exact propagator for a two dimensional inverse quadratic oscillator interacting with a wedge the exact propagator of this dynamic system is derived for a rational wedge for an irrational wedge the proposed propagator can be confirmed by expanding it in terms of eigenfunctions and eigenvalues which agree with those obtained from the corresponding schrodinger equation the results are also valid for an inverse square potential interacting with a wedge finally the authors investigate the classical path s contributions to the propagator for the free particle and the rational wedge case

an efficient method for simulation of noisy coupled multi dimensional oscillators we present an efficient computational method for the study of populations of noisy coupled oscillators by taking a population density approach in which the probability density of observing an oscillator at a point of state space is the primary variable instead of the states of all of the oscillators we are able to seamlessly account for intrinsic noise within the oscillators and global coupling within the population the population is assumed to consist of a large number of oscillators so that the noise process is well sampled over the population our numerical method is able to solve the governing equation even in the challenging case of limit cycle oscillators with a large number of state variables instead of simulating a prohibitive number of oscillators our particle method simulates relatively few particles allowing for the efficient solution of the governing equation

the study of existence of equilibria for generalized games without lower semicontinuity in locally topological vector spaces the aim of this paper is to establish general existence results of maximal elements for l majorized mappings which are in turn used to establish the general existence theorems of equilibria for generalized games resp abstract economics without lower semicontinuity for both constraint and preference mappings correspondences and in which strategic resp commodity spaces may not be compact the set of players resp agents may not be countable and underlying spaces are either finite or infinite dimensional locally topological vector spaces our results unify or improve corresponding results in the literature

on equilibrium problem of abstract economy generalized quasi variational inequality and an optimization problem in locally h convex spaces in this paper the equilibrium existence problem for abstract economies the existence problem for solution of generalized quasi variational inequality and an optimization problem in locally h convex spaces are researched

remarks on generalized quasi equilibrium problems our main aim in this paper is to obtain fundamental existence theorems of quasi equilibrium problems and generalized quasi equilibrium problems of multivalued functions we use a continuous selection theorem to show that the existence theorems of generalized quasi equilibrium problems are simple consequences of the corresponding quasi equilibrium problems we use a generalization of himmelberg s theorem due to the third author for not necessary locally convex topological vector spaces to prove the existence theorems of quasi equilibrium problems for multivalued functions

a variational like inequality for multifunctions with applications abstract we study a variational like inequality problem for a multifunction a certain relevant complementarity problem is defined we establish some existence theorems for the above two problems an application to convex mathematical programming is also shown

on the schrodinger equation for the interaction x2 λx2 1 gx2 the method of obtaining the explicit expressions of all the even odd parity exact solutions of the schrodinger equation for the interaction x2 lambda x2 1 gx2 is discussed when the couplings lambda and g satisfy some specific relations in the general case a simple equation for approximating the energy eigenvalues has been developed

existence of equilibrium for generalized abstract fuzzy economies in this paper we introduce and study a new class of generalized abstract fuzzy economies we prove some new existence theorems of equilibrium and maximal element for generalized abstract fuzzy economies with uncountable number of agents and qualitative fuzzy games respectively

approachability and fixed points for non convex set valued maps this paper studies the approachability of upper semicontinuous nonconvex set valued maps by single valued continuous functions and its applications to fixed point theory the use of approximate selections was initiated by j von neumann 41 in the proof of his well known minimax principle this remarkable property was extended to the normed space setting by cellina s and cellina and lasota 9 in the context of a degree theory for upper semicontinuous convex compact valued maps it also holds for maps with compact contractible values defined on finite polyhedra of iw ma cole11 40 and more generally defined on compact anrs mclennan 39 it was recently taken up by gbrniewicz granas and kryszewski 22 241 in the context of an index theory for non convex maps defined on compact anrs the content of this paper is divided into three parts the first part is devoted to background material concerning the types of spaces and set valued maps studied herein in the second part we define the abstract class d of approachable set valued maps that is the class of maps a x y

