This paper is concerned with the problem of approximating a homeomorphism by piecewise affine homeomorphisms. The main result is as follows: every homeomorphism from a planar domain with a polygonal boundary to R that is globally Hölder continuous of exponent α ∈ (0, 1], and whose inverse is also globally Hölder continuous of exponent α can be approximated in the Hölder norm of exponent β by pi...

We prove a structure theorem for multiplicative functions on the Gaussian integers, showing that every bounded multiplicative function on the Gaussian integers can be decomposed into a term which is approximately periodic and another which has a small U3-Gowers uniformity norm. We apply this to prove partition regularity results over the Gaussian integers for certain equations involving quadrat...

We propose an efficient ADMM method with guarantees for high-dimensional problems. We provide explicit bounds for the sparse optimization problem and the noisy matrix decomposition problem. For sparse optimization, we establish that the modified ADMM method has an optimal regret bound of O(s log d/T ), where s is the sparsity level, d is the data dimension and T is the number of steps. This mat...

For a quantale I, which is unit interval endowed with continuous triangular norm and the Barr extension β¯I of ultrafilter monad to I-Rel, characterization discrete presheaf associated given. It also proved that, when & Łucasiewicz norm, isomorphic saturated prefilter monad, product prime functional ideal submonad monad.

We propose an efficient ADMM method with guarantees for high-dimensional problems. We provide explicit bounds for the sparse optimization problem and the noisy matrix decomposition problem. For sparse optimization, we establish that the modified ADMM method has an optimal regret bound of O(s log d/T ), where s is the sparsity level, d is the data dimension and T is the number of steps. This mat...

This paper constructs tridiagonal random matrix models for general (β > 0) β-Hermite (Gaussian) and β-Laguerre (Wishart) ensembles. These generalize the well-known Gaussian and Wishart models for β = 1, 2, 4. Furthermore, in the cases of the β-Laguerre ensembles, we eliminate the exponent quantization present in the previously known models. We further discuss applications for the new matrix mod...

We consider the following sparse signal recovery (or feature selection) problem: given a design matrix X ∈ Rn×m (m À n) and a noisy observation vector y ∈ R satisfying y = Xβ∗ + 2 where 2 is the noise vector following a Gaussian distribution N(0, σI), how to recover the signal (or parameter vector) β∗ when the signal is sparse? The Dantzig selector has been proposed for sparse signal recovery w...

and Applied Analysis 3 Lemma 5. The operators S α and A α have the following properties: (i) for any fixed t ≥ 0 and any x ∈ X β , one has the operators S α (t) and A α (t) which are linear and bounded operators; that is, for any x ∈ X β , 󵄩󵄩󵄩󵄩Sα (t) x 󵄩󵄩󵄩󵄩β ≤ M‖x‖β, 󵄩󵄩󵄩󵄩Aα (t) x 󵄩󵄩󵄩󵄩β ≤ M Γ (α) ‖x‖ β ; (12) (ii) the operatorsS α (t) andA α (t) are strongly continuous for all t ≥ 0; (iii) S α (...

We study the weak convergence of β- and β′-Delaunay tessellations in Rd−1 that were introduced part I this paper, as β→∞. The limiting stationary simplicial random tessellation, which is called Gaussian-Delaunay characterized terms a space-time paraboloid hull process Rd−1×R. latter object has previously appeared analysis number shocks solution inviscid Burgers’ equation description local asymp...

In this article we study the pair correlation statistic for higher dimensional sequences. We show that any d≥2, strictly increasing sequences (an(1)),…,(an(d)) of natural numbers have metric Poissonian with respect to sup-norm if their joint additive energy is O(N3−δ) δ>0. Further, in dimension two, establish an analogous result 2-norm. As a consequence, it follows ({nα},{n2β}) and ({nα},{[nlog...