Finite Entropy Research Articles

A Frustrated Antipolar Phase Analogous to Classical Spin Liquids.

The study of magnetic frustration in classical spin systems is motivated by the prediction and discovery of classical spin liquid states. These uncommon magnetic phases are characterized by a massive degeneracy of their ground state implying a finite magnetic entropy at zero temperature. While the classical spin liquid state is originally predicted in the Ising triangular lattice antiferromagnet in 1950, this state has never been experimentally observed in any triangular magnets. The discovery of an electric analogue of classical spin liquids on a triangular lattice of uniaxial electric dipoles in EuAl12O19 is reported here. This new type of frustrated antipolar phase is characterized by a highly-degenerate state at low temperature implying an absence of long-range antiferroelectric order, despite short-range antipolar correlations. Its dynamics are governed by a thermally activated process, slowing down upon cooling toward a complete freezing at zerotemperature.

Expansiveness, generators and Lyapunov exponents for random bundle transformations

We generalize Fathi's results by showing that a compact metrizable space admits an fiber expansive homeomorphism if and only if it has a compatible hyperbolic metric. Moreover, we prove that a compact metrizable space admits an fiber expansive homeomorphism if and only if it has a generator in detail. Furthermore, we show that a fiber expansive homeomorphism has finite fiber topological entropy. Finally, we show that fiber Lyapunov exponents for a fiber expansive system are different from zero, indicating that the system presents a chaotic system. Meanwhile, we also prove that negative fiber Lyapunov exponents for compact invariant sets of a dynamical system imply that the compact set is a fiber attractor.

Reducing Bandwidth Usage in CVSLAM: A Novel Approach to Map Point Selection and Efficient Data Compression

In the field of collaborative visual simultaneous localization and mapping (CVSLAM), efficient data communication poses a significant challenge, particularly in environments with limited bandwidth. To address this issue, we introduce a method aimed at reducing communication consumption. Our approach starts with a strategic culling of map points, aiming at maximizing pose-visibility and expanding spatial diversity to effectively eliminate redundant data in CVSLAM. We achieve this by formulating the problem of maximizing pose-visibility and spatial diversity as a minimum-cost maximum-flow graph optimization problem. Subsequently, we apply finite state entropy encoding for the compression of visual information, further alleviating bandwidth constraints. To verify the proposed method, we implement it within a centralized collaborative monocular simultaneous localization and mapping (SLAM) system. Our approach has been tested on publicly available datasets and in real-world scene. The results show a prominent reduction in bandwidth usage by 49% while maintaining mapping accuracy and without introducing additional latency, confirming its effectiveness in a multi-agent system setting.

Entanglement entropy in string compactifications

We consider ℤN orbifolds of Type-II compactifications to four and six dimensions on several Calabi-Yau manifolds in the orbifold limit with the aim to compute the entanglement entropy. The spectrum can contain tachyons in the doubly-twisted sectors which can lead to new infrared divergences for the partition function that are not present in the orbifolds of the uncompactified ten-dimensional theory. We show that all tachyonic contributions in these models admit a resummation and analytic continuation that yields finite entropy in the physical region 0 < N ≤ 1 just as in ten dimensions.

Algebraic Entropy of Path Algebras and Leavitt Path Algebras of Finite Graphs

The Gelfand–Kirillov dimension is a well established quantity to classify the growth of infinite dimensional algebras. In this article we introduce the algebraic entropy for path algebras. For the path algebras, Leavitt path algebras and the path algebra of the extended (double) graph, we compare the Gelfand–Kirillov dimension and the entropy. We show that path algebras over finite graphs can be classified to be of finite dimension, finite Gelfand–Kirillov dimension or finite algebraic entropy. We show indeed how these three quantities are dependent on cycles inside the graph. Moreover we show that the algebraic entropy is conserved under Morita equivalence but perhaps for a different filtration. In addition we give several examples of the entropy in path algebras and Leavitt path algebras.

