Let M be a complete Kähler manifold whose universal covering is biholomorphic to a ball Bm(R0)(0

For a measure μ preserved by a C1+α0 (α0>0) diffeomorphism f and a continuous function ϕ on the manifold, we study the relationship between the exponential growth rate of ∑eSnϕ(x) over the orbits of some periodic points and the free energy hμ(f)+∫ϕdμ (In certain cases, it is equal to the measure theoretic pressure) or the topological pressure. When μ is an ergodic hyperbolic measure, we prove that the exponential growth rate coicides with the free energy (measure theoretic pressure). And we also verify the equality of the exponential growth rate and the topological pressure when the manifold is 2-dimensional. However, for the higher-dimensional manifold, we show an inequality between the exponential growth rate and the topological pressure. For an ergodic hyperbolic measure ω, we also prove that there is a ω-full measured set Λ˜ such that for every f-invariant measure supported on Λ˜, the exponential growth rate equals to the free energy. And moreover, we prove that there is another ω-full measured set Δ˜ such that for every f-invariant measure supported on Δ˜, the exponential growth rate equals to the topological pressure.

We study measurable dependence of measures on a parameter in the following two classical problems: constructing conditional measures and the Kantorovich optimal transportation. For parametric families of measures and mappings we prove the existence of conditional measures measurably depending on the parameter. A particular emphasis is made on the Borel measurability (which cannot be always achieved). Our second main result gives sufficient conditions for the Borel measurability of optimal transports and transportation costs with respect to a parameter in the case where marginal measures and cost functions depend on a parameter. As a corollary we obtain the Borel measurability with respect to the parameter for disintegrations of optimal plans. Finally, we show that the Skorohod parametrization of measures by mappings can be also made measurable with respect to a parameter.

Magnitude-preserving ranking (MPRank) under Tikhonov regularization framework has shown competitive performance on information retrial besides theoretical advantages for computation feasibility and statistical guarantees. In this paper, we further characterize the learning rate and asymptotic bias of MPRank, and then propose a new debiased ranking algorithm. In terms of the operator representation and approximation techniques, we establish their convergence rates and bias characterizations. These theoretical results demonstrate that the new model has smaller asymptotic bias than MPRank, and can achieve the satisfactory convergence rate under appropriate conditions. In addition, some empirical examples are provided to verify the effectiveness of our debiased strategy.

In this paper, the problem of global finite-time stabilization of bilinear control systems by means of homogeneous feedback law is investigated. We prove under some reasonable assumptions on the operators A and B that continuous bounded and discontinuous unbounded feedbacks stabilize globally in finite-time the closed loop system. For illustrative, the examples of heat, Schrödinger and transport equations are considered where homogeneous stabilizing feedback is built for these systems.

The large-time asymptotics of the density matrix solving a drift-diffusion-Poisson model for the spin-polarized electron transport in semiconductors is proved. The equations are analyzed in a bounded domain with initial and Dirichlet boundary conditions. If the relaxation time is sufficiently small and the boundary data is close to the equilibrium state, the density matrix converges exponentially fast to the spinless near-equilibrium steady state. The proof is based on a reformulation of the matrix-valued cross-diffusion equations using spin-up and spin-down densities as well as the perpendicular component of the spin-vector density, which removes the cross-diffusion terms. Key elements of the proof are time-uniform positive lower and upper bounds for the spin-up and spin-down densities, derived from the De Giorgi–Moser iteration method, and estimates of the relative free energy for the spin-up and spin-down densities.

In this paper, we establish the uniquely existence of the global mild solution to the nematic liquid crystal equations in Besov-Morrey spaces. Some self-similarity and large time behavior of the global mild solution are also investigated.

In this paper, unstable metric entropy, unstable topological entropy and unstable pressure for partially hyperbolic endomorphisms are introduced and investigated. A version of Shannon-McMillan-Breiman Theorem is established, and a variational principle is formulated, which gives a relationship between unstable metric entropy and unstable pressure (unstable topological entropy). As an application of the variational principle, some results on the u-equilibrium states are given.

Because of a mistake in the proof of [10, Theorem A - part 1], the main statements of [10] ([10, Theorem 1 - part 1] and [10, Theorem 2]) are not proved in full generality. We provide an alternative proof to such statements.

