Quantum Gravity Phenomenology from the Thermodynamics of Spacetime

This work is based on the formalism developed in the study of the thermodynamics of spacetime used to derive Einstein equations from the proportionality of entropy within an area. When low-energy quantum gravity effects are considered, an extra logarithmic term in the area is added to the entropy expression. Here, we present the derivation of the quantum modified gravitational dynamics from this modified entropy expression and discuss its main features. Furthermore, we outline the application of the modified dynamics to cosmology, suggesting the replacement of the Big Bang singularity with a regular bounce.

Online Learning and Optimization for Revenue Management Problems with Add-on Discounts

We study in this paper a revenue-management problem with add-on discounts. The problem is motivated by the practice in the video game industry by which a retailer offers discounts on selected supportive products (e.g., video games) to customers who have also purchased the core products (e.g., video game consoles). We formulate this problem as an optimization problem to determine the prices of different products and the selection of products for add-on discounts. In the base model, we focus on an independent demand structure. To overcome the computational challenge of this optimization problem, we propose an efficient fully polynomial-time approximation scheme (FPTAS) algorithm that solves the problem approximately to any desired accuracy. Moreover, we consider the problem in the setting in which the retailer has no prior knowledge of the demand functions of different products. To solve this joint learning and optimization problem, we propose an upper confidence bound–based learning algorithm that uses the FPTAS optimization algorithm as a subroutine. We show that our learning algorithm can converge to the optimal algorithm that has access to the true demand functions, and the convergence rate is tight up to a certain logarithmic term. We further show that these results for the independent demand model can be extended to multinomial logit choice models. In addition, we conduct numerical experiments with the real-world transaction data we collect from a popular video gaming brand’s online store on Tmall.com. The experiment results illustrate our learning algorithm’s robust performance and fast convergence in various scenarios. We also compare our algorithm with the optimal policy that does not use any add-on discount. The comparison results show the advantages of using the add-on discount strategy in practice. This paper was accepted by J. George Shanthikumar, big data analytics.

SURFACE LAYER THICKNESS AND ANISOTROPY OF THE SURFACE ENERGY OF CUBIC RUTHENIUM CRYSTALS

В работе рассмотрены вопросы анизотропии поверхностного слоя и анизотропии поверхностной энергии кубических кристаллов рутения. В основе этого рассмотрения лежит эмпирическая модель атомарно-гладких кристаллов, толщина поверхностного слоя которых зависит от одного фундаментального параметра -атомного объема элемента. Расчеты кристаллов рутения показали, что толщина поверхностного слоя кристаллов рутения во всех направлениях не превышает d (I) < 10 нм и они представляют собой наноструктуру. Кристаллы рутенийалюминий, рутенийгафний, рутенийтитан, рутенийцирконий имеют ơ > 3 Дж/м в направлении (100) . Нами рассмотрена задача о диффузии газа в нанометровой пластине рутения. В отличие от классической задачи в полученном уравнении появляется логарифмический член. Это приводит к расходимости в начале координат. Поэтому граничные условия нужно задавать не при x = 0, а при x = d (0) - длине де Бройлевской волны электронов. Только в этом случае имеют смысл классические уравнения диффузии. Существенно также, что, согласно полученному уравнению, диффузии нанопластины зависит как от материала пластины через коэффициент диффузии массивного образца, так и от размерного фактора. В классическом случае такой зависимости нет. Для описания фазовых переходов в наноструктурах предложены различные модели, среди которых можно отметить метод среднего поля Ландау, в котором используется параметр порядка. Мы воспользуемся теорией Ландау, заменяя температуру T на координату h . The paper deals with the anisotropy of the surface layer and the anisotropy of the free surface energy of cubic ruthenium crystals. This consideration is based on an empirical model of atomically smooth crystals, the thickness of the surface layer of which depends on single fundamental parameter - the atomic volume of an element. Calculations of ruthenium crystals showed that the thickness of the surface layer of ruthenium crystals in all directions does not exceed d(I)< 10 nm and they represent a nanostructure. Crystals of ruthenium aluminum, ruthenium hafnium, ruthenium titanium, ruthenium zirconium have ơ > 3 J/m in the (100) direction. We have considered the problem of gas diffusion in a nanometer ruthenium plate. In contrast to the classical problem, a logarithmic term appears in the resulting equation. This leads to divergence at the origin. Therefore, the boundary conditions must be specified not at x = 0, but at x = d (0) - the de Broglie wavelength of electrons. Only in this case the classical diffusion equations are meaningful. It is also important that, according to the obtained equation, the diffusion of the nanoplate depends both on the material of the plate through the diffusion coefficient of the bulk sample and on the size factor. In the classical case, there is no such dependence. Various models have been proposed to describe phase transitions in nanostructures, among which we can mention the Landau mean field method, in which the order parameter is used. We will use Landau's theory, replacing the temperature T with the coordinate h.

Exact solutions and critical behaviour for a linear growth-diffusion equation on a time-dependent domain

A linear growth-diffusion equation is studied in a time-dependent interval whose location and length both vary. We prove conditions on the boundary motion for which the solution can be found in exact form and derive the explicit expression in each case. Next, we prove the precise behaviour near the boundary in a ‘critical’ case: when the endpoints of the interval move in such a way that near the boundary there is neither exponential growth nor decay, but the solution behaves like a power law with respect to time. The proof uses a subsolution based on the Airy function with argument depending on both space and time. Interesting links are observed between this result and Bramson's logarithmic term in the nonlinear FKPP equation on the real line. Each of the main theorems is extended to higher dimensions, with a corresponding result on a ball with a time-dependent radius.