A holographic entanglement entropy at spi

Defining finite entanglement entropy for a subregion in quantum field theory requires the introduction of two logically independent scales: an IR scale that controls the size of the subregion, and a UV cut-off. In AdS/CFT, the IR scale is the AdS lengthscale, the UV cut-off is the bulk radial cut-off, and the subregion is specified by dimensionless angles. This is the data that determines Ryu-Takayanagi surfaces and their areas in AdS/CFT. We argue that in asymptotically flat space there exists the notion of a “spi-subregion” that one can associate to spatial infinity (spi). Even though geometrically quite different from an AdS subregion, this angle data has the crucial feature that it allows an interpretation as a bi-partitioning of spi. Therefore, the area of the RT surface associated to the spi-subregion can be interpreted as the entanglement entropy of the reduced density matrix of the bulk state under this bi-partition, as in AdS/CFT. For symmetric spi-subregions, these RT surfaces are the waists of Asymptotic Causal Diamonds. In empty flat space they reduce to Rindler horizons, and are analogues of the AdS-Rindler horizons of Casini, Huerta & Myers. We connect these results to previous work on minimal surfaces anchored to screens in empty space, but also generalize the discussion to the case where there are black holes in the bulk. The phases of black hole RT surfaces as the spi-subregion is varied, naturally connect with those of black holes (small and large) in AdS. A key observation is that the radial cut-off is associated to an IR scale in flat space — and in fact there are no UV divergences. We argue that this is consistent with previous suggestions that in sub-AdS scales the holographic duality is an IR/IR correspondence and that the degrees of freedom are not those of a local QFT, but those of long strings. Strings are of course, famously UV finite.

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Entropy for Monge-Ampère Measures in the Prescribed Singularities Setting

In this note, we generalize the notion of entropy for potentials in a relative full Monge-Ampère mass $\mathcal{E}(X, \theta, \phi)$, for a model potential $\phi$. We then investigate stability properties of this condition with respect to blow-ups and perturbation of the cohomology class. We also prove a Moser-Trudinger type inequality with general weight and we show that functions with finite entropy lie in a relative energy class $\mathcal{E}^{\frac{n}{n-1}}(X, \theta, \phi)$ (provided $n&gt;1$), while they have the same singularities of $\phi$ when $n=1$.

Shift invariant subspaces in growth spaces and sets of finite entropy

AbstractWe investigate shift invariant subspaces within the realm of analytic functions in the unit disc, whose radial growth is determined by a majorant . Our main result offers a complete characterization of the shift invariant subspaces generated by Nevanlinna class functions within the aforementioned class of growth spaces. Our description is expressed in terms of a certain ‐entropy condition, which naturally arises in connection with boundary zero sets for analytic functions in the unit disc, having modulus of continuity not exceeding on the unit circle. Furthermore, we utilize our findings to establish a structural theorem on approximation in model spaces.

Crossed products, extended phase spaces and the resolution of entanglement singularities

We identify a direct correspondence between the crossed product construction which plays a crucial role in the theory of Type III von Neumann algebras, and the extended phase space construction which restores the integrability of non-zero charges generated by gauge symmetries in the presence of spatial substructures. This correspondence provides a blue-print for resolving singularities which are encountered in the computation of entanglement entropy for subregions in quantum field theories. The extended phase space encodes quantities that would be regarded as ‘pure gauge’ from the perspective of the full theory, but are nevertheless necessary for gluing together, in a path integral sense, physics in different subregions. These quantities are required in order to maintain gauge covariance under such gluings. The crossed product provides a consistent method for incorporating these necessary degrees of freedom into the operator algebra associated with a given subregion. In this way, the extended phase space completes the subregion algebra and subsequently allows for the assignment of a meaningful, finite entropy to states therein.

Grid entropy in last passage percolation — A superadditive critical exponent approach

Working in the setting of i.i.d. last-passage percolation on RD with no assumptions on the underlying edge-weight distribution, we arrive at the notion of grid entropy — a Subadditive Ergodic Theorem limit of the entropies of paths with empirical measures weakly converging to a given target, or equivalently a deterministic critical exponent of canonical order statistics associated with the Levy-Prokhorov metric. This provides a fresh approach to an entropy first developed by Rassoul-Agha and Seppäläinen as a large deviation rate function of empirical measures along paths. In their 2014 paper, variational formulas are developed for the point-to-point/point-to-level Gibbs Free Energies as the convex conjugates of this entropy. We rework these formulas in our new framework and explicitly link our descriptions of grid entropy to theirs. We also improve on a known bound for this entropy by introducing a relative entropy term in the inequality. Furthermore, we show that the set of measures with finite grid entropy coincides with the deterministic set of limit points of empirical measures studied in a 2020 paper by Bates. We partially answer a directed polymer version of a question of Hoffman which was previously tackled in the zero temperature case by Bates.

