Knauf’s degree and monodromy in planar potential scattering

We consider Hamiltonian systems on (T*?², dq ∧ dp) defined by a Hamiltonian function H of the “classical” form H = p ²/2 + V(q). A reasonable decay assumption V(q) → 0, ‖q‖ → ∞, allows one to compare a given distribution of initial conditions at t = ?∞ with their final distribution at t = +∞. To describe this Knauf introduced a topological invariant deg(E), which, for a nontrapping energy E > 0, is given by the degree of the scattering map. For rotationally symmetric potentials V(q) = W(‖q‖), scattering monodromy has been introduced independently as another topological invariant. In the present paper we demonstrate that, in the rotationally symmetric case, Knauf’s degree deg(E) and scattering monodromy are related to one another. Specifically, we show that scattering monodromy is given by the jump of the degree deg(E), which appears when the nontrapping energy E goes from low to high values.     

A Schur complement approach to preconditioning sparse linear least-squares problems with some dense rows

The effectiveness of sparse matrix techniques for directly solving large-scale linear least-squares problems is severely limited if the system matrix A has one or more nearly dense rows. In this paper, we partition the rows of A into sparse rows and dense rows (A_s and A_d) and apply the Schur complement approach. A potential difficulty is that the reduced normal matrix AsTA_s is often rank-deficient, even if A is of full rank. To overcome this, we propose explicitly removing null columns of A_s and then employing a regularization parameter and using the resulting Cholesky factors as a preconditioner for an iterative solver applied to the symmetric indefinite reduced augmented system. We consider complete factorizations as well as incomplete Cholesky factorizations of the shifted reduced normal matrix. Numerical experiments are performed on a range of large least-squares problems arising from practical applications. These demonstrate the effectiveness of the proposed approach when combined with either a sparse parallel direct solver or a robust incomplete Cholesky factorization algorithm.     

A Note on MDS Codes, <Emphasis Type="Italic">n</Emphasis>-Arcs and Complete Designs

A generalized incidence matrix of a design over GF(q) is any matrix obtained from the (0, 1)-incidence matrix by replacing ones with nonzero elements from GF(q). The dimension d _q of a design D over GF(q) is defined as the minimum value of the q-rank of a generalized incidence matrix of D. It is proved that the dimension d _q of the complete design on n points having as blocks all w-subsets, is greater that or equal to n ? w + 1, and the equality d _q = n ? w + 1 holds if and only if there exists an [n, n ? w + 1, w] MDS code over GF(q), or equivalently, an n-arc in PG(w ? 2, q).     

Scheduling Independent Tasks with Deadlines on Semi-identical Processors

Given m semi-identical processors which are parallel processors all working with the same speed but in different time intervals of availability and n independent tasks with deadlines, the problem of constructing a feasible pre-emptive schedule is examined. We present an O (nm log n) time algorithm to construct such a schedule whenever one exists. We show that the number of induced pre-emptions is proportional to the total number of processing intervals and deadlines.     

On collineation groups of finite projective spaces containing a Singer cycle

By a result of Kantor, any subgroup of GL(n, q) containing a Singer cycle normalizes a field extension subgroup. This result has as a consequence a projective analogue, and this paper gives the details of this deduction, showing that any subgroup of PΓL(n, q) containing a projective Singer cycle normalizes the image of a field extension subgroup GL(n/s, q^s) under the canonical homomorphism GL(n, q) → PGL(n, q), for some divisor s of n, and so is contained in the image of ΓL(n/s, q^s) under the canonical homomorphism ΓL(n, q) → PΓL(n, q). The actions of field extension subgroups on V (n, q) are also investigated. In particular, we prove that any field extension subgroup GL(n/s, q^s) of GL(n, q) has a unique orbit on s-dimensional subspaces of V (n, q) of length coprime to q. This orbit is a Desarguesian s-partition of V (n, q).     

On nonabelian composition factors of a finite prime spectrum minimal group

Assume that G is a primitive permutation group on a finite set X, x ∈ X, y ∈ X \ {x}, and G _x,y \(\underline \triangleleft \) G _x . P. Cameron raised the question about the validity of the equality G _x,y = 1 in this case. The author proved earlier that, if soc(G) is not a direct power of an exceptional group of Lie type, then G _x,y = 1. In the present paper, we prove that, if soc(G) is a direct power of an exceptional group of Lie type distinct from E ₆(q), ² E ₆(q), E ₇(q), and E ₈(q), then G _x,y = 1.     

