We discuss the distribution problem of the right zeros of the special quadratic equation(x一β)(x一α)= 0 over noncommutative division rings, and whereby we obtain a newmethod for solving the quadratic equation x￣2 +αx+b=0 over quaternion rings R(i,j,k).

Being different from the existing implementation based on the discrete logarithm problem, the new realization is based on the difficulty of deducing x and y from A and B=x·A·y in a specific monoid (m,·) which is noncommutative.

Let G be a non-abelian group and associate a non-commuting graph ▽(G) with G as follows:the vertex set of ▽(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity.

In this paper,the pure state ranges of non-commuting C~*-algebras are discussed and a representation of the essential pure state range of a pair of elements in the tensor products algebra of non-commuting C~*-algebras are obtained.

Let G be a non-abelian group and associate a non-commuting graph ▽(G) with G as follows:the vertex set of ▽(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity.

Let G be a group of |G:Z(G)| = 4, if G is a nilpotent group, then its sylow subgroups with odd number order are commutative groups, its sylow-2 subgroups are non-commutative groups, and P/Z(G)≌ Z2×Z2.

Let =C_q[x_1~(±1),x_2~(±1)] be the non-commutative torus over the complex field, and A=\C,Der be the Lie algebra of Derivations over . In this paper we study the automorphism group Aut L_q for the Lie algebra L_q=Der+A.

Caiheng Li has proved that[1]: the connected cubic arc-transitive Cayleygraphs on the finite nonabelian simple groups are all normal except A_5, PSL(2,11), M_(11), A_(11), M_(23), A(23) and A_(47).

In this paper, we study the noncommuting graph (G) associated with a finite group G and prove that:Theorem 1 Let p be a prime. If (G)■(D4p), then G■D4p or G■Q4p.

Let f be a nonnegative operator monotone function on [0,∞). Then for all non-commutative Banach function space norms, we obtain that‖f(A)X-Xf(B)‖54‖f(|AX-XB|)‖,where A,B are τ-measurable positive operators, X is a contraction.

A convexity theorem for noncommutative gradient flows

The paper studies serving extensions of noncommutative groups in the sense of the definition transferred naturally from the theory of Abelian groups.

Heredity of radicals in a class of noncommutative Jordan rings

We construct a quantum extension of the Markov semigroup of the classical Bessel process of orderv≥1 to the noncommutative von Neumann algebra ?(L2(0, +∞)) of bounded operators onL2(0, +∞).

A noncommutative identity for the appell polynomials

Let G be a non-abelian group and associate a non-commuting graph ?(G) with G as follows: the vertex set of ?(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity.

This entropic function is then applied to construct an alternative uncertainty relation of two non-commuting observables in quantum physics.

Whereas a pure Bose system is comparatively easy to treat in this way, the presence of Fermi fields involves a complication due to the occurrence of non-commuting quantities in the classical action function.

We give a definition involving only two non-commuting quantities, and work out some illustrative examples.

transformations in which the parameters are functions of the space-time point leads to the necessity of introducingf non-commuting vectors fields.

Let G be a non-abelian group and associate a non-commuting graph ?(G) with G as follows: the vertex set of ?(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity.

It is proved that for any set of finite non-Abelian simple groups there exists a finite simple group as a result of this set.

We know that a group G is functionally complete if and only if it is either trivial or a finite simple non-Abelian group [Ref.

It is shown that for n >amp;gt; 1 these notions and the notion of functional completeness are equivalent, and apart from all finite simple non-Abelian groups, only the trivial group and groups of second order are 1-functionally complete.

The structure of the totally local formation,?∞ formG, generated by a simple non-Abelian groupG is described.

In this article, we have introduced a new Regge trajectory, αq, with vacuum quantum numbers, α9(0) = αp(0) = 1, but αq'(0)≠ αp'(0). αq as well as αp takes part in the high energy non-exchange processes, and they interfere with each other. The consequence of the two vacuum Regge trajectories is equivalent to an effective trajectory αc, the slope of which at zero energy αs'(0) is dependent on the processes. The essential difference of our theory from the old one deduced from a single vacuum trajectory, αp, is...