Freestream Temperature Effects on the Receptivity of Hypersonic Boundary Layer Induced by Finite-Amplitude Pulse Entropy Waves

The unsteady hypersonic flow under finite amplitude pulse entropy perturbation at different freestream temperatures was calculated by direct numerical simulation. The flow response characteristics under the perturbation of entropy waves in freestreaming are analyzed. The temperature effect of freestreaming is studied based on the sensitivity of the boundary layer caused by pulse entropy perturbation. The results show that the higher freestream temperature promotes the first growth of the above third-order modes after leaving the head region, and strongly inhibits the first attenuation. The influence of the freestream temperature on the evolution of the induced disturbance wave is more significant than that on the development of the main flow disturbance waves. Low freestream temperature can suppress the attenuation of the modes below the second order. As the disturbance wave evolves downstream, the frequency band of the finite frequency disturbance wave gradually narrows, and the frequency band narrows faster when the temperature of freestreaming is low than when the temperature of freestreaming is high.

Time reversal of diffusion processes under a finite entropy condition

Time reversal of diffusion processes under a finite entropy condition

A novel finite spectral entropy: Gated term memory unit recursive network integrated with Ladybug Beetle Optimization algorithm for epileptic seizure detection.

Professional medical experts use a visual electroencephalography (EEG) signal for epileptic seizure detection, although this method is time-consuming and highly subject to bias. The majority of previous epileptic detection techniques have poor efficiency, detection performance and also which are unsuited to handle large datasets. In order to solve the aforementioned issues and to assist medical professionals with an advanced technology, a computerized epileptic seizure detection system is essential. Therefore, the proposed work intends to design an automated detection tool for predicting an epileptic seizure from EEG signals. For this purpose, a novel non-linear feature analysis and deep learning algorithms are deployed in this work. Initially, the signal decomposition, filtering and artifacts removal operations are carried out with the use of finite Haar wavelet transformation technique. After that, the finite spectral entropy (FSE) based feature extraction model has been used to extract the time, frequency, and time-frequency features from the normalized signal. Consequently, the novel gated term memory unit recursive network (GTRN) model is employed to predict the given EEG signal as whether healthy or seizure affected including the class with high accuracy. During this process, the recently developed Ladybug Beetle Optimization (LBO) algorithm is used to compute the logistic sigmoid function based on the solution. The purpose of using this algorithm is to simplify the process of classification with increased seizure prediction accuracy and performance. Moreover, the standard and popular benchmark EEG datasets are used to validate and test the results of the proposed FSE-GTRN-LBO mechanism. By leveraging the finite Haar wavelet transformation and FSE-based feature extraction, we can efficiently process EEG signals. The utilization of the GTRN model enables accurate classification of healthy and seizure-affected EEG data. To optimize the classification process further, we integrate the LBO algorithm, streamlining the computation of the logistic sigmoid function. Through comprehensive validation on standard EEG datasets, our proposed FSE-GTRN-LBO mechanism achieves outstanding seizure prediction accuracy and performance, surpassing existing state-of-the-art techniques.

Late-time correlators and complex geodesics in de Sitter space

We study two-point correlation functions of a massive free scalar field in de Sitter space using the heat kernel formalism. Focusing on two operators in conjugate static patches we derive a geodesic approximation to the two-point correlator valid for large mass and at late times. This expression involves a sum over two complex conjugate geodesics that correctly reproduces the large-mass, late-time limit of the exact two-point function in the Bunch-Davies vacuum. The exponential decay of the late-time correlator is associated to the timelike part of the complex geodesics. We emphasize that the late-time exponential decay is in tension with the finite maximal entropy of empty de Sitter space, and we briefly discuss how non-perturbative corrections might resolve this paradox.

