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: $\text{f}{}^1\text{(z}{}_1{}^1\text{,…,z}{}_{\mathrm{n}}{}^1\text{)}$.     

A method of feasible directions using function approximations,with applications to min max problems

This paper presents a demonstrably convergent method of feasible directions for solving the problem min{φ(ξ)| gⁱ(ξ)?0i=1,2,…,m}, which approximates, adaptively, both φ(x) and ▽φ(x). These approximations are necessitated by the fact that in certain problems, such as when $\text{φ(x) =}\text{max}\text{{f(x, y) ¦ y ? Ω}{}_{\mathrm{y}}\text{}}$, a precise evaluation of φ(x) and ▽φ(x) is extremely costly. The adaptive procedure progressively refines the precision of the approximations as an optimum is approached and as a result should be much more efficient than fixed precision algorithms.It is outlined how this new algorithm can be used for solving problems of the form $\text{min}{}_{\mathrm{y}}\text{? Ω}{}_{\mathrm{x}}\text{max}{}_{\mathrm{y}}\text{? Ω}{}_{\mathrm{y}}\text{f(x, y)}$ under the assumption that Ωm_ξ={x|gⁱ(x)?0, j=1,…,s} ∩$\text{R}$ⁿ, Ω_y={y|ζ_i(y)?0, i-1,…,t} ∩ $\text{R}$^m, with f, g^j, ζⁱ continuously differentiable, f(x, ·) concave, ζⁱ convex for $\text{i = 1,…, t,}\text{and}\text{Ω}{}_{\mathrm{x}}\text{, Ω}{}_{\mathrm{y}}$ compact.     

Segmental solutions of Boolean equations

Davio and Deschamps have shown that the solution set, K, of a consistent Boolean equation $\text{?(x}{}_1\text{, …, x}{}_{\mathrm{n}}\text{)=0}$ over a finite Boolean algebra B may be expressed as the union of a collection of subsets of Bⁿ, each of the form {(x₁, …, x_n) | a_i≤x_i≤b_i, a_i?B, b_i?B, i = 1, …, n}. We refer to such subsets of Bⁿ as segments and to the collection as a segmental cover of K. We show that $\text{?(x}{}_1\text{, …, x}{}_{\mathrm{n}}\text{) = 1}$ is consistent if and only if ? can be expressed by one of a class of sum-of-products expressions which we call segmental formulas. The object of this paper is to relate segmental covers of K to segmental formulas for ?.     

Existence of stable equilibrium point for dynamical systems of Volterra type

This paper presents sufficient conditions for the existence of a nonnegative and stable equilibrium point of a dynamical system of Volterra type, (1) $\text{(}\text{d}\text{dt}\text{) x}{}_{\mathrm{i}}\text{(t) = ?x}{}_{\mathrm{i}}\text{(t)[f}{}_{\mathrm{i}}\text{(x}{}_1\text{(t),…, x}{}_{\mathrm{n}}\text{(t)) ? q}{}_{\mathrm{i}}\text{], i = 1,…, n}$, for every q = (q₁,…, q_n)^T?Rⁿ. Results of a nonlinear complementarity problem are applied to obtain the conditions. System (1) has a nonnegative and stable equilibrium point if (i) f(x) = (f₁(x),…,f_n(x))^T is a continuous and differentiable M-function and it satisfies a certain surjectivity property, or (ii), f(x) is continuous and strongly monotone on R₊₀ⁿ.     

Uniqueness and continuous dependence results for solutions of singular differential equations

Solutions of Cauchy problems for the singular equations $\text{u}{}_{\mathrm{tt}}\text{+ (}\text{Ψ(t)}\text{t}\text{) u}{}_{\mathrm{t}}\text{= Mu}$ (in a Hilbert space setting) and $\text{u}{}_{\mathrm{t}}\text{+ Δu +}\text{∑}\text{m}\text{i=1}\text{((}\text{k}{}_{\mathrm{i}}\text{x}{}_{\mathrm{i}}\text{)(}\text{?}{}_{\mathrm{i}}\text{?}{}_{\mathrm{i}}\text{)) + g(t)u=0}\text{in}\text{ω × |0,T)}$, ω={(x₁,…,x_M)ε$\text{R}$^m: 0 < x_i < c_i for each i=1,…,m} are shown to be unique and to depend Hölder continuously on the initial data in suitably chosen measures for 0?t < T < ∞. Logarithmic convexity arguments are used to derive the inequalities from which such results can be deduced.     

