On the rate for uniform strong consistency of empirical distributions of independent nonidentically distributed multivariate random variables

Let X_j = (X_1j ,…, X_pj), j = 1,…, n be n independent random vectors. For x = (x₁ ,…, x_p) in R^p and for α in [0, 1], let F_j${}^1$(x) = αI(X_1j < x₁ ,…, X_pj < x_p) + (1 ? α) I(X_1j ≤ x₁ ,…, X_pj ≤ x_p), where I(A) is the indicator random variable of the event A. Let F_j(x) = E(F_j${}^1$(x)) and D_n = sup_{x, α} max_{1 ≤ N ≤ n} |Σ₀ⁿ(F_j${}^1$(x) ? F_j(x))|. It is shown that P[D_n ≥ L] < 4pL exp{?2(L²n^?1 ? 1)} for each positive integer n and for all L² ≥ n; and, as n → ∞, $\text{D}{}_{\mathrm{n}}\text{= 0((n}\text{log}\text{n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ with probability one.     

Positive definite Hermitian matrices and reproducing kernels

Some quadratic identities associated with positive definite Hermitian matrices are derived by use of the theory of reproducing kernels. For example, the following identity is obtained: Let{A_j}^m_j=1 be N × N positive definite Hermitian matrices. Then, for any complex vector x ∈ $\text{C}$^N, we have the identity

$\text{x}{}^1\text{∑}\text{j=}\text{1}\text{m}\text{A}{}^{-1}{}_{\mathrm{j}}{}^{-1}\text{x = min}\text{∑}\text{j=}\text{1}\text{m}\text{x}{}^1{}_{\mathrm{j}}\text{A}{}_{\mathrm{j}}\text{x}{}_{\mathrm{j}}$

. The minimum is taken here over all the decompositions x =∑^m_j=1x_j. This identity gives, in a sense, a precise converse for an inequality which was derived by T. Ando. Moreover, this paper shows that the sum of two reproducing kernels is naturally related to the harmonic-arithmetic-mean inequality for matrices and also that the geometric-arithmetic-mean inequality for matrices can be naturally interpreted in terms of tensor-product spaces.     

The Dirichlet problem for the Kohn Laplacian on the Heisenberg group,II

Let L = ∑_{j = 1}^mX_j² be sum of squares of vector fields in $\text{R}$ⁿ satisfying a Hörmander condition of order 2: span{X_j, [X_i, X_j]} is the full tangent space at each point. A point x??D of a smooth domain D is characteristic if X₁,…, X_m are all tangent to ?D at x. We prove sharp estimates in non-isotropic Lipschitz classes for the Dirichlet problem near (generic) isolated characteristic points in two special cases: (a) The Grushin operator $\text{?}{}^2\text{?x}{}^2\text{+ x}{}^2\text{?}{}^2\text{?t}{}^2$ in $\text{R}$². (b) The real part of the Kohn Laplacian on the Heisenberg group $\text{∑}{}_{\mathrm{j\; ?\; 1}}{}^{\mathrm{n}}\text{(}\text{?}\text{?x}{}_{\mathrm{j}}\text{+ 2y}{}_{\mathrm{j}}\text{?}\text{?t}\text{)}{}^2\text{+ (}\text{?}\text{?y}{}_{\mathrm{j}}\text{? 2x}{}_{\mathrm{j}}\text{?}\text{?t}\text{)}{}^2$ in $\text{R}$^{2n + 1}. In contrast to non-characteristic points, C^∞ regularity may fail at a characteristic point. The precise order of regularity depends on the shape of ?D at x.     

Temporally inhomogeneous scattering theory

A theory of scattering for the time dependent evolution equations $\text{du}\text{dt}\text{= iH}{}_{\mathrm{j}}\text{(t)u, j = 0, 1}$ (¹) is developed. The wave operators are defined in terms of the evolution operators U_j(t, s), which govern (¹). The scattering operator remains unitary. Sufficient conditions for existence and completeness of the wave operators are obtained; these are the main results. General properties, such as the chain rule and various intertwining relations, are also established. Applications include potential scattering (H₀(t) = ?Δ, Δ denoting the Laplacian, and H₁(t) = ?Δ + q(t, ·)) and scattering for second-order differential operators with coefficients constant in the spatial variable ($\text{H}{}_{\mathrm{j}}\text{(t) = ∑}{}_{\mathrm{m,\; k\; =\; 1}}{}^{\mathrm{n}}\text{a}{}_{\mathrm{mk}}{}^{\mathrm{(j)}}\text{(t)(}\text{?}{}^2\text{?x}{}_{\mathrm{m}}\text{?x}{}_{\mathrm{k}}\text{) + b}{}_{\mathrm{j}}\text{(t)}\text{for}\text{j = 0, 1}$).     

