Sum Of Digits Research Articles

A NOTE ON THE RELATIVE GROWTH RATE OF THE MAXIMAL DIGITS IN LÜROTH EXPANSIONS

This paper is concerned with the growth rate of the maximal digits relative to the rate of approximation of the number by its convergents, as well as relative to the rate of the sum of digits for the Lüroth expansion of an irrational number. The Hausdorff dimension of the sets of points with a given relative growth rate is proved to be full.

Evaluation of the Reliability, Reproducibility and Validity of Digital Orthodontic Measurements Based on Various Digital Models among Young Patients.

The advantages of intraoral model scanning have yielded recent developments. However, few studies have explored the orthodontic clinical use of this technique particularly among young patients. This study aimed to evaluate the reliability, reproducibility and validity of the orthodontic measurements: tooth width, arch length and arch length discrepancy in each digital model obtained by model scanner and intraoral scanner, relative to a plaster model. Arch length measured using two methods: curved arch length (CAL) measured automatically by digital program and sum of sectional liner arch length (SLAL) measured sum of anterior and posterior liner arch lengths. Arch length discrepancy calculated each arch length measurement methods: curved arch length discrepancy (CALD) and sum of sectional liner arch length discrepancy (SLALD). Forty young patients were eligible for the study. A plaster model (P), model-scanned digital model (MSD) and intraoral scanned digital model (ISD) were acquired from each patient. The reliability of the measurements was evaluated using Pearson’s correlation coefficient, while the reproducibility was evaluated using the intraclass correlation coefficient. The validity was assessed by a paired t-test. All measurements measured in P, MSD and ISD exhibited good reliability and reproducibility. Most orthodontic measurements despite of CAL in MSD exhibited high validity. Only the SLAL and SLALD in ISD group differed significantly, despite the good validity of the tooth width, CAL and CALD. The measurements based on the digital program appeared high reliability, reproducibility and accurate than conventional measurement. However, SLAL and SLALD in ISD group appeared shorter because of distortion during intraoral scanning. However, this could be compensated by using digital programed curved arch. Although the validity of SLAL and SLALD in the ISD group differed statistically, the difference is not considered clinically significant. Although MSD and ISD are acceptable for a clinical space analysis, clinicians should be aware of digital model-induced errors.

On the Sum of Digits of the Zeckendorf Representations of Two Consecutive Numbers

On the Sum of Digits of the Zeckendorf Representations of Two Consecutive Numbers

Global extrema of the Delange function, bounds for digital sums and concave functions

The sums are defined by the equality for all positive integers and , where is the sum of digits of the integer written in the system with base . In 1975 Delange generalised Trollope’s formula and proved that , where and is the continuous nowhere differentiable Delange function. We find global extrema of and, using this, obtain a precise bound for the difference . In the case this becomes the bound for binary sums proved by Krüppel in 2008 and also earlier by other authors. We also evaluate the global extrema of some other continuous nowhere differentiable functions. We introduce the natural concave hull of a function and prove a criterion simplifying the evaluation of this hull. Moreover, we introduce the notion of an extreme subargument of a function on a convex set. We show that all points of global maximum of the difference , where the function is strictly concave and some additional conditions hold, are extreme subarguments for . A similar result is obtained for functions of the form . We evaluate the global extrema and find extreme subarguments of the Delange function on the interval . The results in the paper are illustrated by graphs and tables. Bibliography: 16 titles.

Error detection by modified codes with summation of single digits in the arbitrary modulus residue ring

Error detection by modified codes with summation of single digits in the arbitrary modulus residue ring

Thanks for Twenty-Four

The number 24 belongs to a variety of integer sequences: 24 is the factorial of 4; the sum of its proper divisors, 1 + 2 + 3 + 4 + 6 + 8 + 12 = 36, exceeds 24 making it abundant; the sum of digits ...

