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Novi L, von Hardenberg J, Hughes DW, Provenzale A, Spiegel EA. Rapidly rotating Rayleigh-Bénard convection with a tilted axis. Phys Rev E 2019; 99:053116. [PMID: 31212482 DOI: 10.1103/physreve.99.053116] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/06/2019] [Indexed: 11/07/2022]
Abstract
We numerically explore the dynamics of an incompressible fluid heated from below, bounded by free-slip horizontal plates and periodic lateral boundary conditions, subject to rapid rotation about a distant axis that is tilted with respect to the gravity vector. The angle ϕ between the rotation axis and the horizontal plane measures the tilting of the rotation axis; it can be taken as a proxy for latitude if we think of a local Cartesian representation of the convective dynamics in a rotating fluid shell. The results of the simulations indicate the existence of three different convective regimes, depending on the value of ϕ: (1) sheared, intermittent large-scale winds in the direction perpendicular to the plane defined by the gravity and rotation vectors, when rotation is "horizontal" (ϕ=0^{∘}); (2) a large-scale cyclonic vortex tilted along the rotation axis, when the angle between the rotation axis and the gravity vector is relatively small (ϕ between about 45^{∘} and 90^{∘}); and (3) a new intermediate regime characterized by vertically sheared large-scale winds perpendicular to both gravity and rotation. In this regime, the winds are organized in bands that are tilted along the rotation axis, with unit horizontal wave number in the plane defined by gravity and rotation at values of ϕ less than about 60^{∘}. This intermediate solution, studied for the first time in this work, is characterized by weaker vertical heat transport than the cases with large-scale vortices. For intermediate values of ϕ (between about 45^{∘} and 60^{∘}), the banded, sheared solution coexists with the large-scale vortex solution, with different initial conditions leading to one or the other dynamical behavior. A discussion of the possible implications of these results for the dynamics of rapidly rotating planetary atmospheres is provided.
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Affiliation(s)
- L Novi
- Institute of Geosciences and Earth Resources (IGG), National Research Council (CNR), Pisa, 56124, Italy
| | - J von Hardenberg
- Institute of Atmospheric Sciences and Climate (ISAC), National Research Council (CNR), Torino, 10133, Italy
| | - D W Hughes
- Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
| | - A Provenzale
- Institute of Geosciences and Earth Resources (IGG), National Research Council (CNR), Pisa, 56124, Italy
| | - E A Spiegel
- Department of Astronomy, Columbia University, 550 W 120th St, New York, New York 10027, USA
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von Hardenberg J, Goluskin D, Provenzale A, Spiegel EA. Generation of Large-Scale Winds in Horizontally Anisotropic Convection. Phys Rev Lett 2015; 115:134501. [PMID: 26451558 DOI: 10.1103/physrevlett.115.134501] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/16/2015] [Indexed: 06/05/2023]
Abstract
We simulate three-dimensional, horizontally periodic Rayleigh-Bénard convection, confined between free-slip horizontal plates and rotating about a distant horizontal axis. When both the temperature difference between the plates and the rotation rate are sufficiently large, a strong horizontal wind is generated that is perpendicular to both the rotation vector and the gravity vector. The wind is turbulent, large-scale, and vertically sheared. Horizontal anisotropy, engendered here by rotation, appears necessary for such wind generation. Most of the kinetic energy of the flow resides in the wind, and the vertical turbulent heat flux is much lower on average than when there is no wind.
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Affiliation(s)
- J von Hardenberg
- Institute of Atmospheric Sciences, National Research Council, Torino 10133, Italy
| | - D Goluskin
- Mathematics Department, University of Michigan, Ann Arbor, Michigan 48109, USA
- Center for the Study of Complex Systems, University of Michigan, Ann Arbor 48109, Michigan, USA
| | - A Provenzale
- Institute of Geosciences and Earth Resources, National Research Council, Pisa 56124, Italy
| | - E A Spiegel
- Astronomy Department, Columbia University, New York, New York 10027, USA
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Bees MA, Coullet PH, Spiegel EA. On the bifurcation of species. Chaos 2008; 18:043114. [PMID: 19123624 DOI: 10.1063/1.3009196] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/27/2023]
Abstract
We propose and analyze a model of evolution of species based upon a general description of phenotypes in terms of a single quantifiable characteristic. In the model, species spontaneously arise as solitary waves whose members almost never mate with those in other species, according to the rules laid down. The solitary waves in the model bifurcate and we interpret such events as speciation. Our aim in this work is to determine whether a generic mathematical mechanism may be identified with this process of speciation. Indeed, there is such a process in our model: it is the Andronov homoclinic bifurcation. It is robust and is at the heart of the formation of new solitary waves, and thus (in our model) new species.
