On orthogonal arrays of strength 3 and 5 achieving Rao's bound

It is shown that ifA is an orthogonal array (N, n, q, 3) achieving Rao's bound, thenA is either

1. an orthogonal array (2n, n, 2, 3) withn ≡ 0 (mod 4), or
2. an orthogonal array (q ³,q + 2,q, 3) withq even.

This result should be compared with a theorem of P.J. Cameron on extendable symmetric designs. It is also shown that ifA is an orthogonal array (N, n, q, 5) achieving Rao's bound, thenA is either the orthogonal array (32, 6, 2, 5) or the orthogonal array (3⁶, 12, 3, 5).     

On the diophantine equation x 2 − Dy 2 = n

We propose a method to determine the solvability of the diophantine equation x2-Dy2=n for the following two cases:(1) D = pq,where p,q ≡ 1 mod 4 are distinct primes with(q/p)=1 and(p/q)4(q/p)4=-1.(2) D=2p1p2 ··· pm,where pi ≡ 1 mod 8,1≤i≤m are distinct primes and D=r2+s2 with r,s ≡±3 mod 8.     

Die Endlichen Polardreiseitgeometrien

LetG be a finite group which is generated by a subsetS of involutions satisfying the theorem of the three reflections: Ifa,b,x,y,z ∈ S, ab ≠ 1 and ifabx,aby,abz are involutions, thenxyz ∈ S. Assume thatS contains three elements which generate a four-group. IfS contains four elements of which no three have a product of order two, then one of the following occurs.

1. G?PGL(2,n), n≡1 (mod 2).
2. G?PSL(2,n), n≡1 (mod 2) and n≥5.
3. G?PSU(3,16).
4. G/Z(G)?PSL(2,9) with ¦Z(G)¦=3.

     

Infinite family of large complete arcs in PG(2, q n ), with q odd and n > 1 odd

For q odd and n > 1 odd, a new infinite family of large complete arcs K′ in PG(2, q ⁿ ) is constructed from complete arcs K in PG(2, q) which have the following property with respect to an irreducible conic ${\mathcal{C}}$ in PG(2, q): all the points of K not in ${\mathcal{C}}$ are all internal or all external points to ${\mathcal{C}}$ according as q ≡ 1 (mod 4) or q ≡ 3 (mod 4).     

Regular sets of matrices and applications

SupposeA ₁,...,A _s are (1, - 1) matrices of order m satisfying 1 $$A_i A_j = J, i,j \in \left\{ {1,...s} \right\}$$ 2 $$A_i^T A_j = A_j^T A_i = J, i \ne j, i,j \in \left\{ {1,...,s} \right\}$$ 3 $$\sum\limits_{i = 1}^s {(A_i A_i^T = A_i^T A_i ) = 2smI_m } $$ 4 $$JA_i = A_i J = aJ, i \in \left\{ {1,...,s} \right\}, a constant$$ Call A₁,…,A _s ,a regular s- set of matrices of order m if Eq. 1-3 are satisfied and a regular s-set of regular matrices if Eq. 4 is also satisfied, these matrices were first discovered by J. Seberry and A.L. Whiteman in “New Hadamard matrices and conference matrices obtained via Mathon’s construction”, Graphs and Combinatorics, 4(1988), 355-377. In this paper, we prove that

1. if there exist a regular s-set of order m and a regulart-set of order n there exists a regulars-set of ordermn whent =sm
2. if there exist a regular s-set of order m and a regulart-set of order n there exists a regulars-set of ordermn when 2t = sm (m is odd)
3. if there exist a regulars-set of order m and a regulart-set of ordern there exists a regular 2s-set of ordermn whent = 2sm As applications, we prove that if there exist a regulars-set of order m there exists
4. an Hadamard matrices of order4hm whenever there exists an Hadamard matrix of order4h ands =2h
5. Williamson type matrices of ordernm whenever there exists Williamson type matrices of ordern and s = 2n
6. anOD(4mp;ms₁,…,ms_u whenever anOD (4p;s₁,…,s_u)exists and s = 2p
7. a complex Hadamard matrix of order 2cm whenever there exists a complex Hadamard matrix of order 2c ands = 2c

This paper extends and improves results of Seberry and Whiteman giving new classes of Hadamard matrices, Williamson type matrices, orthogonal designs and complex Hadamard matrices.     

