Infinite Limit Research Articles

Krylov complexity in the IP matrix model. Part II

We continue the analysis of the Krylov complexity in the IP matrix model. In a previous paper, [1], for a fundamental operator, it was shown that at zero temperature, the Krylov complexity oscillates and does not grow, but in the infinite temperature limit, the Krylov complexity grows exponentially in time as ∼expOt\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sim \\exp \\left(\\mathcal{O}\\left(\\sqrt{t}\\right)\\right) $$\\end{document}. We study how the Krylov complexity changes from a zero-temperature oscillation to an infinite-temperature exponential growth. At low temperatures, the spectral density is approximated as collections of infinite Wigner semicircles. We showed that this infinite collection of branch cuts yields linear growth to the Lanczos coefficients and gives exponential growth of the Krylov complexity. Thus the IP model for any nonzero temperature shows exponential growth for the Krylov complexity even though the Green function decays by a power law in time. We also study the Lanczos coefficients and the Krylov complexity in the IOP matrix model taking into account the 1/N2 corrections. There, the Lanczos coefficients are constants and the Krylov complexity does not grow exponentially as expected.

Cosmic acceleration and turns in the Swampland

We argue that field trajectories, which lead to cosmic acceleration and feature rapid turns near the boundary of the moduli space, are in the Swampland. We obtain this result by assuming the validity of the Swampland Distance Conjecture (SDC) in the presence of a positive scalar potential and by focusing on hyperbolic spaces, as prototype geometries of infinite distance limits of Calabi-Yau compactifications. We find that, in a quasi-de Sitter space with Hubble rate H and acceleration parameter ϵ, the turning rate Ω is upper bounded such as Ω/H < 𝒪(√(ϵ)). Therefore, field trajectories consistent with the SDC can only have a negligible deviation from geodesics. This has direct implications for the realization and consistency of multi-field scenarios in string theory. Moreover, it implies a tension between asymptotic accelerating expansion, consistent with observations, and the de Sitter conjecture.

Fractional and tempered fractional models for Reynolds-averaged Navier–Stokes equations

Turbulence is a non-local phenomenon and has multiple-scales. Non-locality can be addressed either implicitly or explicitly. Implicitly, by subsequent resolution of all spatio-temporal scales. However, if directly solved for the temporal or spatially averaged fields, a closure problem arises on account of missing information between two points. To solve the closure problem in Reynolds-averaged Navier–Stokes equations (RANS), an eddy-viscosity hypotheses has been a popular modelling choice, where it follows either a linear or nonlinear stress–strain relationship. Here, a non-constant diffusivity is introduced. Such a non-constant diffusivity is also characteristic of non-Fickian diffusion equation addressing anomalous diffusion process. An alternative approach, is a fractional derivative-based diffusion equations. Thus, in the paper, we formulate a fractional stress–strain relationship using variable-order Caputo fractional derivative. This provides new opportunities for future modelling effort. We pedagogically study of our model construction, starting from one-sided model and followed by two-sided model applied to channel, couette and pipe flow. Non-locality at a point is the amalgamation of all the effects, thus we find the two-sided model is physically consistent. Further, our construction can also addresses viscous effects, which is a local process. Thus, our fractional model addresses the amalgamation of local and non-local process. We also show its validity at infinite Reynolds number limit. An expression for the fractional order is also found, thereby solving the closure problem for the considered cases. This study is further extended to tempered fractional calculus, where tempering ensures finite jump lengths, this is an important remark for unbounded flows. Within the context of this paper, we limit ourselves to wall bounded flow. Two tempered definitions are introduced with a smooth and sharp cutoff, by the exponential term and Heaviside function, respectively and we also define the horizon of non-local interactions. We further study the equivalence between the two definitions, as truncating the domain has computational advantages. For the above investigations, we carefully designed algorithms, notably, the pointwise version of fractional physics-informed neural network to find the fractional order as an inverse problem.