A mapping method in inverse Sturm-Liouville problems with singular potentials

In the space L ₂[0, π], the Sturm-Liouville operator L _D(y) = ?y″ + q(x)y with the Dirichlet boundary conditions y(0) = y(π) = 0 is analyzed. The potential q is assumed to be singular; namely, q = σ′, where σ ∈ L ₂[0, π], i.e., q ∈ W ₂ ^?1 [0, π]. The inverse problem of reconstructing the function σ from the spectrum of the operator L _D is solved in the subspace of odd real functions σ(π/2 ? x) = ?σ(π/2 + x). The existence and uniqueness of a solution to this inverse problem is proved. A method is proposed that allows one to solve this problem numerically.     

Schur positivity and the <Emphasis Type="Italic">q</Emphasis>-log-convexity of the Narayana polynomials

We prove two recent conjectures of Liu and Wang by establishing the strong q-log-convexity of the Narayana polynomials, and showing that the Narayana transformation preserves log-convexity. We begin with a formula of Brändén expressing the q-Narayana numbers as a specialization of Schur functions and, by deriving several symmetric function identities, we obtain the necessary Schur-positivity results. In addition, we prove the strong q-log-concavity of the q-Narayana numbers. The q-log-concavity of the q-Narayana numbers N _q (n,k) for fixed k is a special case of a conjecture of McNamara and Sagan on the infinite q-log-concavity of the Gaussian coefficients.     

Seifert manifolds and (1, 1)-knots

The aim of this paper is to investigate the relations between Seifert manifolds and (1, 1)-knots. In particular, we prove that each orientable Seifert manifold with invariants

$\{ Oo,0| - 1;\underbrace {(p,q),...,(p,q)}_{n times},(l,l - 1)\} $

has the fundamental group cyclically presented by G _n ((x ₁ ^q ...x _n ^{q l} x _n ^?p ) and, moreover, it is the n-fold strongly-cyclic covering of the lens space L(|nlq ? p|, q) which is branched over the (1, 1)-knot K(q, q(nl ? 2), p ? 2q, p ? q) if p ≥ 2q and over the (1, 1)-knot K(p? q, 2q ? p, q(nl ? 2), p ? q) if p< 2q.

     

Recognition of some families of finite simple groups by order and set of orders of vanishing elements

Let G be a finite group. An element g ∈ G is called a vanishing element if there exists an irreducible complex character χ of G such that χ(g)= 0. Denote by Vo(G) the set of orders of vanishing elements of G. Ghasemabadi, Iranmanesh, Mavadatpour (2015), in their paper presented the following conjecture: Let G be a finite group and M a finite nonabelian simple group such that Vo(G) = Vo(M) and |G| = |M|. Then G ≌ M. We answer in affirmative this conjecture for M = Sz(q), where q = 2²ⁿ⁺¹ and either q ? 1, \(q - \sqrt {2q} + 1\) or q + \(\sqrt {2q} + 1\) is a prime number, and M = F⁴(q), where q = 2 ⁿ and either q⁴ + 1 or q⁴ ? q² + 1 is a prime number.     

Finite Groups with a Given Set of Character Degrees

Let G be a finite group with {1, q, r, qm, q ² m, rqm} as the character degree set, where r and q are distinct primes and m>1 is an integer not divisible by q or r. We show that G is solvable and the derived length of G equals 3.     

ON THE COADJOINT ORBITS OF MAXIMAL UNIPOTENT SUBGROUPS OF REDUCTIVE GROUPS

Let G be a simple algebraic group defined over an algebraically closed field of characteristic 0 or a good prime for G. Let U be a maximal unipotent subgroup of G and \( \mathfrak{u} \) its Lie algebra. We prove the separability of orbit maps and the connectedness of centralizers for the coadjoint action of U on (certain quotients of) the dual \( \mathfrak{u} \)* of \( \mathfrak{u} \). This leads to a method to give a parametrization of the coadjoint orbits in terms of so-called minimal representatives which form a disjoint union of quasi-affine varieties. Moreover, we obtain an algorithm to explicitly calculate this parametrization which has been used for G of rank at most 8, except E₈.When G is defined and split over the field of q elements, for q the power of a good prime for G, this algorithmic parametrization is used to calculate the number k(U(q); \( \mathfrak{u} \)*(q)) of coadjoint orbits of U(q) on \( \mathfrak{u} \)*(q). Since k(U(q), \( \mathfrak{u} \)*(q)) coincides with the number k(U(q)) of conjugacy classes in U(q), these calculations can be viewed as an extension of the results obtained in [11]. In each case considered here there is a polynomial h(t) with integer coefficients such that for every such q we have k(U(q)) = h(q). We also explain implications of our results for a parametrization of the irreducible complex characters of U(q).     

