We derive an algorithm of optimal complexity which determines whether a given matrix is a Cauchy matrix, and which exactly recovers the Cauchy points defining a Cauchy matrix from the matrix entries. Moreover, we study how to approximate a given matrix by a Cauchy matrix with a particular focus on the recovery of Cauchy points from noisy data. We derive an approximation algorithm of optimal com...

We study entire solutions of the biharmonic heat equation on complete Riemannian manifolds without boundary. provide exponential decay estimates for kernel under assumptions lower bound Ricci curvature and noncollapsing unit balls. And we prove a uniqueness criteria Cauchy problem. As corollaries, conservation law uniform $$L^\infty $$ estimate starting with bounded initial data.

Definition 1.1. Let (X, τ) be a topological space. A subset D ⊆ X is called dense if D ∩O 6= ∅ for every nonempty open set O ⊆ X. X is called separable if X has a countable dense subset. X is called metrizable if there is a metric d on X such that the topology τ is the same as the topology induced by the metric. The metric is called complete if every Cauchy sequence converges in X. Finally, X i...

Escardó and Simpson defined a notion of interval object by a universal property in any category with binary products. The Homotopy Type Theory book defines a higher-inductive notion of reals, and suggests that the interval may satisfy this universal property. We show that this is indeed the case in the category of sets of any universe. We also show that the type of HoTT reals is the least Cauch...

We study the properties of locally uniformly differentiable functions on N, a non-Archimedean field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. In particular, we show that locally uniformly differentiable functions are C1, they include all polynomial functions, and they are closed under addition, multiplication, and composition. Th...

Let κ be an uncountable cardinal with κ = κ. In this paper we introduce Rκ, a Cauchy-complete real closed field of cardinality 2. We will prove that Rκ shares many features with R which have a key role in real analysis and computable analysis. In particular, we will prove that the Intermediate Value Theorem holds for a non-trivial subclass of continuous functions over Rκ. We propose Rκ as a can...

It has been recently proved that analytic capacity, γ, is semiadditive. This result is a consequence of the comparability between γ and γ+, a version of γ originated by bounded Cauchy potentials of positive measures. In this paper we describe the main ideas involved in the proof of this result and we give a complete proof of it in the particular case of the N -th approximation of the corner qua...

Nonlinear cross-correlation equations in finite and semi-infinite intervals are studied by means of Fourier transform and Cauchy integral techniques. The equations are reduced to a bilinear conjugacy problem for two analytic functions on the real axis which can be solved in explicit form. In particular, Engibaryan’s equations from nonlinear factorization theory of Wiener-Hopf integral equations...

We obtain the complete characterization of those domains G ⊂ C which admit the so called estimate of the Cauchy integral, that is to say, ̨̨̨R ∂G f(z) dz ̨̨̨ ≤ C(G) ‖f‖∞ γ(E) for all E ⊂ G and f ∈ H∞(G \ E), where γ(E) is the analytic capacity of E. The corresponding result for continuous functions f and the continuous analytic capacity α(E) is also proved.

A large class of vacuum space-times is constructed in dimension 4+1 from hyperboloidal initial data sets which are not small perturbations of empty space data. These space-times are future geodesically complete, smooth up to their future null infinity I, and extend as vacuum space-times through their Cauchy horizon. Dimensional reduction gives non-vacuum space-times with the same properties in ...