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.     

On the number of groups of a given order

Letting G(n) denote the number of nonisomorphic groups of order n, it is shown that for square-free n, G(n) ≤ ?(n) and G(n) ≤ (log n)^c on a set of positive density. Letting F_k(x) denote the number of n ≤ x for which G(n) = k, it is shown that $\text{F}{}_2\text{(x) =}\text{O(x(}\text{log}{}_4\text{x)}\text{(}\text{log}{}_3\text{x)}{}^2\text{)}$, where log_rx denotes the r-fold iterated logarithm.     

A note on polynomial matrix functions over a finite field

Let F_n denote the ring of n×n matrices over the finite field F=GF(q) and let A(x)=A_Nx^N+ ?+ A₁x+A₀?F_n[x]. A function $\text{?:F}{}_{\mathrm{n}}\text{→F}{}_{\mathrm{n}}$ is called a right polynomial function iff there exists an A(x)?F_n[x] such that $\text{?(B)=A}{}_{\mathrm{N}}\text{B}{}^{\mathrm{N}}\text{+?+A}{}_1\text{B+ A}{}_0$ for every B?F_n. This paper obtains unique representations for and determines the number of right polynomial functions.     

Temporal exponential decay for the Stark effect in atoms

Let $\text{H}{}_1\text{= ?∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{N}}\text{(Δ}{}_{\mathrm{i}}\text{+ V(}\text{x}{}_{\mathrm{i}}\text{)) + ∑}{}_{\mathrm{1\; ?\; i\; <j\; ?\; N}}\text{¦}\text{x}{}_{\mathrm{i}}\text{?}\text{x}{}_{\mathrm{j}}\text{¦}{}^{\mathrm{?1}}\text{, V(}\text{x}{}_{\mathrm{i}}\text{) = N ∝ ¦}\text{x}\text{?}\text{y}\text{¦}{}^{\mathrm{?1}}\text{?(}\text{y}\text{)d}\text{y}$, with ? a normalized Gaussian. Suppose $\text{E}$ ≠ 0 and that $\text{H = H}{}_1\text{+}\text{E}\text{· (∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{N}}\text{x}{}_{\mathrm{i}}\text{)}$ has no eigenfunctions in L²($\text{R}$^3N. If H₁ψ = μψ with μ < infσ_ess(H₁), then (ψ, e^?itHψ) decays exponentially at a rate governed by the positions of the resonances of H.     

Singular perturbation problems for systems of partial differential equations of elliptic type

Elliptic boundary value problems for systems of nonlinear partial differential equations of the form $\text{F}{}_{\mathrm{i}}\text{(x, u}{}_1\text{, u}{}_2\text{,…, u}{}_{\mathrm{N}}\text{,}\text{?u}{}_{\mathrm{i}}\text{?x}{}_{\mathrm{j}}\text{, ?}{}^{\mathrm{p}}{}_{\mathrm{i}}\text{?}{}^2\text{u}{}_{\mathrm{i}}\text{?x}{}_{\mathrm{j}}\text{?x}{}_{\mathrm{k}}\text{) = ?}{}_{\mathrm{i}}\text{(x), x ? R}{}^{\mathrm{n}}$, i = 1(1)N, j, k = 1(1)n, p_i ? 0, ? being a small parameter, with Dirichlet boundary conditions are considered. It is supposed that a formal approximation Z is given which satisfies the boundary conditions and the differential equations upto the order χ(?) = o(1) in some norm. Then, using the theory of differential inequalities, it is shown that under certain conditions the difference between the exact solution u of the boundary value problem and the formal approximation Z, taken in the sense of a suitable norm, can be made small.     

