travelling waves of auto catalytic chemical reaction of general order an elliptic approach abstract in this paper we study the existence and non existence of travelling wave to parabolic system of the form a t a x x a f b b t d b x x a f b with f a degenerate nonlinearity in the context of an auto catalytic chemical reaction a is the density of a chemical species called reactant a b that of another chemical species b called auto catalyst and d d b d a 0 is the ratio of diffusion coefficients d b of b and d a of a respectively such a system also arises from isothermal combustion the nonlinearity is called degenerate since f 0 f 0 0 one case of interest in this article is the propagating wave fronts in an isothermal auto catalytic chemical reaction of order n a n b n 1 b with 1 n 2 and d 1 due to different molecular weights and or sizes of a and b the resulting nonlinearity is f b b n explicit bounds v and v that depend on d are derived such that there is a unique travelling wave of every speed v v and there does not exist any travelling wave of speed v v new to the literature it is shown that v v d when d 1 furthermore when d 1 it is shown rigorously that there exists a v min such that there is a travelling wave of speed v if and only if v v min estimates on v min improve significantly that of early works another case in which two different orders of isothermal auto catalytic chemical reactions are involved is also studied with interesting new results proved

s duality as fourier transform for arbitrary ϵ1 ϵ2 the alday gaiotto tachikawa relations reduce s duality to the modular transformations of conformal blocks it was recently conjectured that for the four point conformal block the modular transform up to the non perturbative contributions can be written in the form of the ordinary fourier transform when 1 2 1 here i extend this conjecture to general values of 1 2 namely i argue that for a properly normalized four point conformal block the s duality is perturbatively given by the fourier transform for arbitrary values of the deformation parameters 1 2 the conjecture is based on explicit perturbative computations in the first few orders of the string coupling constant g2 1 2 and hypermultiplet masses

riemann stieltjes operators and multipliers on q p spaces in the unit ball of c n this paper is devoted to characterizing the riemann stieltjes operators and pointwise multipliers acting on mobius invariant spaces q p which unify bmoa and bloch space in the scale of p the boundedness and compactness of these operators on q p spaces are determined by means of an embedding theorem i e q p spaces boundedly embedded in the non isotropic tent type spaces t q infinity c 2010 elsevier inc all rights reserved

a berndtsson andersson operator solving overline partial equation with wα estimates on convex domains of finite type the carleson measures were first introduced by carleson in order to solve the corona problem for the disc in mathbb c the notion of carleson measure can be generalized to any homogeneous space and were also used in the context of the corona problem in mathbb c n for example by varopoulos amar and andersson and carlsson one of the steps to solve the hp corona problem in a pseudoconvex domain is to solve the overline partial equation for a form satisfying a carleson condition and get norm estimates of the solution in term of the carleson norm of the main goal of this paper is to consider this question in the case of convex domains of finite type and to get estimates linked to the multitype of the domain

analysis of chemical kinetic systems over the entire parameter space i the sal nikov thermokinetic oscillator in this series of papers we re examine using recently developed techniques some chemical kinetic models that have appeared in the literature with a view to obtaining a complete description of all the qualitatively distinct behaviour that the system can exhibit each of the schemes is describable by two coupled ordinary differential equations and contain at most three independent parameters we find that even with these relatively simple chemical schemes there are regions of parameter space in which the systems display behaviour not previously found quite often these regions are small and it seems unlikely that they would be found via classical methods in part i of the series we consider one of the thermally coupled kinetic oscillator models studied by sal nikov he showed that there is a region in parameter space in which the system would be in a state of undamped oscillations because the relevant phase portrait consists of an unstable steady state surrounded by a stable limit cycle our analysis has revealed two further regions in which the phase portraits contain respectively two limit cycles of opposite stability enclosing a stable steady state and three limit cycles of alternating stability surrounding an unstable steady state this latter region is extremely small so much so that it could be reasonably neglected in any predictions made from the model

holomorphic triebel lizorkin spaces abstract the goal of this work is to obtain characterizations of the holomorphic triebel lizorkin spaces in terms of littlewood paley functions admissible area functions complex tangential derivatives and boundary values furthermore we obtain results on duality complex interpolation and traces on submanifolds

