contemporaneous asymmetry in garch processes abstract the paper introduces a new concept of asymmetry contemporaneous asymmetry in conditional heteroskedasticity models we propose an original class of models aimed to capture the leverage effect contemporaneous asymmetry as well as time varying skewness and kurtosis not only past up and down moves have different impacts on the conditional variance but also positive and negative changes are governed by different conditional variances we give conditions for the existence of a second order and strictly stationary solution the paper also provides consistency results on the quasi maximum likelihood estimation finally an empirical analysis on the french cac 40 stock index is proposed

virtual refinements of the vafa witten formula we conjecture a formula for the generating function of virtual chi y genera of moduli spaces of rank 2 sheaves on arbitrary surfaces with holomorphic 2 form specializing the conjecture to minimal surfaces of general type and to virtual euler characteristics we recover part of a formula of c vafa and e witten these virtual chi y genera can be written in terms of descendent donaldson invariants using t mochizuki s formula the latter can be expressed in terms of seiberg witten invariants and certain explicit integrals over hilbert schemes of points these integrals are governed by seven universal functions which are determined by their values on mathbb p 2 and mathbb p 1 times mathbb p 1 using localization we calculate these functions up to some order which allows us to check our conjecture in many cases in an appendix by h nakajima and the first named author the virtual euler characteristic specialization of our conjecture is extended to include mu classes thereby interpolating between vafa witten s formula and witten s conjecture for donaldson invariants

simultaneous reconstruction of the spatially distributed reaction coefficient initial temperature and heat source from temperature measurements at different times abstract in many practical situations concerned with high temperatures pressures loads and or hostile environments certain properties of the physical medium geometry boundary and or initial conditions are not known and their direct measurement can be very inaccurate or even inaccessible in such situations one can adopt an inverse approach and try to infer the unknowns from some extra accessible measurements of other quantities that may be available however the simultaneous identification of several non constant physical properties along with initial and or boundary conditions is very challenging especially when it cannot be decoupled as it combines both nonlinear as well as ill posedness features one such new inverse problem concerning the identification of the space dependent reaction coefficient the initial temperature and the source term from measured temperatures at two instants t 1 t 2 and at the final time t where 0 t 1 t 2 t is investigated in this paper insight into the uniqueness of solution is gained by considering various particular cases moreover as in practice the input temperature data are usually noise polluted due to the errors that are inherently present their influence on the solution of inversion has to be assessed as such the least squares objective functional modelling the gap between the measured and computed data is minimized to obtain the quasi solution to the inverse problem and the frechet gradients are obtained the conjugate gradient method cgm with the fletcher reeves formula is applied to estimate the three unknown coefficients numerically numerical examples are illustrated to show that accurate and stable numerical solutions are obtained using the cgm regularized by the discrepancy principle

a rational route to randomness the concept of adaptively rational equilibrium a r e is introduced agents adapt their beliefs over time by choosing from a finite set of different predictor or expectations functions each predictor is a function of past observations and has a performance or fitness measure which is publicly available agents make a rational choice concerning the predictors based upon their past performance this results in a dynamics across predictor choice which is coupled to the equilibrium dynamics of the endogenous variables as a simple but typical example we consider a cobweb type demand supply model where agents can choose between rational and naive expectations in an unstable market with small positive information costs for rational expectations a high intensity of choice to switch predictors leads to highly irregular equilibrium prices converging to a strange attractor the irregularity of the equilibrium time paths is explained by the existence of a so called homoclinic orbit and its associated complicated dynamical phenomena thus local instability and global complicated dynamics may be a feature of a fully rational notion of equilibrium

from sheaves on p 2 to a generalization of the rademacher expansion moduli spaces of stable coherent sheaves on a surface are of much interest for both mathe matics and physics yoshioka computed generating functions of poincare polynomials of such moduli spaces if the surface is p 2 and the rank of the sheaves is 2 motivated by physical arguments this pa per investigates the modular properties of these generating functions it is shown that these functions can be written in terms of the lerch sum and theta function based on this we prove a conjecture by vafa and witten which expresses the generating functions of euler numbers as a mixed mock mod ular form moreover we derive an exact formula of rademacher type for the fourier coefficients of this function this formula requires a generalization of the classical circle method this is the first example of an exact formula for the fourier coefficients of mixed mock modular forms which is of independent mathematical interest 1 introduction and statement of results in the past interactions between physics and mathematics have led to many interesting results motivated by strong weak coupling duality or s duality in physics this article considers various gen erating functions which appear in the study of moduli spaces of stable coherent sheaves on the projective plane p 2 we express the generating functions of poincare polynomials of moduli spaces of rank 2 sheaves in terms of the lerch sum and theta function which we will recall later using these expressions we prove a conjec ture by vafa and witten 45 for the generating functions of euler numbers these functions appear to be related to ramanujan s mock theta functions and therefore transform almost as weakly holomorphic modular forms i e meromorphic mod ular forms whose poles if there are any may only lie in cusps our second main result is an exact formula for the fourier coefficients of these generating functions that formally resembles the rademacher expansion for the coefficients of weakly holomorphic modular forms moduli spaces of coherent sheaves on a complex surface s receive much attention see for example 28 for an extensive work on such moduli spaces more specifically one is interested in the moduli space m r c1 c 2 of semi stable sheaves of rank r with first chern class c1 and second chern class c2 w e will consider topological invariants of m in particular the poincare polynomial p m s 2d imc m i 0 bi m s i and the euler number m p m 1 w here

