Topological entropy is not lower semi-continuous: small perturbation of the dynamical system can lead to a collapse of entropy. In this note we show that for some special classes of dynamical systems (geodesic flows, Reeb flows, positive contactomorphisms) topological entropy at least is stable in the sense that there exists a nontrivial continuous lower bound, given that a certain homological invariant

In this paper, we introduce a new cohomology theory associated to a strict Lie 2-algebra. This cohomology theory is shown to also extend the classical cohomology theory of Lie algebras; in particular, we show that the second cohomology group classifies an appropriate type of extension.

Mean value formulas are of great importance in the theory of partial differential equations: many very useful results are drawn, for instance, from the well-known equivalence between harmonic functions and mean value properties. In the nonlocal setting of fractional harmonic functions, such an equivalence still holds, and many applications are nowadays available. The nonlinear case, corresponding to

The topology of the moduli space for Lamé functions of degree m is determined: this is a Riemann surface which consists of two connected components when m≥2; we find the Euler characteristics and genera of these components. As a corollary we prove a conjecture of Maier on degrees of Cohn’s polynomials. These results are obtained with the help of a geometric description of these Riemann surfaces, as

In this paper, we use the Leray–Schauder degree theory to study the following singular periodic problems: x″+f(x′)+g(t,x)=s, x(0)−x(T)=0=x′(0)−x′(T), where f:ℝ→ℝ is a continuous function with f(0)=0, function g:ℝ/Tℤ×ℝ+→ℝ is continuous with an attractive singularity at the origin, and s is a constant. We consider the case where the friction term f satisfies a local superlinear growth condition but not

The main goal of this paper is to reveal the symplectic structure related to renormalization of circle maps with breaks. We first show that iterated renormalizations of 𝒞r circle diffeomorphisms with d breaks, r>2, with given size of breaks, converge to an invariant family of piecewise Möbius maps, of dimension 2d. We prove that this invariant family identifies with a relative character varietyχ(π1Σ

We consider homotopy actions of a Lie algebroid on a graded manifold, defined as suitable L∞-algebra morphisms. On the “semi-direct product” we construct a homological vector field that projects to the Lie algebroid. Our main theorem states that this construction is a bijection. Since several classical geometric structures can be described by homological vector fields as above, we can display many

In this paper, we prove the local null controllability property for a nonlinear coupled system of two Korteweg–de Vries equations posed on a bounded interval and with a source term decaying exponentially on t=T. The system was introduced by Gear and Grimshaw to model the interactions of two-dimensional, long, internal gravity waves propagation in a stratified fluid. We address the controllability problem

In this paper, we show the existence of infinitely many symmetric solutions for a cubic Dirac equation in two dimensions, which appears as effective model in systems related to honeycomb structures. Such equation is critical for the Sobolev embedding and solutions are found by variational methods. Moreover, we also prove smoothness and exponential decay at infinity.

In this paper, we consider Serrin’s overdetermined problems in warped product manifolds and we prove Serrin’s type rigidity results by using the P-function approach introduced by Weinberger.

We extend the theory of isospectral reductions of Bunimovich and Webb to infinite graphs, and describe an application of this extension to the problems of existence and approximation of stationary measures on infinite graphs.

In this paper, we study irreducible modules over the mirror Heisenberg–Virasoro algebra 𝔇, which is the semi-direct product of the Virasoro algebra and the twisted Heisenberg algebra. We classify all Harish-Chandra modules over 𝔇, i.e. irreducible modules with finite-dimensional weight spaces. Every such module is either an irreducible highest or an irreducible lowest weight module, or an irreducible

We refine and generalize several interpolation inequalities bounding the Lp norm of a probability density with respect to the reference measure μ by its Sobolev norm and the Kantorovich distance to μ on a smooth weighted Riemannian manifold satisfying CD(0,∞) condition.

