On a lemma of bowen

We are concerned with the sets of quasi generic points in finite symbolic space. We estimate the sizes of the sets by the Billingsley dimension defined by Gibbs measures. A dimension formula of such set is given, which generalizes Bowen's result. An application is given to the level sets of Birkhoff average.     

On a lemma of Crochemore and Rytter

Crochemore and Rytter introduced in 1995 a structural lemma on three squares starting at the same position. This influential lemma has been used by many researchers in the field of periodicities in strings. In particular, Fraenkel and Simpson used it in 1998 to obtain a universal upper bound for the maximum number of distinct squares occurring in a string. We present a generalization of Crochemore and Rytter's lemma by exploiting the combinatorics of two squares starting at the same position.     

On a generalization of the Kalman-Yakubovic lemma

In this paper we generalize the Kalman-Yakubovic lemma to infinite dimensions—or, more precisely, to semigroups of operators over a Hilbert space. The proof differs substantially from the finite-dimensional version and is based on the Paley-Wiener-Helson-Lowdenslager factorization theorem.     

On the Borel-Cantelli lemma

Necessary and sufficient conditions for P(A_n infinitely often) = , [0, 1], are obtained, where {A_n} is a sequence of events such that P(A _n ) = .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 184, pp. 200–207, 1990.     

On the bramble-hilbert lemma

The aim of this paper is to give an improvement of a result by Bramble and Hilbert [2] concerning upper bounds for linear functionals on certain Sobolev spaces by showing that the constants in the estimate are independent of the bounded Lipschitz domain Ω.     

On the Borel-Cantelli lemma

A new variant of the “divergent” part of the Borel-Cantelli lemma for events derived from a Markov chain is given. Further two applications are considered. One of the applications refers to the denumerable Markov chain and the second is a new proof of the “strong” theorem corresponding to the “arc sine law”.     

On a decomposition lemma for positive semi-definite block-matrices

This short note, in part of expository nature, points out several new or recent consequences of a quite nice decomposition for positive semi-definite matrices.     

On a lemma due to Ladyzhenskaya and Solonnikov and some applications

We present some applications of a lemma by Ladyzhenskaya and Solonnikov [Determination of solutions of boundary value problems for stationary Stokes and Navier–Stokes equations having an unbounded Dirichlet integral, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 96 (1980) 117–160 (English Transl.: J. Soviet Math. 21 (1983) 728–761)]. Some other results in that paper referring to stationary Navier–Stokes equations are extended to a non-Newtonian fluid, the so-called micropolar fluid. This model depends on the microrotational viscosity _νr${\nu }_{\mathrm{r}}$ which vanishes for a Navier–Stokes fluid. We use the lemma in full to show that, as _νr${\nu }_{\mathrm{r}}$ tends to zero, the solutions of the Ladyzhenskaya–Solonnikov problem converge to the solutions of the corresponding problem for Navier–Stokes equations. In addition, we obtain a similar convergence regarding the Leray problem for micropolar fluids.