The KMS condition and spectral passivity in group duality

Let (A, G, α) be a C¹-dynamical system, where G is abelian, and let φ be an invariant state. Suppose that there is a neighbourhood Ω of the identity in $\text{G}\text{?}$ and a finite constant κ such that $\text{Π}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{n}}\text{φ(x}{}_{\mathrm{i}}{}^1\text{x}{}_{\mathrm{i}}\text{) ? κ}\text{Π}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{n}}\text{φ(x}{}_{\mathrm{i}}\text{x}{}_{\mathrm{i}}{}^1\text{)}$ whenever x_i lies in a spectral subspace $\text{R}{}^{\mathrm{\alpha }}\text{(Ω}{}_{\mathrm{i}}\text{)}$, where $\text{Ω}{}_1\text{+ … + Ω}{}_{\mathrm{n}}\text{? Ω}$. This condition of complete spectral passivity, together with self-adjointness of the left kernel of φ, ensures that φ satisfies the KMS condition for some one-parameter subgroup of G.     

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

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

.     

Numerical approximation of product integrals

A technique for the numerical approximation of matrix-valued Riemann product integrals is developed. For a ? x < y ? b, I_m(x, y) denotes

$\text{∫}\text{χ}\text{y}\text{∫}\text{χ}\text{v}{}_2\text{?}\text{∫}\text{χ}\text{v}{}_2\text{∏}\text{i=1}\text{m}\text{F(ν}{}_{\mathrm{i}}\text{)dν}{}_1\text{dν}{}_2\text{?dν}{}_{\mathrm{m}}$

, and A_m(x, y) denotes an approximation of I_m(x, y) of the form

$\text{(y?x)}{}^{\mathrm{m}}\text{∑}\text{k=1}\text{n}\text{a}{}_{\mathrm{k}}\text{∏}\text{i=1}\text{m}\text{F(χ}{}_{\mathrm{ik}}\text{)}$

, where a_k and y_ik are fixed numbers for i = 1, 2,…, m and k = 1, 2,…, N and x_ik = x + (y ? x)y_ik. The following result is established. If p is a positive integer, F is a function from the real numbers to the set of w × w matrices with real elements and F⁽¹⁾ exists and is continuous on [a, b], then there exists a bounded interval function H such that, if n, r, and s are positive integers, $\text{(b ? a)}\text{n}\text{= h < 1, x}{}_{\mathrm{i}}\text{= a + hi}\text{for}\text{i = 0, 1,…, n}\text{and}\text{0 < r ? s ? n}$, then

$\text{∏}\text{χ}{}_{\mathrm{r?}}\text{χ}{}_{\mathrm{s}}\text{(I+F dχ)?}\text{∏}\text{i=r}\text{s}\text{I+}\text{∑}\text{j=1}\text{p}\text{I}{}_{\mathrm{j}}\text{(χ}{}_{\mathrm{i?1}}\text{,χ}{}_{\mathrm{i}}\text{)}$

$\text{=h}{}^{\mathrm{p}}\text{H(χ}{}_{\mathrm{r?1}}\text{,χ}{}_{\mathrm{s}}\text{)+O(h}{}^{\mathrm{p+1}}\text{)}$

Further, if F^(j) exists and is continuous on [a, b] for j = 1, 2,…, p + 1 and A is exact for polynomials of degree less than p + 1 ? j for j = 1, 2,…, p, then the preceding result remains valid when A_j is substituted for I_j.     

