One way of characterizing Atle Selberg’s mathematical genius is that he had a “golden touch”. In those domains he thought about in depth, he saw further than generations before him, repeatedly uncovering truths lying below the surface. His breakthroughs on long-standing problems were based on imaginative and novel ideas which, once digested, were appreciated as simple and decisive. The impact o...

The work [1] is generalized to the singularly perturbed nonlinear Schrödinger (NLS) equation of which the regularly perturbed NLS studied in [1] is a mollification. Specifically, the existence of Smale horseshoes and Bernoulli shift dynamics is established in a neighborhood of a symmetric pair of Silnikov homoclinic orbits under certain generic conditions, and the existence of the symmetric pai...

Abstract We present natural-age-grid Galerkin methods for a model of a biological population undergoing aging. We use a mollified birth term in the method and analysis. The error due to mollification is of arbitrary order, depending on the choice of mollifier. The methods in this paper generalize the methods presented in [1], where the approximation space in age was taken to be a discontinuous ...

Given a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound ‖u(t, · )− uε(t, · )‖L1 = O(1)(1 + t) · √ ε |ln ε| on the distance between an exact BV solution u and a viscous approximation uε , letting the viscosity coefficient ε → 0. In the proof, starting from u we construct an approximation of the viscous solution uε by taking a mollification u ∗ φ√...

Abstract. In this paper, we investigate a traffic model similar to the Lighthill–Whitham– Richards model, consisting of a hyperbolic conservation law with a discontinuous, non-convex, piecewise-linear flux. Using Dias and Figueira’s mollification framework, analytical solutions to the corresponding Riemann problem are derived. For certain initial data, these Riemann problems can generate zero w...

The Schauder estimate for the Laplace equation was traditionally built upon the Newton potential theory. Different proofs were found later by Campanato [Ca], in which he introduced the Campanato space; Peetre [P], who used the convolution of functions; Trudinger [T], who used the mollification of functions; and Simon [Si], who used a blowup argument. Also a perturbation argument was found by Sa...

  Background: Nonspecific symptoms are common in all areas of medicine. Patients and caregivers can be frustrated when an illness cannot be reduced to a discrete pathophysiological process that corresponds with the symptoms. We therefore asked the following questions: 1) Which demographic factors and psychological comorbidities are associated with change from an initial diagnosis of nonspecific...

in this paper the problem of minimizing the total transmitting power subject to a fixedtotal bit rate in ofdm systems is considered. upper bounds on transmitting power and bit rate ofeach subcarrier can also be taken into account. in practice, the number of bits of each subcarriershould be integer and nonnegative. in this paper an analytical optimal solution is derived for thecase of assuming t...

In 2013, Masson and Siljander determined a method to prove that the minimal p-weak upper gradient $$g_{f_\varepsilon }$$ for time mollification $$f_\varepsilon $$ , $$\varepsilon >0$$ of parabolic Newton–Sobolev function $$f\in L^p_\mathrm {loc}(0,\tau ;N^{1,p}_\mathrm {loc}(\Omega ))$$ with $$\tau $$\Omega open domain in doubling metric measure space $$(\mathbb {X},d,\mu )$$ supporting weak (1...