High orders of the Weyl expansion for quantum billiards: resurgence of periodic orbits, and the Stokes phenomenon

1994 ◽  
Vol 447 (1931) ◽  
pp. 527-555 ◽  
Keyword(s):  
Smooth Boundary ◽  
Error Function ◽  
High Energy ◽  
Negative Energy ◽  
Stokes Line ◽  
Stokes Phenomenon ◽  
Constant Negative Curvature ◽  
Family Of Curves ◽  
The Laplace Operator ◽  
Weyl Expansion

A formalism is developed for calculating high coefficients c r of the Weyl (high energy) expansion for the trace of the resolvent of the Laplace operator in a domain B with smooth boundary ∂ B The c r are used to test the following conjectures. ( a ) The sequence of c r diverges factorially, controlled by the shortest accessible real or complex periodic geodesic. ( b ) If this is a 2-bounce orbit, it corresponds to the saddle of the chord length function whose contour is first crossed when climbing from the diagonal of the Möbius strip which is the space of chords of B . ( c ) This orbit gives an exponential contribution to the remainder when the Weyl series, truncated at its least term, is subtracted from the resolvent; the exponential switches on smoothly (according to an error function) where it is smallest, that is across the negative energy axis (Stokes line). These conjectures are motivated by recent results in asymptotics. They survive tests for the circle billiard, and for a family of curves with 2 and 3 bulges, where the dominant orbit is not always the shortest and is sometimes complex. For some systems which are not smooth billiards (e. g. a particle on a ring, or in a billiard where ∂ B is a polygon), the Weyl series terminates and so no geodesics are accessible; for a particle on a compact surface of constant negative curvature, only the complex geodesics are accessible from the Weyl series.

Smoothing of the Stokes phenomenon for high-order differential equations

1992 ◽  
Vol 436 (1896) ◽  
pp. 165-186 ◽  
Keyword(s):  
Functional Form ◽  
Arbitrary Order ◽  
Error Function ◽  
Smooth Transition ◽  
Stokes Line ◽  
Parabolic Cylinder ◽  
Stokes Phenomenon ◽  
Function Type ◽  
Parabolic Cylinder Functions ◽  
Class Of Functions

We extend the class of functions for which the smooth transition of a Stokes multiplier across a Stokes line can be rigorously established to functions satisfying a certain differential equation of arbitrary order n . The equation chosen admits solutions of hypergeometric function type which, in the case n = 2, are related to the parabolic cylinder functions. In general, the solutions of this equation involve compound asymptotic expansions, valid in certain sectors of the complex z -plane, with more than one dominant and subdominant series. The functional form of the Stokes multipliers, expressed in terms of an appropriately scaled variable describing transition across a Stokes line, is found to obey the error function smoothing law derived by Berry.

Waves near Stokes lines

1990 ◽  
Vol 427 (1873) ◽  
pp. 265-280 ◽  
Keyword(s):  
Refractive Index ◽  
Error Function ◽  
Divergent Series ◽  
Stokes Line ◽  
Stokes Phenomenon ◽  
Total Change ◽  
Integral Approximation ◽  
Reflected Waves ◽  
Stokes Lines ◽  
Phase Integral

The large- k asymptotics of d 2 u ( z )/d z 2 = k 2 R 2 ( z ) u ( z ) are studied near a Stokes line ( ω ≡ ∫ z z 0 R d z real, where z 0 is a zero of R 2 ( z ), of any order), on which there is greatest disparity between the dominant and subdominant exponential waves in the phase-integral (WKB) approximations. The aim is to establish precisely how the multiplier b _ of the subdominant wave varies across the Stokes line. Although b _ always has a total change proportional to i times the multiplier of the dominant wave (the Stokes phenomenon), the form of the change depends on the convention used to define the two waves. The optimal convention, for which the variation is maximally compact and smooth, is to define them by the phase-integral approximation truncated at its least term, whose order is proportional to k and therefore large (‘asymptotics of asymptotics’). Then the variation of b _ is proportional to the error function of the natural Stokes-crossing variable Im ω √( k /Re ω ). This result is obtained without resumming divergent series (thereby avoiding ‘asymptotics of asymptotics of asymptotics’). An application is given, to the birth of exponentially weak reflected waves in media with smoothly varying refractive index.

scholarly journals Performance Characteristics in Runner of an Impulse Water Turbine with Splitter Blade

Processes ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 303
Author(s):  
Lingdi Tang ◽  
Shouqi Yuan ◽  
Yue Tang ◽  
Zhijun Gao
Keyword(s):  
Energy Conversion ◽  
Flow Direction ◽  
Mechanical Energy ◽  
Experimental Tests ◽  
High Energy ◽  
Negative Energy ◽  
Water Turbine ◽  
Conversion Performance ◽  
Water Turbines ◽  
Current Energy

