Arbitrary Dimension Research Articles

Combinatorics of linear stability for Hamiltonian systems in arbitrary dimension

We address the general problem of studying linear stability and bifurcations of periodic orbits for Hamiltonian systems of arbitrary degrees of freedom. We study the topology of the GIT sequence introduced by the first author and Urs Frauenfelder in [7], in arbitrary dimension. In particular, we note that the combinatorics encoding the linear stability of periodic orbits is governed by a quotient of the associahedron. Our approach gives a topological/combinatorial proof of the classical Krein–Moser theorem, and refines it for the case of symmetric orbits.

Generalization of conformal Hamada operators

AbstractThe six-derivative conformal scalar operator was originally found by Hamada in its critical dimension of spacetime, $$d=6$$ d = 6 . We generalize this construction to arbitrary dimensions d by adding new terms cubic in gravitational curvatures and by changing its coefficients of expansion in various curvature terms. The consequences of global scale-invariance and of infinitesimal local conformal transformations are derived for the form of this generalized operator. The system of linear equations for coefficients is solved giving explicitly the conformal Hamada operator in any d. Some singularities in construction for dimensions $$d=2$$ d = 2 and $$d=4$$ d = 4 are noticed. We also prove a general theorem that a scalar conformal operator with n derivatives in $$d=n-2$$ d = n - 2 dimensions is impossible to construct. Finally, we compare our explicit construction with the one that uses conformal covariant derivatives and conformal curvature tensors. We present new results for operators built with different orders of conformal covariant derivatives.

Designs in finite classical polar spaces

AbstractCombinatorial designs have been studied for nearly 200 years. 50 years ago, Cameron, Delsarte, and Ray-Chaudhury started investigating their q-analogs, also known as subspace designs or designs over finite fields. Designs can be defined analogously in finite classical polar spaces, too. The definition includes the m-regular systems from projective geometry as the special case where the blocks are generators of the polar space. The first nontrivial such designs for $$t > 1$$ t > 1 were found by De Bruyn and Vanhove in 2012, and some more designs appeared recently in the PhD thesis of Lansdown. In this article, we investigate the theory of classical and subspace designs for applicability to designs in polar spaces, explicitly allowing arbitrary block dimensions. In this way, we obtain divisibility conditions on the parameters, derived and residual designs, intersection numbers and an analog of Fisher’s inequality. We classify the parameters of symmetric designs. Furthermore, we conduct a computer search to construct designs of strength $$t=2$$ t = 2 , resulting in designs for more than 140 previously unknown parameter sets in various classical polar spaces over $$\mathbb {F}_2$$ F 2 and $$\mathbb {F}_3$$ F 3 .

Geometric Formulation of Generalized Root-TT[over ¯] Deformations.

We develop a generic geometric formalism that incorporates both TT[over ¯]-like and root-TT[over ¯]-like deformations in arbitrary dimensions. This framework applies to a wide family of stress-energy tensor perturbations and encompasses various well-known field theories. Building upon the recently proposed correspondence between Ricci-based gravity and TT[over ¯]-like deformations, we further extend this duality to include root-TT[over ¯]-like perturbations. This refinement extends the potential applications of our approach and contributes to a deeper exploration of the interplay between stress tensor perturbations and gravitational dynamics. Among the various original outcomes detailed in this Letter, we have also obtained a deformation of the flat Jackiw-Teitelboim gravity action.

Dynamical edge modes and entanglement in Maxwell theory

Previous work on black hole partition functions and entanglement entropy suggests the existence of “edge” degrees of freedom living on the (stretched) horizon. We identify a local and “shrinkable” boundary condition on the stretched horizon that gives rise to such degrees of freedom. They can be interpreted as the Goldstone bosons of gauge transformations supported on the boundary, with the electric field component normal to the boundary as their symplectic conjugate. Applying the covariant phase space formalism for manifolds with boundary, we show that both the symplectic form and Hamiltonian exhibit a bulk-edge split. We then show that the thermal edge partition function is that of a codimension-two ghost compact scalar living on the horizon. In the context of a de Sitter static patch, this agrees with the edge partition functions found by Anninos et al. in arbitrary dimensions. It also yields a 4D entanglement entropy consistent with the conformal anomaly. Generalizing to Proca theory, we find that the prescription of Donnelly and Wall reproduces existing results for its edge partition function, while its classical phase space does not exhibit a bulk-edge split.

