Imaginary Numbers

Mathematicians knew that the multiplication of two negative numbers yields a positive product Thus, not only does+1×+1=+1,but-1×-1=+1. What number,then,multiplied by itself yields -1? To put it another way,what is the square root of -1? Mathematicians can invent the necessary number, call it an imaginary number,and symbolize it as i for     

Structure of 6m±1 Order Imaginary Cube

This paper gives the notions of point matrix and equideviate point matrix.6m±1 orderimaginary cube is obtained by using a kind of spocial point matrix.(m is a nature number)     

A Note on Purely Imaginary Roots of Generalized Kac–Moody Algebras

Abstract

In this paper, we generalize the concept of purely imaginary roots of Kac–Moody algebras to generalized Kac–Moody algebras. Also we give a complete classification of those generalized Kac–Moody algebras with the purely imaginary property. We also define a new class of indefinite non-hyperbolic generalized Kac–Moody algebras called extended hyperbolic generalized Kac–Moody algebras and find that it does not always possess the purely imaginary property whereas the extended hyperbolic Kac–Moody algebras possess the purely imaginary property.     

Error Bounds for Uniform Asymptotic Expansions—Modified Bessel Function of Purely Imaginary Order

The authors modify a method of Olde Daalhuis and Temme for representing the remainder and coefficients in Airy-type expansions of integrals.By using a class of rational functions,they express these quantities in terms of Cauchy-type integrals;these expressions are natural generalizations of integral representations of the coe？cients and the remainders in the Taylor expansions of analytic functions.By using the new representation,a computable error bound for the remainder in the uniform asymptotic expansion of the modified Bessel function of purely imaginary order is derived.     

étale Descent for Two-Primary Algebraic K -Theory of Totally Imaginary Number Fields

We show that the natural map from 2-adic algebraic K-theory to 2-adic étale K-theory induces an isomorphism in positive degrees for rings of 2-integers in totally imaginary number fields. In this sense, 2-adic algebraic K-theory of totally imaginary number fields satisfies étale descent in positive degrees.     

Error Bounds for Uniform Asymptotic Expansions—Modified Bessel Function of Purely Imaginary Order

The authors modify a method of Olde Daalhuis and Temme for representing the remainder and coefficients in Airy-type expansions of integrals. By using a class of rational functions, they express these quantities in terms of Cauchy-type integrals; these expressions are natural generalizations of integral representations of the coefficients and the remainders in the Taylor expansions of analytic functions. By using the new representation, a computable error bound for the remainder in the uniform asymptotic expansion of the modified Bessel function of purely imaginary order is derived.     

2-Sylow Subgroups of K_2O_F for Real Quadratic Field and Class Group for Imaginary Quadratic Field

Inthispaper,westudytherelationbetweenthe2-Syl0w'subgroups0fK,OFf0rrealquadraticfie1dF=Q(N)andthec1assgroupC(E)forimaginaryquadraticfie1dE=Q(In).ForgeneralnumberfieldF,manyauth0rshavecharacterizedthe2-SylowsubgroupstructureofK2OFandobtained2'-rankK2OFformulas(n>1).Especially,forquadraticfieldF,QinHuorong['Jgottheformulas0f4-orderelementsinK2OFonthebasis0fBrowkin-Sch1nzel[2].Inthispaper,wediscussthemfurther.Lemma(Llj,Theorems2.2and3-3)LetF=Q(H),d>2beasquarefreeinteger,andmap0sitive…