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.

The classical Aleksandrov–Bakel’man–Pucci estimate (ABP estimate) for a second-order elliptic operator in nondivergence form is one of the fundamental tools for the bounds of subsolutions. Cabre improved the ABP estimate by replacing a constant factor, the diameter of a given domain, with a geometric character, which can be defined and finite for some unbounded domains. In the proof, Cabre used the

We construct Ionel–Parker’s proposed refinement of the standard relative Gromov–Witten invariants in terms of abelian covers of the symplectic divisor and discuss in what sense it gives rise to invariants. We use it to obtain some vanishing results for the standard relative Gromov–Witten invariants. In a separate paper, we describe to what extent this refinement sharpens the usual symplectic sum formula

We prove that every 2-Segal space is unital.

Let X⊆ℙN be a non-degenerate normal projective variety of codimension e and degree d with isolated ℚ-Gorenstein singularities. We prove that the Castelnuovo–Mumford regularity reg(𝒪X)≤d−e, as predicted by the Eisenbud–Goto regularity conjecture. Such a bound fails for general projective varieties by a recent result of McCullough–Peeva. The main techniques are Noma’s classification of non-degenerate

In this paper, by utilizing a newly established variational principle on convex sets, we provide an existence and multiplicity result for a class of semilinear elliptic problems defined on the whole ℝN with nonlinearities involving linear and superlinear terms. We shall impose no growth restriction on the nonlinear term, and consequently, our problem can be supercritical in the sense of the Sobolev

In this paper, we study solitary wave solutions of the Cauchy problem for Half-wave-Schrödinger equation in the plane. First, we show the existence and the orbital stability of the ground states. Second, we prove that given any speed v, traveling wave solutions exist and converge to the zero wave as the velocity tends to 1. Finally, we solve the Cauchy problem for initial data in Lx2Hys(ℝ2), with s>12

In his PhD thesis [G. Goosen, Oriented 123-tqfts via string-nets and state-sums, PhD thesis, Stellenbosch University, Stellenbosch (2018)], Goosen combined the string-net and the generators-and-relations formalisms for arbitrary once-extended 3-dimensional topological quantum field theories (TQFTs). In this paper, we work this out in detail for the simplest nontrivial example, where the underlying

In this paper, we establish a scale invariant Harnack inequality for the fractional powers of parabolic operators (∂t−ℒ)s, 0

Let G be a stratified homogeneous group with homogeneous dimension Q and whose Lie algebra is generated by the left-invariant vector fields X1,…,Xd1. Let 10. We prove that for any function f∈Ḟqα,p(G) there exists a function F∈L∞(G)∩Ḟqα,p(G) such that ∑i=1k∥Xi(f−F)∥Ḟqα−1,p(G)≤δ∥f∥Ḟqα,p(G),∥F∥L∞(G)+∥F∥Ḟqα,p(G)≤Cδ∥f∥Ḟqα,p(G) where k is the largest integer smaller than min(p,d1) and Cδ is a positive constant

We prove the analyticity of smooth critical points for O’Hara’s knot energies ℰα,p, with p=1 and 2<α<3, subject to a fixed length constraint. This implies, together with the already established regularity results for O’Hara’s knot energies, that bounded energy critical points of ℰα,1 subject to a fixed length constraint are not only C∞ but also analytic. Our approach is based on Cauchy’s method of

This work concerns with polynomial families of real planar vector fields having a monodromic nilpotent singularity. The families considered are those for which the centers are characterized by the existence of a formal inverse integrating factor vanishing at the singularity with a leading term of minimum (1,n)-quasihomogeneous weighted degree, being n the Andreev number of the singularity. These families

We improve known upper bounds for the minimal dispersion of a point set in the unit cube and its inverse in both the periodic and non-periodic settings. Some of our bounds are sharp up to logarithmic factors.

We study spectral problems for integro-differential equations arising in the theory of Gaussian processes similar to the fractional Brownian motion. We generalize the method of Chigansky–Kleptsyna and obtain the two-term eigenvalue asymptotics for such equations. Application to the small ball probabilities in L2-norm is given.

In this paper, we study the global regularity estimates in Lorentz spaces for gradients of solutions to quasilinear elliptic equations with measure data of the form −div(𝒜(x,∇u))=μin Ω,u=0on ∂Ω, where μ is a finite signed Radon measure in Ω, Ω⊂ℝn is a bounded domain such that its complement ℝn∖Ω is uniformly p-thick and 𝒜 is a Carathéodory vector-valued function satisfying growth and monotonicity

We obtain a pair of nontrivial solutions for a class of concave-linear-convex type elliptic problems that are either critical or subcritical. The solutions we find are neither local minimizers nor of mountain pass type in general. They are higher critical points in the sense that they each have a higher critical group that is nontrivial. This fact is crucial for showing that our solutions are nontrivial

We discuss a version of the Chevalley–Eilenberg cohomology in characteristic 2, where the alternating cochains are replaced by symmetric ones.