Isospectral matrices and classical polynomials

The matrices of order n defined, in terms of the n arbitrary numbers x_j, by the formulae $\text{X=}\text{diag}\text{(x}{}_{\mathrm{j}}\text{)}\text{and}\text{Z}{}_{\mathrm{jk}}\text{=δ}{}_{\mathrm{jk}}\text{∑′}\text{l=1}\text{n}\text{(x}{}_{\mathrm{j}}\text{?x}{}_{\mathrm{l}}\text{)}{}^{\mathrm{?1}}\text{+(1?δ}{}_{\mathrm{jk}}\text{(x}{}_{\mathrm{j}}\text{?x}{}_{\mathrm{k}}\text{)}{}^{\mathrm{?1}}$, are representations of the multiplicative operator ξ and of the differential operator d/dξ in a space spanned by the polynomials in ξ of degree less than n. This elementary fact implies a number of remarkable formulae involving these matrices, including novel representations of the classical polynomials.     

Product formulas for semigroups with elliptic generators

Real constant coefficient nth order elliptic operators, Q, which generate strongly continuous semigroups on L²($\text{R}$^k) are analyzed in terms of the elementary generator,

$\text{A = (?n)}{}^{\mathrm{<mtext>(</mtext><mtext>n</mtext><mtext>2</mtext><mtext>\; ?\; 1</mtext>}}\text{)(n!)}{}^{\mathrm{?1}}\text{∑}\text{k}\text{j = 1}\text{?}{}^{\mathrm{n}}\text{?x}{}_{\mathrm{j}}{}^{\mathrm{n}}$

, for n even. Integral operators are defined using the fundamental solutions p_n(x, t) to u_t = Au and using real polynomials q_l,…, q_k on $\text{R}$^m by the formula, for q = (q_l,…, q_k),

$\text{(F(t)?)(x) = ∫}$

_$\text{R}$m

$\text{?(x + q(z)) P}{}_{\mathrm{n}}\text{(z, t)dz}$

. It is determined when, strongly on L²($\text{R}$^k),

$\text{e}{}^{\mathrm{tQ}}\text{=}\text{lim}\text{j → ∞}\text{F}\text{t}\text{j}{}^{\mathrm{j}}$

. If n = 2 or k = 1, this can always be done. Otherwise the symbol of Q must have a special form.     

Subproper splitting for rectangular matrices

In this paper iterative schemes for approximating a solution to a rectangular but consistent linear system Ax = b are studied. Let A?C^{m × n}_r. The splitting A = M ? N is called subproper if R(A) ? R(M) and $\text{R(A}{}^1\text{) ?R(M}{}^1\text{)}$. Consider the iteration $\text{x}{}_{\mathrm{i}}\text{= M}{}^2\text{Nx}{}_{\mathrm{i}}{}_{\mathrm{?1}}\text{+ M}{}^2\text{b}$. We characterize the convergence of this scheme to a solution of the linear system. When A?R^m×ⁿ_r, monotonicity and the concept of subproper regular splitting are used to determine a necessary and a sufficient condition for the scheme to converge to a solution.     

On tensor products of certain group C1-algebras

If A and B are C¹-algebras there is, in general, a multiplicity of C¹-norms on their algebraic tensor product A ⊙ B, including maximal and minimal norms ν and α, respectively. A is said to be nuclear if α and ν coincide, for arbitrary B. The earliest example, due to Takesaki [11], of a nonnuclear C¹-algebra was $\text{C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{)}$, the C¹-algebra generated by the left regular representation of the free group on two generators F₂. It is shown here that W¹-algebras, with the exception of certain finite type I's, are nonnuclear.If $\text{C}{}^1\text{(F}{}_2\text{)}$ is the group C¹-algebra of F₂, there is a canonical homomorphism λ_l of $\text{C}{}^1\text{(F}{}_2\text{)}$ onto $\text{C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{)}$. The principal result of this paper is that there is a norm ζ on $\text{C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{) ⊙ C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{)}$, distinct from α, relative to which the homomorphism $\text{λ ⊙ λ}{}_{\mathrm{l}}\text{: C}{}^1\text{(F}{}_2\text{) ⊙ C}{}^1\text{(F}{}_2\text{) → C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{) ⊙ C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{)}$ is bounded ($\text{C}{}^1\text{(F}{}_2\text{) ⊙ C}{}^1\text{(F}{}_2\text{)}$ being endowed with the norm α). Thus quotients do not, in general, respect the norm α; a consequence of this is that the set of ideals of the α-tensor product of C¹-algebras A and B may properly contain the set of product ideals {$\text{I ? B + A ? J: I ? A, J ? B}$}.Let A and B be C¹-algebras. If A or B is a W¹-algebra there are on A ⊙ B certain C¹-norms, defined recently by Effros and Lance [3], the definitions of which take account of normality. In the final section of the paper it is shown by example that these norms, with α and ν, can be mutually distinct.     