Digit sums and generating functions

We connect a primitive operation from arithmetic -- summing the digits of a base-$B$ integer -- to $q$-series and product generating functions analogous to those in partition theory. We find digit sum generating functions to be intertwined with distinctive classes of "$B$-ary" Lambert series, which themselves enjoy nice transformations. We also consider digit sum Dirichlet series.

On Black-Hole Numbers of Iteration of Digital Sextic Power Sum

Abstract For iteration of six power sum of digits there are two fixed points, one 2-circle, one 3-circle, one 4-circle, one 10-circles and one 30-circle. The probabilities for natural numers falling into each fixed point or circle by several times iterating of six power sum are unbalanced. In it about 48.7% fall into 30-circle, about 39.3% into 10-circle, about 11.4% into 3-circle, the probability of falling into other circles and two fixed points is less than 6‰.

On uniformity of q‐multiplicative sequences

We show that any $q$-multiplicative sequence which is \emph{oscillating} of order $1$, i.e.\ does not correlate with linear phase functions $e^{2\pi i n\alpha}$ ($\alpha \in \mathbb{R})$, is Gowers uniform of all orders, and hence in particular does not correlate with polynomial phase functions $e^{2\pi i p(n)}$ ($p \in \mathbb{R}[x]$). Quantitatively, we show that any $q$-multiplicative sequence which is of Gelfond type of order 1 is automatically of Gelfond type of all orders. Consequently, any such $q$-multiplicative sequence is a good weight for ergodic theorems. We also obtain combinatorial corollaries concerning linear patterns in sets which are described in terms of sums of digits.

Averages of the Dirichlet convolution of the binary digital sum

Averages of the Dirichlet convolution of the binary digital sum

The Tu–Deng Conjecture holds Almost Surely

The Tu–Deng Conjecture is concerned with the sum of digits $w(n)$ of $n$ in base $2$ (the Hamming weight of the binary expansion of $n$) and states the following: assume that $k$ is a positive integer and$1\leqslant t<2^k-1$. Then\[\Bigl \lvert\Bigl\{(a,b)\in\bigl\{0,\ldots,2^k-2\bigr\}^2:a+b\equiv t\bmod 2^k-1, w(a)+w(b)<k\Bigr\}\Bigr \rvert\leqslant 2^{k-1}.\]We prove that the Tu–Deng Conjecture holds almost surely in the following sense: the proportion of $t\in[1,2^k-2]$ such that the above inequality holds approaches $1$ as $k\rightarrow\infty$.Moreover, we prove that the Tu–Deng Conjecture implies a conjecture due to T. W. Cusick concerning the sum of digits of $n$ and $n+t$.

Somme des chiffres et changement de base

Let $s_a(n)$ denote the sum of digits of an integer $n$ in the base $a$ expansion. Answering, in a extended form, a question of Deshouillers, Habsieger, Laishram, and Landreau, we show that, provided $a$ and $b$ are multiplicatively independent, any positive real number is a limit point of the sequence $\{s_b(n)/s_a(n)\}_{n=1}^{\infty}$. We also provide upper and lower bounds for the counting functions of the corresponding subsequences.

The Rudin–Shapiro Sequence and Similar Sequences Are Normal Along Squares

AbstractWe prove that digital sequences modulomalong squares are normal, which covers some prominent sequences, such as the sum of digits in baseqmodulom, the Rudin–Shapiro sequence, and some generalizations. This gives, for any base, a class of explicit normal numbers that can be efficiently generated.

MULTIFRACTAL ANALYSIS OF ONE-DIMENSIONAL BIASED WALKS

For an infinite sequence [Formula: see text] of [Formula: see text] and [Formula: see text] with probability [Formula: see text] and [Formula: see text], we mainly study the multifractal analysis of one-dimensional biased walks. Let [Formula: see text] and [Formula: see text]. The Hausdorff and packing dimensions of the sets [Formula: see text] are [Formula: see text], which is the development of the theorem of Besicovitch [On the sum of digits of real numbers represented in the dyadic system, Math. Ann. 110 (1934) 321–330] on random walk, saying that: For any [Formula: see text], the set [Formula: see text] has Hausdorff dimension [Formula: see text].

