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D. Novikov

Publications and source records attributed to D. Novikov.

At least 19 recordsLinked to original sources
arXiv2024

Least response method to separate CMB spectral distortions from foregrounds

We present a signal-foreground separation algorithm for filtering observational data to extract spectral distortions of the cosmic microwave background (CMB). Our linear method, called the least response method (LRM), is based on the idea of simultaneously minimizing the response to all possible foregrounds with poorly defined spectral shapes and random noise while maintaining a constant response to the signal of interest. This idea was introduced in detail in our previous paper. Here, we have expanded our analysis by taking into consideration all the main foregrounds. We draw a detailed comparison between our approach and the moment internal linear combination method, which is a modification of the internal linear combination technique previously used for CMB anisotropy maps. We demonstrate advantages of LRM and evaluate the prospects for measuring various types of spectral distortions. Besides, we show that LRM suggests the possibility of its improvements if we use an iterative approach with sequential separation and partial subtraction of foreground components from the observed signal. In addition, we estimate the optimal temperature that the telescope's optical system should have in order to detect the chemical type $\mu$ distortions. We present a design of an instrument where, according to our estimates, the optimal contrast between its thermal emission and the CMB allows us to measure such distortions.

astro-ph.CO↗
arXiv2023

Infinitesimal and tangential 16-th Hilbert problem on zero-cycles

In this paper, given two polynomials $f$ and $g$ of one variable and a $0$-cycle $C$ of $f$, we consider the deformation $f+\epsilon g$. We define two functions: the displacement function $\Delta(t,\epsilon)$ and its first order approximation: the abelian integral $M_1(t)$. The infinitesimal and tangential 16-th Hilbert problem for zero-cycles are problems of counting isolated regular zeros of $\Delta(t,\epsilon)$, for $\epsilon$ small, or of $M_1(t)$, respectively. We show that the two problems are not equivalent and find optimal bounds, in function of the degrees of $f$ and $g$, for the infinitesimal and tangential 16-th Hilbert problem on zero-cycles. These two problems are the zero-dimensional analogue of the classical infinitesimal and tangential 16-th Hilbert problems for vector fields in the plane.

math.DS↗
arXiv2015

A tropical analog of Descartes' rule of signs

We prove that for any degree d, there exist (families of) finite sequences a_0, a_1,..., a_d of positive numbers such that, for any real polynomial P of degree d, the number of its real roots is less than or equal to the number of the so-called essential tropical roots of the polynomial obtained from P by multiplication of its coefficients by a_0, a_1,... a_d respectively. In particular, for any real univariate polynomial P of degree d with non-vanishing constant term, we conjecture that one can take a_k = e^{-k^2}, k = 0, ... , d. The latter claim can be thought of as a tropical generalization of Descartes's rule of signs. We settle this conjecture up to degree 4 as well as a weaker statement for arbitrary real polynomials. Additionally we describe an application of the latter conjecture to the classical Karlin problem on zero-diminishing sequences.

math.CA↗
arXiv2008

A direct proof of one Gromov's theorem

We give a new proof of the Gromov theorem: For any $C>0$ and integer $n>1$ there exists a function $Δ_{C,n}$ such that if the Gromov--Hausdorff distance between complete Riemannian $n$-manifolds $V$ and $W$ is not greater than $δ$, absolute values of their sectional curvatures $|K_σ|\leq C$, and their injectivity radii $\geq 1/C$, then the Lipschitz distance between $V$ and $W$ is less than $Δ_{C,n}(δ)$ and $Δ_{C,n}\to 0$ as $δ\to 0$.

math.DG↗
arXiv2007

The three dimensional skeleton: tracing the filamentary structure of the universe

The skeleton formalism aims at extracting and quantifying the filamentary structure of the universe is generalized to 3D density fields; a numerical method for computating a local approximation of the skeleton is presented and validated here on Gaussian random fields. This method manages to trace well the filamentary structure in 3D fields such as given by numerical simulations of the dark matter distribution on large scales and is insensitive to monotonic biasing. Two of its characteristics, namely its length and differential length, are analyzed for Gaussian random fields. Its differential length per unit normalized density contrast scales like the PDF of the underlying density contrast times the total length times a quadratic Edgeworth correction involving the square of the spectral parameter. The total length scales like the inverse square smoothing length, with a scaling factor given by 0.21 (5.28+ n) where n is the power index of the underlying field. This dependency implies that the total length can be used to constrain the shape of the underlying power spectrum, hence the cosmology. Possible applications of the skeleton to galaxy formation and cosmology are discussed. As an illustration, the orientation of the spin of dark halos and the orientation of the flow near the skeleton is computed for dark matter simulations. The flow is laminar along the filaments, while spins of dark halos within 500 kpc of the skeleton are preferentially orthogonal to the direction of the flow at a level of 25%.