a simple model for sustained oscillations in isothermal branched chain or autocatalytic reactions in a well stirred open system i stationary states and local stabilities the behaviour of a simple autocatalytic or branched chain reaction a b rightarrow 2b text rate k 1 ab coupled with a surface controlled decay of the catalyst b rightarrow c text rate k 2 b 1 rb is considered under isothermal well stirred open conditions the c s t r this system has a unique physically acceptable stationary state at any given residence time if the saturation term r is zero this solution is always stable although it may exhibit damped oscillatory responses characteristic of a stable focus small non zero values of r do not introduce qualitative changes however for sufficiently large surface effects such that r k 1 k 2 the stable focus may become unstable at a point of hopf bifurcation this leads to the appearance of sustained oscillations in the concentrations a and b for relatively stable catalysts such that k 2 is non zero but very small compared with k 1 a 0 where a 0 is the concentration of reactant a in the feed to the reactor the governing kinetic equations can be reduced to especially simple asymptotic forms these in turn allow the analytical determination of the point of hopf bifurcation in terms of the residence time t res at which the change in stability occurs numerical solution of the full equations reveals that the asymptotic expressions provide a very good prediction for the dependence of t res on k 1 a 0 k 2 and r even when k 2 approx k 1 a 0 systems which have some autocatalyst b in the feed show an important qualitative change at long residence times there is a second point of hopf bifurcation at which the focal stationary state becomes stable again oscillations die out at this point and in the limit of infinitely long residence times the behaviour approaches that of the corresponding closed system

on the dotsenko fateev representation of the toric conformal blocks we demonstrate that the recent ansatz of maruyoshi and yagi 2010 arxiv 1009 5553 inspired by the original remark of r dijkgraaf and c vafa reproduces the toric conformal blocks in the same sense as the spherical blocks given by the integral representation of mironov et al 2010 j high energy phys jhep02 2010 030 arxiv 0911 5721 and mironov et al 2010 arxiv 1001 0563 with a peculiar choice of open integration contours for screening insertions in other words we provide some evidence that the toric conformal blocks are reproduced by appropriate ensembles not only in the large n limit but also at finite n the check is explicitly performed at the first two levels for the 1 point toric functions generalizations to higher genera are briefly discussed

analysis of chemical kinetic systems over the entire parameter space ii isothermal oscillators two simple but similar isothermal kinetic schemes are studied by using recently developed qualitative techniques with the aim of finding all of the qualitatively distinct behaviour that the system can exhibit the first scheme which can be regarded as one of the simplest isothermal models of interest is based on the quadratic autocatalytic step a b rightarrow 2b text rate k q ab where a and b are the concentrations of a and b respectively combined with a nonlinear termination step of the form b rightarrow c text rate k t b 1 rb where r is some parameter the second scheme is a simplified by treating one parameter as a constant version of a model that has been proposed for the oxidation of carbon monoxide co this scheme has a termination step of the same form as above and like the quadratic model the highest nonlinearity apart from this term is quadratic for the quadratic model we find 10 qualitatively distinct phase portraits and with our choice of bifurcation parameter 22 distinct bifurcation diagrams the most complex phase portraits contain 3 steady states and 2 limit cycles or 2 steady states and 3 limit cycles for the co model we find 8 distinct phase portraits but leading to only 5 separate bifurcation diagrams in both models much of the behaviour occurs over only extremely small regions in parameter space

pointwise multipliers and decomposition theorems in analytic besov spaces for the holomorphic besov spaces mathcal b p s in the unit ball of mathbb c n we study necessary and sufficient conditions on holomorphic functions g 1 g 2 so that the operator f 1 f 2 mapsto g 1f 1 g 2f 2 maps mathcal b p s times mathcal b p s onto mathcal b p s related to this problem we also obtain some characterizations of the pointwise multipliers for these spaces

on the creation growth and extinction of oscillatory solutions for a simple pooled chemical reaction scheme the equations which govern a simple pooled chemical reaction scheme are analysed in detail in terms of a nondimensional parameter mu which represents the amount of the pooled chemical originally present it is shown that there is one finite equilibrium point with a hopf bifurcation occurring at mu 1 the phase plane at infinity is then examined and it is shown that there are equilibrium points at infinity at the positive ends of both axes the nature of which are discussed this together with a knowledge of the global phase portraits for mu ll 1 and mu gg 1 enables the global phase portrait to be constructed for all positive mu from this it emerges that the stable limit cycle created at mu 1 by a hopf bifurcation is destroyed at mu 0 mu 0 1 by an infinite period bifurcation due to the formation of a heteroclinic orbit by the separatrices from the equilibrium points at infinity the form of this heteroclinic orbit is then discussed in detail and in particular i

