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Vladimir Lebedev

Publications and source records attributed to Vladimir Lebedev.

At least 37 records · Page 2Linked to original sources
arXiv2015

Change of variable and the rapidity of decrease of Fourier coefficients

We consider the class $C(T)$ of continuous real-valued functions on the circle. For certain classes of functions naturally characterised by the rapidity of decrease of Fourier coefficients we investigate whether it is possible to bring families of functions in $C(T)$ into these classes by a change of variable. This paper was originally published in Matematicheski\vı Sbornik, 181:8 (1990), 1099--1113 (Russian). The English translation, published in Mathematics of the USSR, Sbornik, 70:2 (1991), 541--555, is to a large extent inconsistent with the original text. Herein the author provides a corrected translation.

math.CA↗
arXiv2015

The Bohr--Pál Theorem and the Sobolev Space $W_2^{1/2}$

The well-known Bohr--Pál theorem asserts that for every continuous real-valued function $f$ on the circle $\mathbb T$ there exists a change of variable, i.e., a homeomorphism $h$ of $\mathbb T$ onto itself, such that the Fourier series of the superposition $f\circ h$ converges uniformly. Subsequent improvements of this result imply that actually there exists a homeomorphism that brings $f$ into the Sobolev space $W_2^{1/2}(\mathbb T)$. This refined version of the Bohr--Pál theorem does not extend to complex-valued functions. We show that if $α<1/2$, then there exists a complex-valued $f$ that satisfies the Lipschitz condition of order $α$ and at the same time has the property that $f\circ h\notin W_2^{1/2}(\mathbb T)$ for every homeomorphism $h$ of $\mathbb T$.

math.CA↗
arXiv2014

Cascading of Fluctuations in Interdependent Energy Infrastructures: Gas-Grid Coupling

The revolution of hydraulic fracturing has dramatically increased the supply and lowered the cost of natural gas in the United States driving an expansion of natural gas-fired generation capacity in many electrical grids. Unrelated to the natural gas expansion, lower capital costs and renewable portfolio standards are driving an expansion of intermittent renewable generation capacity such as wind and photovoltaic generation. These two changes may potentially combine to create new threats to the reliability of these interdependent energy infrastructures. Natural gas-fired generators are often used to balance the fluctuating output of wind generation. However, the time-varying output of these generators results in time-varying natural gas burn rates that impact the pressure in interstate transmission pipelines. Fluctuating pressure impacts the reliability of natural gas deliveries to those same generators and the safety of pipeline operations. We adopt a partial differential equation model of natural gas pipelines and use this model to explore the effect of intermittent wind generation on the fluctuations of pressure in natural gas pipelines. The mean square pressure fluctuations are found to grow linearly in time with points of maximum deviation occurring at the locations of flow reversals.

physics.soc-ph↗
arXiv2014

Universal profile of the vortex condensate in two-dimensional turbulence

An inverse turbulent cascade in a restricted two-dimensional periodic domain leads to the creation of condensate -- a pair of coherent system-size vortices. We perform extensive numerical simulations of this system and carry on detailed theoretical analysis based upon momentum and energy exchanges between the turbulence fluctuations and the mean coherent condensate (vortices). The theory predicts the vortex profile and amplitude which perfectly agree with the numerical data.

physics.flu-dyn↗
arXiv2012

Tail-Constraining Stochastic Linear-Quadratic Control: Large Deviation and Statistical Physics Approach

Standard definition of the stochastic Risk-Sensitive Linear-Quadratic (RS-LQ) control depends on the risk parameter, which is normally left to be set exogenously. We reconsider the classical approach and suggest two alternatives resolving the spurious freedom naturally. One approach consists in seeking for the minimum of the tail of the Probability Distribution Function (PDF) of the cost functional at some large fixed value. Another option suggests to minimize the expectation value of the cost functional under constraint on the value of the PDF tail. Under assumption of the resulting control stability, both problems are reduced to static optimizations over stationary control matrix. The solutions are illustrated on the examples of scalar and 1d chain (string) systems. Large Deviation self-similar asymptotic of the cost functional PDF is analyzed.

cond-mat.stat-mech↗
arXiv2012

On $L^2$ -functions with bounded spectrum

We consider the class $PW(\mathbb R^n)$ of functions in $L^2(\mathbb R^n)$, whose Fourier transform has bounded support. We obtain a description of continuous maps $φ: \mathbb R^m\rightarrow\mathbb R^n$ such that $f\circφ\in PW(\mathbb R^m)$ for every function $f\in PW(\mathbb R^n)$. Only injective affine maps $φ$ have this property.

math.CA↗
arXiv2012

On uniform convergence of Fourier series

We consider the space $U(\mathbb T)$ of all continuous functions on the circle $\mathbb T$ with uniformly convergent Fourier series. We show that if $φ: \mathbb T\rightarrow\mathbb T$ is a continuous piecewise linear but not linear map, then $\|e^{inφ}\|_{U(\mathbb T)}\simeq\log n$.

math.CA↗
arXiv2012

Absolutely convergent Fourier series. An improvement of the Beurling--Helson theorem

