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...

In this paper we investigate the exact peroidic wave shock solutions of the Burgers equations. Our purpose is to describe the asymptotic behavior of the solution in the cauchy problem for viscid equation with small parametr ε and to discuss in particular the case of periodic wave shock. We show that the solution of this problem approaches the shock type solution for the cauchy problem of the in...

Let L be an unbounded linear operator in a real Hilbert space H, a generator of a C0 semigroup, and let g : H → H be a C2 loc nonlinear map. The DSM (dynamical systems method) for solving equation F (v) := Lv+ g(v) = 0 consists of solving the Cauchy problem u̇ = Φ(t, u), u(0) = u0, where Φ is a suitable operator, and proving that i) ∃u(t) ∀t > 0, ii) ∃u(∞), and iii) F (u(∞)) = 0. Conditions on L...

We show that in the analytic category, given a Riemannian metric g on a hypersurface M ⊂ Z and a symmetric tensor W on M , the metric g can be locally extended to a Riemannian Einstein metric on Z with second fundamental form W , provided that g and W satisfy the constraints on M imposed by the contracted Gauss and CodazziMainardi equations. We use this fact to study the Cauchy problem for metr...

Proof. Properties 2 and 3 are left to the reader. For property 1, assume that S is an unbounded compact set. Since S is unbounded, we may select a sequence {vn}n=1 such that ‖vn‖ → 0 as n→∞. Since S is compact, this sequence will have a convergent subsequence, say {vk}k=1, which will still be unbounded. This sequence is Cauchy, so there is a positive integer K for which ‖v`− vm‖ ≤ 1/2 for all `...

Since Cauchy numbers were introduced, various types of Cauchy numbers have been presented. In this paper, we define degenerate Cauchy numbers of the third kind and give some identities for the degenerate Cauchy numbers of the third kind. In addition, we give some relations between four kinds of the degenerate Cauchy numbers, the Daehee numbers and the degenerate Bernoulli numbers.

The symmetric shape of some inequalities between two sequences real numbers generates the same in operator theory. In this paper, we study a new refinement Cauchy–Bunyakovsky–Schwarz inequality for Euclidean spaces and several bounded linear operators on Hilbert space, where mention Bohr’s Bergström’s operators. We present an type operators, by technique monotony sequence. also prove Aczél spac...

We derive strong laws of large numbers and central limit theorems for Bajraktarević, Gini exponential- (also called Beta-type) logarithmic Cauchy quotient means independent identically distributed (i.i.d.) random variables. The a sequence i.i.d. variables behave asymptotically normal with the usual square root scaling just like geometric given Somewhat surprisingly, multiplicative in rather dif...