Souriau’s general covariant formulation of Relativistic Hyperelasticity revisited

We present and modernize Souriau’s 1958 geometric framework for Relativistic continuous media, and enlighten the necessary and the ad hoc modeling choices made since. We focus as much as possible on the Continuum Mechanics point of view and describe the general covariant formulation of Hyperelasticity in (Variational) General Relativity. At this level of generality, different relativistic strain and stress tensors are formulated and discussed. Then, the choice of an observer, through the introduction of a spacetime structure allows us to make deeper insights into the foundations of Continuum Mechanics. We extend Souriau’s calculations, initially performed in the flat Minkowski spacetime, to the Schwarzschild spacetime accounting for gravitation. Finally, we recover the Classical Galilean Hyperelasticity with gravity, as the Newton–Cartan infinite light speed limit of this formulation.

Emergent quantum probability from full quantum dynamics and the role of energy conservation

We propose and study a toy model for the quantum measurements that yield the Born’s rule of quantum probability. In this model, the electrons interact with local photon modes and the photon modes are dissipatively coupled with local photon reservoirs. We treat the interactions of the electrons and photons with full quantum mechanical description, while the dissipative dynamics of the photon modes are treated via the Lindblad master equation. By assigning double quantum dots setup for the electrons coupling with local photons and photonic reservoirs, we show that the Born’s rule of quantum probability can emerge directly from microscopic quantum dynamics. We further discuss how the microscopic quantities such as the electron–photon coupling, detuning, and photon dissipation rate affect the quantum dynamics. Surprisingly, in the infinite long time measurement limit, the energy conservation already dictates the emergence of the Born’s rule of quantum probability. For finite-time measurement, the local photon dissipation rate determines the characteristic time-scale for the completion of the measurement, while other microscopic quantities affect the measurement dynamics. Therefore, in genuine measurements, the measured probability is determined by both the local devices and the quantum mechanical wavefunction.

Quantum phase transitions in the anti-Jaynes-Cummings triangle model

We carefully investigate the comprehensive impact of atom-cavity interaction and artificial magnetic fields on quantum phase transitions of anti-Jaynes-Cummings triangle model in the infinite frequency limit. We discover that ground states of the optical field can be a gapped normal phase (NP) or three kinds of gapless superradiant phases with infinite degeneracy. When the atom-cavity coupling is weak, the optical field is in a NP, which is a vacuum with no photons. Otherwise, it will stay at one of the superradiant phases: a normal superradiant phase without photon currents and another two chiral superradiant phases with opposite photon currents. The former only breaks the continuous U(1) symmetry and its gapless excitations are normal Goldstone modes. Nevertheless, the latter, mainly induced by an external synthetic gauge field, break both the continuous U(1) symmetry and chiral symmetry, thereby corresponding gapless excitations are chiral Goldstone modes. In addition, we also propose a detecting scheme to distinguish these superradiant phases.

Effect of the Internet on Enhancing Marketing among Business Organizations

The profitability of business organizations greatly relies on utilizing the Internet for marketing purposes. Through careful and knowledgeable use of the Internet, organizations can improve various aspects of their operations, mainly marketing. This study examines the Internet's specific impact on business organizations' marketing strategies. This study aims to enhance our understanding of how Internet usage impacts effective Internet management and its potential to boost marketing efforts for businesses in achieving their goals. The study highlights the advantages of using the Internet in business, differentiating organizations that utilize it from those that do not. Additionally, it emphasizes how the Internet can make long-term tasks more manageable and less time-consuming. Using the Internet can take you to levels that may have once been just a fantasy since the beginning of your business. In this paper, we have used the relevant literature mainly about the effects of using the Internet for a business organization, such as gaining more customers, increasing profits, reaching an infinite limit, and finally succeeding. Marketing, Internet, business organization, customer, attract customer.

