The plastic loading of face-centred cubic crystals along directions of symmetry leads to ambiguities in slip system selection. A solution to this problem has been proposed (Fuh & Havner, Proc. R. Soc. Lond. A 427, 193-239 (1989)) by postulating that the operating slip combination is the one which corresponds to the minimum plastic spin (MPS). A comparison is made between the MPS and rate-dependent slip theories. The predictions obtained are identical for the following cases: (i) (110) loading in channel die compression (with one exception) and (ii) all the multiple slip orientations in uniaxial tension. In the case of [100] pure plane strain compression, there is agreement for the four slip-system orientations, but not when six systems are required in the MPS analysis. It is shown that the symmetry of the slip distributions called for by the MPS and rate sensitive approaches is responsible for the broad equivalence between the predictions of the two theories. In each of the examples discussed, the number of operating slip systems is determined by symmetry considerations, leading to the operation of two, four, six or eight slip-systems, rather than by the number of prescribed constraints.
面心立方晶体沿对称方向的塑性载荷导致滑移系统选择的模糊性。已经提出了解决这个问题的方法(Fuh & Havner, Proc. R. Soc)。Lond。A 427,193 -239(1989))，假设操作滑移组合对应于最小塑性自旋(MPS)。将MPS理论与速率相关滑移理论进行了比较。在以下情况下获得的预测是相同的:(i)(110)通道模具压缩中的加载(有一个例外)和(ii)在单轴拉伸中所有的多滑移方向。在[100]纯平面应变压缩的情况下，四个滑移系统的取向是一致的，但在MPS分析中需要六个系统时就不一致了。结果表明，MPS和速率敏感方法所要求的滑移分布的对称性是两种理论预测之间广泛等效的原因。在讨论的每个例子中，操作滑移系统的数量是由对称性考虑决定的，从而导致两个、四个、六个或八个滑移系统的操作，而不是由规定约束的数量决定的。
{"title":"Comparison of the minimum plastic spin and rate sensitive slip theories for loading of symmetrical crystal orientations","authors":"L. Toth, J. Jonas, K. Neale","doi":"10.1098/rspa.1990.0008","DOIUrl":"https://doi.org/10.1098/rspa.1990.0008","url":null,"abstract":"The plastic loading of face-centred cubic crystals along directions of symmetry leads to ambiguities in slip system selection. A solution to this problem has been proposed (Fuh & Havner, Proc. R. Soc. Lond. A 427, 193-239 (1989)) by postulating that the operating slip combination is the one which corresponds to the minimum plastic spin (MPS). A comparison is made between the MPS and rate-dependent slip theories. The predictions obtained are identical for the following cases: (i) (110) loading in channel die compression (with one exception) and (ii) all the multiple slip orientations in uniaxial tension. In the case of [100] pure plane strain compression, there is agreement for the four slip-system orientations, but not when six systems are required in the MPS analysis. It is shown that the symmetry of the slip distributions called for by the MPS and rate sensitive approaches is responsible for the broad equivalence between the predictions of the two theories. In each of the examples discussed, the number of operating slip systems is determined by symmetry considerations, leading to the operation of two, four, six or eight slip-systems, rather than by the number of prescribed constraints.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"20 1","pages":"201 - 219"},"PeriodicalIF":0.0,"publicationDate":"1990-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81732080","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Asymptotic forms far from the source region of waves generated by oscillating sources in a linear homogeneous anisotropic system were derived by a method of proof that requires emendation although the final conclusions remain unchanged. An intermediate asymptotic result, in the form of an integral over that part S+ of the whole real wavenumber surface S on which a certain inequality (related to the radiation condition) is satisfied, needs modification as described in §2; but, as shown in §3, it is the modified form that is correctly estimated as in the final conclusions. Thus the proof is given two necessary emendations that cancel out. A careful analysis in §4 of why they cancel shows that the original intermediate result regains validity if S+, besides including that part of the real wavenumber surface S on which the inequality ∂ω/∂k1 > 0 is satisfied (where ω is frequency and k1 the component of wavenumber in the direction chosen for wave estimation), is considered as being continued on the complex wavenumber surface S, beyond the curve C on which ∂ω/∂k1 = 0, in the negative pure-imaginary k1-direction. This change is required to ensure the proper application of Cauchy’s theorem. Furthermore, the removal of any discontinuity at C prevents the appearance of an additional asymptotic term that would be unavoidably associated with such a singularity. I am grateful to Professor V. A. Borovikov for stimulating me to make these necessary clarifications.