Factorization of fredholm operators on analytic functions

Let Ω be a simply connected domain in the complex plane, and $\text{A(Ω}{}^{\mathrm{n}}\text{)}$, the space of functions which are defined and analytic on $\text{Ω}{}^{\mathrm{n}}$, if K is the operator on elements $\text{u(t, a}{}_1\text{, …, a}{}_{\mathrm{n}}\text{)}\text{of}\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ defined in terms of the kernels k_i(t, s, a₁, …, a_n) in $\text{A(Ω}{}^{\mathrm{n\; +\; 2}}\text{)}$ by is the identity operator on $\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$, then the operator I ? K may be factored in the form (I ? K)(M ? W) = (I ? ΠK)(M ? ΠW). Here, W is an operator on $\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ defined in terms of a kernel w(t, s, a₁, …, a_n) in $\text{A(Ω}{}^{\mathrm{n\; +\; 2}}\text{)}$ by Wu = ∝_an^tw(t, s, a₁, …, a_n) u(s, a₁, …, a_n) ds. ΠW is the operator; ΠWu = ∝_{an ? 1}w(t, s, a₁, …, a_n) u(s, a₁, …, a_n) ds. ΠK is the operator; ΠKu = ∑_{i = 1}^{n ? 1} ∝_ai^tk_i(t, s, a₁, …, a_n) ds + ∝_{an ? 1}^tk_n(t, s, a₁, …, a_n) u(s, a₁, …, a_n) ds. The operator M is of the form m(t, a₁, …, a_n)I, where $\text{m ? A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ and maps elements of $\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ into itself by multiplication. The function m is uniquely derived from K in the following manner. The operator K defines an operator $\text{K}{}^1$ on functions u in $\text{A(Ω}{}^{\mathrm{n\; +\; 2}}\text{)}$, by . A determinant $\text{δ(I ? K}{}^1\text{)}$ of the operator $\text{I ? K}{}^1$ is defined as an element $\text{m}{}^1\text{(t, a}{}_1\text{, …, a}{}_{\mathrm{n\; +\; 1}}\text{)}$ of $\text{A(Ω}{}^{\mathrm{n\; +\; 2}}\text{)}$. This is mapped into $\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ by setting a_{n + 1} = t to give m(t, a₁, …, a_n). The operator I ? ΠK may be factored in similar fashion, giving rise to a chain factorization of I ? K. In some cases all the matrix kernels k_i defining K are separable in the sense that k_i(t, s, a₁, …, a_n) = P_i(t, a₁, …, a_n) Q_i(s, a₁, …, a_n), where P_i is a 1 × p_i matrix and Q_i is a p_i × 1 matrix, each with elements in $\text{A(Ω}{}_{\mathrm{n\; +\; 1}}\text{)}$, explicit formulas are given for the kernels of the factors W. The various results are stated in a form allowing immediate extension to the vector-matrix case.     

Nonuniform right definiteness

The multiparameter eigenvalue problem W_m(λ) x_m = x_m, $\text{W}{}_{\mathrm{m}}\text{(λ) = T}{}_{\mathrm{m}}\text{+}\text{∑}\text{n = 1}\text{k}\text{λ}{}_{\mathrm{n}}\text{V}{}_{\mathrm{mn}}$, m = 1,…, k, where /gl /gE $\text{C}$^k, x_m is a nonzero element of the separable Hilbert space H_m, and T_m and V_mn are compact symmetric is studied. Various properties, including existence and uniqueness, of λ = λⁱ ? $\text{C}$^k for which the i_mth greatest eigenvalue of W_m(λⁱ) equals one are proved. “Right definiteness” is assumed, which means positivity of the determinant with (m, n)th entry (y_m, V_mny_m) for all nonzero y_m?H_m, m = 1 … k. This gives a “Klein oscillation theorem” for systems of an o.d.e. satisfying a definiteness condition that is usefully weaker than in previous such results. An expansion theorem in terms of the corresponding eigenvectors x_mⁱ is also given, thereby connecting the abstract oscillation theory with a result of Atkinson.     

