scholarly journals On the Number of Witnesses in the Miller–Rabin Primality Test

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 890
Author(s):  
Shamil Talgatovich Ishmukhametov ◽  
Bulat Gazinurovich Mubarakov ◽  
Ramilya Gakilevna Rubtsova
Keyword(s):  
Positive Integer ◽  
Euler’S Totient Function ◽  
Number Of Factors ◽  
Primality Test ◽  
Average Probability ◽  
Test Errors ◽  
The Difference

In this paper, we investigate the popular Miller–Rabin primality test and study its effectiveness. The ability of the test to determine prime integers is based on the difference of the number of primality witnesses for composite and prime integers. Let W ( n ) denote the set of all primality witnesses for odd n. By Rabin’s theorem, if n is prime, then each positive integer a < n is a primality witness for n. For composite n, the power of W ( n ) is less than or equal to φ ( n ) / 4 where φ ( n ) is Euler’s Totient function. We derive new exact formulas for the power of W ( n ) depending on the number of factors of tested integers. In addition, we study the average probability of errors in the Miller–Rabin test and show that it decreases when the length of tested integers increases. This allows us to reduce estimations for the probability of the Miller–Rabin test errors and increase its efficiency.

HYPOCHROMIC MICROCYTIC ANEMIA OF INFANCY: IRON DEPLETION AS A FACTOR

PEDIATRICS ◽  
1956 ◽  
Vol 18 (6) ◽  
pp. 959-978
Author(s):  
Hugh W. Josephs
Keyword(s):  
Iron Deficiency ◽  
Severe Infection ◽  
Microcytic Anemia ◽  
Severe Anemia ◽  
Iron Depletion ◽  
Hypochromic Microcytic Anemia ◽  
Premature Babies ◽  
Number Of Factors ◽  
The Difference ◽  
Insight Into

In this work the author has attempted to gain insight into the significance of iron depletion by the use of 4 simple calculations, justification for which is found in recent articles. These are: (a) iron with which the infant is born; (b) iron retained from the food; (c) iron being used by the tissues and therefore unavailable for hemoglobin, and (d) iron combined with the total mass of hemoglobin. With these 4 figures it is possible to estimate the iron still potentially available for use (the "reserves" or "stones"). When the difference between a + b and c + d has reached about zero, depletion is considered to exist. The following characteristics of depletion may be emphasized: Depletion is the result of gain in weight and maximum possible usage of iron. It is therefore a normal result of growth and need not be associated with anemia. As soon as depletion has occurred, the organism is thereafter dependent on current absorption of iron. This is ordinarily sufficient, even with a diet of milk alone, to maintain an adequate concentration of hemoglobin after about 8 to 10 months of age. Severe anemia due to depletion alone is practically confined to premature babies whose relative gain in weight is rapid. Severe anemia in other than premature babies is the result of a number of factors by which iron becomes unavailable or is actually diverted from hemoglobin to storage. Response to iron medication is considerably better in infants with depletion than in those in whom some factor is present that interferes with iron utilization, and which is not connected by the mere giving of iron. The dependence on current absorption, whether the result of depletion or non-availability, introduces a certain precariousness which is apparently characteristic of this time of life. The organism gets along from day to day if nothing happens, but may not be able to meet an emergency, whether this appears as a rapid gain in weight, or a necessity to repair damage done by severe infection. If we consider iron deficiency as the cause of anemia, we can think of deficiency as due to a number of factors of which depletion is only one. The development and characteristics of depletion have been considered in this paper; other factors in iron deficiency will be considered in subsequent papers.

scholarly journals On the 2-adic order of Stirling numbers of the second kind and their differences

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Tamás Lengyel
Keyword(s):  
Positive Integer ◽  
Stirling Numbers ◽  
Bell Polynomials ◽  
Binary Representation ◽  
Exact Order ◽  
Special Cases ◽  
International Audience ◽  
The Difference ◽  
Divisibility Properties ◽  
Positive Integers

International audience Let $n$ and $k$ be positive integers, $d(k)$ and $\nu_2(k)$ denote the number of ones in the binary representation of $k$ and the highest power of two dividing $k$, respectively. De Wannemacker recently proved for the Stirling numbers of the second kind that $\nu_2(S(2^n,k))=d(k)-1, 1\leq k \leq 2^n$. Here we prove that $\nu_2(S(c2^n,k))=d(k)-1, 1\leq k \leq 2^n$, for any positive integer $c$. We improve and extend this statement in some special cases. For the difference, we obtain lower bounds on $\nu_2(S(c2^{n+1}+u,k)-S(c2^n+u,k))$ for any nonnegative integer $u$, make a conjecture on the exact order and, for $u=0$, prove part of it when $k \leq 6$, or $k \geq 5$ and $d(k) \leq 2$. The proofs rely on congruential identities for power series and polynomials related to the Stirling numbers and Bell polynomials, and some divisibility properties.