{"title":"Emendations to a proof in the general three-dimensional theory of oscillating sources of waves","authors":"M. Lighthill","doi":"10.1098/rspa.1990.0003","DOIUrl":"https://doi.org/10.1098/rspa.1990.0003","url":null,"abstract":"Asymptotic forms far from the source region of waves generated by oscillating sources in a linear homogeneous anisotropic system were derived by a method of proof that requires emendation although the final conclusions remain unchanged. An intermediate asymptotic result, in the form of an integral over that part S+ of the whole real wavenumber surface S on which a certain inequality (related to the radiation condition) is satisfied, needs modification as described in §2; but, as shown in §3, it is the modified form that is correctly estimated as in the final conclusions. Thus the proof is given two necessary emendations that cancel out. A careful analysis in §4 of why they cancel shows that the original intermediate result regains validity if S+, besides including that part of the real wavenumber surface S on which the inequality ∂ω/∂k1 > 0 is satisfied (where ω is frequency and k1 the component of wavenumber in the direction chosen for wave estimation), is considered as being continued on the complex wavenumber surface S, beyond the curve C on which ∂ω/∂k1 = 0, in the negative pure-imaginary k1-direction. This change is required to ensure the proper application of Cauchy’s theorem. Furthermore, the removal of any discontinuity at C prevents the appearance of an additional asymptotic term that would be unavoidably associated with such a singularity. I am grateful to Professor V. A. Borovikov for stimulating me to make these necessary clarifications.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"15 17 1","pages":"31 - 42"},"PeriodicalIF":0.0,"publicationDate":"1990-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79428788","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A human brain operates as a pattern of switching. An abstract definition of a quantum mechanical switch is given that allows for the continual random fluctuations in the warm wet environment of the brain. Among several switch-like entities in the brain, I choose to focus on the sodium channel proteins. After explaining what these are, I analyse the ways in which my definition of a quantum switch can be satisfied by portions of such proteins. I calculate the perturbing effects of normal variations in temperature and electric field on the quantum state of such a portion. These are shown to be acceptable within the fluctuations allowed for by my definition. Information processing and unpredictability in the brain are discussed. The ultimate goal underlying the paper is an analysis of quantum measurement theory based on an abstract definition of the physical manifestations of consciousness. The paper is written for physicists with no prior knowledge of neurophysiology, but enough introductory material has also been included to allow neurophysiologists with no prior knowledge of quantum mechanics to follow the central arguments.
{"title":"Quantum theory and the brain","authors":"Matthew J. Donald","doi":"10.1098/rspa.1990.0004","DOIUrl":"https://doi.org/10.1098/rspa.1990.0004","url":null,"abstract":"A human brain operates as a pattern of switching. An abstract definition of a quantum mechanical switch is given that allows for the continual random fluctuations in the warm wet environment of the brain. Among several switch-like entities in the brain, I choose to focus on the sodium channel proteins. After explaining what these are, I analyse the ways in which my definition of a quantum switch can be satisfied by portions of such proteins. I calculate the perturbing effects of normal variations in temperature and electric field on the quantum state of such a portion. These are shown to be acceptable within the fluctuations allowed for by my definition. Information processing and unpredictability in the brain are discussed. The ultimate goal underlying the paper is an analysis of quantum measurement theory based on an abstract definition of the physical manifestations of consciousness. The paper is written for physicists with no prior knowledge of neurophysiology, but enough introductory material has also been included to allow neurophysiologists with no prior knowledge of quantum mechanics to follow the central arguments.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"33 1","pages":"43 - 93"},"PeriodicalIF":0.0,"publicationDate":"1990-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82421490","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A method is described by means of which the characteristic initial value problem can be reduced to the Cauchy problem and examples are given of how it can be used in practice. As an application it is shown that the characteristic initial value problem for the Einstein equations in vacuum or with perfect fluid source is well posed when data are given on two transversely intersecting null hypersurfaces. A new discussion is given of the freely specifiable data for this problem.
