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).     

On quasigroups rich in associative triples

Let G be a group and $\text{G(1)}$ a quasigroup on the same underlying set. Let dist($\text{G, G(1)}$) denote the number of pairs (x, y) ?G² such that $\text{xy ≠ x 1 y}$. For a finite quasigroup Q, n = card(Q), let t = dist(Q) = min dist(G, Q), where G runs through all groups with the same underlying set, and s = s(Q) the number of non-associative triples. Then 4tn?2t²?24t?s?4tn. If 1 ? s < 3n²/32, then 3tn < s holds as well. Let n ? 168 be an even integer and let σ = min s(Q), where Q runs through all non-associative quasigroups of order n. Then σ = 16n?64.     

On a generalization of Minkowski's convex body theorem

Given a lattice $\text{Λ ? R}{}^{\mathrm{n}}$ and a bounded function g(x), x ∈ Rⁿ, vanishing outside of a bounded set, the functions ?(x)$\text{g}\text{?}\text{(x)?}$max_u∈Λg(u +x), ?(x)?Σ_{u∈Λ g(u +x)}, and ?₊(x)?Σ_u∈Λ max_{v∈Λ min {g(v + x); g(u + v + x)}} are defined and periodic mod Λ on Rⁿ. In the paper we prove that ?(x) + ?₊(x) ? 2?(x) ≥ ?(x) + h?₊(x) ? 2?(x) holds for all x ∈ Rⁿ, where h(x) is any “truncation” of g by a constant c ≥ 0, i.e., any function of the form h(x)?g(x) if g(x) ≤ c and h(x)?c if g(x) > c. This inequality easily implies some known estimations in the geometry of numbers due to Rado [1] and Cassels [2]. Moreover, some sharper and more general results are also derived from it. In the paper another inequality of a similar type is also proved.     

Boundary layer methods for certain nonlinear singularly perturbed optimal control problems

We shall examine the control problem consisting of the system $\text{dx}\text{dt}\text{= f}{}_1\text{(x, z, u, t, ?)}$$\text{?(}\text{dz}\text{dt}\text{) = f}{}_2\text{(x, z, u, t, ?)}$ on the interval 0 ? t ? 1 with the initial values x(0, ?) and z(0, ?) prescribed, where the cost functional J(?) = π(x(1, ?), z(1, ?), ?) + ∝₀¹V(x(t, ?), z(t, ?), u(t, ?), t, ?) dt is to be minimized. We shall restrict attention to the special problem where the f_i's are linear in z and u, V is quadratic in z and independent of z when ? = 0, π and V are positive semidefinite functions of x and z, and V is a positive definite function of u. Under appropriate conditions, we shall obtain an asymptotic solution of the problem valid as the small parameter ? tends to zero. The techniques of constructing such asymptotic expansions will be stressed.     

Optimality conditions for fractional minmax programming

Let V ?H be real separable Hilbert spaces. The abstract wave equation u′' + A(t)u = g(u), where u(t) ?V, A(t) maps V to its dual $\text{V}{}^1$, and g is a nonlinear map from the ball S(R₀) = {u?V: ∥u∥ < R₀} into H, is considered. It is assumed that g is locally Lipschitz in S(R₀) and possibly singular at the boundary. Local existence and continuation theorems are established for the Cauchy problem u(0) = u₀?S(R₀), u′(0) = u₁?H. Global existence is shown for g(u) = εφ(u), where φ has a potential and ε is small. Global nonexistence is shown for g(u) = εφ(u), where φ satisfies an abstract convexity property and ε is large.     

An extension of Tutte's 1-factor theorem

We prove the following theorem: Let G be a graph with vertex-set V and ?, g be two integer-valued functions defined on V such that $\text{0}\text{?g(x) ?1??(x)}$ for all x ∈ V. Then G contains a factor F such that $\text{g(x)?d}{}_{\mathrm{F}}\text{(x)??(x)}$ for all x ∈ V if and only if for every subset X of V, $\text{?(X)}$ is at least equal to the number of connected components C of G[V ? X] such that either C = {x} and g(x) = 1, or |C| is odd ?3 and $\text{g(x) = ?(x) =}\text{1}$ for all x ∈ C. Applications are given to certain combinatorial geometries associated with factors of graphs.     

