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[en] Central in the paper are two results on the existence of 'economical' embeddings in a Euclidean space. The first result (Corollary 1.4) states the existence of an embedding with image intersecting the large-dimensional planes in sets of 'controllable' dimension. The second result (Corollary 1.6) proves the existence of maps such that each small-dimensional plane contains 'controllably' many points of the image. Well known results of Noebeling-Pontryagin, Roberts, Hurewicz, Boltyanskii, and Goodsell can be obtained as consequences of these results. Their infinite-dimensional version concerning an embedding in a Hilbert space is also established (Theorem 1.8).
[en] Superluminal behavior has been extensively studied in recent years, especially with regard to the topic of superluminality in the propagation of a signal. Particular interest has been devoted to Bessel-X waves propagation, since some experimental results showed that these waves have both phase and group velocities greater that light velocity c. However, because of the lack of an exact definition of signal velocity, no definite answer about the signal propagation (or velocity of information) has been found. The present Letter is a short note that deals in a general way with this vexed question. By analyzing the field of existence of the Bessel X-pulse in pseudo-Euclidean spacetime, it is possible to give a general description of the propagation, and to overcome the specific question related to a definition of signal velocity
[en] Let Csup(n) be a complex euclidean space. A bounded open connected subset D of Csup(n) is called a bounded domain. Bounded homogeneous domains in Csup(n) are studied. Contents: bounded homogeneous domains; j-Lie algebras; Siegel domains; s-structure on a bounded domain. (author)
[en] We are going to show the link between the ε (∞) Cantorian space and the Hilbert spaces H (∞). In particular, El Naschie's ε (∞) is a physical spacetime, i.e. an infinite dimensional fractal space, where time is spacialized and the transfinite nature manifests itself. El Naschie's Cantorian spacetime is an arena where the physics laws appear at each scale in a self-similar way linked to the resolution of the act of observation. By contrast the Hilbert space H (∞) is a mathematical support, which describes the interaction between the observer and the dynamical system under measurement. The present formulation, which is based on the non-classical Cantorian geometry and topology of spacetime, automatically solves the paradoxical outcome of the two-slit experiment and the so-called particle-wave duality. In particular, measurement (i.e. the observation) is equivalent to a projection of ε (∞) in the Hilbert space built on 3 + 1 Euclidean spacetime. Consequently, the wave-particle duality becomes a mere natural consequence of conducting an experiment in a spacetime with non-classical topological and geometrical structures, while observing and taking measurements in a classical smooth 3 + 1 Euclidean spacetime. In other words, the experimental fact that a wave-particle duality exists is an indirect confirmation of the existence of ε (∞) and a property of the quantum-classical interface. Another direct consequence of the fact that real spacetime is infinite dimensional hierarchical ε (∞) is the existence of scaling law R(N), introduced by the author, which generalizes the Compton wavelength. It gives an answer to the problem of segregation of matter at different scales, and shows the role of fundamental constants such as the speed of light and Plank's constant h in the fundamental lengths scale without invoking the principles of quantum mechanics
[en] We shortly review different methods to obtain the scattering solutions of the Bethe–Salpeter equation in Minkowski space. We emphasize the possibility to obtain the zero energy observables in terms of the Euclidean scattering amplitude. (author)
[en] We investigate the set of completely positive, trace-nonincreasing linear maps acting on the set MN of mixed quantum states of size N. Extremal point of this set of maps are characterized and its volume with respect to the Hilbert-Schmidt (HS) (Euclidean) measure is computed explicitly for an arbitrary N. The spectra of partially reduced rescaled dynamical matrices associated with trace-nonincreasing completely positive maps belong to the N cube inscribed in the set of subnormalized states of size N. As a by-product we derive the measure in MN induced by partial trace of mixed quantum states distributed uniformly with respect to the HS measure in MN2
[en] The paper is devoted to spinors in the theory of relatively and quantum theory; it also expounds the theory of Pythagoras numbers. Based on a historical-mathematical method the geometrical and physical meaning of spinors is clarified. TIt is shown that the Pyphagoras formulae for a right triangle lead naturally to a spinor representation of the Lorentz orthochronous group. The state vector in quantum fermion theory is proved to be a spinor of the Euclidean space, and in quantum boson theory, a spinor of the symplectic space. The practical problem of ancient ages, to draw on the Earth's surface a right angle, appears to be tightly connected with both the main theories of the twentieth century
[en] Gradient semidynamical systems, depending on a parameter (parameters) λ and having a finite number of hyperbolic fixed points, are considered. Under certain conditions on the system, it is shown that the attractor is Hoelder continuous with respect to λ (in the Hausdorff metric). At the same time an estimate, exponential with respect to time, has been obtained for the velocity of the attraction of bounded subsets of the phase space to the attractor. These results are applied to the proof of the convergence of the attractors of an abstract hyperbolic equation, with a small parameter var-epsilon at the second time derivative, to the attractor of the corresponding parabolic equation when var-epsilon searrow 0. 11 refs
[en] We have shown that the Unruh quantization scheme can be realized in Minkowski spacetime in the presence of Bose-Einstein condensate containing infinite average number of particles in the zero boost mode and located basically inside the light cone. Unlike the case of an empty Minkowski spacetime the condensate provides the boundary conditions necessary for the Fulling quantization of the part of the field restricted only to the Rindler wedge of Minkowski spacetime
[en] In the multiverse, as in AdS space, light cones relate bulk points to boundary scales. This holographic UV-IR connection defines a preferred global time cutoff that regulates the divergences of eternal inflation. An entirely different cutoff, the causal patch, arises in the holographic description of black holes. Remarkably, I find evidence that these two regulators define the same probability measure in the multiverse. Initial conditions for the causal patch are controlled by the late-time attractor regime of the global description.