We consider Pogorelov estimates and Liouville-type theorems to parabolic k-Hessian equations of the form −utσk(D2u)=1 in ℝn×(−∞,0]. We derive that any k+1-convex-monotone solution to −utσk(D2u)=1 when u(x,0) satisfies a quadratic growth and 0

Recently, both the bilinear decompositions h1(ℝn)×bmo(ℝn)⊂L1(ℝn)+h∗Φ(ℝn) and h1(ℝn)×bmo(ℝn)⊂L1(ℝn)+hlog(ℝn) were established. In this paper, the authors prove in some sense that the former is sharp, while the latter is not. To this end, the authors first introduce the local Orlicz-slice Hardy space which contains h∗Φ(ℝn), a variant of the local Orlicz Hardy space, introduced by Bonami and Feuto as

We study the zero Dirichlet problem for the equation −Δpu−Δqu=α|u|p−2u+β|u|q−2u in a bounded domain Ω⊂ℝN, with 1

Conformal geodesics are distinguished curves on a conformal manifold, loosely analogous to geodesics of Riemannian geometry. One definition of them is as solutions to a third-order differential equation determined by the conformal structure. There is an alternative description via the tractor calculus. In this article, we give a third description using ideas from holography. A conformal n-manifold

Assume Ω⊂ℝ2 is a planar domain, and u is a locally bounded distributional solution to the elliptic equation −Δu=|x|2βh(x)f(u)in Ω, where β>−1 is a constant, h and f are real analytic functions defined on Ω and the real line ℝ, respectively. We establish asymptotic expansions of u(x) to arbitrary orders near 0, which complements the recent results of Han–Li–Li on the Yamabe equation, Guo–Li–Wanon the

In this work we prove a Berestycki–Lions type result for the following class of problems: −Δ1u+u|u|=f(u)inℝN,u∈BV(ℝN), where Δ1 is the 1-Laplacian operator and f is a continuous function satisfying some technical conditions. Here we apply variational methods by using p-Laplacian problems and taking the limit when p→1+.

We construct families of commutative (super) algebra objects in the category of weight modules for the unrolled restricted quantum group U¯qH(𝔤) of a simple Lie algebra 𝔤 at roots of unity, and study their categories of local modules. We determine their simple modules and derive conditions for these categories being finite, non-degenerate, and ribbon. Motivated by numerous examples in the 𝔤=𝔰𝔩2

We study the existence and properties of birationally equivalent models for elliptically fibered varieties. In particular these have either the structure of Mori fiber spaces or, assuming some standard conjectures, minimal models with a Zariski decomposition compatible with the elliptic fibration. We prove relations between the birational invariants of the elliptically fibered variety, the base of

In this paper, we prove the existence of the C1,1-solution to the classical Neumann problem for the degenerate elliptic Hessian quotient equation σk(D2u)σl(D2u)=f(x) under the condition that f1k−l∈C2(Ω¯).

In this paper, we investigate a local to global principle for Galois cohomology of number fields with coefficients in the Tate module of an abelian variety. In [G. Banaszak and P. Krasoń, On a local to global principle in étale K-groups of curves, J. K-Theory Appl. Algebra Geom. Topol.12 (2013) 183–201], G. Banaszak and the author obtained the sufficient condition for the validity of the local to global

Characteristic classes of Schubert varieties can be used to study the geometry and the combinatorics of homogeneous spaces. We prove a relation between elliptic classes of Schubert varieties on a generalized full flag variety and those on its Langlands dual. This new symmetry is motivated by 3D mirror symmetry, and it is only revealed if Schubert calculus is elevated from cohomology or K theory to

In this paper, we are concerned with the following Dirichlet problem for nonlinear equations involving the fractional p-Laplacian: (−Δ)pαu=f(x,u,∇u),u>0in Ω,u≡0in ℝn∖Ω, where Ω is a bounded or an unbounded domain which is convex in x1-direction, and (−Δ)pα is the fractional p-Laplacian operator defined by (−Δ)pαu(x)=Cn,α,pP.V.∫ℝn|u(x)−u(y)|p−2[u(x)−u(y)]|x−y|n+αpdy. Under some mild assumptions on the

An explicit expression for the cofactor related to an irreducible invariant algebraic curve of a polynomial dynamical system in the plane is derived. A sufficient condition for a polynomial dynamical system in the plane to have a finite number of irreducible invariant algebraic curves is obtained. All these results are applied to Liénard dynamical systems xt=y, yt=−f(x)y−g(x) with degf

In this paper, we prove the existence of at least three nontrivial solutions for the following class of fractional Kirchhoff-type problems: (1+b∥u∥2)[(−Δ)1/2u+V(x)u]=f(u)in Ω,u=0in ℝ∖Ω, where b≥0 is a constant, Ω⊂ℝ is a bounded open interval, V:Ω¯→[0,+∞) is a continuous potential, the nonlinear term f(u) has exponential growth of Trudinger–Moser type, ∥u∥2=12π[u]1,22+∥V1/2u∥22 and [u]1,2 denotes the