On the pronormality of subgroups of odd index in finite simple symplectic groups

A subgroup H of a group G is pronormal if the subgroups H and H ^g are conjugate in 〈H,H ^g 〉 for every g ∈ G. It was conjectured in [1] that a subgroup of a finite simple group having odd index is always pronormal. Recently the authors [2] verified this conjecture for all finite simple groups other than PSL _n (q), PSU _n (q), E ₆(q), ² E ₆(q), where in all cases q is odd and n is not a power of 2, and P Sp_2n (q), where q ≡ ±3 (mod 8). However in [3] the authors proved that when q ≡ ±3 (mod 8) and n ≡ 0 (mod 3), the simple symplectic group P Sp²ⁿ (q) has a nonpronormal subgroup of odd index, thereby refuted the conjecture on pronormality of subgroups of odd index in finite simple groups.The natural extension of this conjecture is the problem of classifying finite nonabelian simple groups in which every subgroup of odd index is pronormal. In this paper we continue to study this problem for the simple symplectic groups P Sp_2n (q) with q ≡ ±3 (mod 8) (if the last condition is not satisfied, then subgroups of odd index are pronormal). We prove that whenever n is not of the form 2 ^m or 2 ^m (2^2k +1), this group has a nonpronormal subgroup of odd index. If n = 2 ^m , then we show that all subgroups of P Sp_2n (q) of odd index are pronormal. The question of pronormality of subgroups of odd index in P Sp_2n (q) is still open when n = 2 ^m (2^2k + 1) and q ≡ ±3 (mod 8).     

An embedding theorem for a weighted space of Sobolev type and correct solvability of the Sturm-Liouville equation

We consider the weighted space W ₁ ⁽²⁾ (?,q) of Sobolev type $$W_1^{(2)} (\mathbb{R},q) = \left\{ {y \in A_{loc}^{(1)} (\mathbb{R}):\left\| {y''} \right\|_{L_1 (\mathbb{R})} + \left\| {qy} \right\|_{L_1 (\mathbb{R})} < \infty } \right\} $$ and the equation $$ - y''(x) + q(x)y(x) = f(x),x \in \mathbb{R} $$ Here f ε L ₁(?) and 0 ? q ∈ L ₁ ^loc (?). We prove the following:

1. The problems of embedding W ₁ ⁽²⁾ (?q) ? L ₁(?) and of correct solvability of (1) in L ₁(?) are equivalent
2. an embedding W ₁ ⁽²⁾ (?,q) ? L ₁(?) exists if and only if $$\exists a > 0:\mathop {\inf }\limits_{x \in R} \int_{x - a}^{x + a} {q(t)dt > 0} $$

     

Free quadratic bialgebra

In this paper, we obtain the following main theorem for a free quadratic bialgebraJ:

1. Forp≠0,J is a pointed cosemisimple coalgebra. Forp=0,J is a hyperalgebra.
2. Forp≠0 andq≠0,J has antipodeS iffp·q+2=0 andS(x)=x. Forp=0 orq=0,J has antipode andS(x)=×.
3. All leftJ ^*-modules are rational.

Also, we give some applications in homological theory and algebraicK-theory.     

