an elementary derivation of the routh hurwitz criterion in most undergraduate texts on control systems the routh hurwitz criterion is usually introduced as a mechanical algorithm for determining the hurwitz stability of a real polynomial unlike many other stability criteria such as the nyquist criterion root locus etc no attempt whatsoever is made to even allude to a proof of the routh hurwitz criterion recent results using the hermite biehler theorem have however succeeded in providing a simple derivation of routh s algorithm for determining the hurwitz stability or otherwise of a given real polynomial however this derivation fails to capture the fact that routh s algorithm can also be used to count the number of open right half plane roots of a given polynomial this paper shows that by using appropriately generalized versions of the hermite biehler theorem it is possible to provide a simple derivation of the routh hurwitz criterion which also captures its unstable root counting capability

some hecke rogers type identities abstract recently a new partition function associated with ramanujan s third order mock theta function q was discovered and subsequently its overpartition analogue was introduced in this paper we prove an intriguing identity arising from the study of that analogue which involves a double series of hecke rogers type in addition two further identities will be given our proofs heavily rely on formulas from the work of liu 12

on the complete classification of unitary n 2 minimal superconformal field theories aiming at a complete classification of unitary n 2 minimal models where the assumption of space time supersymmetry has been dropped it is shown that each modular invariant candidate partition function of such a theory is indeed the partition function of a fully fledged unitary n 2 minimal model subject to the assumptions that orbifolding is a physical process and that the space time supersymmetric mathcal a mathcal d mathcal e models are physical a family of models constructed via orbifoldings of either the diagonal model or of the space time supersymmetric exceptional models then demonstrates that there exists a unitary n 2 minimal model for every one of the allowed partition functions in the list obtained from gannon s work gannon in nucl phys b 491 659 688 1997

the w n minimal model classification we first rigourously establish for any n 2 that the toroidal modular invariant partition functions for the not necessarily unitary w n p q minimal models biject onto a well defined subset of those of the su n su n wess zumino witten theories at level p n q n this permits considerable simplifications to the proof of the cappelli itzykson zuber classification of virasoro minimal models more important we obtain from this the complete classification of all modular invariants for the w 3 p q minimal models all should be realised by rational conformal field theories previously only those for the unitary models i e w 3 p p 1 were classified for all n our correspondence yields for free an extensive list of w n p q modular invariants the w 3 modular invariants like the virasoro minimal models all factorise into su 3 modular invariants but this fails in general for larger n we also classify the su 3 su 3 modular invariants and find there a new infinite series of exceptionals

deterministic evolutionary game dynamics in finite populations evolutionary game dynamics describes the spreading of successful strategies in a population of reproducing individuals typically the microscopic definition of strategy spreading is stochastic such that the dynamics becomes deterministic only in infinitely large populations here we present a microscopic birth death process that has a fully deterministic strong selection limit in well mixed populations of any size additionally under weak selection from this process the frequency dependent moran process is recovered this makes it a natural extension of the usual evolutionary dynamics under weak selection we find simple expressions for the fixation probabilities and average fixation times of the process in evolutionary games with two players and two strategies for cyclic games with two players and three strategies we show that the resulting deterministic dynamics crucially depends on the initial condition in a nontrivial way

spin and wedge representations of infinite dimensional lie algebras and groups abstract we suggest a purely algebraic construction of the spin representation of an infinite dimensional orthogonal lie algebra sections 1 and 2 and a corresponding group section 4 from this we deduce a construction of all level one highest weight representations of orthogonal affine lie algebras in terms of creation and annihilation operators on an infinite dimensional grassmann algebra section 3 we also give a similar construction of the level one representations of the general linear affine lie algebra in an infinite dimensional wedge space along these lines we construct the corresponding representations of the universal central extension of the group sln k t t 1 in spaces of sections of line bundles over infinite dimensional homogeneous spaces section 5

