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We establish a theory of $Q$-valued functions minimizing a suitable generalization of the Dirichlet integral. In a second paper the theory will be used to approximate efficiently area minimizing currents $\mathrm{mod}(p)$ when $p=2Q$, and to establish a first general partial regularity theorem for every $p$ in any dimension and codimension., 37 pages. First part of a two-papers work aimed at establishing a first general partial regularity theory for area minimizing currents modulo p, for any p and in any dimension and codimension. v3 is the final version, to appear on Comm. Pure Appl. Math

In this paper, we show how to use the framework of mod-Gaussian convergence in order to study the fluctuations of certain models of random graphs, of random permutations and of random integer partitions. We prove that, in these three frameworks, a generic homogeneous observable of a generic random model is mod-Gaussian under an appropriate renormalisation. This implies a central limit theorem with an extended zone of normality, a moderate deviation principle, an estimate of the speed of convergence, a local limit theorem and a concentration inequality. The universal asymptotic behavior of the observables of these models gives rise to a notion of mod-Gaussian moduli space., Comment: New version: the paper has been slightly shortened, and a few references were added. 52 pages, 13 figures

Let R be a commutative noetherian ring. Lindo and Pande have recently posed the question asking when every ideal of R is isomorphic to some trace ideal of R. This paper studies this question and gives several answers. In particular, a complete answer is given in the case where R is local: it is proved in this paper that every ideal of R is isomorphic to a trace ideal if and only if R is an artinian Gorenstein ring, or a 1-dimensional hypersurface with multiplicity at most 2, or a unique factorization domain.

The paper deals with boundedness problems of integral operators in weighted grand Bochner-Lebesgue spaces. We will treat both cases: when a weight function appears as a multiplier in the definition of the norm, or when it defines the absolute continuous measure of integration. Along with the diagonal case we deal with the off-diagonal case. To get the appropriate result for the Hardy-Littlewood maximal operator we rely on the reasonable bound of the sharp constant in the Buckley type theorem which is also derived in the paper.

The purpose of this document is to connect two maps related to certain graphs embedded in the disc. The first is Postnikov's boundary measurement map, which combines partition functions of matchings in the graph into a map from an algebraic torus to an open positroid variety in a Grassmannian. The second is a rational map from the open positroid variety to an algebraic torus, given by certain Plucker coordinates which are expected to be a cluster in a cluster structure. This paper clarifies the relationship between these two maps, which has been ambiguous since they were introduced by Postnikov in 2001. The missing ingredient supplied by this paper is a twist automorphism of the open positroid variety, which takes the target of the boundary measurement map to the domain of the (conjectural) cluster. Among other applications, this provides an inverse to the boundary measurement map, as well as Laurent formulas for twists of Plucker coordinates.

In this paper we prove a sharp distortion property of the Cassinian metric under Mobius transformations of a punctured ball onto another punctured ball. The paper also deals with discussion on local convexity properties of the Cassinian metric balls in proper subdomains of Rn. Inclusion properties of the Cassinian metric balls with other hyperbolic-type metric balls are also investigated. In particular, several conjectures are also stated in response to sharpness of the inclusion relations.

This paper is devoted to one-dimensional interpolation Gagliardo-Nirenberg-Sobolev inequalities. We study how various notions of duality, transport and monotonicity of functionals along flows defined by some nonlinear diffusion equations apply. We start by reducing the inequality to a much simpler dual variational problem using mass transportation theory. Our second main result is devoted to the construction of a Lyapunov functional associated with a nonlinear diffusion equation, that provides an alternative proof of the inequality. The key observation is that the inequality on the line is equivalent to Sobolev's inequality on the sphere, at least when the dimension is an integer, or to the critical interpolation inequality for the ultraspherical operator in the general case. The time derivative of the functional along the flow is itself very interesting. It explains the machinery of some rigidity estimates for nonlinear elliptic equations and shows how eigenvalues of a linearized problem enter in the computations. Notions of gradient flows are then discussed for various notions of distances. Throughout this paper we shall deal with two classes of inequalities corresponding either to p>2 or to p

We give a possible generalization of Lutz twist to all dimensions. This reproves the fact that every contact manifold can be given a non-fillable contact structure and also shows great flexibility in the manifolds that can be realized as cores of overtwisted families. We moreover show that $R^{2n+1}$ has at least three distinct contact structures. This version of the paper contains both the texts of the published version of the paper together with an Erratum to the published version appended to the end., Comment: 19 pages, 2 figures, and erratum to the published version of the paper