In this manuscript we consider the stability of periodic solutions to Lambda-Omega lattice dynamical systems. In particular, we show that an appropriate ansatz casts the lattice dynamical system as an infinite-dimensional fast-slow differential equation. In a neighborhood of the periodic solution an invariant slow manifold is proven to exist, and that this slow manifold is uniformly exponentially attracting. The dynamics of solutions on the slow manifold become significantly more complicated and require a more delicate treatment. We present sufficient conditions to guarantee convergence on the slow manifold which is algebraic, as opposed to exponential, in the slow-time variable. Of particular interest to our work in this manuscript is the stability of a rotating wave solution, recently found to exist in the Lambda-Omega systems studied herein.

We develop the counterpart of weak KAM theory for potential mean field games. This allows to describe the long time behavior of time-dependent potential mean field game systems. Our main result is the existence of a limit, as time tends to infinity, of the value function of an optimal control problem stated in the space of measures. In addition, we show a mean field limit for the ergodic constant associated with the corresponding Hamilton-Jacobi equation.

We consider a population structured by a space variable and a phenotypical trait, submitted to dispersion, mutations, growth and nonlocal competition. This population is facing an environmental gradient: the optimal trait for survival depends linearly on the spatial variable. The survival or extinction depends on the sign of an underlying principal eigenvalue. We investigate the survival case when the initial data satisfies a so-called heavy tail condition in the space-trait plane. Under these assumptions, we show that the solution propagates in the favorable direction of survival by accelerating. We derive some precise estimates on the location of the level sets corresponding to the total population in the space variable, regardless of their traits. Our analysis also reveals that the orientation of the initial heavy tail is of crucial importance.

We show the existence of solution in the maximal Lp−Lq regularity framework to a class of symmetric parabolic problems on a uniformly C2 domain in Rn. Our approach consist in showing R - boundedness of families of solution operators to corresponding resolvent problems first in the whole space, then in half-space, perturbed half-space and finally, using localization arguments, on the domain. Assuming additionally boundedness of the domain we also show exponential decay of the solution. In particular, our approach does not require assuming a priori the uniform Lopatinskii - Shapiro condition.

In this work we give an asymptotic lower bound for the Hilbert number for real planar polynomial differential systems. This lower bound equals, up to leading order, to the best existing one, but the method we provide is new and involves slow-fast systems. The construction strongly relies on generalized Liénard systems.

We develop techniques to capture the effect of transport on the long-term dynamics of small, localized initial data in nonlinearly coupled reaction-diffusion-advection equations on the real line. It is well-known that quadratic or cubic nonlinearities in such systems can lead to growth of small, localized initial data and even finite time blow-up. We show that, if the components exhibit different velocities, then quadratic or cubic mixed-terms, i.e. terms with nontrivial contributions from both components, are harmless. We establish global existence and diffusive Gaussian-like decay for exponentially and algebraically localized initial conditions allowing for quadratic and cubic mixed-terms. Our proof relies on a nonlinear iteration scheme that employs pointwise estimates. The situation becomes very delicate if other quadratic or cubic terms are present in the system. We provide an example where a quadratic mixed-term and a Burgers'-type coupling can compensate for a cubic term due to differences in velocities.

We consider a simplified chemotaxis model of tumor angiogenesis, described by a Keller-Segel system on the two dimensional infinite cylindrical domain (x,y)∈R×Sλ, where Sλ is the circle of perimeter λ>0. The domain models a virtual channel where newly generated blood vessels toward the vascular endothelial growth factor will be located. The system is known to allow planar traveling wave solutions of an invading type. In this paper, we establish the nonlinear stability of these traveling invading waves when chemical diffusion is present if λ is sufficiently small. The same result for the corresponding system in one-dimension was obtained by Li-Li-Wang (2014) [17]. Our result solves the problem remained open in Chae-Choi-Kang-Lee (2018) [3] at which only linear stability of the planar traveling waves was obtained under certain artificial assumption.

The FitzHugh-Nagumo (FHN) equation is a simplified model of a nerve axon. We explore the controllability of this model using a localized interior control only for the first equation. The Linearized system is not null controllable using a localized interior control since the spectrum of the linearized system has an accumulation point though it is approximate controllable. We show that the solution of the FHN equation fails to be globally approximate controllable in a given time. But it is possible to move from any steady state to any other steady state arbitrarily close after some appropriate time by a localized interior control, provided that both steady states are in the same connected component of the set of steady states. Finally we make some additional remarks and comments and we mention some open questions for our system. For the sake of completeness, we give the details of the existence, uniqueness and uniform bound of the solution in Appendix.

We study the asymptotic properties of the trajectories of a discrete-time random dynamical system in an infinite-dimensional Hilbert space. Under some natural assumptions on the model, we establish a multiplicative ergodic theorem with an exponential rate of convergence. The assumptions are satisfied for a large class of parabolic PDEs, including the 2D Navier–Stokes and complex Ginzburg–Landau equations perturbed by a non-degenerate bounded random kick force. As a consequence of this ergodic theorem, we derive some new results on the statistical properties of the trajectories of the underlying random dynamical system. In particular, we obtain large deviations principle for the occupation measures and the analyticity of the pressure function in a setting where the system is not irreducible. The proof relies on a refined version of the uniform Feller property combined with some contraction and bootstrap arguments.

We study the zero surface tension limit of three-dimensional interfacial Darcy flow. We start with a proof of well-posedness of three-dimensional interfacial Darcy flow for any positive value of the surface tension coefficient. The primary tool for this well-posedness proof is an energy estimate. The time of existence for these solutions will, in general, go to zero with the surface tension parameter. However, in the case that a stability condition is satisfied by the initial data, we prove an additional energy estimate, establishing that the time of existence can be made uniform in the surface tension parameter. Then, an additional estimate allows the limit to be taken as surface tension vanishes, demonstrating that three-dimensional interfacial Darcy flow without surface tension is the limit of three-dimensional interfacial Darcy flow with surface tension as surface tension vanishes. This provides a new proof of existence of solutions for the problem without surface tension.

In this paper, we study some overdetermined boundary value problems for anisotropic elliptic PDEs in a bounded domain Ω in a convex cone of Rn. By using some integral identities and maximum principle, we prove the corresponding Wulff shape characterizations, which includes the classical Serrin's overdetermined boundary value problem.

We consider the Gierer-Meinhardt system with small inhibitor diffusivity and very small activator diffusivity on a compact two-dimensional Riemannian manifold without boundary. We study steady state solutions which are far from spatial homogeneity. We construct two different spike clusters, each consisting of two spikes, which both approach the same nondegenerate local maximum point of the Gaussian curvature. We show that one of these spike clusters is stable, the other one is unstable.

We study the behavior of various Lyapunov functionals (relative entropies) along the solutions of a family of nonlinear drift-diffusion-reaction equations coming from statistical mechanics and population dynamics. These equations can be viewed as gradient flows over the space of Radon measures equipped with the Hellinger-Kantorovich distance. The driving functionals of the gradient flows are not assumed to be geodesically convex or semi-convex. We prove new isoperimetric-type functional inequalities, allowing us to control the relative entropies by their productions, which yields the exponential decay of the relative entropies.

The persistence of periodic, grazing and impact solutions is studied for periodically perturbed ordinary differential equations with impacts. An approach of the Poincaré-Adronov-Melnikov method is applied. It is based on introducing and studying an appropriate impact Poincaré mapping. Examples are presented to illustrate theoretical results.

We study the null controllability of linear shadow models for reaction-diffusion systems arising as singular limits when the diffusivity of some of the components is very high. This leads to a coupled system where one component solves a parabolic partial differential equation (PDE) and the other one an ordinary differential equation (ODE). We analyze these shadow systems from a controllability perspective and prove two types of results. First, by employing Carleman inequalities and ODE arguments, we prove that the null controllability of the shadow model holds. This result, together with the effectiveness of the controls to control the original dynamics, is illustrated by numerical simulations. We also obtain a uniform Carleman estimate for the reaction-diffusion equations which allows to obtain the null control for the shadow system as a limit when the diffusivity tends to infinity in one of the equations.

In this paper, we study complex isochronous center problem for cubic complex planar vector fields, which are assumed to be Z2-equivariant with two symmetric centers. Such integrable systems can be classified as 11 cases. A complete classification is given on the complex isochronous centers and proven to have a total of 54 cases. All the algebraic conditions for the 54 cases are derived and, moreover, all the corresponding linearization transformations are obtained. This problem for the Z2-equivariant with two symmetric centers has been completely solved.

We establish a Sturm–Liouville theorem for quadratic operator pencils with matrix-valued potentials counting their unstable real roots, with applications to stability of waves. Such pencils arise, for example, in reduction of eigenvalue systems to higher-order scalar problems.