New characterization of <Emphasis Type="Italic">S</Emphasis><Subscript>4</Subscript>(<Emphasis Type="Italic">q</Emphasis>) by its noncommuting graph

Let G be a nonabelian group, and associate the noncommuting 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 S ₄(q) be the projective symplectic simple group, where q is a prime power. We prove that if G is a group with ?(G) ? ?(S ₄(q)) then G ? S ₄(q).     

Blow-up of <Emphasis Type="Italic">p</Emphasis>-Laplacian evolution equations with variable source power

We study the blow-up and/or global existence of the following p-Laplacian evolution equation with variable source power

$${s_j} = {\beta _j} + \overline {{\beta _{n - j}}}p$$

where Ω is either a bounded domain or the whole space ? ^N , q(x) is a positive and continuous function defined in Ω with 0 < q _? = inf q(x) ? q(x) ? sup q(x) = q+ < ∞. It is demonstrated that the equation with variable source power has much richer dynamics with interesting phenomena which depends on the interplay of q(x) and the structure of spatial domain Ω, compared with the case of constant source power. For the case that Ω is a bounded domain, the exponent p ? 1 plays a crucial role. If q+ > p ? 1, there exist blow-up solutions, while if q ₊ < p ? 1, all the solutions are global. If q _? > p ? 1, there exist global solutions, while for given q _? < p ? 1 < q ₊, there exist some function q(x) and Ω such that all nontrivial solutions will blow up, which is called the Fujita phenomenon. For the case Ω = ? ^N , the Fujita phenomenon occurs if 1 < q _? ? q ₊ ? p ? 1 + p/N, while if q _? > p ? 1 + p/N, there exist global solutions.

     

Hochschild cohomology ring modulo nilpotence of a one point extension of a quiver algebra defined by two cycles and a quantum-like relation

We consider a one point extension algebra B of a quiver algebra A _q over a field k defined by two cycles and a quantum-like relation depending on a nonzero element q in k. We determine the Hochschild cohomology ring of B modulo nilpotence and show that if q is a root of unity, then B is a counterexample to Snashall-Solberg’s conjecture.     

On recognizability of some finite simple orthogonal groups by spectrum

It is proved that, if G is a finite group that has the same set of element orders as the simple group D _p (q), where p is prime, p ≥ 5 and q ∈ {2, 3, 5}, then the commutator group of G/F(G) is isomorphic to D _p (q), the subgroup F(G) is equal to 1 for q = 5 and to O _q (G) for q ∈ {2, 3}, F(G) ≤ G′, and |G/G′| ≤ 2.     

Diffraction of sound pulses by a rigid cylinder in an inhomogeneous medium

We consider the problem of scattering of two-dimensional sound pulses by a rigid circular cylinder embedded in a cylindrically stratified inhomogeneous medium. The line source is parallel to the axis of the cylinder. It is assumed that the velocity of soundc is given byc ^?1=pr ^q, wherep andq are real constants andp>0. The method of dual integral transformation developed by Friedlander is used. The solution in terms of pulse propagation modes gives the diffracted pulse and the method of steepest descents yields the geometrical acoustic field.     

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).     

The Einstein-Kähler metric on the third Cartan-Hartogs domain

In this paper we discuss the Einstein-Kahler metric on the third Cartan-Hartogs domain Y111（n, q; K）. Firstly we get the complete Einstein Kahler metric with explicit form on Y111（n, q; K） in the case of K=q/2 ＋ 1/q-1. Secondly we obtain the holomorphic sectional curvature under this metric and get the sharp estimate for this holomorphic curvature. Finally we prove that the complete Einstein-Kahler metric is equivalent to the Bergman metric on Y111（n, q; K） in case of K=q/2＋1/q-1.