zero varieties for the nevanlinna class in convex domains of finite type in mathbb c n with x ak each ak being counted according to its multiplicity in this case too it is well known that conversely the blaschke condition implies that the sequence x is the zero sequence of a function f e n q or even a bounded holomorphic function the question whether in dimension n 1 the blaschke condition b is sufficient for the divisor x xk nk to be defined by a function f e n q has been one of the main topics in this area the first result was independently obtained by henkin 11 and skoda 14 who proved it for strictly pseudoconvex domains with some obvious topological conditions also some partial results

a simple model for sustained oscillations in isothermal branched chain or autocatalytic reactions in a well stirred open system iii multiple stationary states and hopf bifurcations a detailed and thorough numerical investigation of the dynamical behaviour of the simple prototype scheme begin align tag 1 text quadratic autocatalytic a b rightarrow 2b text rate k 1 ab tag 2 text catalyst decay quad b rightarrow c text rate k 2b 1 rb end align reveals a remarkable range of complexity reaction occurs in a well stirred continuously fed isothermal open system the chemical engineers cstr such a model is appropriate to autocatalytic or branched chain chemical systems including some aspects of the oxidations of hydrogen or of carbon monoxide similar rate laws may arise in biochemical situations the important parameters for this system are the relative concentration of a and b in the inflow the relative rates of reactions 1 and 2 the mean residence time and the saturation term r the system shows multiple stationary states the dependence of the stationary state concentration of a on the residence time may show any one of five patterns unique breaking wave mushroom isola or isola breaking wave depending on the other parameter values sustained oscillatory reaction is also possible this begins at hopf bifurcation points which may be either supercritical soft excitation or subcritical hard excitation with sudden jumps to relatively large oscillations degenerate bifurcations of different types have also been located and tracked through the parameter space by numerical path following techniques developed during this investigation the full range of behaviour for this four parameter system may be compactly presented in terms of these curves of degenerate points in a new way introduced here

q vertex operator from 5d nekrasov function the five dimensional agt correspondence implies the connection between the q deformed virasoro block and the 5d nekrasov partition function in this paper we determine a q deformation of the four point block in the coulomb gas representation from the 5d nekrasov function and obtain an expression of the q deformed vertex operator if we use only one kind of the q vertex operators one of the insertion points of them must be modified in order to hold the 2d 5d correspondence

pointwise multipliers and decomposition theorems in f q s in this paper we give a characterization of the pointwise multipliers of the holomorphic triebel lizorkin spaces f q s and we apply these results to prove a corona decomposition theorem for these spaces

exact wkb and abelianization for the t_3 equation we describe the exact wkb method from the point of view of abelianization both for schrodinger operators and for their higher order analogues opers the main new example which we consider is the t 3 equation an order 3 equation on the thrice punctured sphere with regular singularities at the punctures in this case the exact wkb analysis leads to consideration of a new sort of darboux coordinate system on a moduli space of flat mathrm sl 3 connections we give the simplest example of such a coordinate system and verify numerically that in these coordinates the monodromy of the t 3 equation has the expected asymptotic properties we also briefly revisit the schrodinger equation with cubic potential and the mathieu equation from the point of view of abelianization

quantum hitchin systems via beta deformed matrix models we study the quantization of hitchin systems in terms of beta deformations of generalized matrix models related to conformal blocks of liouville theory on punctured riemann surfaces we show that in a suitable limit corresponding to the nekrasov shatashvili one the loop equations of the matrix model reproduce the hamiltonians of the quantum hitchin system on the sphere and the torus with marked points the eigenvalues of these hamiltonians are shown to be the epsilon 1 deformation of the chiral observables of the corresponding mathcal n 2 four dimensional gauge theory moreover we find the exact wave functions in terms of the matrix model representation of the conformal blocks with degenerate field insertions

