On the rational topological complexity of coformal elliptic spaces

We establish some upper and lower bounds of the rational topological complexity for certain classes of elliptic spaces. Our techniques permit us in particular to show that the rational topological complexity coincides with the dimension of the rational homotopy for some special families of coformal elliptic spaces.     

On infinite real trace rational languages of maximum topological complexity

We consider the set ℝ^ω(Γ, D) of infinite real traces, over a dependence alphabet (Γ,D) with no isolated letter, equipped with the topology induced by the prefix metric. We prove that all rational languages of infinite real traces are analytic sets. We also reprove that there exist some rational languages of infinite real traces that are analytic but non-Borel sets; in fact, these sets are even Σ ₁ ¹ -complete, hence have maximum possible topological complexity. For this purpose, we give an example of a Σ ₁ ¹ -complete language that is fundamentally different from the known example of a Σ ₁ ¹ -complete infinitary rational relation given by Finkel (2003). Bibliography: 35 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 316, 2004, pp. 205–223.     

On the rational homotopy type of function spaces

The main result of this paper is the construction of a minimal model for the function space of continuous functions from a finite type, finite dimensional space to a finite type, nilpotent space in terms of minimal models for and . For the component containing the constant map, in positive dimensions. When is formal, there is a simple formula for the differential of the minimal model in terms of the differential of the minimal model for and the coproduct of . We also give a version of the main result for the space of cross sections of a fibration.

     

On multiplicative independence of rational function iterates

We give lower bounds for the degree of multiplicative combinations of iterates of rational functions (with certain exceptions) over a general field, establishing the multiplicative independence of said iterates. This leads to a generalisation of Gao’s method for constructing elements in the finite field $${mathbb {F}}_{q^n}$$ whose orders are larger than any polynomial in n when n becomes large. Additionally, we discuss the finiteness of polynomials which translate a given finite set of polynomials to become multiplicatively dependent.     

Evaluation of a rational function

Evaluation of a given rational function can be carried out in various ways, such as by Horner's method, directly through a continued fraction representation, or by a subsequent process that uses three-term recurrences. Four such algorithms are summarized, and their relative performances when they are implemented using a parallel architecture (a four-node transputer system) is investigated.     

On the complexity of submodular function minimisation on diamonds

Let (L;?,?) be a finite lattice and let n be a positive integer. A function f:Lⁿ→R is said to be submodular if for all . In this article we study submodular functions when L is a diamond. Given oracle access to f we are interested in finding such that as efficiently as possible. We establish

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  a min–max theorem, which states that the minimum of the submodular function is equal to the maximum of a certain function defined over a certain polyhedron; and

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  a good characterisation of the minimisation problem, i.e., we show that given an oracle for computing a submodular f:Ln→Z and an integer m such that , there is a proof of this fact which can be verified in time polynomial in n and ; and

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  a pseudopolynomial-time algorithm for the minimisation problem, i.e., given an oracle for computing a submodular f:Ln→Z one can find in time bounded by a polynomial in n and .

     

On the complexity of finding a local minimizer of a quadratic function over a polytope

Mathematical Programming - We show that unless P=NP, there cannot be a polynomial-time algorithm that finds a point within Euclidean distance $$c^n$$ (for any constant $$c ge 0$$ ) of a local...     

The bounded complexity function versus the unbounded complexity function

We construct two point-wise periodic flows which are equivalent, such that all the complexity functions of one flow are bounded while the other flow has an unbounded complexity function.     

On the limit set at infinity of a gradient trajectory of a semialgebraic function

Given any C² semialgebraic function f defined on a non-bounded open set of Rn, we prove that the limit of the secants at infinity of a non-bounded trajectory of the gradient of f does exist. As a corollary we find a new sufficient condition to ensure the trivialisation by the gradient flow of f nearby a regular asymptotic critical value at infinity.     

Estimates of the derivative of a rational function

We investigate some regular rational functions on the real axis, with fixed poles, conjugate in pairs, of half-integer order. An extremal equality is obtained, which generalizes one of L. V. Taikov's results for trigonometric polynomials.Translated from Matematicheskie Zametki, Vol. 13, No. 4, pp. 493–498, April, 1973.     

Motivic Milnor fibers of a rational function

Let P and Q be two complex polynomials and f be the induced rational function. In this Note we define a motivic Milnor fiber of the germ of f at an indeterminacy point x for a value a, a motivic Milnor fiber of f for a value a and finally motivic bifurcation sets.     

An estimate for the modulus of a rational function

The paper presents sharp inequalities for the moduli of rational functions under certain constraints on the modulus of the independent variable. These inequalities supplement some results of Govil, Mohapatra, and Dubinin. Bibliography: 6 titles.     

Expected values for the rational complexity of finite binary sequences

2-Adic complexity plays an important role in cryptology. It measures the difficulty of outputting a binary sequence using a feedback with carry shift register. This paper studies the 2-adic complexity of finite sequences by investigating the corresponding rational complexity whose logarithm to the base 2 is just equal to the 2-adic complexity. Experiments show that the logarithm to the base 2 of the expected values for rational complexity is a good approximation to the expected values for the 2-adic complexity. Both a nontrivial lower bound and a nontrivial upper bound on the expected values for the rational complexity of finite sequences are given in the paper. In particular, the lower bound is much better than the upper bound.     

Computation complexity of a ternary linear function

The computation complexity is shown to equal 18 in the class of generalized (ternary) ??-schemes of a ternary counter of order 3 that depend on tree variables.     

On an optimal control problem with an integral functional of a rational control function

We present the solution of an optimization problem with integral performance functional that is nonlinear with respect to the control and contains a discounting parameter in the class of programmed controls under two-sided control constraints. The optimal control is found in the form of a function of time (a program). On the basis of the theoretical results, we perform numerical experiments with model and real data.