de Sitter space as a Glauber-Sudarshan state

Glauber-Sudarshan states, sometimes simply referred to as Glauber states, or alternatively as coherent and squeezed-coherent states, are interesting states in the configuration spaces of any quantum field theories, that closely resemble classical trajectories in space-time. In this paper, we identify four-dimensional de Sitter space as a coherent state over a supersymmetric Minkowski vacuum. Although such an identification is not new, what is new however is the claim that this is realizable in full string theory, but only in conjunction with temporally varying degrees of freedom and quantum corrections resulting from them. Furthermore, fluctuations over the de Sitter space is governed by a generalized graviton (and flux)-added coherent state, also known as the Agarwal-Tara state. The realization of de Sitter space as a state, and not as a vacuum, resolves many issues associated with its entropy, zero-point energy and trans-Planckian censorship, amongst other things.

Forward triplets and topological entropy on trees

<p style='text-indent:20px;'>We provide a new and very simple criterion of positive topological entropy for tree maps. We prove that a tree map <inline-formula><tex-math id="M1">\begin{document}$ f $\end{document}</tex-math></inline-formula> has positive entropy if and only if some iterate <inline-formula><tex-math id="M2">\begin{document}$ f^k $\end{document}</tex-math></inline-formula> has a periodic orbit with three aligned points consecutive in time, that is, a triplet <inline-formula><tex-math id="M3">\begin{document}$ (a,b,c) $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M4">\begin{document}$ f^k(a) = b $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ f^k(b) = c $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ b $\end{document}</tex-math></inline-formula> belongs to the interior of the unique interval connecting <inline-formula><tex-math id="M7">\begin{document}$ a $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ c $\end{document}</tex-math></inline-formula> (a <i>forward triplet</i> of <inline-formula><tex-math id="M9">\begin{document}$ f^k $\end{document}</tex-math></inline-formula>). We also prove a new criterion of entropy zero for simplicial <inline-formula><tex-math id="M10">\begin{document}$ n $\end{document}</tex-math></inline-formula>-periodic patterns <inline-formula><tex-math id="M11">\begin{document}$ P $\end{document}</tex-math></inline-formula> based on the non existence of forward triplets of <inline-formula><tex-math id="M12">\begin{document}$ f^k $\end{document}</tex-math></inline-formula> for any <inline-formula><tex-math id="M13">\begin{document}$ 1\le k&lt;n $\end{document}</tex-math></inline-formula> inside <inline-formula><tex-math id="M14">\begin{document}$ P $\end{document}</tex-math></inline-formula>. Finally, we study the set <inline-formula><tex-math id="M15">\begin{document}$ \mathcal{X}_n $\end{document}</tex-math></inline-formula> of all <inline-formula><tex-math id="M16">\begin{document}$ n $\end{document}</tex-math></inline-formula>-periodic patterns <inline-formula><tex-math id="M17">\begin{document}$ P $\end{document}</tex-math></inline-formula> that have a forward triplet inside <inline-formula><tex-math id="M18">\begin{document}$ P $\end{document}</tex-math></inline-formula>. For any <inline-formula><tex-math id="M19">\begin{document}$ n $\end{document}</tex-math></inline-formula>, we define a pattern that attains the minimum entropy in <inline-formula><tex-math id="M20">\begin{document}$ \mathcal{X}_n $\end{document}</tex-math></inline-formula> and prove that this entropy is the unique real root in <inline-formula><tex-math id="M21">\begin{document}$ (1,\infty) $\end{document}</tex-math></inline-formula> of the polynomial <inline-formula><tex-math id="M22">\begin{document}$ x^n-2x-1 $\end{document}</tex-math></inline-formula>.</p>

A Random Matrix Approach to Absorption in Free Products

This paper gives a free entropy theoretic perspective on amenable absorption results for free products of tracial von Neumann algebras. In particular, we give the 1st free entropy proof of Popa’s famous result that the generator MASA in a free group factor is maximal amenable, and we partially recover Houdayer’s results on amenable absorption and Gamma stability. Moreover, we give a unified approach to all these results using $1$-bounded entropy. We show that if ${\mathcal{M}} = {\mathcal{P}} * {\mathcal{Q}}$, then ${\mathcal{P}}$ absorbs any subalgebra of ${\mathcal{M}}$ that intersects it diffusely and that has $1$-bounded entropy zero (which includes amenable and property Gamma algebras as well as many others). In fact, for a subalgebra ${\mathcal{P}} \leq{\mathcal{M}}$ to have this absorption property, it suffices for ${\mathcal{M}}$ to admit random matrix models that have exponential concentration of measure and that “simulate” the conditional expectation onto ${\mathcal{P}}$.