The impulse water turbine is a promising energy conversion device that can be used as mechanical power or a micro hydro generator, and its application can effectively ease the current energy crisis. This paper aims to clarify the mechanism of liquid acting on runner blades, the hydraulic performance, and energy conversion characteristics in the runner domain of an impulse water turbine with a splitter blade by using experimental tests and numerical simulations. The runner was divided into seven areas along the flow direction, and the power variation in the runner domain was analyzed to reflect its energy conversion characteristics. The obtained results indicate that the critical area of the runner for doing the work is in the front half of the blades, while the rear area of the blades does relatively little work and even consumes the mechanical energy of the runner to produce negative work. The high energy area is concentrated in the flow passage facing the nozzle. The energy is gradually evenly distributed from the runner inlet to the runner outlet, and the negative energy caused by flow separation with high probability is gradually reduced. The clarification of the energy conversion performance is of great significance to improve the design of impulse water turbines.

scholarly journals Negative energy states in the Reissner–Nordström metric

2021 ◽  
pp. 2150120
Author(s):  
O. B. Zaslavskii
Keyword(s):  
Turning Point ◽  
High Energy ◽  
Negative Energy ◽  
Infinite Chain ◽  
White Hole ◽  
Negative Energy States ◽  
Energy Formula ◽  
High Energy Collisions ◽  
Energy States

We consider electrogeodesics on which the energy [Formula: see text] in the Reissner–Nordström metric. It is shown that outside the horizon there is exactly one turning point inside the ergoregion for such particles. This entails that such a particle passes through an infinite chain of black–white hole regions or terminates in the singularity. These properties are relevant for two scenarios of high energy collisions in which the presence of white holes is essential.

Body condition at calving and the performance of dairy cows in early lactation under Australian conditions: a review

2001 ◽  
Vol 41 (6) ◽  
pp. 823 ◽  
Author(s):  
C. R. Stockdale
Keyword(s):  
Dairy Cows ◽  
Body Condition ◽  
Dietary Protein ◽  
High Energy ◽  
Negative Energy ◽  
Considerable Proportion ◽  
Early Lactation ◽  
Protein Supplements ◽  
Grazed Pasture ◽  
Feeding Experiments

With the current increases in genetic merit and feeding occurring at farm level, dairy cows are under increasing nutritional stress in early lactation. Cows obtain their energy at this time from the feeds they eat and from body reserves. The relationship between body condition at calving and productivity of dairy cows has been reviewed, with particular emphasis on interactions between body condition and nutrition in early lactation. Recent research on the influence of body condition at calving on subsequent milk productivity, conducted mainly in the United Kingdom with complete diets fed indoors, has produced results in apparent conflict with the previous results from southern Australia and New Zealand where cows grazed pasture. In particular, the overseas research suggests considerably less advantage to improvements in body condition than had been previously thought. It is concluded that more information is needed concerning the interaction between body condition at calving and nutrition in early lactation, with dietary energy and protein both being important. There is a suggestion that, when complete diets are fed, it is better to achieve high energy concentrations in post-calving diets by the use of high-fibre concentrates with a fat supplement, rather than with high-starch concentrates. This has implications for dairying in Australia, since cereal grains are the major energy supplement used on many farms in early lactation and recent research has indicated that immediate marginal milk production responses to the use of concentrates may be poorer with fat cows than with thin cows. Reports from controlled feeding experiments indicate that fat cows need more dietary protein than thin cows and undegradable dietary protein might be of more concern than rumen degradable protein. However, in dairy systems where pasture is a considerable proportion of the diet, benefits of supplying specific undegradable dietary protein supplements still need to be established. Recent research has suggested that pasture appears to provide considerable quantities of undegradable dietary protein, even though the crude protein in pasture is potentially highly degradable in the rumen. Body condition at calving may also affect subsequent reproductive performance. This is due to its association with the degree of negative energy balance occurring in early lactation and because fat cows may be more susceptible to metabolic disease(s). While the mechanisms involved are probably quite complex, increases in animal productivity will generate more stress in cows at a time of their annual cycle when stress needs to be minimised. Further understanding is required to link the relevancy of overseas research to Australian dairy farming conditions where pasture is a key input.

scholarly journals The annihilation of fast positrons by electrons in the K-shell

