Third order efficiency of conditional tests for exponential families

Let P_η, η = (θ, γ) ∈ Θ × Γ ? $\text{R}$ × $\text{R}$^k, be a (k + 1)-dimensional exponential family. Let $\text{?}{}_{\mathrm{n}}{}^1$, n ∈ $\text{N}$, be an optimal similar test for the hypothesis {P_(θ,γ)ⁿ: γ ∈ Γ} (θ ∈ Θ fixed) against alternatives P_(θ1,γ1)ⁿ, θ₁ > θ, γ₁ ∈ Γ. It is shown that (?_n¹)_{n∈$\text{N}$} is third order efficient in the class of all test-sequences that are asymptotically similar of level α + o(n^?1) (locally uniformly in the nuisance parameter γ).     

Diagonal norm hermitian matrices

If v is a norm on $\text{C}$ⁿ, let H(v) denote the set of all norm-Hermitians in $\text{C}$ⁿⁿ. Let S be a subset of the set of real diagonal matrices D. Then there exists a norm v such that S=H(v) (or S = H(v)∩D) if and only if S contains the identity and S is a subspace of D with a basis consisting of rational vectors. As a corollary, it is shown that, for a diagonable matrix h with distinct eigenvalues λ₁,…, λ_r, r?n, there is a norm v such that h ∈ H(v), but h^s?H(v), for some integer s, if and only if λ₂–λ₁,…, λ_r–λ₁ are linearly dependent over the rationals. It is also shown that the set of all norms v, for which H(v) consists of all real multiples of the identity, is an open, dense subset, in a natural metric, of the set of all norms.     

Injectivity and operator spaces

Let B(H) be the bounded operators on a Hilbert space H. A linear subspace R ? B(H) is said to be an operator system if 1 ?R and R is self-adjoint. Consider the category $\text{b}$ of operator systems and completely positive linear maps. R ∈ $\text{C}$ is said to be injective if given A ? B, A, B ∈ $\text{C}$, each map A → R extends to B. Then each injective operator system is isomorphic to a conditionally complete C¹-algebra. Injective von Neumann algebras R are characterized by any one of the following: (1) a relative interpolation property, (2) a finite “projectivity” property, (3) letting M_m = B(C^m), each map R → N ? M_m has approximate factorizations R → M_n → N, (4) letting K be the orthogonal complement of an operator system N ? M_m, each map $\text{M}{}_{\mathrm{m}}\text{K}\text{→ R}$ has approximate factorizations $\text{M}{}_{\mathrm{m}}\text{K}\text{→ M}{}_{\mathrm{n}}\text{→ R}$. Analogous characterizations are found for certain classes of C¹-algebras.     

On admissible shifts of generalized white noises

Let $\text{S}$ be the Schwartz space of rapidly decreasing real functions. The dual space $\text{S}$¹ consists of the tempered distributions and the relation $\text{S}$ ? L²($\text{R}$) ? $\text{S}$¹ holds. Let γ be the Gaussian white noise on $\text{S}$¹ with the characteristic functional $\text{γ}\text{(ξ) =}\text{exp}\text{{?∥ξ∥}{}^2\text{/2}}$, ξ ∈ $\text{S}$, where ∥·∥ is the L²($\text{R}$)-norm. Let ν be the Poisson white noise on $\text{S}$¹ with the characteristic functional $\text{ν}\text{(ξ)}$ = exp?_$\text{R}$∫_$\text{R}$ {[exp(iξ(t)u)] ? 1 ? (1 + u²)^?1(iξ(t)u)} dη(u)dt), ξ ∈ $\text{S}$, where the Lévy measure is assumed to satisfy the condition ∫_$\text{R}$u²dη(u) < ∞. It is proved that γ1ν has the same dichotomy property for shifts as the Gaussian white noise, i.e., for any ω ∈ $\text{S}$¹, the shift $\text{(γ1ν)}{}_{\mathrm{\omega }}$ of γ1ν by ω and γ1ν are either equivalent or orthogonal. They are equivalent if and only if when ω ∈ L²($\text{R}$) and the Radon-Nikodym derivative is derived. It is also proved that for the Poisson white noice ν_ω is orthogonal to ν for any non-zero ω in $\text{S}$¹.     

A note on a theorem of heath

We propose a generalization of Heath's theorem that semi-metric spaces with point-countable bases are developable: A semi-metrizable space X is developabale if (and only if) there is on it a σ-discrete family $\text{C}\text{=?}{}_{\mathrm{m?N}}\text{C}{}_{\mathrm{m}}$ of closed sets, interior-preserving over each member C of which is a countable family {$\text{D}$_n(C): n ∈ N} of collections of open sets such that if U is a neighbourhood of ξ∈X, then there are such a Γ∈$\text{C}$ and such a v∈ N that ξ ? Γ and ξ∈ int ∩ (D: ξ: D∈$\text{D}$_v(Γ))?U.     

The lattice of subfields of a radical extension

Let x^m ? a be irreducible over F with char $\text{F?m}$ and let α be a root of x^m ? a. The purpose of this paper is to study the lattice of subfields of $\text{F(α)}\text{F}$ and to this end $\text{C(}\text{F(α)}\text{F}\text{, k)}$ is defined to be the number of subfields of F(α) of degree k over $\text{F. C(}\text{F(α)}\text{F}\text{, p}{}^{\mathrm{n}}\text{)}$ is explicitly determined for p a prime and the following structure theorem for the lattice of subfields is proved. Let N be the maximal normal subfield of F(α) over F and set n = |N : F|, then $\text{C(}\text{F(α)}\text{F}\text{, k) = C(}\text{F(α)}\text{F}\text{, (k, n)) = C(}\text{N}\text{F}\text{, (k, n))}$. The irreducible binomials x^s ? b, x^s ? c are said to be equivalent if there exist roots β^s = b, γ^s = a such that F(β) = F(γ). All the mutually inequivalent binomials which have roots in F(α) are determined. Finally these results are applied to the study of normal binomials and those irreducible binomials x^2e ? a which are normal over F (char F ≠ 2) together with their Galois groups are characterized.     

On the analytic continuation of a certain Dirichlet series

A Dirichlet series associated with a positive definite form of degree δ in n variables is defined by

$\text{D}{}_{\mathrm{F}}\text{(s,p,α)}\text{=}\text{∑}\text{α∈Z}{}^{\mathrm{n}}\text{?{0}}\text{F(α)}{}^{\mathrm{?s}}\text{e(ρF(α)+〈α, α〉)}$

where ? ∈ $\text{Q}$, α ∈ $\text{Q}$ⁿ, 〈x, y〉 = x₁y₁ + ? + x_ny_n, e(a) = exp (2πia) for a ∈ $\text{R}$, and s = σ + ti is a complex number. The author proves that: (1) D_F(s, ?, α) has analytic continuation into the whole s-plane, (2) D_F(s, ?, α), ? ≠ 0, is a meromorphic function with at most a simple pole at $\text{s =}\text{n}\text{δ}$. The residue at $\text{s =}\text{n}\text{δ}$ is given explicitly. (3) ? = 0, α ? $\text{Z}$ⁿ, D_F(s, 0, α) is analytic for $\text{α>,}\text{n}\text{(δ ? 1)}$.     

Concrete model theory for a class of operators with unitary part

Let m and v_t, 0 ? t ? 2π be measures on T = [0, 2π] with m smooth. Consider the direct integral $\text{H}$ = ⊕L²(v^t) dm(t) and the operator $\text{(L?)(t, λ) = e}{}^{\mathrm{?i\lambda }}\text{?(t, λ) ? 2e}{}^{\mathrm{?i\lambda }}\text{∫}{}_{\mathrm{t}}{}^{\mathrm{2\pi }}\text{∫}{}_{\mathrm{T}}\text{?(s, x) e(s, t) dv}{}_{\mathrm{s}}\text{(x) dm(s)}$ on $\text{H}$, where e(s, t) = exp ∫_s^t ∫_Tdv_λ(θ) dm(λ). Let μ_t be the measure defined by $\text{∫}{}_{\mathrm{T}}\text{?(x) dμ}{}_{\mathrm{t}}\text{(x) = ∫}{}_0{}^{\mathrm{t}}\text{∫}{}_{\mathrm{T}}\text{?(x) dv}{}_{\mathrm{s}}\text{dm(s)}$ for all continuous ?, and let ?_t(z) = exp[?∫ (e^iθ + z)(e^iθ ? z)^?1dμ_t(gq)]. Call {v_t} regular iff for all $\text{t, ¦?}{}_{\mathrm{t}}\text{(e}{}^{\mathrm{i\theta }}\text{)¦ = ¦?}{}_{\mathrm{2\pi }}\text{(e}{}^{\mathrm{i\theta }}\text{)¦}$ for 1 a.e.     