population density approach for discrete mrna distributions in generalized switching models for stochastic gene expression we present a generalization of a population density approach for modeling and analysis of stochastic gene expression in the model the gene of interest fluctuates stochastically between an inactive state in which transcription cannot occur and an active state in which discrete transcription events occur and the individual mrna molecules are degraded stochastically in an independent manner this sort of model in simplest form with exponential dwell times has been used to explain experimental estimates of the discrete distribution of random mrna copy number in our generalization the random dwell times in the inactive and active states t 0 and t 1 respectively are independent random variables drawn from any specified distributions consequently the probability per unit time of switching out of a state depends on the time since entering that state our method exploits a connection between the fully discrete random process and a related continuous process we present numerical methods for computing steady state mrna distributions and an analytical derivation of the mrna autocovariance function we find that empirical estimates of the steady state mrna probability mass function from monte carlo simulations of laboratory data do not allow one to distinguish between underlying models with exponential and nonexponential dwell times in some relevant parameter regimes however in these parameter regimes and where the autocovariance function has negative lobes the autocovariance function disambiguates the two types of models our results strongly suggest that temporal data beyond the autocovariance function is required in general to characterize gene switching

indecomposable generalisations of the standard angular momentum representations of so 3 indecomposable representations of the lie algebra of the rotation group so 3 are studied on the heisenberg weyl basis which is the basis for a subspace of a quotient space of the space of the universal enveloping algebra of the heisenberg weyl algebra the author obtains an infinite dimensional indecomposable representation which induces the finite dimensional representation on certain quotient spaces and subduces the infinite dimensional indecomposable representations on invariant subspaces the standard angular momentum representations of so 3 are given as special cases

an intersection theorem formultivalued maps and applications a new intersection theorem for multivalued maps is obtained this new theorem requires the maps involved to satisfy a weaker compactness condition and generalizes known results applications of this new theorem are given to the existence of maximal and greatest elements for strict and weak relations and to minimax inequalities

classification of oscillators in the heisenberg matrix representation for an arbitrary finite and not too singular presumably phenomenological superposition of potentials rdelta with rational exponents we solve the old problem of conversion of the corresponding differential schrodinger bound state problem into its matrix equivalent with the minimalized number l of non zero diagonals the construction with proofs is performed via the standard power series method in its non hermitean matrix sometimes called hill determinant version all the simplest and in this sense algebraizable interactions are then displayed up to l 5 an example of application in perturbation theory is included

on equilibria of non compact generalized games abstract a fixed point theorem is first proved from which a theorem on the existence of maximal elements for lc majorized correspondences defined on a non compact set is obtained next an equilibrium theorem for non compact qualitative games with lc majorized correspondences defined on a non compact strategy set is proved and is applied to prove an equilibrium theorem for non compact generalized games with lc majorized correspondences defined on a non compact strategy set our results generalize several previously published corresponding results

low temperature behaviour of the six state clock model with competing interactions the phase diagram of the six state clock model on a three dimensional lattice with first and second neighbour competing interactions along one direction is analysed using a systematic low temperature expansion carried to all orders where necessary a transfer matrix method simplifies the configurational analysis we find that an infinite number of commensurate phases are stable near a zero temperature multiphase point a richer hierarchy of mixed phases is found to be stable down to zero temperature than in the annni model the system corresponds to an xy model with infinite hexagonal anisotropy and may be relevant to rare earth multilayers

quasi variational inequalities without the concavity assumption this paper generalizes a foundational quasi variationalinequality by relaxing the 0 diagonal concavity condition the approach adopted in this paper is based on continuous selection type arguments and hence it is quite different from the approach used in the literature thus it enables us to prose the existence of equilibrium of the constrained noncooperative games without assuming the quasi convexity of loss functions

generalized abstract economy and systems of generalized vector quasi equilibrium problems the present paper is in two fold the first fold is devoted to the existence theory of equilibria for generalized abstract economy with a lower semicontinuous constraint correspondence and a fuzzy constraint correspondence defined on a noncompact nonparacompact strategy set in the second fold we consider systems of generalized vector quasi equilibrium problems for multivalued maps for short sgvqeps which contain systems of vector quasi equilibrium problems systems of generalized mixed vector quasi variational inequalities and debreu type equilibrium problems for vector valued functions as special cases by using the results of first fold we establish some existence results for solutions of sgvqeps

spectral solutions to stochastic models of gene expression with bursts and regulation signal processing molecules inside cells are often present at low copy number which necessitates probabilistic models to account for intrinsic noise probability distributions have traditionally been found using simulation based approaches which then require estimating the distributions from many samples here we present in detail an alternative method for directly calculating a probability distribution by expanding in the natural eigenfunctions of the governing equation which is linear we apply the resulting spectral method to three general models of stochastic gene expression a single gene with multiple expression states often used as a model of bursting in the limit of two states a gene regulatory cascade and a combined model of bursting and regulation in all cases we find either analytic results or numerical prescriptions that greatly outperform simulations in efficiency and accuracy in the last case we show that bimodal response in the limit of slow switching is not only possible but optimal in terms of information transmission