Definition and characterization of multivariate negative binomial distribution

The probability generating function (pgf) of an n-variate negative binomial distribution is defined to be [β(s₁,…,s_n)]^?k where β is a polynomial of degree n being linear in each s_i and k > 0. This definition gives rise to two characterizations of negative binomial distributions. An n-variate linear exponential distribution with the probability function h(x₁,…,x_n)$\text{exp}\text{(}\text{Σ}{}_{\mathrm{i=1}}{}^{\mathrm{n}}\text{θ}{}_{\mathrm{i}}\text{x}{}_{\mathrm{i}}\text{)}\text{f(θ}{}_1\text{,…,θ}{}_{\mathrm{n}}\text{)}$ is negative binomial if and only if its univariate marginals are negative binomial. Let S_t, t = 1,…, m, be subsets of {s₁,…, s_n} with empty ∩_t=1^mS_t. Then an n-variate pgf is of a negative binomial if and only if for all s in S_t being fixed the function is of the form of the pgf of a negative binomial in other s's and this is true for all t.     

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.     

Monotone Splitting of matrices

Given the iterative scheme xⁱ⁺¹ = BTxⁱ + r where B, T are fixed n×:n real matrices, r a fixed real n-vector and xⁱ a real n-vector we investigate the convergence and monotonicity of schemes of the type

$\text{v}{}^{\mathrm{i+1}}\text{w}{}^{\mathrm{i+1}}\text{=}\text{B}\text{O}\text{O}\text{B}\text{S}{}_{11}\text{?S}{}_{12}\text{?S}{}_{21}\text{S}{}_{22}\text{v}{}^{\mathrm{i}}\text{w}{}^{\mathrm{i}}\text{+}\text{r}\text{r}$

, where S_ij are n×:n real matrices related to T. The n-vector iterates vⁱ,wⁱ bracket in a certain sense solutions x of x = BTx + r. We also give necessary and sufficient conditions for the monotonicity of the original iterative scheme itself xⁱ⁺¹ = BTxⁱ + r. This leads to monotonici results for classical iterative schemes such as the Jacobi, Gauss-Seidel, and successive overrelaxation methods.     

“Integer-making” theorems

Let X = {x₁, x₂,…} be a finite set and associate to every x_i a real number α_i. Let f(n) [g (n)] be the least value such that given any family $\text{F}$ of subsets of X having maximum degree n [cardinality n], one can find integers α_i, i=1,2,… so that α_i ? α_i|<1 and

$\text{∑}\text{x}{}_{\mathrm{i\; ?\; E}}\text{a}{}_{\mathrm{i}}\text{?}\text{∑}\text{x}{}_{\mathrm{i\; ?\; E}}\text{α}{}_{\mathrm{i}}\text{≤?(n)}\text{∑}\text{x}{}_{\mathrm{i\; ?\; E}}\text{a}{}_{\mathrm{i}}\text{?}\text{∑}\text{x}{}_{\mathrm{i\; ?\; E}}\text{α}{}_{\mathrm{i}}\text{≤}\text{g(n)}$

for all $\text{E ?}\text{F}$. We prove

$\text{f(n)≤n ? 1 and g(n)≤c(n}\text{log}\text{n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$

.     

Multiplicativity of lp norms for matrices. II

For 1 ? p ? ∞, let

$\text{|A|}{}_{\mathrm{p}}\text{=}\text{Σ}\text{i=1}\text{m}\text{Σ}\text{j=1}\text{n}\text{, |α}{}_{\mathrm{ij}}\text{|}{}^{\mathrm{p}}{}^{\mathrm{<mtext>1</mtext><mtext>p</mtext>}}$

, be the l_p norm of an m × n complex A = (α_ij) ?C_{m × n}. The main purpose of this paper is to find, for any p, q ? 1, the best (smallest) possible constants τ(m, k, n, p, q) and σ(m, k, n, p, q) for which inequalities of the form

$\text{|AB|}{}_{\mathrm{p}}\text{? τ(m, k, n, p, q) |A|}{}_{\mathrm{p}}\text{|B|}{}_{\mathrm{q}}\text{, |AB|}{}_{\mathrm{p}}\text{? σ (m, k, n, p, q)|A|}{}_{\mathrm{q}}\text{|B|}{}_{\mathrm{p}}$

hold for all A?C_{m × k}, B?C_{k × n}. This leads to upper bounds for inner products on C^k and for ordinary l_p operator norms on C_{m × n}.     

