Infinitely many Gerretsen-Blundon style quadratic inequalities, all strongest in Blundon’s sense

Blundon has proved that if R, r and s are respectively the circumradius, the inradius and the semiperimeter of a triangle, then the strongest possible inequalities of the form q(R, r) ≤ s ² ≤ Q(R, r) that hold for all triangles becoming equalities for the equilaterals where q, Q real quadratic forms, occur for the Gerretsen forms q _B (R, r) = 16Rr ? 5r ² and Q _B (R, r) = 4R ² + 4Rr + 3r ²; strongest in the sense that if Q is a quadratic form and s ² ≤ Q(R, r) ≤ Q _B (R, r) for all triangles then Q(R, r) = Q _B (R, r), and similarly for q _B (R, r). In this paper we prove that Q _B (resp. q _B ) is just one of infinitely many forms that appear as minimal (resp. maximal) elements in the partial order induced by the comparability relation in a certain set of forms, and we conclude that all these minimal forms are strongest in Blundon’s sense. We actually find all possible such strongest forms. Moreover we find all possible quadratic forms q, Q for which q(R, r) ≤ s ² ≤ Q(R, r) for all triangles and which hold as equalities for the equilaterals.     

Positive values of non-homogeneous indefinite quadratic forms II

The minimum Γ_{r, r + 1} of positive values of non-homogeneous indefinite quadratic forms of type (r, r + 1), r ≥ 2, is determined. For r = 2, an isolation theorem is proved which is used in the result for r ≥ 3. Some asymmetric inequalities for forms of type (2, 2) needed for these results are also obtained.     

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).     

A Galerkin Approximation for the Initial-Value Problem for Linear Second-Order Differential Equations

In this paper, a Galerkin type algorithm is given for the numerical solution of L(x)=(r(t)x'(t))'-p(t)x(t)=g(t); x(a)=x_a, x'(a)=x'_a, where r (t)>f0, and Spline hat functions form the approximating basis. Using the related quadratic form, a two-step difference equation is derived for the numerical solutions. A discrete Gronwall type lemma is then used to show that the error at the node points satisfies e_k=0(h²). If e(t) is the error function on a?t?b; it is also shown (in a variety of norms) that e(t)?Ch² and e'(t)?C₁h. Test case runs are also included. A (one step) Richardson or Rhomberg type procedure is used to show that e^R_k=0(h⁴). Thus our results are comparable to Runge-Kutta with half the function evaluations.     

Density of positive rank fibers in elliptic fibrations

A question of Mazur asks whether for any non-constant elliptic fibration {Er}_r∈Q, the set {r∈Q:rank(Er(Q))>0}, if infinite, is dense in R (with respect to the Euclidean topology). This has been proved to be true for the family of quadratic twists of a fixed elliptic curve by a quadratic or a cubic polynomial. Here we settle Mazur's question affirmatively for the general quadratic and cubic fibrations. Moreover we show that our method works when Q is replaced by any real number field.     

On the h-triangles of sequentially (S r ) simplicial complexes via algebraic shifting

Recently, Haghighi, Terai, Yassemi, and Zaare-Nahandi introduced the notion of a sequentially (S _r ) simplicial complex. This notion gives a generalization of two properties for simplicial complexes: being sequentially Cohen–Macaulay and satisfying Serre’s condition (S _r ). Let Δ be a (d?1)-dimensional simplicial complex with Γ(Δ) as its algebraic shifting. Also let (h _i,j (Δ))_{0≤j≤i≤d} be the h-triangle of Δ and (h _i,j (Γ(Δ)))_{0≤j≤i≤d} be the h-triangle of Γ(Δ). In this paper, it is shown that for a Δ being sequentially (S _r ) and for every i and j with 0≤j≤i≤r?1, the equality h _i,j (Δ)=h _i,j (Γ(Δ)) holds true.     

Formes antihermitiennes devenant hyperboliques sur un corps de déploiement

Let H=(a,b)_F be a division quaternion algebra over a field F of characteristic not 2. Denote by τ the canonical involution on H and by K a splitting field of H. If h is a skew-hermitian form over (H,τ) then, by extension of scalars to K and by Morita equivalence, we obtain a quadratic form h_K over K. This gives a map of Witt groups ρ:W⁻¹(H,τ)→W(K) induced by ρ(h)=h_K. When K is a generic splitting field of H we prove in this note that the map ρ is injective.     