In this paper we study speed selection for traveling wavefronts of the lattice Lotka-Volterra competition model. For the linear speed selection, by constructing new types of upper solutions to the system, we widely extend the results in the literature. We prove that, for the nonlinear speed selection, the wavefront of the first species decays with a faster rate at the far end. This enables us to construct novel lower solutions to establish the existence of pushed wavefronts, a topic that has been understudied. We raise a new conjecture related to the classical Hosono's version of the diffusive system and our numerical simulations help to confirm it, while our rigorous results only provide a partial answer.

Our goal is to study the uniqueness of bounded entropy solutions for a multidimensional conservation law including a non-Lipschitz convection term and a diffusion term of non-local porous medium type. The non-locality is given by a fractional power of the Laplace operator. For a wide class of nonlinearities, the L1-contraction principle is established, despite the fact that the “finite-infinite” speed of propagation (Alibaud (2007) [1]) cannot be exploited in our framework; existence is deduced with perturbation arguments. The method of proof, adapted from Andreianov and Maliki (2010) [9], requires a careful analysis of the action of the fractional laplacian on truncations of radial powers.

In this paper, we first prove some propositions of Sobolev spaces defined on a locally finite graph G=(V,E), which are fundamental when dealing with equations on graphs under the variational framework. Then we consider a nonlinear biharmonic equationΔ2u−Δu+(λa+1)u=|u|p−2u on G=(V,E). Under some suitable assumptions, we prove that for any λ>1 and p>2, the equation admits a ground state solution uλ. Moreover, we prove that as λ→+∞, the solutions uλ converge to a solution of the equation{Δ2u−Δu+u=|u|p−2u,inΩ,u=0,on∂Ω, where Ω={x∈V:a(x)=0} is the potential well and ∂Ω denotes the boundary of Ω.

We study in a strip of R2 a combustion model of flame propagation with stepwise temperature kinetics and zero-order reaction, characterized by two free interfaces, respectively the ignition and the trailing fronts. The latter interface presents an additional difficulty because the non-degeneracy condition is not met. We turn the system to a fully nonlinear problem which is thoroughly investigated. When the width ℓ of the strip is sufficiently large, we prove the existence of a critical value Lec of the Lewis number Le, such that the one-dimensional, planar, solution is unstable for 0

In this paper, we prove a time dependent lower bound on density in the optimal order O(1/(1+t)) for the general smooth nonisentropic flow of compressible Euler equations.

We prove the finite time extinction property (u(t)≡0 on Ω for any t⩾T⋆, for some T⋆>0) for solutions of the nonlinear Schrödinger problem iut+Δu+a|u|−(1−m)u=f(t,x), on a bounded domain Ω of RN, N⩽3, a∈C with Im(a)>0 (the damping case) and under the crucial assumptions 0

We study an integro-difference equation that describes the spatial dynamics of a species in a shifting habitat. The growth function is nondecreasing in density and space for a given time, and shifts at a constant speed c. The spreading speeds for the model were previously studied. The contribution of the current paper is to provide sharp conditions for existence of forced traveling waves with speed c. We show the existence of traveling waves with zero value at ∞ or −∞ for c in different value ranges determined by the spreading speeds. We also show the existence of a traveling wave with any speed c for the case that the species can grow everywhere. Our results demonstrate the existence of different types of traveling waves with the same speed.

We consider the following nonlinear Neumann problem{−Δu−γu|x|2+μu=|u|2s⁎−2u|x|s in BR⊂RN,N≥3∂u∂ν=0 on ∂BR where γ<γ‾:=(N−2)24, 0

In this paper, we consider the asymptotic stability of rarefaction wave to the equilibrium diffusion limit equations without viscosity from radiation hydrodynamic. The present pressure includes a fourth order term about the absolute temperature from radiation effect as well as the ideal polytropic part, which brings the main difficulty to prove the asymptotic stability of the rarefaction wave. To overcome it, we impose an additional restriction condition on the density and the temperature at the far field, see (1.14). This condition is sufficient to achieve the a priori estimates of the solutions.

It is shown that the classical as well as quasilinear Keller-Segel systems with non-degenerate diffusion possess for given T-periodic and sufficiently small forcing functions a unique, strong T-time periodic solution. The proof given relies on the existence of strong T-periodic solutions for the linearized system, its characterization in terms of maximal Lp-regularity of the underlying operator and a quasilinear version of the Arendt-Bu Theorem. The latter is of independent interest and yields the existence of strong T-periodic solutions to general quasilinear evolution equations under suitable conditions on the operators and the forcing terms.