Tilings of amenable groups

We prove that for any infinite countable amenable group G, any {\varepsilon>0} and any finite subset {K\subset G} , there exists a tiling (partition of G into finite “tiles” using only finitely many “shapes”), where all the tiles are {(K,\varepsilon)} -invariant. Moreover, our tiling has topological entropy zero (i.e., subexponential complexity of patterns). As an application, we construct a free action of G (in the sense that the mappings, associated to elements of G other than the unit, have no fixed points) on a zero-dimensional space, such that the topological entropy of this action is zero.

The problems of numerical inequalities estimates

The purpose of this paper is to compare the shortcomings of the widely used inequality coefficients that appear when working with real (ie, knowingly incomplete) data and searching for alternative quantitative methods for describing inequalities that lack these shortcomings.Research methods:– consideration of an extensive range of as full as possible real data on the population distribution by income, expenditure, property (ie data on the economic structure of society);– revealing the specific shortcomings of these data on the economic structure of society, finding out which information is missing or presented disproportionately;– comparison of the values of the most widely used indices of inequality calculated on real data on the economic structure, with a view to establishing the suitability of these indicators for problems of inequality estimation;– development of an index of inequality that adequately describes the real economic structure of society.Research data:– official data of Rosstat and the Federal Tax Service on incomes of Russian citizens;– specialized sites of announcements about the prices for real estate and cars;– Credit Suisse Research Institute data on the distribution of Russian citizens by property level;– Forbes data on income and wealth of the richest people in Russia.It is shown that the income data are essentially incomplete and fragmentary – the width of the income range (i.e., the income of therichest member of society) is known, but the filling of rich cohorts is not known, since the incomes of the richest members of society are hidden.We proposed the next (criteria as) requirements for an inequality index:– possibility of calculating the index of inequality for arbitrary quantization;– invariance of the value of the inequality index for different quantization of the same data;– sensitivity of the index to the width of the income range.It is noted that only the exponential function describes societies with high social inequality enough well (the intensity of the exponential distribution is more than 10).For the presented population distributions, the next indices of inequality are calculated:– decile coefficient of funds;– Gini coefficient;– Pareto index;– indicators of total entropy (zero, first or Tayle index, and second orders);– the ratio of maximum income (property value) to the modal;– intensity of exponential distribution.It is shown, that:– the value of the Pareto index does not have a unique relationship with the inequality;– the coefficients of the funds (decile, quintile, etc.) are not computable for arbitrary quantization, and therefore are unsuitable for comparing data from various sources and have different quantization;– The Gini index requires complete data on the rich;– from all considered criteria of inequality the first three indicators of the total entropy, as well as the ratio of maximum income (property) to the modal strongly depend on data quantization.Therefore they are unsuitable for comparison data from various sources with different quantization. It is concluded that the intensity of the exponential distribution does not possess the listed disadvantages and can be recommended as an index of inequality.

Aperiodicity at the boundary of chaos

We consider the dynamical properties of $C^{\infty }$-variations of the flow on an aperiodic Kuperberg plug $\mathbb{K}$. Our main result is that there exists a smooth one-parameter family of plugs $\mathbb{K}_{\unicode[STIX]{x1D716}}$ for $\unicode[STIX]{x1D716}\in (-a,a)$ and $a<1$, such that: (1) the plug $\mathbb{K}_{0}=\mathbb{K}$ is a generic Kuperberg plug; (2) for $\unicode[STIX]{x1D716}<0$, the flow in the plug $\mathbb{K}_{\unicode[STIX]{x1D716}}$ has two periodic orbits that bound an invariant cylinder, all other orbits of the flow are wandering, and the flow has topological entropy zero; (3) for $\unicode[STIX]{x1D716}>0$, the flow in the plug $\mathbb{K}_{\unicode[STIX]{x1D716}}$ has positive topological entropy, and an abundance of periodic orbits.

TRIPLE-HORIZON SPHERICALLY SYMMETRIC SPACETIME AND HOLOGRAPHIC PRINCIPLE

We present a family of spherically symmetric spacetimes, specified by the density profile of a vacuum dark energy, which have the same global structure as the de Sitter spacetime but the reduced symmetry which leads to a time-evolving and spatially inhomogeneous cosmological term. It connects smoothly two de Sitter vacua with different values of cosmological constant and corresponds to anisotropic vacuum dark fluid defined by symmetry of its stress–energy tensor which is invariant under the radial boosts. This family contains a special class distinguished by dynamics of evaporation of a cosmological horizon which evolves to the triple horizon with the finite entropy, zero temperature, zero curvature, infinite positive specific heat and infinite scrambling time. Nonzero value of the cosmological constant in the triple-horizon spacetime is tightly fixed by quantum dynamics of evaporation of the cosmological horizon.