Study of Some Fundamental Projective Quadrics Associated with a Standard Pseudo-Hermitian Space H p,q

This self-contained short note deals with the study of the properties of some real projective compact quadrics associated with a a standard pseudo-hermitian space H _p,q , namely [(Q(p, q))\tilde], [(Q_2p+1,1)\tilde], [(Q_1,2q+1)\tilde], [(H_p,q)\tilde].  [(Q(p, q))\tilde]{\widetilde{Q(p, q)}, \widetilde{Q_{2p+1,1}}, \widetilde{Q_{1,2q+1}}, \widetilde{H_{p,q}}. \, \widetilde{Q(p, q)}} is the (2n – 2) real projective quadric diffeomorphic to (S ^2p–1 × S ^2q–1)/Z ₂. inside the real projective space P(E ₁), where E ₁ is the real 2n-dimensional space subordinate to H _p,q . The properties of [(Q(p, q))\tilde]{\widetilde{Q(p, q)}} are investigated. [(H_p,q)\tilde]{\widetilde{H_p,q}} is the real (2n – 3)-dimensional compact manifold-(projective quadric)- associated with H _p,q , inside the complex projective space P(H _p,q ), diffeomorphic to (S ^2p–1 × S ^2q–1)/S ¹. The properties of [(H_p,q)\tilde]{\widetilde{H_{p,q}}} are studied. [(Q_2p+1,1)\tilde]{\widetilde{Q_{2p+1,1}}} is a 2p-dimensional standard real projective quadric, and [(Q_1,2q+1)\tilde]{\widetilde{Q_{1,2q+1}}} is another standard 2q-dimensional projective quadric. [(Q_2p+1,1)\tilde] è[(Q_1,2q+1)\tilde]{\widetilde{Q_{2p+1,1}} \cup \widetilde{Q_{1,2q+1}}}, union of two compact quadrics plays a part in the understanding of the "special pseudo-unitary conformal compactification" of H _p,q . It is shown how a distribution y → D _y , where y ? H\{0},H{y \in H\backslash\{0\},H} being the isotropic cone of H _p,q allows to [(H_p+1,q+1)\tilde]{\widetilde{H_{p+1,q+1}}} to be considered as a "special pseudo-unitary conformal compactified" of H _p,q × R. The following results precise the presentation given in [1,c].     

Three-dimensional Yang-Mills-Higgs equations in gauge-invariant variables

A formulation of three-dimensionalSU(2) andSO(1,2) Yang-Mills theories in the gauge-invariant variablesG _ij=tr^* F _i ^* F_j is proposed. It is shown that the tensorG _ij satisfies three-dimensional Einstein equations with simple right-hand side andG _ij playing the role of a metric. This result is generalized to the case of a system of Yang-Mills-Higgs fields.Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 94, No. 1, pp. 66–75, January, 1993.     

A note on “Constructing matrices with prescribed off-diagonal submatrix and invariant polynomials”

Let F be any field and let B a matrix of F^q×p. Zaballa found necessary and sufficient conditions for the existence of a matrix A=[A_ij]_i,j∈{1,2}∈F^(p+q)×(p+q) with prescribed similarity class and such that A₂₁=B. In an earlier paper [A. Borobia, R. Canogar, Constructing matrices with prescribed off-diagonal submatrix and invariant polynomials, Linear Algebra Appl. 424 (2007) 615-633] we obtained, for fields of characteristic different from 2, a finite step algorithm to construct A when it exists. In this short note we extend the algorithm to any field.     

