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Vladimir Lebedev

Publications and source records attributed to Vladimir Lebedev.

At least 19 recordsLinked to original sources
arXiv2025

The Sobolev space $W_2^{1/2}$: Simultaneous improvement of functions by a homeomorphism of the circle

It is known that for every continuous real-valued function $f$ on the circle $\mathbb T=\mathbb R/2\pi\mathbb Z$ there exists a change of variable, i.e., a self-homeomorphism $h$ of $\mathbb T$, such that the superposition $f\circ h$ is in the Sobolev space $W_2^{1/2}(\mathbb T)$. We obtain new results on simultaneous improvement of functions by a single change of variable in relation to the space $W_2^{1/2}(\mathbb T)$. The main result is as follows: there does not exist a self-homeomorphism $h$ of $\mathbb T$ such that $f\circ h\in W_2^{1/2}(\mathbb T)$ for every $f\in \mathrm{Lip}_{1/2}(\mathbb T)$. Here $\mathrm{Lip}_{1/2}(\mathbb T)$ is the class of all functions on $\mathbb T$ satisfying the Lipschitz condition of order $1/2$.

math.CA↗
arXiv2024

Molecular sorting on a fluctuating membrane

Molecular sorting in biological membranes is essential for proper cellular function. It also plays a crucial role in the budding of enveloped viruses from host cells. We recently proposed that this process is driven by phase separation, where the formation and growth of sorting domains depend primarily on direct intermolecular interactions. In addition to these, Casimir-like forces -- arising from entropic effects in fluctuating membranes -- may also play a significant role in the molecular distillation process. Here, using a combination of theoretical analysis and numerical simulations, we explore how Casimir-like forces between rigid membrane inclusions contribute to sorting, particularly in the biologically relevant regime where direct intermolecular interactions are weak. Our results show that these forces enhance molecular distillation by reducing the critical radius for the formation of new sorting domains and facilitating the capture of molecules within these domains. We identify the relative rigidity of the membrane and supermolecular domains as a key parameter controlling molecular sorting efficiency, offering new insights into the physical principles underlying molecular sorting in biological systems.

physics.bio-ph↗
arXiv2023

On extension to Fourier transforms

For $n$-point sets $K$ in an LCA group $Γ$ we obtain an estimate for the norm of "the best" extension operator from the space $l^\infty(K)$ of bounded functions on $K$ to the space $A(Γ)$ of Fourier transforms. As a simple consequence our estimate implies that if $K$ is an infinite closed subset of $Γ$ then there does not exist a bounded linear extension operator from the space $C_0(K)$ of continuous functions on $K$ vanishing at infinity to $A(Γ)$. The latter result generalizes a result by Graham who considered the case of compact subsets $K$.

math.FA↗
arXiv2023

Correcting One Error in Non-Binary Channels with Feedback

In this paper, the problem of correction of a single error in $q$-ary symmetric channel with noiseless feedback is considered. We propose an algorithm to construct codes with feedback inductively. For all prime power $q$ we prove that two instances of feedback are sufficient to transmit over the $q$-ary symmetric channel the same number of messages as in the case of complete feedback. Our other contribution is the construction of codes with one-time feedback with the same parameters as Hamming codes for $q$ that is not a prime power. We also construct single-error-correcting codes with one-time feedback of size $q^{n-2}$ for arbitrary $q$ and $n\leq q+1$, which can be seen as an analog for Reed-Solomon codes.

cs.IT↗
arXiv2023

Correcting one error in channels with feedback

We address the problem of correcting a single error in an arbitrary discrete memoryless channel with error-free instantaneous feedback. For the case of a one-time feedback, we propose a method for constructing optimal transmission strategies. The obtained result allows us to prove that for a binary channel, two feedbacks are sufficient to transmit the same number of messages as in the case of complete feedback. We also apply the developed techniques to a binary asymmetric channel to construct transmission strategies for small lengths.