Skew Howe duality and limit shapes of Young diagrams

AbstractWe consider the skew Howe duality for the action of certain dual pairs of Lie groups on the exterior algebra as a probability measure on Young diagrams by the decomposition into the sum of irreducible representations. We prove a combinatorial version of this skew Howe duality for the pairs , , , and using crystal bases, which allows us to interpret the skew Howe duality as a natural consequence of lattice paths on lozenge tilings of certain partial hexagonal domains. The ‐representation multiplicity is given as a determinant formula using the Lindström–Gessel–Viennot lemma and as a product formula. These admit natural ‐analogs that we show equals the ‐dimension of a ‐representation (up to an overall factor of ), giving a refined version of the combinatorial skew Howe duality. Using these product formulas (at ), we take the infinite rank limit and prove that the diagrams converge uniformly to the limit shape.

On Some New Properties for the Gamma Functions

In this paper will be presented some new properties in form of the inequalities related to gamma function, product of two gamma functions and some of them will be presented in the form of infinite series and limit where are included the Euler number and the Euler-Mascheroni constant.

Transition between metastable equilibria: Applications to binary-choice games.

Transitions between metastable equilibria in the low-temperature phase of dynamical Ising game with activity spillover are studied in the infinite time limit. It is shown that exponential enhancement due to activity spillover, which takes place in finite-time transitions, is absent in the infinite time limit. In order to demonstrate that, the analytical description for infinite time trajectory is developed. An analytical approach to estimate the probability of transition between metastable equilibria in the infinite time limit is introduced and its results are compared with those of kinetic Monte Carlo simulation. Our study sheds light on the dynamics of the Ising game and has implications for the understanding of transitions between metastable states in complex systems.

The asymptotic weak gravity conjecture in M-theory

The tower Weak Gravity Conjecture predicts infinitely many super-extremal states along every ray in the charge lattice of a consistent quantum gravity theory. We show this far-reaching claim in five-dimensional compactifications of M-theory on Calabi-Yau 3-folds for gauge groups with a weak coupling limit. We first characterize the possible weak coupling limits, building on an earlier classification of infinite distance limits in the Kähler moduli space of M-theory compactifications. We find that weakly coupled gauge groups are associated to curves on the compactification space contained in generic fibers or in fibers degenerating at finite distance in their moduli space. These always admit an interpretation as a Kaluza-Klein or winding U(1) in a dual frame or as part of a dual perturbative heterotic gauge group, in agreement with the Emergent String Conjecture. Using the connection between Donaldson-Thomas invariants and Noether-Lefschetz theory, we then show that every ray in the associated charge lattice either supports a tower of BPS states or of non-BPS states, and prove that these satisfy the super-extremality condition, at least in the weak coupling regime.

Effects of intramolecular chain conformation on the hydration and miscibility of polyethylene glycol in water studied by means of polymer reference interaction site model theory.

To examine the conventional idea that the gauche conformation of the OCCO dihedral angle promotes the dissolution of polyethylene glycol (PEG) in water through strong hydration, the thermodynamic properties of liquid mixtures of PEG and water were studied by means of polymer reference interaction site model (PRISM) theory. The intramolecular correlation functions required as input for PRISM theory were calculated by the generator matrix method, accompanied by changes in the distribution of dihedral angles. In the infinite dilution limit, the increased probability of gauche conformation of the OCCO dihedral angles stabilizes the hydration of PEG through enhanced hydrogen bonding between the ether oxygen of PEG and water. The mixing Gibbs energies of the liquid mixtures were also calculated in the whole concentration range based on the Gibbs-Duhem equation, as per our recent proposal. A liquid-liquid phase separation was observed when all the dihedral angles of PEG were in the trans conformation; for the liquid mixture to be miscible in the whole concentration range, the introduction of the OCCO gauche conformation was found to be indispensable. The above theoretical results support the conventional idea that the OCCO gauche conformation is important for the high miscibility of PEG and water.

