scholarly journals Distribution of shapes of orthogonal lattices

2017 ◽  
Vol 39 (06) ◽  
pp. 1531-1607
Author(s):  
MANFRED EINSIEDLER ◽  
RENÉ RÜHR ◽  
PHILIPP WIRTH
Keyword(s):  
Product Space ◽  
Error Term ◽  
Orthogonal Complement ◽  
Dimensional Sphere ◽  
Recent Advances ◽  
Unipotent Flows

It was recently shown by Aka, Einsiedler and Shapira that if $d>2$ , the set of primitive vectors on large spheres when projected to the $(d-1)$ -dimensional sphere coupled with the shape of the lattice in their orthogonal complement equidistribute in the product space of the sphere with the space of shapes of $(d-1)$ -dimensional lattices. Specifically, for $d=3,4,5$ some congruence conditions are assumed. By using recent advances in the theory of unipotent flows, we effectivize the dynamical proof to remove those conditions for $d=4,5$ . It also follows that equidistribution takes place with a polynomial error term with respect to the length of the primitive points.

scholarly journals SECOND MOMENTS IN THE GENERALIZED GAUSS CIRCLE PROBLEM

2018 ◽  
Vol 6 ◽  
Author(s):  
THOMAS A. HULSE ◽  
CHAN IEONG KUAN ◽  
DAVID LOWRY-DUDA ◽  
ALEXANDER WALKER
Keyword(s):  
Error Term ◽  
Power Saving ◽  
Lattice Points ◽  
Dimensional Sphere ◽  
Mean Square ◽  
Circle Problem ◽  
Second Moments ◽  
The Laplace Transform ◽  
Error Terms ◽  
Gauss Circle Problem

The generalized Gauss circle problem concerns the lattice point discrepancy of large spheres. We study the Dirichlet series associated to$P_{k}(n)^{2}$, where$P_{k}(n)$is the discrepancy between the volume of the$k$-dimensional sphere of radius$\sqrt{n}$and the number of integer lattice points contained in that sphere. We prove asymptotics with improved power-saving error terms for smoothed sums, including$\sum P_{k}(n)^{2}e^{-n/X}$and the Laplace transform$\int _{0}^{\infty }P_{k}(t)^{2}e^{-t/X}\,dt$, in dimensions$k\geqslant 3$. We also obtain main terms and power-saving error terms for the sharp sums$\sum _{n\leqslant X}P_{k}(n)^{2}$, along with similar results for the sharp integral$\int _{0}^{X}P_{3}(t)^{2}\,dt$. This includes producing the first power-saving error term in mean square for the dimension-3 Gauss circle problem.

scholarly journals Some Properties of Fuzzy Inner Product Space

2021 ◽  
pp. 2384-2392
Author(s):  
Jehad R. Kider
Keyword(s):  
Normed Space ◽  
Product Space ◽  
Closed Subspace ◽  
Cross Product ◽  
Inner Product Space ◽  
Inner Product ◽  
Orthogonal Complement ◽  
Fuzzy Normed Space ◽  
Direct Sum ◽  
New Type

     Our goal in the present paper is to introduce a new type of fuzzy inner product space. After that, to illustrate this notion, some examples are introduced. Then we prove that that every fuzzy inner product space is a fuzzy normed space. We also prove that the cross product of two fuzzy inner spaces is again a fuzzy inner product space. Next, we prove that the fuzzy inner product is a non decreasing function. Finally, if U is a fuzzy complete fuzzy inner product space and D is a fuzzy closed subspace of U, then we prove that U can be written as a direct sum of D and the fuzzy orthogonal complement    of D.

scholarly journals On a Type of Subgroups of a Compact Lie Group

1951 ◽  
Vol 2 ◽  
pp. 1-15 ◽  
Author(s):  
Yozô Matsushima
Keyword(s):  
Lie Group ◽  
Product Space ◽  
Closed Subgroup ◽  
The Other ◽  
Dimensional Sphere ◽  
Compact Lie Group ◽  
Coset Space ◽  
Fibre Bundles ◽  
Other Hand ◽  
Interesting Paper

Let G be a connected compact Lie group and H a connected closed subgroup. Then H is an orientable submanifold of G and we may consider H as a cycle in G. In his interesting paper on the topology of group manifolds H. Samelson has proved that, if H is not homologous to 0, then the homology ring of the coset space G/H is isomorphic to the homology ring of a product space of odd dimensional spheres and the homology ring of G is isomorphic to that of the product of the spaces H and G/H. On the other hand, in a recent investigation of fibre bundles’ T. Kudo has shown that, if the homology ring of the coset space G/H is isomorphic to that of an odd dimensional sphere, then H is not homologous to 0.