the exact eigenfunctions and eigenvalues of a particle in a box obtained using gaussian wavepacket dynamics exact eigenfunctions for a particle in a box are obtained using gaussian wavepacket dynamics the eigenfunctions are obtained by propagating without approximation an infinite set of gaussian wavepackets that collectively satisfy the boundary conditions of the problem being coherent states appropriate to this problem the method of images is applied to enforce these boundary conditions this technique may be applied to the quantum billiard problem whenever the particle is confined to any open or closed region that tessellates space regardless of the dimension of the region also it is shown that the use of frozen gaussians along with the de leon heller spectral quantisation method gives the exact solution for the one dimensional problem as well as for the above multi dimensional problems provided the components of the momentum of the wavepackets are chosen at random

on the reduction of n fold tensor products in su 2 a map carrying irreducible representations of su 2 into n fold tensor product spaces of su 2 is constructed it is shown that the multiplicity of an su 2 representation in the n fold tensor product space is given by certain gelfand patterns

on the hill determinant method the conditions of applicability of the hill determinant method to the potentials v r v1r v2r2 and v x x2 lambda x2 1 gx2 are investigated it is shown that the procedure leads to the actual eigenvalues in the former case provided the ansatz is properly chosen but it completely fails to yield the right answer in the latter one

a selection theorem for quasi lower semicontinuous mappings in hyperconvex spaces we prove a continuous selection theorem for quasi lower semicontinuous mappings with values that are closed sub admissible subsets of a hyperconvex metric space and apply this result to obtain fixed point theorems in these spaces

new fixed point theorems for two maps and applications to problems on sets with convex sections and minimax inequalities abstract new fixed point theorems for two maps defined on product spaces are obtained these new results only require one of them to satisfy a noncompactness condition previous results required each map to satisfy a noncompactness condition applications of our results are given to intersection problems for transfer closed maps to problems on sets with convex sections and to minimax inequalities of sion type our results generalize many well known results

differential realizations boson fermion realizations of the spl 2 1 superalgebra and its representations differential realizations of the spl 2 1 superalgebra on the spaces of homogeneous and inhomogeneous polynomials and the corresponding boson fermion realizations are studied the new indecomposable and irreducible representations of the spl 2 1 superalgebra are given on subspaces and quotient spaces of the universal enveloping algebra of heisenberg weyl superalgebra all the finite dimensional irreducible representations of the spl 2 1 superalgebra are naturally obtained as special cases

equilibria and maximal elements of abstract fuzzy economies and qualitative fuzzy games the purpose of this paper is to introduce the concepts of abstract fuzzy economies and qualitative fuzzy games and to use the connectedness of topological spaces without linear structures to study the existence problems of fuzzy equilibrium and fuzzy maximal elements of abstract fuzzy economies and qualitative fuzzy games the results presented in this paper are new in the cases of ordinary abstract economies and ordinary qualitative games

on the approximation of upper semi continuous correspondences and the equilibriums of generalized games in section 1 we explain some of the definitions and terminology that we use in section 2 we prove several theorems concerning the approximation of upper semi continuous correspondences having for range a locally convex space theorems 1 and 2 and corollary 1 generalize certain approximation theorems by g haddad 16 pp 1352213541 g haddad and j m lasry 17 pp 299 3001 and j p aubin and a cellina 2 pp 86891 see also f s de blasi lo and j m lasry and r robert 21 although some of the details of the proofs of theorems 1 and 2 are new the basic ideas are taken from the above mentioned papers theorem 3 shows that the correspondences in the approximating families can be chosen so that they are regular see section 1 this theorem see the remark following its proof contains a classical result of m hukuhara 18 pp 56573 theorem 3 is proved using theorem 2 and propositions 1 and 4 variants of propositions 2 3 and 4 were given in 193 in section 3 we give among others theorem 5 which concerns the existence of equilibriums of generalized games abstract economies the main purpose of this result is to replace the continuity hypothesis in the w shafer h sonnenschein equilibrium theorem for generalized games by an upper semi continuity one the proof of theorem 5 is based on the results on the approximation of upper semi continuous correspondences obtained in section 2 theorem 5 us used in section 4 in section 4 we establish theorems 6 and 7 these theorems show that certain statements concerning the equilibrium of generalized games are equivalent to certain statements concerning minimax inequalities of k fan type theorems 1 2 3 5 6 and 7 are the main results of this paper 267 0022 247x 88 3 00