Molecular dynamics study of fission product behaviour in molten [formula omitted]-[formula omitted] salt

The solvation of fission products in fuel salts and the effects due to their accumulation are important factors to understand in molten salt reactor design. In this contribution, we use ab initio Molecular Dynamics (AIMD) and polarizable ion model (PIM) based MD to study the structural and dynamic characteristics of the fission products (Cs, I and Xe) in the model fuel salt ThF4-LiF with different compositions at the infinite dilution limit. We first develop or refine existing PIM potentials to better reproduce the results of AIMD simulations. The AIMD and PIM-MD results are then compared in terms of the structures present in the salt, and the dynamics of their coordination. The results of this study indicate that, at 1200 K and 1400 K the fission products studied here have weak interactions with all composition ranges of fuel salt. This is evident, as first coordination shell associations being generally short-lived, as determined by cage correlation times. The diffusion coefficients determined through PIM-MD match the AIMD results reasonably well. The three fission products studied here are diffuse at similar rates; however, the Cs+ diffusion is slightly lower likely due to its more tightly bound associated F−. With all solutes, the diffusion coefficients decrease with a greater ThF4 fraction. Additionally, the behaviour of these solutes was studied at finite concentration ranges, up to 4 atom %. Here, it was found that the presence of Cs+ has little effect on the density, heat capacity and viscosity. However, these properties were affected by higher I+ and Xe concentrations.

Large population limit of the spectrum of killed birth-and-death processes

We consider a general class of birth-and-death processes with state space {0,1,2,3,…} which describes the size of a population going eventually to extinction with probability one. We obtain the complete spectrum of the generator of the process killed at 0 in the large population limit, that is, we scale the process by a parameter K, and take the limit K→+∞. We assume that the differential equation dx/dt=b(x)−d(x) describing the infinite population limit (in any finite-time interval) has a repulsive fixed point at 0, and an attractive fixed point x⁎>0. We prove that, asymptotically, the spectrum is the superposition of two spectra. One is the spectrum of the generator of an Ornstein-Uhlenbeck process, which is n(b′(x⁎)−d′(x⁎)), n≥0. The other one is the spectrum of a continuous-time binary branching process conditioned on non-extinction, and is given by n(d′(0)−b′(0)), n≥1. A major difficulty is that different scales and function spaces are involved. We work at the level of the eigenfunctions that we split over different regions, and study their asymptotic dependence on K in each region. In particular, we prove that the spectral gap goes to min⁡{b′(0)−d′(0),d′(x⁎)−b′(x⁎)}. This work complements a previous work of ours in which we studied the approximation of the quasi-stationary distribution and of the mean time to extinction.

Comparison of different hydrological and stability assumptions for physically-based modeling of shallow landslides

Physically-based models are reliable techniques for landslide susceptibility assessment, therefore, several models have been proposed in the literature. In this study, two different hydrological models (FSLAM and TRIGRS) and two slope stability calculation methods (infinite slope and limit equilibrium method with spherical failure surface in SCOOPS3D) were compared. The rainfall-triggered shallow landslides that occurred in Kaptanpaşa, Rize (Turkey) in 2017 were selected as a case study. The main focus is investigating the impact of landslide morphology on the success of the slope stability models. For this purpose, landslides in the inventory were grouped according to their morphological characteristics, such as the landslide depth, length, width and their ratios, and their compatibility with model assumptions (e.g. the infinite slope assumption). Then, the model performances in different landslide types were evaluated separately. To calibrate the model parameters a new time-saving, semi-automated calibration strategy, based on a sensitivity analysis, is proposed. Concretely, stability parameters c′∅′Z were calibrated via a MATLAB code by considering the area under the curve (AUC) and extreme conditions: saturated and initial conditions. Overall, SCOOPS3D is deduced to have better performance than infinite slope analysis. However, the accuracy of SCOOPS3D is revealed to decrease for landslides with a high ratio of length/depth and width/depth. From the comparison of hydrological models, FSLAM yielded more plausible results than TRIGRS with the available data. It is demonstrated that the morphological features of the landslides significantly influence the performance of the physically-based stability models. Furthermore, it is highlighted that the resolution of the Digital Elevation Model in comparison to the width and length of the landslides in an inventory can be influential in the performance of the physical model. 10 m resolution of DEM is deemed to be suitable for the study area judging the predictive performance in small landslides.