We consider the space $A(\mathbb T)$ of all continuous functions $f$ on the circle $\mathbb T$ such that the sequence of Fourier coefficients $\hat{f}=\{\hat{f}(k), ~k \in \mathbb Z\}$ belongs to $l^1(\mathbb Z)$. The norm on $A(\mathbb T)$ is defined by $\|f\|_{A(\mathbb T)}=\|\hat{f}\|_{l^1(\mathbb Z)}$. According to the known Beurling--Helson theorem, if $ϕ: \mathbb T\rightarrow\mathbb T$ is a continuous mapping such that $\|e^{inϕ}\|_{A(\mathbb T)}=O(1), ~n\in\mathbb Z,$ then $ϕ$ is linear. It was conjectured by Kahane that the same conclusion about $ϕ$ is true under the assumption that $\|e^{inϕ}\|_{A(\mathbb T)}=o(\log |n|)$. We show that if $\|e^{inϕ}\|_{A(\mathbb T)}=o((\log\log |n|/\log\log\log |n|)^{1/12})$ then $ϕ$ is linear.

math.CA↗
arXiv2012

Estimates in Beurling--Helson type theorems. Multidimensional case

We consider the spaces $A_p(\mathbb T^m)$ of functions $f$ on the $m$ -dimensional torus $\mathbb T^m$ such that the sequence of the Fourier coefficients $\hat{f}=\{\hat{f}(k), ~k \in \mathbb Z^m\}$ belongs to $l^p(\mathbb Z^m), ~1\leq p<2$. The norm on $A_p(\mathbb T^m)$ is defined by $\|f\|_{A_p(\mathbb T^m)}=\|\hat{f}\|_{l^p(\mathbb Z^m)}$. We study the rate of growth of the norms $\|e^{iλφ}\|_{A_p(\mathbb T^m)}$ as $|λ|\rightarrow \infty, ~λ\in\mathbb R,$ for $C^1$ -smooth real functions $φ$ on $\mathbb T^m$ (the one-dimensional case was investigated by the author earlier). The lower estimates that we obtain have direct analogues for the spaces $A_p(\mathbb R^m)$.

math.CA↗
arXiv2011

Quantitative estimates in Beurling--Helson type theorems

We consider the spaces $A_p(\mathbb T)$ of functions $f$ on the circle $\mathbb T$ such that the sequence of Fourier coefficients $\fu{\f}=\{\fu{\f}(k), ~k \in \mathbb Z\}$ belongs to $l^p, ~1\leq p<2$. The norm on $A_p(\mathbb T)$ is defined by $\|f\|_{A_p}=\|\fu{\f}\nolinebreak\|_{l^p}$. We study the rate of growth of the norms $\|e^{iλφ}\|_{A_p}$ as $|λ|\rightarrow \infty, ~λ\in\mathbb R,$ for $C^1$ -smooth real functions $φ$ on $\mathbb T$. The results have natural applications to the problem on changes of variable in the spaces $A_p(\mathbb T)$.

math.CA↗
arXiv2010

Vorticity statistics in the direct cascade of two-dimensional turbulence

For the steady-state direct cascade of two-dimensional Navier-Stokes turbulence, we derive analytically the probability of strong vorticity fluctuations. The probability density function (pdf) of the vorticity coarse-grained over a scale in the inertial interval is shown to have a universal self-similar form, in distinction from other known direct cascades. The tails of the pdf are found to be exponential.

nlin.CD↗
arXiv2003

Periodic and Quasi-Periodic Compensation Strategies of Extreme Outages caused by Polarization Mode Dispersion and Amplifier Noise

Effect of birefringent disorder on the Bit Error Rate (BER) in an optical fiber telecommunication system subject to amplifier noise may lead to extreme outages, related to anomalously large values of BER. We analyze the Probability Distribution Function (PDF) of BER for various strategies of Polarization Mode Dispersion (PMD) compensation. A compensation method is proposed that is capable of more efficient extreme outages suppression, which leads to substantial improvement of the fiber system performance.

physics.optics↗
arXiv2003

Compensation for Extreme Outages caused by Polarization Mode Dispersion and Amplifier noise

Joint effect of weak birefringent disorder and amplifier noise on transmission in optical fiber communication systems appears to be strong. The probability of an extreme outage that corresponds to anomalously large values of Bit Error Rate (BER) is perceptible. We analyze the dependence of the Probability Distribution Function (PDF) of BER on the first-order and also higher-order PMD compensation schemes.

physics.optics↗
arXiv2003

Extreme Outages due to Polarization Mode Dispersion

We investigate the dependence of the bit-error-rate (BER) caused by amplifier noise in a linear optical fiber telecommunication system on the fiber birefringence. We show that the probability distribution function (PDF) of BER obtained by averaging over many realizations of birefringent disorder has an extended tail corresponding to anomalously large values of BER. We specifically discuss the dependence of the tail on such details of the pulse detection at the fiber output as "setting the clock" and filtering procedures.

physics.optics↗
arXiv2003

Probability of anomalously large Bit-Error-Rate in long haul optical transmission

We consider a linear model of optical pulse transmission through fiber with birefringent disorder in the presence of amplifier noise. Both disorder and noise are assumed to be weak, i.e. the average bit-error rate (BER) is small. The probability of rare violent events leading to the values of BER much larger than its typical value is estimated. We show that the probability distribution has a long algebraic tail.

cond-mat.stat-mech↗