the application of singularity theory to isothermal autocatalytic open systems the elementary scheme a mb m 1 b in a well stirred open system isothermal c s t r is determined via singularity theory methods these methods allow all of the possible patterns for the dependence of the stationary state extent of conversion x on the residence time da i e the x da bifurcation diagrams to be identified and located the cubic rate law has a cusp singularity separating diagrams with a unique dependence of x on da from those which display a simple s shaped hysterisis loop with multiple stationary states this behaviour is qualitatively similar to that shown by a simple exothermic reaction in an adiabatic c s t r i e the two systems are contact equivalent if the autocatalyst is not infinitely stable but instead undergoes a simple decay b c rate k2b a wider range of bifurcation diagrams is possible with isolas or mushroom patterns these arise as the system is unfolded from its winged cusp singularity by varying parameters such as the catalyst lifetime and inflow concentration it is shown that these are also the only patterns possible for a generalized order of autocatalysis i e for a rate proportional to abm with m taking any value greater than one integral or non integral the ranges of the above parameters over which t he different responses are found are also given analytically for the general m these parameters cannot however give a complete unfolding so certain additional bifurcation diagrams that are found for the exothermic reaction in a nonisothermal non adiabatic reactor which also has a winged cusp singularity are not found in the autocatalytic system

parameter space analysis pattern sensitivity and model comparison for turning and stationary flow distributed waves fds a new type of instability in coupled reaction diffusion advection systems is analysed in a one dimensional domain this instability arising due to the combined action of flow and diffusion creates spatially periodic stationary waves termed flow and diffusion distributed structures fds here we show via linear stability analysis that fds are predicted in a considerably wider domain and are more robust in the parameter domain than the classical turing instability patterns fds also represent a natural extension of the recently discovered flow distributed oscillations fdo nonlinear bifurcation analysis and numerical simulations in one dimensional spatial domains show that fds also have much richer solution behaviour than turing structures in the framework presented here turing structures can be viewed as a particular instance of fds we conclude that fds should be more easily obtainable in chemical systems than turing and fdo structures and that they may play a potentially important role in biological pattern formation

nekrasov functions from exact bohr sommerfeld periods the case of su n we suggested in 2009 that the nekrasov function with one non vanishing deformation parameter is obtained by the standard seiberg witten sw contour integral construction the only difference is that the sw differential pdx is substituted by its quantized version for the corresponding integrable system and contour integrals become exact monodromies of the wavefunction this provides an explicit formulation of the earlier guess by nekrasov and shatashvili in 2009 in this paper we successfully check this suggestion in the first order in 2 and the first order in instanton expansion for the su n model where the consistency of the so deformed sw equations is already non trivial

integrating over higgs branches we develop some useful techniques for integrating over higgs branches in supersymmetric theories with 4 and 8 supercharges in particular we define a regularized volume for hyperkahler quotients we evaluate this volume for certain ale and alf spaces in terms of the hyperkahler periods we also reduce these volumes for a large class of hyperkahler quotients to simpler integrals these quotients include complex coadjoint orbits instanton moduli spaces on 4 and ale manifolds hitchin spaces and moduli spaces of parabolic higgs bundles on riemann surfaces in the case of hitchin spaces the evaluation of the volume reduces to a summation over solutions of bethe ansatz equations for the non linear schrodinger system we discuss some applications of our results

a simple model for sustained oscillations in isothermal branched chain or autocatalytic reactions in a well stirred open system ii limit cycles and non stationary states the oscillatory patterns of behaviour exhibited by the simple kinetic scheme a b rightarrow 2b text rate k 1 ab b rightarrow c text rate k 2 b 1 rb are examined in detail for systems with slowly decaying catalysts such that k 2 simeq k 1 a 2 0 a reduced asymptotic form of the governing equations allows a full analytical treatment oscillations begin as the residence time is increased through a point of hopf bifurcation tau res the bifurcation is always supercritical with the amplitudes of the concentration variations increasing from infinitesimally small values the amplitudes grow initially as tau res tau res frac 1 2 and tend to finite limiting magnitudes at very long residence times at this limit the oscillations in a have a saw tooth waveform with b varying in a pulse like manner numerical solutions of the full kinetic equations for a non zero inflow of the catalyst b reveal how the system approaches a second hopf bifurcation as the residence time is increased the oscillatory amplitude now reaches a maximum and then decays back to zero the applicability of this model to real chemical systems is discussed with particular reference to the gas phase oxidation of carbon monoxide