In this article, we have introduced a new Regge trajectory, αq, with vacuum quantum numbers, α9(0) = αp(0) = 1, but αq'(0)≠ αp'(0). αq as well as αp takes part in the high energy non-exchange processes, and they interfere with each other. The consequence of the two vacuum Regge trajectories is equivalent to an effective trajectory αc, the slope of which at zero energy αs'(0) is dependent on the processes. The essential difference of our theory from the old one deduced from a single vacuum trajectory, αp, is that αz'(0) will be different for different processes. This does not conflict with the newest experiments.

From the determination of exchangeable and non-exchangeable potassium ofthe different soils in Jiangsu Province and the experiments of potash fertilizationon cotton,it was preliminarily established that bleached whitish soils in ChenjiangDistrict,the sandy and sandy loom soils along the Yangtze River and the sandysoils in Xuzhow-Huaiying District are deficient in available potassium(exchange-able K 10—20 jin/mu,non-exchangeable K 65—200 jin/mu),the K supply ofthe permeable paddy soils in Suzhow Distriet is also...

From the determination of exchangeable and non-exchangeable potassium ofthe different soils in Jiangsu Province and the experiments of potash fertilizationon cotton,it was preliminarily established that bleached whitish soils in ChenjiangDistrict,the sandy and sandy loom soils along the Yangtze River and the sandysoils in Xuzhow-Huaiying District are deficient in available potassium(exchange-able K 10—20 jin/mu,non-exchangeable K 65—200 jin/mu),the K supply ofthe permeable paddy soils in Suzhow Distriet is also low(exchangeable K 18—24jin/mu,non-exchangeable K 60—80 jin/mu),and the soils in The Lower RiverBasin and along the coast line are rich in K supply(exchangeable K 40—70 jin/mu,non-exchangeable K 200—300 jin/mu).The results from the cotton potash fertilizaton during 1973—75 had shownthat when the amounts of exchangeable K were less than 25 jin per mu,theapplication of potash fertilizer had a marked eliect in increasing cotton yield.Potash fertilization lessened the wilting disease and prevented the prematurity.Potash fertilization lessened the root diseases during the seedling stage toensure the stand in the field.Potash fertilization increased the boll wight,promoted boll opening and im-proved the fibre quality.The diagnostic technique of potassium on cotton during the square stage wasa very useful tool to decide whether the potash fertilizer should be side-dressed.

In this paper, the kinetics of shell-progressive reaction in a solution of finite volume is investigated. A more general finite solution volume condition is obtained from material balance. The theoretical treatment supplements what is lacking in the mathematical model proposed by previous authors, who assume that there is no invasion of electrolyte over that of the fixed ionic groups in the resin, and the concentration of counter-ion in solution is independent of reaction time. The description of shell-progressive...

In this paper, the kinetics of shell-progressive reaction in a solution of finite volume is investigated. A more general finite solution volume condition is obtained from material balance. The theoretical treatment supplements what is lacking in the mathematical model proposed by previous authors, who assume that there is no invasion of electrolyte over that of the fixed ionic groups in the resin, and the concentration of counter-ion in solution is independent of reaction time. The description of shell-progressive reaction in ion exchange resin is now more nearly perfect and definite than was before. Some numerical calculations have been done with a computer. On the basis of the great number of calculations, it may finally be concluded that(l) shell-progressive reaction is different from that occurs simultaneously throughout resin beads in the dependence of reaction rate on various factors; (2) as long as the solution volume is finite, there is a necessary condition for the application of the theoretical expressions deduced by former authors; (3) under the ordinary conditions of finite solution volume, the fractional conversion of the resin can be determined experimentally from the concentrations of counter-ion in solution.