{"title":"Reduction of the characteristic initial value problem to the Cauchy problem and its applications to the Einstein equations","authors":"A. Rendall","doi":"10.1098/rspa.1990.0009","DOIUrl":"https://doi.org/10.1098/rspa.1990.0009","url":null,"abstract":"A method is described by means of which the characteristic initial value problem can be reduced to the Cauchy problem and examples are given of how it can be used in practice. As an application it is shown that the characteristic initial value problem for the Einstein equations in vacuum or with perfect fluid source is well posed when data are given on two transversely intersecting null hypersurfaces. A new discussion is given of the freely specifiable data for this problem.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"1 1","pages":"221 - 239"},"PeriodicalIF":0.0,"publicationDate":"1990-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91158836","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present various techniques for the asymptotic expansions of generalized functions. We show that the moment asymptotic expansions hold for a very wide variety of kernels such as generalized functions of rapid decay and rapid oscillations. We do not use Mellin transform techniques as done by previous authors in the field. Instead, we introduce a direct approach that not only solves the one-dimensional problems but also applies to various multidimensional integrals and oscillatory kernels as well. This approach also helps in the development of various asymptotic series arising in diverse fields of mathematics and physics. We find that the asymptotic expansions of generalized functions depend on the selection of suitable spaces of test functions. Accordingly, we have exercised special care in classifying the spaces and the distributions defined on them. Furthermore, we use the theory of topological tensor products to obtain the expansions of vector-valued distributions. We present several examples to illustrate that many classical results follow in a simple manner. For instance, we derive from our results the asymptotic expansions of certain series considered by Ramanujan.
{"title":"A distributional theory for asymptotic expansions","authors":"Ricardo Estrada, R. Kanwal","doi":"10.1098/rspa.1990.0041","DOIUrl":"https://doi.org/10.1098/rspa.1990.0041","url":null,"abstract":"We present various techniques for the asymptotic expansions of generalized functions. We show that the moment asymptotic expansions hold for a very wide variety of kernels such as generalized functions of rapid decay and rapid oscillations. We do not use Mellin transform techniques as done by previous authors in the field. Instead, we introduce a direct approach that not only solves the one-dimensional problems but also applies to various multidimensional integrals and oscillatory kernels as well. This approach also helps in the development of various asymptotic series arising in diverse fields of mathematics and physics. We find that the asymptotic expansions of generalized functions depend on the selection of suitable spaces of test functions. Accordingly, we have exercised special care in classifying the spaces and the distributions defined on them. Furthermore, we use the theory of topological tensor products to obtain the expansions of vector-valued distributions. We present several examples to illustrate that many classical results follow in a simple manner. For instance, we derive from our results the asymptotic expansions of certain series considered by Ramanujan.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"15 1","pages":"399 - 430"},"PeriodicalIF":0.0,"publicationDate":"1990-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91251366","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
O. Zienkiewicz, Y. Xie, B. Schrefler, A. Ledesma, N. Bićanić
Negative pore pressures existing in semi-saturated conditions provide a substantial ‘cohesion’ of the soil. This cohesion is of importance in the dynamic response of embankments and dams. The paper extends the formulation presented in part I to problems of semi-saturated behaviour with the assumption of free air ingress. An approximate reconstruction of the failure of the lower San Fernando dam during the 1971 earthquake is presented.
