scholarly journals Moments and oscillations of exponential sums related to cusp forms

2016 ◽  
Vol 162 (3) ◽  
pp. 479-506 ◽  
Author(s):  
ESA V. VESALAINEN
Keyword(s):  
Exponential Sums ◽  
Upper Bounds ◽  
Cusp Forms ◽  
Average Order ◽  
Small Denominators ◽  
Moment Estimates ◽  
Holomorphic Cusp Forms ◽  
Order Of Magnitude ◽  
The Mean ◽  
The Right

AbstractWe consider large values of long linear exponential sums involving Fourier coefficients of holomorphic cusp forms. The sums we consider involve rational linear twistse(nh/k) with sufficiently small denominators. We prove both pointwise upper bounds and bounds for the frequency of large values. In particular, thek-aspect is treated. As an application we obtain upper bounds for all the moments of the sums in question. We also give the asymptotics with the right main term for fourth moments.We also consider the mean square of very short sums, proving that on average short linear sums with rational additive twists exhibit square root cancellation. This result is also proved in a slightly sharper form.Finally, the consideration of moment estimates for both long and short exponential sums culminates in a result concerning the oscillation of the long linear sums. Essentially, this result says that for a positive proportion of time, such a sum stays in fairly long intervals, where its order of magnitude does not drop below the average order of magnitude and where its argument is in a given interval of length 3π/2+ϵ and so cannot wind around the origin.

ON THE ERROR TERM IN THE APPROXIMATE FUNCTIONAL EQUATION FOR EXPONENTIAL SUMS RELATED TO CUSP FORMS

2008 ◽  
Vol 04 (05) ◽  
pp. 747-756 ◽  
Author(s):  
ANNE-MARIA ERNVALL-HYTÖNEN
Keyword(s):  
Functional Equation ◽  
Upper Bound ◽  
Error Term ◽  
Exponential Sums ◽  
Cusp Forms ◽  
Approximate Functional Equation ◽  
Holomorphic Cusp Forms

We give a proof for the approximate functional equation for exponential sums related to holomorphic cusp forms and derive an upper bound for the error term.

scholarly journals A relation between Fourier coefficients of holomorphic cusp forms and exponential sums

2009 ◽  
Vol 86 (100) ◽  
pp. 97-105 ◽  
Author(s):  
Anne-Maria Ernvall-Hytönen
Keyword(s):  
Fourier Coefficients ◽  
Exponential Sums ◽  
Cusp Forms ◽  
Lehmer Conjecture ◽  
Holomorphic Cusp Forms

We consider certain specific exponential sums related to holomorphic cusp forms, give a reformulation for the Lehmer conjecture and prove that certain exponential sums of Fourier coefficients of holomorphic cusp forms contain information on other similar non-overlapping exponential sums. Also, we prove an Omega result for short sums of Fourier coefficients.

BOUNDS FOR A MEAN VALUE OF CHARACTER SUMS

2008 ◽  
Vol 04 (02) ◽  
pp. 249-293 ◽  
Author(s):  
NIGEL WATT
Keyword(s):  
Prime Factor ◽  
Upper Bounds ◽  
Mean Value ◽  
Cusp Forms ◽  
Smooth Functions ◽  
Mean Values ◽  
Carmichael Numbers ◽  
Dirichlet Characters ◽  
Dirichlet Polynomials ◽  
The Mean

We obtain new upper bounds for the mean squared modulus of sums ∑n∈ℕAnχ(n), where the sequence (An) is fixed and the variable χ belongs to the set of non-principal Dirichlet characters for some modulus q. It is assumed that, for some M, some complex sequence (cm) satisfying cm = 0 for m ∉ (M/2,M], and some α(x) and β(y) (smooth functions with compact support), one has An = ∑uvm = n α(u)β(v)cm (n ∈ ℕ). There is a natural analogy between the bounds obtained and bounds on mean values of Dirichlet polynomials previously obtained by Deshouillers and Iwaniec. Our proofs make use of results from the spectral theory of automorphic functions, including the bound of Kim and Sarnak for the eigenvalues of Hecke operators acting on certain spaces of Maass cusp forms. The results depend on the size of P, the largest prime factor of q, and improve as log q(P) is diminished. In separate work, Harman has given an application of our results to the theory of Carmichael numbers.