Attainable sets for linear stochastic control systems

Let x_t^u(w) be the solution process of the n-dimensional stochastic differential equation dx_t^u = [A(t)x_t^u + B(t) u(t)] dt + C(t) dW_t, where A(t), B(t), C(t) are matrix functions, W_t is a n-dimensional Brownian motion and u is an admissable control function. For fixed ? ? 0 and 1 ? δ ? 0, we say that x?Rⁿ is (?, δ) attainable if there exists an admissable control u such that P{x_t^u?S_?(x)} ? δ, where S_?(x) is the closed ?-ball in Rⁿ centered at x. The set of all (?, δ) attainable points is denoted by $\text{A}$(t). In this paper, we derive various properties of $\text{A}$(t) in terms of K(t), the attainable set of the deterministic control system $\text{x}\text{?}\text{= A(t)x + B(t)u}$. As well a stochastic bang-bang principle is established and three examples presented.     

Existence and comparison theorems for nonlinear diffusion systems

In this paper we prove existence, uniqueness, and regularity results for systems of nonlinear second order parabolic equations with boundary conditions of the Dirichlet, Neumann, and regular oblique derivative types. Let K(t) consist of all functions (v¹(x), v²(x),…, v^m(x)) from $\text{Ω ? R}{}^{\mathrm{n}}$ into R^m which satisfy ψⁱ(x, t) ? vⁱ(x) ? θⁱ(x, t) for all $\text{x ? Ω}\text{and}\text{1 ? i ? m}$, where ψⁱand θⁱ are extended real-valued functions on $\text{}\text{?}\text{gW × [0, T)}$. We find conditions which will ensure that a solution U(x, t) ≡ (u¹(x, t), u²(x, t),…, u^m(x, t)) which satisfies U(x, 0) ?K(0) will also satisfy U(x, t) ?K(t) for all 0 ? t < T. This result, which has some similarity to the Gronwall Inequality, is then used to prove a global existence theorem.     

Polynomial sums over automorphs of a positive definite binary quadratic form

Let P(X) be a homogeneous polynomial in X = (x, y), Q(X) a positive definite integral binary quadratic form, and G the group of integral automorphs of Q(X). Let A(m) = {N ∈ $\text{Z}$ × $\text{Z}$ : Q(N) = m}. It is shown that if Σ_N∈A(m)P(N) = 0 for each m = 1, 2, 3,… then Σ_U∈GP(UX) ≡ 0.     

Arrangements in unitary and orthogonal geometry over finite fields

Let V be an n-dimensional vector space over F_q. Let Φ be a Hermitian form with respect to an automorphism σ with σ² = 1. If σ = 1 assume that q is odd. Let $\text{A}$ be the arrangement of hyperplanes of V which are non-isotropic with respect to Φ, and let L be the intersection lattice of $\text{A}$. We prove that the characteristic polynomial of L has n ? v roots 1, q,…, q^{n ? v? 1} where v is the Witt index of Φ.     

Finitely generated submodules of differentiable functions II

Let [$\text{E}$(Ω)]^p be the Cartesian product of the space of real-valued infinitely differentiable functions on a connected open set Ω in $\text{R}$ⁿ with itself p-times. The finitely generated submodules of [$\text{E}$(Ω)]^p are of the form im(F) where F: [$\text{E}$(Ω)]^q → [$\text{E}$(Ω)]^p is a p × q matrix of infinitely differentiable functions on Ω. Let $\text{r =}\text{max}\text{{}\text{rank}\text{(F(x)): x ? Ω}}$. The main results of the present paper are that for Ω ? $\text{R}$ⁿ, if the finitely generated submodule im(F) is closed in [$\text{E}$(Ω)]^p, then for every x?ω with rank(F(x)) < r there exists an r × r sub-matrix A of F such that x is a zero of finite order of det(A), and for Ω ? $\text{R}$¹ the converse also holds.     

Entire solutions blowing up at infinity for semilinear elliptic systems

We consider the system Δu=p(x)g(v), Δv=q(x)f(u) in $\text{R}{}^{\mathrm{N}}$, where f,g are positive and non-decreasing functions on (0,∞) satisfying the Keller–Osserman condition and we establish the existence of positive solutions that blow-up at infinity.     

Quaternionic compositum genus

A quaternionic field over the rationals contains three quadratic subfields with a compositum genus relation of the type described in the author's paper in Volume 9 of this journal, involving the representation of a prime as norm in these subfields. These representations had previously been only partially exlored by the transfer of class structure from the rational to the quadratic fields. Here a full exposition is given by constructing the Artin characters when the subfields are $\text{Q}$(2^1/2), $\text{Q}$(q^1/2), and $\text{Q}$(2q)^1/2 (q prime). A special role belongs to q = A² + 32b².     