Potenzieren und Wurzelziehen in rationalen Quaternionenalgebren

LetF be a field not of characteristic 2 andQ =F +F i +F j +F k the quaternion algebra overF whereij = -ji =k andi ² = α andj ² = β with 0 ≠ α, β ∈F fixed. (IfF = ℝ and α = β = - 1 thenQ is the division algebra of the Hamilton quaternions.) IfF = ℚ and Q is a division algebra then by embedding certain quadratic number fields inQ we derive an efficient formula to compute the powers of any quaternion. This formula is even true in general and reads as follows. If a, a₁, a₂, a₃ ∈F andn ∈ ℕ then where ω ig a square root of αa₁ ² + βa ₂ ² - αβa ₃ ² in or overF and andA ₀ =na ^n-1. With the help of this formula and related ones we are able to solve the equationX ⁿ =q for arbitrary quaternionsq and positive integers n in case ofF = ℝ and hence in case ofF ⊂ ℝ as well. IfF = ℝ then the total number of all solutions equals 0, 1, 2, 4,n or ∞. (4 is possible even whenn < 4.) In case ofF = ℚ, which we are primarily interested in, there are always either at most six or infinitely many solutions. Further, for everyq ≠ 0 there is at most one solution provided thatn is odd and not divisible by 3. The questions when there are infinitely many solutions and when there are none can always be decided by checking simple conditions on the radicandq ifF = ℝ. ForF = ℚ the two questions are comprehensively investigatet in a natural connection with ternary and quaternary quadratic rational forms. Finally, by applying some of our theorems on powers and roots of quate-rions we also obtain several nice results in matrix theory. For example, for every k ∈ ℤ the mappingA ↦A ^k on the group of all nonsingular 2-by-2 matrices over ℚ is injective if and only ifk is odd and not divisible by 3.

     

Spectrum of Ornstein-Uhlenbeck Operators in L Spaces with Respect to Invariant Measures

Let A=∑_i,j=1^Nq_ijD_ij+∑_i,j=1^Nb_ijx_jD_i be a possibly degenerate Ornstein-Uhlenbeck operator in R^N and assume that the associated Markov semigroup has an invariant measure μ. We compute the spectrum of A in L_μ^p for 1?p<∞.     

Moments of square-integrable functions

This paper is motivated by [2], where we have given necessary and sufficient conditions for a given basis P in the space of polynomials to be orthogonal with respect to the measure ϱdφ for a certain function ϱ ϵ L₂(dφ). Let P = {p_i: i = 0, 1, …}, p₀ = 1. Then the conditions are (1) a multivariate analog of the three-term recurrence relation holds, see Section 4 for details; and (2) {q_i = ∑_{j = 0}^∞ c_ij P_j, i = 0, 1, …} is a φ-orthonormal basis in the space of polynomials for some coefficients c_ij such that ∑_{i = 0}^∞ c_i0² <-∞. This paper provides an algebraic condition (a condition on the coefficients c_i0) such that ϱ satisfies ∥p∥ <B, Bϵ (0, ∞], and has a cone-positivity property. In particular, our results imply that ϱ is nonnegative a.e. if ∑_{i = 0}^∞ c_i0² < ∞ and ∑_{jϵ Sk} c_j0 q_j defines nonnegative polynomials for certain finite sets S₁, S₂, … of integers.     

On the ternary goldbach problem with primes in independent arithmetic progressions

We show that for every fixed A > 0 and θ > 0 there is a ϑ = ϑ(A, θ) > 0 with the following property. Let n be odd and sufficiently large, and let Q ₁ = Q ₂:= n ^1/2(log n)^−ϑ and Q ₃:= (log n) ^θ . Then for all q ₃ ≦ Q ₃, all reduced residues a ₃ mod q ₃, almost all q ₂ ≦ Q ₂, all admissible residues a ₂ mod q ₂, almost all q ₁ ≦ Q ₁ and all admissible residues a ₁ mod q ₁, there exists a representation n = p ₁ + p ₂ + p ₃ with primes p _i ≡ a _i (q _i ), i = 1, 2, 3.      