scholarly journals Spatial transportation networks with transfer costs: asymptotic optimality of hub-and-spoke models

2008 ◽  
Vol 145 (2) ◽  
pp. 471-487 ◽  
Author(s):  
DAVID J. ALDOUS
Keyword(s):  
Unit Area ◽  
Asymptotic Optimality ◽  
Upper Bounds ◽  
Average Density ◽  
Worst Case ◽  
Hub And Spoke ◽  
Optimal Network ◽  
Route Length ◽  
Order Of Magnitude ◽  
The Mean

AbstractConsider networks on n vertices at average density 1 per unit area. We seek a network that minimizes total length subject to some constraint on journey times, averaged over source-destination pairs. Suppose journey times depend on both route-length and number of hops. Then for the constraint corresponding to an average of 3 hops, the length of the optimal network scales as n13/10. Alternatively, constraining the average number of hops to be 2 forces the network length to grow slightly faster than order n3/2. Finally, if we require the network length to be O(n) then the mean number of hops grows as order log log n. Each result is an upper bound in the worst case (of vertex positions), and a lower bound under randomness or equidistribution assumptions. The upper bounds arise in simple hub and spoke models, which are therefore optimal in an order of magnitude sense.

DMSA-scintigraphy in paediatrics: Is the evaluation of the geometric mean necessary for the calculation of the differential renal function?

2001 ◽  
Vol 40 (04) ◽  
pp. 107-110 ◽  
Author(s):  
B. Roßmüller ◽  
S. Alalp ◽  
S. Fischer ◽  
S. Dresel ◽  
K. Hahn ◽  
...  
Keyword(s):  
Renal Function ◽  
Geometric Mean ◽  
Region Of Interest ◽  
Posterior View ◽  
Renal Abnormality ◽  
Anterior View ◽  
Differential Renal Function ◽  
The Mean ◽  
The Right ◽  
To Receive

SummaryFor assessment of differential renal function (PF) by means of static renal scintigraphy with Tc-99m-dimer-captosuccinic acid (DMSA) the calculation of the geometric mean of counts from the anterior and posterior view is recommended. Aim of this retrospective study was to find out, if the anterior view is necessary to receive an accurate differential renal function by calculating the geometric mean compared to calculating PF using the counts of the posterior view only. Methods: 164 DMSA-scans of 151 children (86 f, 65 m) aged 16 d to 16 a (4.7 ± 3.9 a) were reviewed. The scans were performed using a dual head gamma camera (Picker Prism 2000 XP, low energy ultra high resolution collimator, matrix 256 x 256,300 kcts/view, Zoom: 1.6-2.0). Background corrected values from both kidneys anterior and posterior were obtained. Using region of interest technique PF was calculated using the counts of the dorsal view and compared with the calculated geometric mean [SQR(Ctsdors x Ctsventr]. Results: The differential function of the right kidney was significantly less when compared to the calculation of the geometric mean (p<0.01). The mean difference between the PFgeom and the PFdors was 1.5 ± 1.4%. A difference > 5% (5.0-9.5%) was obtained in only 6/164 scans (3.7%). Three of 6 patients presented with an underestimated PFdors due to dystopic kidneys on the left side in 2 patients and on the right side in one patient. The other 3 patients with a difference >5% did not show any renal abnormality. Conclusion: The calculation of the PF from the posterior view only will give an underestimated value of the right kidney compared to the calculation of the geometric mean. This effect is not relevant for the calculation of the differntial renal function in orthotopic kidneys, so that in these cases the anterior view is not necesssary. However, geometric mean calculation to obtain reliable values for differential renal function should be applied in cases with an obvious anatomical abnormality.