We determine relations between the Euler obstruction of a map f and the Chern obstruction of a collection of 1-forms associated to f . We obtain explicit applications in singularity theory. On one hand, in [10], Grulha defines a generalization of the Euler obstruction of a holomorphic function to the case of a map-germ f : (V, 0) → (C, 0) where (V, 0) is the germ of a reduced equidimensional complex analytic variety of dimension d ≥ p. The main result in that paper is a formula that computes the Euler obstruction of f in terms of the Euler obstruction for p-frames as defined by Brasselet, Seade and Suwa in [3]. On the other hand, Ebeling and Gusein-Zade introduced in [4] the notion of Chern obstruction for singular spaces using collections of differential 1-forms. This number is well defined for any germ of reduced equidimensional complex analytic space, and the authors provide a formula for the Chern obstruction in terms of intersection number. One of the main results of the present paper is that the Euler obstruction of a map can be written as a Chern obstruction, therefore as an intersection multiplicity.

In a landmark paper, D.Y. Kleinbock and G.A. Margulis established the fundamental Baker-Sprindzuk conjecture on homogeneous Diophantine approximation on manifolds. Subsequently, there has been dramatic progress in this area of research. However, the techniques developed to date do not seem to be applicable to inhomogeneous approximation. Consequently, the theory of inhomogeneous Diophantine approximation on manifolds remains essentially non-existent. In this paper we develop an approach that enables us to transfer homogeneous statements to inhomogeneous ones. This is rather surprising as the inhomogeneous theory contains the homogeneous theory and so is more general. As a consequence, we establish the inhomogeneous analogue of the Baker-Sprindzuk conjecture. Furthermore, we prove a complete inhomogeneous version of the profound theorem of Kleinbock, Lindenstrauss & Weiss on the extremality of friendly measures. The results obtained in this paper constitute the first step towards developing a coherent inhomogeneous theory for manifolds in line with the homogeneous theory., 37 pages: a final section on further developments has been added

For a smooth projective curve, the cycles of e-secant k-planes are among the most studied objects in classical enumerative geometry and there are well-known formulas due to Castelnuovo, Cayley and MacDonald concerning them. Despite various attempts, surprisingly little is known about the enumerative validity of such formulas. The aim of this paper is to completely clarify this problem in the case of the generic curve C of given genus. Using degeneration techniques and a few facts about the birational geometry of moduli spaces of stable pointed curves we determine precisely under which conditions the cycle of e-secant k-planes in non-empty and we compute its dimension. We also precisely determine the dimension of the variety of linear series on C carrying e-secant k-planes. In a different direction, in the last part of the paper we study the distribution of ramification points of the powers of a line bundle on C having prescribed ramification at a given point., Comment: 25 pages. Numerous changes suggested by the referee, several proofs explained in more detail. To appear in the Journal of the London Mathematical Society

Twenty-five years ago, Conway and Norton published their remarkable paper `Monstrous Moonshine', proposing a completely unexpected relationship between finite simple groups and modular functions. This paper reviews the progress made in broadening and understanding that relationship., 33 pages. 10 references added; minor changes to text

In the present paper, we construct a cusped hyperbolic $4$-manifold with all cusp sections homeomorphic to the Hantzsche-Wendt manifold, which is a rational homology sphere. By a result of Gol\'enia and Moroianu, the Laplacian on $2$-forms on such a manifold has purely discrete spectrum. This shows that one of the main results of Mazzeo and Phillips from 1990 cannot hold without additional assumptions on the homology of the cusps. This also answers a question by Gol\'enia and Moroianu from 2012. We also correct and refine the incomplete classification of compact orientable flat $3$-manifolds arising from cube colourings provided earlier by the last two authors., Comment: 15 pages, 1 figure, 1 table; SageMath worksheets available at https://github.com/sashakolpakov/24-cell-colouring

In this paper, we further Meirong Zhang, et al.'s work by computing the number of weighted eigenvalues for Sturm-Liouville equations, equipped with general integrable potentials and Dirac weights, under Dirichlet boundary condition. We show that, for a Sturm-Liouville equation with a general integrable potential, if its weight is a positive linear combination of $n$ Dirac Delta functions, then it has at most $n$ (may be less than $n$, or even be $0$) distinct real Dirichlet eigenvalues, or every complex number is a Dirichlet eigenvalue; in particular, under some sharp condition, the number of Dirichlet eigenvalues is exactly $n$. Our main method is to introduce the concepts of characteristics matrix and characteristics polynomial for Sturm-Liouville problem with Dirac weights, and put forward a general and direct algorithm used for computing eigenvalues. As an application, a class of inverse Dirichelt problems for Sturm-Liouville equations involving single Dirac distribution weights is studied., 23 pages

