Using symmetric function theory, we study the cycle structure and increasing subsequence structure of permutations after iterations of various shuffling methods. We emphasize the role of Cauchy type identities and variations of the Robinson-Schensted-Knuth correspondence.

We solve the Cauchy problem for the Korteweg–de Vries equation with initial conditions which are steplike Schwartz-type perturbations of finitegap potentials under the assumption that the mutual spectral bands either coincide or are disjoint.

We give here a simple proof of weighted logarithmic Sobolev inequality, for example for Cauchy type measures, with optimal weight, sharpening results of BobkovLedoux [12]. Some consequences are also discussed.

Given function f holomorphic at infinity, the n-th diagonal Padé approximant to f , say [n/n]f , is a rational function of type (n,n) that has the highest order of contact with f at infinity. Equivalently, [n/n]f is the n-th convergent of the continued fraction representing f at infinity. BernsteinSzegő theorem provides an explicit non-asymptotic formula for [n/n]f and all n large enough in the...

The computability of the solution operator of the Cauchy problem for the Fifth-order CamassaHolm equation is studied in this paper. Firstly, a nonlinear map KR : H → C (R;H (R)) is defined from the initial value φ to the solution u. Then we used the relevant knowledge of type-2 theory of effectivity, functional analysis and Sobolev space to prove that when s > (6 √ 10− 17) / 4, the solution ope...

To every Darboux integrable system there is an associated Lie group G which is a fundamental invariant of the system and which we call the Vessiot group. This article shows that solving the Cauchy problem for a Darboux integrable partial differential equation can be reduced to solving an equation of Lie type for the Vessiot group G. If the Vessiot group G is solvable then the Cauchy problem can...

A Sine function approach is used to derive a new Hunter type quadrature rule for the evaluation of Cauchy principal value integrals of certain analytic functions. Integration over a general arc in the complex plane is considered. Special treatment is given to integrals over the interval (-1, 1). It is shown that the quadrature error is of order 0(e~ ), where N is the number of nodes used, and w...

In this paper, we present a few recent existence results via variational approach for the Cauchy problem dy dt (t) + A(t)y(t) ∋ f(t), y(0) = y0, t ∈ [0, T ], where A(t) : V → V ′ is a nonlinear maximal monotone operator of subgradient type in a dual pair (V, V ′) of reflexive Banach spaces. In this case, the above Cauchy problem reduces to a convex optimization problem via Brezis–Ekeland device...

Every infinitely divisible law defines a convolution semigroup that solves an abstract Cauchy problem. In the fractional Cauchy problem, we replace the first order time derivative by a fractional derivative. Solutions to fractional Cauchy problems are obtained by subordinating the solution to the original Cauchy problem. Fractional Cauchy problems are useful in physics to model anomalous diffus...

Chruściel and Galloway constructed a Cauchy horizon that is nondifferentiable on a dense set. We prove that in a certain class of Cauchy horizons densely nondifferentiable Cauchy horizons are generic. We show that our class of densely nondifferentiable Cauchy horizons implies the existence of densely nondifferentiable Cauchy horizons arising from partial Cauchy surfaces and also the existence o...