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## Depth as an Extra Spatial Dimension and its Implications for Cosmology and Gravity Theory

I develop the idea that there exists a special dimension of depth, or of scale. The depth dimension is physically real and extends from the bottom micro-level to the ultimate macro-level of the Universe. The depth dimension, or the scales axis, complements the standard three spatial dimensions. I discuss the tentative qualities of the depth dimension and the universal arrangement of matter along this dimension. I suggest that all matter in the Universe, at least in the present cosmological epoch, is in joint downward motion along the depth dimension. The joint downward motion manifests itself in the universal contraction of matter. The opposite direction of motion, upward the dimension, would cause the expansion of matter. The contraction of matter is a primary factor, whereas the shrinking of space in the vicinity of matter is a derivative phenomenon. The observed expansion of the Universe is explained by the fact that celestial bodies become smaller due to matter contraction, while the overall space remains predominantly intact. Thus, relative to the contracting material bodies, the total span of cosmic space appears to be becoming vaster. I attempt to explain how the contraction of matter engenders the effect of universal gravity. I use over thirty animated and graphical color visualizations in the text to make the explanation of the proposed ideas more lucid.

The aim of this paper is to consider some problems with evaluation of the impact of high frequency trading on market liquidity. The first part is devoted to difficulties of disentangling the impact of high frequency on market liquidity from other relevant factors. The remainder of the paper is intended to discuss some issues affecting the evaluation of the influence of high frequency trading on particular aspects of market liquidity.

The article analyzes the writing of Alexander Koiré «From the closed world to the infinite universe» in relation to several achievements of modern physics connected to fundamental concept of space. The evolution of other certain scientific concepts, such as in particular, the «vacuum» and «gravity», also comes under review. The author reaches the key point that the some modern physical theories are again based on the idea of the closed universe.

Several aspects of the transformation of ideas about the concept of space and its properties are being analyzed through a historical perspective. The research results in the particular conclusions on these ideas development in the context of modern science.

En révélant l’influence de l’interprétation de Koyré dans les lectures que fait Henry de la philosophie boehmienne, je propose d’interroger la position henryenne qui, en s’appuyant sur les concepts classiques de l’originaire et de l’authentique, refuse au monde – et par là à toute connaissance théorique – le statut de vérité. Je me demande donc s’il est suffisant d’exclure chaque tentative d’expression de la manifestation de l’Absolu. Peut-être faut-il relire le projet de la phénoménologie comme une science descriptive plutôt que prescriptive ?

The monograph "Philosophy of Physics: towards New Principles of New Knowledge" is intended both for those who are just discovering the world of philosophy of science in the most fundamental field – physics, and for specialists who are professionally engaged in the topic and are interested in the most relevant research. The author analyzes a number of key problems of modern physics and cosmology, offers original interpretations and solutions, and also discusses the prospects for the development of science in the context of attempts to create a "Theory of Everything". The book focuses on the physical theories of the multiverse, the new principles of scientific knowledge derived from these theories, and the connection between consciousness and concrete physical reality.

My goal is to conceive how the reality would look like for hypothetical creatures that supposedly perceive on time scales much faster or much slower that of us humans. To attain the goal, I propose modelling in two steps. At step we have to single out a uni“ed parameter that sets time scale of perception. Changing substantially the value of the parameter would mean changing scale. argue that the required parameter is duration of discrete perceptive frames, snapshots, whose sequencing constitutes perceptive process. I show that different standard durations of perceptive frames is the ground for differences in perceptive time scales of various animals. Abnormally changed duration of perceptive frames is the cause of the effect of distorted subjective time observed by humans under some conditions. Now comes step two of the modelling. By inserting some arbitrary duration of a perceptive frame, we set a hypothetical scale and thus emulate viewpoint for virtual observation of the reality in a wider or narrower angle embracing events in time. Like changing lenses of a microscope, viewing reality different temporal scales makes certain features of reality manifested, others veiled. These are, in particular, features of life. If we observe an object in an inappropriate interval, we may not notice the very essence of a process it is undergoing.

In this paper we consider choice problems under the assumption that the preferences of the decision maker are expressed in the form of a parametric partial weak order without assuming the existence of any value function. We investigate both the sensitivity (stability) of each non-dominated solution with respect to the changes of parameters of this order, and the sensitivity of the set of non-dominated solutions as a whole to similar changes. We show that this type of sensitivity analysis can be performed by employing techniques of linear programming.

An algorithm is implemented in the article for finding the independence number of a n-vertex graph from the class Free({P5,C5, Kp}) in time O(np+O(1)).

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.