cs.IT↗
arXiv2022

Non-adaptive and two-stage coding over the Z-channel

In this paper, we developed new coding strategies for the Z-channel. In particular, we look at the case with two-stage encoding. In this case, the encoder uses noiseless feedback once and adjusts the further encoding strategy based on the previous partial output of the channel. Nevertheless, the developed codes improve the known results with full feedback for small length and 1 error. A tool for the two-stage strategy is the development of a new optimality condition for non-adaptive codes.

cs.IT↗
arXiv2022

Two-stage coding over the Z-channel

In this paper, we discuss two-stage encoding algorithms capable of correcting a fraction of asymmetric errors. Suppose that the encoder transmits $n$ binary symbols $(x_1,\ldots,x_n)$ one-by-one over the Z-channel, in which a 1 is received only if a 1 is transmitted. At some designated moment, say $n_1$, the encoder uses noiseless feedback and adjusts further encoding strategy based on the partial output of the channel $(y_1,\ldots,y_{n_1})$. The goal is to transmit error-free as much information as possible under the assumption that the total number of errors inflicted by the Z-channel is limited by $τn$, $0<τ<1$. We propose an encoding strategy that uses a list-decodable code at the first stage and a high-error low-rate code at the second stage. This strategy and our converse result yield that there is a sharp transition at $τ=\max\limits_{0<w<1}\frac{w + w^3}{1+4w^3}\approx 0.44$ from positive rate to zero rate for two-stage encoding strategies. As side results, we derive bounds on the size of list-decodable codes for the Z-channel and prove that for a fraction $1/4+ε$ of asymmetric errors, an error-correcting code contains at most $O(ε^{-3/2})$ codewords.

cs.IT↗
arXiv2021

Tame semicascades and cascades generated by affine self-mappings of the $d$-torus

We give a complete characterization of the affine self-mappings $φ$ of the torus $\mathbb T^d$ that generate Köhler-tame semicascades and cascades. Namely, we show that the semicascade generated by $φ$ is tame if and only if the matrix $A$ of $φ$ satisfies $A^p=A^q$, where $p$ and $q$ are some nonnegative integers, $p\neq q$. For cascades the corresponding condition has the form $A^m=I$, where $m$ is some positive integer and $I$ is the identity matrix.

math.CA↗
arXiv2021

Coding with Noiseless Feedback over the Z-channel

In this paper, we consider encoding strategies for the Z-channel with noiseless feedback. We analyze the combinatorial setting where the maximum number of errors inflicted by an adversary is proportional to the number of transmissions, which goes to infinity. Without feedback, it is known that the rate of optimal asymmetric-error-correcting codes for the error fraction $τ\ge 1/4$ vanishes as the blocklength grows. In this paper, we give an efficient feedback encoding scheme with $n$ transmissions that achieves a positive rate for any fraction of errors $τ<1$ and $n\to\infty$. Additionally, we state an upper bound on the rate of asymptotically long feedback asymmetric error-correcting codes.

cs.IT↗
arXiv2020

Algorithms for $q$-ary Error-Correcting Codes with Limited Magnitude and Feedback

Berlekamp and Zigangirov completely determined the capacity error function for binary error correcting codes with noiseless feedback. It is still an unsolved problem if the upper bound for the capacity error function in the non-binary case of Ahlswede, Lebedev, and Deppe is sharp. We consider wraparound channels with limited magnitude and noiseless feedback. We completely determine the capacity error function for all $q$-ary wraparound channels with a magnitude of level $r$. All of our algorithms use partial noiseless feedback. Furthermore, a special case of the problem is equivalent to Shannon's zero-error problem.

cs.IT↗
arXiv2020

How to apply the rubber method for channels with feedback

We give an overview of applications of the rubber method. The rubber method is a coding algorithm that was developed in 2005 by Ahlswede, Deppe and Lebedev for channels with feedback. It was a big surprise that an encoding strategy that reserves one symbol exclusively as a rubber can sometimes reach the capacity of a channel. Since then, generalizations of the algorithm have been developed. These generalizations can be used for special channels. There are also other ideas for modifying the rubber method.

cs.IT↗
arXiv2020

Bounds for the capacity error function for unidirectional channels with noiseless feedback