scholarly journals Statistics on Lattice Walks and q-Lassalle Numbers

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Lenny Tevlin
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Integer Coefficients ◽  
Major Index ◽  
Lattice Walks ◽  
International Audience ◽  
The Difference ◽  
Positive Integers

International audience This paper contains two results. First, I propose a $q$-generalization of a certain sequence of positive integers, related to Catalan numbers, introduced by Zeilberger, see Lassalle (2010). These $q$-integers are palindromic polynomials in $q$ with positive integer coefficients. The positivity depends on the positivity of a certain difference of products of $q$-binomial coefficients.To this end, I introduce a new inversion/major statistics on lattice walks. The difference in $q$-binomial coefficients is then seen as a generating function of weighted walks that remain in the upper half-plan. Cet document contient deux résultats. Tout d’abord, je vous propose un $q$-generalization d’une certaine séquence de nombres entiers positifs, liés à nombres de Catalan, introduites par Zeilberger (Lassalle, 2010). Ces $q$-integers sont des polynômes palindromiques à $q$ à coefficients entiers positifs. La positivité dépend de la positivité d’une certaine différence de produits de $q$-coefficients binomial.Pour ce faire, je vous présente une nouvelle inversion/major index sur les chemins du réseau. La différence de $q$-binomial coefficients est alors considérée comme une fonction de génération de trajets pondérés qui restent dans le demi-plan supérieur.

Het Aanpassen Van Informatieve Schoolboekteksten In Groep Vijf Van Het Basisonderwijs

1991 ◽  
Vol 39 ◽  
pp. 31-45
Author(s):  
Mark Wouters
Keyword(s):  
Text Comprehension ◽  
Great Part ◽  
Number Of Factors ◽  
Level Of Difficulty ◽  
The Difference

This article describes research which has focused on the level of difficulty of texts and questions as an explanation for the difference in text comprehension between non-native and native pupils. The degree to which text comprehension in native and non-native pupils could be facilitated by means of controlling texts and questions was investigated. From the results the following conclusions were drawn: 1) Non-native pupils scored significantly lower than native Dutch pupils in all tests that dealt with text comprehension; 2) When the test questions were controlled by changing a number of factors which supposedly made understanding more difficult, the scores of both non-native and native pupils improved significantly; 3) When the texts of the tests were rewitten, the non-native pupils improved in their scores and the native pupils did not; 4) When both the texts and questions were controlled, the results seemed to show that the non-native pupils had caught up for a great part where they were found to have "lagged behind".

scholarly journals A lower bound for a remainder term associated with the sum of digits function

1991 ◽  
Vol 34 (1) ◽  
pp. 121-142 ◽  
Author(s):  
D. M. E. Foster
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Upper Bound ◽  
Remainder Term ◽  
Fixed Integer ◽  
Sum Of Digits ◽  
Sum Of Digits Function ◽  
The Difference

For a fixed integer q≧2, every positive integer k = Σr≧0ar(q, k)qr where each ar(q, k)∈{0,1,2,…, q−1}. The sum of digits function α(q, k) Σr≧0ar(q, k) behaves rather erratically but on averaging has a uniform behaviour. In particular if , where n>1, then it is well known that A(q, n)∼½((q − 1)/log q)n logn as n → ∞. For odd values of q, a lower bound is now obtained for the difference 2S(q, n) = A(q, n)−½(q − 1))[log n/log q, where [log n/log q] denotes the greatest integer ≦log n /log q. This complements an upper bound already found.

scholarly journals Estimation of the Contribution of Intrinsic Currents to Motoneuron Firing Based on Paired Motoneuron Discharge Records in the Decerebrate Cat

2008 ◽  
Vol 100 (1) ◽  
pp. 292-303 ◽  
Author(s):  
Randall K. Powers ◽  
Paul Nardelli ◽  
T. C. Cope
Keyword(s):  
Firing Rate ◽  
Motor Units ◽  
Unit Discharge ◽  
Decerebrate Cat ◽  
Persistent Inward Currents ◽  
Quantitative Estimates ◽  
Number Of Factors ◽  
Inward Currents ◽  
The Difference ◽  
Synaptic Drive

Motoneuron activation is strongly influenced by persistent inward currents (PICs) flowing through voltage-sensitive channels. PIC characteristics and their contribution to the control of motoneuron firing rate have been extensively described in reduced animal preparations, but their contribution to rate modulation in human motoneurons is controversial. It has recently been proposed that the analysis of discharge records of a simultaneously recorded pair of motor units can be used to make quantitative estimates of the PIC contribution, based on the assumption that the firing rate of an early recruited (reporter) unit can be used as a measure of the synaptic drive to a later recruited (test) unit. If the test unit's discharge is augmented by PICs, less synaptic drive will be required to sustain discharge than required to initially recruit it, and the difference in reporter unit discharge (Δ F) at test recruitment and de-recruitment is a measure of the size of the PIC contribution. We applied this analysis to discharge records of pairs of motoneurons in the decerebrate cat preparation, in which motoneuron PICs have been well-characterized and are known to be prominent. Mean Δ F values were positive in 58/63 pairs, and were significantly greater than zero in 40/63 pairs, as would be expected based on PIC characteristics recorded in this preparation. However, several lines of evidence suggest that the Δ F value obtained in a particular motoneuron pair may depend on a number of factors other than the PIC contribution to firing rate.