approximate selection theorems and their applications abstract we prove three approximate selection theorems and give an improved version of the michael selection theorem as their applications new fixed point theorems and equilibrium theorems for generalized games are established

remarks on non standard statistics the theory of non standard statistics is reviewed and placed in a co homological setting some general mathematical results are shown to be relevant the specific heat of a two particle molecular gas is calculated as a function of temperature and the statistics determining parameter it shows strong behaviour close to the fermion case the second virial coefficient is calculated by a slightly improved method and its behaviour is compared with that of the vacuum energy of a quantum field interacting with a flux tube

some applications of variational inequalities in nonsmooth analysis abstract in this paper we apply some results on variational inequalities to the clarke generalized gradient and the generalized second order directional derivative f x u v and obtain some variational inequalities of them

michael s selection theorem under an assumption weaker than lower semicontinuous in h spaces a new continuous selection theorem is proved which unifies and generalizes some known results

branching processes in the annni model the mean field equations of the simple cubic annni model are studied on finite lattices the results are consistent with the sequence of distinct commensurate phases 2k 13 k 1 2 3 springing from the multiphase point found using low temperature series expansions moreover evidence for new structure combination branching processes is presented which generate phases of type 2l3 m 2l 13 n or 23l m 23l 1 n where l m and n are integers

on the classical theory of ordinary linear differential equations of the second order and the schrodinger equation for power law potentials the power law potentials in the schrodinger equation solved recently are shown to come from the classical treatment of the singularities of a linear second order differential equation this allows one to enlarge the class of solvable power law potentials

the dirichlet to robin transform a simple transformation converts a function satisfying a dirichlet boundary condition to a function satisfying a robin generalized neumann condition under sufficiently restrictive conditions one can arrange that the old function is the solution of a solvable differential equation and the new one solves an interesting differential equation in the simplest cases this observation enables the exact construction of the green functions for the wave heat and schrodinger problems with a robin boundary condition the resulting physical picture is that the field can exchange energy with the boundary and a delayed reflection from the boundary results in more general situations the method allows at least approximate and local construction of the appropriate reflected solutions and hence a classical path analysis of the green functions and the associated spectral information by this method we solve the wave equation on an interval with one robin and one dirichlet endpoint and thence derive several variants of a gutzwiller type expansion for the density of eigenvalues the variants are consistent except for an interesting subtlety of distributional convergence that affects only the neighbourhood of zero in the frequency variable

new results on equilibria of abstract fuzzy economies abstract in this paper we consider the model of an abstract fuzzy economy and we obtain new equilibrium theorems for the situations when the involved correspondences satisfy properties which are weaker than upper semicontinuity in our case the constraint or preference correspondences are almost weakly upper semicontinuous or have the e upper semicontinuous selection property the obtained results either generalize or improve corresponding ones present in the literature

on the existence of maximal elements fixed points and equilibria of generalized games in a fuzzy environment we give new sufficient conditions for the existence of fuzzy maximal elements fuzzy fixed points and equilibria of generalized fuzzy games in our results a key role is played by the existence of upper semicontinuous selection maps of set valued functions that we obtain for the class of generalized transfer open lower sections maps

existence of solutions for quasi equilibrium problems in noncompact topological spaces abstract in this paper a new fan browder type fixed point theorem is proved under noncompact setting of general topological spaces by applying the fixed point theorem several new existence theorems of solutions for quasi equilibrium problems are proved under noncompact setting of topological spaces these theorems improve and generalize a number of important known results in literature

distribution profiles and their dynamic transition in stochastic gene transcription abstract gene transcription is a central process in life in single cells it is a random process since many regulatory molecules are present at low copy numbers the randomness is best characterized by the mass function p m t the probability that there are m mrna copies at time t in one cell however even for the classical two state model of stochastic gene transcription the analytical form of p m t has remained elusive and its analysis has relied on numerical simulation or empirically defined hypergeometric functions in this work we have endeavored to express p m t as simple mathematical functions of the rate constants defined in the two state model for this purpose we have transformed the master equations of p m t into several special types of second order ordinary differential equations and solved their corresponding initial value problems successfully this has helped us derive the exact forms of p m t and explore its property in mathematical detail we have discovered a striking phenomenon that p m t could peak at any prescribed mrna number m at any prescribed time t over a large set of system parameters we have also proved that single transcription systems could generate three modes of distributions at different time points and depicted the dynamic transition among these distributions

