arXiv subjects
Publications and source records attributed to Dmitry Novikov.
The paper is devoted to a generalization of static and dynamic mathematical models of behavior with explicitly stated reflexive models of agents' decision-making. Reflexion is considered as agent's beliefs about nature, opponents' beliefs and opponents' decision-making principles in the framework of game theory, collective behavior theory and learning models.
We consider small polynomial deformations of integrable systems of the form $dF=0$, $F\in\mathbb{C}[x,y]$ and the first nonzero term $M_μ$ of the displacement function $Δ(t,ε)=\sum_{i=μ}M_i(t)ε^i$ along a cycle $γ(t)\in F^{-1}(t)$. It is known that $M_μ$ is an iterated integral of length at most $μ$. The bound $μ$ depends on the deformation of $dF$. In this paper we give a universal bound for the length of the iterated integral expressing the first nonzero term $M_μ$ depending only on the topology of the unperturbed system $dF=0$. The result generalizes the result of Gavrilov and Iliev providing a sufficient condition for $M_μ$ to be given by an abelian integral i.e. by an iterated integral of length $1$. We conjecture that our bound is optimal.
We consider the structure ${\mathbb R}^{\mathrm{RE}}$ obtained from $({\mathbb R},<,+,\cdot)$ by adjoining the restricted exponential and sine functions. We prove Wilkie's conjecture for sets definable in this structure: the number of rational points of height $H$ in the transcendental part of any definable set is bounded by a polynomial in $\log H$. We also prove two refined conjectures due to Pila concerning the density of algebraic points from a fixed number field, or with a fixed algebraic degree, for ${\mathbb R}^{\mathrm{RE}}$-definable sets.
We present a complex analytic proof of the Pila-Wilkie theorem for subanalytic sets. In particular, we replace the use of $C^r$-smooth parametrizations by a variant of Weierstrass division.
The \emph{Noetherian class} is a wide class of functions defined in terms of polynomial partial differential equations. It includes functions appearing naturally in various branches of mathematics (exponential, elliptic, modular, etc.). A conjecture by Khovanskii states that the \emph{local} geometry of sets defined using Noetherian equations admits effective estimates analogous to the effective \emph{global} bounds of algebraic geometry. We make a major step in the development of the theory of Noetherian functions by providing an effective upper bound for the local number of isolated solutions of a Noetherian system of equations depending on a parameter $ε$, which remains valid even when the system degenerates at $ε=0$. An estimate of this sort has played the key role in the development of the theory of Pfaffian functions, and is expected to lead to similar results in the Noetherian setting. We illustrate this by deducing from our main result an effective form of the Lojasiewicz inequality for Noetherian functions.
In this note we show that a linear ordinary differential equation with polynomial coefficients is globally non-oscillating in $\mathbb{C} P^1$ if and only if it is Fuchsian, and at every its singular point any two distinct characteristic exponents have distinct real parts. As a byproduct of our study, we obtain a new explicit upper bound for the number of zeros of exponential polynomials in a horizontal strip.
For functions of a single complex variable, points of multiplicity greater than $k$ are characterized by the vanishing of the first $k$ derivatives. There are various quantitative generalizations of this statement, showing that for functions that are in some sense close to having multiplicity greater than $k$, the first $k$ derivatives must be small. In this paper we aim to generalize this situation to the multi-dimensional setting. We define a class of differential operators, the \emph{multiplicity operators}, which act on maps from $\C^n$ to $\C^n$ and satisfy properties analogous to those described above. We demonstrate the usefulness of the construction by applying it to some problems in the theory of Noetherian functions.
We introduce and discuss a new class of (multivalued analytic) transcendental functions which still share with algebraic functions the property that the number of their isolated zeros can be explicitly counted. On the other hand, this class is sufficiently rich to include all periods (integral of rational forms over algebraic cycles).
We provide a partial answer to the following problem: \emph{give an effective upper bound on the multiplicity of non-isolated common zero of a tuple of Noetherian functions}. More precisely, consider a foliation defined by two commuting polynomial vector fields $V_1,V_2$ in $\C^n$, and $p$ a nonsingular point of the foliation. Denote by $\cL$ the leaf passing through $p$, and let $F,G\in\C[X]$ be two polynomials. Assume that $F\onL=0,G\onL=0$ have several common branches. We provide an effective procedure which allows to bound from above multipllicity of intersection of remaining branches of $F\onL=0$ with $G\onL=0$ in terms of the degrees and dimensions only.