{"title":"Static and dynamic behaviour of soils: a rational approach to quantitative solutions. II. Semi-saturated problems","authors":"O. Zienkiewicz, Y. Xie, B. Schrefler, A. Ledesma, N. Bićanić","doi":"10.1098/rspa.1990.0062","DOIUrl":"https://doi.org/10.1098/rspa.1990.0062","url":null,"abstract":"Negative pore pressures existing in semi-saturated conditions provide a substantial ‘cohesion’ of the soil. This cohesion is of importance in the dynamic response of embankments and dams. The paper extends the formulation presented in part I to problems of semi-saturated behaviour with the assumption of free air ingress. An approximate reconstruction of the failure of the lower San Fernando dam during the 1971 earthquake is presented.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"53 1","pages":"311 - 321"},"PeriodicalIF":0.0,"publicationDate":"1990-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89163171","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A unique discrete form of the Navier-Stokes equations for unsteady, three-dimensional, incompressible flow has been used to study vortex breakdown numerically. A Burgers-type vortex was introduced along the central axis of the computational domain, and allowed to evolve in space and time. By varying the strength of the vortex and the free stream axial velocity distribution, using a previously developed Rossby number criterion as a guide, the location and size of the vortex breakdown region was controlled. While the boundaries of the vortex breakdown bubble appear to be nominally symmetric, the internal flow field is not. Consequently, the mechanisms for mixing and entrainment required to sustain the bubble region are different from those suggested by earlier axisymmetric models. Results presented in this study, for a Reynolds number of 200, are in good qualitative agreement with higher Reynolds number experimental observations, and a variety of plots have been presented to help illuminate the fluid physics.
{"title":"The structure and dynamics of bubble-type vortex breakdown","authors":"R. Spall, T. Gatski, R. Ash","doi":"10.1098/rspa.1990.0076","DOIUrl":"https://doi.org/10.1098/rspa.1990.0076","url":null,"abstract":"A unique discrete form of the Navier-Stokes equations for unsteady, three-dimensional, incompressible flow has been used to study vortex breakdown numerically. A Burgers-type vortex was introduced along the central axis of the computational domain, and allowed to evolve in space and time. By varying the strength of the vortex and the free stream axial velocity distribution, using a previously developed Rossby number criterion as a guide, the location and size of the vortex breakdown region was controlled. While the boundaries of the vortex breakdown bubble appear to be nominally symmetric, the internal flow field is not. Consequently, the mechanisms for mixing and entrainment required to sustain the bubble region are different from those suggested by earlier axisymmetric models. Results presented in this study, for a Reynolds number of 200, are in good qualitative agreement with higher Reynolds number experimental observations, and a variety of plots have been presented to help illuminate the fluid physics.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"3 1","pages":"613 - 637"},"PeriodicalIF":0.0,"publicationDate":"1990-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79794806","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
When a metastable, damped oscillator is driven by strong periodic forcing, the catchment basin of constrained motions in the space of the starting conditions {x(0),ẋ(0)} develops a fractal boundary associated with a homoclinic tangling of the governing invariant manifolds. The four-dimensional basin in the phase-control space spanned by {x(0), ẋ)(0), F, ω}, where F is the magnitude and ω the frequency of the excitation, will likewise acquire a fractal boundary, and we here explore the engineering significance of the control cross section corresponding, for example, to x(0) = ẋ(0) = 0. The fractal boundary in this section is a failure locus for a mechanical or electrical system subjected, while resting in its ambient equilibrium state, to a sudden pulse of excitation. We assess here the relative magnitude of the uncertainties implied by this fractal structure for the optimal escape from a universal cubic potential well. Absolute and transient basins are examined, giving control-space maps analogous to familiar pictures of the Mandelbrot set. Generalizing from this prototype study, it is argued that in engineering design, against boat capsize or earthquake damage, for example, a study of safe basins should augment, and perhaps entirely replace, conventional analysis of the steady-state attracting solutions.