a method for the complete qualitative analysis of two coupled ordinary differential equations dependent on three parameters in this paper recent advances in bifurcation theory are specialized to systems describable by two coupled ordinary differential equations odes containing at most three independent parameters for such systems by plotting in the relevant parameter plane the locus of successively degenerate singular points a complete description of all the qualitatively distinct behaviour of the system can be obtained the description is in terms of phase portraits and bifurcation diagrams even though much use is made of existing results obtained via local analyses the results of this technique cover the entire parameter space furthermore because the information is built up in successive stages the question of whether the parameters universally unfold a given degeneracy does not arise this can mean a major saving in effort particularly for degenerate hopf points finally if as is often the case the parameters appear in the system in a simple way the procedure can be applied analytically because the variables which will appear nonlinearly can be used to parametrize the relevant loci

mathcal w algebras and surface operators in mathcal n 2 gauge theories a general class of algebras can be constructed from the affine sl n algebra by quantum drinfeld sokolov reduction and are classified by partitions of n surface operators in an su n 4d gauge theory are also classified by partitions of n we argue that instanton partition functions of gauge theories in the presence of a surface operator can also be computed from the corresponding algebra we test this proposal by analysing the polyakov bershadsky algebra obtaining results that are in agreement with the known partition functions for su 3 gauge theories with a so called simple surface operator as a byproduct our proposal implies relations between and algebras

weighted and boundary l p estimates for solutions of the equation on lineally convex domains of finite type and applications we obtain sharp weighted estimates for solutions of the equation u f in a lineally convex domain of finite type precisely we obtain estimates in the spaces l p being the distance to the boundary with two different types of hypothesis on the form f first if the data f belongs to l p 1 we have a mixed gain on the index p and the exponent secondly we obtain a similar estimate when the data f satisfies an apropriate anisotropic l p estimate with weight 1 moreover we extend those results to 1 and obtain l p and bmo estimates these results allow us to extend the l p regularity results for weighted bergman projection obtained in cdm14b for convex domains to more general weights

the evolution of travelling waves in fractional order autocatalysis with decay i permanent form travelling waves this paper considers a model isothermal autocatalytic reaction scheme with termination when the reaction orders m n are fractional i e 0 n and k in 0 k c where kc depends upon m and n it is further established that these wavefronts are of excitable rather than fisher kolmogorov type and have only semi infinite support that is the reaction is totally dormant ahead of the wavefront these features are in contrast to the case when m n geq 1 as studied by merkin needham and scott proc roy soc london ser a 424 1989 pp 187 209 merkin and needham j engrg math 23 1989 pp 343 356 and needham and merkin phil trans roy soc london ser a 337 1991 pp 261 274 it should be noted that the corresponding reaction diffusion equations are singular in the sense that the reaction and decay terms are not lipsch

the evolution of traveling waves in a simple isothermal chemical system modeling quadratic autocatalysis with strong decay abstract in this paper we study a reaction diffusion system for an isothermal chemical reaction scheme governed by a quadratic autocatalytic step a b 2 b and a decay step b c where a b and c are the reactant the autocatalyst and the inner product respectively previous numerical studies and experimental evidences demonstrate that if the autocatalyst is introduced locally into this autocatalytic reaction system where the reactant a initially distributes uniformly in the whole space then a pair of waves will be generated and will propagate outwards from the initial reaction zone one crucial feature of this phenomenon is that for the strong decay case the formation of waves is independent of the amount of the autocatalyst b introduced into the system it is this phenomenon of kpp type which we would like to address in this paper to study the propagation of reactant and autocatalyst analytically we first use the tail behavior of waves to construct a pair of generalized super sub solutions for the approximate system of the autocatalytic reaction system note that the autocatalytic reaction system does not enjoy comparison principle together with a family of truncated problems we can establish the existence of a family of traveling waves with the minimal speed second we use this pair of generalized super sub solutions to show that the propagation of waves is fully determined by the rate of decay of the initial data at infinity in the sense of aronson weinberger formulation which in turn confirms the aforementioned numerical and experimental results

estimates of solutions of the h p and bmoa corona problem we prove new sharper estimates of solutions to the h p corona problem in strictly pseudoconvex domains in particular we show that the constant is independent of the number of generators we also obtain sharper estimates for solutions to the bmoa corona problem the proofs also lead to new results about the taylor spectrum of analytic toeplitz operators on h p and bmoa