Finite Index Operators on Surfaces

We consider differential operators L acting on functions on a Riemannian surface, Σ, of the form $$L = \Delta+ V -a K,$$ where Δ is the Laplacian of Σ, K is the Gaussian curvature, a is a positive constant, and V∈C ^∞(Σ). Such operators L arise as the stability operator of Σ immersed in a Riemannian three-manifold with constant mean curvature (for particular choices of V and a). We assume L is nonpositive acting on functions compactly supported on Σ. If the potential, V:=c+P with c a nonnegative constant, verifies either an integrability condition, i.e., P∈L ¹(Σ) and P is nonpositive, or a decay condition with respect to a point p ₀∈Σ, i.e., |P(q)|≤M/d(p ₀,q) (where d is the distance function in Σ), we control the topology and conformal type of Σ. Moreover, we establish a Distance Lemma. We apply such results to complete oriented stable H-surfaces immersed in a Killing submersion. In particular, for stable H-surfaces in a simply-connected homogeneous space with 4-dimensional isometry group, we obtain:

• There are no complete stable H-surfaces Σ??²×?, H>1/2, so that either $K_{e}^{+}:=\max \left \{0,K_{e}\right \} \in L^{1} (\Sigma)$ or there exist a point p ₀∈Σ and a constant M so that |K _e (q)|≤M/d(p ₀,q); here K _e denotes the extrinsic curvature of Σ.
• Let $\Sigma\subset \mathbb{E}(\kappa, \tau)$ , τ≠0, be an oriented complete stable H-surface so that either ν ²∈L ¹(Σ) and 4H ²+κ≥0, or there exist a point p ₀∈Σ and a constant M so that |ν(q)|²≤M/d(p ₀,q) and 4H ²+κ>0. Then:
• In $\mathbb{S}^{3}_{\text{Berger}}$ , there are no such a stable H-surfaces.
• In Nil₃, H=0 and Σ is either a vertical plane (i.e., a vertical cylinder over a straight line in ?²) or an entire vertical graph.
• In $\widetilde{\mathrm{PSL}(2,\mathbb{R})}$ , $H=\sqrt{-\kappa }/2$ and Σ is either a vertical horocylinder (i.e., a vertical cylinder over a horocycle in ?²(κ)) or an entire graph.

     

Plane structures in thermal runaway

We consider the problem

1. u _t=u _xx+e ^u whenx ∈ ?,t > 0,
2. u(x, 0) =u ₀(x) whenx ∈ ?,

whereu ₀(x) is continuous, nonnegative and bounded. Equation (1) appears as a limit case in the analysis of combustion of a one-dimensional solid fuel. It is known that solutions of (1), (2) blow-up in a finite timeT, a phenomenon often referred to as thermal runaway. In this paper we prove the existence of blow-up profiles which are flatter than those previously observed. We also derive the asymptotic profile ofu(x, T) near its blow-up points, which are shown to be isolated.     

Rows and diagonals of the Walsh array for entire functions with smooth Maclaurin series coefficients

Let \(f(z): = \sum\nolimits_{j = 0}^\infty {a_j z^J } \) be entire, witha _j≠0,j large enough, \(\lim _{J \to \infty } a_{j + 1} /a_J = 0\) , and, for someq∈C, \(q_j : = a_{j - 1} a_{j + 1} /a_j^2 \to q\) asj→∞. LetE _mn(f; r) denote the error in best rational approximation off in the uniform norm on |z‖≤r, by rational functions of type (m, n). We study the behavior ofE _mn(f; r) asm and/orn→∞. For example, whenq above is not a root of unity, or whenq is a root of unity, butq _m has a certain asymptotic expansion asm→∞, then we show that, for each fixed positive integern, ,m→∞. In particular, this applies to the Mittag-Leffler functions \(f(z): = \sum\nolimits_{j = 0}^\infty {z^j /\Gamma (1 + j/\lambda )} \) and to \(f(z): = \sum\nolimits_{j = 0}^\infty {z^j /(j!)^{I/\lambda } } \) , λ>0. When |q‖<1, we also handle the diagonal case, showing, for example, that ,n→∞. Under mild additional conditions, we show that we can replace 1+0(1) ⁿ by 1+0(1). In all cases we show that the poles of the best approximants approach ∞ asm→∞.     

