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By
Pritsker, Igor; Ramachandran, Koushik
We consider the zero distribution of random polynomials of the form
$$P_n(z) = \sum _{k=0}^n a_k B_k(z)$$
, where
$$\{a_k\}_{k=0}^{\infty }$$
are nontrivial i.i.d. complex random variables with mean 0 and finite variance. Polynomials
$$\{B_k\}_{k=0}^{\infty }$$
are selected from a standard basis such as Szegő, Bergman, or Faber polynomials associated with a Jordan domain G whose boundary is
$$C^{2, \alpha }$$
smooth. We show that the zero counting measures of
$$P_n$$
converge almost surely to the equilibrium measure on the boundary of G. We also show that if
$$\{a_k\}_{k=0}^{\infty }$$
are i.i.d. random variables, and the domain G has analytic boundary, then for a random series of the form
$$f(z) =\sum _{k=0}^{\infty }a_k B_k(z),$$
$$\partial {G}$$
is almost surely the natural boundary for f(z).
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By
Palmieri, Alessandro
2 Citations
We consider a semilinear wave equation with scaleinvariant damping and mass and power nonlinearity. For this model we prove some global (in time) existence results in odd spatial dimension n, under the assumption that the multiplicative constants μ and ν^{2}, which appear in the coefficients of the damping and of the mass terms, respectively, satisfy an interplay condition which makes the model somehow “wavelike”. Combining these global existence results with a recently proved blowup result, we will find as critical exponent for the considered model the largest between suitable shifts of the Strauss exponent and of Fujita exponent, respectively. Besides, the competition among these two kind of exponents shows how the interrelationship between μ and ν^{2} determines the possible transition from a “hyperboliclike” to a “paraboliclike” model. Nevertheless, in the case n ≥ 3 we will restrict our considerations to the radial symmetric case.
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By
Aral, Ali; Inoan, Daniela ; Raşa, Ioan
1 Citations
This paper is a natural continuation of Acar et al. (Mediterr J Math 14:6, 2017,
https://doi.org/10.1007/s0000901608047
) where Szász–Mirakyan operators preserving exponential functions are defined. As a first result, we show that the sequence of the norms of the operators, acting on weighted spaces having different weights, is uniformly bounded. Then, we prove Korovkin type approximation theorems through exponential weighted convergence. The uniform weighted approximation errors of the operators and their derivatives are characterized for exponential weights. Furthermore we give a Voronovskaya type theorem for the derivative of the operators.
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By
Fernando, José F. ; Ueno, Carlos
We prove constructively that: The complement
$${{\mathbb {R}}}^n{\setminus }\mathcal {K}$$
of anndimensional unbounded convex polyhedron
$$\mathcal {K}\subset {{\mathbb {R}}}^n$$
and the complement
$${{\mathbb {R}}}^n{\setminus }{\text {Int}}(\mathcal {K})$$
of its interior are polynomial images of
$${{\mathbb {R}}}^n$$
whenever
$$\mathcal {K}$$
does not disconnect
$${{\mathbb {R}}}^n$$
. The case of a compact convex polyhedron and the case of convex polyhedra of small dimension were approached by the authors in previous works. The techniques here are more sophisticated than those corresponding to the compact case and require rational separation results for tuples of variables, which have interest by their own and can be applied to separate certain types of (noncompact) semialgebraic sets.
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By
Matthes, Nils
We characterize Zagier’s generating series of extended period polynomials of normalized Hecke eigenforms for
$${{\,\mathrm{PSL}\,}}_2(\mathbb {Z})$$
in terms of the period relations and existence of a suitable factorization. For this, we prove a characterization of the Kronecker function as the “fundamental solution” of the Fay identity.
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By
Butters, Jerry; Henle, Jim
In the second, third, and fourth box scores appearing in this paper, the number for LOB (left on base) for Mudville should be 6, not 4.
By
Lenells, Jonatan; Quirchmayr, Ronald
According to its Lax pair formulation, the nonlinear Schrödinger (NLS) equation can be expressed as the compatibility condition of two linear ordinary differential equations with an analytic dependence on a complex parameter. The first of these equations—often referred to as the xpart of the Lax pair—can be rewritten as an eigenvalue problem for a Zakharov–Shabat operator. The spectral analysis of this operator is crucial for the solution of the initial value problem for the NLS equation via inverse scattering techniques. For spaceperiodic solutions, this leads to the existence of a Birkhoff normal form, which beautifully exhibits the structure of NLS as an infinitedimensional completely integrable system. In this paper, we take the crucial steps towards developing an analogous picture for timeperiodic solutions by performing a spectral analysis of the tpart of the Lax pair with a periodic potential.
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By
Wu, Weili; Zhang, Zhao; Du, DingZhu
The nonsubmodular optimization is a hot research topic in the study of nonlinear combinatorial optimizations. We discuss several approaches to deal with such optimization problems, including supermodular degree, curvature, algorithms based on DS decomposition, and sandwich method.
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By
Föll, Fabian; Pandey, Sandeep; Chu, Xu; Munz, ClausDieter; Laurien, Eckart; Weigand, Bernhard
Show all (6)
Supercritical fluids are suggested as one of the potential candidates for the next generation nuclear reactor by Generation IV nuclear forum to improve the thermal efficiency. But, supercritical fluids suffer from the deteriorated heat transfer under certain conditions. This deteriorated heat transfer phenomenon is a result of a peculiar attenuation of turbulence within the flow. This peculiarity is difficult to predict by conventional turbulence modeling. Therefore, direct numerical simulations were used in the past employing the lowMach assumption with a finite volume code. As a next step, we extend the discontinuous Galerkin spectral element method for direct numerical simulation of supercritical carbon dioxide. The higherorder of accuracy and fully compressible code improve the fidelity of the simulations. A computationally robust and efficient implementation of the equation of state was used which is based on adaptive mesh refinement. The objective of this report is to demonstrate the usage of the code in complex flow. Therefore, channel geometry is adopted and simulations were conducted at different Mach numbers to observe the effects of compressibility in the supercritical fluid regime. The isothermal boundary conditions were used at the walls of the channel. The mean profile of pressure, density and temperature are drastically affected by the Mach number variation. In the end, scalability tests were conducted and code shows a very good parallel scalability up to 12,000 cores.
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