This paper deals with the mathematics of the Markowitz theory of portfolio management. Let E and V be two homogeneous functions defined on ℝⁿ, the first linear, the other positive definite quadratic. Furthermore let Δ be a simplex contained in ℝⁿ (the set of admissible portfolios), for example Δ : x₁+ ... + xₙ = 1, $x_i\ge 0$. Our goal is to investigate the properties of the restricted mappings (V,E):Δ → ℝ² (the so called Markowitz mappings) and to classify them. We introduce the notion of a generic model...

Let K be an ordered field and R its real closure. A semipolynomial will be defined as a function from Rn to R obtained by composition of polynomial functions and the absolute value. Every semipolynomial can be defined as a straight-line program containing only instructions with the following type: polynomial, absolute value, sup and inf and such a program will be called a semipolynomial expression. It will be proved, using the ordinary real positivstellensatz, a general real positivstellensatz concerning...

Let K,R be an algebraically closed field (of characteristic zero) and a real closed field respectively with K=R(√(-1)). We show that every K-analytic set definable in an o-minimal expansion of R can be locally approximated by a sequence of K-Nash sets.

Compositional data, multivariate observations that hold only relative information, need a special treatment while performing statistical analysis, with respect to the simplex as their sample space ([Aitchison, J.: The Statistical Analysis of Compositional Data. Chapman and Hall, London, 1986.], [Aitchison, J., Greenacre, M.: Biplots of compositional data. Applied Statistics 51 (2002), 375–392.], [Buccianti, A., Mateu-Figueras, G., Pawlowsky-Glahn, V. (eds): Compositional data analysis in the geosciences:...

We establish, for smooth projective real curves, an analogue of the classical Clifford inequality known for complex curves. We also study the cases when equality holds.

Let ${ℳ}_g$ be the moduli space of smooth complex projective curves of genus $g$. Here we prove that the subset of ${ℳ}_g$ formed by all curves for which some Brill-Noether locus has dimension larger than the expected one has codimension at least two in ${ℳ}_g$. As an application we show that if $X\in {ℳ}_g$ is defined over $ℝ$, then there exists a low degree pencil $uX\to {\mathbb{P}}^1$ defined over $ℝ$.

Let f: ℝⁿ → ℝ be a polynomial function of degree d with f(0) = 0 and ∇f(0) = 0. Łojasiewicz’s gradient inequality states that there exist C > 0 and ϱ ∈ (0,1) such that ${\left|\nabla f\right|\ge C\left|f\right|}^{\varrho }$ in a neighbourhood of the origin. We prove that the smallest such exponent ϱ is not greater than $1-R{(n,d)}^{-1}$ with $R(n,d)=d{(3d-3)}^{n-1}$.

We investigate several extension properties of Fréchet differentiable functions defined on closed sets for o-minimal expansions of real closed fields.

In a paper written in 1876 [4], Felix Klein gave a formula relating the number of real flexes of a generic real plane projective curve to the number of real bitangents at non-real points and the degree, which shows in particular that the number of real flexes cannot exceed one third of the total number of flexes. We show that Klein's arguments can be made rigorous using a little of the theory of singularities of maps, justifying in particular his resort to explicit examples.

Let ${ℳ}_0^{ℝ}$ be the moduli space of smooth real cubic surfaces. We show that each of its components admits a real hyperbolic structure. More precisely, one can remove some lower-dimensional geodesic subspaces from a real hyperbolic space $H^4$ and form the quotient by an arithmetic group to obtain an orbifold isomorphic to a component of the moduli space. There are five components. For each we describe the corresponding lattices in $\mathrm{PO}(4,1)$. We also derive several new and several old results on the topology of ${ℳ}_0^{ℝ}$....

We construct invariants under deformation of real symplectic four-manifolds. These invariants are obtained by counting three different kinds of real rational $J$-holomorphic curves which realize a given homology class and pass through a given real configuration of (the appropriate number of) points. These curves are cuspidal curves, reducible curves and curves with a prescribed tangent line at some real point of the configuration. They are counted with respect to some sign defined by the parity of...

A mapping $F:ℝⁿ\to {ℝ}^m$ is called overdetermined if m > n. We prove that the calculations of both the local and global Łojasiewicz exponent of a real overdetermined polynomial mapping $F:ℝⁿ\to {ℝ}^m$ can be reduced to the case m = n.

We consider some variants of Łojasiewicz inequalities for the class of subsets of Euclidean spaces definable from addition, multiplication and exponentiation : Łojasiewicz-type inequalities, global Łojasiewicz inequalities with or without parameters. The rationality of Łojasiewicz’s exponents for this class is also proved.