continuous selection and fixed point theorems in this paper we obtain continuous selection theorems with range in a g convex space or in a space related to a g convex space and noncompact domain some applications to fixed point theorems and coincidence theorems are given we obtain some new results as well as generalizations of many known results

a further generalization of yannelis prabhakar s continuous selection theorem and its applications abstract in this article we first prove one improved version of the yannelis prabhakar continuous selection theorem and next as its applications a fixed point theorem in noncompact product spaces a nonempty intersection theorem some existence theorems of solutions for the generalized quasi variational inequalities and some equilibrium existence theorems for the abstract economies are given

system of coincidence theorems with applications in this paper we establish systems of coincidence theorems from which solution of system of inequalities and system of minimax theorems was established in this paper

exact propagators for moving hard wall potentials the semiclassical approximation has been used to evaluate the exact propagators for two one dimensional quantum systems a a free particle interacting with one hard wall potential moving with constant velocity and b a free particle inside a rigid box with one wall moving uniformly in time the results derived from these propagators are in agreement with those of the corresponding schrodinger equations

the unique labelling of states via mackey s theorem a new approach to finding unique labels for many electron states in multicentre systems is described and a particular example eight electrons in s states on eight centres is worked out in detail this approach is shown to provide a labelling with a more direct physical interpretation than has previously been found

study of a class of non polynomial oscillator potentials we develop a variational method to obtain accurate bounds for the eigenenergies of h v in arbitrary dimensions n 1 where v r is the nonpolynomial oscillator potential the variational bounds are compared with results previously obtained in the literature an infinite set of exact solutions is also obtained and used as a source of comparison eigenvalues

boson fermion realisation of indecomposable representations for lie superalgebras lndecomposable representations of lie superalgebras are studied on quotient spaces of the universal enveloping algebra of the heisenberg weyl superalgebra b b f j e by boson fermion realisation these representations are constructed from certain types of indecomposable representations of the heisenberg weyl superalgebra and induce usual irreducible representations on invariant subspaces of a quotient space as a physically significant example the explicit form of the boson fermion realisation of indecomposable representations of the classical lie superalgebra su 211 are obtained and discussed in this paper

on existence and essential components of the solution set for the system of vector quasi equilibrium problems abstract in this paper we study the existence and essential components of the solution set for the system of vector quasi equilibrium problems and establish some new results by defining the best reply map

fixed point and maximal element theorems with applications to abstract economies and minimax inequalities in this paper we establish some fixed point theorems for a family of multivalued maps under mild conditions by using our fixed point theorems we derive some maximal element theorems for a particular family of multivalued maps namely the condensing multivalued maps as applications of our results we prove some general equilibrium existence theorems in the generalized abstract economies with preference correspondences further applications of our results are also given to minimax inequalities for a family of functions

existence of equilibria for generalized games without paracompactness in this paper we first prove an existence theorem of maximal elements without any paracompactness then we establish an existence theorem of equilibrium points of generalized games with infinite many players and without any paracompactness by applying our maximal element existence theorem our equilibrium existence theorem answers the two questions raised by kim 6 in the affirmative

indecomposable representations of the poincare algebra the authors consider indecomposable representations of the poincare algebra iso 3 1 on the space omega omega omega h of its universal enveloping algebra a master representation is obtained on omega which induces representations on k the invariant subalgebra of translations and on omega and omega these representations are discussed in particular in view of finite dimensional indecomposable representations of iso 3 1 the approach taken is analogous to the approach chosen by the authors in their analysis of indecomposable representations of the lorentz algebra so 3 1 1983 thus under restriction of iso 3 1 to so 3 1 the earlier results are recovered the interpretation of the finite dimensional indecomposable representations of iso 3 1 then follows easily as a coupling of a finite number of irreducible so 3 1 representations to an indecomposable iso 3 1 representation with the dimension of the irreducible representations strictly increasing or strictly decreasing the bases for the finite dimensional indecomposable iso 3 1 representations are explicitly determined and thus also their matrix elements via the inducing representations a formula for their dimensionalities is obtained the methods employed are purely algebraic and follow the line of work of jacobson and dixmier 1962 1978

a generalized fixed point theorem and equilibrium point of an abstract economy a generalization of an earlier fixed point theorem to a noncompact case is proved and has been applied to prove the existence of equilibrium points of abstract economies

