The numerical range of a tridiagonal operator

Let r be a real number and A a tridiagonal operator defined by Aej=ej−1+rjej+1, j=1,2,…, where {e₁,e₂,…} is the standard orthonormal basis for ?²(N). Such tridiagonal operators arise in Rogers-Ramanujan identities. In this paper, we study the numerical ranges of these tridiagonal operators and finite-dimensional tridiagonal matrices. In particular, when r=−1, the numerical range of the finite-dimensional tridiagonal matrix is the convex hull of two explicit ellipses. Applying the result, we obtain that the numerical range of the tridiagonal operator is the square

     

The inverse positivity of perturbed tridiagonal M-matrices

A well-known property of an M-matrix M is that the inverse is element-wise non-negative, which we write as M^-1?0. In this paper, we consider element-wise perturbations of non-symmetric tridiagonal M-matrices and obtain sufficient bounds on the perturbations so that the non-negative inverse persists. These bounds improve the bounds recently given by Kennedy and Haynes [Inverse positivity of perturbed tridiagonal M-matrices, Linear Algebra Appl. 430 (2009) 2312-2323]. In particular, when perturbing the second diagonals (elements (l,l+2) and (l,l-2)) of M, these sufficient bounds are shown to be the actual maximum allowable perturbations. Numerical examples are given to demonstrate the effectiveness of our estimates.     

Galerkin's method for operator equations with nonnegative index — With application to Cauchy singular integral equations

We consider solving operator equation (1) Hu + Ku = f, where H and K are bounded linear operators between two real Hilbert spaces H₁ and H₂. Operator H is assumed to have a finite-dimensional nullspace N(H) and a bounded right inverse H¹:H₂ → H₁ and K is compact. It follows that dim(N(H + K)) = dim(N(H)), so that to obtain uniqueness the m additional conditions (2) 〈u,φ_k〉₁ = b_k, k=1, 2, h.,dim(N(H)) = m are imposed, where the {φ_k}_{k = 1}^m are an orthonormal basis for N(H). To solve (1) and (2), these equations are converted to an equivalent equation of the second kind to which Galerkin's method is applied using the basis $\text{{φ}{}_1\text{, φ}{}_2\text{, …, φ}{}_{\mathrm{m}}\text{, ¦φ}{}_{\mathrm{m\; +\; 1}}\text{,…, φ}{}_{\mathrm{n}}\text{,…}}$. It was shown that this method is equivalent to the method of weighted residuals when $\text{H}{}^1\text{= H}{}^1$ (the adjoint of H). The results are applied to obtain convergence proofs of some numerical methods for solving several classes of Cauchy singular integral equations whose kernels are only square integrable.     

Two equivalent measures on weighted hypergraphs

Let H=(N,E,w) be a hypergraph with a node set N={0,1,…,n-1}, a hyperedge set E⊆2^N, and real edge-weights w(e) for e∈E. Given a convex n-gon P in the plane with vertices x₀,x₁,…,x_n-1 which are arranged in this order clockwisely, let each node i∈N correspond to the vertex x_i and define the area A_P(H) of H on P by the sum of the weighted areas of convex hulls for all hyperedges in H. For 0?i<j<k?n-1, a convex three-cut C(i,j,k) of N is {{i,…,j-1}, {j,…,k-1}, {k,…,n-1,0,…,i-1}} and its size c_H(i,j,k) in H is defined as the sum of weights of edges e∈E such that e contains at least one node from each of {i,…,j-1}, {j,…,k-1} and {k,…,n-1,0,…,i-1}. We show that the following two conditions are equivalent:

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A_P(H)?A_P(H^′) for all convex n-gons P.

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c_H(i,j,k)?c_H^′(i,j,k) for all convex three-cuts C(i,j,k).

From this property, a polynomial time algorithm for determining whether or not given weighted hypergraphs H and H^′ satisfy “A_P(H)?A_P(H^′) for all convex n-gons P” is immediately obtained.     

Numerical experiments on the condition number of the interpolation matrices for radial basis functions

Through numerical experiments, we examine the condition numbers of the interpolation matrix for many species of radial basis functions (RBFs), mostly on uniform grids. For most RBF species that give infinite order accuracy when interpolating smooth f(x)—Gaussians, sech's and Inverse Quadratics—the condition number κ(α,N) rapidly asymptotes to a limit κasymp(α) that is independent of N and depends only on α, the inverse width relative to the grid spacing. Multiquadrics are an exception in that the condition number for fixed α grows as N². For all four, there is growth proportional to an exponential of 1/α (1/α² for Gaussians). For splines and thin-plate splines, which contain no width parameter, the condition numbers grows asymptotically as a power of N—a large power as the order of the RBF increases. Random grids typically increase the condition number (for fixed RBF width) by orders of magnitude. The quasi-random, low discrepancy Halton grid may, however, have a lower condition number than a uniform grid of the same size.     