vortex generation in the rsp game on the triangular lattice abstract a new model of population dynamics on lattices is proposed the model consists of players on lattice points each of which plays the rsp game with neighboring players each player copies the next hand from the hand of the neighbouring player with the maximum point the model exhibits a steady pattern with pairs of vortices and sinks on the triangular lattice it is shown that the stationary vortex is due to the frustrations on the triangular lattice a frustration is the three sided situation where each of the three players around a triangle chooses the rock the scissors and the paper respectively

the routh hurwitz stability criterion revisited the case of multiple poles on imaginary axis this technical note presents a relation between the number of zero rows rows with all zero elements in the routh array and the multiplicity of j axis poles the main result of the technical note is that the existence of more than one zero row in the routh array amounts to instability of the system irrespective of any sign change in the first column

simple wzw currents a complete classification of simple currents of wzw theories is obtained the proof is based on an analysis of the quantum dimensions of the primary fields simple currents are precisely the primaries with unit quantum dimension for wzw theories explicit formulae for the quantum dimensions can be derived so that an identification of the fields with unit quantum dimension is possible

the routh hurwitz array and realization of characteristic polynomials we show that the routh hurwitz array of a given characteristic polynomial provides all the information required to realize the polynomial using an electrical circuit this new interpretation also lends to an intuitive proof of the routh hurwitz stability criterion

a commutant of βγ system associated to the highest weight module v4 of sl 2 c analogous to the commutant in the theory of associative algebras one can construct a new subalgebra of a vertex algebra known as the vertex algebra commutant in this paper for the highest weight module v4 of lie algebra sl 2 c we describe a commutant of system s v4 by giving a finite set of generators and operator product expansion relations among generators

representations of mock theta functions abstract motivated by the works of liu we provide a unified approach to find appell lerch series and hecke type series representations for mock theta functions we establish a number of parameterized identities with two parameters a and b specializing the choices of a b we not only give various known and new representations for the mock theta functions of orders 2 3 5 6 and 8 but also present many other interesting identities we find that some mock theta functions of different orders are related to each other in the sense that their representations can be deduced from the same a b parameterized identity furthermore we introduce the concept of false appell lerch series we then express the appell lerch series false appell lerch series and hecke type series in this paper using the building blocks m x q z and f a b c x y q introduced by hickerson and mortenson as well as m x q z and f a b c x y q introduced in this paper we also show the equivalences of our new representations for several mock theta functions and the known representations

quantum r matrix for g2 and a solvable 175 vertex model the quantum r matrix for cartan s exceptional simple lie algebra g2 in its seven dimensional minimal representation is presented the r matrix is determined through the irreducible decomposition of the tensor modules over the quantised universal enveloping algebra uq g2 this defines a new solvable seven state 175 vertex model on the planar square lattice whose boltzmann weights satisfy the yang baxter equation for the equivalent face model the local state probability is obtained in terms of a relative of a level 1 string function related to the affine lie algebra pair b3 1 contains implies g2 1

routh hurwitz test under vanishing leading array elements a slight modification to construct routh s array is presented to cope with the special case of vanishing leading array elements by means of the modified array root locations of a polynomial can readily be determined through counting first column sign changes as is with the normal routh s test without the necessity of adopting the e approach

a note on the routh hurwitz test this note demonstrates how the e method in the original routh hurwitz test can be applied when the routh array contains a row with zero leftmost element together with an all zero row the complete root distribution is determined by applying the criterion only once to the given polynomial d s without either factoring out a common divisor or shifting the imaginary axis since the test may become computationally tedious due to the inclusion of cumbersome e terms the recently reported method of shamash 1 is used to simplify the computational scheme this method being restricted to the case where d s has purely imaginary roots constituting the whole set of roots of the greatest common even divisor is here refined and generalized to accommodate all cases in which all zero rows appear

mathematics underlying the f theory heterotic string duality in eight dimensions we give an analytic description of the moduli space of classical vacua for f theory compactified on elliptic k3 surfaces on open tubular regions near the two type ii boundary divisors the structure of these open sets is related to the total spaces of certain holomorphic theta fibrations over the corresponding boundary divisors as the two type ii divisors can be naturally identified as moduli spaces of elliptic curves and flat g bundles with g e8 e8 or spin 32 one is led to an analytic isomorphism between these open domains and regions of the moduli spaces of heterotic string theory compactified on the two torus corresponding to large volumes of the torus this provides a proof for the classical version of f theory heterotic string duality in eight dimensions a description of the type ii boundary points in terms of elliptic stable k3 surfaces is also given

three manifold invariants from chern simons field theory with arbitrary semi simple gauge groups invariants for framed links in s 3 obtained from chern simons gauge field theory based on an arbitrary gauge group semi simple have been used to construct a three manifold invariant this is a generalization of a similar construction developed earlier for su 2 chern simons theory the procedure exploits a theorem of lickorish and wallace and also those of kirby fenn and rourke which relate three manifolds to surgeries on framed unoriented links the invariant is an appropriate linear combination of framed link invariants which does not change under kirby calculus this combination does not see the relative orientation of the component knots the invariant is related to the partition function of chern simons theory this thus provides an efficient method of evaluating the partition function for these field theories as some examples explicit computations of these manifold invariants for a few three manifolds have been done

property t for kac moody groups over rings abstract let r be a finitely generated commutative ring with 1 let a be an indecomposable 2 spherical generalized cartan matrix of size at least 2 and m m a the largest absolute value of a non diagonal entry of a we prove that there exists an integer n n a such that the kac moody group g a r has property t whenever r has no proper ideals of index less than n and all positive integers less than or equal to m are invertible in r

a simple proof of the routh test an elementary proof of the classic routh method for counting the number of left half plane and right half plane zeros of a real coefficient polynomial p sub n s of degree n is given such a proof refers to the polynomials p sub i s of degree i spl les n formed from the entries of the rows of order i and i 1 of the relevant routh array in particular it is based on the consideration of an auxiliary polynomial p sub i s q linearly dependent on a real parameter q which reduces to either polynomial p sub i s or to polynomial p sub i 1 s for particular values of q in this way it is easy to show that i 1 zeroes of p sub i s lie in the same half plane as the zeros of p sub i s and the remaining zero lies in the left or in the right half plane according to the sign of the ratio of the leading coefficients of p sub i s and p sub i 1 s by successively applying this property to all pairs of polynomials in the sequence starting from p sub o s and p sub 1 s the standard rule for determining the zero distribution of p sub n s is immediately derived

bursts in three strategy evolutionary ordinal potential games on a square lattice abstract evolutionary potential games provide an analogy for equilibrium statistical physics under suitable conditions that is broken if cyclic components are added using monte carlo simulations we study the effect of weak cyclic perturbations on the equilibrium in a system of players located on a square lattice while the dynamics is controlled by a logit rule the pair interaction is composed of a symmetric three strategy coordination game of unit strength a weak self dependent term and a cyclic rock paper scissors component the self dependent component favors the first rock strategy which dominates the system behavior at low noises if the strength of the cyclic component is below a threshold value in the opposite case the predator of the first strategy paper rules the behavior at low noises while the first strategy can occasionally form growing domains bursts diminished after the appearance of the third one the variation of noise modifies the invasion velocities as well as the rate of nucleation processes and yields drastic changes in the strategy frequencies fluctuations average payoffs and spatio temporal patterns

quantumr matrices related to the spin representations ofbn anddn we present the explicit form of the trigonometricr matrices related to the spin representations of the simple lie algebrasx n b n d n we conjecture that one dimensional configuration sums of the corresponding vertex models in the face formulation are the string functions ofx 1 modules

