The equivalence between ( AB )†= B † A † and other mixed-type reverse-order laws

The standard reverse-order law for the Moore–Penrose inverse of a matrix product is (AB)^??=?B ^? A ^?. The purpose of this article is to give a set of equivalences of this reverse-order law and other mixed-type reverse-order laws for the Moore–Penrose inverse of matrix products.     

Explicit solutions to the reverse order law (AB)+ = B-mrA-lr

Shinozaki and Sibuya have shown that the Moore-Penrose inverse (AB)⁺ can always be expressed as B^-A^- for generalized inverses A^- and B^- of matrices A and B, respectively. In this paper, explicit solutions B^-_mr and A^-_lr to (AB)⁺ = B^-_mrA^-_lr are given. A class of solutions is obtained which is related to an equation of Greville, and expressions for the general solutions are presented.     

Weak solution of (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0))u′(0) = ku(0), r(L, u(L))u′ (L) = hu(L) and k, h are suitable elements of [0, ∞]

We consider weak solutions to the nonlinear boundary value problem (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0)) u′(0) = ku(0), r(L, u(L)) u′(L) = hu(L) and k, h are suitable elements of [0, ∞]. In addition to studying some new boundary conditions, we also relax the constraints on r(x, u) and (Fu)(x). r(x, u) > 0 may have a countable set of jump discontinuities in u and r(x, u)^?1?L_q((0, L) × (0, p)). F is an operator from a suitable set of functions to a subset of L_p(0, L) which have nonnegative values. F includes, among others, examples of the form (Fu)(x) = (1 ? H(x ? x₀)) u(x₀), (Fu)(x) = ∫_x^Lf(y, u(y)) dy where f(y, u) may have a countable set of jump discontinuities in u or F may be chosen so that (Fu)′(x) = ? g(x, u(x)) u′(x) ? q(x) u(x) ? f(x, u(x)) where q is a distributional derivative of an L₂(0, L) function.     

Exact solutions of nonlinear dispersive K(m, n) model with variable coefficients

The variable-coefficient Korteweg-de Vries (KdV) equation with additional terms contributed from the inhomogeneity in the axial direction and the strong transverse confinement of the condense was presented to describe the dynamics of nonlinear excitations in trapped quasi-one-dimensional Bose-Einstein condensates with repulsive atom-atom interactions. To understand the role of nonlinear dispersion in this variable-coefficient model, we introduce and study a new variable-coefficient KdV with nonlinear dispersion (called vc-K(m, n) equation). With the aid of symbolic computation, we obtain its compacton-like solutions and solitary pattern-like solutions. Moreover, we also present some conservation laws for both vc-K⁺(n, n) equation and vc-K⁻(n, n) equation.     

Eigenvalue bounds for AB+BA, with A,B positive definite matrices

Bounds are derived for the eigenvalues of the Hermitian matrix C given by C=AB+BA, where A and B are positive definite, Hermitian, complex matrices. A sufficient condition is given for C to be positive definite.     

L(μ)-averaging domains

In this paper we first introduce L^s(μ)-averaging domains which are generalizations of L^s-averaging domains introduced by S.G. Staples. We characterize L^s(μ)-averaging domains using the quasihyperbolic metric. As applications, we prove norm inequalities for conjugate A-harmonic tensors in L^s(μ)-averaging domains which can be considered as generalizations of the Hardy and Littlewood theorem for conjugate harmonic functions. Finally, we give applications to quasiconformal and quasiregular mappings.     

Soliton solutions of a BBM(m, n) equation with generalized evolution

We consider a BBM(m, n) equation which is a generalization of the celebrated Benjamin-Bona-Mahony equation with generalized evolution term. By using two solitary wave ansatze in terms of sech^p(x) and tanh^p(x) functions, we find exact analytical bright and dark soliton solutions for the considered model. The physical parameters in the soliton solutions are obtained as function of the dependent model coefficients. The conditions of existence of solitons are presented. Note that, it is always useful and desirable to construct exact analytical solutions especially soliton-type envelope for the understanding of most nonlinear physical phenomena.     

