Short intervals asymptotic formulae for binary problems with prime powers

AbstractWe improve some results in our paper [A. Languasco and A. Zaccagnini, ‘Short intervals asymptotic formulae for binary problems with prime powers’, J. Théor. Nombres Bordeaux30 (2018) 609–635] about the asymptotic formulae in short intervals for the average number of representations of integers of the forms $n=p_{1}^{\ell _{1}}+p_{2}^{\ell _{2}}$ and $n=p^{\ell _{1}}+m^{\ell _{2}}$, where $\ell _{1},\ell _{2}\geq 2$ are fixed integers, $p,p_{1},p_{2}$ are prime numbers and $m$ is an integer. We also remark that the techniques here used let us prove that a suitable asymptotic formula for the average number of representations of integers $n=\sum _{i=1}^{s}p_{i}^{\ell }$, where $s$, $\ell$ are two integers such that $2\leq s\leq \ell -1$, $\ell \geq 3$ and $p_{i}$, $i=1,\ldots ,s$, are prime numbers, holds in short intervals.

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On the asymptotic formulae for some multiplicative functions in short intervals

International Journal of Number Theory ◽

10.1142/s1793042115400102 ◽

2015 ◽

Vol 11 (05) ◽

pp. 1571-1587 ◽

Cited By ~ 1

Author(s):

Alisa Sedunova

Keyword(s):

Short Interval ◽

Multiplicative Functions ◽

Density Estimates ◽

Mean Values ◽

Short Intervals ◽

Asymptotic Formulae ◽

Free Region ◽

Complex Integration ◽

The Mean ◽

Riemann Ζ Function

We are going to study the mean values of some multiplicative functions connected with the divisor function in short interval of summation. The asymptotics for such mean values will be proved. Considering instead of well-known multiplicative functions, their inverses lead to very weak results of application of standard methods of complex integration. In order to get better estimations, we propose another method which uses as its main tools the density estimates and zero-free region for Riemann ζ-function and Dirichlet L-functions.

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On two sums related to the Lehmer problem over short intervals

AIMS Mathematics ◽

10.3934/math.2021681 ◽

2021 ◽

Vol 6 (11) ◽

pp. 11723-11732

Author(s):

Yanbo Song ◽

Keyword(s):

Asymptotic Formula ◽

Error Term ◽

Short Intervals ◽

Asymptotic Formulae ◽

Lehmer Problem ◽

Lehmer Numbers

<abstract><p>In this article, we study sums related to the Lehmer problem over short intervals, and give two asymptotic formulae for them. The original Lehmer problem is to count the numbers coprime to a prime such that the number and the its number theoretical inverse are in different parities in some intervals. The numbers which satisfy these conditions are called Lehmer numbers. It prompts a series of investigations, such as the investigation of the error term in the asymptotic formula. Many scholars investigate the generalized Lehmer problems and get a lot of results. We follow the trend of these investigations and generalize the Lehmer problem.</p></abstract>

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Artificial satellite orbit computations

Symposium - International Astronomical Union ◽

10.1017/s0074180900105662 ◽

1966 ◽

Vol 25 ◽

pp. 363-371

Author(s):

P. Sconzo

Keyword(s):

Artificial Satellite ◽

Satellite Orbit ◽

Artificial Satellites ◽

Orbital Elements ◽

Differential Correction ◽

Gravitational Effects ◽

Short Intervals ◽

The Subject ◽

Introductory Discussion ◽

Orbit Computation

In this paper an orbit computation program for artificial satellites is presented. This program is operational and it has already been used to compute the orbits of several satellites.After an introductory discussion on the subject of artificial satellite orbit computations, the features of this program are thoroughly explained. In order to achieve the representation of the orbital elements over short intervals of time a drag-free perturbation theory coupled with a differential correction procedure is used, while the long range behavior is obtained empirically. The empirical treatment of the non-gravitational effects upon the satellite motion seems to be very satisfactory. Numerical analysis procedures supporting this treatment and experience gained in using our program are also objects of discussion.

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The Influence of Ascorbic Acid on Platelet Structure and Function

Thrombosis and Haemostasis ◽

10.1055/s-0038-1651445 ◽

1975 ◽

Vol 34 (01) ◽

pp. 050-062

Author(s):

Dale H Cowan ◽

Richard C Graham ◽

Patricia Shook ◽

Ronda Griffin

Keyword(s):

Ascorbic Acid ◽

Open Channel ◽

Channel System ◽

Short Intervals ◽

Platelet Survival ◽

Artifactual Changes ◽

Survival Studies ◽

And Function ◽

Induced Aggregation ◽

Platelet Structure

SummaryTo determine the effect on platelet behavior of transient exposure of platelets to ascorbic acid, studies of platelet function and ultrastructure were done before exposure to ascorbic acid at pH 6.5, during exposure to pH 6.5, and after restoration of pH to pre-acidifìcation levels. The effect of ascorbic acid (A. A.) was compared to that of HCl and citric acid (C. A.). ADP- and collagen-induced aggregation of normal platelets were significantly impaired by both A. A. and C. A. but were less affected by HCl. The release of 14C-serotonin was significantly reduced by each agent. The ultra-structure of normal platelets brought to pH 6.5 by A.A. was normal. After neutralization, there was marked dilatation of the open channel system and loss of the disc shape. When platelets were brought to pH 6.5 by A. A., then neutralized, the aggregates which formed after stimulation by ADP or collagen were smaller than normal, the platelets were less closely approximated, and degranulation was less complete. The data show that exposure of platelets to ascorbic acid for short intervals impairs their function when measured after restoration of pH to levels compatible with maximal responses. Platelet survival studies using autologous platelets labelled with 51Cr in the presence or absence of ascorbic acid showed that the recovery of normal platelets was unaffected by ascorbic acid, whereas recovery of platelets from patients with idiopathic thrombocytopenic purpura, idiopathic thrombocythemia, and alcohol-related thrombocytopenia was markedly reduced. The injury resulting from the use of ascorbic acid in preparing platelets for studies of platelet survival in patients with disorders affecting platelets may impair the recovery of the cells, resulting in artifactual changes in the survival studies.