semi infinite weil complex and the virasoro algebra we define a semi infinite analogue of the weil algebra associated an infinite dimensional lie algebra it can be used for the definition of semi infinite characteristic classes by analogy with the chern weil construction the second term of a spectral sequence of this weil complex consists of the semi infinite cohomology of the lie algebra with coefficients in its adjoint semi infinite symmetric powers we compute this cohomology for the virasoro algebra this is just the brst cohomology of the bosonic system with central charge 26 we give a complete description of the fock representations of this bosonic system as modules over the virasoro algebra using friedan martinec shenker bosonization we derive a combinatorial identity from this result

three and four state rock paper scissors games with diffusion cyclic dominance of three species is a commonly occurring interaction dynamics often denoted the rock paper scissors rps game such a type of interactions is known to promote species coexistence here we generalize recent results of reichenbach et al nature london 448 1046 2007 of a four state variant of the rps game we show that spiral formation takes place only without a conservation law for the total density nevertheless in general fast diffusion can destroy species coexistence we also generalize the four state model to slightly varying reaction rates this is shown both analytically and numerically not to change pattern formation or the effective wavelength of the spirals and therefore not to alter the qualitative properties of the crossover to extinction

and as vertex operator extensions of dual affine algebras we discover a realisation of the affine lie superalgebra and of the exceptional affine superalgebra as vertex operator extensions of two algebras with dual levels and an auxiliary level 1 algebra the duality relation between the levels is we construct the representation of on a sum of tensor products of and modules and decompose it into a direct sum over the spectral flow orbit this decomposition gives rise to character identities which we also derive the extension of the construction to is traced to the properties of embeddings into and their relation with the dual pairs conversely we show how the representations are constructed from representations

virasoro character identities and artin l functions some identities between unitary minimal virasoro characters at levels m 3 4 5 are shown to arise as a consequence of relations between artin l functions of different quadratic fields the definitions and concepts of number theory necessary to present the theta function identities which can be derived from these relations are introduced a new infinite family of identities between virasoro characters at level 3 and level m 4a 2 for a odd and 1 4a 2 a 2 p where p is prime is obtained as a by product

automatic filters for the detection of coherent structure in spatiotemporal systems most current methods for identifying coherent structures in spatially extended systems rely on prior information about the form which those structures take here we present two approaches to automatically filter the changing configurations of spatial dynamical systems and extract coherent structures one local sensitivity filtering is a modification of the local lyapunov exponent approach suitable to cellular automata and other discrete spatial systems the other local statistical complexity filtering calculates the amount of information needed for optimal prediction of the system s behavior in the vicinity of a given point by examining the changing spatiotemporal distributions of these quantities we can find the coherent structures in a variety of pattern forming cellular automata without needing to guess or postulate the form of that structure we apply both filters to elementary and cyclical cellular automata eca and cca and find that they readily identify particles domains and other more complicated structures we compare the results from eca with earlier ones based upon the theory of formal languages and the results from cca with a more traditional approach based on an order parameter and free energy while sensitivity and statistical complexity are equally adept at uncovering structure they are based on different system properties dynamical and probabilistic respectively and provide complementary information

meromorphic c 24 conformal field theories modular invariant conformal field theories with just one primary field and central charge c 24 are considered it has been shown previously that if the chiral algebra of such a theory contains spin 1 currents it is either the leech lattice cft or it contains a kac moody sub algebra with total central charge 24 in this paper all meromorphic modular invariant combinations of the allowed kac moody combinations are obtained the result suggests the existence of 71 meromorphic c 24 theories including the 41 that were already known

three state potts model in combination with the rock scissors paper game we study a three state potts model extended by allowing cyclic dominance between the states as exemplified in the rock scissors paper game monte carlo simulations are performed on a square lattice while varying the temperature and the strength of cyclic dominance it is shown that the critical phase transition from the disordered state to the ordered one is destroyed by the cyclic dominance which yields a self organizing pattern even at low temperatures the differences and similarities are discussed between the present model and half filled driven lattice gases with repulsive interaction