The models introduced in this paper describe a uniform distribution of plant stems competing for sunlight. The shape of each stem, and the density of leaves, are designed in order to maximize the captured sunlight, subject to a cost for transporting water and nutrients from the root to all the leaves. Given the intensity of light, depending on the height above ground, we first solve the optimization problem determining the best possible shape for a single stem. We then study a competitive equilibrium among a large number of similar plants, where the shape of each stem is optimal given the shade produced by all others. Uniqueness of equilibria is proved by analyzing the two-point boundary value problem for a system of ODEs derived from the necessary conditions for optimality.

In this paper, we mainly study a two-species competition model in a one-dimensional advective homogeneous environment, where the two species are identical except their diffusion rates. One interesting feature of the model is that the boundary condition at the downstream end represents a net loss of individuals, which is tuned by a parameter b to measure the magnitude of the loss. When the upstream end has the no-flux condition, Lou and Zhou (2015) [11] have confirmed that large diffusion rate is more favorable when 0≤b<1. Here we consider the case where the upstream end has the free-flow condition, which means that the upstream end is linked to a lake. We firstly investigate the corresponding single species model. Here we establish the existence and uniqueness of positive steady states. Then for the two-species model, we find that the parameter b can be regarded as a bifurcation parameter. Precisely, when 0≤b<1, large diffusion rate is more favorable while when b>1, small diffusion rate is selected (if exists). When b=1, the system is degenerate in the sense that there is a compact global attractor consisting of a continuum of steady states.

This paper is concerned with the asymptotic behavior of solutions of the non-autonomous reaction-diffusion equations driven by nonlinear colored noise defined on unbounded domains. The nonlinear drift and diffusion terms are assumed to be continuous but not necessarily Lipschitz continuous which leads to the non-uniqueness of solutions. We prove the existence and uniqueness of pullback random attractors for the multi-valued non-autonomous cocycles generated by the solution operators. The measurability of the random attractors is established by the method based on the weak upper semicontinuity of the solutions. The asymptotic compactness of the solutions is derived by Ball’s idea of energy equations in order to overcome the non-compactness of Sobolev embeddings on unbounded domains.

We prove that, in any infinite dimensional or asymptotic Teichmüller space, the angles between Teichmüller geodesic rays issuing from a common point, defined by using the law of cosines, do not always exist. As a consequence, any infinite dimensional or asymptotic Teichmüller space equipped with the Teichmüller metric is not a CAT(κ) space for any κ∈R. We also establish a sufficient condition for the angles not to always exist in a Finsler manifold, and apply it to study the Hilbert metrics.

In the present paper, it is shown that the large amplitude viscous shock wave is nonlinearly stable for isentropic Navier-Stokes equations, in which the pressure could be general and includes γ-law, and the viscosity coefficient is a smooth function of density. The strength of shock wave could be arbitrarily large. The proof is given by introducing a new variable, which can formulate the original system into a new one, and the elementary energy method introduced in [21].

In this paper, we establish analyticity of solutions to the barotropic compressible Navier-Stokes equations describing the motion of the density ρ and the velocity field u in R3. We assume that ρ0 is a small perturbation of 1 and (1−1/ρ0,u0) are analytic in Besov spaces with analyticity radius ω>0. We show that the corresponding solutions are analytic globally in time when (1−1/ρ0,u0) are sufficiently small. To do this, we introduce the exponential operator e(ω−θ(t))D acting on (1−1/ρ,u), where D is the differential operator whose Fourier symbol is given by |ξ|1=|ξ1|+|ξ2|+|ξ3| and θ(t) is chosen to satisfy θ(t)<ω globally in time.

In this paper, we first introduce a new function space MHθ,p whose norm is given by the ℓp-sum of modulated Hθ-norms of a given function. In particular, when θ<−12, we show that the space MHθ,p agrees with the modulation space M2,p(R) on the real line and the Fourier-Lebesgue space FLp(T) on the circle. We use this equivalence of the norms and the Galilean symmetry to adapt the conserved quantities constructed by Killip-Vişan-Zhang to the modulation space and Fourier-Lebesgue space setting. By applying the scaling symmetry, we then prove global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation (NLS) in almost critical spaces. More precisely, we show that the cubic NLS on R is globally well-posed in M2,p(R) for any p<∞, while the renormalized cubic NLS on T is globally well-posed in FLp(T) for any p<∞. In Appendix, we also establish analogous global-in-time bounds for the modified KdV equation (mKdV) in the modulation spaces on the real line and in the Fourier-Lebesgue spaces on the circle. An additional key ingredient of the proof in this case is a Galilean transform which converts the mKdV to the mKdV-NLS equation.