Transitive A 6-invariant k-arcs in PG(2, q)

For q = p ^r with a prime p ≥ 7 such that ${q \equiv 1}$ or 19 (mod 30), the desarguesian projective plane PG(2, q) of order q has a unique conjugacy class of projectivity groups isomorphic to the alternating group A ₆ of degree 6. For a projectivity group ${\Gamma \cong A_6}$ of PG(2, q), we investigate the geometric properties of the (unique) Γ-orbit ${\mathcal{O}}$ of size 90 such that the 1-point stabilizer of Γ in its action on ${\mathcal O}$ is a cyclic group of order 4. Here ${\mathcal O}$ lies either in PG(2, q) or in PG(2, q ²) according as 3 is a square or a non-square element in GF(q). We show that if q ≥ 349 and q ≠ 421, then ${\mathcal O}$ is a 90-arc, which turns out to be complete for q = 349, 409, 529, 601,661. Interestingly, ${\mathcal O}$ is the smallest known complete arc in PG(2,601) and in PG(2,661). Computations are carried out by MAGMA.     

On a certain type of nonlinear differential equations admitting transcendental meromorphic solutions

We study the differential equations w ²+R(z)(w ^(k))² = Q(z), where R(z),Q(z) are nonzero rational functions. We prove

1. if the differential equation w ²+R(z)(w′)² = Q(z), where R(z), Q(z) are nonzero rational functions, admits a transcendental meromorphic solution f, then Q ≡ C (constant), the multiplicities of the zeros of R(z) are no greater than 2 and f(z) = √C cos α(z), where α(z) is a primitive of $\tfrac{1} {{\sqrt {R(z)} }}$ such that √C cos α(z) is a transcendental meromorphic function.
2. if the differential equation w ² + R(z)(w ^(k))² = Q(z), where k ? 2 is an integer and R,Q are nonzero rational functions, admits a transcendental meromorphic solution f, then k is an odd integer, Q ≡ C (constant), R(z) ≡ A (constant) and f(z) = √C cos (az + b), where $a^{2k} = \tfrac{1} {A}$ .

     

Two identities related to dirichlet character of polynomials

Let q be a positive integer, χ denote any Dirichlet character mod q. For any integer m with (m, q) = 1, we define a sum C(χ, k,m; q) analogous to high-dimensional Kloosterman sums as follows: , where a · ā ≡ 1 mod q. The main purpose of this paper is to use elementary methods and properties of Gauss sums to study the computational problem of the absolute value |C(χ, k,m; q)|, and give two interesting identities for it.     

Complex Banach spaces whose duals areL 1-spaces

We prove that for a complex Banach spaceA the following properties are equivalent:

1. A ^* is isometric to anL ₁(μ)-space;
2. every family of 4 balls inA with the weak intersection property has a non-empty intersection;
3. every family of 4 balls inA such that any 3 of them have a non-empty intersection, has a non-empty intersection.

     

On lines missing polyhedral sets in 3-space

We show some combinatorial and algorithmic results concerning finite sets of lines and terrains in 3-space. Our main results include:

1. An $O(n^3 2^{c\sqrt {\log n} } )$ upper bound on the worst-case complexity of the set of lines that can be translated to infinity without intersecting a given finite set ofn lines, wherec is a suitable constant. This bound is almost tight.
2. AnO(n ^1.5+ε) randomized expected time algorithm that tests whether a directionv exists along which a set ofn red lines can be translated away from a set ofn blue lines without collisions. ε>0 is an arbitrary small but fixed constant.
3. An $O(n^3 2^{c\sqrt {\log n} } )$ upper bound on the worst-case complexity of theenvelope of lines above a terrain withn edges, wherec is a suitable constant.
4. An algorithm for computing the intersection of two polyhedral terrains in 3-space withn total edges in timeO(n ^4/3+ε+k ^1/3 n ^1+ε+klog² n), wherek is the size of the output, and ε>0 is an arbitrary small but fixed constant. This algorithm improves on the best previous result of Chazelleet al. [5].

