MORE CONSTRUCTIONS OF APPROXIMATELY MUTUALLY UNBIASED BASES

2015 ◽  
Vol 93 (2) ◽  
pp. 211-222
Author(s):  
XIWANG CAO ◽  
WUN-SENG CHOU
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Prime Number ◽  
Number Field ◽  
Complex Number ◽  
Character Sums ◽  
Mutually Unbiased Bases ◽  
Complex Number Field

Let $m$ be a positive integer and $p$ a prime number. We prove the orthogonality of some character sums over the finite field $\mathbb{F}_{p^{m}}$ or over a subset of a finite field and use this to construct some new approximately mutually unbiased bases of dimension $p^{m}$ over the complex number field $\mathbb{C}$, especially with $p=2$.

scholarly journals A Simple Cocyclic Jacket Matrices

2008 ◽  
Vol 2008 ◽  
pp. 1-18 ◽  
Author(s):  
Moon Ho Lee ◽  
Gui-Liang Feng ◽  
Zhu Chen
Keyword(s):  
Finite Field ◽  
Number Field ◽  
Complex Number ◽  
Close Relation ◽  
Theoretical Physics ◽  
Unitary Matrices ◽  
New Class ◽  
Hand Tool ◽  
The Common ◽  
Complex Number Field

We present a new class of cocyclic Jacket matrices over complex number field with any size. We also construct cocyclic Jacket matrices over the finite field. Such kind of matrices has close relation with unitary matrices which are a first hand tool in solving many problems in mathematical and theoretical physics. Based on the analysis of the relation between cocyclic Jacket matrices and unitary matrices, the common method for factorizing these two kinds of matrices is presented.

scholarly journals Type II degenerations of K3 surfaces

1985 ◽  
Vol 99 ◽  
pp. 11-30 ◽  
Author(s):  
Shigeyuki Kondo
Keyword(s):  
Number Field ◽  
Complex Manifold ◽  
Complex Number ◽  
Three Dimensional ◽  
K3 Surfaces ◽  
Type Ii ◽  
Holomorphic Map ◽  
Canonical Class ◽  
Proper Holomorphic Map ◽  
Complex Number Field

A degeneration of K3 surfaces (over the complex number field) is a proper holomorphic map π: X→Δ from a three dimensional complex manifold to a disc, such that, for t ≠ 0, the fibres Xt = π-1(t) are smooth K3 surfaces (i.e. surfaces Xt with trivial canonical class KXt = 0 and dim H1(Xt, Oxt) = 0).

On the Nonemptiness of the Adjoint Linear System of Polarized Manifolds

1998 ◽  
Vol 41 (3) ◽  
pp. 267-278 ◽  
Author(s):  
Yoshiaki Fukuma
Keyword(s):  
Linear System ◽  
Number Field ◽  
Complex Number ◽  
Complex Number Field

AbstractLet (X, L) be a polarized manifold over the complex number field with dim X = n. In this paper, we consider a conjecture of M. C. Beltrametti and A. J. Sommese and we obtain that this conjecture is true if n = 3 and h0(L) ≥ 2, or dim Bs |L| ≤ 0 for any n ≥ 3. Moreover we can generalize the result of Sommese.

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

Castelnuovo-Mumford regularity bound for smooth threefolds in ℙ5 and extremal examples

1999 ◽  
Vol 1999 (509) ◽  
pp. 21-34
Author(s):  
Si-Jong Kwak
Keyword(s):  
Complete Intersection ◽  
Number Field ◽  
Complex Number ◽  
Smooth Surfaces ◽  
Veronese Surface ◽  
Optimal Bound ◽  
Integral Curves ◽  
Complex Number Field

Abstract Let X be a nondegenerate integral subscheme of dimension n and degree d in ℙN defined over the complex number field ℂ. X is said to be k-regular if Hi(ℙN, ℐX (k – i)) = 0 for all i ≧ 1, where ℐX is the sheaf of ideals of ℐℙN and Castelnuovo-Mumford regularity reg(X) of X is defined as the least such k. There is a well-known conjecture concerning k-regularity: reg(X) ≦ deg(X) – codim(X) + 1. This regularity conjecture including the classification of borderline examples was verified for integral curves (Castelnuovo, Gruson, Lazarsfeld and Peskine), and an optimal bound was also obtained for smooth surfaces (Pinkham, Lazarsfeld). It will be shown here that reg(X) ≦ deg(X) – 1 for smooth threefolds X in ℙ5 and that the only extremal cases are the rational cubic scroll and the complete intersection of two quadrics. Furthermore, every smooth threefold X in ℙ5 is k-normal for all k ≧ deg(X) – 4, which is the optimal bound as the Palatini 3-fold of degree 7 shows. The same bound also holds for smooth regular surfaces in ℙ4 other than for the Veronese surface.

scholarly journals Some inequalities in Pseudo-Hilbert spaces

2011 ◽  
Vol 42 (4) ◽  
pp. 483-492
Author(s):  
Loredana Ciurdariu
Keyword(s):  
Linear Space ◽  
Number Field ◽  
Hilbert Spaces ◽  
Complex Number ◽  
Normed Linear Space ◽  
Normed Algebra ◽  
The Real ◽  
Complex Number Field

The aim of this paper is to obtain new versions of the reverse of the generalized triangle inequalities given in \cite{SSDNA}, %[4],and \cite{SSDPR} %[5] if the pair $(a_i,x_i),\;i\in\{1,\ldots,n\}$ from Theorem 1 of \cite{SSDNA} %[4] belongs to ${\mathbb C}\times\mathcal H $, where $\mathcal H$ is a Loynes $Z$-space instead of ${\mathbb K}\times X$, $X$ being a normed linear space and ${\mathbb K}$ is the field of scalars. By comparison, in \cite{SSDNA} %[4] the pair $(a_i,x_i),\;i\in\{1,\ldots,n\}$ belongs to $A^2$, where $A$ is a normed algebra over the real or complex number field ${\mathbb K}.$ The results will be given in Theorem 1, Theorem 3, Remark 2 and Corollary 3 which represent other interesting variants of Theorem 2.1, Remark 2.2, Theorem 3.2 and Theorem 3.4., see \cite{SSDNA}. %[4].

scholarly journals The ring of invariants of matrices

1986 ◽  
Vol 104 ◽  
pp. 149-161 ◽  
Author(s):  
Yasuo Teranishi
Keyword(s):  
Vector Space ◽  
Number Field ◽  
Complex Number ◽  
Rational Representation ◽  
Invertible Matrices ◽  
Complex Number Field

We denote by M(n) the space of all n × n-matrices with their coefficients in the complex number field C and by G the group of invertible matrices GL(n, C). Let W = M(n)i be the vector space of l-tuples of n × ra-matrices. We denote by ρ: G → GL(W) a rational representation of G defined as follows:if S ∈ G, A(i) ∈ M(n) (i = 1, 2, …, l).

scholarly journals Analytic Log Picard Varieties

2008 ◽  
Vol 191 ◽  
pp. 149-180 ◽  
Author(s):  
Takeshi Kajiwara ◽  
Kazuya Kato ◽  
Chikara Nakayama
Keyword(s):  
Number Field ◽  
Complex Number ◽  
Basic Properties ◽  
Picard Variety ◽  
Complex Number Field

AbstractWe introduce a log Picard variety over the complex number field by the method of log geometry in the sense of Fontaine-Illusie, and study its basic properties, especially, its relationship with the group of log version of m-torsors.

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