In this article, we study some Kramers–Fokker–Planck operators with a polynomial potential $V(q)$ of degree greater than two having quadratic limiting behaviour. This work provides an accurate global subelliptic estimate for Kramers–Fokker–Planck operators under some conditions imposed on the potential.

In this paper we present a classification of a class of globally subanalytic CMC surfaces in ℝ3 that generalizes the recent classification made by Barbosa and do Carmo in 2016. We show that a globally subanalytic CMC surface in ℝ3 with isolated singularities and a suitable condition of local connectedness is a plane or a finite union of round spheres and right circular cylinders touching at the singularities. As a consequence, we obtain that a globally subanalytic CMC surface in ℝ3 that is a topological manifold does not have isolated singularities. It is also proved that a connected closed globally subanalytic CMC surface in ℝ3 with isolated singularities which is locally Lipschitz normally embedded needs to be a plane or a round sphere or a right circular cylinder. A result in the case of non-isolated singularities is also presented. It also presented some results on regularity of semialgebraic sets and, in particular, it proved a real version of Mumford's Theorem on regularity of normal complex analytic surfaces and a result about C1 regularity of minimal varieties.

We show quantifier elimination theorems for real closed valued fields with separated analytic structure and overconvergent analytic structure in their natural one-sorted languages and deduce that such structures are weakly o-minimal. We also provide a short proof that algebraically closed valued fields with separated analytic structure (in any rank) are C-minimal.

We present a definable smooth version of the Thom transversality theorem. We show further that the set of non-transverse definable smooth maps is nowhere dense in the definable smooth topology. Finally, we prove a definable version of a theorem of Trotman which says that the Whitney (a)-regularity of a stratification is necessary and sufficient for the stability of transversality.

We consider homogeneous multiaffine polynomials whose coefficients are the Plücker coordinates of a point $V$ of the Grassmannian. We show that such a polynomial is stable (with respect to the upper half plane) if and only if $V$ is in the totally nonnegative part of the Grassmannian. To prove this, we consider an action of matrices on multiaffine polynomials. We show that a matrix $A$ preserves stability of polynomials if and only if $A$ is totally nonnegative. The proofs are applications of classical theory of totally nonnegative matrices, and the generalized Pólya-Schur theory of Borcea and Brändén.

The detection of the bifurcation set of polynomial mapping ℝn → ℝp, n ⩾ p, in more than two variables remains an unsolved problem. In this note we provide a solution for n = p + 1 ⩾ 3.

An orbitope is the convex hull of an orbit of a compact group acting linearly on a vector space. These highly symmetric convex bodies lie at the crossroads of several fields, including convex geometry, algebraic geometry, and optimization. We present a self-contained theory of orbitopes, with particular emphasis on instances arising from the groups SO(n) and O(n); these include Schur–Horn orbitopes, tautological orbitopes, Carathéodory orbitopes, Veronese orbitopes, and Grassmann orbitopes. We study their face lattices, algebraic boundaries, and representations as spectrahedra or projected spectrahedra.

Natural sufficient conditions for a polynomial to have a local minimum at a point are considered. These conditions tend to hold with probability 1. It is shown that polynomials satisfying these conditions at each minimum point have nice presentations in terms of sums of squares. Applications are given to optimization on a compact set and also to global optimization. In many cases, there are degree bounds for such presentations. These bounds are of theoretical interest, but they appear to be too large to be of much practical use at present. In the final section, other more concrete degree bounds are obtained which ensure at least that the feasible set of solutions is not empty.

In this paper we present Thom’s transversality theorem in o-minimal structures (a generalization of semialgebraic and subanalytic geometry). There are no restrictions on the differentiability class and the dimensions of manifolds involved in comparison withthe general case.

We study natural $*$-valuations, $*$-places and graded $*$-rings associated with $*$-ordered rings. We prove that the natural $*$-valuation is always quasi-Ore and is even quasi-commutative (i.e., the corresponding graded $*$-ring is commutative), provided the ring contains an imaginary unit. Furthermore, it is proved that the graded $*$-ring is isomorphic to a twisted semigroup algebra. Our results are applied to answer a question of Cimprič regarding $*$-orderability of quantum groups.

It is proved that each mod 2 homology class of a compact Nash manifold can be represented by a semialgebraic arc-symmetric subset.

We prove a generalization to the context of real geometry of an intersection formula for the vanishing cycle functor, which in the complex context is due to Dubson, Lê, Ginsburg and Sabbah (after a conjecture of Deligne). It is also a generalization of similar results of Kashiwara and Schapira, where these authors work with a suitable assumption about the micro-support of the corresponding constructible complex of sheaves. We only use a similar assumption about the support of the corresponding characteristic cycle so that our result can be formulated in the language of constructible functions and Lagrangian cycles.

Let Ω = (ω1,…,ωn-k) be differential 1-forms with polynomial coefficients in Rn. A Pfaffian manifold of Ω is by definition a maximal integral k-manifold of Ω. It is shown that the number of homeomorphism classes of all Pfaffian manifolds of Rolle Type of Ω is finite and, moreover, bounded by a computable function in variables n, k and the degree of ω1,…, ωn-k. Finiteness is proved also in any o-minimal structure.

We give also an example of a semi-algebraic C1 differential form on a semi-algebraic C2 3-manifold whose Pfaffian manifolds have homeomorphism classes of the cardinality of continuum. Hence the cardinality of all manifolds is the continuum (not countable).

This paper develops a refinement of Lasserre's algorithm for optimizing a polynomial on a basic closed semialgebraic set via semidefinite programming and addresses an open question concerning the duality gap. It is shown that, under certain natural stability assumptions, the problem of optimization on a basic closed set reduces to the compact case.

The paper considers the relationship between positive polynomials, sums of squares and the multi-dimensional moment problem in the general context of basic closed semi-algebraic sets in real $n$-space. The emphasis is on the non-compact case and on quadratic module representations as opposed to quadratic preordering presentations. The paper clarifies the relationship between known results on the algebraic side and on the functional-analytic side and extends these results in a variety of ways.

A generalization of Schmüdgen’s Positivstellensatz is given which holds for any basic closed semialgebraic set in ${{\mathbb{R}}^{n}}$ (compact or not). The proof is an extension of Wörmann’s proof.

A preordering $T$ is constructed in the polynomial ring $A=\mathbb{R}[{{t}_{1}},{{t}_{2}},...]$ (countablymany variables) with the following two properties: (1) For each $f\,\in \,A$ there exists an integer $N$ such that $-\,N\,\le \,f\left( p \right)\,\le \,N$ holds for all $P\in \text{Spe}{{\text{r}}_{T}}(A)$. (2) For all $f\,\in \,A$, if $N+f,N-f\in T$ for some integer $N$, then $f\,\in \,\mathbb{R}$. This is in sharp contrast with the Schmüdgen-Wörmann result that for any preordering $T$ in a finitely generated $\mathbb{R}$-algebra $A$, if property (1) holds, then for any $f\in A,f>0\,\text{on}\,\text{Spe}{{\text{r}}_{T}}(A)\Rightarrow f\in T$. Also, adjoining to $A$ the square roots of the generators of $T$ yields a larger ring $C$ with these same two properties but with $\sum{{{C}^{2}}}$ (the set of sums of squares) as the preordering.

Let V be an irreducible nonsingular algebraic surface, Y ⊂ V be an algebraic curve and P a point of Y. Suppose a sign distribution is given locally in a neighbourhood of P on some connected components of V — Y. We give an algorithmic criterion to decide whether this sign distribution is induced by a regular function or not. As an application, this criterion enables one to decide whether two semialgebraic sets can be locally separated or not.