vertical d4 d2 d0 bound states on k3 fibrations and modularity an explicit formula is derived for the generating function of vertical d4 d2 d0 bound states on smooth k3 fibered calabi yau threefolds generalizing previous results of gholampour and sheshmani it is also shown that this formula satisfies strong modularity properties as predicted by string theory this leads to a new construction of vector valued modular forms which exhibit some of the features of a generalized hecke transform

vafa witten theory and iterated integrals of modular forms vafa witten vw theory is a topologically twisted version of mathcal n 4 supersymmetric yang mills theory s duality suggests that the partition function of vw theory with gauge group su n transforms as a modular form under duality transformations interestingly vafa and witten demonstrated the presence of a modular anomaly when the theory has gauge group su 2 and is considered on the complex projective plane mathbb p 2 this modular anomaly could be expressed as an integral of a modular form and also be traded for a holomorphic anomaly we demonstrate that the modular anomaly for gauge group su 3 involves an iterated integral of modular forms moreover the modular anomaly for su 3 can be traded for a holomorphic anomaly which is shown to factor into a product of the partition functions for lower rank gauge groups the su 3 partition function is mathematically an example of a mock modular form of depth two

relations among dirichlet series whose coefficients are class numbers of binary cubic forms we study the zeta functions for the space of binary cubic forms introduced by shintani the zeta function is defined for each invariant lattice we classify the invariant lattices and investigate explicit relationships between the zeta functions associated with those lattices we also study the analytic properties of the zeta functions and rewrite shintani s functional equation in a self dual form using an explicit relation

uniqueness and stability in determining the heat radiative coefficient the initial temperature and a boundary coefficient in a parabolic equation we consider an inverse parabolic problem we prove that the heat radiative coefficient the initial temperature and a boundary coefficient can be simultaneously determined from the final overdetermination provided that the heat radiative coefficient is a priori known in a small subdomain moreover we establish a stability estimate for this inverse problem

stability estimate for the hyperbolic inverse boundary value problem by local dirichlet to neumann map abstract in this paper we consider the stability of the inverse problem of determining a function q x in a wave equation t 2 u u q x u 0 in a bounded smooth domain in r n from boundary observations this information is enclosed in the hyperbolic dynamic dirichlet to neumann map associated to the solutions to the wave equation we prove in the case of n 2 that q x is uniquely determined by the range restricted to a subboundary of the dirichlet to neumann map whose stability is a type of double logarithm

on the existence of an inverse problem in this paper we consider the existence of the solution to the inverse problem to identify q u satisfying tu i aij x t ju bi x t iu q x c x t u f x t vu 0 u t 0 g x u t t z x we use the galerkin method to reduce the above mentioned problem to a finite dimensional problem and then use the degree theory to obtain the existence of the inverse problem when the unknown parameter q is of definite signs

alternative models for stock price dynamics abstract this paper evaluates the role of various volatility specifications such as multiple stochastic volatility sv factors and jump components in appropriate modeling of equity return distributions we use estimation technology that facilitates nonnested model comparisons and use a long data set which provides rich information about the conditional and unconditional distribution of returns we consider two broad families of models 1 the multifactor loglinear family and 2 the affine jump family both classes of models have attracted much attention in the derivatives and econometrics literatures there are various tradeoffs in considering such diverse specifications if pure diffusion sv models are chosen over jump diffusions it has important implications for hedging strategies if logarithmic models are chosen over affine ones it may seriously complicate option pricing comparing many different specifications of pure diffusion multifactor models and jump diffusion models we find that 1 log linear models have to be extended to two factors with feedback in the mean reverting factor 2 affine models have to have a jump in returns stochastic volatility or probably both models 1 and 2 are observationally equivalent on the data set in hand in either 1 or 2 the key is that the volatility can move violently as we obtain models with comparable empirical fit one must make a choice based on arguments other than statistical goodness of fit criteria the considerations include facility to price options to hedge and parsimony the affine specification with jumps in volatility might therefore be preferred because of the closed form derivatives prices