Therapeutic Effects of Hab Shabyar on Open-Angle Glaucoma: A Double-Blinded, Randomized, Placebo-Controlled Trial

2019 ◽  
Vol 8 ◽  
pp. 1218
Author(s):  
Ebrahim Khalil BaniHabib ◽  
Ali Mostafai ◽  
Seyyed Mohammad Bagher Fazljou ◽  
Ghadir Mohammdi
Keyword(s):  
Placebo Group ◽  
Intraocular Pressure ◽  
Open Angle Glaucoma ◽  
Controlled Trial ◽  
Intervention Group ◽  
Therapeutic Effects ◽  
Open Angle ◽  
The Mean ◽  
The Right ◽  
Double Blinded

Background: Open-angle glaucoma (OAG) is one of the leading causes of blindness worldwide. This study evaluates the therapeutic effects of hab shabyar in patients with open-angle glaucoma. Materials and Methods: In this clinical randomized controlled trial, 50 patients with OAG were randomized into two groups. The intervention group was received a drop of timolol plus 500 mg of hab shabyar every 12 hours. The placebo group was received a drop of timolol every 12 hours plus 500 mg of wheat germ as a placebo. The intraocular pressure in patients with OAG was measured in each group and compared at before the intervention (t1), one month (t2), and two months (t3) after the intervention. Results: The mean decrease in intraocular pressure for the right eye at three times in the intervention group was statistically significant, but the mean decrease in the placebo group was not significant. Similar results were obtained for the left eye at t1 when compared to t3. The patients in the intervention group expressed more satisfaction than the patients in the placebo group (P≤0.001). Conclusion: Our study demonstrated that consumption of timolol plus hab shabyar instead of consuming of timolol alone was probably more effective for reducing intraocular pressure in patients with OAG.[GMJ.2019;In press:e1218]

Flow cytometric identification and isolation of hypertrophic type II pneumocytes after partial pneumonectomy

1989 ◽  
Vol 257 (3) ◽  
pp. C528-C536 ◽  
Author(s):  
B. D. Uhal ◽  
S. R. Rannels ◽  
D. E. Rannels
Keyword(s):  
Gradient Centrifugation ◽  
Type Ii Cells ◽  
Type Ii ◽  
Acridine Orange Staining ◽  
Flow Cytometric ◽  
Percoll Density Gradient Centrifugation ◽  
Type Ii Pneumocytes ◽  
The Mean ◽  
Cellular Dna ◽  
The Right

Type II pneumocytes were isolated by either Percoll density gradient centrifugation or by immunoglobulin G (IgG) panning from the lungs of normal rats and the right lung of rats subjected to left pneumonectomy. Cells were studied at 7- (pnx-7) and 15- (pnx-15) days postoperative, times during and after, respectively, rapid compensatory growth of the right lung. Acridine orange staining permitted resolution of type II cells from contaminants on the basis of high red fluorescence (greater than 590 nm). Simultaneous measurement of forward-angle light scatter (FALS) suggested a shift of pnx-7 cells toward greater size, which was reversed in pnx-15 cells. By Percoll gradient isolation, approximately 15% of pnx-7 cells analyzed were above the mean FALS of control cells. In contrast, approximately 30% of the pnx-7 cells isolated by IgG panning were above the mean FALS of corresponding control cells. Biochemical analyses of pnx-7 cells separated by cell sorting into "high FALS" and "low FALS" subgroups revealed that high FALS type II cells contained 50% more protein (P less than 0.05) and 140% more RNA (P less than 0.01) than low FALS cells, with no significant change in cellular DNA content. These data are consistent with previous studies of type II cells isolated from the lungs of pneumonectomized animals and confirm the presence of hypertrophic cells in these preparations. They provide a foundation from which to design further flow cytometric studies of the role of hypertrophic type II pneumocytes in compensatory lung growth.

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