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Affiliation(s)
- M A Bees
- Department of Mathematics, University of Glasgow, Glasgow G128QW, United Kingdom.
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Balmforth NJ, Spiegel EA. Pattern dynamics in a checkerboard map. Chaos 2004; 14:784-792. [PMID: 15446989 DOI: 10.1063/1.1784752] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
Differential equations often have solutions in the forms of trains of coherent structures such as pulses and antipulses. For such systems, the methods of singular perturbation theory permit the derivation of pattern maps that predict the sequence of spacings between successive pulses. Here we apply such a procedure to cases where two distinct kinds of pulse (or antipulse) may coexist in the system. In that case, direct application of the method leads to multivalued maps that make for complicated descriptions, especially when the succession of pulse types becomes chaotic. We show how this description may be simplified by using maps arrayed in checkerboard style to provide causal descriptions of both the successions of pulse spacings and the order in which the different kinds of pulse go by.
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Affiliation(s)
- N J Balmforth
- Departments of Mathematics and Earth and Ocean Science, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada
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Chen X, Rao H, Spiegel EA. Continuum description of rarefied gas dynamics. I. Derivation from kinetic theory. Phys Rev E Stat Nonlin Soft Matter Phys 2001; 64:046308. [PMID: 11690147 DOI: 10.1103/physreve.64.046308] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/30/2000] [Revised: 04/16/2001] [Indexed: 05/23/2023]
Abstract
We describe an asymptotic procedure for deriving continuum equations from the kinetic theory of a simple gas. As in the works of Hilbert, of Chapman, and of Enskog, we expand in the mean flight time of the constituent particles of the gas, but we do not adopt the Chapman-Enskog device of simplifying the formulas at each order by using results from previous orders. In this way, we are able to derive a new set of fluid dynamical equations from kinetic theory, as we illustrate here for the relaxation model for monatomic gases. We obtain a stress tensor that contains a dynamical pressure term (or bulk viscosity) that is process dependent and our heat current depends on the gradients of both temperature and density. On account of these features, the equations apply to a greater range of Knudsen number (the ratio of mean free path to macroscopic scale) than do the Navier-Stokes equations, as we see in the accompanying paper. In the limit of vanishing Knudsen number, our equations reduce to the usual Navier-Stokes equations with no bulk viscosity.
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Affiliation(s)
- X Chen
- Department of Astronomy, Columbia University, New York, New York 10027, USA
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Chen X, Rao H, Spiegel EA. Continuum description of rarefied gas dynamics. II. The propagation of ultrasound. Phys Rev E Stat Nonlin Soft Matter Phys 2001; 64:046309. [PMID: 11690148 DOI: 10.1103/physreve.64.046309] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/30/2000] [Revised: 04/16/2001] [Indexed: 05/23/2023]
Abstract
The equations of fluid dynamics developed in Paper I [X. Chen, H. Rao, and E. A. Spiegel, Phys. Rev. E 64, 46308 (2001)] are applied to the study of the propagation of ultrasound waves. There is good agreement between the predicted propagation speed and experimental results for a wide range of Knudsen numbers.
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Affiliation(s)
- X Chen
- Department of Astronomy, Columbia University, New York, New York 10027, USA
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Abstract
If we treat the galaxies in published redshift catalogues as point sets, we may determine the generalized dimensions of these sets by standard means, outlined here. For galaxy separations up to about 5 Mpc, we find the dimensions of the CfA galaxy set to be about 1.2, with only a modest indication of multifractality. For larger scales, out to about 30 Mpc, there is also good scaling with a dimension of about 1.8. For even larger scales, the data seem too sparse to be conclusive, but we fmd that the dimension is climbing as the scales increase. We report simulations that suggest a rationalization of such measurements, namely that in the intermediate range the scaling behavior is dominated by flat structures (pancakes) and that the results on the smallest scales are a reflection of the formation of density singularities.
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Affiliation(s)
- G Murante
- Istituto di Cosmogeofisica, Turin, Italy
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Abstract
We study the weakly nonlinear evolution of acoustic instability of a plane-parallel polytrope with thermal dissipation in the form of Newton's law of cooling. The most unstable horizontal wavenumbers form a band around zero and this permits the development of a nonlinear pattern theory leading to a complex Ginzburg-Landau equation (CGLE). Numerical solutions for a subcritical, quintic CGLE produce vertically oscillating, localized structures that resemble the oscillons observed in recent experiments of vibrated granular material.