astro-ph↗
arXiv2003

Oscillation of Fourier Integrals with a spectral gap

Suppose that Fourier transform of a function f is zero on the interval [-a,a]. We prove that the lower density of sign changes of f is at least a/pi, provided that f is a locally integrable temperate distribution in the sense of Beurling, with non-quasianalytic weight. We construct an example showing that the last condition cannot be omitted.

math.CA↗
arXiv2002

Lectures on meromorphic flat connections

These notes form an extended version of a minicourse delivered in Universite de Montreal (June 2002) within the framework of a NATO workshop ``Normal Forms, Bifurcations and Finiteness Problems in Differential Equations''. The focus is on Poincare--Dulac theory of ``Fuchsian'' (logarithmic) singularities of integrable systems, with applications to problems on zeros of Abelian integrals in view.

math.CA↗
arXiv2002

Modules of Abelian integrals and Picard-Fuchs systems

We give a simple proof of an isomorphism between the two $\mathbb{C}[t]$-modules: the module of relative cohomologies $Λ^2/dH\land Λ^1$ and the module of Abelian integrals corresponding to a regular at infinity polynomial $H$ in two variables. Using this isomorphism, we prove existence and deduce some properties of the corresponding Picard-Fuchs system.

math.DS↗
arXiv2002

Convex-concave body in $\mathbb{R}P^3$ contains a line

We define a class of $L$-convex-concave subsets of $\mathbb{R}P^3$, where $L$ is a projective line in $\mathbb{R}P^3$. These are sets whose sections by any plane containing $L$ are convex and concavely depend on this plane. We prove a version of Arnold hypothesis for these sets, namely we prove that each such set contains a line.

math.DG↗
arXiv2002

On affine hypersurfaces with everywhere nondegenerate Second Quadratic Form

Consider a closed connected hypersurface in $\mathbb{R}^n$ with constant signature (k,l) of the second quadratic form, and approaching a quadratic cone at infinity. This hypersurface divides $\mathbb{R}^n$ into two pieces. We prove that one of them contains a k-dimensional subspace, and another contains a l-dimensional subspace, thus proving an affine version of Arnold hypothesis. We construct an example of a surface of negative curvature in $\mathbb{R}^3$ with slightly different asymptotical behavior for which the previous claim is wrong.

math.DG↗
arXiv2002

L-convex-concave sets in real projective space and L-duality

We define a class of L-convex-concave subsets of $\Bbb{R}P^n$, where L is a projective subspace of dimension l in $\Bbb{R}P^n$. These are sets whose sections by any (l+1)-dimensional space L' containing L are convex and concavely depend on L'. We introduce an L-duality for these sets, and prove that the L-dual to an L-convex-concave set is an $L^*$-convex-concave subset of $(\Bbb RP^n)^*$. We discuss a version of Arnold hypothesis for these sets and prove that it is true (or wrong) for an L-convex-concave set and its L-dual simultaneously.

math.DG↗
arXiv2000

Estimation of Signal and Noise Parameters from CMB Polarization Observations

We propose a technique for determination of the spectral parameters of the cosmological signal and pixel noise using observational data on CMB polarization without any additional assumptions. We introduce the notion of so called crossing points in the observational maps and derive the theoretical dependence of the total number of crossing points at different level of the polarization. Finally, we use the statistics of the signal in the vicinities of the singular points, where the polarization of the pure CMB vanishes to correct the final result.

astro-ph↗
arXiv2000

Peculiarities of anisotropy and polarization as an indicator of noises in the CMB maps

We discuss some new problems of the modern cosmology which arose after the BOOMERANG and MAXIMA-1 successful missions. Statistics of high peaks of the CMB anisotropy is analyzed and we discuss possible inner structure of such peaks in the observational data of future MAP and PLANCK missions. We have investigated geometrical and statistical properties of the CMB polarization around such high isolated peaks of anisotropy in the presence of a polarized pixel noise and point sources. The structure of polarization fields in the vicinity of singular points with zero polarization is very sensitive to the level of pixel noises and point sources in the CMB maps.

astro-ph↗
arXiv2000

Amplitude-Phase Analysis of Cosmic Microwave Background maps

We propose a novel method for the extraction of unresolved point sources from CMB maps. This method is based on the analysis of the phase distribution of the Fourier components for the observed signal and unlike most other methods of denoising does not require any significant assumptions about the expected CMB signal. The aim of our paper is to show how, using our algorithm, the contribution from point sources can be separated from the resulting signal on all scales. We believe that this technique is potentially a very powerful tool for extracting this type of noise from future high resolution maps.

astro-ph↗
arXiv2000

Power filtration of CMB observational data

We propose a power filter Gp for linear reconstruction of the CMB signal from observational maps. This Gp filter preserves the power spectrum of the CMB signal in contrast to the Wiener filter which diminishes the power spectrum of the reconstructed CMB signal. We demonstrate how peak statistics and a cluster analysis can be used to estimate the probability of the presence of a CMB signal in observational records. The efficiency of the Gp filter is demonstrated on a toy model of an observational record consisting of a CMB signal and noise in the form of foreground point sources.

astro-ph↗