In digital systems such as fiber optical communications, the ratio between probability of errors of type $1\to 0$ and $0 \to 1$ can be large. Practically, one can assume that only one type of error can occur. These errors arecalled asymmetric. Unidirectional errors differ from asymmetric type of errors; here both $1 \to 0$ and $0 \to 1$ type of errors are possible, but in any submittedcodeword all the errors are of the same type. This can be generalized for the $q$-ary case. We consider $q$-ary unidirectional channels with feedback and give bounds for the capacity error function. It turns out that the bounds depend on the parity of the alphabet $q$. Furthermore, we show that for feedback, the capacity error function for the binary asymmetric channel is different from the symmetric channel. This is in contrast to the behavior of the function without feedback.

cs.IT↗
arXiv2019

Statistical properties of the laser beam propagating in a turbulent medium

We examine statistical properties of a laser beam propagating in a turbulent medium. We prove that the intensity fluctuations at large propagation distances possess Gaussian probability density function and establish quantitative criteria for realizing the Gaussian statistics depending on the laser propagation distance, the laser beam waist, the laser frequency and the turbulence strength. We calculate explicitly the laser envelope pair correlation function and corrections to its higher order correlation functions breaking Gaussianity. We discuss also statistical properties of the brightest spots in the speckle pattern.

cond-mat.stat-mech↗
arXiv2019

Homeomorphic Changes of Variable and Fourier Multipliers

We consider the algebras $M_p$ of Fourier multipliers and show that every bounded continuous function $f$ on $\mathbb R^d$ can be transformed by an appropriate homeomorphic change of variable into a function that belongs to $M_p(\mathbb R^d)$ for all $p$, $1<p<\infty$. Moreover, under certain assumptions on a family $K$ of continuous functions, one change of variable will suffice for all $f\in K$. A similar result holds for functions on the torus $\mathbb T^d$. This may be contrasted with the known result on the Wiener algebra, related to Luzin's rearrangement problem.

math.CA↗
arXiv2018

Universality in statistics of Stokes flow over no-slip wall with random roughness

Stochastic roughness is widespread feature of natural surfaces and is an inherent by-product of most fabrication techniques. In view of rapid development of microfluidics, the important question is how this inevitable evil affects the low-Reynolds flows which are common for micro-devices. Moreover, one could potentially turn the flaw into a virtue and control the flow properties by means of specially "tuned" random roughness. In this paper we investigate theoretically the statistics of fluctuations in fluid velocity produced by the waviness irregularities at the surface of a no-slip wall. Particular emphasis is laid on the issue of the universality of our findings.

physics.flu-dyn↗
arXiv2017

Quantitative aspects of the Beurling--Helson theorem: Phase functions of a special form

We consider the space $A(\mathbb{T}^d)$ of absolutely convergent Fourier series on the torus $\mathbb{T}^d$. The norm on $A(\mathbb{T}^d)$ is naturally defined by $\|f\|_{A}=\|\widehat{f}\|_{l^1}$, where $\widehat{f}$ is the Fourier transform of a function $f$. For real functions $φ$ of a certain special form on $\mathbb T^d, \,d\geq 2,$ we obtain lower bounds for the norms $\|e^{iλφ}\|_A$ as $λ\rightarrow\infty$. In particular, we show that if $φ(x, y)=a(x)|y|$ for $|y|\leqπ$, where $a\in A(\mathbb{T})$ is an arbitrary nonconstant real function, then $\|e^{iλφ}\|_{A(\mathbb{T}^2)}\gtrsim |λ|$.

math.CA↗
arXiv2015

Structure of coherent vortices generated by the inverse cascade of two-dimensional turbulence in a finite box

We discuss structure and geometrical characteristics of coherent vortices appearing as a result of the inverse cascade in the two-dimensional turbulence in a finite box. We demonstrate that the universal velocity profile, established previously, corresponds to the passive regime of flow fluctuations. We find the vortex core radius and the vortex size and argue that an amount of the vortices generated in the box depends on the system parameters.

nlin.CD↗