On the Kato square root problem

In the infinite-dimensional separable complex Hilbert space we construct new abstract examples of unbounded maximal accretive and maximal sectorial operators B for which $\mathrm{dom}{B}^{\frac{1}{2}}\ne \mathrm{dom}{B}^{?\frac{1}{2}}$. New criterions for the equality are established.     

An integer based square root algorithm

We propose a fast integer based method for computing square roots of floating point numbers. This implies high accuracy and robustness, since no precision will be lost during the computation. Only integer addition and shifts are necessary to obtain the square root. Comparisons made with the modified Newton method indicate that the suggested method is twice as fast for computing floating point square roots.     

Predator-prey dynamics with square root functional responses

A predator-prey model is considered in which a modified Lotka-Volterra interaction term is used as the functional response of the predator to the prey. The interaction term is proportional to the square root of the prey population, which appropriately models systems in which the prey exhibits strong herd structure implying that the predator generally interacts with the prey along the outer corridor of the herd. Because of the square root term, the solution behavior near the origin is more subtle and interesting than standard models and makes sense ecologically.     

A recurrence relation for the square root

The behavior of the sequence x_{n + 1} = x_n(3N − x_n²)/2N is studied for N > 0 and varying real x₀. When 0 < x₀ < (3N)^1/2 the sequence converges quadratically to N^1/2. When x₀ > (5N)^1/2 the sequence oscillates infinitely. There is an increasing sequence β_r, with β₋₁ = (3N)^1/2 which converges to (5N)^1/2 and is such that when β_r < x₀ < β_{r + 1} the sequence {x_n} converges to (−1)^rN^1/2. For x₀ = 0, β₋₁, β₀,… the sequence converges to 0. For x₀ = (5N)^1/2 the sequence oscillates: x_n = (−1)ⁿ(5N)^1/2. The behavior for negative x₀ is obtained by symmetry.     

Kato's square root problem in Banach spaces

Let L be an elliptic differential operator with bounded measurable coefficients, acting in Bochner spaces Lp(Rn;X) of X -valued functions on Rn. We characterize Kato's square root estimates and the H^∞-functional calculus of L in terms of R-boundedness properties of the resolvent of L, when X is a Banach function lattice with the UMD property, or a noncommutative Lp space. To do so, we develop various vector-valued analogues of classical objects in Harmonic Analysis, including a maximal function for Bochner spaces. In the special case X=C, we get a new approach to the Lp theory of square roots of elliptic operators, as well as an Lp version of Carleson's inequality.     

On duality for square root convex programs

Conjugate function theory is used to develop dual programs for nonseparable convex programs involving the square root function. This function arises naturally in finance when one measures the risk of a portfolio by its variance–covariance matrix, in stochastic programming under chance constraints and in location theory.     

Stable iterations for the matrix square root

Any matrix with no nonpositive real eigenvalues has a unique square root for which every eigenvalue lies in the open right half-plane. A link between the matrix sign function and this square root is exploited to derive both old and new iterations for the square root from iterations for the sign function. One new iteration is a quadratically convergent Schulz iteration based entirely on matrix multiplication; it converges only locally, but can be used to compute the square root of any nonsingular M-matrix. A new Padé iteration well suited to parallel implementation is also derived and its properties explained. Iterative methods for the matrix square root are notorious for suffering from numerical instability. It is shown that apparently innocuous algorithmic modifications to the Padé iteration can lead to instability, and a perturbation analysis is given to provide some explanation. Numerical experiments are included and advice is offered on the choice of iterative method for computing the matrix square root.     

Why the square root function is not linear

Six proofs are given for the fact that for each integer n?2, the nth root function, viewed as a function from the set of non-negative real numbers to itself, is not linear. If p is a prime number, then ?/p? is characterized, up to isomorphism, as the only integral domain D of characteristic p such that D admits a pth root function D→D which is linear. The first part of this note could find classroom use in courses on precalculus or calculus; the second part, in courses on abstract algebra.     

Growth of power series with square root gaps

For entire functionsf whose power series have Hadamard gaps with ratio ≥1+α>1, Gaier has shown that the condition |f(x)|≤e ^x forx≥0 implies |f(z)|≤C _αe^|z| (*) for allz. Here the result is extended to the case of square root gaps, that is, , with , where α>0. Smaller gaps cannot work. In connection with his proof of the general high indices theorem for Borel summability, Gaier had shown that square root gaps imply . Having such an estimate, one can adapt Pitt’s Tauberian method for the restricted Borel high indices theorem to show that, in fact, , which implies (*). The author also states an equivalent distance formula involving monomialsx ^pke^−xinL ^∞ (0, ∞).     

The unique square root of a positive semidefinite matrix

An easy way to present the uniqueness of the square root of a positive semidefinite matrix is given.     

The monotonic property of the generalized root mean square

The link between a differential equation and its initial conditions is not quite as simple as usually imagined. Several examples illustrate the complications that can arise.     

An extension of the square root law of TCP

Using probabilistic scaling methods, we extend the square root law of TCP to schemes which may not be of the AIMD type. Our results offer insight in the relationship between throughput and loss rate, and the time scale on which losses take place. Similar results are shown to hold in scenarios where dependencies between losses occur.     

Generalized concavity and duality with a square root term

Duality results for a class of nondifferentiable mathematical programming problems are given. These results allow for the weakening of the usual convexity conditions required for duality to hold. A pair of symmetric and self dual nondifferentiable programs under weaker convexity conditions are also given. A subgradient symmetric duality is proposed and its limitations discussed. Finally, a pair of nondifferentiable mathematical programs containing arbitrary norms is presented.     

Planar graphs with square or cube root are four-colorable

A square (cube) coloring of a graph is an assignment of colors to its points so that no two points of distance less than or equal to two (three) get the same color. We prove that planar graphs with a square or cube root are four colorable using a square (cube) coloring of their square (cube) root.     

Stability of a functional equation for square root spirals

By using an idea of Heuvers, Moak and Boursaw [1], we will prove a Hyers-Ulam-Rassias stability (or a general Hyers-Ulam stability) of the functional equation (1), which is closely related to the square root spiral.     

On the square root of a positive selfadjoint operator

We provide a short, elementary proof of the existence and uniqueness of the square root in the context of unbounded positive selfadjoint operators on real or complex Hilbert spaces.     

Every synaptic algebra has the monotone square root property

A synaptic algebra is a common generalization of several ordered algebraic structures based on algebras of self-adjoint operators, including the self-adjoint part of an AW\(^{*}\)-algebra. In this paper we prove that a synaptic algebra A has the monotone square root property, i.e., if \(0\le a,b\in A\), then \(a\le b \Rightarrow a^{1/2}\le b^{1/2}\).     

The square root base of polynomials in Clifford analysis

In this paper the problem of taking the square root of bases of special monogenic polynomials is studied, thus leading to a number of results under some additional conditions of associated infinite matrices, related essentially to the so-called algebraicness of these matrices.     

A one parameter method for the matrix inverse square root

This paper is motivated by the paper [3], where an iterative method for the computation of a matrix inverse square root was considered. We suggest a generalization of the method in [3]. We give some sufficient conditions for the convergence of this method, and its numerical stabillity property is investigated. Numerical examples showing that sometimes our generalization converges faster than the methods in [3] are presented.