Thermodynamic effects of Bardeen black hole surrounded by perfect fluid dark matter under general uncertainty principle

In this paper, the thermodynamics of Bardeen black hole surrounded by perfect fluid dark matter is investigated. We calculate the analytical expresses of corresponding thermodynamic variables, e.g. the Hawking temperature, entropy of the black hole. In addition, we derive the heat capacity to analyze the thermal stability of the black hole. We also compute the rate of emission in terms of photons through tunneling. By numerical method, an obvious phase transition behavior is found. Furthermore, according to the general uncertainty principle, we study the quantum corrections to these thermodynamic quantities and obtain the quantum-corrected entropy containing the logarithmic term. At last, we investigate the effects of the magnetic charge g, the dark matter parameter k and the generalized uncertainty principle parameter α on the thermodynamics of Bardeen black hole surrounded by perfect fluid dark matter under general uncertainty principle.

Entanglement measures in a nonequilibrium steady state: Exact results in one dimension

Entanglement plays a prominent role in the study of condensed matter many-body systems: Entanglement measures not only quantify the possible use of these systems in quantum information protocols, but also shed light on their physics. However, exact analytical results remain scarce, especially for systems out of equilibrium. In this work we examine a paradigmatic one-dimensional fermionic system that consists of a uniform tight-binding chain with an arbitrary scattering region near its center, which is subject to a DC bias voltage at zero temperature. The system is thus held in a current-carrying nonequilibrium steady state, which can nevertheless be described by a pure quantum state. Using a generalization of the Fisher-Hartwig conjecture, we present an exact calculation of the bipartite entanglement entropy of a subsystem with its complement, and show that the scaling of entanglement with the length of the subsystem is highly unusual, containing both a volume-law linear term and a logarithmic term. The linear term is related to imperfect transmission due to scattering, and provides a generalization of the Levitov-Lesovik full counting statistics formula. The logarithmic term arises from the Fermi discontinuities in the distribution function. Our analysis also produces an exact expression for the particle-number-resolved entanglement. We find that although to leading order entanglement equipartition applies, the first term breaking it grows with the size of the subsystem, a novel behavior not observed in previously studied systems. We apply our general results to a concrete model of a tight-binding chain with a single impurity site, and show that the analytical expressions are in good agreement with numerical calculations. The analytical results are further generalized to accommodate the case of multiple scattering regions.

Convergence analysis of online learning algorithm with two-stage step size

Online learning is a classical algorithm for optimization problems. Due to its low computational cost, it has been widely used in many aspects of machine learning and statistical learning. Its convergence performance depends heavily on the step size. In this paper, a two-stage step size is proposed for the unregularized online learning algorithm, based on reproducing Kernels. Theoretically, we prove that, such an algorithm can achieve a nearly min–max convergence rate, up to some logarithmic term, without any capacity condition.

Quantum corrected thermodynamics of nonlinearly charged BTZ black holes in massive gravity’s rainbow

In this paper, we consider three-dimensional massive gravity’s rainbow and obtain black hole solutions in three different cases of Born–Infeld, logarithmic, and exponential theories of nonlinear electrodynamics. We discuss the horizon structure and geometrical properties. Then, we study thermodynamics of these models by considering the first-order quantum correction effects, which appear as a logarithmic term in the black hole entropy. We discuss such effects on the black hole stability and phase transitions. We find that due to the quantum corrections, the second-order phase transition happens in Born–Infeld and logarithmic models. We obtain the modified first law of black hole thermodynamics in the presence of logarithmic corrections.

Einstein–Gauss–Bonnet Gravity with Nonlinear Electrodynamics: Entropy, Energy Emission, Quasinormal Modes and Deflection Angle

The logarithmic correction to Bekenshtein–Hawking entropy in the framework of 4D Einstein–Gauss–Bonnet gravity coupled with nonlinear electrodynamics is obtained. We explore the black hole solution with the spherically symmetric metric. The logarithmic term in the entropy has a structure similar to the entropy correction in the semi-classical Einstein equations. The energy emission rate of black holes and energy conditions are studied. The quasinormal modes of a test scalar field are investigated. The gravitational lensing of light around BHs was studied. We calculated the deflection angle for some model parameters.

Regularity and monotonicity for solutions to a continuum model of epitaxial growth with nonlocal elastic effects

We study the following parabolic nonlocal 4-th order degenerate equation: u t = - [ 2 ⁢ π ⁢ H ⁢ ( u x ) + ln ⁡ ( u x ⁢ x + a ) + 3 2 ⁢ ( u x ⁢ x + a ) 2 ] x ⁢ x , u_{t}=-\Bigl{[}2\pi H(u_{x})+\ln(u_{xx}+a)+\frac{3}{2}(u_{xx}+a)^{2}\Bigr{]}_{% xx}, arising from the epitaxial growth on crystalline materials. Here H denotes the Hilbert transform, and a > 0 {a>0} is a given parameter. By relying on the theory of gradient flows, we first prove the global existence of a variational inequality solution with a general initial datum. Furthermore, to obtain a global strong solution, the main difficulty is the singularity of the logarithmic term when u x ⁢ x + a {u_{xx}+a} approaches zero. Thus we show that, if the initial datum u 0 {u_{0}} is such that ( u 0 ) x ⁢ x + a {(u_{0})_{xx}+a} is uniformly bounded away from zero, then such property is preserved for all positive times. Finally, we will prove several higher regularity results for this global strong solution. These finer properties provide a rigorous justification for the global-in-time monotone solution to the epitaxial growth model with nonlocal elastic effects on vicinal surface.