The main object of the present paper is to consider the binomial transforms for Horadam quaternion sequences. We gave new formulas for recurrence relation, generating function, Binet formula and some basic identities for the binomial sequence of Horadam quaternions. Working with Horadam quaternions, we have found the most general formula that includes all binomial transforms with recurrence relation from the second order. In the last part, we determined the recurrence relation for this new type of quaternion by working with the iterated binomial transform, which is a different type of binomial transform. WOS:000638679800001 2-s2.0-85104113068

The mod two cohomology of the three connective covering of S3 has the form F2[X2n] ⊗ E(Sq1X2n) where x2n is in degree 2n and n = 2. If F denotes the homotopy theoretic fibre of the map S3 → B2S1 of degree 2, then the mod 2 cohomology of F is also of the same form for n = 1. Notice (cf. Section 7 of the present paper) that the existence of spaces whose cohomology has this form for high values of n would immediately provide Arf invariant elements in the stable stem. Hence, it is worthwhile to determine for what values of n the above algebra can be realized as the mod2 cohomology of some space. The purpose of this paper is to construct a further example of a space with such a cohomology algebra for n = 4 and to show that no other values of n are admissible. More precisely, we prove the following.

The Markov evolution is studied of an infinite age-structured population of migrants arriving in and departing from a continuous habitat $X \subseteq\mathds{R}^d$ -- at random and independently of each other. Each population member is characterized by its age $a\geq 0$ (time of presence in the population) and location $x\in X$. The population states are probability measures on the space of the corresponding marked configurations. The result of the paper is constructing the evolution $\mu_0 \to \mu_t$ of such states by solving a standard Fokker-Planck equation for this models. We also found a stationary state $\mu$ existing if the emigration rate is separated away from zero. It is then shown that $\mu_t$ weakly converges to $\mu$ as $t\to +\infty$.

The main focus of this paper is on study of s‐numbers for a higher order Sobolev embedding on an interval and also for the corresponding Hardy‐type operator. Upper and lower estimates for approximation and Bernstein numbers are described and their relations with results for lower order Sobolev embeddings are discussed.

In this paper, we define and study Gevrey spaces associated with a Hormander family of (globally defined) vector fields and its corresponding sub‐Laplacian. We show some natural relations between the various Gevrey spaces in this setting on general manifolds, and more particular properties on Lie groups with polynomial growth of the volume. In the case of the Heisenberg group and of SU(2), we show that all our descriptions coincide.

The paper is on the vanishing topology of singular Milnor fibres of holomorphic families of arbitrary square, symmetric and skew-symmetric matrices with sufficiently many parameters. We define vanishing cycles on such fibres, prove an extended form of the Damon-Pike $\mu=\tau$ conjecture about the families of a special type, and make first steps towards understading of the monodromy of matrix singularities. We also prove a Lyashko-Looijenga type theorem for simple matrix families, and point out a surprising relationship between certain Shephard-Todd groups, simple odd functions and a sporadic part of the Bruce-Tari simple matrix classification.

The main goal of this paper is to study some properties of an extension of valuations from classical invariants. More specifically, we consider a valued field $(K,\nu)$ and an extension $\omega$ of $\nu$ to a finite extension $L$ of $K$. Then we study when the valuation ring of $\omega$ is essentially finitely generated over the valuation ring of $\nu$. We present a necessary condition in terms of classic invariants of the extension by Hagen Knaf and show that in some particular cases, this condition is also sufficient. We also study when the corresponding extension of graded algebras is finitely generated. For this problem we present an equivalent condition (which is weaker than the one for the finite generation of the valuation rings).

In this paper we consider a class of chemotaxis models with two arbitrary constitutive functions g(u) and f(v). After having performed a complete symmetry group classification with respect to them the reduced systems are derived. By considering g(u) of the logistic form wide classes of exact solutions are found.