Cyclic quartic fields and genus theory of their subfields

Let k = $\text{Q}$(√u) (u ≠ 1 squarefree), K any possible cyclic quartic field containing k. A close relation is established between K and the genus group of k. In particular: (1) Each K can be written uniquely as K = $\text{Q}$(√vwη), where η is fixed in k and satisfies η ? 1, (η) = $\text{U}$²√u, |$\text{U}$²| = |(√u)|, (v, u) = 1, v ∈ $\text{Z}$ is squarefree, w|u, 0 < w < √u. Thus if u ≠ a² + b², there is no K ? k. If u = a² + b² then for each fixed v there are 2^{g ? 1}K ? k, where g is the number of prime divisors of u. (2) $\text{K}\text{k}$ has a relative integral basis (RIB) (i.e., O_K is free over O_k) iff N(ε₀) = ?1 and w = 1, where ε₀ is the fundamental unit of k, (or, equivalently, iff K = $\text{Q}$(√vε₀√u), (v, u) = 1). (3) A RIB is constructed explicitly whenever it exists. (4) disc(K) is given. In particular, the following results are special cases of (2): (i) Narkiewicz showed in 1974 that $\text{K}\text{k}$ has a RIB if u is a prime; (ii) Edgar and Peterson (J. Number Theory12 (1980), 77–83) showed that for u composite there is at least one K ? k having no RIB. Besides, it follows from (4) that the classification and integral basis of K given by Albert (Ann. of Math.31 (1930), 381–418) are wrong.     

Isolated singularities of some semilinear elliptic equations

Let Ω be an open subset of $\text{R}{}^{\mathrm{<mi\; mathvariant="italic"\; is="true">N</mi>}}$, N ? 3, containing 0. We consider the solutions of ?Δu(x) + g(u(x)) = f(x) in Ω-{0}, where g is nondecreasing and f is bounded and we study the possible singularities at 0: when u(x) = o(|x|^{1 ? N}) we prove that u is isotropic near 0 and show that either it is a C¹ function in Ω (removable singularity) or |x|^{N ? 2}u(x) → c, c ≠ 0 (weak singularity) or |x|^{N ? 2} |u(x) |→ + ∞ (strong singularity). We also characterize the g's for which solutions with a weak singularity exist and improve a previous removability result of H. Brézis and L. Véron (Arch. Rational Mech. Anal.23 (1979), 153–166).     

On the density of various classes of groups

Let $\text{T}$ be a subset of the set of all isomorphism classes of finite groups. We consider the number F_$\text{g}$(x) of positive integers n≤x such that all groups of order n lie in $\text{T}$. When $\text{T}$ consists of the isomorphism classes of all finite groups of any of the following types, we obtain an asymptotic formula for F_$\text{g}$(x): cyclic groups, abelian groups, nilpotent groups, supersolvable groups, and solvable groups. In the course of the arguments, we also obtain, for almost all n, a lower bound for the number of groups of a given order n.     

The number of positive integers ≤x and free of prime factors >y

The number defined by the title is denoted by Ψ(x, y). Let $\text{u =}\text{log}\text{x}\text{log}\text{y}$ and let ?(u) be the function determined by ?(u) = 1, 0 ≤ u ≤ 1, u?′(u) = ? ?(u ? 1), u > 1. We prove the following:Theorem. For x sufficiently large and log y ≥ (log log x)², Ψ(x,y) ? x?(u) while for 1 + log log x ≤ log y ≤ (log log x)², and ε > 0, $\text{Ψ(x, y) ?}{}_{\mathrm{\epsilon }}\text{x?(u)}\text{exp}\text{(?u e}\text{xp}\text{(?(}\text{log}\text{y)}{}^{\mathrm{<mtext>(</mtext><mtext>3</mtext><mtext>5</mtext><mtext>\; ?\; \epsilon )</mtext>}}\text{))}$.The proof uses a weighted lower approximation to Ψ(x, y), a reinterpretation of this sum in probability terminology, and ultimately large-deviation methods plus the Berry-Esseen theorem.     

A global genericity theorem for bifurcations in variational problems

Let C be a Banach space, H a Hilbert space, and let F(C,H) be the space of C^∞ functions f: C × H → R having Fredholm second derivative with respect to x at each (c, x) ?C × H for which $\text{D?}{}_{\mathrm{c}}\text{(x) = 0}$; here we write $\text{?}{}_{\mathrm{c}}\text{(x)}$ for $\text{?(c, x)}$. Say ? is of standard type if at all critical points of ?_c it is locally equivalent (as an unfolding) to a quadratic form Q plus an elementary catastrophe on the kernel of Q. It is proved that if f?F (A × B, H) satisfies a certain ‘general position’ condition, and dim B ? 5, then for most a?A the function f_o?F(B,H) is of standard type. Using this it is shown that those f?F(B,H) of standard type form an open dense set in F(B,H) with the Whitney topology. Thus both results are Hilbert-space versions of Thom's theorem for catastrophes in $\text{R}$ⁿ.