Thin-wall vacuum decay in the presence of a compact dimension

We study the problem of false vacuum decay in arbitrary dimensions, in the presence of gravity, and compute the transition probability within the thin-wall approximation, generalising the results of Coleman and de Luccia. In the particular case of one compact dimension, we present explicit formulae for the Euclidean Bounce configuration that drives the transition from a de Sitter to Minkowski or from a Minkowski to anti-de Sitter vacua.

Unified theory for the scaling of the crossover between strong and weak disorder behaviors of optimal paths and directed or undirected polymers in disordered media.

In this paper, we are concerned with the crossover between strong disorder (SD) and weak disorder (WD) behaviors in three well-known problems that involve minimal paths: directed polymers (directed paths with fixed starting point and length), optimal paths (undirected paths with a fixed end-to-end or spanning distance), and undirected polymers (undirected paths with a fixed starting point and length). We present a unified theoretical framework from which we can easily deduce the scaling of the crossover point of each problem in an arbitrary dimension. Our theory is based on the fact that the SD limit behavior of these systems is closely related to the corresponding percolation problem. As a result, the properties of those minimal paths are completely controlled by the so-called red bonds of percolation theory. Our model is first addressed numerically and then approximated by a two-term approach. This approach provides us with an analytical expression that seems to be reasonably accurate. The results are in perfect agreement with our simulations and with most of the results reported in related works. Our research also leads us to propose this crossover point as a universal measure of the disorder strength in each case. Interestingly, that measure depends on both the statistical properties of the disorder and the topological properties of the network.

Hierarchical Control for Partially Feasible Tasks With Arbitrary Dimensions: Stability Analysis for the Tracking Case

Hierarchical Control for Partially Feasible Tasks With Arbitrary Dimensions: Stability Analysis for the Tracking Case

Generalized N-Dimensional Effective Temperature for Cryogenic Systems in Accelerator Physics

Investigations into the properties of generalized effective temperature are conducted across arbitrary dimensions. Maxwell–Boltzmann distribution is displayed for one, two, and three dimensions, with effective temperatures expressed for each dimension. The energy density of blackbody radiation is examined as a function of dimensionality. Effective temperatures for non-uniform temperature distributions in one, two, three, and higher dimensions are presented, with generalizations extended to arbitrary dimensions. Furthermore, the application of generalized effective temperature is explored not only for linearly non-uniform temperature distributions but also for scenarios involving the volume fraction of two distinct temperature distributions. The effective temperature is determined for a cryogenic system supplied with both liquid nitrogen and liquid helium. This effective temperature is applied to the Coefficient of Performance (COP) in cryogenic systems and can also be applied to high-energy accelerator physics, including high-dimensional physics.

Temporal multilayer structures in discrete physical systems towards arbitrary-dimensional non-Abelian Aharonov-Bohm interferences

Temporal modulation recently draws great attentions in wave manipulations, with which one can introduce the concept of temporal multilayer structure, a temporal counterpart of spatially multilayer configurations. This kind of multilayer structure holds temporal interfaces in the time domain, which provides additional flexibility in temporal operations. Here we take this opportunity and propose to simulate a non-Abelian gauge field with a temporal multilayer structure in the discrete physical system. Two basic temporal operations, i.e., the folding/unfolding operation and the phase shift operation are used to design such a temporal multilayer structure, which hence can support noncommutative operations to realize the non-Abelian Aharonov-Bohm interference in the time domain. A two-/three-dimensional non-Abelian gauge field can be built, which may be further extended to higher dimensions. Our work therefore provides a unique platform enabling generalization of non-Abelian physics to arbitrary dimensions and offers a method for wave manipulations with temporal band engineering.

