ترغب بنشر مسار تعليمي؟ اضغط هنا

Quadratic Equation over Associative D-Algebra

555   0   0.0 ( 0 )
 نشر من قبل Aleks Kleyn
 تاريخ النشر 2015
  مجال البحث
والبحث باللغة English
 تأليف Aleks Kleyn




اسأل ChatGPT حول البحث

In this paper, I treat quadratic equation over associative $D$-algebra. In quaternion algebra $H$, the equation $x^2=a$ has either $2$ roots, or infinitely many roots. Since $ain R$, $a<0$, then the equation has infinitely many roots. Otherwise, the equation has roots $x_1$, $x_2$, $x_2=-x_1$. I considered different forms of the Vietes theorem and a possibility to apply the method of completing the square. In quaternion algebra, there exists quadratic equation which either has $1$ root, or has no roots.



قيم البحث

اقرأ أيضاً

146 - Aleks Kleyn 2014
From the symmetry between definitions of left and right divisors in associative $D$-algebra $A$, the possibility to define quotient as $Aotimes A$-number follows. In the paper, I considered division and division with remainder. I considered also definition of prime $A$-number.
257 - Aleks Kleyn 2018
In the book, I considered differential equations of order $1$ over Banach $D$Hyph algebra: differential equation solved with respect to the derivative; exact differential equation; linear homogeneous equation. I considered examples of differential eq uations in quaternion algebra. In order to study homogeneous system of linear differential equations, I considered vector space over division $D$-algebra, solving of linear equations over division $D$-algebra and the theory of eigenvalues in non commutative division $D$-algebra.
110 - Aleks Kleyn 2015
Module is effective representation of ring in Abelian group. Linear map of module over commutative ring is morphism of corresponding representation. This definition is the main subject of the book. To consider this definition from more general poin t of view I started the book from consideration of Cartesian product of representations. Polymorphism of representations is a map of Cartesian product of representations which is a morphism of representations with respect to each separate independent variable. Reduced morphism of representations allows us to simplify the study of morphisms of representations. However a representation has to satisfy specific requirements for existence of reduced polymomorphism of representations. It is possible that Abelian group is only $Omega$-algebra, such that representation in this algebra admits polymorphism of representations. However, today, this statement has not been proved. Multiplicative $Omega$-group is $Omega$-algebra in which product is defined. The definition of tensor product of representations of Abelian multiplicative $Omega$-group is based on properties of reduced polymorphism of representations of Abelian multiplicative $Omega$-group. Since an algebra is a module in which the product is defined, then we can use this theory to study linear map of algebra. For instance, we can study the set of linear transformations of $D$-algebra $A$ as representation of algebra $Aotimes A$ in algebra $A$.
155 - Aleks Kleyn 2016
Let $A$, $B$ be Banach $D$-algebras. The map $f:Arightarrow B$ is called differentiable on the set $Usubset A$, if at every point $xin U$ the increment of map $f$ can be represented as $$f(x+dx)-f(x) =frac{d f(x)}{d x}circ dx +o(dx)$$ where $$frac{d f(x)}{d x}:Arightarrow B$$ is linear map and $o:Arightarrow B$ is such continuous map that $$lim_{arightarrow 0}frac{|o(a)|_B}{|a|_A}=0$$ Linear map $displaystylefrac{d f(x)}{d x}$ is called derivative of map $f$. I considered differential forms in Banach Algebra. Differential form $omegainmathcal{LA}(D;Arightarrow B)$ is defined by map $g:Arightarrow Botimes B$, $omega=gcirc dx$. If the map $g$, is derivative of the map $f:Arightarrow B$, then the map $f$ is called indefinite integral of the map $g$ $$f(x)=int g(x)circ dx=intomega$$ Then, for any $A$-numbers $a$, $b$, we define definite integral by the equality $$int_a^bomega=int_{gamma}omega$$ for any path $gamma$ from $a$ to $b$.
74 - N. A. Carella 2020
Let $p$ be a large prime, and let $kll log p$. A new proof of the existence of any pattern of $k$ consecutive quadratic residues and quadratic nonresidues is introduced in this note. Further, an application to the least quadratic nonresidues $n_p$ modulo $p$ shows that $n_pll (log p)(log log p)$.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا