Extremal traces on some group-invariant C1-algebras

Let $\text{U}$ be a UHF-algebra of Glimm type n^∞, and {α_g: g?G} a strongly continuous group of ¹-automorphisms of product type on $\text{U}$, for G compact. Let $\text{U}$^α be the C¹-subalgebra of fixed elements of $\text{U}$. We show that any extremal normalized trace on $\text{U}$^α arises as the restriction of a symmetric product state ? on $\text{U}$ of the form ? = ?_k?1 ω. As an example we classify the extremal traces on $\text{U}$^α for the case G = SU(n), α_g = ?_{k ? 1} Ad(g).     

Phase reconstruction from magnitude of band-limited multidimensional signals

In this paper, the problem of phase reconstruction from magnitude of multidimensional band-limited functions is considered. It is shown that any irreducible band-limited function f(z₁…,z_n), z_i ? $\text{C}$, i=1, …, n, is uniquely determined from the magnitude of f(x₁…,x_n): | f(x₁…,x_n)|, x_i ? $\text{R}$, i=1,…, n, except for (1) linear shifts: ^{i(α1z1+…+α_n2n+β}), β, α_i?$\text{R}$, i=1,…, n; and (2) conjugation: $\text{f}{}^1\text{(z}{}_1{}^1\text{,…,z}{}_{\mathrm{n}}{}^1\text{)}$.     

Asymptotic independence of distributions of normalized order statistics of the underlying probability measure

Let n denote the sample size, and let r_i ∈ {1,…,n} fulfill the conditions r_i ? r_i?1 ≥ 5 for i = 1,…,k. It is proved that the joint normalized distribution of the order statistics Z_ri:n, i = 1,…,k, is independent of the underlying probability measure up to a remainder term of order $\text{O((}\text{k}\text{n}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$. A counterexample shows that, as far as central order statistics are concerned, this remainder term is not of the order $\text{O((}\text{k}\text{n}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ if r_i ? r_i?1 = 1 for i = 2,…,k.     

Some Ramsey numbers for directed graphs

Given k directed graphs G₁,…,G_k the Ramsey number R(G₁,…, G_k) is the smallest integer n such that for any partition (U₁,…,U_k) of the arcs of the complete symmetric directed graph K_n, there exists an integer i such that the partial graph generated by U₁ contains G₁ as a subgraph. In the article we give a necessary and sufficient condition for the existence of Ramsey numbers, and, when they exist an upper bound function. We also give exact values for some classes of graphs. Our main result is: $\text{R(}\text{P}{}_{\mathrm{n}}\text{,….}\text{P}{}_{\mathrm{nk-1}}\text{, G) = n}{}_1\text{…n}{}_{\mathrm{k-1}}\text{(p-1) + 1}$, where G is a hamltonian directed graph with p vertices and $\text{P}{}_{\mathrm{n}}{}_{\mathrm{i}}$ denotes the directed path of length n_t     

A generalization of the Kruskal-Katona theorem

Let k_n ? k_n?1 ? … ? k₁ be positive integers and let ($\text{i}\text{j}$) denote the coefficient of xⁱ in $\text{Π}\text{r=1}\text{j}\text{(1 + x + x}{}^2\text{+ … + x}{}^{\mathrm{kr}}\text{)}$. For given integers l, m, where 1 ? l ? k_n + k_n?1 + … + k₁ and $\text{1 ? m ? (}\text{n}\text{n}\text{)}$, it is shown that there exist unique integers m(l), m(l ? 1),…, m(t), satisfying certain conditions, for which $\text{m = (}\text{m(l)}\text{l}\text{+ (}\text{m(l?1)}\text{l?1}\text{) + … + (}\text{m(t)}\text{t}\text{)}$. Moreover, any m l-subsets of a multiset with k_i elements of type i, i = 1, 2,…, n, will contain at least $\text{(}\text{m(l)}\text{l?1}\text{) + (}\text{m(l?1)}\text{l?2}\text{) + … + (}\text{m(t)}\text{t?1}$ different (l ? 1)-subsets. This result has been anticipated by Greene and Kleitman, but the formulation there is not completely correct. If k₁ = 1, the numbers ($\text{j}\text{i}$) are binomial coefficients and the result is the Kruskal-Katona theorem.     

Decomposable bilinear numerical radii

Let A be an n-square normal matrix over $\text{C}$, and Q_{m, n} be the set of strictly increasing integer sequences of length m chosen from 1,…, n. For α,β∈Q_{m, n} denote by A[α|β] the submatrix obtained from A by using rows numbered α and columns numbered β. For k∈{0,1,…,m} write z.sfnc;α∩β|=k if there exists a rearrangement of 1,…,m, say i₁,…,i_k, i_k+1,…,i_m, such that α(i_j)=β(i_j), j=1,…,k, and {α(i_k+1),…,α(i_m)};∩{β(i_k+1),…,β(i_m)}=ø. Let

be the group of n-square unitary matrices. Define the nonnegative number

$\text{?}{}_{\mathrm{k}}\text{(A)=}\text{max}\text{U∈}\textcolor{red}{}\text{|}\text{det}\text{(U}{}^1\text{AU) [α|β]|}$

, where |α∩β|=k. Theorem 1 establishes a bound for ?_k(A), 0?k<m?1, in terms of a classical variational inequality due to Fermat. Let A be positive semidefinite Hermitian, n?2m. Theorem 2 leads to an interlacing inequality which, in the case n=4, m=2, resolves in the affirmative the conjecture that

$\text{?}{}_{\mathrm{m}}\text{(A)??}{}_{\mathrm{m?1}}\text{(A)????}{}_0\text{(A)}$

.     

On maximal antichains containing no set and its complement

Let 1?k₁?k₂?…?k_n be integers and let S denote the set of all vectors x = (x₁, …, x_n with integral coordinates satisfying 0?x_i?k_i, i = 1,2, …, n; equivalently, S is the set of all subsets of a multiset consisting of k_i elements of type i, i = 1,2, …, n. A subset X of S is an antichain if and only if for any two vectors x and y in X the inequalities x_i?y_i, i = 1,2, …, n, do not all hold. For an arbitrary subset H of S, (i)H denotes the subset of H consisting of vectors with component sum i, i = 0, 1, 2, …, K, where K = k₁ + k₂ + …k_n. |H| denotes the number of vectors in H, and the complement of a vector x?S is (k₁-x₁, k₂-x₂, …, k_n -x_n). What is the maximal cardinality of an antichain containing no vector and its complement? The answer is obtained as a corollary of the following theorem: if X is an antichain, K is even and$\text{|(}\text{1}\text{2}\text{K)X|}$ does not exceed the number of vectors in $\text{(}\text{1}\text{2}\text{K)S}$ with first coordinate different from k₁, then

$\text{∑}\text{i=0}\text{K}\text{i≠}\text{1}\text{2}\text{K}\text{|(i)X|}\text{|(i)S|}\text{+}\text{|(}\text{1}\text{2}\text{K)X|}\text{|(}\text{1}\text{2}\text{K-1)S|}\text{?1}$

.     

Enumeration of pairs of permutations

Let π = (a₁, a₂, …, a_n), ? = (b₁, b₂, …, b_n) be two permutations of $\text{Z}{}_{\mathrm{n}}\text{= {1, 2, …, n}}$. A rise of π is pair a_i, a_i+1 with a_i < a_i+1; a fall is a pair a_i, a_i+1 with a_i > a_i+1. Thus, for i = 1, 2, …, n ? 1, the two pairs a_i, a_i+1; b_i, b_i+1 are either both rises, both falls, the first a rise and the second a fall or the first a fall and the second a rise. These possibilities are denoted by RR, FF, RF, FR. The paper is concerned with the enumeration of pairs π, p with a given number of RR, FF, RF, FR. In particular if ω_n denotes the number of pairs with RR forbidden, it is proved that $\text{∑}{}^{\mathrm{\infty }}{}_0\text{ω}{}_{\mathrm{n}}\text{z}{}^{\mathrm{n}}\text{n!n!}\text{=}\text{1}\text{?(z)}$, $\text{?(z) = ∑}{}^{\mathrm{\infty }}{}_0\text{(-1)}{}^{\mathrm{n}}\text{z}{}^{\mathrm{n}}\text{n!n!}$. More precisely if ω(n, k) denotes the number of pairs π, p with exactly k occurences of RR(or FF, RF, FR) then $\text{1 + ∑}{}^{\mathrm{\infty }}{}_{\mathrm{n=1}}\text{z}{}^{\mathrm{n}}\text{n!n!}$$\text{∑}{}^{\mathrm{n?1}}{}_{\mathrm{k=0}}\text{ω(n, k)x}{}^{\mathrm{k}}\text{=}\text{(1 ? x)}\text{(?(z(1 ? x)) ? x)}$.     

Three problems on the lengths of increasing runs

Let U₁, U₂,… be a sequence of independent, uniform (0, 1) r.v.'s and let R₁, R₂,… be the lengths of increasing runs of {U_i}, i.e., X₁=R₁=inf{i:U_i+1<U_i},…, X_n=R₁+R₂+?+R_n=inf{i:i>X_n?1,U_i+1<U_i}. The first theorem states that the sequence $\text{(}\text{3}\text{2}\text{n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(X}{}_{\mathrm{n}}\text{?2n)}$ can be approximated by a Wiener process in strong sense.Let τ(n) be the largest integer for which R₁+R₂+?+R_τ(n)?n, $\text{R}{}^1{}_{\mathrm{n}}\text{=n?(R}{}_1\text{+R}{}_2\text{+?+R}{}_{\mathrm{\tau (n)}}\text{)}$ and $\text{M}{}_{\mathrm{n}}\text{=}\text{max}\text{{R}{}_1\text{,R}{}_2\text{,…,R}{}_{\mathrm{\tau (n)}}\text{,R}{}^1{}_{\mathrm{n}}\text{}}$. Here M_n is the length of the longest increasing block. A strong theorem is given to characterize the limit behaviour of M_n.The limit distribution of the lengths of increasing runs is our third problem.     

A survey of published programs for best approximation

We list and discuss published programs for best approximation of functions by linear and nonlinear families in all standard forms.In this note we list and discuss the published programs for obtaining best approximations. Let X be a set on which we wish to approximate. Most sets will be finite (an equivalent terms is discrete). Let 6 6 be a norm on the continuous functions on X. Let G be a familiy of continuous functions on X. For a given basis {φ₁…,φ_n}, the linear family G is the set of all functions of the form

$\text{L(A,x)=}\text{∑}\text{k=1}\text{n}\text{a}{}_{\mathrm{k}}\text{φ}{}_{\mathrm{k}}\text{(x)}$

The problem of best approximation is given a continuous function f, to find g¹ to minimize e(g) = ∥f-g∥ over g?G. Such g¹ is called a best approximation to f.Discrete linear approximation problems are sometimes formulated as solution of an overdetermined system of linear equations Ax = b with respect to a norm where a_ij = φ_j(x_i) and b_i = f(x)_i).     

On the closability of some positive definite symmetric differential forms on c0∞(ω)

Let $\text{S}$ be a Dirichlet form in $\text{L}{}^2\text{(Ω; m)}$, where Ω is an open subset of $\text{R}$ⁿ, n ? 2, and m a Radon measure on Ω; for each integer k with 1 ? k < n, let $\text{S}$_k be a Dirichlet form on some k-dimensional submanifold $\text{Ω}{}_{\mathrm{k}}$ of Ω. The paper is devoted to the study of the closability of the forms E with domain $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ and defined by: $\text{(?,g)=}$$\text{E}$$\text{(?, g)+}\text{∑}\text{i}\text{p}\text{=1}$$\text{E}$k_i where 1 ? k_p < ? < n, and where , g^k_i denote restrictions of ?, g in $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ to . Conditions are given for E to be closable if, for each i = 1,…, p, one has k_i = n ? i. Other conditions are given for E to be nonclosable if, for some i, k_i < n ? i.     

Some improvements in Neuberger's iteration procedure for solving partial differential equations

If m and n are positive integers then let S(m, n) denote the linear space over R whose elements are the real-valued symmetric n-linear functions on E_m with operations defined in the usual way. If $\text{U}$ is a function from some sphere S in E_m to R then let $\text{U}$⁽ⁱ⁾(x) denote the ith Frechet derivative of $\text{U}$ at x. In general $\text{U}$⁽ⁱ⁾(x)∈S(m,i). In the paper “An Iterative Method for Solving Nonlinear Partial Differential Equations” [Advances in Math. 19 (1976), 245–265] Neuberger presents an iterative procedure for solving a partial differential equation of the form

$\text{AU}{}^{\mathrm{n}}\text{(x)=F(x, U(x), U′(x),…,U}{}^{\mathrm{k}}\text{(x))}$

, where k > n, $\text{U}$ is the unknown from some sphere in E_m to R, A is a linear functional on S(m, n), and F is analytic. The defect in the theory presented there was that in order to prove that the iterates converged to a solution $\text{U}$ the condition $\text{k ?}\text{n}\text{2}$ was needed. In this paper an iteration procedure that is a slight variation on Neuberger's procedure is considered. Although the condition $\text{k ?}\text{n}\text{2}$ cannot as yet be eliminated, it is shown that in a broad class of cases depending upon the nature of the functional A the restriction $\text{k ?}\text{n}\text{2}$ may be replaced by the restriction $\text{k ?}\text{3n}\text{4}$.     

A subdeterminant inequality for normal matrices

Let A be an n × n normal matrix over $\text{C}$, and Q_m, _n be the set of strictly increasing integer sequences of length m chosen from 1,…,n. For α, β ? Q_m, _n denote by A[α|β] the submatrix obtained from A by using rows numbered α and columns numbered β. For k ? {0, 1,…, m} we write |α∩β| = k if there exists a rearrangement of 1,…, m, say i₁,…, i_k, i_k+1,…, i_m, such that α(i_j) = β(i_j), i = 1,…, k, and {α(i_k+1),…, α(i_m) } ∩ {β(i_k+1),…, β(i_m) } = ?. A new bound for |detA[α|β ]| is obtained in terms of the eigenvalues of A when 2m = n and |α∩β| = 0.Let $\text{U}$_n be the group of n × n unitary matrices. Define the nonnegative number

where | α ∩ β| = k. It is proved that

Let A be semidefinite hermitian. We conjecture that ρ₀(A) ? ρ₁(A) ? ··· ? ρ_m(A). These inequalities have been tested by machine calculations.     

On a class of relatively prime sequences

For each natural number n, let a₀(n) = n, and if a₀(n),…,a_i(n) have already been defined, let a_i+1(n) > a_i(n) be minimal with (a_i+1(n), a₀(n) … a_i(n)) = 1. Let g(n) be the largest a_i(n) not a prime or the square of a prime. We show that g(n) ～ n and that $\text{g(n) > n + cn}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{log}\text{(n)}$ for some c > 0. The true order of magnitude of g(n) ? n seems to be connected with the fine distribution of prime numbers. We also show that “most” a_i(n) that are not primes or squares of primes are products of two distinct primes. A result of independent interest comes of one of our proofs: For every sufficiently large n there is a prime $\text{p < n}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ with [$\text{n}\text{p}$] composite.     

Spectra of Cayley graphs

By a result of L. Lovász, the determination of the spectrum of any graph with transitive automorphism group easily reduces to that of some Cayley graph.We derive an expression for the spectrum of the Cayley graph X(G,H) in terms of irreducible characters of the group G:

for any natural number t, where ξ_i is an irreducible character (over $\text{C}$), of degree n_i , and λ_i,1 ,…, λ_i,ni are eigenvalues of X(G, H), each one n_i times. (σn_i² = n = | G | is the total'number of eigenvalues.) Using this formula for t = 1,…, n_i one can obtain a polynomial of degree n_i whose roots are λ_i,1,…,λ_i,ni. The results are formulated for directed graphs with colored edges. We apply the results to dihedral groups and prove the existence of k nonisomorphic Cayley graphs of D_p with the same spectrum provided p > 64k, prime.     

Enumeration of rises and falls by position

Let π=(π₁, π₂,…,π_n) denote a permutation of Z_n = {1, 2,…, n}. The pair (π_i, π_i+1) is a rise if π_i<π_i+1 or a fall if π_i>π_i+1. Also a conventional rise is counted at the beginning of π and a conventional fall at the end. Let k be a fixed integer ≥ 1. The rise π_i,π_i+1 is said to be in a in a j (mod k) position if i ≡ j (mod k); similarly for a fall. The conventional rise at the beginning is in a 0 (mod k) position, while the conventional fall at the end is in an n (mod k) position. Let $\text{P}{}_{\mathrm{n}}\text{≡P}{}_{\mathrm{n}}\text{(r}{}_0\text{,…,r}{}_{\mathrm{k}}\text{?1,?}{}_0\text{,…,?;}{}_{\mathrm{k}}\text{?1)}$ denote the number of permutations having r_i rises i (mod k) positions and ?;_i falls in i (mod k) positions. A generating function for P_n is obtained. In particular, for k = 2 the generating function is quite explicit and also, for certain special cases when k = 4.     

Matrix extensions of Liouville-Dirichlet-type integrals

The Dirichlet integral provides a formula for the volume over the k-dimensional simplex ω={x₁,…,x_k: x_i?0, i=1,…,k, s?∑^k₁x_i?T}. This integral was extended by Liouville. The present paper provides a matrix analog where now the region becomes $\text{Ω={V}{}_1\text{,…,V}{}_{\mathrm{k}}\text{: V}{}_{\mathrm{i}}\text{>0, i=1,…,k, 0?∑V}{}_{\mathrm{i}}\text{?t}}$, where now each V_i is a p×p symmetric matrix and A?B means that A?B is positive semidefinite.     

Duality in hypercomplex function theory

Let $\text{A}$ be the Clifford algebra constructed over a quadratic n-dimensional real vector space with orthogonal basis {e₁,…, e_n}, and e₀ be the identity of $\text{A}$. Furthermore, let M_k(Ω;$\text{A}$) be the set of $\text{A}$-valued functions defined in an open subset Ω of R^m+1 (1 ? m ? n) which satisfy D^kf = 0 in Ω, where D is the generalized Cauchy-Riemann operator $\text{D = ∑}{}_{\mathrm{i\; =\; 0}}{}^{\mathrm{m}}\text{e}{}_{\mathrm{i}}\text{(}\text{?}\text{?x}{}_{\mathrm{i}}\text{)}$ and k? N. The aim of this paper is to characterize the dual and bidual of M_k(Ω;$\text{A}$). It is proved that, if M_k(Ω;$\text{A}$) is provided with the topology of uniform compact convergence, then its strong dual is topologically isomorphic to an inductive limit space of Fréchet modules, which in its turn admits M_k(Ω;$\text{A}$) as its dual. In this way, classical results about the spaces of holomorphic functions and analytic functionals are generalized.     

Periodic-type functionals and their extensions

Let Ω denote a simply connected domain in the complex plane and let $\text{K[Ω]}$ be the collection of all entire functions of exponential type whose Laplace transforms are analytic on Ω′, the complement of Ω with respect to the sphere. Define a sequence of functionals $\text{{L}{}_{\mathrm{n}}\text{}}\text{on}\text{K[Ω]}$ by $\text{L}{}_{\mathrm{n}}\text{(f) =}\text{1}\text{2πi}\text{∝}{}_{\mathrm{\Gamma }}\text{g}{}_{\mathrm{n}}\text{(ζ) F(ζ) dζ}$, where F denotes the Laplace transform of f, Γ ? Ω is a simple closed contour chosen so that F is analytic outside and on Ω, and g_n is analytic on Ω. The specific functionals considered by this paper are patterned after the Lidstone functions, L_2n(f) = f⁽²ⁿ⁾(0) and L_{2n + 1}(f) = f⁽²ⁿ⁾(1), in that their sequence of generating functions {g_n} are “periodic.” Set g_{pn + k}(ζ) = h_k(ζ) ζ^pn, where p is a positive integer and each h_k (k = 0, 1,…, p ? 1) is analytic on Ω. We find necessary and sufficient conditions for $\text{f ∈ k[Ω]}\text{with}\text{L}{}_{\mathrm{n}}\text{(f) = 0 (n = 0, 1,…)}$. DeMar previously was able to find necessary conditions [7]. Next, we generalize {L_n} in several ways and find corresponding necessary and sufficient conditions.     

The behaviour of (k,…,kn,n−ik)ci/i! is asymptotically normal

Let N(n,i) = (k,…,kⁿ,n?ik)c^i/i, i = O.…,[n/k]. We prove that the random variable X_n such that $\text{P(X}{}_{\mathrm{<mtext>n</mtext>}}\text{= i) =}\text{N(n, i)}\text{Σ}{}_{\mathrm{<mtext>j</mtext>}}\text{N(n, j)}$ has asymptotically (n → ∞) a normal distribution and we give some combinatorial applications of this result.We also improve a result of Godsil [3] dealing with matchings in graph.