Notes on differential calculus in topological linear spaces, II

1975 ◽  
Vol 20 (2) ◽  
pp. 245-252 ◽  
Author(s):  
S. Yamamuro
Keyword(s):  
Real Number ◽  
Number Field ◽  
Open Subset ◽  
Differential Calculus ◽  
Hausdorff Spaces ◽  
Real Numbers ◽  
Linear Spaces ◽  
Topological Linear ◽  
Locally Convex ◽  
Real Number Field

Throughout this note, let E and F be locally convex Hausdorff spaces over the real number field R. We denote real numbers by Greek letters. The sets of all continuous semi-norms on E and F will be denoted by P(E) and P(F) respectively, and A will always stand for an open subset of E.

scholarly journals Notes on differential calculus in topological linear spaces, III

1976 ◽  
Vol 21 (1) ◽  
pp. 88-95
Author(s):  
S. Yamamuro
Keyword(s):  
Real Number ◽  
Number Field ◽  
Open Subset ◽  
Differential Calculus ◽  
Hausdorff Spaces ◽  
Real Numbers ◽  
Linear Spaces ◽  
Topological Linear ◽  
Locally Convex ◽  
Real Number Field

Throughout this note, let E, F and G be locally convex Hausdorff spaces over the real number field R. We denote real numbers by Greek letters. The sets of all continuous semi-norms on E and F will be denoted by P(E) and P(F) respectively, and A will always stand for an open subset of E.

scholarly journals Monotone mappings in topological linear spaces

1965 ◽  
Vol 5 (1) ◽  
pp. 25-35
Author(s):  
Sadayuki Yamamuro
Keyword(s):  
Real Number ◽  
Linear Space ◽  
Number Field ◽  
Open Subset ◽  
Zero Element ◽  
Monotone Mappings ◽  
Topological Linear Space ◽  
Linear Spaces ◽  
Topological Linear ◽  
Real Number Field

Let E be a topological linear space over the real number field. Throughout of this paper, we denote by G an open subset of E, by ∂G the boundary of G and by the closure of G. The totality of all circled open neighbourhoods of the zero element denoted by U.

scholarly journals Lower bounds for Maass forms on semisimple groups

2020 ◽  
Vol 156 (5) ◽  
pp. 959-1003
Author(s):  
Farrell Brumley ◽  
Simon Marshall
Keyword(s):  
Real Number ◽  
Number Field ◽  
Lower Bounds ◽  
Positive Curvature ◽  
Semisimple Group ◽  
Maass Forms ◽  
Semisimple Groups ◽  
Cartan Involution ◽  
Totally Real ◽  
Real Number Field

Let $G$ be an anisotropic semisimple group over a totally real number field $F$. Suppose that $G$ is compact at all but one infinite place $v_{0}$. In addition, suppose that $G_{v_{0}}$ is $\mathbb{R}$-almost simple, not split, and has a Cartan involution defined over $F$. If $Y$ is a congruence arithmetic manifold of non-positive curvature associated with $G$, we prove that there exists a sequence of Laplace eigenfunctions on $Y$ whose sup norms grow like a power of the eigenvalue.

Phased Bessel functions

2008 ◽  
Vol 86 (7) ◽  
pp. 863-870 ◽  
Author(s):  
X Hu ◽  
H Wang ◽  
D -S Guo
Keyword(s):  
Real Number ◽  
Number Field ◽  
Complex Number ◽  
Bessel Functions ◽  
Natural Extension ◽  
State Transitions ◽  
Photon State ◽  
Analytical Extension ◽  
Complex Number Field ◽  
Real Number Field

In the study of photon-state transitions, we found a natural extension of the first kind of Bessel functions that extends both the range and domain of the Bessel functions from the real number field to the complex number field. We term the extended Bessel functions as phased Bessel functions. This extension is completely different from the traditional “analytical extension”. The new complex Bessel functions satisfy addition, subtraction, and recurrence theorems in a complex range and a complex domain. These theorems provide short cuts in calculations. The single-phased Bessel functions are generalized to multiple-phased Bessel functions to describe various photon-state transitions.PACS Nos.: 02.30.Gp, 32.80.Rm, 42.50.Hz

scholarly journals On Real Irreducible Representations of Lie Algebras

1959 ◽  
Vol 14 ◽  
pp. 59-83 ◽  
Author(s):  
Nagayoshi Iwahori
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Number Field ◽  
Irreducible Representations ◽  
The Real ◽  
Representations Of Lie Algebras ◽  
Real Matrices ◽  
Real Number Field

Let us consider the following two problems:Problem A. Let g be a given Lie algebra over the real number field R. Then find all real, irreducible representations of g.Problem B. Let n be a given positive integer. Then find all irreducible subalgebras of the Lie algebra ôí(w, R) of all real matrices of degree n.

scholarly journals Simultaneous Asymptotic Diophantine Approximations to a Basis of a Real Number Field

1971 ◽  
Vol 42 ◽  
pp. 79-87 ◽  
Author(s):  
William W. Adams
Keyword(s):  
Real Number ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Diophantine Approximations ◽  
Real Number Field

The purpose of this paper is to prove the following result.Theorem 1. Let K be a real algebraic number field of degree m = n + 1. Let 1, β1, …, βn be a basis of K.

On some representations of the real number field

2011 ◽  
Vol 50 (2) ◽  
pp. 189-190
Author(s):  
A. S. Morozov
Keyword(s):  
Real Number ◽  
Number Field ◽  
The Real ◽  
Real Number Field
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