In this paper, we consider discrete groups in ${\rm PGL}_d(\mathbb{R})$ acting convex co-compactly on a properly convex domain in real projective space. For such groups, we establish an analogue of the well known flat torus theorem for ${\rm CAT}(0)$ spaces., Minor revisions, to appear in the Journal of the London Mathematical Society. 20 pages. Comments welcome

In this paper we present a unified simple approach to anisotropic Hardy inequalities in various settings. We consider Hardy inequalities which involve a Finsler distance from a point or from the boundary of a domain. The sharpness and the non-attainability of the constants in the inequalities are also proved., 31 pages. We add "Note added to Proof" in Introduction and several references

In this paper, we consider some generalized holomorphic maps between pseudo-Hermitian manifolds and Hermitian manifolds. By Bochner formulas and comparison theorems, we establish related Schwarz type results. As corollaries, Liouville theorem and little Picard theorem for basic CR functions are deduced. Finally, we study CR Carath\'eodory pseudodistance on CR manifolds., Comment: 16 pages

Let $\mathscr J$ be the set of inner functions whose derivative lies in the Nevanlinna class. In this paper, we discuss a natural topology on $\mathscr J$ where $F_n \to F$ if the critical structures of $F_n$ converge to the critical structure of $F$. We show that this occurs precisely when the critical structures of the $F_n$ are uniformly concentrated on Korenblum stars. The proof uses Liouville's correspondence between holomorphic self-maps of the unit disk and solutions of the Gauss curvature equation. Building on the works of Korenblum and Roberts, we show that this topology also governs the behaviour of invariant subspaces of a weighted Bergman space which are generated by a single inner function., 45 pages

In this paper, the weak Harris theorem developed in \cite{HMS11} is illustrated by using a straightforward Wasserstein coupling, which implies the exponential ergodicity of the functional solutions to a range of neutral type SDEs with infinite length of memory. A concrete example is presented to illustrate the main result.

We present a new proof of the sphere covering inequality in the spirit of comparison geometry, and as a byproduct we find another sphere covering inequality which can be viewed as the dual of the original one. We also prove sphere covering inequalities on surfaces satisfying general isoperimetric inequalities, and discuss their applications to elliptic equations with exponential nonlinearities in dimension two. The approach in this paper extends, improves, and unifies several inequalities about solutions of elliptic equations with exponential nonlinearities.

In this paper, we obtain a boundary Schwarz lemma for C1 (pluriharmonic, holomorphic) mappings from the unit polydisc Un in Cn to irreducible bounded symmetric domains realized as the unit ball BX of an N‐dimensional simple JB*‐triple X. In particular, we obtain a version of the boundary Schwarz lemma for C1 (pluriharmonic, holomorphic) mappings from Un into the Euclidean unit ball BN in CN. These results are generalizations of recent results regarding boundary Schwarz lemma in higher dimensions.

ith respect to the analytic‐algebraic dichotomy, the theory of Siegel modular forms of half‐integral weight is lopsided; the analytic theory is strong, whereas the algebraic lags behind. In this paper, we capitalise on this to establish the fundamental object needed for the analytic side of the Iwasawa main conjecture — the p ‐adic L ‐function obtained by interpolating the complex L ‐function at special values. This is achieved through the Rankin–Selberg method and the explicit Fourier expansion of non‐holomorphic Siegel Eisenstein series. The construction of the p ‐stabilisation in this setting is also of independent interest.

In this paper, we are concerned with the Hardy–Henon equations −Δu=|x|aupandΔ2u=|x|aupwith a∈R and p∈(0,1]. Inspired by Serrin and Zou [25], we prove Liouville theorems for nonnegative solutions to the above Hardy–Henon equations (Theorem 1.1 and Theorem 1.3), that is, the unique nonnegative solution is u≡0.

In this paper we prove transference inequalities for regular and uniform Diophantine exponents in the weighted setting. Our results generalize the corresponding inequalities that exist in the `non-weighted' case.

We study how small is the set of critical values of the distance function from a compact (resp. closed) set in the plane or in a connected complete two-dimensional Riemannian manifold. We show that for a compact set, the set of critical values is compact and Lebesgue null (which is a known result) and that it has "locally" (away from 0) bounded sum of square roots of lengths of gaps (components of the complement). In the planar case, these conditions of local smallness are shown to be optimal. These results improve and generalize those of Fu (1985) and of our earlier paper from 2012. We also find an optimal condition for the smallness of the whole set of critical values of a planar compact set.