In this paper, we investigate the multiplicity of positive solutions for a class of Schrödinger–Poisson systems with concave and convex nonlinearities as follows: −Δu+λV(x)u+μϕu=a(x)|u|p−2u+b(x)|u|q−2uin ℝ3,−Δϕ=u2in ℝ3, where λ,μ>0 are two parameters, 1

In this paper, we consider the Hénon problem in the ball with Dirichlet boundary conditions. We study the asymptotic profile of radial solutions and then deduce the exact computation of their Morse index when the exponent p is close to 1. Next we focus on the planar case and describe the asymptotic profile of some solutions which minimize the energy among functions which are invariant for reflection

In this paper, we prove the existence of an extremal function for the Adams–Moser–Trudinger inequality on the Sobolev space H0m(Ω), where Ω is any bounded, smooth, open subset of ℝ2m, m≥1. Moreover, we extend this result to improved versions of Adams’ inequality of Adimurthi-Druet type. Our strategy is based on blow-up analysis for sequences of subcritical extremals and introduces several new techniques

Hecke–Kiselman monoids HKΘ and their algebras K[HKΘ], over a field K, associated to finite oriented graphs Θ are studied. In the case Θ is a cycle of length n≥3, a hierarchy of certain unexpected structures of matrix type is discovered within the monoid Cn=HKΘ and this hierarchy is used to describe the structure and the properties of the algebra K[Cn]. In particular, it is shown that K[Cn] is a right

We prove that the singularities of the R-matrix R(k) of the minimal quantization of the adjoint representation of the Yangian Y(𝔤) of a finite dimensional simple Lie algebra 𝔤 are the opposite of the roots of the monic polynomial p(k) entering in the OPE expansions of quantum fields of conformal weight 3/2 of the universal minimal affine W-algebra at level k attached to 𝔤.

This paper analyzes the superlinear indefinite prescribed mean curvature problem −div∇u/1+|∇u|2=λa(x)h(u)in Ω,u=0on ∂Ω, where Ω is a bounded domain in ℝN with a regular boundary ∂Ω, h∈C0(ℝ) satisfies h(s)∼sp, as s→0+, p>1 being an exponent with p0 represents a parameter, and a∈C0(Ω¯) is a sign-changing function. The main result establishes the existence of positive regular solutions when λ is sufficiently

In this work, we study a system of Schrödinger equations involving nonlinearities with quadratic growth. We establish sharp criterion concerned with the dichotomy global existence versus blow-up in finite time. Such a criterion is given in terms of the ground state solutions associated with the corresponding elliptic system, which in turn are obtained by applying variational methods. By using the

For singular mean field equations defined on a compact Riemann surface, we prove the uniqueness of bubbling solutions if some blowup points coincide with the singularities of the Dirac data. If the strength of the Dirac mass at each blowup point is not a multiple of 4π, we prove that bubbling solutions are unique. This paper extends previous results of Lin-Yan [C. S. Lin and S. S. Yan, On the mean

The Chow rings of hyperKähler varieties are conjectured to have a particularly rich structure. In this paper, we focus on the locally complete family of double EPW sextics and establish some properties of their Chow rings. First, we prove a Beauville–Voisin type theorem for zero-cycles on double EPW sextics; precisely, we show that the codimension-4 part of the subring of the Chow ring of a double

This paper is a work in progress on Bloch’s conjecture asserting the vanishing of the Pontryagin product of a p codimensional cycle on an abelian variety by p+1 zero cycles of degree zero. We prove an infinitesimal version of the conjecture and we discuss in particular, the case of 3-dimensional cycles.

In this note, we study the maximal perimeter of a convex set in ℝn with respect to various classes of measures. Firstly, we show that for a probability measure μ on ℝn, satisfying very mild assumptions, there exists a convex set of μ-perimeter at least CnVar|X|4𝔼|X|. This implies, in particular, that for any isotropic log-concave measure μ, one may find a convex set of μ-perimeter of order n18. Secondly

We investigate how the motive of hyper-Kähler varieties is controlled by weight-2 (or surface-like) motives via tensor operations. In the first part, we study the Voevodsky motive of singular moduli spaces of semistable sheaves on K3 and abelian surfaces as well as the Chow motive of their crepant resolutions, when they exist. We show that these motives are in the tensor subcategory generated by the