scholarly journals Continuous functions of bounded nth variation

1969 ◽  
Vol 16 (3) ◽  
pp. 205-214
Author(s):  
Gavin Brown
Keyword(s):  
Continuous Function ◽  
Positive Integer ◽  
Decomposition Theorem ◽  
Continuous Functions ◽  
Unit Interval ◽  
Jordan Decomposition ◽  
The Real ◽  
Elementary Construction ◽  
The Difference

Let n be a positive integer. We give an elementary construction for the nth variation, Vn(f), of a real valued continuous function f and prove an analogue of the classical Jordan decomposition theorem. In fact, let C[0, 1] denote the real valued continuous functions on the closed unit interval, let An denote the semi-algebra of non-negative functions in C[0, 1] whose first n differences are non-negative, and let Sn denote the difference algebra An - An. We show that Sn is precisely that subset of C[0, 1] on which Vn(f)<∞. (Theorem 1).

Suicidal Attempts in an Outer Region of Metropolitan Melbourne and in a Provincial Region of Victoria

1975 ◽  
Vol 9 (4) ◽  
pp. 255-261 ◽  
Author(s):  
A. Kessell ◽  
A. Nicholson ◽  
G. Graves ◽  
J. Krupinski
Keyword(s):  
Attempted Suicide ◽  
Psychiatric Care ◽  
Outer Region ◽  
Incidence Rates ◽  
Seasonal Distribution ◽  
Distribution Method ◽  
Number Of Factors ◽  
Suicidal Attempts ◽  
The Difference

A two year study of attempted suicide was undertaken in two Victorian regions. Incidence rates of 16.7 and 22.1 per 10,000 population were found in the metropolitan and provincial regions respectively. A number of factors were considered and found relevant to the difference in incidence. In addition, other parameters including multiple attempts, seasonal distribution, method used, association with alcohol and referral to psychiatric care were considered. Our findings are compared with those of earlier studies reported in Australia.

Studies on pH, number of protozoa and microbial ATP concentrations in rumen-incubated nylon bags with different pore sizes

1984 ◽  
Vol 102 (2) ◽  
pp. 501-504 ◽  
Author(s):  
J. E. Lindberg ◽  
A. Kaspersson ◽  
P. Ciszuk
Keyword(s):  
Particulate Matter ◽  
Great Influence ◽  
Combined Effect ◽  
Pore Sizes ◽  
Number Of Factors ◽  
The Difference ◽  
Selection Of ◽  
In Sacco

A number of factors affect predictions of rumen degradability of feedingstuffs when using thein saccotechnique (Lindberg, 1983). One factor which exerts a great influence is the bag cloth aperture (Lindberg & Knutson, 1981; Lindberg & Varvikko, 1982). Without doubt part of the difference in degradation between pore sizes can be explained by differences in particulate matter losses (Lindberg & Knutsson, 1981; Lindberg & Varvikko, 1982). It has, however, also been suggested that the differences between pore sizes are due to the combined effect of differences in liquid exchange between the bags and the rumen contents and on the selection of microbes entering the bags (Lindberg & Varvikko, 1982).

scholarly journals Evidence for the s count macrogene

2015 ◽  
Author(s):  
Russell Schexnayder
Keyword(s):  
Cerebral Cortex ◽  
Positive Integer ◽  
Brain Size ◽  
Inbred Mice ◽  
Mouse Strains ◽  
Major Genes ◽  
Quantum Variable ◽  
Second Phases ◽  
The Difference ◽  
Repeat Count

Background: A macrogene is defined here as a gene on which successive mutations incrementing a repeat count produces successive punctuated evolutionary events in species that are homogeneous for it. The set of repeat count on the asp (abnormal spindle) family of gene is thought to affect brain size in mammals. Corticogenesis requires two integer valued (quantum) variables, the f and s counts, to determine the number of division cycles during the first and second phases, respectively, of neuron production in the cerebral cortex. Quantum ‘extra’ neuron theory hypothesizes that increments in a quantum variable, the n count, cause punctuated encephalization events in species that are homogenous for it. There is evidence in six pairs of inbred mice strains for one or more major genes affecting brain size. Results: The s count is probably equal to the n count plus a positive integer. The calculated n counts are different in three of the four pairs of strains studied where encephalization data has been previously published. Five different n counts have been found in eleven mouse strains. The difference between the n counts of humans and mice is about 25. Conclusions: Encephalization in mammals may be caused by a macrogene that determines the s count. This theory can be tested by determining the s counts of the various mice strains. However, the asp family of gene is probably not the s count macrogene because the difference in the asp counts of humans and mice of 13 (= 74 – 61) is much smaller than the difference in their s counts of around 25.

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