On approximation by random Lüroth expansions

In this paper, we employ a random dynamical systems approach to study generalized Lüroth series expansions of numbers in the unit interval. We prove that for each [Formula: see text] with [Formula: see text] Lebesgue almost all numbers in [Formula: see text] have uncountably many universal generalized Lüroth series expansions with digits less than or equal to [Formula: see text], so expansions in which each possible block of digits occurs. In particular this means that Lebesgue almost all [Formula: see text] have uncountably many universal generalized Lüroth series expansions using finitely many digits only. For [Formula: see text] we show that typically the speed of convergence to an irrational number [Formula: see text] of the corresponding sequence of Lüroth approximants is equal to that of the standard Lüroth approximants. For other rational values of [Formula: see text] we use stationary measures to study the typical speed of convergence of the approximants and the digit frequencies.

Relative convergence speed of Lüroth expansions and Hausdorff dimension

For any [Formula: see text] in [Formula: see text], let [Formula: see text] be the Lüroth expansion of [Formula: see text]. In this paper, we study the relative convergence speed of its convergents [Formula: see text] to the rate of growth of digits in the Lüroth expansion of an irrational number. For any [Formula: see text] in [Formula: see text], the sets [Formula: see text] and [Formula: see text] are proved to be of same Hausdorff dimension [Formula: see text]. Furthermore, for any [Formula: see text] in [Formula: see text] with [Formula: see text], the Hausdorff dimension of the set [Formula: see text] [Formula: see text] is proved to be either [Formula: see text] or [Formula: see text] according as [Formula: see text] or not.

DNA Structure and the Golden Ratio Revisited

B-DNA, the informational molecule for life on earth, appears to contain ratios structured around the irrational number 1.618…, often known as the “golden ratio”. This occurs in the ratio of the length:width of one turn of the helix; the ratio of the spacing of the two helices; and in the axial structure of the molecule which has ten-fold rotational symmetry. That this occurs in the information-carrying molecule for life is unexpected, and suggests the action of some process. What this process might be is unclear, but it is central to any understanding of the formation of DNA, and so life.

A DIMENSIONAL RESULT ON THE PRODUCT OF CONSECUTIVE PARTIAL QUOTIENTS IN CONTINUED FRACTIONS

This paper is concerned with the growth rate of the product of consecutive partial quotients relative to the denominator of the convergent for the continued fraction expansion of an irrational number. More precisely, given a natural number $m,$ we determine the Hausdorff dimension of the following set: $$ \begin{align*} E_m(\tau)=\bigg\{x\in [0,1): \limsup\limits_{n\rightarrow\infty}\frac{\log (a_n(x)a_{n+1}(x)\cdots a_{n+m}(x))}{\log q_n(x)}=\tau\bigg\}, \end{align*} $$ where $\tau $ is a nonnegative number. This extends the dimensional result of Dirichlet nonimprovable sets (when $m=1$ ) shown by Hussain, Kleinbock, Wadleigh and Wang.

Presburger Arithmetic with algebraic scalar multiplications

We consider Presburger arithmetic (PA) extended by scalar multiplication by an algebraic irrational number $\alpha$, and call this extension $\alpha$-Presburger arithmetic ($\alpha$-PA). We show that the complexity of deciding sentences in $\alpha$-PA is substantially harder than in PA. Indeed, when $\alpha$ is quadratic and $r\geq 4$, deciding $\alpha$-PA sentences with $r$ alternating quantifier blocks and at most $c\ r$ variables and inequalities requires space at least $K 2^{\cdot^{\cdot^{\cdot^{2^{C\ell(S)}}}}}$ (tower of height $r-3$), where the constants $c, K, C>0$ only depend on $\alpha$, and $\ell(S)$ is the length of the given $\alpha$-PA sentence $S$. Furthermore deciding $\exists^{6}\forall^{4}\exists^{11}$ $\alpha$-PA sentences with at most $k$ inequalities is PSPACE-hard, where $k$ is another constant depending only on~$\alpha$. When $\alpha$ is non-quadratic, already four alternating quantifier blocks suffice for undecidability of $\alpha$-PA sentences.

