenfunctions of some two dimensional complex potentials are found

feigin frenkel center in types b c and d for each simple lie algebra mathfrak g consider the corresponding affine vertex algebra v mathrm crit mathfrak g at the critical level the center of this vertex algebra is a commutative associative algebra whose structure was described by a remarkable theorem of feigin and frenkel about two decades ago however only recently simple formulas for the generators of the center were found for the lie algebras of type a following talalaev s discovery of explicit higher gaudin hamiltonians we give explicit formulas for generators of the centers of the affine vertex algebras v mathrm crit mathfrak g associated with the simple lie algebras mathfrak g of types b c and d the construction relies on the schur weyl duality involving the brauer algebra and the generators are expressed as weighted traces over tensor spaces and equivalently as traces over the spaces of singular vectors for the action of the lie algebra mathfrak sl 2 in the context of howe duality this leads to explicit constructions of commutative subalgebras of the universal enveloping algebras mathrm u mathfrak g t and mathrm u mathfrak g and to higher order hamiltonians in the gaudin model associated with each lie algebra mathfrak g we also introduce analogues of the bethe subalgebras of the yangians mathrm y mathfrak g and show that their graded images coincide with the respective commutative subalgebras of mathrm u mathfrak g t

some properties of eigenvalues and eigenfunctions of the cubic oscillator with imaginary coupling constant comparison between the exact value of the spectral zeta function zh 1 5 6 5 3 2cos 5 2 1 5 3 5 and the results of numeric and wkb calculations supports the conjecture by daniel bessis 1995 private communication that all the eigenvalues of this pt invariant hamiltonian are real for one dimensional schrodinger operators with complex potentials having a monotonic imaginary part the eigenfunctions and the imaginary parts of their logarithmic derivatives have no real zeros

summation of power series by continued exponentials it is proposed that a power series may be summed analytically continued outside its radius of convergence by converting it to a continued exponential which is a structure of the form a0 exp a1z exp a2zexp a3z exp a4z the continued exponential coefficients ai for a given function f z are determined by equating the taylor coefficients of the continued exponential with those of f z the coefficients ai have a combinatoric interpretation the nth taylor coefficient enumerates all n 1 vertex tree graphs whose vertex amplitudes are ai continued exponentials have remarkable convergence properties when a power series has a nonzero radius of convergence the corresponding continued exponential often converges in a heart shaped region whose cusp is determined by the nearest zero or singularity of the function being approximated the convergence region contains and is much larger than the circle of convergence of the power series outside the complex plane is divided up into an elabor

non hermitian harmonic oscillator with discrete complex or real spectrum for non unitary squeeze operators in the present paper we enlarge the method of debergh beckers and szafraniec for non hermitian hamiltonians with complex parameters and obtain real eigenvalues for the case of non unitary squeeze operators with two complex parameters of non hermitian harmonic oscillator we obtain discrete complex spectrum and for special values of the complex parameters the spectrum is discrete real and positive even though the corresponding operators are not pt symmetric

new quasi exactly solvable sextic polynomial potentials a hamiltonian is said to be quasi exactly solvable qes if some of the energy levels and the corresponding eigenfunctions can be calculated exactly and in a closed form an entirely new class of qes hamiltonians having sextic polynomial potentials is constructed these new hamiltonians are different from the sextic qes hamiltonians in the literature because their eigenfunctions obey symmetry rather than hermitian boundary conditions these new hamiltonians present a novel problem that is not encountered when the hamiltonian is hermitian it is necessary to distinguish between the parametric region of unbroken symmetry in which all of the eigenvalues are real and the region of broken symmetry in which some of the eigenvalues are complex the precise location of the boundary between these two regions is determined numerically using the extrapolation techniques and analytically using the wkb analysis