An umbral calculus for polynomials characterizing U(n) tensor operators

After the change of variables Δ_i = γ_i ? δ_i and x_{i,i + 1} = δ_i ? δ_{i + 1} we show that the invariant polynomials _μG⁽ⁿ⁾_q(, Δ_i, ; , x_i,i+1,) characterizing U(n) tensor operators 〈p, q,…, q, 0,…, 0〉 become an integral linear combination of Schur functions S_λ(γ ? δ) in the symbol γ ? δ, where γ ? δ denotes the difference of the two sets of variables {γ₁ ,…, γ_n} and {δ₁ ,…, δ_n}. We obtain a similar result for the yet more general bisymmetric polynomials ^m_μG⁽ⁿ⁾_q(γ₁ ,…, γ_n; δ₁ ,…, δ_m). Making use of properties of skew Schur functions $\text{S}{}_{\mathrm{<mtext>\lambda </mtext><mtext>\rho </mtext>}}$ and S_λ(γ ? δ) we put together an umbral calculus for ^m_μG⁽ⁿ⁾_q(γ; δ). That is, working entirely with polynomials, we uniquely determine ^m_μG⁽ⁿ⁾_q(γ; δ) from ^m_μG⁽ⁿ⁾_{q ? 1}(γ; δ) and combinatorial rules involving Ferrers diagrams (i.e., partitions), provided that n ≥ (μ + 1)q. (This restriction does not interfere with writing the general case of ^m_μG⁽ⁿ⁾_q(γ; δ) as a linear combination of S_λ(γ ? δ).) As an application we deduce “conjugation” symmetry for ⁿ_μG⁽ⁿ⁾_q(γ; δ) from “transposition” symmetry by showing that these two symmetries are equivalent.     

Uniqueness of norm on L(G) and C(G) when G is a compact group

Let G be a compact group. If the trivial representation of G is not weakly contained in the left regular representation of G on L₀²(G) and X is either L^p(G) for 1<p?∞ or C(G), then we show that every complete norm |·| on X that makes translations from (X,|·|) into itself continuous is equivalent to ||·||_p or ||·||_∞ respectively. If 1<p?∞ and every left invariant linear functional on L^p(G) is a constant multiple of the Haar integral, then we show that every complete norm |·| on L^p(G) that makes translations from (L^p(G),|·|) into itself continuous and that makes the map t?L_t from G into bounded is equivalent to ||·||_p.     

Translation-invariant linear forms on C0(G), C(G), Lp(G) for noncompact groups

Suppose G is a locally compact noncompact group. For abelian such G's, it is shown in this paper that L¹(G), C(G), and L^∞(G) always have discontinuous translation-invariant linear forms(TILF's) while C₀(G) and L^p(G) for 1 < p < ∞ have such forms if and only if $\text{G}\text{H}$ is a torsion group for some open σ-compact subgroup H of $\text{G}$. For σ-compact amenable G's, all the above spaces have discontinuous left TILF's.     

Factorization of Q(h(T)(x)) over a finite field where Q(x) is irreducible and h(T)(x) is linear—I

Let GF(q) be the finite field of order q, let Q(x) be an irreducible polynomial in GF(q)(x), and let h(T)(x) be a linear polynomial in GF(q)[x], where T:x→x^q. We use properties of the linear operator h(T) to give conditions for Q(h(T)(x)) to have a root of arbitrary degree k over GF(q), and we describe how to count the irreducible factors of Q(h(T)(x)) of degree k over GF(q). In addition we compare our results with those Ore which count the number of irreducible factors belonging to a linear polynomial having index k.     

An elementary operator with log-hyponormal, p-hyponormal entries

A Hilbert space operator A∈B(H) is p-hyponormal, A∈(p-H), if |A^∗|^2p?|A|^2p; an invertible operator A∈B(H) is log-hyponormal, A∈(?-H), if log(TT^∗)?log(T^∗T). Let d_AB=δ_AB or ?_AB, where δ_AB∈B(B(H)) is the generalised derivation δ_AB(X)=AX-XB and ?_AB∈B(B(H)) is the elementary operator ?_AB(X)=AXB-X. It is proved that if A,B^∗∈(?-H)∪(p-H), then, for all complex λ, , the ascent of (d_AB-λ)?1, and d_AB satisfies the range-kernel orthogonality inequality ‖X‖?‖X-(d_AB-λ)Y‖ for all X∈(d_AB-λ)^-1(0) and Y∈B(H). Furthermore, isolated points of σ(d_AB) are simple poles of the resolvent of d_AB. A version of the elementary operator E(X)=A₁XA₂-B₁XB₂ and perturbations of d_AB by quasi-nilpotent operators are considered, and Weyl’s theorem is proved for d_AB.     

Weak barrelledness for C(X) spaces

Buchwalter and Schmets reconciled C_c(X) and C_p(X) spaces with most of the weak barrelledness conditions of 1973, but could not determine if -barrelled ⇔ ?^∞-barrelled for C_c(X). The areas grew apart. Full reconciliation with the fourteen conditions adopted by Saxon and Sánchez Ruiz needs their 1997 characterization of Ruess' property (L), which allows us to reduce the C_c(X) problem to its 1973 status and solve it by carefully translating the topology of Kunen (1980) and van Mill (1982) to find the example that eluded Buchwalter and Schmets. The more tractable C_p(X) readily partitions the conditions into just two equivalence classes, the same as for metrizable locally convex spaces, instead of the five required for C_c(X) spaces. Our paper elicits others, soon to appear, that analytically characterize when the Tychonov space X is pseudocompact, or Warner bounded, or when C_c(X) is a df-space (Jarchow's 1981 question).     