We prove C1,α regularity (in the parabolic sense) for the viscosity solution of a boundary obstacle problem with a fully nonlinear parabolic equation in the interior. Following the method which was first introduced for the harmonic case by L. Caffarelli in 1979, we extend the results of I. Athanasopoulos (1982) who studied the linear parabolic case and the results of E. Milakis and L. Silvestre (2008)

Let M=G/H be an (n+1)-dimensional homogeneous manifold and Jk(n,M)=:Jk be the manifold of k-jets of hypersurfaces of M. The Lie group G acts naturally on each Jk. A G-invariant partial differential equation of order k for hypersurfaces of M (i.e., with n independent variables and 1 dependent one) is defined as a G-invariant hypersurface ℰ⊂Jk. We describe a general method for constructing such invariant

Assume that Φ:𝕄n(ℂ)→𝕄n(ℂ) is a superoperator which preserves hermiticity. We give an algorithm determining whether Φ preserves semipositivity (we call Φpositive in this case). Our approach to the problem has a model-theoretic nature, namely, we apply techniques of quantifier elimination theory for real numbers. An approach based on these techniques seems to be the only one that allows to decide whether

We determine the largest rate of exponential decay at infinity for non-trivial solutions to the Dirac equation 𝒟nψ+𝕍ψ=0in ℝn, being 𝒟n the massless Dirac operator in dimension n≥2 and 𝕍 a (possibly non-Hermitian) matrix-valued perturbation such that |𝕍(x)|∼|x|−𝜖 at infinity, for −∞<𝜖<1. Also, we show that our results are sharp for n∈{2,3}, providing explicit examples of solutions that have the

A quadruple of Lie groups (G,L,K,H), where G is a compact semisimple Lie group, H⊂K⊂L are closed subgroups of G, and the related Casimir constants satisfy certain appropriate conditions, is called a basic quadruple. A basic quadruple is called Einstein if the Killing form metrics on the coset spaces G/H, G/K and G/L are all Einstein. In this paper, we first give a complete classification of the Einstein

Let (M,𝜃) be a compact, connected, strictly pseudo-convex CR manifold. In this paper, we give some properties of the CR Yamabe Operator L𝜃. We present an upper bound for the Second CR Yamabe Invariant, when the First CR Yamabe Invariant is negative, and the existence of a minimizer for the Second CR Yamabe Invariant, under some conditions.

We study parafermion vertex algebras N−3/2(𝔰𝔩(2)) and N−3/2(𝔰𝔩(3)). Using the isomorphism between N−3/2(𝔰𝔩(3)) and the logarithmic vertex algebra 𝒲0(2)A2 from [D. Adamović, A realization of certain modules for the N=4 superconformal algebra and the affine Lie algebra A2(1), Transform. Groups21(2) (2016) 299–327], we show that these parafermion vertex algebras are infinite direct sums of irreducible

Let π:𝒳→Δ be a holomorphic family of compact complex manifolds over an open disk in ℂ. If the fiber π−1(t) for each nonzero t in an uncountable subset B of Δ is Moishezon and the reference fiber X0 satisfies the local deformation invariance for Hodge number of type (0,1) or admits a strongly Gauduchon metric introduced by D. Popovici, then X0 is still Moishezon. We also obtain a bimeromorphic embedding

Motivated by the limiting behavior of an explicit class of compact ancient curve shortening flows, by adapting the work of Colding–Minicozzi [11], we prove codimension bounds for ancient mean curvature flows by their tangent flow at −∞. In the case of the m-covered circle, we apply this bound to prove a strong rigidity theorem. Furthermore, we extend this paradigm by showing that under the assumption

To compute the unique formal normal form of families of vector fields with nilpotent linear part, we choose a basis of the Lie algebra consisting of orbits under the action of the nilpotent linear part. This creates a new problem: to find explicit formulas for the structure constants in this new basis. These are well known in the 2D case, and recently expressions were found for the 3D case by ad hoc

