scholarly journals Symmetric double bubbles in the Grushin plane

2019 ◽  
Vol 25 ◽  
pp. 77
Author(s):  
Valentina Franceschi ◽  
Giorgio Stefani

We address the double bubble problem for the anisotropic Grushin perimeter Pα, α ≥ 0, and the Lebesgue measure in ℝ2, in the case of two equal volumes. We assume that the contact interface between the bubbles lies on either the vertical or the horizontal axis. We first prove existence of minimizers via the direct method by symmetrization arguments and then characterize them in terms of the given area by first variation techniques. Even though no regularity theory is available in this setting, we prove that angles at which minimal boundaries intersect satisfy the standard 120-degree rule up to a suitable change of coordinates. While for α = 0 the Grushin perimeter reduces to the Euclidean one and both minimizers coincide with the symmetric double bubble found in Foisy et al. [Pacific J. Math. 159 (1993) 47–59], for α = 1 vertical interface minimizers have Grushin perimeter strictly greater than horizontal interface minimizers. As the latter ones are obtained by translating and dilating the Grushin isoperimetric set found in Monti and Morbidelli [J. Geom. Anal. 14 (2004) 355–368], we conjecture that they solve the double bubble problem with no assumptions on the contact interface.

1993 ◽  
Vol 26 (6) ◽  
pp. 787-794 ◽  
Author(s):  
M. A. Singh ◽  
S. S. Ghosh ◽  
R. F. Shannon

A direct (i.e. noniterative) method for desmearing the beam-height effect in small-angle X-ray scattering is discussed. The method is applicable to rectangular collimation systems with arbitrary beam-height intensity profiles. The process involves the construction of an upper-triangular matrix of terms containing the resolution information. A straightforward back-substitution process can then be used to determine the ideal pinhole-collimated curve for any experimental curve obtained with the given resolution. The principal advantage of the method lies in its simplicity, which facilitates an examination of the propagation of random errors through the desmearing process. A comparison between the direct method and the iterative approach of Glatter [J. Appl. Cryst. (1974), 7, 147–153] is made to illustrate the efficiency of the technique.


The memoir consists of four sections. In the first section, the theory of the residues obtained by applying the process of the common measure to two algebraical functions is discussed. It is shown that a certain superfluous or allotrious factor enters into each, the value of which, in terms of the leading coefficients of the residues in their simplified form, is determined; and the simplified residues themselves are subsequently obtained from the given functions by a direct method.


1994 ◽  
Vol 31 (03) ◽  
pp. 797-815 ◽  
Author(s):  
Søren Asmussen ◽  
Reuven Y. Rubinstein ◽  
Chia-Li Wang

In this paper we obtain some new theoretical and numerial results on estimation of small steady-state probabilities in regenerative queueing models by using the likelihood ratio (score function) method, which is based on a change of the probability measure. For simple GI/G/1 queues, this amounts to simulating the regenerative cycles by a suitable change of the interarrival and service time distribution, typically corresponding to a reference traffic intensity ρ 0 which is < 1 but larger than the given one ρ. For the M/M/1 queue, the resulting gain of efficiency is calculated explicitly and shown to be considerable. Simulation results are presented indicating that similar conclusions hold for gradient estimates and in more general queueing models like queueing networks.


2014 ◽  
Vol 25 (01) ◽  
pp. 1450011 ◽  
Author(s):  
GERARDO R. CHACÓN ◽  
RENATO COLUCCI ◽  
HUMBERTO RAFEIRO ◽  
ANDRÉS VARGAS

We study the existence of minimizers of a regularized non-convex functional in the context of variable exponent Sobolev spaces by application of the direct method in the calculus of variations. The results are new even in the framework of classical Lebesgue spaces.


1960 ◽  
Vol 106 (443) ◽  
pp. 531-536 ◽  
Author(s):  
J. R. Smythies ◽  
C. K. Levy

There are, in the main, two principal methods of biological research into the problem of causation of schizophrenia. In the first, or direct method, studies are made of the body fluids and of the metabolism of schizophrenics in the hope of turning up some toxic factor or error of metabolism. Such studies are currently under way in a number of centres and, although many claims as to positive findings have been made, none of these has as yet been substantiated and the majority have actually been refuted by subsequent analysis. These studies have been directed by two main hypotheses of the causation of the illness described in a previous communication (14). Now it is clear that one great difficulty entailed in using this approach is that our basic knowledge of neuro-chemistry and neuropharmacology is still extremely limited. The very complexity of the neurochemical processes of the brain would seem to offer a large number of possible loci of metabolic disturbance, any of which might be responsible for the onset of schizophrenia. Thus, we can say that the probable results of all such investigations at present will be purely negative, i.e. we will discover that the given metabolism of injected adrenaline, serotonin, etc., is normal in the illness. Thus, the second, or indirect, method becomes increasingly important. This consists of two parts. The first is implied merely by the general statement that the advance of neurochemistry and neuropharmacology may disclose other areas of function in which specific hypotheses of the causation of the illness may be constructed. The second consists of a detailed study of the precise mode of action of known psychotomimetic agents. Knowledge of the mode of action of mescaline and LSD-25 in detail would pinpoint areas for research in the metabolism of schizophrenics. Unfortunately, however, very little seems to be known at present of the mode of action of these drugs. Most of the work that has been done has been carried out on LSD-25 (see e.g. the annotated bibliography put out by Sandoz Inc., Hanover, New Jersey, 1958). However, the important field of the structure-activity relationships (SAR) of mescaline seems to have been strangely neglected. The research programme here would include the synthesis of a number of analogues of mescaline according to systematic principles to determine the role the details of its molecular structure play in the mode of action. These analogues would be subjected to a number of tests—psychopharmacological, neurophysiological and neurochemical—comparison between the results of which would give important information not only of the SAR of mescaline but also of the details of its mode of action and the interrelationships between behavioural, neurophysiological and neurochemical factors involved that would help us explain the former effects in terms of the latter, which reduction is the main aim of biological science. The present paper reviews the little that is known at present about the SAR of mescaline and then reports some new studies in the psychopharmacology of this field.