Porous medium equation and cross-diffusion systems as limit of nonlocal interaction

This paper studies the derivation of the quadratic porous medium equation and a class of cross-diffusion systems from nonlocal interactions. We prove convergence of solutions of a nonlocal interaction equation, resp. system, to solutions of the quadratic porous medium equation, resp. cross-diffusion system, in the limit of a localising interaction kernel. The analysis is carried out at the level of the (nonlocal) partial differential equations and we use gradient flow techniques to derive bounds on energy, second order moments, and logarithmic entropy. The dissipation of the latter yields sufficient regularity to obtain compactness results and pass to the limit in the localised convolutions. The strategy we propose relies on a discretisation scheme, which can be slightly modified in order to extend our result to PDEs without gradient flow structure. In particular, it does not require convexity of the associated energies. Our analysis allows to treat the case of limiting weak solutions of the non-viscous porous medium equation at relevant low regularity, assuming the initial value to have finite energy and entropy.

From Classical to Quantum: Uniform Continuity Bounds on Entropies in Infinite Dimensions

We prove a variety of improved uniform continuity bounds for entropies of both classical random variables on an infinite state space and of quantum states of infinite-dimensional systems. We obtain the first tight continuity estimate on the Shannon entropy of random variables with a countably infinite alphabet. The proof relies on a new mean-constrained Fano-type inequality. We then employ this classical result to derive a tight energy-constrained continuity bound for the von Neumann entropy. To deal with more general entropies in infinite dimensions, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e.g</i> . α-Rényi and α-Tsallis entropies, we develop a novel approximation scheme based on operator Hölder continuity estimates. Finally, we settle an open problem raised by Shirokov [1], [2] regarding the characterisation of states with finite entropy.

Frequency Analysis of EEG Microstate Sequences in Wakefulness and NREM Sleep

The majority of EEG microstate analyses concern wakefulness, and the existing sleep studies have focused on changes in spatial microstate properties and on microstate transitions between adjacent time points, the shortest available time scale. We present a more extensive time series analysis of unsmoothed EEG microstate sequences in wakefulness and non-REM sleep stages across many time scales. Very short time scales are assessed with Markov tests, intermediate time scales by the entropy rate and long time scales by a spectral analysis which identifies characteristic microstate frequencies. During the descent from wakefulness to sleep stage N3, we find that the increasing mean microstate duration is a gradual phenomenon explained by a continuous slowing of microstate dynamics as described by the relaxation time of the transition probability matrix. The finite entropy rate, which considers longer microstate histories, shows that microstate sequences become more predictable (less random) with decreasing vigilance level. Accordingly, the Markov property is absent in wakefulness but in sleep stage N3, 10/19 subjects have microstate sequences compatible with a second-order Markov process. A spectral microstate analysis is performed by comparing the time-lagged mutual information coefficients of microstate sequences with the autocorrelation function of the underlying EEG. We find periodic microstate behavior in all vigilance states, linked to alpha frequencies in wakefulness, theta activity in N1, sleep spindle frequencies in N2, and in the delta frequency band in N3. In summary, we show that EEG microstates are a dynamic phenomenon with oscillatory properties that slow down in sleep and are coupled to specific EEG frequencies across several sleep stages.

An Excess Entropy Approach to Classify Long-Term and Short-Term Memory Stationary Time Series

Long-term memory behavior is one of the most important phenomena that has appeared in the time series analysis. Different from most definitions of second-order properties, an excess entropy approach is developed for stationary time series to classify long-term and short-term memory. A stationary sequence with finite block entropy is long-term memory if its excess entropy is infinite. The simulation results are graphically demonstrated after some theoretical results are simply presented by various stochastic sequences. Such an approach has advantages over the traditional ways that the excess entropy of stationary sequence with finite block entropy is invariant under instantaneous one-to-one transformation, and that it only requires very weak moment conditions rather than second-order moment conditions and thus can be applied to distinguish the LTM behavior of stationary sequences with unbounded second moment (e.g., heavy tail distribution). Finally, several applications on real data are exhibited.

Islands in proliferating de Sitter spaces

We study two-dimensional de Sitter universe which evolves and proliferates according to the Ginsparg-Perry-Bousso-Hawking mechanism, using Jackiw-Teitelboim gravity coupled to conformal matter. Black holes are generated by quantum gravity effects from pure de Sitter space and then evaporate to yield multiple disjoint de Sitter spaces. The back-reaction from the matter CFT is taken into account for the dilaton as a function of the temperature of the CFT. We discuss the evaporation of black holes and calculate the finite temperature entropy of an inflating region using the island formula. We find that the island moves towards the apparent horizon of the black hole as the temperature increases. The results are applied to the case of multiple evaporating black holes, for which we suggest multiple islands.