Massive and massless neutrinos on unbalanced seesaws

The observation of neutrino oscillations requires new physics beyond the standard model (SM).A SM-like gauge theory with p lepton families can be extended by introducing q heavy right-handed Majorana neutrinos but preserving its SU(2)L x U(1)y gauge symmetry.The overall neutrino mass matrix M turns out to be a symmetric (p+q) x (p+q) matrix.Given p＞q,the rank of M is in general equal to 2q,corresponding to 2q non-zero mass eigenvalues.The existence of (p-q) massless left-handed Majorana neutrinos is an exact consequence of the model,independent of the usual approximation made in deriving the Type-I seesaw relation between the effective p x p light Majorana neutrino mass matrix M,and the q x q heavy Majorana neutrino mass matrix MR.In other words,the numbers of massive left- and right-handed neutrinos are fairly matched.A good example to illustrate this "seesaw fair play rule"is the minimal seesaw model with p ＝ 3 and q ＝ 2,in which one masslese neutrino sits on the unbalanced seesaw.     

最小方差无失真响应浊音谱建模方法研究

讨论了在语音处理中广泛使用的线性预测模型,分析了线性预测模型在建模语音谱包络方面存在缺点,并分析了线性预测模型过高估计浊音谐波频率处能量以及随着阶数增加线性预测谱逐渐恶化的原因。在此基础上,提出了最小方差无失真响应(MVDR)建模方法,并讨论了通过Toeplitz矩阵的Cholesky分解确定MVDR滤波器系数的方法,通过与线性预测方法进行比较,发现最小方差无失真响应滤波器能提供一个更好的原始语音包络。     

Sequential Semidefinite Program for Maximum Robustness Design of Structures under Load Uncertainty

A robust structural optimization scheme as well as an optimization algorithm are presented based on the robustness function. Under the uncertainties of the external forces based on the info-gap model, the maximization of the robustness function is formulated as an optimization problem with infinitely many constraints. By using the quadratic embedding technique of uncertainty and the S-procedure, we reformulate the problem into a nonlinear semidefinite programming problem. A sequential semidefinite programming method is proposed which has a global convergent property. It is shown through numerical examples that optimum designs of various linear elastic structures can be found without difficulty.The authors are grateful to the Associate Editor and two anonymous referees for handling the paper efficiently as well as for helpful comments and suggestions.     

有限秩的幂零p-群的p-自同构

设G是一个有限秩的幂零p-群,α和β是G的两个p-自同构,记I= ((αβ(g))(βα(g))-1)|g∈G),则(i)当I是有限循环群时,α和β生成一个有限P-群; (ii)当I是拟循环p-群时,α和β生成一个可解的剩余有限P-群,它是有限生成的无挠幂零群被有限p-群的扩张．     

Generalized double affine Hecke algebras of rank 1 and quantized del Pezzo surfaces

Let D be a simply laced Dynkin diagram of rank r whose affinization has the shape of a star (i.e., D₄,E₆,E₇,E₈). To such a diagram one can attach a group G whose generators correspond to the legs of the affinization, have orders equal to the leg lengths plus 1, and the product of the generators is 1. The group G is then a 2-dimensional crystallographic group: G=Z??Z², where ? is 2, 3, 4, and 6, respectively. In this paper, we define a flat deformation H(t,q) of the group algebra C[G] of this group, by replacing the relations saying that the generators have prescribed orders by their deformations, saying that the generators satisfy monic polynomial equations of these orders with arbitrary roots (which are deformation parameters). The algebra H(t,q) for D₄ is the Cherednik algebra of type C^∨C₁, which was studied by Noumi, Sahi, and Stokman, and controls Askey-Wilson polynomials. We prove that H(t,q) is the universal deformation of the twisted group algebra of G, and that this deformation is compatible with certain filtrations on C[G]. We also show that if q is a root of unity, then for generic t the algebra H(t,q) is an Azumaya algebra, and its center is the function algebra on an affine del Pezzo surface. For generic q, the spherical subalgebra eH(t,q)e provides a quantization of such surfaces. We also discuss connections of H(t,q) with preprojective algebras and Painlevé VI.     

Undominated d.c. Decompositions of Quadratic Functions and Applications to Branch-and-Bound Approaches

In this paper we analyze difference-of-convex (d.c.) decompositions for indefinite quadratic functions. Given a quadratic function, there are many possible ways to decompose it as a difference of two convex quadratic functions. Some decompositions are dominated, in the sense that other decompositions exist with a lower curvature. Obviously, undominated decompositions are of particular interest. We provide three different characterizations of such decompositions, and show that there is an infinity of undominated decompositions for indefinite quadratic functions. Moreover, two different procedures will be suggested to find an undominated decomposition starting from a generic one. Finally, we address applications where undominated d.c.d.s may be helpful: in particular, we show how to improve bounds in branch-and-bound procedures for quadratic optimization problems.     

Kernels and <Emphasis Type="Italic">p</Emphasis>-Kernels of <Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">r</Emphasis></Superscript>-ary 1-Perfect Codes

The rank of a q-ary code C is the dimension of the subspace spanned by C. The kernel of a q-ary code C of length n can be defined as the set of all translations leaving C invariant. Some relations between the rank and the dimension of the kernel of q-ary 1-perfect codes, over as well as over the prime field , are established. Q-ary 1-perfect codes of length n=(q^m − 1)/(q − 1) with different kernel dimensions using switching constructions are constructed and some upper and lower bounds for the dimension of the kernel, once the rank is given, are established.Communicated by: I.F. Blake     

The Minimum Rank of a 3 × 3 Partial Block Matrix

After recalling the definition and some basic properties of finite hypergroups—a notion introduced in a recent paper by one of the authors—several non-trivial examples of such hypergroups are constructed. The examples typically consist of n n×n matrices, each of which is an appropriate polynomial in a certain tri-diagonal matrix. The crucial result required in the construction is the following: ‘let A be the matrix with ones on the super-and sub-diagonals, and with main diagonal given by a ₁…a _n which are non-negative integers that form either a non-decreasing or a symmetric unimodal sequence; then A_k =P_k (A) is a non-negative matrix, where p_k denotes the characteristic polynomial of the top k× k principal submatrix of A, for k=1,…,n. The matrices A_k as well as the eigenvalues of A, are explicitly described in some special cases, such as (i) a_i =0 for all ior (ii) a_i =0 for i<n and a_n =1. Characters ot finite abelian hypergroups are defined, and that naturally leads to harmonic analysis on such hypergroups.     

Possible numbers of ones in the squares of 0–1 matrices with a given rank

For a symmetric 0–1 matrix A, we give the number of ones in A ² when rank(A) = 1, 2, and give the maximal number of ones in A ² when rank(A) = k (3 ≤ k ≤ n). The sufficient and necessary condition under which the maximal number is achieved is also obtained. For generic 0–1 matrices, we only study the cases of rank 1 and rank 2.