In this paper we consider a spectral problem for ordinary differential equations of fourth order with the spectral parameter contained in three of the boundary conditions. We study the oscillatory properties of the eigenfunctions and, using these properties, we obtain sufficient conditions for the system of eigenfunctions of the problem in question to form a basis in the space Lp(0,1),1

In this paper, we establish the existence of a 1-parameter family of spatially inhomogeneous radially symmetric classical self-similar solutions to a Cauchy problem for a semi-linear parabolic PDE with non-Lipschitz nonlinearity and trivial initial data. Specifically we establish well-posedness for an associated initial value problem for a singular two-dimensional non-autonomous dynamical system with non-Lipschitz nonlinearity. Additionally, we establish that solutions to the initial value problem converge algebraically to the origin and oscillate as η→∞.

In this note, by analyzing the behavior at infinity of the matrix symbol of an invariant operator P with respect to a fixed elliptic operator, we obtain a necessary and sufficient condition to guarantee that P is globally hypoelliptic. As an application, we obtain the characterization of global hypoellipticity on compact Lie groups and examples on the sphere and the torus. We also investigate relations between the global hypoellipticity of P and global subelliptic estimates.

In this paper, we obtain the boundary pointwise C1,α and C2,α regularity for viscosity solutions of fully nonlinear elliptic equations. That is, if ∂Ω is C1,α (or C2,α) at x0∈∂Ω, the solution is C1,α (or C2,α) at x0. Our results are new even for the Laplace equation. Moreover, our proofs are simple.

In the present study, we consider the chemotaxis system with logistic-type superlinear degradation{∂tu1=τ1Δu1−χ1∇⋅(u1∇v)+λ1u1−μ1u1k1,x∈Ω,t>0,∂tu2=τ2Δu2−χ2∇⋅(u2∇v)+λ2u2−μ2u2k2,x∈Ω,t>0,0=Δv−γv+α1u1+α2u2,x∈Ω,t>0, under the homogeneous Neumann boundary condition, where γ>0, τi>0, χi>0, λi∈R, μi>0 , αi>0 (i=1,2). Consider an arbitrary ball Ω=BR(0)⊂Rn,n≥3,R>0, when ki>1(i=1,2), it is shown that for any parameter kˆ=max⁡{k1,k2} satisfieskˆ<{76ifn∈{3,4},1+12(n−1)ifn≥5, there exist nonnegative radially symmetric initial data under suitable conditions such that the corresponding solutions blow up in finite time in the sense thatlim supt↗Tmax(‖u1(⋅,t)‖L∞(Ω)+‖u2(⋅,t)‖L∞(Ω))=∞for some0

In this paper, we present new upper and lower bounds for the Mills ratio of the standard Gaussian law. Several different methods are used to derive these new bounds. One of the methods reproduces the bounds of several different authors in previous works as special cases and is a very general method that produces many new bounds. One of the bounds can be written in terms of hyperbolic sine and inverse hyperbolic sine functions. Some of the bounds involve exponential functions and are improved versions of previously proposed bounds or are improved versions of the new bounds introduced earlier in this paper. Some results from reliability theory and Jensen's inequality are used to improve determinantal inequalities. Some open problems are discussed, and conjectures are made.

The PROJ method for pricing European options on one underlying asset was proposed by J.Lars Kirkby and then was applied to price Bermudan and Asian options. In this paper, we extend the method to higher dimensions, especially two-dimensions in which some exotic options can be priced. Our method does not rely on a-priori truncation of the integration range and exhibits excellent performance compared with other state-of-the-art methods, particularly for fatter-tailed short maturity models. We also discuss the errors introduced in each approximation and give corresponding error bounds. Numerical results on implementation of this method to price for popular two-assets options, under both the geometric Brownian Motion and Variance-Gamma dynamics, demonstrate remarkable accuracy and robustness.