On Formal Maps Between Generic Submanifolds in Complex Space

Let H:(M,p)→(M ^′,p ^′) be a formal mapping between two germs of real-analytic generic submanifolds in ? ^N with nonvanishing Jacobian. Assuming M to be minimal at p and M ^′ holomorphically nondegenerate at p ^′, we prove the convergence of the mapping H. As a consequence, we obtain a new convergence result for arbitrary formal maps between real-analytic hypersurfaces when the target does not contain any holomorphic curve. In the case when both M and M ^′ are hypersurfaces, we also prove the convergence of the associated reflection function when M is assumed to be only minimal. This allows us to derive a new Artin type approximation theorem for formal maps of generic full rank.     

Norm estimates related to self-commulators

For a given self-adjoint matrix H with ∥H∥ = 1 and tr(H) = 0, we consider the number γ(H) which is defined to be the minimum of ∥T∥² for those T satisfying [T¹,T] = H. We show that 1 ⩽ γ(H) ⩽ 2 and that γ(H) is close to 2 if H is suitably chosen.     

Square functions and <Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript> calculus on subspaces of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> and on Hardy spaces

Let X be a (closed) subspace of L^p with 1≤p<∞, and let A be any sectorial operator on X. We consider associated square functions on X, of the form and we show that if A admits a bounded H^∞ functional calculus on X, then these square functions are equivalent to the original norm of X. Then we deduce a similar result when X=H¹(ℝ^N) is the usual Hardy space, for an appropriate choice of || ||_F. For example if N=1, the right choice is the sum for h ∈ H¹(ℝ), where H denotes the Hilbert transform.     

The Estermann cubic problem with almost equal summands

We prove an asymptotic formula for the number of representations of a sufficiently large natural number N as the sum of two primes p ₁ and p ₂ and the cube of a natural numbermsatisfying the conditions |p _i ? N/3| ≤ H, |m ³ ? N/3| ≤ H, H ≥ N ^5/6 ? ¹⁰.     

Stability of nonlocal difference schemes in a subspace

We consider a family of two-layer difference schemes for the heat equation with nonlocal boundary conditions containing the parameter γ. In some interval γ ∈ (1, γ ₊), the spectrum of the main difference operator contains a unique eigenvalue λ ₀ in the left complex half-plane, while the remaining eigenvalues λ ₁, λ ₂, …, λ _N?1 lie in the right half-plane. The corresponding grid space H _N is represented as the direct sum H _N = H ₀⊕H _N?1 of a one-dimensional subspace and the subspace H _N?1 that is the linear span of eigenvectors µ⁽¹⁾, µ⁽²⁾, …, µ^(N?1). We introduce the notion of stability in the subspace H _N?1 and derive a stability criterion.     

Numerical shadows: Measures and densities on the numerical range

For any operator M acting on an N-dimensional Hilbert space H_N we introduce its numerical shadow, which is a probability measure on the complex plane supported by the numerical range of M. The shadow of M at point z is defined as the probability that the inner product (Mu, u) is equal to z, where u stands for a random complex vector from H_N, satisfying ||u||=1. In the case of N=2 the numerical shadow of a non-normal operator can be interpreted as a shadow of a hollow sphere projected on a plane. A similar interpretation is provided also for higher dimensions. For a hermitian M its numerical shadow forms a probability distribution on the real axis which is shown to be a one dimensional B-spline. In the case of a normal M the numerical shadow corresponds to a shadow of a transparent solid simplex in R^N-1 onto the complex plane. Numerical shadow is found explicitly for Jordan matrices J_N, direct sums of matrices and in all cases where the shadow is rotation invariant. Results concerning the moments of shadow measures play an important role. A general technique to study numerical shadow via the Cartesian decomposition is described, and a link of the numerical shadow of an operator to its higher-rank numerical range is emphasized.     

Approximating eigenfunctions of Fredholm operators in Banach spaces

A singular Fredholm operator A is perturbed by an operator of finite rank to obtain an invertible operator B. Theory previously developed for A and B in Hilbert spaces is extended here to Banach spaces. The operator B^?1 is used to construct independent elements in the null spaces N(A), N(A²),…, N(A^k), for some positive integer k, and a basis for N(A) and N(A²). The theory is used to compute approximations to eigenfunctions and generalized eigenfunctions of integral operators.     

Extremal edge problems for graphs with given hyperoctahedral automorphism group

For k > 1, let H_k denote the hyperoctahedral group S_k[S₂] of order 2^kk!. An (H_k, n)- graph is a graph on n vertices with automorphism group abstractly isomorphic to H_k. For each k an (H_k, n)-graph exists precisely when n ? 2k; for each n ? 2k the minimum and maximum number of edges possible for such graphs are determined. The analogous results for connected (H_k, n)-graphs are also obtained.     

On subword decomposition and balanced polynomials

Let H(x) be a monic polynomial over a finite field F=GF(q). Denote by Na(n) the number of coefficients in Hn which are equal to an element a∈F, and by G the set of elements a∈F^× such that Na(n)>0 for some n. We study the relationship between the numbers (Na(n))_a∈G and the patterns in the base q representation of n. This enables us to prove that for “most” n's we have Na(n)≈Nb(n), a,b∈G. Considering the case H=x+1, we provide new results on Pascal's triangle modulo a prime. We also provide analogous results for the triangle of Stirling numbers of the first kind.     

Exponential sets and their geometric motions

Let us call an “exponential set” in a C*-algebraA any set consisting of the exponentialse ^X of all the self-adjoint elementsX of a subspaceH ofA. For example, ifH = A the resulting exponential setG ⁺ consists of all the positive invertible elements ofA, and all other exponential sets are contained in G⁺. An exponential setC ? G⁺ inherits the geometric structure of the space G⁺ when the defining subspaceH has suitable properties. Here we investigate reasonable conditions onH that permit, for example, reduction of the canonical connection of G⁺ toC. As a consequence, in these cases the setC has a rich family of motions that are “rigid” for the geometry of G⁺. In particular we find thatC itself operates on C by the actionL _g a = (g^?1)*ag^? of the groupG of all invertible elements ofA in G⁺, and that the subgroup generated byC is transitive. Similarly, in several cases the productscu withc ε C andu unitary form a closed Lie subgroup ofG that acts onC, withC contained in it. This is the case forH, the space of elements of trace zero, when there is a trace. The conditions onH are all additions to the following basic situation:H is the kernel of a (bounded linear) projection Φ:A → A. For example, ifH is closed under triple brackets [X, [Y, Z]] then parallel transport in G⁺ along geodesics inC through 1 ∈C preserves vectors tangent toC. Similarly, if the symmetric part of [e ^X Ye ^?X ,Z] is inH for allX, Y, Z ∈H ^s thenC is “geodesically convex” in the sense that geodesics tangent toC stay inC. The most interesting cases correspond to a conditional expectation. Two additional conditions produce the groups described in the first paragraph: the case of a Z₂-graded C*-algebra with Φ the projection on the elements of degree 0 (which is automatically a conditional expectation) and the case of a conditional expectation such that the anti-symmetric part ofe ^X Ye ^?Y is in the range of Φ wheneverX, Y are self-adjoint and Φ(X)= Φ(Y) = 0. This is verified for example in the case of central traces.     

On the small sieve. II. sifting by composite numbers

For any set A of natural numbers let F(x, A) denote the number of natural numbers up to x that are divisible by no element of A and let H(x, K) be the maximum of F(x, A) when A runs over the sets not containing 1 and having a sum of reciprocals not greater than K. A logarithmic asymptotic formula is given for H(x, K)—in particular it shows H(x, K) < x^ε for K > K₀(ε)—and some related problems are discussed.     

The number of edge 3-colorings of a planar cubic graph as a permanent

In this paper, we establish that the number of edge 3-colorings of a finite planar cubic graph G, i.e., 3-colorings of its interchange graph H, is equal to 2^?N|Permanent(A)|, where N is the number of edges of G, and A is the 2N × 2N square matrix formed by repeating each row of the N × 2N vertex-directed edge incidence matrix of H (with an arbitrary orientation). As the number of 4-colorings of a finite maximal (fully triangulated) planar graph is equal to four times the number of edge 3-colorings of its planar cubic dual, this result gives a formula for the number of 4-colorings of a finite maximal planar graph.     

Order estimates for irreducible projections in L2 of a nilmanifold

Let π be an irreducible representation occurring in $\text{L}$²(Г?N), where N is a nilpotent Lie group and Γ is a discrete, cocompact subgroup. The projection onto the π-equivariant subspace is given by convolution against a distribution D_π. For certain π, we obtain an estimate on the order of D_π. The condition on π involves an extension of the “canonical objects” associated to elements of the Kirillov orbit of π; there does not appear to be an example in the literature where it is not satisfied.     

How many integer homogeneous polynomials at small coprime integers have value of a univariate polynomial?

Let p(y) = p _m y ^m + p _m?1 y ^m?1 + ?+ p ₀ ?? $ \mathbb{Z} $ [y] be a polynomial of degree m > 0 in an integer variable. We estimate the number of times it equals some homogeneous polynomial in two variables with integer coefficients, degree at most n, and Euclidean norm at most N evaluated at a pair of small coprime integers (we count this number with the occurring multiplicities). For pairs of coprime integers of absolute value at most $ H<N/\sqrt{n} $ , this estimate is ?? _n,p (H)N ^n+1/m + O(N ^n+1/m?1 H ³ + N ⁿ H ²), where ?? _n,p (H) does not depend on N.     

The disengagement algorithm or a new generalization of the exclusion algorithm

Let us consider a finite inf semilattice G with a set 1 of internal binary operations 1₁, isotono, satisfying certain conditions of no dispersion, of increasing and of substitution, and so that the greatest lower bound is distributive relatively to 1₁. A finite subset A of G being given, this article gives a method for enumerating the maximal elements of the sub-algebra A¹ generated by A with regard to 1, when A¹ is finite. This method, called disengagement algorithm, allows to examine each element once; it generalizes an algorithm giving the maximal n-rectangles of a part of a product of distributive lattices algorithm which already generalized a conjecture of Tison in Boolean algebra. Two applications are developed.