The tools used to obtain these results include Plücker coordinates of lines, random sampling, and polarity transformations in 3-space.     

Group inverse for a class of 2×2 anti-triangular block matrices over skew fields

Suppose K is a skew field. Let K ^m×n denote the set of all m×n matrices over K. In this paper, we give necessary and sufficient conditions for the existence and explicit representations of the group inverses of the block matrices in the following three cases, respectively:

1. $\mathrm{rank}(S)=\mathrm{rank}(B^{\pi}A)$ ;
2. $\mathrm{rank}(S)=\mathrm{rank}(AB^{\pi})$ ;
3. $\mathrm{rank}(S)=\mathrm{rank}(B^{\pi}A)=\mathrm{rank}(AB^{\pi})$ ,

where A,B,C??K ^n×n , B ^# exists, R(B)=R(C), N(B)=N(C) and S=B ^?? AB ^?? . The paper??s conclusions generalized some related results of Zhao and Bu (Electron. J. Linear Algebra 21:63?C75, 2010).     

Extension of a theorem of Cauchy and Jacobi

Let q and p be prime with q = a² + b² ≡ 1 (mod 4), a ≡ 1 (mod 4), and p = qf + 1. In the nineteenth century Cauchy (Mém. Inst. France17 (1840), 249–768) and Jacobi (J. für Math.30 (1846), 166–182) generalized the work of earlier authors, who had determined certain binomial coefficients (mod p) (see H. J. S. Smith, “Report on the Theory of Numbers,” Chelsea, 1964), by determining two products of factorials given by Π_kkf! (mod p = qf + 1) where k runs through the quadratic residues and the quadratic non-residues (mod q), respectively. These determinations are given in terms of parameters in representations of p^h or of 4p^h by binary quadratic forms. A remarkable feature of these results is the fact that the exponent h coincides with the class number of the related quadratic field. In this paper C. R. Mathews' (Invent. Math.54 (1979), 23–52) recent explicit evaluation of the quartic Gauss sum is used to determine four products of factorials (mod p = qf + 1, q ≡ 5 (mod 8) > 5), given by Π_kkf! where k runs through the quartic residues (mod q) and the three cosets which may be formed with respect to this subgroup. These determinations appear to be considerably more difficult. They are given in terms of parameters in representations of 16p^h by quaternary quadratic forms. Stickelberger's theorem is required to determine the exponent h which is shown to be closely related to the class number of the imaginary quartic field Q(i√2q + 2a√q), q = a² + b² ≡ 5 (mod 8), a odd.     

ДИФФЕРЕНцИАльНыЕ сВ ОИстВА ИжВИВАУЩИхсь ФУНкцИИ

A continuous real valued function defined on an intervalD is called crinkly iff the setf ^?1(У) ∩I is uncountable for each interval \(I \subseteqq D\) and number \(y \in (\mathop {\inf }\limits_I f,\mathop {\sup }\limits_I f)\) . The main result of the paper consists in the following assertion. Let the closed segment [0, 1] be represented as a union of four measurable, mutually nonintersecting setsE ₁,Е ₂,E ₃,E ₄. Then, for each functionH(δ) such thatH(δ)→ + ∞ andδH(δ)→0 asδ→0, there exists a crinkly functionf possessing the following five properties:

1. a.e. onE ₁:D ⁺ f(x)=D-f(x)=+∞,D ₊ f(x)=D?f(x)=?∞;
2. a.e. onE ₂:D ⁺ f(x)=+∞,D?f(x)=?∞,D ₊f(x)=D-f(x)=0;
3. a.e. onE ₃:D ₊ f(x)=?∞,D ^? f(x)=+∞,D ⁺ f(x)=D?f(x)=0;
4. a.e. onE ₄:Df(x)=0;
5. the modulus of continuityΩ off on [0, 1] satisfies $$\omega (\delta ,f,[0,1]) \leqq \delta H(\delta ).$$