The birth and death processes with zero as their absorbing barrier

LetE=(0, 1,...), Q ^b=(q_ij), i, j=0, 1, ..., whereq _{i, i–1}=a_i, q_{i, i+1}=b_i, q_ii=–(a_i+b_i), q_ij=0, when|i–j|>1. a ₀=0, b₀=b>0, a_i, b_i>0 (i>0). Lettingb=0 inQ ^b, we get the matrixQ ⁰.The time homogeneous Markov processX ^b ={x ^b (t,w), 0t< ^b (w)} (X ⁰={x⁰(t,w), 0t<⁰(w)}), withQ ^b (Q ⁰, respectively) as its density matrix and withE as its state space, is calledQ ^b (Q ⁰, respectively) process in this paper.Q ^b andQ ⁰ processes are all called the birth and death processes, with zero being the reflecting barrier ofQ ^b processes, the absorbing barrier ofQ ⁰ processes.AllQ ^b processes have been constructed by both probability and analytical methods (Wang [2], Yang [1]). In this paper, theQ ⁰ processes are imbedded intoQ ^b processes and all theQ ⁰ processes are directly constructed from theQ ^b processes. The main results are:Letb>0 be arbitrarily fixed, then there is a one to one correspondence between theQ ⁰ processes and theQ ^b processes (in the sense of transition probability).TheQ ⁰ process is unique iffR ^*=. SupposingR<, then:IfX ⁰={x⁰(t,w), 0t<⁰(w)} is a non-minimalQ ⁰ process, then its eigensequence (p, q, r _n, n–1) satisfies § 4(7).Conversely, let a non-negative number sequence (p, q, r _n, n–1) satisfying § 4(7) be arbitrarily given, then there exists a unique non-minimalQ ⁰ processX ⁰ with eigensequence (p, q, r _n, n–1). The Laplace transform of the transition probability (p _ij ⁰ (t)) ofX ⁰ is determined by § 4(15). X ⁰ is honest iffr _–1=0.X ⁰ satisfies the forward equation iffp=0.     

The Number of Permutation Polynomials of a Given Degree Over a Finite Field

We relate the number of permutation polynomials in F_q[x] of degree d≤q−2 to the solutions (x₁,x₂,…,x_q) of a system of linear equations over F_q, with the added restriction that x_i≠0 and x_i≠x_j whenever i≠j. Using this we find an expression for the number of permutation polynomials of degree p−2 in F_p[x] in terms of the permanent of a Vandermonde matrix whose entries are the primitive pth roots of unity. This leads to nontrivial bounds for the number of such permutation polynomials. We provide numerical examples to illustrate our method and indicate how our results can be generalised to polynomials of other degrees.     

Extensions of band matrices with band inverses

Let R_ij be a given set of μ_i× μ_j matrices for i, j=1,…, n and |i?j| ?m, where 0?m?n?1. Necessary and sufficient conditions are established for the existence and uniqueness of an invertible block matrix =[F_ij], i,j=1,…, n, such that F_ij=R_ij for |i?j|?m, F admits either a left or right block triangular factorization, and (F^?1)_ij=0 for |i?j|>m. The well-known conditions for an invertible block matrix to admit a block triangular factorization emerge for the particular choice m=n?1. The special case in which the given R_ij are positive definite (in an appropriate sense) is explored in detail, and an inequality which corresponds to Burg's maximal entropy inequality in the theory of covariance extension is deduced. The block Toeplitz case is also studied.     

Higher Power Residue Codes

For certain primeslandp, and characters χ:F*_p→F*_l², we construct codesWof lengthp+ 1 overF_l²which are linear overF_l, but not overF_l², and which are invariant under a monomial action of the group SL(2,p). We consider the cases of cubic and quartic characters in detail and use theWto construct linear codes overF_lin these cases.     

Sufficient conditions for the existence of a left quotient ring of a ring decomposed into a direct sum of left ideals

Let R=P₁⊕P₂⊕…⊕P_n be a decomposition of a ring into a direct sum of indecomposable left ideals. Assume that these ideals possess the following properties: (1) any nonzero homomorphisms ϕ: P_i→P_j is a monomorphism; (2) if subideals Q₁, Q₂ of the ideal P_j are isomorphic to the ideal P_i, then there exists a subideal Q₃⊆Q₁⊎Q₂, which is also isomorphic to P_i. It is proved that, under these asumptions, a left quotient ring of the ring R exists. This left quotient ring inherits properties(1), (2) and satisfies condition (3): any nonzero homomorphism ϕ: P_i→P_i is an automorphism of the ideal P_i. Bibliography: 2 titles. Translated fromZapiski Nauchnyk Seminarov POMI, Vol. 227, 1995, pp. 9–14.     

