We investigate a Maclaurin inequality for vectors and its connection to an Aleksandrov-type inequality for parallelepipeds.

A W1,p-metric on an n-dimensional closed Riemannian manifold naturally induces a distance function, provided p is sufficiently close to n. If a sequence of metrics gk converges in W1,p to a limit metric g, then the corresponding distance functions dgk subconverge to a limit distance function d, which satisfies d≤dg. As an application, we show that the above convergence result applies to a sequence

We study existence and uniqueness of solutions of (E1) −Δu+μ|x|2u+g(u)=ν in Ω, u=λ on ∂Ω, where Ω⊂ℝ+N is a bounded smooth domain such that 0∈∂Ω, μ≥−N24 is a constant, g a continuous nondecreasing function satisfying some integral growth condition and ν and λ two Radon measures, respectively, in Ω and on ∂Ω. We show that the situation differs considerably according the measure is concentrated at 0 or

This is the second of a series of papers devoted to the study of relaxed highest-weight modules over affine vertex algebras and W-algebras. The first [K. Kawasetsu and D. Ridout, Relaxed highest-weight modules I: Rank 1 cases, Commun. Math. Phys.368 (2019) 627–663, arXiv:1803.01989 [math.RT]] studied the simple “rank-1” affine vertex superalgebras Lk(𝔰𝔩2) and Lk(𝔬𝔰𝔭(1|2)), with the main results

In this paper, we study the following mean field type equation: (MF)−Δgu=ϱKeu∫ΣKeudVg−1inΣ, where (Σ,g) is a closed oriented surface of unit volume Volg(Σ) = 1, K positive smooth function and ϱ=8πm, m∈ℕ. Building on the critical points at infinity approach initiated in [M. Ahmedou, M. Ben Ayed and M. Lucia, On a resonant mean field type equation: A “critical point at infinity” approach, Discrete Contin

We prove that if (C,0) is a reduced curve germ on a rational surface singularity (X,0) then its delta invariant can be recovered by a concrete expression associated with the embedded topological type of the pair C⊂X. Furthermore, we also identify it with another (a priori) embedded analytic invariant, which is motivated by the theory of adjoint ideals. Finally, we connect our formulae with the local

In this paper, we show that given a set of lengths of closed geodesics, there are only finitely many convex cocompact hyperbolic 3-manifolds with that specified length spectrum with multiplicity, homotopy equivalent to a given 3-manifold without a handlebody factor, up to orientation preserving isometries.

A version of the Hodge–Riemann relations for valuations was recently conjectured and proved in several special cases by [J. Kotrbatý, On Hodge–Riemann relations for translation-invariant valuations, preprint (2020), arXiv:2009.00310]. The Lefschetz operator considered there arises as either the product or the convolution with the mixed volume of several Euclidean balls. Here we prove that in (co-)degree

We extend to the spatial case a technique of integration of the close encounters formulated by Tullio Levi-Civita for the planar restricted three-body problem. We consider the Hamiltonian introduced in the Kustaanheimo–Stiefel regularization and construct a complete integral of the related Hamilton–Jacobi equation by means of a series convergent in a neighborhood of the collisions with the primary

Using Floer-theoretic methods, we prove that the non-existence of an exotic symplectomorphism on the standard symplectic ball, 𝔹2n, implies a rather strict topological condition on the free contact circle actions on the standard contact sphere, 𝕊2n−1. We also prove an analogue for a Liouville domain and contact circle actions on its boundary. Applications include results concerning the symplectic

1 We prove the generalized Margulis lemma with a uniform index bound on an Alexandrov n-space X with curvature bounded below, i.e. small loops at p∈X generate a subgroup of the fundamental group of the unit ball B1(p) that contains a nilpotent subgroup of index ≤w(n), where w(n) is a constant depending only on the dimension n. The proof is based on the main ideas of V. Kapovitch, A. Petrunin and W

We study the geometry of centrally symmetric random polytopes, generated by N independent copies of a random vector X taking values in ℝn. We show that under minimal assumptions on X, for N≳n and with high probability, the polytope contains a deterministic set that is naturally associated with the random vector — namely, the polar of a certain floating body. This solves the long-standing question on

We show that direct limit completions of vertex tensor categories inherit vertex and braided tensor category structures, under conditions that hold for example for all known Virasoro and affine Lie algebra tensor categories. A consequence is that the theory of vertex operator (super)algebra extensions also applies to infinite-order extensions. As an application, we relate rigid and non-degenerate vertex

This paper deals with the second-order nonlinear differential equation (a(t)|x′|p(t)−2x′)′=b(t)|x|q(t)−2x involving p(t)-Laplacian. The existence and the uniqueness of nonoscillatory solutions of this equation in certain classes, which are related with integral conditions, are studied. Moreover, a minimal set for solutions of this equation is introduced as an extension of the concept of principal solutions

We study the ramified Prym map 𝒫g,r→𝒜g−1+r2δ which assigns to a ramified double cover of a smooth irreducible curve of genus g ramified in r points the Prym variety of the covering. We focus on the six cases where the dimension of the source is strictly greater than the dimension of the target giving a geometric description of the generic fiber. We also give an explicit example of a totally geodesic

We study images of equilibrium (Gibbs) states for a class of non-invertible transformations associated to conformal iterated function systems (IFSs) with overlaps 𝒮. We prove exact dimensionality for these image measures, and find a dimension formula using their overlap numbers. In particular, we obtain a geometric formula for the dimension of self-conformal measures for IFSs with overlaps, in terms

We give a generalization of Meeks–Yau’s celebrated embeddedness result for the solutions of the Plateau problem for extreme curves.