simultaneous reconstruction of space dependent heat transfer coefficients and initial temperature abstract many complex physical phenomena and engineering systems e g in heat exchanges reflux condensers combustion chambers nuclear vessels etc due to the high temperatures high pressures hostile environment involved possess certain properties which are inaccessible to measure and therefore their influence determination using inverse analysis is very important and desirable in this spirit the purpose of this paper is to mathematically formulate and analyse a new inverse problem in which given measurements of temperature at two different instants it is required to obtain the space dependent heat transfer coefficients htcs and the initial temperature this simultaneous identification is challenging since it is both nonlinear and ill posed the uniqueness of solution is established based on the max min principle for parabolic equations and the contraction mapping principle for the existence and uniqueness of a fixed point the novel inverse mathematical model that is proposed offers appropriate scientific guidance to the polymer heat transfer processing industry as to which data to measure provide in order to be able to reliably determine the desirable htcs along with the initial temperature which is in general unknown furthermore for the reconstruction the surface htc is determined separately whilst the variational formulation is introduced for the simultaneous determination of the domain htc and the initial temperature the frechet gradient of the minimizing objective functional is derived the numerical reconstruction process is based on the conjugate gradient method cgm regularized by the discrepancy principle accurate and stable numerical solutions are obtained even in the presence of noise in the input temperature data since noisy data are invented the study models realistic practical situations in which temperature measurements recorded using sensors or thermocouples are inherently contaminated with random noise

fourier mukai transform on abelian surfaces we study moduli spaces of stable sheaves on abelian surfaces whose mukai vectors are related by a cohomological fourier mukai transform we show that there is a fourier mukai transform inducing a birational map between them

financial asset returns direction of change forecasting and volatility dynamics we consider three sets of phenomena that feature prominently in the financial economics literature 1 conditional mean dependence or lack thereof in asset returns 2 dependence and hence forecastability in asset return signs and 3 dependence and hence forecastability in asset return volatilities we show that they are very much interrelated and explore the relationships in detail among other things we show that 1 volatility dependence produces sign dependence so long as expected returns are nonzero so that one should expect sign dependence given the overwhelming evidence of volatility dependence 2 it is statistically possible to have sign dependence without conditional mean dependence 3 sign dependence is not likely to be found via analysis of sign autocorrelations runs tests or traditional market timing tests because of the special nonlinear nature of sign dependence so that traditional market timing tests are best viewed as tests for sign dependence arising from variation in expected returns rather than from variation in volatility or higher moments 4 sign dependence is not likely to be found in very high frequency e g daily or very low frequency e g annual returns instead it is more likely to be found at intermediate return horizons and 5 the link between volatility dependence and sign dependence remains intact in conditionally non gaussian environments for example with time varying conditional skewness and or kurtosis

s duality and refined bps indices whenever available refined bps indices provide considerably more information on the spectrum of bps states than their unrefined version extending earlier work on the modularity of generalized donaldson thomas invariants counting d4 d2 d0 brane bound states in type iia strings on a calabi yau threefold mathfrak y we construct the modular completion of generating functions of refined bps indices supported on a divisor class although for compact mathfrak y the refined indices are not protected switching on the refinement considerably simplifies the construction of the modular completion furthermore it leads to a non commutative analogue of the tba equations which suggests a quantization of the moduli space consistent with s duality in contrast for a local cy threefold given by the canonical bundle over a complex surface s refined dt invariants are well defined and equal to vafa witten invariants of s our construction provides a modular completion of the generating function of these refined invariants for arbitrary rank in cases where all reducible components of the divisor class are collinear which occurs e g when b 2 mathfrak y 1 or in the local case we show that the holomorphic anomaly equation satisfied by the completed generating function truncates at quadratic order in the local case it agrees with an earlier proposal by minahan et al for unrefined invariants and extends it to the refined level using the afore mentioned non commutative structure finally we show that these general predictions reproduce known results for u 2 and u 3 vafa witten theory on mathrm p 2 and make them explicit for u 4

adiabatic limits of η invariants and the meyer functions we construct a function on the orbifold fundamental group of the moduli space of smooth theta divisors which we call the meyer function for smooth theta divisors in the construction we use the adiabatic limits of the invariants of the mapping torus of theta divisors we shall prove that the meyer function for smooth theta divisors cobounds the signature cocycle and we determine the values of the meyer function for the dehn twists in particular we give an analytic construction of the meyer function of genus two

atiyah class and sheaf counting on local calabi yau fourfolds abstract we discuss donaldson thomas dt invariants of torsion sheaves with 2 dimensional support on a smooth projective surface in an ambient non compact calabi yau fourfold given by the total space of a rank 2 bundle on the surface we prove that in certain cases when the rank 2 bundle is chosen appropriately the universal truncated atiyah class of these codimension 2 sheaves reduces to one defined over the moduli space of such sheaves realized as torsion codimension 1 sheaves in a noncompact divisor threefold embedded in the ambient fourfold such reduction property of universal atiyah class enables us to relate our fourfold dt theory to a reduced dt theory of a threefold and subsequently then to the moduli spaces of sheaves on the base surface using results in 15 16 we finally make predictions about modularity of such fourfold invariants when the base surface is an elliptic k3