Unitary similarity of algebras generated by pairs of orthoprojectors

It is shown that the unitary similarity of two matrix algebras generated by pairs of orthoprojectors { P₁, Q₁} and {P₂,Q₂} can be verified by comparing the traces of P₁, Q₁, and (P₁Q₁)ⁱ, i = 1, 2, …, n, with those of P₂, Q₂, and (P₂Q₂)ⁱ. The conditions of the unitary similarity of two matrices with quadratic minimal polynomials presented in [A. George and Kh. D. Ikramov, Unitary similarity of matrices with quadratic minimal polynomials, Linear Algebra Appl., 349, 11–16 (2002)] are refined. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 5–14.     

Maximum IPP Codes of Length 3

Let Q be an alphabet with q elements. For any code C over Q of length n and for any two codewords a = (a ₁, . . . , a _n ) and b = (b ₁, . . . , b _n ) in C, let ${D({\bf a, b}) = \{(x_1, . . . , x_n) \in {Q^n} : {x_i} \in \{a_i, b_i\}\,{\rm for}\,1 \leq i \leq n\}}Let Q be an alphabet with q elements. For any code C over Q of length n and for any two codewords a = (a ₁, . . . , a _n ) and b = (b ₁, . . . , b _n ) in C, let D(a, b) = {(x₁, . . . , x_n) ? Qⁿ : x_i ? {a_i, b_i} for 1 ￡ i ￡ n}{D({\bf a, b}) = \{(x_1, . . . , x_n) \in {Q^n} : {x_i} \in \{a_i, b_i\}\,{\rm for}\,1 \leq i \leq n\}}. Let C^* = è_{a, b ? C}D(a, b){C^* = {{\bigcup}_{\rm {a,\,b}\in{C}}}D({\bf a, b})}. The code C is said to have the identifiable parent property (IPP) if, for any x ? C^*{{\rm {\bf x}} \in C^*}, ?_{x ? D(a, b)}{a, b} 1 ?{{\bigcap}_{{\rm x}{\in}D({\rm a,\,b})}\{{\bf a, b}\}\neq \emptyset} . Codes with the IPP were introduced by Hollmann et al [J. Combin. Theory Ser. A 82 (1998) 21–133]. Let F(n, q) = max{|C|: C is a q-ary code of length n with the IPP}.T? and Safavi-Naini [SIAM J. Discrete Math. 17 (2004) 548–570] showed that 3q + 6 - 6 é?{q+1}ù ￡ F(3, q) ￡ 3q + 6 - é6 ?{q+1}ù{3q + 6 - 6 \lceil\sqrt{q+1}\rceil \leq F(3, q) \leq 3q + 6 - \lceil 6 \sqrt{q+1}\rceil}, and determined F (3, q) precisely when q ≤ 48 or when q can be expressed as r ² + 2r or r ² + 3r +2 for r ≥ 2. In this paper, we establish a precise formula of F(3, q) for q ≥ 24. Moreover, we construct IPP codes of size F(3, q) for q ≥ 24 and show that, for any such code C and any x ? C^*{{\rm {\bf x}} \in C^*}, one can find, in constant time, a ? C{{\rm {\bf a}} \in C} such that if x ? D (c, d){{\rm {\bf x}} \in D ({\bf c, d})} then a ? {c, d}{{\rm {\bf a}} \in \{{\rm {\bf c, d}}\}}.     

A Load-Balanced Network with Two Servers

A load-balanced network with two queues Q ₁ and Q ₂ is considered. Each queue receives a Poisson stream of customers at rate _i , i=1,2. In addition, a Poisson stream of rate arrives to the system; the customers from this stream join the shorter of two queues. After being served in the ith queue, i=1,2, customers leave the system with probability 1–p _i ^*, join the jth queue with probability p(i,j), j=1,2, and choose the shortest of two queues with probability p(i,{1,2}). We establish necessary and sufficient conditions for stability of the system.     