We consider the focusing nonlinear Klein–Gordon (NLKG) equation ∂ttu−Δu+u−|u|p−1u=0,(t,x)∈ℝ×ℝd for 1≤d≤5 and p>2 subcritical for the Ḣ1 norm. In this paper, we show the existence of a solution u(t) of the equation such that u(t)−∑k=1,2Qk(t)H1+∂tu(t)L2→0as t→+∞, where Qk(t,x) are two solitary waves of the equation with translations zk:ℝ→ℝd satisfying |z1(t)−z2(t)|∼2log(t)as t→+∞. This behavior is due

Our aim is to study the large time asymptotics of solutions to the fourth-order nonlinear Schrödinger equation in two space dimensions i∂tu+14Δ2u=λ|u|2u,t>0,x∈ℝ2,u(0,x)=u0(x),x∈ℝ2, where λ>0. We show that the nonlinearity has a dissipative character, so the solutions obtain more rapid time decay rate comparing with the corresponding linear case, if we assume the nonzero total mass condition ∫ℝu0(x)dx≠0

We consider the problem of minimizing or maximizing the quantity λ(Ø)Tq(Ø) on the class of open sets of prescribed Lebesgue measure. Here q>0 is fixed, λ(Ø) denotes the first eigenvalue of the Dirichlet Laplacian on H01(Ø), while T(Ø) is the torsional rigidity of Ø. The optimization problem above is considered in the class of all domainsØ, in the class of convex domainsØ, and in the class of thin domains

In this paper, we study two deformation procedures for quantum groups: deformations by twists, that we call “comultiplication twisting”, as they modify the coalgebra structure, while keeping the algebra one — and deformations by 2-cocycle, that we call “multiplication twisting”, as they deform the algebra structure, but save the coalgebra one. We deal with quantized universal enveloping algebras (in

Assuming that the differential field (K,δ) is differentially large, in the sense of [León Sánchez and Tressl, Differentially large fields, preprint (2020); arXiv:2005.00888], and “bounded” as a field, we prove that for any linear differential algebraic group G over K, the differential Galois (or constrained) cohomology set Hδ1(K,G) is finite. This applies, among other things, to closed ordered differential

In this paper, we consider a free boundary model in one space dimension which describes the spreading of a species subject to climate change, where favorable environment is shifting away with a constant speed c>0 and replaced by a deteriorated yet still favorable environment. We obtain two threshold speeds c1σ∗. Moreover, in the last case, while the spreading front propagates with asymptotic speed

We consider the deformation theory of two kinds of geometric objects: foliations on one hand, pre-symplectic forms on the other. For each of them, we prove that the geometric notion of equivalence given by isotopies agrees with the algebraic notion of gauge equivalence obtained from the L∞-algebras governing these deformation problems.

Let Rj denote the jth Riesz transform on ℝn. We prove that there exists an absolute constant C>0 such that |{|Rjf|>λ}|≤C1λ∥f∥L1(ℝn)+supν|{|Rjν|>λ}| for any λ>0 and f∈L1(ℝn), where the above supremum is taken over measures of the form ν=∑k=1Nakδck for N∈ℕ, ck∈ℝn, and ak∈ℝ+ with ∑k=1Nak≤16∥f∥L1(ℝn). This shows that to establish dimensional estimates for the weak-type (1,1) inequality for the Riesz transforms

The aim of this paper is to define a new type of cohomology for multiplicative Hom-Leibniz algebras which control deformations of Hom-Leibniz algebras. The cohomology and the associated deformation theory for Hom-Leibniz algebras as developed here are also extended to equivariant context, under the presence of finite group actions on Hom-Leibniz algebras.

In this paper, we shall classify all positive solutions of Δu=aup on the upper half space H=ℝ+n with nonlinear boundary condition ∂u/∂t=buq on ∂H for parameters a>0 and b<0. We will prove that for p≥(n+2)/(n−2),1≤q(n+2)/(n−2), 1≤q≤n/(n−2) (and n≥3) all positive solutions are functions of last variable; for p=(n+2)/(n−2),q=n/(n−2) (and n≥3) positive solutions must be either some functions depending

In this paper, we introduce the notion of Lévy α-stable distribution within the discrete setting. Using this notion, a subordination principle is proved, which relates a sequence of solution operators — given by a discrete C-semigroup — for the abstract Cauchy problem of first order in discrete-time, with a sequence of solution operators for the abstract Cauchy problem of fractional order 0<α<1 in

Motivated by the study of rationally connected fibrations, we study different notions of birationally simple fibrations. Our main result is the construction of maximal Chow constant and cohomologically constant fibrations. This paper is largely self-contained and we prove a number of basic properties of these fibrations. One application is to the classification of “rationalizations of singularities

This paper deals with a nonlocal diffusion elliptic eigenvalue problem. Specifically, the diffusion of the unknown variable at a point of the domain depends on its value in a neighborhood of the point. We apply bifurcation arguments and appropriate approximation to obtain our results. Some applications to the population dynamics will be given.