Rekurrente zahlentheoretische Funktionen in mehreren Variablen

Some results of C. Methfessel (Multiplicative and additive recurrent sequences, Arch. Math. 63, 321–328 (1994)) on recurrent multiplicative arithmetical functions are generalized to the case of several variables. As ?[X ₁,…, X _r ] for r ≧ 2 is no principal ideal domain it is necessary to prove some elementary algebraic facts on the calculation of recurrent functions in several variables. The main tool is a vector space duality between ?[X ₁,…, X _r ] and the space of all functions f : ?^r → ?. This gives a correspondence between spaces of recurrent functions and ideals in ?[X ₁,…, X _r ]. The practical calculation can be done using Groebner bases. It is proved that a multiplicative function g in r variables which satisfies enough recurrences is of the form g (n ₁…, n _r )=n ₁ ^l … n _r ^rl h (n ₁ …, n _r ), with h multiplicative and periodic in each variable.     

Some optimization problems with applications to canonical correlations and sphericity tests

Optimization problems are connected with maximization of three functions, namely, geometric mean, arithmetic mean and harmonic mean of the eigenvalues of (X′ΣX)^?1ΣY(Y′ΣY)^?1Y′ΣX, where Σ is positive definite, X and Y are p × r and p × s matrices of ranks r and s (≥r), respectively, and X′Y = 0. Some interpretations of these functions are given. It is shown that the maximum values of these functions are obtained at the same point given by X = (h₁ + ?₁h_p, …, h_r + ?_rh_p?r+1) and $\text{Y = (h}{}_1\text{? ?}{}_1\text{h}{}_{\mathrm{p}}\text{, …, h}{}_{\mathrm{r}}\text{? ?}{}_{\mathrm{r}}\text{h}{}_{\mathrm{p?r+1}}\text{,}\text{Y}{}_{\mathrm{r+1}}\text{, …,}\text{Y}{}_{\mathrm{s}}\text{)}$, where h₁, …, h_p are the eigenvectors of Σ corresponding to the eigenvalues λ₁ ≥ λ₂ ≥ … ≥ λ_p > 0, ?_j = +1 or ?1 for j = 1,2,…, r and $\text{Y}{}_{\mathrm{r+1}}\text{, …,}\text{Y}{}_{\mathrm{s}}$, are linear functions of h_r+1,…, h_p?r. These results are extended to intermediate stationary values. They are utilized in obtaining the inequalities for canonical correlations θ₁,…,θ_r and they are given by expressions (3.8)–(3.10). Further, some new union-intersection test procedures for testing the sphericity hypothesis are given through test statistics (3.11)–(3.13).     

On the generalized lower bound conjecture for polytopes and spheres

In 1971, McMullen and Walkup posed the following conjecture, which is called the generalized lower bound conjecture: If P is a simplicial d-polytope then its h-vector (h ₀, h ₁, …, h _d ) satisfies $ {h_0}\leq {h_1}\leq \ldots \leq {h_{{\left\lfloor {{d \left/ {2} \right.}} \right\rfloor }}} $ . Moreover, if h _r?1 = h _r for some $ r\leq \frac{1}{2}d $ then P can be triangulated without introducing simplices of dimension ≤d ? r. The first part of the conjecture was solved by Stanley in 1980 using the hard Lefschetz theorem for projective toric varieties. In this paper, we give a proof of the remaining part of the conjecture. In addition, we generalize this result to a certain class of simplicial spheres, namely those admitting the weak Lefschetz property.     

Asymptotic behavior of number fields with prescribed ℓ-class numbers

Let K be a cyclic Galois extension of the rational numbers Q of degree ?, where ? is a prime number. Let h_? denote the order of the Sylow ?-subgroup of the ideal class group of K. If h_? = ?^s(s ≥ 0), it is known that the number of (finite) primes that ramify in K/Q is at most s + 1 (or s + 2 if K is real quadratic). This paper shows that “most” of these fields K with h_? = ?^s have exactly s + 1 ramified primes (or s + 2 ramified primes if K is real quadratic). Furthermore the Sylow ?-subgroup of the ideal class group is elementary abelian when h_? = ?^s and there are s + 1 ramified primes (or s + 2 ramified primes if K is real quadratic).     

Equivalent conditions for noncentral generalized Laplacianness and independence of matrix quadratic forms

Let Y be an n×p multivariate normal random matrix with general covariance Σ_Y and W be a symmetric matrix. In the present article, the property that a matrix quadratic form Y^′WY is distributed as a difference of two independent (noncentral) Wishart random matrices is called the (noncentral) generalized Laplacianness (GL). Then a set of algebraic results are obtained which will give the necessary and sufficient conditions for the (noncentral) GL of a matrix quadratic form. Further, two extensions of Cochran’s theorem concerning the (noncentral) GL and independence of a family of matrix quadratic forms are developed.     