solving processes for a system of first order fuzzy differential equations abstract in this third paper of a series of reports on fuzzy differential equations we consider the system of first order inhomogeneous fuzzy differential equations d d t x t a t x t f t 1 with x 0 x 0 where d x t t is an n dimensional vector of first same order or reverse order derived functions of an n dimensional vector x t x 1 t x n t t of unknown fuzzy set valued functions that is d dt x t d dt x 1 t d dt x n t t f t is an n dimensional vector f 1 t f n t t of known fuzzy set valued functions a t is an n n matrix of known real functions we introduce the time domain and frequency domain methods for the solutions of the system of first order fuzzy differential equations 1 the solving processes of time domain and frequency domain for the system of first order fuzzy differential equations with constant coefficients and variable coefficients are put forward one example is considered in order to demonstrate the rationality and validity of the methods the work provides an indispensable mathematical tool for setting up the theories of fuzzy stochastic differential equations 7 fuzzy dynamical systems 3 fuzzy random vibration 8 fuzzy stochastic dynamical systems 10 11 14 16 and fuzzy stochastic systems 17 19

non hermitian susy hydrogen like hamiltonians with real spectra it is shown that the radial part of the hydrogen hamiltonian factorizes as the product of two not mutually adjoint first order differential operators plus a complex constant t he 1 susy approach is used to construct non hermitian operators with hydrogen spectra other non hermitian hamiltonians are shown to admit an extra complex energy at n ew self adjoint hydrogen like hamiltonians are also derived by using a 2 susy transformation with complex conjugate pairs

another realization of the category of modules over the small quantum group abstract let g be a semi simple simply connected lie algebra and let ul be the corresponding quantum group with divided powers where l is an even order root of unity let in addition ul ul be the corresponding small quantum group in this paper we establish the following relation between the categories of representations of ul and ul we show that the category of ul modules is naturally equivalent to the category of ul modules which have a hecke eigen property with respect to representations lifted by means of the quantum frobenius map u l u g where g is the langlands dual lie algebra this description allows to express the regular linkage class in the category ul mod in terms of perverse sheaves on the affine flag variety with a hecke eigen property moreover it can serve as a basis to the program to understand the connection between the category ul mod and the category of representations of the corresponding affine algebra at the critical level

non hermitian hamiltonian versus e 0 localized states we analyze the zero energy solutions of a two dimensional system which undergoes a non radial symmetric complex potential v r by virtue of the coherent states concept the localized states are constructed and the consequences of the imaginary part of the potential are found both analytically and schematically

d modules on the affine grassmannian and representations of affine kac moody algebras let mathfrak g be a simple lie algebra for a level thought of as a symmetric mathfrak g invariant form of mathfrak g let hat mathfrak g kappa be the corresponding affine kac moody algebra let grg be the affine grassmannian of mathfrak g and let d grg mod be the category of twisted right d modules on grg by taking global sections of a d module we obtain a functor d grg mod mathfrak g mod it is known that this functor is exact and faithful when is less than critical or irrational in this paper we show that the functor is also exact and faithful when is the critical level

eigenspectra of a complex coupled harmonic potential in three dimensions within the framework of extended complex phase space approach characterized by position and momentum coordinates we investigate the quasi exact solutions of the schrodinger equation for a coupled harmonic potential and its variants in three dimensions for this purpose ansatz method is employed and nature of the eigenvalues and eigenfunctions is determined by the analyticity property of the eigenfunctions alone the energy eigenvalue is real for the real coupling parameters and becomes complex if the coupling parameters are complex however in case of complex coupling parameters the imaginary component of energy eigenvalue reduces to zero if the p t symmetric condition is satisfied thus a non hermitian hamiltonian possesses real eigenvalue if it is p t symmetric

symmetric quantum theory defined in a krein space we provide a mathematical framework for symmetric quantum theory which is applicable irrespective of whether a system is defined on or a complex contour whether symmetry is unbroken and so on the linear space in which symmetric quantum theory is naturally defined is a krein space constructed by introducing an indefinite metric into a hilbert space composed of square integrable complex functions in a complex contour we show that in this krein space every symmetric operator is hermitian if and only if it has transposition symmetry as well from which the characteristic properties of the symmetric hamiltonians found in the literature follow some possible ways to construct physical theories are discussed within the restriction to the class k h