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Management of Haemophilia A with Antibodies - The Effect of Combined Treatment with Factor VIII, Hydrocortisone and Cyclophosphamide

Thrombosis and Haemostasis ◽

10.1055/s-0038-1660131 ◽

1985 ◽

Vol 54 (04) ◽

pp. 776-779 ◽

Cited By ~ 13

Author(s):

U Hedner ◽

L Tengborn

Keyword(s):

Factor Viii ◽

Combined Treatment ◽

Factor Ix ◽

Treatment Schedule ◽

High Antibody ◽

Anamnestic Response ◽

Short Intervals ◽

Low Responders ◽

Antibody Levels ◽

High Responders

SummaryImmune tolerance has by several methods been induced in haemophiliacs with antibodies. A conversion of “high responders” into “low responders” was previously reported after repeated moderate factor IX doses over periods of 7-10 days in combination with cyclophosphamide and steroids in two patients with haemophilia B and inhibitors. This paper reports similar results in a heamophilia A patient by giving factor VIII, cyclophosphamide, and steroids during relatively short periods of time (7-8 days). The anamnestic response markedly decreased already following the first treatment and never exceeded a level of 1 u/ml (˜ 3 BU/ml) even when boosted with ordinary factor VIII doses for only 3 days. It is concluded that the markedly decreased secondary antibody response is most probably the result of factor VIII given at short intervals (twice a day) for periods of up to about one week when given in combination with cyclophosphamide and steroids. The same effect may be achieved by other methods. The treatment schedule suggested in the present paper is, however, simple and avoids long periods of high antibody levels. Furthermore, the total factor VIII dose used is lower than suggested in most other treatment schedules, which makes the treatment substantially less expensive.

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Mit Initialfemeln zum Plenterwald (Essay) | The creation of selection forest using initial femel cut (essay)

Schweizerische Zeitschrift fur Forstwesen ◽

10.3188/szf.2009.0137 ◽

2009 ◽

Vol 160 (6) ◽

pp. 137-143

Author(s):

Rudi Kynast

Keyword(s):

Mixed Forest ◽

Point Of View ◽

Age Group ◽

Black Forest ◽

Crown Height ◽

Short Intervals ◽

Total Growth ◽

Cost Calculations ◽

Selection Forest ◽

Individual Trees

Although selection forests have clear advantages over age-group forests in view of their total growth performance, their net product and their stability, not to mention the sustainability of their beneficial effect, the proportion of this type of forest is insignificantly small in Germany and also in mixed forest in the mountains. It is therefore all the more surprising that scarcely any discernable efforts have been made to increase the proportion of selection forests. For a conversion, an alternative model for the treatment of the stands is adopted, whereby it is no longer the encouragement of the growth to maturity of individual trees in the stand which is aimed for, but rather the transformation of the whole stand to a selection forest using available stand elements and elements created by an early initiation of regeneration. Based on his experience in the forestry district of Kirchzarten in the Black Forest, Germany, the author describes the procedure for a successful conversion. This is to be started as soon as possible, that is to say when the crown height of the trees is about 18 metres and with corresponding usable dimensions, using small group shelter-wood cuts, a so-called initial femel cut. To get the conversion started it is advisable to remove whole groups of predominantly badly situated and overgrown trees. The stand will be additionally structured later through further interventions at short intervals. In the process, here and there really well situated trees will actually be left to stand solitar y, in other places w hol e self-cont aine d groups will b e created and else where valuabl e mixed s tand elements will be selected for permanent preservation, this in order to create a situation in which there are about 35 overstorey trees per hectare. On the basis of his own cost calculations, the author comes to the conclusion that the conversion is, from a financial point of view, superior compared with the age-group forest in that it brings higher proceeds more quickly and more often.

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Asymptotic behavior of repeated eigenvalues of perturbed self-adjoint elliptic operators

Asymptotic Analysis ◽

10.3233/asy-201669 ◽

2021 ◽

pp. 1-16

Author(s):

Alexander Dabrowski

Keyword(s):

General Type ◽

Differential Operators ◽

Laplacian Operator ◽

Variational Characterization ◽

Repeated Eigenvalues ◽

Direct Extension ◽

Asymptotic Formulae ◽

Elliptic Differential Operators ◽

Domain Perturbations ◽

Boundary Deformation

A variational characterization for the shift of eigenvalues caused by a general type of perturbation is derived for second order self-adjoint elliptic differential operators. This result allows the direct extension of asymptotic formulae from simple eigenvalues to repeated ones. Some examples of particular interest are presented theoretically and numerically for the Laplacian operator for the following domain perturbations: excision of a small hole, local change of conductivity, small boundary deformation.

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