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Affiliation(s)
- O M Umurhan
- Department of Astronomy, Columbia University, New York, New York 10027, USA
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Abstract
We study the longitudinal instabilities of two interpenetrating fluids interacting only through gravity. When one of the constituents is of relatively low density, it is possible to have a band of unstable wavenumbers well separated from those involved in the usual Jeans instability. If the initial streaming is large enough, and there is no linear instability, the indefinite sign of the free energy has the possible consequence of explosive interactions between positive and negative energy modes in the nonlinear regime. The effect of dissipation on the negative energy modes is also examined.
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Affiliation(s)
- A R Casti
- Department of Applied Physics, Columbia University, New York, New York 10027, USA
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Abstract
The present distribution of galaxies in space is a remnant of their formation and interaction. On a large enough scale, we may represent the galaxies as a set of points and quantify the structures in this set by its generalized dimensions [Beck and Schlogl, Thermodynamics of Chaotic Systems (Cambridge University Press, Cambridge, 1986); Paladin and Vulpiani, Phys. Rep. 156, 147 (1987)]. The results of such evaluation are often taken to be evidence of a fractal (or multifractal) distribution of galaxies. However, those results, for some scales, may also reveal the presence of singularities formed in the gravitational processes that produce structure in the galaxy distribution. To try to make some decision about this issue, we look for the more subtle galactic lacunarity. We believe that this quantity is discernible in the currently available data and that it provides important evidence on the galaxy formation process. (c) 1997 American Institute of Physics.
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Affiliation(s)
- A. Provenzale
- Istituto di Cosmogeofisica, Corso Fiume 4, I-10133 Turin, Italy
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Balmforth NJ, Spiegel EA, Tresser C. Checkerboard maps. Chaos 1995; 5:216-226. [PMID: 12780176 DOI: 10.1063/1.166071] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
When a map has one positive Lyapunov exponent, its attractors often look like multidimensional, Cantorial plates of spaghetti. What saves the situation is that there is a deterministic jumping from strand to strand. We propose to approximate such attractors as finite sets of K suitably prescribed curves, each parametrized by an interval. The action of the map on each attractor is then approximated by a map that takes a set of curves into itself, and we graph it on a KxK checkerboard as a discontinuous one-dimensional map that captures the quantitative dynamics of the original system when K is sufficiently large. (c) 1995 American Institute of Physics.
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Affiliation(s)
- N. J. Balmforth
- Department of Astronomy, Columbia University, New York, New York 10027IBM T. J. Watson Laboratories, Yorktown Heights, New York 10598
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Balmforth NJ, Spiegel EA, Tresser C. Topological entropy of one-dimensional maps: Approximations and bounds. Phys Rev Lett 1994; 72:80-83. [PMID: 10055571 DOI: 10.1103/physrevlett.72.80] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Abstract
Wavetrains of impulses in homogeneous excitable media relax during propagation toward constant-speed patterns. Here we present a study of this relaxation process. Starting with the basic reaction-diffusion or cable equations, we derive kinematics for the trajectories of widely spaced impulses in the form of ordinary differential equations for the set of times at which impulses arrive at a given point in space. Stability criteria derived from these equations allow us to determine the possible asymptotic forms of propagating trains. When the recovery after excitation is monotonic, only one stable train exists for a given propagation speed. In the case of an oscillatory recovery, however, many stable trains are possible. This essential difference between monotonic and oscillatory recoveries manifests itself in qualitatively distinct relaxational behaviors.
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Affiliation(s)
- C Elphick
- Physics Department, Universidad Técnica F. Santa María, Chile
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Spiegel EA. [3 decades of stereotaxic neurosurgery (stereo-encephalotomy)]. GAC MED MEX 1984; 120:339-45. [PMID: 6399276] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [MESH Headings] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/20/2023] Open
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Spiegel EA. Should the brain be regarded as a computer? Appl Neurophysiol 1983; 46:7-10. [PMID: 6670865 DOI: 10.1159/000101233] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/21/2023]
Abstract
In recent years analogies between thinking processes and computer mechanisms have been pointed out. There exist, however, also considerable differences between higher cerebral activities and computers. It seems, therefore, hazardous to consider the human brain exclusively as a computer.
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Spiegel EA. Observations and perspectives related to subcortical procedures. Appl Neurophysiol 1982; 45:1-7. [PMID: 7036852 DOI: 10.1159/000101570] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/23/2023]
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Spiegel EA. Remarks to Hassler's basal ganglia historical perspective. Appl Neurophysiol 1980; 43:37-9. [PMID: 7193429 DOI: 10.1159/000102231] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/23/2023]
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Abstract
Bioconvection in suspensions of Tetrahymena pyriformis and Crypthecodinium cohnii is described and 2 new patterns, the toroid and the cat's-eye, which appear in shallow suspensions of C. cohnii, are reported. Except in very dense cultures, bioconvection does not arise unless the depth of the suspensions or the mean concentration exceed certain critical values, other things being equal. A mathematical model describing the hydrodynamics of suspension of negatively geotactic microorganisms is described which predicts the existence of critical depths and concentrations. The equations presented admit solutions describing the "polka-dot" patterns seen at low organism concentration in suspensions slightly deeper than the critical value. The discussion here is limited to the case of fairly dilute suspensions, but the basic approach can be applied also to richer cultures.