Light ray fluctuation and lattice refinement of simplicial quantum gravity

In several approaches of non-perturbative quantum gravity, a major outstanding problem is to obtain results valid at the infinite lattice refinement limit. Working with Lorentzian simplicial quantum gravity, we compute light ray fluctuation probabilities in 3D and 4D across different lattices. In a simplified refined box model with the Einstein–Hilbert action, numerical results show that lattice refinement does not simply suppress or simply enhance light ray fluctuations, but actually drives very wide and very narrow light probability distributions towards intermediate ones. A comparison across lattices and across couplings reveals numerical hints at a lattice refinement fixed point associated with a universality class of couplings. The results fit the intuition that quantum spacetime fluctuations reflected by light ray fluctuations start wild microscopically and become mild macroscopically. The refined box model is limited by the assumption of a rigid frame at all scales. The present results suggest further studies around the zero-coupling limit to relax the simplifying assumptions of the model.

Affine algebras at infinite distance limits in the Heterotic String

We analyze the boundaries of the moduli spaces of compactifications of the heterotic string on Td, making particular emphasis on d = 2 and its F-theory dual. We compute the OPE algebras as we approach all the infinite distance limits that correspond to (possibly partial) decompactification limits in some dual frame. When decompactifying k directions, we find infinite towers of states becoming light that enhance the algebra arising at a given point in the moduli space of the Td−k compactification to its k-loop version, where the central extensions are given by the k KK vectors. For T2 compactifications, we reproduce all the affine algebras that arise in the F-theory dual, and show all the towers explicitly, including some that are not manifest in the F-theory counterparts. Furthermore, we construct the affine SO(32) algebra arising in the full decompactification limit, both in the heterotic and in the F-theory sides, showing that not only affine algebras of exceptional type arise in the latter.

APPROXIMATIONS OF THE THEORIES OF STRUCTURES WITH ONE EQUIVALENCE RELATION

Recently, various methods similar to the “transfer principle” have been rapidly developing, where one property of a structure or pieces of this structure is satisfied in all infinite structures or in another algebraic structure. Such methods include smoothly approximable structures, holographic structures, almost sure theories, and pseudofinite structures approximable by finite structures. Pseudofinite structures are mathematical structures that resemble finite structures but are not actually finite. They are important in various areas of mathematics, including model theory and algebraic geometry. Pseudofinite structures are a fascinating area of mathematical logic that bridge the gap between finite and infinite structures. They allow studying infinite structures in ways that resemble finite structures, and they provide a connection to various other concepts in model theory. Further studying pseudofinite structures will continue to reveal new insights and applications in mathematics and beyond. Pseudofinite theory is a branch of mathematical logic that studies structures that are similar in some ways to finite structures, but can be infinitely large in other ways. It is an area of research that lies at the intersection of model theory and number theory and deals with infinite structures that share some properties with finite structures, such as having only finitely many elements up to isomorphism. A. Lachlan introduced the concept of smoothly approximable structures in order to change the direction of analysis from finite to infinite, that is, to classify large finite structures that seem to be smooth approximations to an infinite limit. The theory of pseudofinite structures is particularly relevant for studying equivalence relations. In this paper, we study the model-theoretic property of the theory of equivalence relations, in particular, the property of smooth approximability. Let L = {E}, where Е is an equivalence relation. We prove that an any ω-categorical L-structure M is smoothly approximable. We also prove that any infinite L-structure M is pseudofinite.

Celestial Supersymmetry

We discuss supersymmetric Yang-Mills theory coupled to dilatons in the framework of celestial holography. We show that in the presence of point-like dilaton sources, the CCFT operators associated with the gauge supermultiplet acquire a simple, factorized form. They factorize into the holomorphic (super)current part and the exponential “light” operators of Liouville theory, in the infinite central charge limit. The current sector exhibits (1,0) supersymmetry, thus implementing spacetime supersymmetry in CCFT.

Quantum Entropies and Decoherence for the Multiparticle Quantum Arnol'd Cat.

I study the scaling behavior in the physical parameters of dynamical entropies, classical and quantum, in a specifically devised model of collision-induced decoherence in a chaotic system. The treatment is fully canonical and no approximations are involved or infinite limits taken. I present this model in a detailed way, in order to clarify my views in the debate about the nature, definition, and relevance of quantum chaos.