existence of equilibrium and separation in generalized games abstract the purpose of this paper is to give new existence theorems of equilibrium and separation in generalized games with an uncountable number of agents

equilibria with discontinuous preferences new fixed point theorems abstract we prove new equilibrium existence results for games and economies with discontinuous and non ordered preferences to do so we introduce the notion of continuous inclusion property and prove new fixed point theorems which extend and generalize the results of fan 17 glicksberg 20 browder 7 and gale and mas colell 19 our results also extend the previous work of yannelis 47 and wu and shen 46

on the equivalence of annni model polytypes formed by square wave modulation and branching mechanisms it is proved that long period structures formed by a square wave modulation of a lattice are identical to those of stable phases found in the annni model the proof is based on a continued fraction expansion of the modulation half period producing structural formulae shown to be equivalent to those obtained by a structure combination branching process the same structures have been observed experimentally by high resolution transmission electron microscopy in certain ordered alloys

on the existence of equilibrium for abstract economies abstract in this paper several existence theorems for abstract economies have been established they are natural modifications strict generalizations when strategy spaces are metrizable of the main results of yannelis and prabhakar 23 tulcea 22 and ding kim and tan 10 for infinite dimensional casas for the finite dimensional case we show that the topological conditions on correspondences in shafer and sonnenschein s result 20 can be much weaker along the proofs several propositions are established which are of independent interest we also show that many existence results in fixed point theory maximization theory and generalized quasi variational inequalities can be encompassed

exact form of green functions for segmented potentials we derive the exact green functions for segmented potentials i e for potentials of the form where each has compact support the expressions depend explicitly on the wavefunctions quantum amplitudes e g transmission and reflection coefficients for the individual s we also show that the exact green functions are in fact generalizations of the usual semiclassical formula finally the possibility of applying the method to a general potential is briefly discussed

a generalization of browder s fixed point theorem with applications abstract in this paper the concept of c compact mapping is introduced a generalization of browder s fixed point theorem and some equivalence forms are given as applications the existence of solutions for some variational inequalities and monotone operator equations is discussed

generalized quasi variational inequalities in locally convex topological vector spaces abstract let e be a hausdorff topological vector space and x e an arbitrary nonempty set denote by e the dual space of e and the pairing between e and e by w x for w e and x e given a point to set map s x 2x and a point to set map t x 2e the generalized quasi variational inequality problem gqvi is to find a point y s y and a point u t y such that re u y x 0 for all x s y by using the ky fan minimax principle or its generalized version as a tool some general theorems on solutions of the gqvi in locally convex hausdorff topological vector spaces are obtained which include a fixed point theorem due to ky fan and i l glicksberg and two different multivalued versions of the hartman stampacchia variational inequality

selecting families and their applications deguire and lassonde p deguire m lassonde familles selectantes topol methods nonlinear anal 5 1995 261 269 extend the concept of continuous selection and introduce the notion of selecting family for a family of set valued mappings in this paper we first establish a new existence theorem of selecting families the existence of the selecting families will be then used in order to obtain several fixed component theorems of fan browder type an intersection result a maximal element theorem for a family of set valued mappings and a minimax inequality

distribution modes and their corresponding parameter regions in stochastic gene transcription gene transcription is a stochastic process manifested by heterogeneous distribution of messenger rna mrna copy numbers in a population of isogenic cells the most panoramic information for the stochasticity is captured by the mass function p m defined as the probability that there are m mrna molecules for a gene of interest in one cell although the two state model of stochastic gene transcription has been the most prevailing mathematical framework to interpret gene expression data and gain mechanistic insights into the process of transcription several fundamental questions on p m remain open including these what types of the distribution can the two state model generate given the values of the system parameters can we predict the distribution type our major aim in this paper is to tackle these important questions analytically we prove that the two state model can generate and generate only the decaying unimodal or bimodal steady state distributions we also decompose the system parameter space into simply connected regions each of which corresponds to exactly one of the three distributions

kkm and nash equilibria type theorems in topological ordered spaces abstract in this paper we obatin a generalized kkm theorem a generalized fan browder fixed theorem and an existence theorem of nash equilibria in topological ordered spaces

numerical solution of the mean field theory of the annni model in an external field a comparison of two methods it is shown how the solution of the mean field equations for the annni model can be reduced to the iteration of an area preserving two dimensional mapping if an external magnetic field is applied this method is analysed in detail and compared with the exact numerical solution of the mean field equations for simple periodic magnetisations