scholarly journals Bounds for discrete moments of Weyl sums and applications

2020 ◽  
Vol 194 (1) ◽  
pp. 1-28
Author(s):  
Karin Halupczok
Keyword(s):  
2009 ◽  
Vol 59 (3) ◽  
Author(s):  
Jörn Steuding

AbstractOn the basis of the Random Matrix Theory-model several interesting conjectures for the Riemann zeta-function were made during the recent past, in particular, asymptotic formulae for the 2kth continuous and discrete moments of the zeta-function on the critical line, $$ \frac{1} {T}\int\limits_0^T {|\zeta (\tfrac{1} {2} + it)|^{2k} dt} and \frac{1} {{N(T)}}\sum\limits_{0 < \gamma \leqslant {\rm T}} {|\zeta (\tfrac{1} {2} + i(\gamma + \tfrac{\alpha } {L}))|^{2k} } $$, by Conrey, Keating et al. and Hughes, respectively. These conjectures are known to be true only for a few values of k and, even under assumption of the Riemann hypothesis, estimates of the expected order of magnitude are only proved for a limited range of k. We put the discrete moment for k = 1, 2 in relation with the corresponding continuous moment for the derivative of Hardy’s Z-function. This leads to upper bounds for the discrete moments which are off the predicted order by a factor of log T.


Author(s):  
Gaber Hassan ◽  
Khalid M. Hosny ◽  
R. M. Farouk ◽  
Ahmed M. Alzohairy

One of the most often used techniques to represent color images is quaternion algebra. This study introduces the quaternion Krawtchouk moments, QKrMs, as a new set of moments to represent color images. Krawtchouk moments (KrMs) represent one type of discrete moments. QKrMs use traditional Krawtchouk moments of each color channel to describe color images. This new set of moments is defined by using orthogonal polynomials called the Krawtchouk polynomials. The stability against the translation, rotation, and scaling transformations for QKrMs is discussed. The performance of the proposed QKrMs is evaluated against other discrete quaternion moments for image reconstruction capability, toughness against various types of noise, invariance to similarity transformations, color face image recognition, and CPU elapsed times.


2010 ◽  
Vol 171-172 ◽  
pp. 518-522
Author(s):  
Shi Long Wang ◽  
Yu Ru Xu ◽  
Lei Wan ◽  
Xu Dong Tang

in the complex underwater environment, underwater images are taken by special underwater CCD camera and its S/N is low and the edge is fuzzy. For the four types of characteristic underwater targets, the novel moments called relative boundary moments are proposed, and the affine invariants of discrete moments are constructed. With scale, translating and rotating invariance, the moments can be used as the descriptors of the samples. Experimental results show that compared with the traditional regional moments, the new moment invariants not only can reduce the calculation in data processing to a large extent, but also improve the robustness and timeliness for engineering applications. When applying to the practical engineering, that is particularly approval for AUV to complete a certain mission.


2011 ◽  
Vol 8 (6) ◽  
pp. 10095-10123 ◽  
Author(s):  
A. R. Mangel ◽  
S. M. J. Moysey ◽  
J. C. Ryan ◽  
J. A. Tarbutton

Abstract. A lab scale infiltration experiment was conducted to evaluate the use of transient multi-offset ground-penetrating radar (GPR) data for characterizing dynamic hydrologic events in the vadose zone. A unique GPR data acquisition setup allowed sets of 21 traces at different offsets to be recorded every 30 s during a 3 h infiltration experiment. The result is a rich GPR data cube that can be viewed as multi-offset gathers at discrete moments in time or as common offset images that track changes in the GPR arrivals over the course of the experiment. These data allows us to continuously resolve the depth to soil boundaries while simultaneously tracking changes in wave velocity, which are strongly associated with soil water content variations. During the experiment the average volumetric water content estimated in the tank ranged between 10–30% with discrepancies between the GPR results, moisture probe data, and 1-D numerical modeling on the order of 3–5% (vol vol−1), though the patterns of the estimated water content over time were consistent for both wetting and drying cycles. Relative errors in the estimated depth to a soil boundary located 60 cm from the surface of the tank were typically on the order of 2% over the course of the experiment. During the period when a wetting front migrated downward through the tank, however, errors in the estimated depth of this boundary were as high as 25%, primarily as a result of wave interference between arrivals associated with the wetting front and soil boundary. Given that our analysis assumed one-dimensional, vertical infiltration, this high error could also suggest that more exhaustive GPR data and comprehensive analysis methods are needed to accurately image non-uniform flow produced during periods of intense infiltration. Regardless, we were able to track the movement of the wetting front through the tank and found a reasonably good correlation with in-situ water content measurements. We conclude that transient multi-offset GPR data are capable of quantitatively monitoring dynamic soil hydrologic processes.


Author(s):  
Suzanne Knight

In this chapter the author takes up the use of narrative inquiry within a secondary English language arts methods course. She focuses on two discrete moments that took place during one class session, where she and her students shared and discussed personal narratives. In particular, she explores the pedagogy that might be required to support a group of pre-service teachers’ work to become a connected knowing group, including the disruptive nature of vulnerability and risk taking.


Author(s):  
Ben Bridges ◽  
Sarah Osterhoudt

Broadly, landscapes can be considered terrains of connectivity. Landscapes encompass wild, cultivated, urban, feral, and fallow spaces, as well as the human and nonhuman entities who inhabit and shape them. Memory refers to the past as it exists in the present, bridging temporally discrete moments through the intentional or unintentional act of remembering. Memory studies, from the view of anthropology, include explorations of individual forms of remembrance, as well as the collective, heterogenous ways of marking, interpreting, and erasing the past. Taken together, landscapes and memory co-constitute one another: landscapes store, depict, and evoke memories while memories recall, revise, and shape landscapes. Knowledge and power are inevitably wrapped up in the relationship, and anthropologists have investigated the manifest ways such forces emerge through human acts of cultivation, commemoration, nostalgia, and forgetting. Because landscapes and memory appear in both physical and immaterial forms, the social constructs, cultural expressions, and human and nonhuman relationships on which they are based generate rich material for anthropological study. While landscape and memory are surely topics independently worthy of study, undertaking the two in tandem elucidates the intertwining threads that bind together space and time; such studies interrogate realms of personal meaning and political power while simultaneously highlighting dynamic processes of adaptation, improvisation, and erasure.


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