scholarly journals A Brunn-Minkowski inequality for the integer lattice

2001 ◽  
Vol 353 (10) ◽  
pp. 3995-4024 ◽  
Author(s):  
R. J. Gardner ◽  
P. Gronchi
Keyword(s):  
Minkowski Inequality ◽  
Integer Lattice

scholarly journals New Generalizations and Results in Shift-Invariant Subspaces of Mixed-Norm Lebesgue Spaces \({L_{\vec{p}}(\mathbb{R}^d)}\)

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 227 ◽  
Author(s):  
Junjian Zhao ◽  
Wei-Shih Du ◽  
Yasong Chen
Keyword(s):  
Invariant Subspace ◽  
Type Inequality ◽  
Invariant Subspaces ◽  
Minkowski Inequality ◽  
Hölder Inequality ◽  
Stability Theorem ◽  
Mixed Norm ◽  
Lebesgue Spaces

In this paper, we establish new generalizations and results in shift-invariant subspaces of mixed-norm Lebesgue spaces Lp→(Rd). We obtain a mixed-norm Hölder inequality, a mixed-norm Minkowski inequality, a mixed-norm convolution inequality, a convolution-Hölder type inequality and a stability theorem to mixed-norm case in the setting of shift-invariant subspace of Lp→(Rd). Our new results unify and refine the existing results in the literature.

scholarly journals Routing sets in the integer lattice

2007 ◽  
Vol 155 (11) ◽  
pp. 1384-1394 ◽  
Author(s):  
Peter Hamburger ◽  
Robert Vandell ◽  
Matt Walsh
Keyword(s):  
Integer Lattice

CELLULAR NEURAL NETWORKS: ASYMMETRIC SPACE-DEPENDENT TEMPLATES, MOSAIC PATTERNS, AND SPATIAL CHAOS

2004 ◽  
Vol 14 (08) ◽  
pp. 2655-2665 ◽  
Author(s):  
LARRY TURYN
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Hexagonal Lattice ◽  
Cellular Neural Network ◽  
Cellular Neural Networks ◽  
Integer Lattice ◽  
Spatial Entropy ◽  
Spatial Chaos

We consider a Cellular Neural Network (CNN), with a bias term, on the integer lattice ℤ2in the plane ℝ2. Space-dependent, asymmetric couplings (templates) appropriate for CNN in the hexagonal lattice on ℝ2are studied. We characterize the mosaic patterns and study their spatial entropy. It appears that for this problem, asymmetry of the template has a more robust effect on the spatial entropy than does the sign of a parameter in the templates.

Prescribing Centro-Affine Curvature From One Convex Body to Another

2020 ◽  
Author(s):  
Alina Stancu
Keyword(s):  
Convex Body ◽  
Convex Bodies ◽  
Minkowski Inequality ◽  
Curvature Flow ◽  
Constant Volume ◽  
Convex Hypersurface ◽  
Strictly Convex ◽  
Initial Hypersurface ◽  
Affine Curvature ◽  
Convex Hypersurfaces

Abstract We study a curvature flow on smooth, closed, strictly convex hypersurfaces in $\mathbb{R}^n$, which commutes with the action of $SL(n)$. The flow shrinks the initial hypersurface to a point that, if rescaled to enclose a domain of constant volume, is a smooth, closed, strictly convex hypersurface in $\mathbb{R}^n$ with centro-affine curvature proportional, but not always equal, to the centro-affine curvature of a fixed hypersurface. We outline some consequences of this result for the geometry of convex bodies and the logarithmic Minkowski inequality.

Integer lattice method for generation of (quasi-)periodic optical coherent lattices

2021 ◽  
Author(s):  
Dmitry Kouznetsov ◽  
Qingzhong Deng ◽  
Ongun Arisev ◽  
Andim Stassen ◽  
Pol Van Dorpe ◽  
...  
Keyword(s):  
Integer Lattice ◽  
Lattice Method
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