integral representation with weights ii division and interpolation let f be a r m matrix of holomorphic functions that is generically surjective we provide explicit integral representation of holomorphic such that f provided that is holomorphic and annihilates a certain residue current with support on the set where f is not surjective we also consider formulas for interpolation as applications we obtain generalizations of various results previously known for the case r 1

an h2 corona theorem on the bidisk for infinitely many functions we extend the h 2 corona theorem on the bidisk to infinitely many functions the main tool is a linear algebra result which simplifies the bookkeeping for estimates of solutions

bmo estimates for the h bn corona problem we study the h bn corona problem j 1nfjgj h and show it is always possible to find solutions f that belong to bmoa bn for any n 1 including infinitely many generators n this theorem improves upon both a 2000 result of andersson and carlsson and the classical 1977 result of varopoulos the former result obtains solutions for strictly pseudoconvex domains in the larger space h bmoa with n while the latter result obtains bmoa bn solutions for just n 2 generators with h 1 our method of proof is to solve problems and to exploit the connection between bmo functions and carleson measures for h2 bn key to this is the exact structure of the kernels that solve the equation for 0 q forms as well as new estimates for iterates of these operators a generalization to multiplier algebras of besov sobolev spaces is also given

some new function spaces and their applications to harmonic analysis abstract in this paper a family of spaces is introduced which seems well adapted for the study of a variety of questions related to harmonic analysis and its applications these spaces are the tent spaces they provide the natural setting for the study of such things as maximal functions the relevant space here is t p and also square functions where the space t2p is relevant as such these spaces lead to unifications and simplifications of some basic techniques in harmonic analysis thus they are closely related to lp and hardy spaces important parts of whose theory become corollaries of the description of tent spaces also as proc conf harmonic analysis cortona lect notes in math vol 992 springer verlag berlin new york 1983 already indicated where these spaces first appeared explicitly the tent spaces can be used to simplify some of the results related to the cauchy integral on lipschitz curves and multilinear analysis in retrospect one can recognize that various ideas important for tent spaces had been used if only implicitly for quite some time here one should mention carleson s inequality its simplifications and extensions the theory of hardy spaces and atomic decompositions

addendum to on the variation in the cohomology of the symplectic form of the reduced phase space here we have used the notation rlx for the inner product of the vector field x with the form a and ix m x j m for the x component of a in an earlier paper 4 it was shown that the push forward j dm of the liouville measure dm on m under the momentum mapping j is a piecewise polynomial measure on moreover in case x has isolated isolated zeros on m an explicit formula for the integral

behavior of the bergman projection on the diederich fornaess worm w 1 introduction in this paper we show that the bergman projection operator for certain smooth bounded pseudoconvex domains does not preserve smoothness as measured by sobolev norms let q be a smooth bounded pseudoconvex domain in c and let p t denote the orthogonal projection from l2 f onto the bergman subspace b ff l2 f3 r with respect to the weighted norm i fll j olf z l 2 e t lzll2dv l 2 let wk t denote the sobolev space consisting of functions whose derivatives of order to k g2 on the other hand there is a large collection of results implying that for certain types of domains the unweighted bergman projection p pr preserves w k for all k 0 see fk for the strictly pseudoconvex case for results on weakly pseudoconvex domains the reader may consult ko2 ca si and the recent bstl ch as well as the references cited therein most of these results are focused on the a neumann operator rather than p see bst2 for the connection also except for kol positive results in this area are typically valid for any choice of smooth positive weight function on 0 the question of whether or not p is similarly well behaved for all weakly pseudoconvex domains has remained open for many years in this paper we show that this is not the case in fact when q is the so called worm domain of diederich and forna ss then p does not map w to w k when k zr total amount of winding this latter quantity is explained in section 4 below where the construction of the worm is reviewed and the main result is proved the proof depends on computations for a piecewise levi flat model domain depending in turn on certain one dimensional computations these are treated in sections 3 and 2 respectively section 5 contains additional remarks and questions

path following methods in bifurcation problems abstract analytical methods of bifurcation theory and numerical path following methods complement each other in many investigations of nonlinear systems this brief state of the art review summarises the contributions to a workshop held at the university of leeds on 12 15th january 1988