Optimal iterative processes for root-finding

Letf ₀(x) be a function of one variable with a simple zero atr ₀. An iteration scheme is said to be locally convergent if, for some initial approximationsx ₁, ...,x _s nearr ₀ and all functionsf which are sufficiently close (in a certain sense) tof ₀, the scheme generates a sequence {x _k} which lies nearr ₀ and converges to a zeror off. The order of convergence of the scheme is the infimum of the order of convergence of {x _k} for all such functionsf. We study iteration schemes which are locally convergent and use only evaluations off,f, ...,f ^[d] atx ₁, ...,x _k–1 to determinex _k, and we show that no such scheme has order greater thand+2. This bound is the best possible, for it is attained by certain schemes based on polynomial interpolation.This work was supported (in part) by the Office of Naval Research under contract numbers N0014-69-C-0023 and N0014-71-C-0112.     

A study of the van der Waerden conjecture and its generalizations

It Is shown here that a permanent of n× ndoubly stochastic matrix is not less than 1/ n!. The proof of this inequality follows from the study of a generalized van der Waerden problem on the set of doubly stochastic matrices.     

A characterization of transform absolute norms

Let 6x6 be a norm on Cⁿ. For A?M_n(C) denote by 6A6 the induced norm of A, by ?(A) the spectral radius of A. It is proved here that if inf_p6PAP^?16 = ?(A), then 6x6 is a transform absolute norm.     

Extremal eigenvalue problems defined for certain classes of functions

Archive for Rational Mechanics and Analysis -     

A duality theorem on a pair of simultaneous functional equations

Given P and Q convex compact sets in R^kandR^s, respectively, and u a continuous real valued function on P × Q, we consider the following pair of dual problems: Problem I—Minimize ? so that ?: $\text{P × Q → R}\text{and}\text{? ?}\text{Cav}{}_{\mathrm{p}}\text{Vex}{}_{\mathrm{q}}\text{×}\text{max}\text{(u, ?)}$. Problem II—Maximize g so that g: P × Q → R and g ? Vex_q × Cav_pmin(u, g). Here Cav_p is the operation of concavification of a function with respect to the variable p?P (for each fixed q?Q). Similarly, Vex_q is the operation of convexification with respect to q?Q. Maximum and minimum are taken here in the partial ordering of pointwise comparison: $\text{? ? g}\text{means}\text{?(p, q) ? g(p, q) ?(p, q) ? P × Q}$. It is proved here that both problems have the same solution which is also the unique simultaneous solution of the following pair of functional equations: (i) $\text{? =}\text{Vex}{}_{\mathrm{q}}\text{max}\text{(u, ?)}$. (ii) $\text{? =}\text{Cav}{}_{\mathrm{p}}\text{min}\text{(u, ?)}$. The problem arises in game theory, but the proof here is purely analytical and makes no use of game-theoretical concepts.     

Gravothermal collapse of self-interacting dark matter halos and the origin of massive black holes

Black hole formation is an inevitable consequence of relativistic core collapse following the gravothermal catastrophe in self-interacting dark matter (SIDM) halos. Very massive SIDM halos form supermassive black holes (SMBHs) > or about 10(6)M(middle dot in circle) directly. Smaller halos believed to form by redshift z = 5 produce seed black holes of (10(2)-10(3))M(middle dot in circle) which can merge and/or accrete to reach the observational SMBH range. This scenario for SMBH formation requires no baryons, no prior star formation, and no other black hole seed mechanism.     

Experimental identification of nonpointlike dark-matter candidates

We show that direct dark-matter detection experiments can distinguish between pointlike and nonpointlike dark-matter candidates. The shape of the nuclear recoil-energy spectrum from pointlike dark-matter particles, e.g., neutralinos, is determined by the velocity distribution of dark matter in the galactic halo and by nuclear form factors. Typical cross sections of nonpointlike dark matter, for example, Q-balls, have a new form factor, which decreases rapidly with the recoil energy. A signal from nonpointlike dark matter is expected to peak near the experimental threshold and to fall off rapidly at higher energies.