In this paper we deal with the following class of quasilinear elliptic problems −Δpu=λ|u|p−2u+u+p∗−1+g(x,u+)+f(x)inΩ,u=0on∂Ω,where 1≤p

In this paper we prove sufficient conditions for the Fredholm property of a non-smooth pseudodifferential operator $P$ which symbol is in a Holder space with respect to the spatial variable. As a main ingredient for the proof we use a suitable symbol-smoothing.

In this paper, we study certain interesting and useful properties of incomplete ? ?functions. The incomplete ? ?function is an extension of the ? ?function. We find several useful classical integral transforms of these functions. Further, we examine the fractional calculus with the incomplete ? ?functions and point out several special cases. Finally, we give the applications of incomplete ? ?functions in detecting glucose supply in human blood.

The paper extends some well-known results for analytic functions onto solutions of the Vekua equation $\partial _{\overline{z}}W=aW+b\overline{W}$ regarding the existence and construction of the Bergman kernel and of the corresponding Bergman projection operator.

In this paper, we study the existence and multiplicity results of nontrivial positive solutions to a quasilinear elliptic equation in $\RN$, when $N\geq2$, as \begin{equation} \Lp u+u^{p-1}=\lambda\hspace{0.2mm}k(x)u^{r-1}-h(x)u^{q-1}.\nonumber \end{equation} Here, $h(x),k(x)>0$ are Lebesgue measurable functions, $10$ is a parameter.

International audience; In the paper, the authors introduce a notion "multivariate exponential polynomials" which generalize exponential numbers and polynomials, establish explicit formulas, inversion formulas, and recurrence relations for multivariate exponential polynomials in terms of the Stirling numbers of the first and second kinds with the help of the Fàa di Bruno formula, two identities for the Bell polynomials of the second kind, and the inversion theorem for the Stirling numbers of the first and second kinds, construct some determinantal inequalities and product inequalities for multivariate exponential polynomials with the aid of some properties of completely monotonic functions and other known results, derive the logarithmic convexity and logarithmic concavity for multivariate exponential polynomials, and finally find an application of multi-variate exponential polynomials to white noise distribution theory by confirming that multivariate exponential polynomials satisfy conditions for sequences required in white noise distribution theory.

In this paper, we study the generalized Lebesgue-Nagell equation \[ x^2+7^{2k+1}=y^n. \] This is the last case of equations of the form $x^2+q^{2k+1}=y^n$ with $k\geq0$ and $q>0$ where $\mathbb{Q}(\sqrt{-q})$ has class number one. Our proof is based on the modular method and the symplectic argument.

In this paper we study the relationship between two fundamental regularity properties of an $s$-dimensional Calder\'{o}n-Zygmund operator (CZO) acting on a Borel measure $\mu$ in $\mathbb{R}^d$, with $s\in (0,d)$. In the classical case when $s=d$ and $\mu$ is equal to the Lebesgue measure, Calder\'on and Zygmund showed that if a CZO is bounded in $L^2$ then the principal value integral exists almost everywhere. However, there are by now several examples showing that this implication may fail for lower-dimensional kernels and measures, even when the CZO has a homogeneous kernel consisting of spherical harmonics. We introduce sharp geometric conditions on $\mu$, in terms of certain scaled transportation distances, which ensure that an extension of the Calder\'{o}n-Zygmund theorem holds. These conditions are necessary and sufficient in the cases of the Riesz transform and the Huovinen transform. Our techniques build upon prior work by Mattila and Verdera, and incorporate the machinery of symmetric measures, introduced to the area by Mattila., Comment: 32 pages

International audience; Orbital varieties are the irreducible components of the intersection between a nilpotent orbit and a Borel subalgebra of the Lie algebra of a reductive group. There is a geometric correspondence between orbital varieties and irreducible components of Springer fibers. In type A, a construction due to Richardson implies that every nilpotent orbit contains at least one smooth orbital variety and every Springer fiber contains at least one smooth component. In this paper, we show that this property is also true for the other classical cases. Our proof uses the interpretation of Springer fibers as varieties of isotropic flags and van Leeuwen's parametrization of their components in terms of domino tableaux. In the exceptional cases, smooth orbital varieties do not arise in every nilpotent orbit, as already noted by Spaltenstein. We however give a (nonexhaustive) list of nilpotent orbits which have this property. Our treatment of exceptional cases relies on an induction procedure for orbital varieties, similar to the induction procedure for nilpotent orbits.