Recent Advances in High Dispersion Spectroscopy of Globular Cluster Stars

1988 ◽  
Vol 132 ◽  
pp. 525-530
Author(s):  
Raffaele G. Gratton
Keyword(s):  
Globular Cluster ◽  
Globular Clusters ◽  
High Dispersion ◽  
Photographic Material ◽  
Recent Advances ◽  
Metal Abundances ◽  
Ccd Detectors ◽  
Major Progress

The use CCD detectors has allowed a major progress in abundance derivations for globular cluster stars in the last years. Abundances deduced from high dispersion spectra now correlates well with other abundance indicators. I discuss some problems concerning the derivation of accurate metal abundances for globular clusters using high dispersion spectra from both the old photographic and the most recent CCD data. The discrepant low abundances found by Cohen (1980), from photographic material for M71 giants, are found to be due to the use of too high microturbulences.

Chelation-assisted transition metal-catalysed C–H chalcogenylations

2020 ◽  
Vol 7 (8) ◽  
pp. 1022-1060 ◽  
Author(s):  
Wenbo Ma ◽  
Nikolaos Kaplaneris ◽  
Xinyue Fang ◽  
Linghui Gu ◽  
Ruhuai Mei ◽  
...  
Keyword(s):  
Transition Metal ◽  
Recent Advances ◽  
Site Selectivity ◽  
Transition Metal Catalyzed ◽  
Metal Catalyzed

This review summarizes recent advances in C–S and C–Se formations via transition metal-catalyzed C–H functionalization utilizing directing groups to control the site-selectivity.

Nucleosome dynamics

2006 ◽  
Vol 73 ◽  
pp. 109-119 ◽  
Author(s):  
Chris Stockdale ◽  
Michael Bruno ◽  
Helder Ferreira ◽  
Elisa Garcia-Wilson ◽  
Nicola Wiechens ◽  
...  
Keyword(s):  
High Resolution ◽  
Chromatin Structure ◽  
Current Knowledge ◽  
Dynamic Properties ◽  
Structure And Dynamics ◽  
Nucleosome Dynamics ◽  
Sharp Focus ◽  
Recent Advances ◽  
Insight Into ◽  
Chromatin Structure And Dynamics

In the 30 years since the discovery of the nucleosome, our picture of it has come into sharp focus. The recent high-resolution structures have provided a wealth of insight into the function of the nucleosome, but they are inherently static. Our current knowledge of how nucleosomes can be reconfigured dynamically is at a much earlier stage. Here, recent advances in the understanding of chromatin structure and dynamics are highlighted. The ways in which different modes of nucleosome reconfiguration are likely to influence each other are discussed, and some of the factors likely to regulate the dynamic properties of nucleosomes are considered.

Recent Advances in the Treatment of Pulmonary Tuberculosis

1950 ◽  
Vol 34 (5) ◽  
pp. 1363-1380
Author(s):  
Theodore L. Badger ◽  
William E. Patton
Keyword(s):  
Pulmonary Tuberculosis ◽  
Recent Advances

Recent Advances in Immunology with Specific Reference to Otolaryngology

1985 ◽  
Vol 18 (4) ◽  
pp. 821-832
Author(s):  
Robert Naderto
Keyword(s):  
Specific Reference ◽  
Recent Advances

Recent Advances in Thyroid Imaging

1990 ◽  
Vol 23 (2) ◽  
pp. 251-270
Author(s):  
Martin P. Sandler, MD ◽  
James A. Patton ◽  
Robert H. Ossoff
Keyword(s):  
Thyroid Imaging ◽  
Recent Advances
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