A rapid and low-cost method to fabricate well of the well (WOW) dishes with arbitrary 3D microwell shapes for improved embryo culture

Despite its high cost, the success rate for in vitro fertilization (IVF) remains < 33% in humans, driving the need for new techniques to improve embryo culture outcomes. The well-of-the-well (WOW) culture system is a platform for in-vitro mammalian embryo culture that has been shown to enhance the developmental competence of embryos and clinical pregnancy rates in humans. However, discovery and testing of the best design for optimal embryo culture quality is hindered by the lack of a method to flexibly produce WOW dishes of various designs. Here, we present a low-cost and simple method to fabricate WOW dishes with microwells of arbitrary shapes and dimensions. We use a low-cost 3D printing service to fabricate a poly(dimethylsiloxane) (PDMS)-based WOW insert that can be paired with a standard in vitro fertilization (IVF) dish to create WOW dishes with new microwell shapes, including pyramidal and hemispherical designs. We validate the fabrication quality of the WOW inserts and demonstrate the utility of the assembled WOW dishes for observation and grading of mouse embryo quality. Moreover, our results indicate that WOW dishes with hemispherical microwells result in better culture outcomes than traditional flat-bottomed IVF dishes and those with other microwell shapes, including the semi-elliptical microwells used in commercial WOW dishes. The proposed fabrication strategy thus provides a way to rapidly fabricate and test new WOW dishes that may bolster IVF success rates.

Loading conditions for self‐organization in the BML model with stochastic direction choice

A dynamical system is considered such that, in this system, particles move on a toroidal lattice of the dimension according to a version of the rule of particle movement in Biham–Middleton–Levine traffic model. We introduce a stochastic case with direction choice for particles. Particles of the first type move along rows, and the particles of the second type move along columns. The goal is to find conditions of self‐organization system for any lattice dimension. We have proved that the BML model as a dynamical system is a special case of Buslaev nets. This equivalence allows us to use of Buslaev net analysis techniques to investigate the BML model. In Buslaev nets conception, the self‐organization property of the system corresponds to the existence of velocity single point spectrum equal to 1. In the paper, we consider the model version when one notable aspect is that a particle may change its type. Exactly, we assume a constant probability that a particle changes type at each step. In the case where , the system corresponds to the classical version of the BML model. We define a state of the system where all particles continue to move indefinitely, in both the present and the future, as a state of free movement. A sufficient condition for the system to result in a state of free movement from any initial state (condition for self‐organization) has been found. This condition is that the number of particles be not greater than half the greatest common divisor of the numbers . It has been proved that, if , and whether or and there are both at least one particle of the first type and at least particle of the second type, then a necessary condition for a state of free movement to exist is the greatest common divisor of and be not less than 3. The theorems are formulated in terms of the algebraic structures and terms of the dynamical system. The spectrum of particle velocities has been found for the net . This approach allows us to hope that the spectrum of dynamical system can be studied for an arbitrary dimension of the net.

Measurement-induced phase transitions by matrix product states scaling

We study the time evolution of long quantum spin chains subjected to continuous monitoring projected on matrix product states (MPS) with fixed bond dimension, by means of the Time-Dependent Variational Principle (TDVP) algorithm. The latter gives an effective unitary evolution which approximates the real quantum evolution up to the projection error. We show that such error displays, at large times, a phase transition in the monitoring strength, which can be well detected by scaling analysis with relatively low values of bond dimensions. Moreover, in the presence of U(1) global spin charge, we show the existence of a charge-sharpening transition well separated from the entanglement transition which we detect by studying the charge fluctuations of a local subpart of the system at large times. Our work shows that quantum monitored dynamics projected on MPS manifolds contains relevant information on measurement-induced phase transitions and provides a new method to identify measured-induced phase transitions in systems of arbitrary dimensions and sizes. Published by the American Physical Society 2024

Density of the boundary regular set of 2d area minimizing currents with arbitrary codimension and multiplicity

In the present work, we consider area minimizing currents in the general setting of arbitrary codimension and arbitrary boundary multiplicity. We study the boundary regularity of 2d area minimizing currents, beyond that, several results are stated in the more general context of (C0,α0,r0)-almost area minimizing currents of arbitrary dimension m and arbitrary codimension taking the boundary with arbitrary multiplicity. Furthermore, we do not consider any type of convex barrier assumption on the boundary, in our main regularity result which states that the regular set, which includes one-sided and two-sided points, of any 2d area minimizing current T is an open dense set in the boundary.