Rings of integral quaternions

A quaternion order derived from an integral ternary quadratic form f = Σa_ijx_ix_j of discriminant d = 4 det (a_ij) is m-maximal if m is not divisible by any prime p such that p² | d, or p 6; d and c_p = 1. If R is m-maximal and m is a product p₁ … p_r of primes, then any primitive element α of R has unique right-divisor ideals of each norm p₁ … p_k (k = 1, …, r). This generalizes Lipschitz's ninety-year-old theorem. We characterize m-maximal orders, study their ideals, and show how the preceding result yields formulas for the number of representations of integers by certain quaternary quadratic forms.     

Band matrices with Toeplitz inverses

It is shown that a square band matrix H=(h_ij) with h_ij=0 for j? i>r and i?j>s, where r+s is less than the order of the matrix, has a Toeplitz inverse if and only if it has a special structure characterized by two polynomials of degrees r and s, respectively.     

Birational composition of quadratic forms over a local field

Let f(X) and g(Y) be nondegenerate quadratic forms of dimensions m and n, respectively, over K, char K ≠ 2. The problem of birational composition of f(X) and g(Y) is considered: When is the product f(X) · g(Y) birationally equivalent over K to a quadratic form h(Z) over K of dimension m + n? The solution of the birational composition problem for anisotropic quadratic forms over K in the case of m = n = 2 is given. The main result of the paper is the complete solution of the birational composition problem for forms f(X) and g(Y) over a local field P, char P ≠ 2.     

Nonlinear eigenvalue problems and spherical fibrations of Banach spaces

Eigenvalue problems of the form g′(v) = λh′(v) are considered, with the normalizations g(v) = r or h(v) = r, where g and h are real-valued C¹ functions on a real Banach space which are invariant under a periodic linear isometry. Theorems are proved on the existence of solutions λ(r), v(r), and on their dependence upon the normalization constant r > 0. In particular, the relation, as r → 0, of λ(r), v(r) to solutions of the linearized problem g″(0)v = λh″(0)v is discussed. The theorems are applied to elliptic problems for Euler-Lagrange operators corresponding to multiple integral functionals on closed subspaces of Sobolev spaces.     

The parabolic map f1/e(z)=(1/e)e

We consider the exponential maps ?_λ : ? → ? defined by the formula ?_λ (z) = λe^z, λ(0,1/e]. Let J_r(?_λ) be the subset of the Julia set consisting of points that do not escape to infinity under forward iterates of ?. Our main result is that the function λh_λ :=HD(J_r(?_λ),)), λ(0, 1/e], is continuons at the point 1/e. As a preparation for this result we deal with the map ?₁/e itself. We prove that the h₁/e-dimensional Hausdorff measure of J_r(?₁/e) is positive and finite on each horizontal strip, and that the h₁/e-dimensional packing measure of J_r(?_λ) is locally infinite at each point of J_r(?_λ). Our main technical devices are formed by the, associated with ?_λ, maps F_λ defined on some strip P of height 2π and also associated with them tonformal measures.     

h-Cobordisms between simply connected 4-manifolds

h-cobordisms between simply connected 4-manifolds are studied. It is shown that most inertial h-cobordisms have a handle decomposition with one 2-handle and one 3-handle, and h-cobordisms between nondiffeomorphic manifolds have handle decompositions with the minimal number of handles consistent with a diffeomorphism between the stabilized ends. Also the number of distinct h-cobordisms between two fixed manifolds is described in terms of isomorphisms of their quadratic forms. These results are applied to Dolgachev surfaces and the Kummer surface using recent work of Donaldson, Friedman and Morgan, and Matumoto.     

Dihedral Congruence Primes and Class Fields of Real Quadratic Fields

We show that for a real quadratic field F the dihedral congruence primes with respect to F for cusp forms of weight k and quadratic nebentypus are essentially the primes dividing expressions of the form ε^k−1₊±1 where ε₊ is a totally positive fundamental unit of F. This extends work of Hida. Our results allows us to identify a family of (ray) class fields of F which are generated by torsion points on modular abelian varieties.     

A modularity criterion for Klein forms, with an application to modular forms of level 13

We find some modularity criterion for a product of Klein forms of the congruence subgroup Γ₁(N) (Theorem 2.6) and, as its application, construct a basis of the space of modular forms for Γ₁(13) of weight 2 (Example 3.4). In the process we face with an interesting property about the coefficients of certain theta function from a quadratic form and prove it conditionally by applying Hecke operators (Proposition 4.3).