Low energy scattering with a nontrivial pion

An earlier calculation in a generalized linear sigma model showed that the well-known current algebra formula for low energy pion pion scattering held even though the massless Nambu Goldstone pion contained a small admixture of a two-quark two-antiquark field. Here we turn on the pion mass and note that the current algebra formula no longer holds exactly. We discuss this small deviation and also study the effects of an SU(3) symmetric quark mass type term on the masses and mixings of the eight SU(3) multiplets in the model. We calculate the s wave scattering lengths, including the beyond current algebra theorem corrections due to the scalar mesons, and observe that the model can fit the data well. In the process, we uncover the way in which linear sigma models give controlled corrections (due to the presence of scalar mesons) to the current algebra scattering formula. Such a feature is commonly thought to exist only in the non-linear sigma model approach.Comment: 15 pages, 8 figure

The splitting process in free probability theory

Free cumulants were introduced by Speicher as a proper analog of classical cumulants in Voiculescu's theory of free probability. The relation between free moments and free cumulants is usually described in terms of Moebius calculus over the lattice of non-crossing partitions. In this work we explore another approach to free cumulants and to their combinatorial study using a combinatorial Hopf algebra structure on the linear span of non-crossing partitions. The generating series of free moments is seen as a character on this Hopf algebra. It is characterized by solving a linear fixed point equation that relates it to the generating series of free cumulants. These phenomena are explained through a process similar to (though different from) the arborification process familiar in the theory of dynamical systems, and originating in Cayley's work

GRAM-SCHMIDT SUPER ORTHOGONALIZATION PROCESS FOR SUPER LINEAR ALGEBRA

Gram-Schmidt Process is a method to transform an arbitrary basis into an orthogonal basis then normalize the orthogonal basis vectors to obtain an orthonormal basis. This process is so important and has many uses in applications of mathematics, particularly linear algebra and numerical analysis. Super linear algebra is an extension of linear algebra, in the which talks about the super matrices, super vectors up to super basis, super orthogonal basis and super diagonalization on a super inner product super spaces. It will be discussed a process to construct an arbitrary basis into an super orthogonal and orthonormal basis for super inner- product super spaces. The modification of the Gram-Schmidt Process to construct an super orthogonal and orthonormal basis, namely Gram-Schmidt Orthogonalization Process for Super Super Linear Algebra

Hidden Symmetries of Stochastic Models

In the matrix product states approach to $n$ species diffusion processes the stationary probability distribution is expressed as a matrix product state with respect to a quadratic algebra determined by the dynamics of the process. The quadratic algebra defines a noncommutative space with a $SU_q(n)$ quantum group action as its symmetry. Boundary processes amount to the appearance of parameter dependent linear terms in the algebraic relations and lead to a reduction of the $SU_q(n)$ symmetry. We argue that the boundary operators of the asymmetric simple exclusion process generate a tridiagonal algebra whose irriducible representations are expressed in terms of the Askey-Wilson polynomials. The Askey-Wilson algebra arises as a symmetry of the boundary problem and allows to solve the model exactly.Comment: This is a contribution to the Proc. of the O'Raifeartaigh Symposium on Non-Perturbative and Symmetry Methods in Field Theory (June 2006, Budapest, Hungary), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Creating And Solving Model Of Linear Equation Through The Balance At Junior Secondary Class

Algebra is one of the most difficult subject felt by most students and this strand is just started given to the students at early junior secondary school. Infact, if we can manage the algebra subject wisely and attractively, it can be predicted that the students’ conceptual understanding algebra would be relatively improved. A study was conducted to the Year 7 students at a Junior Secondary of Laboratory School of UPI. The class was set to learn the linear equation topic by using balance (scales). Through a weighing activity, the students were able to design linear equation models. They followed a guidelines created by the teacher and researcher. The students were not only able to create linear equation models, but also were able to solve mathematical model of linear equation. By using steps of balance (weighing), finally the students were able to know the weight of a hidden quantity. A number of teachers were involved in an observation activity which were designed in a lesson study context. Information related to the students’ reaction as well as the teachers’ reaction toward the implementation of creating and designing models of linear equation. The information were analysed qualitatively. The results indicate that introducing the linear equation through the scale (balance) were responded positively by the students. A brief interview with the students indicated that the students fluently could solve linear equation, and find the value of variable which infact as a weight variable. This variable seemed to be the weight of hidden variable as the solution of the linear equation. Moreover, the students were able to interpret the process of weighing to the form of linear equation, since then the students solved it and found the solution of the problem. While other teachers as observers at the lesson gave comments that the model teacher had practiced the concept of linear equation by using unusual way of teaching. Intuitively they solved the linear equation by using step by step of weighing process and determined how much weight of an object. The process of weighing and thinking are parallel to solving a linear equation. Data of test results regarding the linear equation indicated that the students’ understanding of linear equation improved. The researchers recommend to use the balance (scales) as an alternative to teach the topic of linear equation. Keywords: Balance, realistic, and lesson study

On bases of centres of Iwahori-Hecke algebras of the symmetric group

Using norms, the second author constructed a basis for the centre of the Hecke algebra of the symmetric group over \Q[\xi] in 1990. An integral "minimal" basis was later given by the first author in 1999, following work of Geck and Rouquier. In principle one can then write elements of the norm basis as integral linear combinations of minimal basis elements. In this paper we find an explicit non-recursive expression for the coefficients appearing in these linear combinations. These coefficients are expressed in terms of readily computable numbers involving orders of symmetric groups and conjugacy classes. In the process of establishing this main theorem, we prove the following items of independent interest: a result on the projection of the norms onto parabolic subalgebras, the existence of an inner product on the Hecke algebra with some interesting properties, and the existence of a partial ordering on the norms.Comment: 29 pages. To appear J. Algebra. Original version January 200