F169. BRAIN CONNECTIVITY DURING PSYCHOLOGICAL STRESS IN PATIENTS WITH SCHIZOPHRENIA

BackgroundIt is commonly accepted that in most patients with schizophrenia external factors act on genetic predisposition to produce active psychotic symptoms. In fact, we showed that patients with schizophrenia have an abnormal brain activation and peripheral autonomic response to psychological stress. We sought to characterize the brain connectivity networks of such response in schizophrenia.MethodsWe studied the pattern of brain connectivity in relation to mental arithmetic stress paradigm in 21 patients and 21 healthy subjects aged 18 to 50 years, using 3T-fMRI. A period of 6 minutes of resting state acquisition (PRE) were followed by a block design with three 1-minute CONTROL task (one digit sum), 1 minute STRESS task (two digit subtraction) and 1 minute rest after task (POST). Pairwise Pearson correlations were calculated between 90 regions of interest. Data were analyzed with MATLAB and SPSS software.ResultsPatients with schizophrenia showed a lower connectivity network between fronto-temporal limbic areas compared with control subjects during control and stress task.Moreover, we observed a great variability of link density during resting state in patients but not in controls, and it diminishes in response to task.DiscussionPatients present abnormalities in networks related to stress response showing an alteration in fronto-temporal connectivity, and a poor and random modulation of these networks at rest. Current and previous findings suggest abnormal fronto-temporal connectivity that ultimately would lead to psychotic symptoms emergency in response to an environmental stressor and, even, could be related to hypervigilance and misattribution feeding into the paranoid cognition characteristic of patients with schizophrenia.

On the distribution of the sum of digits of sums $$a+b$$ a + b

Let \({\mathcal {A}}\), \({\mathcal {B}}\) be large subsets of \(\{1,\ldots ,N\}\). We study the distribution of the sum of binary digits of the sums \(a+b\) with \((a,b)\in {\mathcal {A}}\times {\mathcal {B}}\).

On the sum of digits of special sequences in finite fields

In $$\mathbb {F}_q$$ , Dartyge and Sarkozy introduced the notion of digits and studied some properties of the sum of digits function. We will provide sharp estimates for the number of elements of special sequences of $$\mathbb {F}_q$$ whose sum of digits is prescribed. Such special sequences of particular interest include the set of n-th powers for each $$n\ge 1$$ and the set of elements of order d in $$\mathbb {F}_q^*$$ for each divisor d of $$q-1$$ . We provide an optimal estimate for the number of squares whose sum of digits is prescribed. Our methods combine A. Weil bounds with character sums, Gaussian sums and exponential sums.

Power-Sum Denominators

The power sum 1n + 2n +… + xn has been of interest to mathematicians since classical times. Johann Faulhaber, Jacob Bernoulli, and others who followed expressed power sums as polynomials in x of degree n + 1 with rational coefficients. Here, we consider the denominators of these polynomials and prove some of their properties. A remarkable one is that such a denominator equals n + 1 times the squarefree product of certain primes p obeying the condition that the sum of the base-p digits of n + 1 is at least p. As an application, we derive a squarefree product formula for the denominators of the Bernoulli polynomials.

Some Properties of the Fibonacci Sequence on an Infinite Alphabet

The infinite Fibonacci sequence $\mathbf{F}$, which is an extension of the classic Fibonacci sequence to the infinite alphabet $\mathbb{N}$, is the fixed point of the morphism $\phi$: $(2i)\mapsto (2i)(2i+1)$ and $(2i+1)\mapsto (2i+2)$ for all $i\in\mathbb{N}$. In this paper, we study the growth order and digit sum of $\mathbf{F}$ and give several decompositions of $\mathbf{F}$ using singular words.

On the Digits of Sumsets

AbstractLet 𝒜 andbe large subsets of {1,… , N}. We study the number of pairs (a, b) ∊ 𝒜 ×such that the sum of binary digits of a + b is fixed.