Nonhomogeneous dirichlet problem for elliptic operators with axially discontinuous coefficients

Si studia, in un cilindro, il problema di Dirichlet per l'equazione ellittica del II ordine: $\text{L}{}_{\mathrm{\alpha }}\text{u = ?}$, dove $\text{L}{}_{\mathrm{\alpha }}\text{=}\text{αΔ + (1 ? 3α)∑}{}_{\mathrm{ij\; =\; 1}}{}^2\text{x}{}_{\mathrm{i}}\text{x}{}_{\mathrm{j}}\text{(x}{}_1{}^2\text{+ x}{}_2{}^2\text{)}{}^{\mathrm{?1}}\text{?}{}^2\text{?x}{}_{\mathrm{i}}\text{?x}{}_{\mathrm{j}}\text{, α ? (0,}\text{1}\text{3}\text{]}$è l'operatore a coefficienti discontinui sull'asse x₃ già introdotto da N. Ural'tseva per mostrare che l'equazione considerata può non avere soluzione nello spazio di Sobolev W^2,p(p > 2) per qualche f?L^p. In questo lavoro si danno limitazioni a priori e teoremi di esistenza e unicità in W^2,p quando p varia in un intervallo (p₁(α), p₂(α)), dipendente dalla costante di ellitticità α. Se p = p₂(α) le limitazioni a priori cadono: l'esempio è quello di Ural'tseva.     

On the probability that k positive integers are relatively prime II

Given a set S of positive integers let $\text{Z}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(t)}$ denote the number of k-tuples 〈m₁, …, m_k〉 for which $\text{m}{}_{\mathrm{i}}\text{∈ S ? [1, t]}$ and (m₁, …, m_k) = 1. Also let $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n)}$ denote the probability that k integers, chosen at random from $\text{S ? [1, n]}$, are relatively prime. It is shown that if P = {p₁, …, p_r} is a finite set of primes and S = {m : (m, p₁ … p_r) = 1}, then $\text{Z}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(t) = (td(S))}{}^{\mathrm{k}}\text{Π}{}_{\mathrm{\nu ?P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{) + O(t}{}^{\mathrm{k?1}}\text{)}$ if k ≥ 3 and $\text{Z}{}_2{}^{\mathrm{S}}\text{(t) = (td(S))}{}^2\text{Π}{}_{\mathrm{p?P}}\text{(1 ?}\text{1}\text{p}{}^2\text{) + O(t}\text{log}\text{t)}$ where d(S) denotes the natural density of S. From this result it follows immediately that $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n) → Π}{}_{\mathrm{p?P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{) = (ζ(k))}{}^{\mathrm{?1}}\text{Π}{}_{\mathrm{p\in P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{)}{}^{\mathrm{?1}}$ as n → ∞. This result generalizes an earlier result of the author's where $\text{P = ?}$ and S is then the whole set of positive integers. It is also shown that if S = {p₁^x₁ … p_r^x_r : x_i = 0, 1, 2,…}, then $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n) → 0}$ as n → ∞.     

Infinite-variate wide-sense Markov processes and functional analysis for bounded operator-forming vectors

Let p, q be arbitrary parameter sets, and let $\text{H}$ be a Hilbert space. We say that x = (xⁱ)_i?q, xⁱ ? $\text{H}$, is a bounded operator-forming vector (?$\text{H}$_F^q) if the Gram matrix 〈x, x〉 = [(xⁱ, x^j)]_i?q,j?q is the matrix of a bounded (necessarily ≥ 0) operator on $\text{l}{}_{\mathrm{q}}{}^2$, the Hilbert space of square-summable complex-valued functions on q. Let A be p × q, i.e., let A be a linear operator from $\text{l}{}_{\mathrm{q}}{}^2$ to $\text{l}{}_{\mathrm{p}}{}^2$. Then exists a linear operator ǎ from (the Banach space) $\text{H}$_F^q to $\text{H}$_F^p on $\text{D}$(A) = {x:x ? $\text{H}$_F^q, $\text{A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is p × q bounded on $\text{l}{}_{\mathrm{q}}{}^2$} such that y = ǎx satisfies y^j?σ(x) = {space spanned by the xⁱ}, 〈y, x〉 = A〈x, x〉 and $\text{〈y, y〉 = A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}{}^1$. This is a generalization of our earlier [J. Multivariate Anal.4 (1974), 166–209; 6 (1976), 538–571] results for the case of a spectral measure concentrated on one point. We apply these tools to investigate q-variate wide-sense Markov processes.     