Generating random correlation matrices based on partial correlations

A d-dimensional positive definite correlation matrix R=(ρ_ij) can be parametrized in terms of the correlations ρ_i,i+1 for i=1,…,d-1, and the partial correlations ρ_{ij|i+1,…j-1} for j-i2. These parameters can independently take values in the interval (-1,1). Hence we can generate a random positive definite correlation matrix by choosing independent distributions F_ij, 1i<jd, for these parameters. We obtain conditions on the F_ij so that the joint density of (ρ_ij) is proportional to a power of det(R) and hence independent of the order of indices defining the sequence of partial correlations. As a special case, we have a simple construction for generating R that is uniform over the space of positive definite correlation matrices. As a byproduct, we determine the volume of the set of correlation matrices in -dimensional space. To prove our results, we obtain a simple remarkable identity which expresses det(R) as a function of ρ_i,i+1 for i=1,…,d-1, and ρ_{ij|i+1,…j-1} for j-i2.     

An application of L-functions of imaginary quadratic fields

Let F₁(x, y),…, F_2h+1(x, y) be the representatives of equivalent classes of positive definite binary quadratic forms of discriminant ?q (q is a prime such that q ≡ 3 mod 4) with integer coefficients, then the number of integer solutions of F_i(x, y) = n (i = 1,…, 2h + 1) can be calculated for each natural number n using L-functions of imaginary quadratic field $\text{Q}$((?q)^1/2).     

Geometric progressions in sumsets over finite fields

Given two sets A, B í \Bbb F_q^d{\cal A}, {\cal B}\subseteq {\Bbb F}_q^d , the set of d dimensional vectors over the finite field \Bbb F_q{\Bbb F}_q with q elements, we show that the sumset A+B = {a+b | a ? A, b ? B}{\cal A}+{\cal B} = \{{\bf a}+{\bf b}\ \vert\ {\bf a} \in {\cal A}, {\bf b} \in {\cal B}\} contains a geometric progression of length k of the form vΛ ^j , where j = 0,…, k − 1, with a nonzero vector v ? \Bbb F_q^d{\bf v} \in {\Bbb F}_q^d and a nonsingular d × d matrix Λ whenever # A # B 3 20 q^2d-2/k\# {\cal A} \# {\cal B} \ge 20 q^{2d-2/k} . We also consider some modifications of this problem including the question of the existence of elements of sumsets on algebraic varieties.     

The hypermetric cone is polyhedral

The hypermetric coneH _n is the cone in the spaceR ^n(n–1)/2 of all vectorsd=(d _ij)_1i<jn satisfying the hypermetric inequalities: –_1ijn z _j z _j d _ij 0 for all integer vectorsz inZ ⁿ with –_1in z _i =1. We explore connections of the hypermetric cone with quadratic forms and the geometry of numbers (empty spheres andL-polytopes in lattices). As an application, we show that the hypermetric coneH _n is polyhedral.     

Points surrounding the origin

Suppose d > 2, n > d+1, and we have a set P of n points in d-dimensional Euclidean space. Then P contains a subset Q of d points such that for any p ∈ P, the convex hull of Q∪{p} does not contain the origin in its interior. We also show that for non-empty, finite point sets A ₁, ..., A _d+1 in ℝ ^d , if the origin is contained in the convex hull of A _i ∪ A _j for all 1≤i<j≤d+1, then there is a simplex S containing the origin such that |S∩A _i |=1 for every 1≤i≤d+1. This is a generalization of Bárány’s colored Carathéodory theorem, and in a dual version, it gives a spherical version of Lovász’ colored Helly theorem. Dedicated to Imre Bárány, Gábor Fejes Tóth, László Lovász, and Endre Makai on the occasion of their sixtieth birthdays. Supported by the Norwegian research council project number: 166618, and BK 21 Project, KAIST. Part of the research was conducted while visiting the Courant Institute of Mathematical Sciences. Supported by NSF Grant CCF-05-14079, and by grants from NSA, PSC-CUNY, the Hungarian Research Foundation OTKA, and BSF.