The first and third year data releases from the WMAP provide evidence of an anomalous Cold Spot (CS) at galactic latitude b=-57deg and longitude l=209deg. We have examined the properties of the CS in some detail in order to assess its cosmological significance. We have performed a cluster analysis of the local extrema in the CMB signal to show that the CS is actually associated with a large group of extrema rather than just one. In the light of this we have re-examined the properties of the WMAP ILC and co-added "cleaned" WCM maps, which have previously been used for the analysis of the properties of the signal in the vicinity of the CS. These two maps have remarkably similar properties on equal latitude rings for |b|>30deg, as well as in the vicinity of the CS. We have also checked the idea that the CMB signal has a non-Gaussian tail, localized in the low multipole components of the signal. For each ring we apply a linear filter with characteristic scale R, dividing the CMB signal in two parts: the filtered part, with characteristic scale above that of the filter R, and the difference between the initial and filtered signal. Using the filter scale as a variable, we can maximize the skewness and kurtosis of the smoothed signal and minimize these statistics for the difference between initial and filtered signal. We have discovered that the shape of the CS is formed primarily by the components of the CMB signal represented by multipoles between 10<=L<=20, with a corresponding angular scale about 5-10 degs. This signal leads to modulation of the whole CMB sky, clearly seen at |b|>30deg in both the ILC and WCM maps, rather than a single localized feature. After subtraction of this modulation, the remaining part of the CMB signal appears to be consistent with statistical homogeneity and Gaussianity.
We single out a large class of semisimple singularities with the property that all roots of the Poincaré polynomial of the Lie algebra of derivations of the corresponding suitably (not necessarily quasihomogeneously) graded moduli algebra lie on the unit circle; for a still larger class there might occur exactly four roots outside the unit circle.
We study pseudo-Abelian integrals associated with polynomial deformations of slow-fast Darboux integrable systems. Under some assumptions we prove local boundedness of the number of their zeros.
We provide an effective uniform upper bond for the number of zeros of the first non-vanishing Melnikov function of a polynomial perturbations of a planar polynomial Hamiltonian vector field. The bound depends on degrees of the field and of the perturbation, and on the order $k$ of the Melnikov function. The generic case $k=1$ was considered by Binyamini, Novikov and Yakovenko (\cite{BNY-Inf16}). The bound follows from an effective construction of the Gauss-Manin connection for iterated integrals.
We prove that the number of limit cycles generated by a small non-conservative perturbation of a Hamiltonian polynomial vector field on the plane, is bounded by a double exponential of the degree of the fields. This solves the long-standing tangential Hilbert 16th problem. The proof uses only the fact that Abelian integrals of a given degree are horizontal sections of a regular flat meromorphic connection (Gauss-Manin connection) with a quasiunipotent monodromy group.
We consider an integrable polynomial system with generalized Darboux first integral H_0. We assume that it defines a family of real cycles in a region bounded by a polycycle. To any polynomial form ηone can associate the pseudo-abelian integrals I(h), which is the first order term of the displacement function of the system perturbed by η. We consider Darboux first integrals unfolding H_0 (and its saddle-nodes) and pseudo-abelian integrals associated to these unfoldings. Under genericity assumptions we show the existence of a uniform local bound for the number of zeros of these pseudo-abelian integrals. The result is part of a program to extend Varchenko-Khovanskii's theorem from abelian integrals to pseudo-abelian integrals and prove the existence of a bound for the number of their zeros in function of the degree of the polynomial system only.
Let $Π$ be an open, relatively compact period annulus of real analytic vector field $X_0$ on an analytic surface. We prove that the maximal number of limit cycles which bifurcate from $Π$ under a given multi-parameter analytic deformation $X_λ$ of $X_0$ is finite, provided that $X_0$ is either Hamiltonian, or generic Darbouxian vector field.
We prove an existential finiteness Varchenko-Khovanskii type result for integrals of rational 1-forms over the level curves of Darbouxian integrals.
We discuss the problem of finding an upper bound for the number of equilibrium points of a potential of several fixed point charges in R^n. This question goes back to J.C.Maxwell and M.Morse. Using fewnomial theory we show that for a given number of charges there exists an upper bound independent on the dimension, and show it to be 12 for three charges. We conjecture the exact upper bound for a given configuration of nonnegative charges in terms of its Voronoi diagram, and prove it asymptotically.