Asymptotic properties of general symmetric hyperbolic systems

Results on partition of energy and on energy decay are derived for solutions of the Cauchy problem $\text{?u}\text{?t}\text{+ ∑}{}_{\mathrm{j\; =\; 1}}{}^{\mathrm{n}}\text{A}{}_{\mathrm{j}}\text{?u}\text{?x}{}_{\mathrm{j}}\text{= 0, u(0, x) = ?(x)}$. Here the A_j's are constant, k × k Hermitian matrices, x = (x₁,…, x_n), t represents time, and u = u(t, x) is a k-vector. It is shown that the energy of Mu approaches a limit $\text{E}{}_{\mathrm{M}}\text{(?)}\text{as}\text{¦ t ¦ → ∞}$, where M is an arbitrary matrix; that there exists a sufficiently large subspace of data ?, which is invariant under the solution group U₀(t) and such that $\text{U}{}_0\text{(t)? = 0}\text{for}\text{¦ x ¦ ? a ¦ t ¦ ? R, a}\text{and}\text{R}$ depending on ? and that the local energy of nonstatic solutions decays as $\text{¦ t ¦ → ∞}$. More refined results on energy decay are also given and the existence of wave operators is established, considering a perturbed equation $\text{E(x)}\text{?u}\text{?t}\text{+ ∑}{}_{\mathrm{j\; =\; 1}}{}^{\mathrm{n}}\text{A}{}_{\mathrm{j}}\text{?u}\text{?x}{}_{\mathrm{j}}\text{= 0,}\text{where}\text{¦ E(x) ? I ¦ = O(¦ x ¦}{}^{\mathrm{?1\; ?\; ?}}\text{)}$ at infinity.     

A new form of trigonometric orthogonality and Gaussian-type quadrature

Gauss's (2n+1)-point trigonometric interpolation formula, based upon f(x_i), i = 1(1)2n+1, gives a trigonometric sum of the n^th order, S_2n+1(x = a₀ + ∑_jⁿ = 1^(a_jcos jx + b_jsin jx), which may be integrated to provide formulas for either direct quadrature or stepwise integration of differential equations having periodic (or near-periodic) solutions. An “orthogonal” trigonometric sum S_2r+1(x) is one that satisfies

$\text{∫}{}_{\mathrm{a}}{}^{\mathrm{b}}\text{S}{}_{\mathrm{2r+1}}\text{(x)S}{}_{\mathrm{2r\prime +1}}\text{(x)dx=0, r′<r}$

and two other arbitrarily imposable conditions needed to make S_2r1(x) unique. Two proofs are given of a fundamental factor theorem for any S_2n+1(x) (somewhat different from that for polynomials) from which we derive 2r-point Gaussian-type quadrature formulas, r = [n/2] + 1, which are exact for any S_4r?1(x). We have

$\text{∫}{}_{\mathrm{a}}{}^{\mathrm{b}}\text{S}{}_{\mathrm{4r?1(x)}}\text{dx=∑}{}_{\mathrm{j=1}}{}^{\mathrm{2r}}\text{A}{}_{\mathrm{j}}\text{S}{}_{\mathrm{4r?1}}\text{(x}{}_{\mathrm{j}}\text{)}$

where the nodes x_j, j = 1(1)2r, are the zeros of the orthogonal S_2r+1(x). It is proven that A_j > 0 and that 2r-1 of the nodes must lie within the interval [a,b], and the remaining node (which may or may not be in [a,b]) must be real. Unlike Legendre polynomials, any [a′,b′] other than a translation of [a,b], requires different and unrelated sets of nodes and weights. Gaussian-type quadrature formulas are applicable to the numerical integration of the Gauss (2n+1)-point interpolation formulas, with extra efficiency when the latter are expressed in barycentric form. S_2r+1(x), x_jandA_j, j = 1(1)2r, were calculated for [a,b] = [0, π/4], 2r = 2 and 4, to single-precision accuracy.