Adjoint functor theorems give necessary and sufficient conditions for a functor to admit an adjoint. In this paper, we prove general adjoint functor theorems for functors between ∞‐categories. One of our main results is an ∞‐categorical generalization of Freyd's classical General Adjoint Functor Theorem. As an application of this result, we recover Lurie's adjoint functor theorems for presentable ∞‐categories. We also discuss the comparison between adjunctions of ∞‐categories and homotopy adjunctions, and give a treatment of Brown representability for ∞‐categories based on Heller's purely categorical formulation of the classical Brown representability theorem.

In this paper, we propose a new approach to Cwikel estimates both for the Euclidean space and for the noncommutative Euclidean space.

In this paper we show rigidity results for super-solutions to fully nonlinear elliptic conformally invariant equations on subdomains of the standard $n$-sphere $\mathbb S^n$ under suitable conditions along the boundary. We emphasize that our results do not assume concavity assumption on the fully nonlinear equations we will work with. This proves rigidity for compact connected locally conformally flat manifolds $(M,g)$ with boundary such that the eigenvalues of the Schouten tensor satisfy a fully nonlinear elliptic inequality and whose boundary is isometric to a geodesic sphere $\partial D(r)$, where $D(r)$ denotes a geodesic ball of radius $r\in (0,\pi/2]$ in $\mathbb S^n$, and totally umbilical with mean curvature bounded below by the mean curvature of this geodesic sphere. Under the above conditions, $(M,g)$ must be isometric to the closed geodesic ball $\overline{D(r)}$. As a side product, in dimension $2$ our methods provide a new proof to Toponogov's Theorem about the rigidity of compact surfaces carrying a shortest simple geodesic. Roughly speaking, Toponogov's Theorem is equivalent to a rigidity theorem for spherical caps in the Hyperbolic three-space $\mathbb H^3$. In fact, we extend it to obtain rigidity for super-solutions to certain Monge-Amp\`ere equations.

In this paper, we consider a non‐smooth atomic decomposition by using a smooth atomic decomposition. Applying the non‐smooth atomic decomposition, a local means characterization and a quarkonical decomposition, we obtain a pointwise multiplier and a trace operator for generalized Besov–Morrey spaces and generalized Triebel–Lizorkin–Morrey spaces on the whole space. We also develop the theory of those spaces on domains. We consider an extension operator and a trace operator on the upper half space and on compact oriented Riemannian manifolds.

Suppose that T1 is a Calderon–Zygmund operator with isotropic homogeneity and T2 is a Calderon–Zygmund operator with non‐isotropic homogeneity. In this note, the boundedness of the composition operator T1∘T2 on the Hardy space is presented. The results in this paper extend earlier related results on convolution operators to non‐convolution setting.

In this paper, we study the controllability of an impulsive fractional differential equation with infinite state-dependent delay in an arbitrary Banach space. We apply semigroup theory and Schaefer fixed point theorem. As an application, we include an example to illustrate the theory.

In this paper we use the steerable special relativistic (space-time) Fourier transform (SFT), and relate the classical convolution of the algebra for space-time $Cl(3,1)$-valued signals over the space-time vector space $\R^{3,1}$, with the (equally steerable) Mustard convolution. A Mustard convolution can be expressed in the spectral domain as the point wise product of the SFTs of the factor functions. In full generality do we express the classical convolution of space-time signals in terms of finite linear combinations of Mustard convolutions, and vice versa the Mustard convolution of space-time signals in terms of finite linear combinations of classical convolutions.

In this paper, we generalize all the results obtained on para‐Kahler Lie algebras in [3] to para‐Kahler Lie algebroids. In particular, we study exact para‐Kahler Lie algebroids as a generalization of exact para‐Kahler Lie algebras. This study leads to a natural generalization of pseudo‐Hessian manifolds, we call them contravariant pseudo‐Hessian manifolds. Contravariant pseudo‐Hessian manifolds have many similarities with Poisson manifolds. We explore these similarities which, among others, leads to a powerful machinery to build examples of non trivial pseudo‐Hessian structures. Namely, we will show that given a finite dimensional commutative and associative algebra (A,.), the orbits of the action Φ of (A,+) on A∗ given by Φ(a,μ)=exp(La∗)(μ) are pseudo‐Hessian manifolds, where La(b)=a.b. We illustrate this result by considering many examples of associative commutative algebras and show that the resulting pseudo‐Hessian manifolds are very interesting.