1934 ◽  
Vol 146 (859) ◽  
pp. 723-736 ◽  
Keyword(s):  
Kinetic Energy ◽  
Cosmic Radiation ◽  
Nuclear Charge ◽  
Structure Constant ◽  
Photoelectric Effect ◽  
High Energy ◽  
Negative Energy ◽  
Fine Structure Constant ◽  
Quantum Process ◽  
High Energies

The probability of the simultaneous of a positron and an electron, with the emission of two quanta of radiation, has been calculated by Dirac and several other authors. From considerations of energy and momentum it follows that an electron and positron can only annihilate one another with the emission of one quantum of radiation in the presence of a third body. An electron bound in an atom could, therefore, annihilate a positron, represented by a hole on the Dirac theory, by jumping into a state of negative energy which happens to be free, the nucleus taking up the extra momentum. The process is now mathematically analogous to the photoelectric transitions to states of negative energy in the sense that the matrix elements concerned are the same, and we might expect that the effect would be most important for the electrons in the K-shell. Fermi and Uhlenbeck have calculated the process approximately, for the condition where the kinetic energy of the positron is of the order of magnitude of the ionization energy of the K-shell. The result they obtained was very small compared with the two quantum process, which is to be explained by the fact that for these small energies, the positron does not get near the nucleus. In view of the fact that positrons of energies of the order 100 mc 2 occur in considerable quantities in the showers produced by cosmic radiation, and that the primary cosmic radiation itself may consist, in part, of positrons, it becomes of interest to calculate the cross-section for the annihilation of positrons of high energy by electrons in the K-shell, and their absorption in matter, and also to compare this process with the two quantum process for high energies. In the photoelectric effect for hard γ -rays, the electron the electron leaves the atom in states of different angular momentum (described by the azimuthal quantum number l ), and the terms which give the largest contribution are roughly those for which l is of the order of the energy of the γ -ray in terms of mc 2 . For high energies, therefore, a calculation by the method of Hulme, in which the last step is carried out numerically, is out of the question, and we must find some approximate method of effecting a summation. We shall use an adaptation of Sauter's method, in which we shall treat as small the product of the fine structure constant and the nuclear charge. This method may be expected to give a good approximation for small nuclear charge. Our method has the further restriction that it is valid only when the kinetic energy of the positron is not small compared with mc 2 .

Energy balance and body composition during US Army special forces training

2013 ◽  
Vol 38 (4) ◽  
pp. 396-400 ◽  
Author(s):  
Lee M. Margolis ◽  
Jennifer Rood ◽  
Catherine Champagne ◽  
Andrew J. Young ◽  
John W. Castellani
Keyword(s):  
Body Composition ◽  
Energy Balance ◽  
Body Mass ◽  
Skinfold Thickness ◽  
High Energy ◽  
Negative Energy ◽  
Fat Free Mass ◽  
Energy Deficit ◽  
Special Forces ◽  
Small Unit

Small Unit Tactics (SUT) is a 64-day phase of the Special Forces Qualification Course designed to simulate real-world combat operations. Assessing the metabolic and physiological responses of such intense training allows greater insights into nutritional requirements of soldiers during combat. The purpose of this study was to examine energy balance around specific training events, as well as changes in body mass and composition. Data were collected from 4 groups of soldiers (n = 36) across 10-day periods. Participants were 28 ± 5 years old, 177 ± 6 cm tall, and weighed 83 ± 7 kg. Doubly labeled water (D218O) was used to assess energy expenditure. Energy intake was calculated by subtracting energy in uneaten foods from known energy in distributed foods in individually packaged combat rations or in the dining facility. Body composition was estimated from skinfold thickness measurements on days 0 and 64 of the course. Simulated urban combat elicited that largest energy deficit (11.3 ± 2.3 MJ·day−1 (2700 ± 550 kcal·day−1); p < 0.05), and reduction in body mass (3.3 ± 1.9 kg; p < 0.05), during SUT, while energy balance was maintained during weapons familiarization training and platoon size raids. Over the entire course body mass decreased by 4.2 ± 3.7 kg (p < 0.01), with fat mass decreasing by 2.8 ± 2.0 kg (p < 0.01) and fat-free mass decreasing by 1.4 ± 2.8 kg (p < 0.05). The overall reduction in body mass suggests that soldiers were in a negative energy balance during SUT, with high energy deficit being observed during strenuous field training.

scholarly journals Regulation of serum leptin and its role in the hyperphagia of lactation in the rat

2003 ◽  
Vol 176 (2) ◽  
pp. 193-203 ◽  
Author(s):  
RG Denis ◽  
G Williams ◽  
RG Vernon
Keyword(s):  
Energy Balance ◽  
Food Intake ◽  
Milk Production ◽  
Litter Size ◽  
High Energy ◽  
Negative Energy ◽  
Negative Energy Balance ◽  
Calorie Intake ◽  
Serum Leptin ◽  
Nocturnal Rise

