Rational Multipliers Research Articles

Rational maps with rational multipliers

In this article, we show that every rational map whose multipliers all lie in a given number field is a power map, a Chebyshev map or a Lattès map. This strengthens a conjecture by Milnor concerning rational maps with integer multipliers, which was recently proved by Ji and Xie.

Noncommutative rational Clark measures

Abstract We characterize the noncommutative Aleksandrov–Clark measures and the minimal realization formulas of contractive and, in particular, isometric noncommutative rational multipliers of the Fock space. Here, the full Fock space over $\mathbb {C} ^d$ is defined as the Hilbert space of square-summable power series in several noncommuting (NC) formal variables, and we interpret this space as the noncommutative and multivariable analogue of the Hardy space of square-summable Taylor series in the complex unit disk. We further obtain analogues of several classical results in Aleksandrov–Clark measure theory for noncommutative and contractive rational multipliers. Noncommutative measures are defined as positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz algebra, the unital $C^*$ -algebra generated by the left creation operators on the full Fock space. Our results demonstrate that there is a fundamental relationship between NC Hardy space theory, representation theory of the Cuntz–Toeplitz and Cuntz algebras, and the emerging field of noncommutative rational functions.

Unicritical polynomial maps with rational multipliers

In this article, we prove that every unicritical polynomial map that has only rational multipliers is either a power map or a Chebyshev map. This provides some evidence in support of a conjecture by Milnor concerning rational maps whose multipliers are all integers.

Non-commutative rational functions in the full Fock space

A rational function belongs to the Hardy space, H 2 H^2 , of square-summable power series if and only if it is bounded in the complex unit disk. Any such rational function is necessarily analytic in a disk of radius greater than one. The inner-outer factorization of a rational function r ∈ H 2 \mathfrak {r} \in H^2 is particularly simple: The inner factor of r \mathfrak {r} is a (finite) Blaschke product and (hence) both the inner and outer factors are again rational. We extend these and other basic facts on rational functions in H 2 H^2 to the full Fock space over C d \mathbb {C} ^d , identified as the non-commutative (NC) Hardy space of square-summable power series in several NC variables. In particular, we characterize when an NC rational function belongs to the Fock space, we prove analogues of classical results for inner-outer factorizations of NC rational functions and NC polynomials, and we obtain spectral results for NC rational multipliers.

0.3–4.3 GHz Frequency-Accurate Fractional-$N$ Frequency Synthesizer With Integrated VCO and Nested Mixed-Radix Digital $\Delta$-$\Sigma$ Modulator-Based Divider Controller

If the modulus of the digital delta-sigma modulator (DΔΣM) in a fractional- N frequency synthesizer is a power of two, then the output frequency is constrained to be a rational multiple of the phase detector frequency (f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PD</sub> ), where the denominator of the rational multiplier is a power of two. If the required output frequency is not related to f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PD</sub> in this way, one is forced to approximate the ratio by using a small programmable modulus DΔΣM or a very large power-of-two modulus. Both of these solutions involve additional hardware. Furthermore, the programmable modulus solution can suffer from spurs, while the large power of two lacks accuracy. This paper presents a new solution, based on mixed-radix algebra, where the required ratio is formed by combining two different moduli. The programmable modulus solves the accuracy problem, while the large power-of-two modulus minimizes the spur content. In addition, the phase detector can be clocked at high speed. This paper explains the theoretical foundations of the method, elaborates a design methodology, and presents measured results for an 0.18 μm SiGe BiCMOS prototype.

Algebraic trigonometric values at rational multipliers of π

The problem of finding all algebraic values of α ∈ [−1; 1] when arccos α arcsin α and arctan α are rational multiples of π is solved. The values of such α of degree less than five are explicitly determined.

Investor Rationality and House Price Bubbles: Berlin and the German Reunification

Abstract We analyze the behavior of investors in the Berlin rental apartment house market over the years 1980-2004. Using constant-quality multipliers ( price- rent ratios), we reject the hypothesis that multipliers in the market were set in a rational manner. Supported by narrative evidence, we conjecture that investors misjudged the economic effects of the German reunification. To examine this, we employ a stylized structural economic model and analyze the effects of shocks on rational multipliers. It seems that investors confused the reunification with a permanent supply side shock to the economy. By basing their investment decisions on this misjudgement, investors behaved irrationally, but in a very uncertain and unprecedented environment.

Rational multiplier IQCs for uncertain time-delays and LMI stability conditions

Describes a set of delay-dependent integral quadratic constraint (IQC) stability conditions for time-delay uncertainty. The IQCs are linearly parameterized in terms of a pair of rational stability multipliers, each active over one of a pair of complementary frequency intervals. Using the finite-frequency positive real lemma, each of these finite-frequency IQC conditions are shown to be equivalent to a frequency-independent linear matrix inequality condition, thereby dispensing with the need for frequency-sweeping.

Rational multipliers and analytical properties of their transforms

Rational multipliers and analytical properties of their transforms

μ-Analysis and Synthesis with Rational Multipliers

This paper is concerned with analysis and control synthesis for systems subject to structured uncertainties. The techniques here are based on passivity and multiplier theories. A brief survey of the theory is presented together with the related tools, all in the LMI vein, leading to effective analysis and synthesis approaches. A particular emphasis is placed on the use of rational multipliers which have substantial advantages over polynomial ones. The techniques are discussed with regard to two non-trivial examples: a flexible uncertain system and a missile airframe.

Central Value and P-E Estimation—A Formula Approach

THE CRUCIAL PHASE OF SECURITY EVALUATION is in selection of the price-earnings multiplier. A developing thin supply of good-grade equities, and an increasing need to carefully appraise the growth factor, suggest this problem of the P-E multiplier is not to become easier. Past-applied techniques in arriving at rational multipliers have been chiefly based upon relative value or vague historical precedent, and have been highly affected by the coloration of near-term optimism or pessimism. The erratic behaviour of P-E multipliers in the past demonstrates the inadequacy of these nonobjective approaches. A search for a superior analytical and objective technique occasions this presentation of a mathematical formula approach. A mathematical formula approach to estimating proper P-E multipliers has theoretical justification. Price is a reflection of earnings and earnings outlook. Earnings outlook is related to time, amount and growth of base capital, and the rate at which capital may be profitably employed. And as long as stock investment remains an optional decision, stock price-earning multipliers should be related to existing money rates. The many factors expressed in the foregoing may be combined to fit a rational and suitable mathematical formula. Such a formula stems from the basic assumption that a proper P-E multiplier for a stock should be one that affords a future return of stock principal and interim dividends equal to bond principal and interim yield on U. S. Government obligations. Such a multiplier assumes the possibility of superior stock performance to be offset by the additional risk incurred. In equational terms this may be expressed as: