We consider a random permutation drawn from the set of 132-avoiding permutations of length n and show that the number of occurrences of another pattern σ has a limit distribution, after scaling by n where λ(σ) is the length of σ plus the number of descents. The limit is not normal, and can be expressed as a functional of a Brownian excursion. Moments can be found by recursion.

The masses of the scalar (σ) and vector (ω) mesons and their widths in nuclear matter are studied in detail using the quark-meson coupling (QMC) model. It is shown that above normal nuclear matter density the effects of σ-ω mixing and the decay of the σ into 2π change the scalar meson mass considerably, while the vector meson mass remains near its mean-field value. We compare the QMC results wi...

The Schnorr-Stimm dichotomy theorem (Schnorr and Stimm, 1972) concerns finite-state gamblers that bet on infinite sequences of symbols taken from a finite alphabet Σ. asserts that, for any such sequence S, the following two things are true. (1) If S is not normal in sense Borel (meaning every strings equal length appear with asymptotic frequency S), then there gambler wins money at an infinitel...

We derive an a priori C2,α estimate for solutions of the fully non-linear elliptic equation F(D2u) = 0, provided the level set Σ = {M | F(M) = 0} satisfies: (a) Σ∩ {M | TrM = t} is strictly convex for all constants t; (b) the angle between the identity matrix I and the normal Fij to Σ is strictly positive on the non-convex part of Σ. Moreover, we do not need any convexity assumption on F in the...

In this paper, we are concerned with conditions under which [p(T)]⁎=p‾(T⁎), where p(z) is a one-variable complex polynomial, and T an unbounded, densely defined, linear operator. Then, deal the validity of identities σ(AB)=σ(BA), A B two unbounded operators. The equations (TT⁎)⁎=TT⁎ (T⁎T)⁎=T⁎T, defined closable operator, also studied. particular interest will be paid to equation T⁎T=p(T) its va...

The law of errors is the first example of a theory of fluctuations. It deals with sums of a large numberN of values σ1, . . . , σN occurring randomly with probability p(σ) equal for each pair of opposite values (i.e. p(σ) = p(−σ)), hence with 0 average. If the possible values of each σ are finitely many (and at least two) their sum can be of an order of magnitude as large as the number N , howe...