We consider the set $R_n$ of rational functions of degree at most $n$ with no poles on the unit circle and its subclass $R_{n,r}$ consisting of rational functions without poles in the annulus $r<|z|<1/r$. We discuss an approach based on the model space theory which brings some integral representations for functions in $R_n$ and their derivatives. Using this approach we obtain $L^p$-analogs of several classical inequalities for rational functions including the inequalities by P. Borwein and T. Erdelyi, the Spijker Lemma and S.M. Nikolskii's inequalities. These inequalities are shown to be asymptotically sharp as $n$ tends to infinity and the poles of the rational functions approach the unit circle.

Given a rectangle in the real Euclidean n-dimensional space and two maps f and g defined on it and taking values in a metric semigroup, we introduce the notion of the total joint variation TV(f , g) of these maps. This extends similar notions considered by Hildebrandt (1963) [17], Leonov (1998) [18], Chistyakov (2003, 2005) [5,8] and the authors (2010). We prove the following irregular pointwise selection principle in terms of the total joint variation: if a sequence of maps {fj}∞ j=1 from the rectangle into a metric semigroup is pointwise precompact and lim supj,k→∞ TV(fj, fk) is finite, then it admits a pointwise convergent subsequence (whose limit may be a highly irregular, e.g., everywhere discontinuous, map). This result generalizes some recent pointwise selection principles for real functions and maps of several real variables.

Under certain initial conditions, we prove the existence of set-valued selectors of univariate compact-valued multifunctions of bounded (Jordan) variation when the notion of variation is defined taking into account only the Pompeiu asymmetric excess between compact sets from the target metric space. For this, we study subtle properties of the directional variations. We show by examples that all assumptions in the main existence result are essential. As an application, we establish the existence of set-valued solutions X(t) of bounded variation to the functional inclusion of the form X(t) ⊂F(t, X(t)) satisfying the initial condition X(t_0) =X_0.

The Nevanlinna theory is used to derive the general structure of transcendental meromorphic solutions for a wide class of autonomous nonlinear ordinary differential equations with two dominant monomials. An algebro-geometric method, which enables one to obtain these solutions explicitly, is described. New simply periodic solutions of the Lorenz system are obtained.

We show that a measure on the real line, that is the image of a log-concave measure under a polynomial of degree d, possesses a density from the Nikolskii–Besov class of fractional order 1/d. This result is used to prove an estimate for the total variation distance between such measures in terms of the Fortet–Mourier distance.

In this work we derive an inversion formula for the Laplace transform of a density observed on a curve in the complex domain, which generalizes the well known Post– Widder formula. We establish convergence of our inversion method and derive the corresponding convergence rates for the case of a Laplace transform of a smooth density. As an application we consider the problem of statistical inference for variance-mean mixture models. We construct a nonparametric estimator for the mixing density based on the generalized Post–Widder formula, derive bounds for its root mean square error and give a brief numerical example.

We consider the algebras M_p of Fourier multipliers and show that for every bounded continuous function f on R^d there exists a self-homeomorphism h of R^d such that the superposition foh$ is in M_p(R^d) for all p, 1<p<\infty. Moreover, under certain assumptions on a family K of continuous functions, one h will suffice for all f\in K. A similar result holds for functions on the torus T^d. This may be contrasted with the known solution of Luzin's problem related to the Wiener algebra.

We consider a family of non-autonomous second-order differential equations, which generalizes the Liénard equation. We explicitly find the necessary and sufficient conditions for members of this family of equations to admit quadratic, with the respect to the first derivative, first integrals. We show that these conditions are equivalent to the conditions for equations in the family under consideration to possess Lax representations. This provides a connection between the existence of a quadratic first integral and a Lax representation for the studied dissipative differential equations, which may be considered as an analogue to the theorem that connects Lax integrability and Arnold–Liouville integrability of Hamiltonian systems. We illustrate our results by several examples of dissipative equations, including generalizations of the Van der Pol and Duffing equations, each of which have both a quadratic first integral and a Lax representation.

Given a=(a1,…,an), b=(b1,…,bn)∈Rn with ab componentwise and a map f from the rectangle Iab=[a1,b1]×⋯×[an,bn] into a metric semigroup M=(M,d,+), denote by TV(f,Iab) the Hildebrandt–Leonov total variation of f on Iab, which has been recently studied in [V.V. Chistyakov, Yu.V. Tretyachenko, Maps of several variables of finite total variation. I, J. Math. Anal. Appl. (2010), submitted for publication]. The following Helly-type pointwise selection principle is proved: If a sequence{fj}j∈Nof maps fromIabinto M is such that the closure in M of the set{fj(x)}j∈Nis compact for eachx∈IabandC≡supj∈NTV(fj,Iab)is finite, then there exists a subsequence of{fj}j∈N, which converges pointwise onIabto a map f such thatTV(f,Iab)⩽C. A variant of this result is established concerning the weak pointwise convergence when values of maps lie in a reflexive Banach space (M,‖⋅‖) with separable dual M∗.

Given two points *a*=(*a*_{1},…,*a**n*) and *b*=(*b*_{1},…,*b**n*) from *R**n* with *a*<*b* componentwise and a map *f* from the rectangle into a metric semigroup *M*=(*M*,*d*,+), we study properties of the *total variation* of *f* on introduced by the first author in [V.V. Chistyakov, A selection principle for mappings of bounded variation of several variables, in: Real Analysis Exchange 27th Summer Symposium, Opava, Czech Republic, 2003, pp. 217–222] such as the additivity, generalized triangle inequality and sequential lower semicontinuity. This extends the classical properties of C. Jordan's total variation (*n*=1) and the corresponding properties of the total variation in the sense of Hildebrandt [T.H. Hildebrandt, Introduction to the Theory of Integration, Academic Press, 1963] (*n*=2) and Leonov [A.S. Leonov, On the total variation for functions of several variables and a multidimensional analog of Helly's selection principle, Math. Notes 63 (1998) 61–71] (*n*∈*N*) for real-valued functions of *n* variables.

The Liénard equation is used in various applications. Therefore, constructing general analytical solutions of this equation is an important problem. Here we study connections between the Liénard equation and some equations from the Painlevé–Gambier classification. We show that with the help of such connections one can construct general analytical solutions of the Liénard equation's subfamilies. In particular, we find three new integrable families of the Liénard equation. We also propose and discuss an approach for finding one-parameter families of closed-form analytical solutions of the Liénard equation.

We study spectral properties of one-dimensional singular nonselfadjoint perturbations of an unbounded selfadjoint operator and give criteria for the possibility to remove the whole spectrum by a perturbation of this type. A counterpart of our results for the case of bounded operators provides a complete description of compact selfadjoint operators whose rank one perturbation is a Volterra operator.

We introduce a pseudometric TV on the set M^X of all functions mapping a rectangle X on the plane R^2 into a metric space M, called the total joint variation. We prove that if two sequences {fj} and {gj} of functions from M^X are such that {fj} is pointwise precompact on X, {gj} is pointwise convergent on X with the limit g∈M^X, and the limit superior of TV(fj, gj) as j→∞ is finite, then a subsequence of {fj} converges pointwise on X to a function f∈M^X such that TV(f, g ) is finite. One more pointwise selection theorem is given in terms of total ε-variations (ε >0), which are approximations of the total variation as ε →0.

We study extensions of Sobolev and BV functions on infinite-dimensional domains. Along with some positive results we present a negative solution of the long-standing problem of existence of Sobolev extensions of functions in Gaussian Sobolev spaces from a convex domain to the whole space.