We study the ground state problem of the nonlinear Schrödinger functional with a mass-critical inhomogeneous nonlinear term. We provide the optimal condition for the existence of ground states and show that in the critical focusing regime there is a universal blow-up profile given by the unique optimizer of a Gagliardo-Nirenberg interpolation inequality.

In this note prove the following Berwald-type inequality, showing that for any integrable log-concave function f:Rn→[0,∞) and any concave function h:L→[0,∞), where L={(x,t)∈Rn×[0,∞):f(x)≥e−t‖f‖∞}, thenp→(1Γ(1+p)∫Le−tdtdx∫Lhp(x,t)e−tdtdx)1p is decreasing in p∈(−1,∞), extending the range of p where the monotonicity is known to hold true. As an application of this extension, we will provide a new proof of a functional form of Zhang's reverse Petty projection inequality, recently obtained in [3].

We derive Mehler–Heine type asymptotic expansions for the Krawtchouk polynomials Kn(x;p,N) with n=O(N) and x=o(N). These formulas provide good approximations for Kn(x;p,N) and determine the asymptotic limit of their zeros as n→∞.

In this work, we mainly explore the L2-regularity for the second component of the solutions to linear backward stochastic heat equations, which is crucial to obtain the convergence of the numerical solutions. As an application, we provide the convergence rate for time-discretized Galerkin approximation of these equations.

Let G be a non-compact locally compact group with a continuous submultiplicative weight function ω such that ω(e)=1 and ω is diagonally bounded with bound K≥1. When G is σ-compact, we show that ⌊K⌋+1 many points in the spectrum of LUC(ω−1) are enough to determine the topological centre of LUC(ω−1)⁎ and that ⌊K⌋+2 many points in the spectrum of L∞(ω−1) are enough to determine the topological centre of L1(ω)⁎⁎ when G is in addition a SIN-group. We deduce that the topological centre of LUC(ω−1)⁎ is the weighted measure algebra M(ω) and that of C0(ω−1)⊥ is trivial for any locally compact group. The topological centre of L1(ω)⁎⁎ is L1(ω) and that of of L0∞(ω)⊥ is trivial for any non-compact locally compact SIN-group. The same techniques apply and lead to similar results when G is a weakly cancellative right cancellative discrete semigroup.

We study the Yamabe flow on a Riemannian manifold of dimension m≥3 minus a closed submanifold of dimension n and prove that there exists an instantaneously complete solution if and only if n>m−22. In the remaining cases 0≤n≤m−22 including the borderline case, we show that the removability of the n-dimensional singularity is necessarily preserved along the Yamabe flow. In particular, the flow must remain geodesically incomplete as long as it exists. This is contrasted with the two-dimensional case, where instantaneously complete solutions always exist.

Let M be an n-dimensional smooth oriented complete embedded minimal hypersurface in Rn+1 with Euclidean volume growth. We show that if the image under the Gauss map of M avoids some neighborhood of a half-equator, then M must be an affine hyperplane.

This note contains a new characterization of modulation spaces Mmp(Rn), 1≤p≤∞, by symplectic rotations. Precisely, instead to measure the time-frequency content of a function by using translations and modulations of a fixed window as building blocks, we use translations and metaplectic operators corresponding to symplectic rotations. Technically, this amounts to replace, in the computation of the Mmp(Rn)-norm, the integral in the time-frequency plane with an integral on Rn×U(2n,R) with respect to a suitable measure, U(2n,R) being the group of symplectic rotations. More conceptually, we are considering a sort of polar coordinates in the time-frequency plane. To have invariance under symplectic rotations we choose a Gaussian as suitable window function. We also provide a similar (and easier) characterization with the group U(2n,R) being reduced to the n-dimensional torus Tn.

In this note we extend to the quasi-reflexive setting the result of F. Baudier, N. Kalton and G. Lancien concerning the non-embeddability of the family of countably branching trees into reflexive Banach spaces whose Szlenk index and Szlenk index from the dual are both equal to the first infinite ordinal ω. In particular we show that the family of countably branching trees does neither embed into the James space Jp nor into its dual space Jp⁎ for p∈(1,∞).

We define the spaces of Schwartz functions, tempered functions and tempered distributions on manifolds definable in polynomially bounded o-minimal structures. We show that all the classical properties that these spaces have in the Nash category, as first studied in Fokko du Cloux's work, also hold in this generalized setting. We also show that on manifolds definable in o-minimal structures that are not polynomially bounded, such a theory can not be constructed. We present some possible applications, mainly in representation theory.