Zeros of the functions f(n) = Σi=0(−1)i(in−2i)

The main result of this paper is the following: the only zeros of the title function are at n = 3 and n = 12. This is achieved by means of the recursion function for f(n), viz. F(x) = x³ ? x ? 1 which has only one real root w. This turns out to be the fundamental unit of Q(w). From the norm equation of the units, N(w) = x³ + y³ + z³ ? 3xyz + 2x²z + xz² ? xy² ? yz² = 1, and the negative powers of w which are of binary form, the result follows. The paper concludes with two remarkable combinatorial identities.     

Sp(2N)-covers for self-contragredient supercuspidal representations of GL(N)

Let F be a non-archimedean local field of odd residual characteristic. Let (J,τ) be a maximal simple type in GLN(F) for the inertial class [GLN(F),π]_GLN(F) of a self-contragredient supercuspidal irreducible representation π of GLN(F). Identify GLN(F) to the standard Siegel Levi subgroup in Sp2N(F). We construct, in Sp2N(F), a type for the inertial class [GLN(F),π]_Sp2N(F), as a Sp2N(F)-cover of (J,τ), strongly related to the GL2N(F)-cover of (J×J,τ⊗τ) in GL2N(F) constructed by Bushnell and Kutzko and which induces to a simple type in GL2N(F). In the process, we show that if τ has positive level, then the maximal simple type (J,τ) may be attached to a simple stratum [A,n,0,β] where the field F[β] is a quadratic extension of F[β²], and to a simple character θ in C(A,0,β) Galois conjugate of its inverse.     

Some results concerning Lp(μ)-spaces

It is proved that L^p(μ) does not have an unconditional basis if the cardinality of L^p(μ) is sufficiently large and μ is a finite measure. It is also shown that L^p(μ) has a weaker kind of basis for arbitrary μ and 1 < p < ∞. A new truncation lemma concerning sequences in L^p equivalent to the usual l^p-basis is given. This lemma is used in solving the problem of when l^p(Γ) imbeds in L^r(μ) for uncountable sets Γ and finite measures μ. It may also be used to give a nonprobabilistic proof of the fact (due to Schwartz-Kwapien) that there exist non-q-absolutely summing operators from L^∞to L^qfor 2 < q < ∞. It is again used in proving that basic sequences in L^p equivalent to the usual l^p-basis admit subsequences with a complemented linear span. Other applications of the techniques introduced are also given.     

Soliton solutions for (2 + 1)-dimensional and (3 + 1)-dimensional K(m, n) equations

We consider the nonlinear dispersive K(m,n) equation with the generalized evolution term and derive analytical expressions for some conserved quantities. By using a solitary wave ansatz in the form of sech^p function, we obtain exact bright soliton solutions for (2 + 1)-dimensional and (3 + 1)-dimensional K(m,n) equations with the generalized evolution terms. The results are then generalized to multi-dimensional K(m,n) equations in the presence of the generalized evolution term. An extended form of the K(m,n) equation with perturbation term is investigated. Exact bright soliton solution for the proposed K(m,n) equation having higher-order nonlinear term is determined. The physical parameters in the soliton solutions are obtained as function of the dependent model coefficients.     

Decay estimates for oscillatory integrals with polynomial phase in terms of p and p or in terms of p and p

We obtain estimates for certain oscillatory integrals with polynomial (degree n) phase, p(t). These estimates are stated in terms of differences between the roots, real or complex, of p⁽ⁿ⁻³⁾(t) and p⁽ⁿ⁻²⁾(t) or between p⁽ⁿ⁻²⁾(t) and p⁽ⁿ⁻¹⁾(t). The sharpness of these results is also explored. This result is a partial generalization of the results found in [J. Math. Anal. Appl. 280 (2003) 424].     

Factorization of Q(h(T)(x)) over a finite field,where Q(x) is irreducible and h(T)(x) is linear II

Let GF(q) be the finite field of order q, let Q(x) be an irreducible polynomial in GF(qⁱ)[x], and let H(x) = h(T)(x) be a linear polynomial in GF(q)[x]. We give the degrees of the irreducible factors of Q(H(x)) in GF(qⁱ)[x], and the number of irreducible factors of each degree. We consider the special cases when H(x) is a trace function, and when h(x) is cyclotomic. Finally, we give several examples.