The distribution and average order of the coefficients of Dedekind ζ functions

Let K/$\text{Q}$ be an algebraic number field and ζ_K(s) be the associated Dedekind ζ function. A quantitative estimate is proved which shows that the average order of the coefficients of ζ_k^m(s) (for m ∈ $\text{Z}$⁺) arises from infrequent occurrences of very large values of these coefficients. This leads to new Ω-estimates for the associated error terms, improving results of Szegö and Walfisz.     

Prime interchange graphs of classes of matrices of zeros and ones

Let R = (r₁,…, r_m) and S = (s₁,…, s_n) be nonnegative integral vectors, and let $\text{U}$(R, S) denote the class of all m × n matrices of 0's and 1's having row sum vector R and column sum vector S. An invariant position of $\text{U}$(R, S) is a position whose entry is the same for all matrices in $\text{U}$(R, S). The interchange graph G(R, S) is the graph where the vertices are the matrices in $\text{U}$(R, S) and where two matrices are joined by an edge provided they differ by an interchange. We prove that when 1 ≤ r_i ≤ n ? 1 (i = 1,…, m) and 1 ≤ s_j ≤ m ? 1 (j = 1,…, n), G(R, S) is prime if and only if $\text{U}$(R, S) has no invariant positions.     

On solutions of “equations in symmetric groups”

Let x?S_n, the symmetric group on n symbols. Let θ? Aut(S_n) and let the automorphim order of x with respect to θ be defined by

$\text{γθ(x)=}\text{min}\text{{k:x xθ xθ}{}^2\text{? xθ}{}^{\mathrm{k?1}}\text{=1}}$

where xθ is the image of x under θ. Let α_g? Aut(S_n) denote conjugation by the element g?S_n. Let where s and k are positive integers and $\text{a}\text{b}$ denotes a divides b. Further h(s, k : n) ≡ b(1; s, k : n), where 1 denotes the identity automorphim. If g?S_n let c = f(g, s) denote the number of symbols in g which are in cycles of length not dividing the integer s, and let g_s denote the product of all cycles in g whose lengths do not divide s. Then g_s moves c symbols. The main results proved are: (1) recursion: if n ? c + 1 and t = n ? c ? 1 then $\text{b(g; s, 1:n)=∑}{}_{\mathrm{<mtext>i</mtext><mtext>s</mtext>}}\text{b(g; s, 1:n?1)(}{}^{\mathrm{t}}{}_{\mathrm{i?1}}\text{(i?1)!}$ (2) reduction: b(g; s, 1 : c)h(s, 1 : i) = b(g; s, 1 : i + c); (3) distribution: let D(θ, n) ≡ {(k, b) : k?Z⁺ and b = b(θ; 1, k : n) ≠ 0}; then D(θ, m) = D(φ, m) ∨ m ? N = N(θ, φ) iff θ is conjugate to φ; (4) evaluation: the number of cycles in g_s^s of any given length is smaller than the smallest prime dividing s iff b(g_s; s, 1 : c) = 1. If g = (12 … p^m)^t and $\text{sk}\text{p}{}^{\mathrm{m}}$ then $\text{b(g;s,k:p}{}^{\mathrm{m}}\text{) {}{}^0{}_{\mathrm{\pm 1}}\text{(}\text{mod}\text{p)}$.     

Spectral properties of zeroth-order pseudodifferential operators

Let Q be a self-adjoint, classical, zeroth order pseudodifferential operator on a compact manifold X with a fixed smooth measure dx. We use microlocal techniques to study the spectrum and spectral family, {E_S}_{S∈$\text{R}$} as a bounded operator on L²(X, dx).Using theorems of Weyl (Rend. Circ. Mat. Palermo, 27 (1909), 373–392) and Kato (“Perturbation Theory for Linear Operators,” Springer-Verlag, 1976) on spectra of perturbed operators we observe that the essential spectrum and the absolutely continuous spectrum of Q are determined by a finite number of terms in the symbol expansion. In particular Spec_ESSQ = range(q(x, ξ)) where q is the principal symbol of Q. Turning the attention to the spectral family {E_S}_{S∈$\text{R}$}, it is shown that if $\text{dE}\text{ds}$ is considered as a distribution on $\text{R}$×X×X it is in fact a Lagrangian distribution near the set $\text{{σ=0}?}\text{T}{}^1\text{(}\text{R}\text{×X×X)}\text{0}$ where (s, x, y, σ, ξ,η) are coordinates on T¹($\text{R}$×X×X) induced by the coordinates (s, x, y) on $\text{R}$×X×X. This leads to an easy proof that$\text{?(Q)}$ is a pseudodifferential operator if ?∈C^∞($\text{R}$) and to some results on the microlocal character of E_s. Finally, a look at the wavefront set of $\text{dE}\text{ds}$ leads to a conjecture about the existence of absolutely continuous spectrum in terms of a condition on q(x, ξ).     