Author(s):  
Marco Pozzetta

For a smooth closed embedded planar curve, we consider the minimization problem of the Willmore energy among immersed surfaces of a given genus greater than 1 having the given curve as boundary, without any prescription on the conormal. By general lower bound estimates, in case the curve is a circle we prove that such problem is equivalent if restricted to embedded surfaces, we prove that do not exist minimizers, and we calculate the infimum. Then we study the case in which the genus is 1 and the competitors are restricted to a suitable class of varifolds including embedded surfaces, and we prove that the non-existence of minimizers implies a lower bound on the infimum; therefore we use such criterion in order to explicitly find an infinite family of curves for which such problem does have minimizers in the corresponding class of varifolds.


2020 ◽  
Vol 142 (4) ◽  
Author(s):  
P. C. Zhao ◽  
Y. T. Hu ◽  
L. L. Wang ◽  
Z. W. Liu

Abstract The plastic deformation during flash-butt welding (FBW) and its effects on weld quality are investigated by using numerical and experimental methods. The electro-thermo-mechanical coupling model of FBW is validated by comparing the calculated temperature and plastic deformation to measured one, obtaining reasonable agreement. The calculation results reveal that a thin liquid metal film forms at the contact interface during accelerating flash stage to provide temperature conditions for upsetting. The length of liquid metal (including burning and expelled losses) is 29.7 mm for one piece pipeline tube under the given condition. The stress and strain at contact surface are both almost zero at the initial stage of upsetting due to the thin liquid metal film existing at the contact interface, and they rapidly increase to 58.0 MPa and 17.7, respectively, while the liquid metal are excluded from the contact interface between two tubes to be welded. The maximum plastic deformation is 18.1 mm at the given condition under the action of upsetting force. The experimental results illustrate that the microstructure of X65 FBW joints consists of massive ferrite, grain boundary pre-eutectic ferrite, pearlite, and widmannstatten, while the microstructure in heat-affected zone is fine ferrite and pearlite. The coarse grain size and gray spots in the butt joint severely decrease the tension strength and impact toughness.


1994 ◽  
Vol 31 (3) ◽  
pp. 797-815 ◽  
Author(s):  
Søren Asmussen ◽  
Reuven Y. Rubinstein ◽  
Chia-Li Wang

In this paper we obtain some new theoretical and numerial results on estimation of small steady-state probabilities in regenerative queueing models by using the likelihood ratio (score function) method, which is based on a change of the probability measure. For simple GI/G/1 queues, this amounts to simulating the regenerative cycles by a suitable change of the interarrival and service time distribution, typically corresponding to a reference traffic intensity ρ0 which is < 1 but larger than the given one ρ. For the M/M/1 queue, the resulting gain of efficiency is calculated explicitly and shown to be considerable. Simulation results are presented indicating that similar conclusions hold for gradient estimates and in more general queueing models like queueing networks.


2013 ◽  
Vol 22 (7) ◽  
pp. 970 ◽  
Author(s):  
Jorge C. S. André ◽  
João C. Gonçalves ◽  
Gilberto C. Vaz ◽  
Domingos X. Viegas

Laboratory fire tests were performed in still air, for variable inclinations (10°, 15°) and fuel bed dimensions (1.28×2.50–3.0×4.6m2), with homogeneous fuel beds of pine needles and pine wood excelsior. The fire ignition was made at a point, along a closed line with no fuel inside and along a straight edge of the fuel bed. The tests were recorded with an infrared camera and various techniques were developed to implement direct and indirect empirical methods of construction of the ‘orientation function’ of the fire in the given fuel bed and ambient conditions, showing how the rate of spread of a steady straight fire front depends on its orientation on the terrain. The direct method uses a set of straight fire fronts with various orientations whereas the indirect method uses essentially a point ignited fire front. Contrary to what is assumed in BehavePlus model, the orientation function is observed to depend significantly on the properties of the fuel bed. In all tests with closed fire fronts, the full steadiness required by the indirect method was not achieved (namely, at the head of the front), although, for slope angle of 10°, the corresponding errors induced on the orientation function were small.


2018 ◽  
Vol 6 (1) ◽  
pp. 64-88
Author(s):  
Eliot Bongiovanni ◽  
Leonardo Di Giosia ◽  
Alejandro Diaz ◽  
Jahangir Habib ◽  
Arjun Kakkar ◽  
...  

Abstract The classic double bubble theorem says that the least-perimeter way to enclose and separate two prescribed volumes in ℝN is the standard double bubble. We seek the optimal double bubble in ℝN with density, which we assume to be strictly log-convex. For N = 1 we show that the solution is sometimes two contiguous intervals and sometimes three contiguous intervals. In higher dimensions we think that the solution is sometimes a standard double bubble and sometimes concentric spheres (e.g. for one volume small and the other large).


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