The known integral transforms of Funk–Radon type are applied to manifolds which have algebraic structure (planes, spheres, ellipsoids, hyperboloids etc.). A variety of new exact reconstructions is described in this paper for integral transforms of Funk–Radon type on smooth hypersurfaces Xn properly embedded in space Rn+1 which is endowed with an additional structure.

We investigate the behavior near zero of the integrated density of states for random Schrödinger operators Φ(−Δ)+Vω in L2(ℝd), d≥1, where Φ is a complete Bernstein function such that for some α∈(0,2], one has Φ(λ)≍λα/2, λ↘0, and Vω(x)=∑i∈ℤdqi(ω)W(x−i) is a random nonnegative alloy-type potential with compactly supported single site potential W. We prove that there are constants C,C̃,D,D̃>0 such that

The lens space Lp,q is the orbit space of a ℤp-action on the 3-sphere. We investigate polynomials of two complex variables that are invariant under this action, and thus define links in Lp,q. We study properties of these links, and their relationship with the classical algebraic links. We prove that all algebraic links in lens spaces are fibered, and obtain results about their Seifert genus. We find

In this paper, we analyze evolution problems associated to homogenous operators. We show that they have an homogenous associated semigroup of solutions that must satisfy some sharp estimates when acting on homogenous spaces and on the associated fractional power spaces. These sharp estimates are determined by the homogeneity alone. We also consider fractional diffusion problems and Schrödinger type

In this paper, we consider the following critical equation: −Δu+V(y)u=K(y)uN+2N−2,u>0,u∈H1(ℝN), where (y′,y″)∈ℝ2×ℝN−2, V(|y′|,y″) and K(|y′|,y″) are two nonnegative and bounded functions. Using a finite-dimensional reduction argument and local Pohozaev type of identities, we show that if N≥5, K(r,y″) has a stable critical point (r0,y0″) with r0>0,K(r0,y0″)>0 and B1:=V(r0,y0″)∫ℝNU0,12dy−ΔK(r0,y0″)2∗N∫ℝN|y|2U0

An AF algebra 𝔄 is said to be an AFℓ algebra if the Murray–von Neumann order of its projections is a lattice. Many, if not most, of the interesting classes of AF algebras existing in the literature are AFℓ algebras. We construct an algorithm which, on input a finite presentation (by generators and relations) of the Elliott semigroup of an AFℓ algebra 𝔄, generates a Bratteli diagram of 𝔄. We generalize

Let p∈(1,∞), ρ∈(2,∞) and W be a matrix Ap weight. In this paper, we introduce a version of variation 𝒱ρ(𝒯n,∗) for matrix Calderón–Zygmund operators with modulus of continuity satisfying the Dini condition. We then obtain the Lp(W)-boundedness of 𝒱ρ(𝒯n,∗) with norm ∥𝒱ρ(𝒯n,∗)∥Lp(W)→Lp(W)≤C[W]Ap1+1p−1−1p by first proving a sparse domination of the variation of the scalar Calderón–Zygmund operator

We consider the Cauchy problem of a third order in time nonlinear equation known as the Jordan–Moore–Gibson–Thompson (JMGT) equation arising in acoustics as an alternative model to the well-known Kuznetsov equation. We show a local existence result in appropriate function spaces, and, using the energy method together with a bootstrap argument, we prove a global existence result for small data, without

In this paper, by means of minimax techniques involving Cerami sequences, we prove the existence of at least one pair of positive solutions for a Hamiltonian system of Schrödinger equations in ℝN with potentials vanishing at infinity and subcritical nonlinearities which are superlinear at the origin and at infinity. We establish new estimates to prove the boundedness of a Cerami sequence.

The pointwise gradient estimate for weak solution pairs to the stationary Stokes system with Dini-BMO coefficients is established via the Havin–Maz’ya–Wolff type nonlinear potential of the nonhomogeneous term. In addition, we present a pointwise bound for the weak solutions under no extra regularity assumption on the coefficients.