Bounds for the uniform deviation of empirical measures

If X₁,…,X_n are independent identically distributed R^d-valued random vectors with probability measure μ and empirical probability measure μ_n, and if $\text{a}$ is a subset of the Borel sets on R^d, then we show that P{sup_{A∈$\text{a}$}|μ_n(A)?μ(A)|≥ε} ≤ cs($\text{a}$, n²)e^?2n∈2, where c is an explicitly given constant, and s($\text{a}$, n) is the maximum over all (x₁,…,x_n) ∈ R^dn of the number of different sets in {{x₁…,x_n}∩A|A ∈$\text{a}$}. The bound strengthens a result due to Vapnik and Chervonenkis.     

Minimum range sequences of all k-subsets of a set

The purpose of this paper is to find sequential orderings of the class S_k of all k-element subsets of the v element set V = {1, 2, …, υ} with certain minimizing properties. A sequential ordering of S_k is just a numbering of the (^v_k) k-element subsets of V, where the first one is s₁, the second s₂ and so on. Let f(s) denote the position of the k-element subset s in the ordering. For each element e of V the range R_f(e) of e in the sequence f is given by

$\text{R}{}_{\mathrm{?}}\text{(}\text{e}\text{) =}\text{max}\text{e}\text{?}\text{s}\text{,}\text{s}\text{′}\text{|?(}\text{s}\text{)??(}\text{s}\text{′}\text{)z.sfnc;}$

The total range R_f of a sequence f is

$\text{R}{}_{\mathrm{?}}\text{=}\text{∑}\text{e}\text{?}\text{v}\text{R?(}\text{e}\text{)}$

and the bottleneck range B_f of a sequence f is

$\text{R}{}_{\mathrm{?}}\text{=}\text{max}\text{r}\text{?}\text{v}\text{R?(}\text{e}\text{)}$

Let F denote the set of all sequences (for fixed values of v and k). In this paper we show there exist sequences f that simultaneously minimize R_f and B_f, give a construction of such sequences for arbitrary v and k, and give exact formulae for min_f? _FR_f and min_f?FB_f as functions of v and k.These problems arise in experimental designs involving human subjects and in the linking between two stages of switches in multistage switching networks.     

On the solutions in the large of the two-dimensional flow of a nonviscous incompressible fluid

The Euler equations (1.1) for the motion of a nonviscous imcompressible fluid in a plane domain Ω are studied. Let E be the Banach space defined in (1.4), let the initial data v₀ belong to E, and let the external forces f(t) belong to L_loc¹(R; E). In Theorem 1.1 the strong continuity and the global boundedness of the (unique) solution v(t) are proved, and in Theorem 1.2 the strong-continuous dependence of v on the data v₀ and f is proved. In particular the vorticity rot v(t) is a continuous function in $\text{}\text{?}$gW, for every t ? R, if and only if this property holds for one value of t. In Theorem 1.3 some properties for the associated group of nonlinear operators S(t). are stated. Finally, in Theorem 1.4 a quite general sufficient condition is given on the data in order to get classical solutions.     

Some permutation problems

Put Z_n = {1, 2,…, n} and let π denote an arbitrary permutation of Z_n. Problem I. Let π = (π(1), π(2), …, π(n)). π has an up, down, or fixed point at a according as a < π(a), a > π(a), or a = π(a). Let $\text{A}\text{(r, s, t)}$ be the number of π ∈ Z_n with r ups, s downs, and t fixed points. Problem II. Consider the triple π^?1(a), a, π(a). Let R denote an up and F a down of π and let B(n, r, s) denote the number of π ∈ Z_n with r occurrences of π^?1(a)RaRπ(a) and s occurrences of π^?1(a)FaFπ(a). Generating functions are obtained for each enumerant as well as for a refinement of the second. In each case use is made of the cycle structure of permutations.     

Limit sets of dynamical systems on the space of probability measures

Let m be a dynamical system on the space of probability measures $\text{M}$₁(R^d), and let Λ + (?) be the positive limit set for ? ∈ $\text{M}$₁(R^d), where ? has compact support K ?R^d. The main result of this paper states that support of Λ⁺(?) ?

,support of Λ + (δ_x), where δ_x is the Dirac measure at point x.