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Spiegel EA. [In memoriam Henry T. Wycis]. Neurochirurgia (Stuttg) 1972; 15:193. [PMID: 4566979] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [MESH Headings] [Subscribe] [Scholar Register] [Indexed: 01/11/2023]
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Abstract
The effect of an imposed vertical magnetic field on convective transfer in a horizontal Boussinesq layer of fluid heated from below is studied in the mean field approximation. Solutions are found over a wide range of conditions, for free boundaries, by a combination of numerical and analytic techniques. Quantitative estimates are made of the significant modifications to the heat transfer which are brought about by the presence of the magnetic field. It is found that the general properties of nonlinear steady cellular convection seem to persist in the face of magnetic inhibition.
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Szekely EG, Zivanovic D, Spiegel EA. The influence of stimulation of the hypothalamus and of the dorsomedial nucleus upon experimental bradykinesia and akinesia. J Neurol Sci 1969; 9:255-60. [PMID: 4899239 DOI: 10.1016/0022-510x(69)90075-6] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/12/2023]
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Szekely EG, Zivanovic D, Spiegel EA. The influence of stimulation of the hypothalamus and of the dorsomedial nucleus upon experimental brady- and akinesia. Confin Neurol 1969; 31:79. [PMID: 5394754 DOI: 10.1159/000103466] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/15/2023]
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Wycis HT, Spiegel EA, Szekely EG, Ditunno JF, Hayes JP, Schwartz H. Effect of deafferentation of an arm upon the tremor of the opposite arm. Confin Neurol 1969; 31:73-8. [PMID: 4900491 DOI: 10.1159/000103465] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [What about the content of this article? (0)] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/12/2023]
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Spiegel EA, Egyed J, Szekely EG. Cortical responses to rotation. II. Responses recorded at the onset of rotation from the second somatic sensory and posterior areas. Acta Otolaryngol 1968; 66:261-72. [PMID: 5706036 DOI: 10.3109/00016486809126293] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/16/2023]
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Spiegel EA, Wycis HT. The possible role of the hypothalamus in Parkinsonian brady- and akinesia. A working hypothesis. Confin Neurol 1967; 29:262-4. [PMID: 4968476 DOI: 10.1159/000103737] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [What about the content of this article? (0)] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/13/2023]
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Wycis HT, Baird HW, Spiegel EA. Long range results following pallidotomy and pallidoamygdalotomy in certain types of convulsive disorders. Confin Neurol 1966; 27:114-20. [PMID: 4959509 DOI: 10.1159/000103940] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/13/2023]
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Spiegel EA, Wycis HT, Szekely EG, Constantinovici A, Egyed JJ, Gildenberg P, Lehman R, Werthan M. Role of the caudate nucleus in parkinsonian bradykinesia. (Effect of dopamine and of stereotaxic caudate lesions). Confin Neurol 1965; 26:336-41. [PMID: 5329837 DOI: 10.1159/000104049] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/14/2023]
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Spiegel EA, Zanes C, Wycis HT. Treatment of Parkinsonian speech disturbances refractory to stereotaxic procedures. Confin Neurol 1965; 26:458. [PMID: 5329854 DOI: 10.1159/000104069] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/14/2023]
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Szekely EG, Egyed JJ, Jacoby CG, Moffet R, Spiegel EA. High frequency coagulation by means of a stylet electrode under temperature control. Confin Neurol 1965; 26:146-52. [PMID: 5329809 DOI: 10.1159/000104017] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/14/2023]
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Spiegel EA, Spiegel-Adolf M, Wycis HT. Chromatolytic Effect of Cerebrospinal Fluid Following Cerebral Concussion. Science 1947. [DOI: 10.1126/science.105.2721.208.b] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/02/2022]
Affiliation(s)
- E. A. Spiegel
- Departments of Experimental Neurology and Colloid Chemistry, Temple University School of Medicine, Philadelphia
| | - M. Spiegel-Adolf
- Departments of Experimental Neurology and Colloid Chemistry, Temple University School of Medicine, Philadelphia
| | - H. T. Wycis
- Departments of Experimental Neurology and Colloid Chemistry, Temple University School of Medicine, Philadelphia
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