We prove that the Reeb space of a proper definable map $$f:X \rightarrow Y$$ in an arbitrary o-minimal expansion real closed field is realizable as quotient. This result can be seen analog Stein factorization morphisms algebraic geometry. also show Betti numbers f arbitrarily large compared to those X, unlike special case graphs manifolds. Nevertheless, when semi-algebraic and X bounded, we sin...

this paper extends the scope of algebraic computation based on a non standard × to the more basic case of a non standard +, where standard means associative and commutative. two physically meaningful examples of a non standard + are provided by the observation of motion in special relativity, from either outside (3d) or inside (2d or more), we revisit the “gyro”-theory of ungar to present the m...

in this paper, a vector version of the intermediate value theorem is established. the main theorem of this article can be considered as an improvement of the main results have been appeared in [textit{on fixed point theorems for monotone increasing vector valued mappings via scalarizing}, positivity, 19 (2) (2015) 333-340] with containing the uniqueness, convergent of each iteration to the fixe...

Homology theories for algebraic varieties are often constructed using simplicial sets of algebraic cycles. For example, Bloch’s higher Chow groups and motivic cohomology, as defined by Suslin and Voevodsky [17], are given in this fashion. In this paper, we construct homology groups Hi(X,G), where G is an algebraic group and X is a variety, by considering cycles on the simplicial scheme BG×X, an...

The Dirac quantization ‘procedure’ for constrained systems is well known to have many subtleties and ambiguities. Within this ill-defined framework, we explore the generality of a particular interpretation of the Dirac procedure known as refined algebraic quantization. We find technical conditions under which refined algebraic quantization can reproduce the general implementation of the Dirac s...

Let X be an Artin stack (always assumed to have quasi-compact and separated diagonal over SpecZ; cf. [2, §1.3]). A coarse moduli space for X is a map π : X → X to an algebraic space such that (i) π is initial among maps from X to algebraic spaces (note that the category of maps from X to an algebraic space is discrete), and (ii) for every algebraically closed field k the map [X (k)]→ X(k) is bi...

A first example of a connection between transcendental numbers and complex dynamics is the following. Let p and q be polynomials with complex coefficients of the same degree. A classical result of Böttcher states that p and q are locally conjugates in a neighborhood of∞: there exists a function f , conformal in a neighborhood of infinity, such that f(p(z)) = q(f(z)). Under suitable assumptions,...

Let k be a field. Among the algebraic invariants associated to k are the Milnor Kgroups, one for each integer n ≥ 0. These abelian groups were first defined (but not so named) by Milnor in the context of quadratic forms [Mi]. The definition is completely algebraic; nevertheless, a beautiful geometric connection with Bloch’s higher Chow groups was discovered by Nesterenko-Suslin [NS] and Totaro ...

We examine various versions of oriented cohomology and BorelMoore homology theories in algebraic geometry and put these two together in the setting of an “oriented duality theory”, a generalization of Bloch-Ogus twisted duality theory. We apply this to give a Borel-Moore homology version MGL ∗,∗ of Voevodsky’s MGL-theory, and a natural map θ : Ω∗ → MGL 2∗,∗ , where Ω∗ is the algebraic cobordism...

Topological quantum field theory led recently to a spectacular progress in numerical algebraic geometry. It was shown that generating functions of certain char-actertistic numbers of modular spaces of stable algebraic curves with labelled points satisfy remarkable differential equations of KP–type (E. Witten, M. Kontsevich). In a later series of developments, this was generalized, partly conjec...