Stability of certain relations of three unitaries in C∗-algebras

We show that if [Formula: see text] is an irrational number in [Formula: see text], [Formula: see text] and [Formula: see text] are in [Formula: see text] [Formula: see text] is a matrix of infinite order in SL[Formula: see text], either tr[Formula: see text] or tr[Formula: see text] and the greatest common divisor of the entries in [Formula: see text] is one, then for any [Formula: see text] there exists [Formula: see text] satisfying the following: For any unital simple separable [Formula: see text]-algebra [Formula: see text] with tracial rank at most one, any three unitaries [Formula: see text] in [Formula: see text], if [Formula: see text] satisfy certain trace conditions and [Formula: see text] then there exists a triple of unitaries [Formula: see text] in [Formula: see text] such that [Formula: see text] [Formula: see text]

Quick Estimation of Periodic Signal Parameters from One-bit Measurements

<div> <div> <div> <p>Estimation of periodic signals, based on quantized data, is a topic of general interest in the area of instrumentation and measurement. While several methods are available, new applications require low-power, low-complexity, and adequate estimation accuracy. In this paper, we consider the simplest possible quantization, that is binary quantization, and describe a technique to estimate the parameters of a sampled periodic signal, using a fast algorithm. By neglecting the possibility that the sampling process is triggered by some signal-derived event, sampling is assumed to be asynchronous, that is the ratio between the signal and the sampling periods is defined to be an irrational number. To preserve enough information at the quantizer output, additive Gaussian input noise is assumed as the information encoding mechanism. With respect to published techniques addressing the same problem, the proposed approach does not rely on the numerical estimation of the maximum likelihood function, but provides solutions that are very closed to this estimate. At the same time, since the main estimator is based on matrix inversion, it proves to be less time-consuming than the numerical maximization of the likelihood function, especially when solving problems with a large number of parameters. The estimation procedure is described in detail and validated using both simulation and experimental results. The estimator performance limitations are also highlighted. </p> </div> </div> </div>

Quick Estimation of Periodic Signal Parameters from One-bit Measurements

<div> <div> <div> <p>Estimation of periodic signals, based on quantized data, is a topic of general interest in the area of instrumentation and measurement. While several methods are available, new applications require low-power, low-complexity, and adequate estimation accuracy. In this paper, we consider the simplest possible quantization, that is binary quantization, and describe a technique to estimate the parameters of a sampled periodic signal, using a fast algorithm. By neglecting the possibility that the sampling process is triggered by some signal-derived event, sampling is assumed to be asynchronous, that is the ratio between the signal and the sampling periods is defined to be an irrational number. To preserve enough information at the quantizer output, additive Gaussian input noise is assumed as the information encoding mechanism. With respect to published techniques addressing the same problem, the proposed approach does not rely on the numerical estimation of the maximum likelihood function, but provides solutions that are very closed to this estimate. At the same time, since the main estimator is based on matrix inversion, it proves to be less time-consuming than the numerical maximization of the likelihood function, especially when solving problems with a large number of parameters. The estimation procedure is described in detail and validated using both simulation and experimental results. The estimator performance limitations are also highlighted. </p> </div> </div> </div>

Simple cyclic covers of the plane and the Seshadri constants of some general hypersurfaces in weighted projective space

Let $$X \subset {\mathbb P}(1,1,1,m)$$ X ⊂ P ( 1 , 1 , 1 , m ) be a general hypersurface of degree md for some for $$d\ge 2$$ d ≥ 2 and $$m\ge 3$$ m ≥ 3 . We prove that the Seshadri constant $$\varepsilon ( {\mathcal O}_X(1), x)$$ ε ( O X ( 1 ) , x ) at a general point $$x\in X$$ x ∈ X lies in the interval $$\left[ \sqrt{d}- \frac{d}{m}, \sqrt{d}\right] $$ d - d m , d and thus approaches the possibly irrational number $$\sqrt{d}$$ d as m grows. The main step is a detailed study of the case where X is a simple cyclic cover of the plane.

On the Irrationality and Transcendence of Rational Powers of e

A number that can’t be expressed as the ratio of two integers is called an irrational number. Euler and Lambert were the first mathematicians to prove the irrationality and transcendence of e. Since then there have been many other proofs of irrationality and transcendence of e and generalizations of that proof to rational powers of e. In this article we review various proofs of irrationality and transcendence of rational powers of e founded by mathematicians over the time.