The factors regulating serum leptin concentration and its relationship to the hyperphagia of lactation have been investigated in rats. Lactation results in hypoleptinaemia and loss, or at least marked attenuation, of the nocturnal rise in serum leptin. Litter removal resulted in a fall in food intake and restoration of the nocturnal rise in serum leptin. Returning the litter to the mother after a 48-h absence increased food intake and began to reinitiate milk production, but the nocturnal serum leptin levels were still increased at 48 h after litter restoration. Adjusting litter size to four, eight, ten or fourteen pups at parturition resulted in different rates of litter growth and food intake during the subsequent lactation, but had no effect on the degree of hypoleptinaemia. Reducing litter size from ten to four pups at mid-lactation resulted in a transient increase in both serum leptin and pup growth rate, while food intake fell to a level found in rats suckling four pups throughout lactation. Reducing milk production by injection of bromocriptine increased serum leptin, but did not restore the nocturnal rise in serum leptin; food intake decreased, but remained much higher than in non-lactating rats. Feeding a varied, high-energy diet resulted in a decrease in the weight of food ingested, but no change in calorie intake, and had no effect on the hypoleptinaemia. These studies suggested that the hypoleptinaemia of lactating rats is due to negative energy balance, but the loss of the nocturnal rise in serum leptin is due to the suckling stimulus. The negative energy balance of lactation does not appear to be caused by a physical constraint on food intake. While the hypoleptinaemia should facilitate the hyperphagia of lactation, other orexigenic signals must also be involved.

Overlapping Stokes smoothings: survival of the error function and canonical catastrophe integrals

1994 ◽  
Vol 444 (1920) ◽  
pp. 201-216 ◽  
Keyword(s):  
Saddle Point ◽  
Catastrophe Theory ◽  
Asymptotic Theory ◽  
Error Function ◽  
Large Parameter ◽  
Stokes Phenomenon ◽  
Numerical Illustration ◽  
Distant Cluster ◽  
Uniform Approximations ◽  
Common Framework

We derive doubly uniform approximations for the remainder in the optimally truncated saddle-point expansion for an integral containing a large parameter. Double uniformity means that the formulae remain valid while distant saddles responsible for the divergence of the expansion coalesce and separate (as described by catastrophe theory) and while the subdominant exponentials they contribute switch on and off (as described by the error-function smoothing of the Stokes phenomenon). Two sorts of asymptotic singularity are thereby united in a common framework. The formula for the remainder incorporates both the Stokes error function and the canonical catastrophe integrals. A numerical illustration is given, in which the distant cluster contains two saddles; the asymptotic theory gives an accurate description of the details of the fractional remainder, even when this is of order exp ( –36).

Managing the grazing dairy cow through the transition period: a review

2015 ◽  
Vol 55 (7) ◽  
pp. 936 ◽  
Author(s):  
J. K. Kay ◽  
J. J. Loor ◽  
A. Heiser ◽  
J. McGowan ◽  
J. R. Roche
Keyword(s):  
Milk Production ◽  
Dairy Cow ◽  
Transition Period ◽  
Body Condition Score ◽  
Management Strategies ◽  
Molecular Data ◽  
High Energy ◽  
Negative Energy ◽  
Health Indices ◽  
Grazing Dairy Cows

The transition period of the dairy cow generally refers to the last three weeks of gestation and the first three weeks of lactation. During this period, the dairy cow faces numerous physiological challenges, requiring both homeostatic and homeorhetic changes to support the demands of lactation. Management strategies to achieve a successful transition have developed over many decades. Historically, these strategies focussed on achieving high energy intakes pre-calving in an attempt to improve post-calving metabolism; however, more recent research has indicated that this approach may not be appropriate. Physiological and molecular data have indicated that imposing a slight negative energy balance (EBAL) pre-calving can improve post-calving EBAL, metabolic health indices and milk production. It was hypothesised that the challenges of the transition period would be less in a grazing system than in an intensive confinement system, due to the lower milk production and the difference in population density and, therefore, pathogen exposure. However, the molecular and immunological responses to the change of state are similar in magnitude in a moderate-yielding pasture-fed cow and in a high-yielding cow fed a total mixed ration. The collective data point to a peripartum immunosuppression, which is affected by body condition score and feeding level. This review will outline the literature and provide an assessment of the most recent transition cow management for grazing dairy cows.

Sign in / Sign up

Export Citation Format

Share Document