inverse routh table construction and stability of delay equations abstract based on an inversion of the routh table construction a unimodular characterization of all hurwitz polynomials is obtained in parameter space the hurwitz polynomials of degree n correspond to the positive 2 n tant the method is then used to construct classes of stable continuous time delay difference equations and delay differential equations of neutral type by a suitable limiting process

nonabelian orbifolds and the boson fermion correspondence for a positive integerl divisible by 8 there is a bosonic holomorphic vertex operator algebra voa v gamma l associated to the spin lattice l for a broad class of finite groupsg of automorphisms of v gamma l we prove the existence and uniqueness of irreducibleg twisted v gamma l modules and establish the modular invariance of the partition functionsz g h for commuting elements ing in particular for any finite group there are infinitely many holomorphic voas admittingg for which these properties hold the proof is facilitated by a boson fermion correspondence which gives a voa isomorphism between v gamma l and a certain fermionic construction and which extends work of frenkel and others

the semi infinite cohomology of affine lie algebras we study the semi infinite or brst cohomology of affine lie algebras in detail this cohomology is relevant in the brst approach to gauged wznw models our main result is to prove necessary and sufficient conditions on ghost numbers and weights for non trivial elements in the cohomology in particular we prove the existence of an infinite sequence of elements in the cohomology for non zero ghost numbers this will imply that the brst approach to the topological wznw model admits many more states than a conventional coset construction this conclusion also applies to some non topological models

extended table for eliminating the singularities in routh s array a simple procedure for eliminating the classical epsilon method introduced by the routh hurwitz test is given the procedure extends the construction of the routh table in the special case of vanishing leading array elements without introducing complementary multiplications it is shown that it is always possible to associate with any polynomial g an extended routh table the number of sign changes in the first column of the table equals the number of zeros of g with positive real part

representations of affine kac moody algebras and the affine scalar product simple procedures are given for finding all the dominant weights in a highest weight representation of an affine algebra for finding the weyl orbit of an arbitrary weight and for determining whether or not any given weight is in any given representation a simple definition of congruency is given that applies to all affine algebras the standard indefinite scalar product is generalized the generalization is used in the procedures

analysis of quasi dynamic ordinary differential equations and the quasi dynamic replicator abstract we study the mathematical properties of the quasi dynamic ordinary differential equations defined empirically in chen et al 2019 in particular we show how the allometric scaling mentioned in that work emerges naturally from the generalized lotka volterra model under the quasi dynamic ordinary differential equations paradigm we then define and study the proportional quasi dynamic ordinary differential equations and discuss the relationship of this equation system to both the classical and discrete time replicator dynamics we prove asymptotic properties of these systems for large and small populations and show that there exist populations for which the proportion of the population varies cyclically as a function of total logarithmic population size

chaotic eigenfunctions in phase space we study individual eigenstates of quantized area preserving maps on the 2 torus which are classically chaotic in order to analyze their semiclassical behavior we use the bargmann husimi representations for quantum states as well as their stellar parametrization which encodes states through a minimal set of points in phase space the constellation of zeros of the husimi density we rigorously prove that a semiclassical uniform distribution of husimi densities on the torus entails a similar equidistribution for the corresponding constellations we deduce from this property a universal behavior for the phase patterns of chaotic bargmann eigenfunctions which is reminiscent of the wkb approximation for eigenstates of integrable systems though in a weaker sense in order to obtain more precise information on chaotic eigenconstellations we then model their properties by ensembles of random states generalizing former results on the 2 sphere to the torus geometry this approach yields statistical predictions for the constellations which fit quite well the chaotic data we finally observe that specific dynamical information e g the presence of high peaks like scars in husimi densities can be recovered from the knowledge of a few long wavelength fourier coefficients which therefore appear as valuable order parameters at the level of individual chaotic eigenfunctions