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31.
32.
Paul W. Vos 《Annals of the Institute of Statistical Mathematics》1989,41(3):429-450
Many applications of Amari's dual geometries involve one or more submanifolds imbedded in a supermanifold. In the differential geometry literature, there is a set of equations that describe relationships between invariant quantities on the submanifold and supermanifold when the Riemannian connection is used. We extend these equations to statistical manifolds, manifolds on which a pair of dual connections is defined. The invariant quantities found in these equations include the mean curvature and the statistical curvature which are used in statistical calculations involving such topics as information loss and efficiency. As an application of one of these equations, the Bartlett correction is interpreted in terms of curvatures and other invariant quantities. 相似文献
33.
The last decade of research on κ-Minkowski noncommutative spacetime has been strongly characterized by a controversy concerning the speed of propagation of massless particles. Most arguments suggested that this speed should depend on the momentum of the particle strongly enough to be of interest for some ongoing experimental studies. But the only explicit derivations of worldlines in κ-Minkowski predicted no momentum dependence for the speed of massless particles. We return to this controversy equipped with the recent understanding that in some quantum spacetimes coincidences of events assessed by an observer who is distant from the events can be artifactual. We therefore set up our investigation in such a way that we never rely on the assessment of coincidences of events by distant observers. This allows us to verify explicitly that in κ-Minkowski simultaneously-emitted massless particles of different momentum are detected at different times, and establish a linear dependence of the detection times on momentum. 相似文献
34.
35.
Let the isoclines of a function u be the level lines of the function θ = arg(Du). Formulas for the curvature and the length of isocline lines in terms of the curvatures k, j of the level curves and of the steepest descent lines of u are given. The special case when all isoclines are straight lines is studied: in this case the steepest descent lines bend proportionally to the level lines; the support function of the level lines is linear function on the isoclines parameterized by the level values, possibly changing them. This characterization gives a new proof of a property of the developable surfaces found in [A. Fialkow, Geometric characterization of invariant partial differential equations, Amer. J. Math. 59(4) (1937), pp. 833–844]. When u is in the class of quasi convex functions, the L p norm of the length function I θ of the isoclines has minimizers with isoclines straight lines; the same occurs for other functionals on u depending on k, j. For a strictly regular quasi convex function isoclines may have arbitrarily large length and arbitrarily large L 1 norm of I θ. 相似文献
36.
Bryce DeWitt 《General Relativity and Gravitation》2009,41(2):413-419
Bryce DeWitt was one of the great pioneers of quantum gravity. This unpublished lecture gives his recent views on the topic,
which we believe will be of great interest not only to researchers involved in modern attempts to quantize Einstein’s theory,
but also to a much wider audience. It is the first installment of a book “The Pursuit of Quantum Gravity 1946–2004; Memoirs
of Bryce DeWitt” that Cecile DeWitt is preparing. We would like to thank her for the permission to publish this lecture separately
in General Relativity and Gravitation. Readers who have unpublished material such as letters from Bryce, and would be willing
to send copies to Cecile, are hereby invited to do so. She would be very grateful. G.F.R. Ellis, H. Nicolai (Editors-in-chief).
This article was found in Bryce’s files without references and without an indication of its purpose.
References and the name of a previously anonymous student have been supplied by Cecile DeWitt and Brandon DiNunno.
Bryce DeWitt (1923–2004).
An erratum to this article can be found at 相似文献
37.
Edmundo M. Monte 《International Journal of Theoretical Physics》2009,48(2):409-413
The braneworlds models were inspired partly by Kaluza-Klein’s theory, where both the gravitational and the gauge fields are
obtained from the geometry of a higher dimensional space. The positive aspects of these models consist in perspectives of
modifications it could bring in to particle physics, such as: unification in a TeV scale, quantum gravity in this scale and
deviation of Newton’s law for small distances. One of the principles of these models is to suppose that all space-times can
be embedded in a bulk of higher dimension. The main result in these notes is a theorem showing a mathematical inconsistency
of the Randall-Sundrum braneworld model, namely that the Schwarzschild space-time cannot be embedded locally and isometrically
in a five dimensional bulk with constant curvature (for example AdS-5). From the point of view of semi-Riemannian geometry
this last result represents a serious restriction to the Randall-Sundrum’s braneworld model. 相似文献
38.
Yu-Chung Chang 《Monatshefte für Mathematik》2009,158(1):1-22
In this paper we consider a compact oriented hypersurface M
n
with constant mean curvature H and two distinct principal curvatures λ and μ with multiplicities (n − m) and m, respectively, immersed in the unit sphere S
n+1. Denote by the trace free part of the second fundamental form of M
n
, and Φ be the square of the length of . We obtain two integral formulas by using Φ and the polynomial . Assume that B
H,m
is the square of the positive root of P
H,m
(x) = 0. We show that if M
n
is a compact oriented hypersurface immersed in the sphere S
n+1 with constant mean curvatures H having two distinct principal curvatures λ and μ then either or . In particular, M
n
is the hypersurface .
相似文献
39.
In this paper we obtain a height estimate concerning compact space-like hypersurfaces Σ
n
immersed with some positive constant r-mean curvature into an (n + 1)-dimensional Lorentzian product space , and whose boundary is contained into a slice {t} × M
n
. By considering the hyperbolic caps of the Lorentz–Minkowski space , we show that our estimate is sharp. Furthermore, we apply this estimate to study the complete space-like hypersurfaces
immersed with some positive constant r-mean curvature into a Lorentzian product space. For instance, when the ambient space–time is spatially closed, we show that
such hypersurfaces must satisfy the topological property of having more than one end which constitutes a necessary condition
for their existence. 相似文献
40.