{"title":"Fractal control boundaries of driven oscillators and their relevance to safe engineering design","authors":"J. Thompson, M. Soliman","doi":"10.1098/rspa.1990.0022","DOIUrl":"https://doi.org/10.1098/rspa.1990.0022","url":null,"abstract":"When a metastable, damped oscillator is driven by strong periodic forcing, the catchment basin of constrained motions in the space of the starting conditions {x(0),ẋ(0)} develops a fractal boundary associated with a homoclinic tangling of the governing invariant manifolds. The four-dimensional basin in the phase-control space spanned by {x(0), ẋ)(0), F, ω}, where F is the magnitude and ω the frequency of the excitation, will likewise acquire a fractal boundary, and we here explore the engineering significance of the control cross section corresponding, for example, to x(0) = ẋ(0) = 0. The fractal boundary in this section is a failure locus for a mechanical or electrical system subjected, while resting in its ambient equilibrium state, to a sudden pulse of excitation. We assess here the relative magnitude of the uncertainties implied by this fractal structure for the optimal escape from a universal cubic potential well. Absolute and transient basins are examined, giving control-space maps analogous to familiar pictures of the Mandelbrot set. Generalizing from this prototype study, it is argued that in engineering design, against boat capsize or earthquake damage, for example, a study of safe basins should augment, and perhaps entirely replace, conventional analysis of the steady-state attracting solutions.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"11 1","pages":"1 - 13"},"PeriodicalIF":0.0,"publicationDate":"1990-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88667338","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Numerical methods are used to study the 4, 6-cell exchange process in the Taylor vortex problem, with particular reference to the homotopy devised by Schaeffer. The homotopy describes a transformation between two models, one incorporating periodic boundary conditions and so referring to flows in an infinite annulus, the other with realistic boundary conditions. Our calculations indicate that the former model is more complicated than previously suspected and lead to a better understanding of the consequences of Schaeffer’s device.
{"title":"A numerical investigation of the Schaeffer homotopy in the problem of Taylor‒Couette flows","authors":"D. K. Anson, K. Cliffe","doi":"10.1098/rspa.1989.0128","DOIUrl":"https://doi.org/10.1098/rspa.1989.0128","url":null,"abstract":"Numerical methods are used to study the 4, 6-cell exchange process in the Taylor vortex problem, with particular reference to the homotopy devised by Schaeffer. The homotopy describes a transformation between two models, one incorporating periodic boundary conditions and so referring to flows in an infinite annulus, the other with realistic boundary conditions. Our calculations indicate that the former model is more complicated than previously suspected and lead to a better understanding of the consequences of Schaeffer’s device.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"3 1","pages":"331 - 342"},"PeriodicalIF":0.0,"publicationDate":"1989-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78856591","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The method of the sequential completion has been used efficiently to define products of distributions. Some results for products of distributions on R1 have been given by J. Mikusinski, B. Fisher and Cheng Lin Zhi et al. The main aim of this paper is to extend the method to the case for several variables. A number of products such as δ(r) ᴑ δ(s), xr + ᴑ δ(s) and x-r ᴑ δ(s), etc., are considered.
顺序补全的方法被有效地用于定义分布的乘积。J. Mikusinski, B. Fisher和程林志等人给出了R1上分布积的一些结果。本文的主要目的是将该方法推广到多个变量的情况。考虑了许多产品，如δ(r)ᴑδ(s)， xr +ᴑδ(s)和x-rᴑδ(s)等。
{"title":"Several products of distributions on Rm","authors":"C. Zhi, B. Fisher","doi":"10.1098/rspa.1989.0133","DOIUrl":"https://doi.org/10.1098/rspa.1989.0133","url":null,"abstract":"The method of the sequential completion has been used efficiently to define products of distributions. Some results for products of distributions on R1 have been given by J. Mikusinski, B. Fisher and Cheng Lin Zhi et al. The main aim of this paper is to extend the method to the case for several variables. A number of products such as δ(r) ᴑ δ(s), xr + ᴑ δ(s) and x-r ᴑ δ(s), etc., are considered.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"10 1","pages":"425 - 439"},"PeriodicalIF":0.0,"publicationDate":"1989-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78921861","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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