scholarly journals Exact upper bounds of norms of polynomials, splines, and their derivatives on arbitrary interval

2021 ◽  
Vol 19 ◽  
pp. 67
Author(s):  
V.A. Kofanov
Keyword(s):  
Finite Interval ◽  
Upper Bounds ◽  
Arbitrary Interval ◽  
Sharp Estimates

We establish sharp estimates of the $L_q$-norms on any finite interval for the polynomials and splines and their derivatives with the help of local $L_p$-norms of these polynomials and splines.

scholarly journals Sharp estimates and homogenization of the control cost of the heat equation on large domains

2020 ◽  
Vol 26 ◽  
pp. 54 ◽  
Author(s):  
Ivica Nakić ◽  
Matthias Täufer ◽  
Martin Tautenhahn ◽  
Ivan Veselić
Keyword(s):  
Heat Equation ◽  
Uncertainty Relation ◽  
Unbounded Domains ◽  
Upper Bounds ◽  
Heat Evolution ◽  
Control Cost ◽  
Potential Term ◽  
Sharp Estimates ◽  
Control Set ◽  
Spectral Inequality

We prove new bounds on the control cost for the abstract heat equation, assuming a spectral inequality or uncertainty relation for spectral projectors. In particular, we specify quantitatively how upper bounds on the control cost depend on the constants in the spectral inequality. This is then applied to the heat flow on bounded and unbounded domains modeled by a Schrödinger semigroup. This means that the heat evolution generator is allowed to contain a potential term. The observability/control set is assumed to obey an equidistribution or a thickness condition, depending on the context. Complementary lower bounds and examples show that our control cost estimates are sharp in certain asymptotic regimes. One of these is dubbed homogenization regime and corresponds to the situation where the control set becomes more and more evenly distributed throughout the domain while its density remains constant.

scholarly journals Sharp estimates on the first Dirichlet eigenvalue of nonlinear elliptic operators via maximum principle

2018 ◽  
Vol 9 (1) ◽  
pp. 278-291 ◽  
Author(s):  
Francesco Della Pietra ◽  
Giuseppina di Blasio ◽  
Nunzia Gavitone
Keyword(s):  
Maximum Principle ◽  
Function Method ◽  
Upper Bounds ◽  
Elliptic Operators ◽  
Lower And Upper Bounds ◽  
Dirichlet Eigenvalue ◽  
Sharp Estimates ◽  
Nonlinear Elliptic ◽  
Suitable Function ◽  
First Dirichlet Eigenvalue

Abstract In this paper, we study optimal lower and upper bounds for functionals involving the first Dirichlet eigenvalue {\lambda_{F}(p,\Omega)} of the anisotropic p-Laplacian, {1<p<+\infty} . Our aim is to enhance, by means of the {\mathcal{P}} -function method, how it is possible to get several sharp estimates for {\lambda_{F}(p,\Omega)} in terms of several geometric quantities associated to the domain. The {\mathcal{P}} -function method is based on a maximum principle for a suitable function involving the eigenfunction and its gradient.

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

Energy of spherical fuzzy graphs

2020 ◽  
pp. 321-332
Author(s):  
S. Yahya Mohamed ◽  
A. Mohamed Ali
Keyword(s):  
Adjacency Matrix ◽  
Upper Bounds ◽  
Fuzzy Graph ◽  
Lower And Upper Bounds ◽  
Fuzzy Graphs

In this paper, the notion of energy extended to spherical fuzzy graph. The adjacency matrix of a spherical fuzzy graph is defined and we compute the energy of a spherical fuzzy graph as the sum of absolute values of eigenvalues of the adjacency matrix of the spherical fuzzy graph. Also, the lower and upper bounds for the energy of spherical fuzzy graphs are obtained.

Sign in / Sign up

Export Citation Format

Share Document