A well-known theorem of Kulikov, Persson and Pinkham states that a degeneration of a family of K3-surfaces with trivial monodromy can be completed to a smooth family. We give a simple example that an analogous statement does not hold for Calabi–Yau threefolds.

We consider the series expansion of the Lp-Hardy inequality of [G. Barbatis, S. Filippas and A. Tertikas, Series expansion for Lp Hardy inequalities, Indiana Univ. Math. J.52 (2003) 171–190], in the particular case where the distance is taken from an interior point of a bounded domain in ℝn and 1n we improve it by adding as a remainder term the optimally weighted Hölder seminorm, extending the Hardy–Morrey

We build blowing-up solutions for a supercritical perturbation of the Yamabe problem on manifolds with boundary, provided the dimension of the manifold is n≥7 and the trace-free part of the second fundamental form is nonzero everywhere on the boundary.

We study compactness of embeddings of Sobolev-type spaces of arbitrary integer order into function spaces on domains in ℝn with respect to upper Ahlfors regular measures ν, that is, Borel measures whose decay on balls is dominated by a power of their radius. Sobolev-type spaces as well as target spaces considered in this paper are built upon general rearrangement-invariant function norms. Several sufficient

We are concerned with the multiplicity of positive solutions for the singular superlinear and subcritical Schrödinger equation −Δu+V(x)u=λa(x)u−γ+b(x)upin ℝN, beyond the Nehari extremal value, as defined in [Y. Il’yasov, On extreme values of Nehari manifold via nonlinear Rayleigh’s quotient, Topol. Methods Nonlinear Anal.49 (2017) 683–714], when the potential b∈L∞(ℝN) may change its sign, 00 is a real

We characterize the smallest codimension components of the Hodge locus of smooth degree d hypersurfaces of the projective space ℙn+1 of even dimension n, passing through the Fermat variety (with d≠3,4,6). They correspond to the locus of hypersurfaces containing a linear algebraic cycle of dimension n2. Furthermore, we prove that among all the local Hodge loci associated to a nonlinear cycle passing

Frobenius reciprocity asserts that induction from a subgroup and restriction to it are adjoint functors in categories of unitary G-modules. In the 1980s, Guillemin and Sternberg established a parallel property of Hamiltonian G-spaces, which (as we show) unfortunately fails to mirror the situation where more than one G-module “quantizes” a given Hamiltonian G-space. This paper offers evidence that the

In this paper, we obtain local derivative estimates of Shi-type for the heat equation coupled to the Ricci flow. As applications, in part combining with Kuang’s work, we extend some results of Zhang and Bamler–Zhang including distance distortion estimates and a backward pseudolocality theorem for Ricci flow on compact manifolds to the noncompact case.

In this paper, we are concerned with the following integral system ui(x)=∫ℝnui+1pi+1(y)|x−y|n−αdy,i=1,2,…,m−1,um(x)=∫ℝnu1p1(y)|x−y|n−αdy,m≥1,n≥1, where ui>0, 0<α1 (i=1,2,…,m). When m∈{1,2}, such an integral system is associated with the best constants of the Hardy–Littlewood–Sobolev inequality. Chen, Li and their cooperators obtained optimal integrability intervals of the finite energy solutions by

We prove existence of solutions to the Schrödinger–Poisson–Proca system, and stability of this system, in the electro-magneto-static regime in closed 3-dimensional Riemannian manifolds.

We investigate non-existence of non-negative dead core solutions for the problem |Du|γF(x,D2u)+a(x)uq=0in Ω,u=0on ∂Ω. Here, Ω⊂ℝN is a bounded smooth domain, F is a fully nonlinear elliptic operator, a:Ω→ℝ is a sign-changing weight, γ≥0, and 0

We introduce the notion of exotic periodic points of a meromorphic self-map. We then establish the expected asymptotic for the number of isolated or exotic periodic points for holomorphic self-maps with a simple action on the cohomology groups on a compact Kähler manifold.

We establish nonexistence results for complete spacelike translating solitons immersed in a Lorentzian product space ℝ1×ℙn, under suitable curvature constraints on the curvatures of the Riemannian base ℙn. In particular, we obtain Calabi–Bernstein type results for entire translating graphs constructed over ℙn. For this, we prove a version of the Omori–Yau’s maximum principle for complete spacelike

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