Complementing and exactly covering sequences

The following is proved (in a slightly more general setting): Let α₁, …, α_m be positive real, γ₁, …, γ_m real, and suppose that the system [nα_i + γ_i], i = 1, …, m, n = 1, 2, …, contains every positive integer exactly once (= a complementing system). Then $\text{α}{}_{\mathrm{i}}\text{α}{}_{\mathrm{j}}$ is an integer for some i ≠ j in each of the following cases: (i) m = 3 and m = 4; (ii) m = 5 if all α_i but one are integers; (iii) m ? 5, two of the α_i are integers, at least one of them prime; (iv) m ? 5 and α_n ? 2ⁿ for n = 1, 2, …, m ? 4.For proving (iv), a method of reduction is developed which, given a complementing system of m sequences, leads under certain conditions to a derived complementing system of m ? 1 sequences.     

A determinant of symmetric forms

Let Δ(α + β) = |H_λ2?r+1| where H_r is the complete symmetric function in (α₁ + β₁), (α₂ + β₂), …, (α_n + β_n). It is proved that Δ(α + β) ? Δ(α) + Δ(β). This inequality is generalised for certain symmetric functions defined by Littlewood. Let . Then we prove that Ω(α + β) ? Ω(α) + Ω(β). Here λ₁, λ₂, λ₃, …, λ_n is a partition such that λ_n > λ_n?1 > ··· > λ₂ > λ₁.     

Decomposability of symmetric multilinear functions

Let V denote a finite dimensional vector space over a field K of characteristic 0, let T_n(V) denote the vector space whose elements are the K-valued n-linear functions on V, and let S_n(V) denote the subspace of T_n(V) whose members are the fully symmetric members of T_n(V). If $\text{L}$_n denotes the symmetric group on {1,2,…,n} then we define the projection $\text{P}{}_{\mathrm{<mtext>L</mtext>}}\text{: T}{}_{\mathrm{n}}\text{(V) → S}{}_{\mathrm{n}}\text{(V)}$ by the formula , where P_σ : T_n(V) → T_n(V) is defined so that P_σ(A)(y₁,y₂,…,y_n = A(y_σ(1),y_σ(2),…,y_σ(n)) for each A?T_n(V) and y_i?V, 1 ? i ? n. If $\text{x}{}_{\mathrm{i}}\text{? V}{}^1\text{, 1 ? i ? n}$, then x₁?x₂? … ?x_n denotes the member of T_n(V) such that $\text{(x}{}_1\text{?x}{}_2\text{· ? ? ?x}{}_{\mathrm{n}}\text{)(y}{}_1\text{,y}{}_2\text{,…,y}{}_{\mathrm{n}}\text{) = П}{}^{\mathrm{n}}{}_{\mathrm{i=1}}\text{x}{}_{\mathrm{i}}\text{(y}{}_{\mathrm{i}}\text{)}$ for each y₁ ,₂,…,y_n in V, and x₁·x₂… x_n denotes $\text{P}{}_{\mathrm{<mtext>L</mtext>}}\text{(x}{}_1\text{?x}{}_2\text{? … ?x}{}_{\mathrm{n}}\text{)}$. If B? S_n(V) and there exists $\text{x}{}_{\mathrm{i}}\text{? V}{}^1\text{, 1 ? i ? n}$, such that B = x₁·x₂…x_n, then B is said to be decomposable. We present two sets of necessary and sufficient conditions for a member B of S_n(V) to be decomposable. One of these sets is valid for an arbitrary field of characteristic zero, while the other requires that K = R or C.