General Relations between Stress Fluctuations and Viscoelasticity in Amorphous Polymer and Glass-Forming Systems.

Mechanical stress governs the dynamics of viscoelastic polymer systems and supercooled glass-forming fluids. It was recently established that liquids with long terminal relaxation times are characterized by transiently frozen stress fields, which, moreover, exhibit long-range correlations contributing to the dynamically heterogeneous nature of such systems. Recent studies show that stress correlations and relaxation elastic moduli are intimately related in isotropic viscoelastic systems. However, the origin of these relations (involving spatially resolved material relaxation functions) is non-trivial: some relations are based on the fluctuation-dissipation theorem (FDT), while others involve approximations. Generalizing our recent results on 2D systems, we here rigorously derive three exact FDT relations (already established in our recent investigations and, partially, in classical studies) between spatio-temporal stress correlations and generalized relaxation moduli, and a couple of new exact relations. We also derive several new approximate relations valid in the hydrodynamic regime, taking into account the effects of thermal conductivity and composition fluctuations for arbitrary space dimension. One approximate relation was heuristically obtained in our previous studies and verified using our extended simulation data on two-dimensional (2D) glass-forming systems. As a result, we provide the means to obtain, in any spatial dimension, all stress-correlation functions in terms of relaxation moduli and vice versa. The new approximate relations are tested using simulation data on 2D systems of polydisperse Lennard-Jones particles.

Noisy Landau-Streater and Werner-Holevo channels in arbitrary dimensions

Noisy Landau-Streater and Werner-Holevo channels in arbitrary dimensions

An approach to modeling the spatial interaction of point sets on integer regular grids

The paper demonstrates a generalized approach to modeling the interaction of points and point sets on integer regular grids in metric spaces. The modeling process is based on the definition of spatial relations in "Point-Set" and "Set-Set" pairs, which allows to determine the spatial interaction between arbitrary geometric objects, defined as grid-based geometric models (GGM), which represent point sets on regular grids, generalized schemes of spatial interaction are developed in the work. As a result of the work, software was developed for conducting computer experiments on simulation of interaction in metric spaces of arbitrary dimensions and visualization of their results. The developed approach was tested on data sets of different dimensions. Figs.: 10. Refs.: 11 titles.

Entanglement entropy via double-cone regularization

This paper proposes an alternative regularization method for handling the ultraviolet behavior of entanglement entropy. Utilizing an iε prescription in the Euclidean double cone geometry, it accurately reproduces the universal behavior of entanglement entropy. The method is demonstrated in the free boson theory in arbitrary dimensions and two-dimensional conformal field theories. The findings highlight the effectiveness of the iε regularization method in addressing ultraviolet issues in quantum field theory and gravity, suggesting potential applications to other calculable quantities. Published by the American Physical Society 2024

Explicit Parameterizations of Ortho-Symplectic Matrices in R4

Starting from the very first principles we derive explicit parameterizations of the ortho-symplectic matrices in the real four-dimensional Euclidean space. These matrices depend on a set of four real parameters which splits naturally as a union of the real line and the three-dimensional space. It turns out that each of these sets is associated with a separate Lie algebra which after exponentiations generates Lie groups that commute between themselves. Besides, by making use of the Cayley and Fedorov maps, we have arrived at alternative realizations of the ortho-symplectic matrices in four dimensions. Finally, relying on the fundamental structure results in Lie group theory we have derived one more explicit parameterization of these matrices which suggests that the obtained earlier results can be viewed as a universal method for building the representations of the unitary groups in arbitrary dimension.

The collisional particle-in-cell method for the Vlasov–Maxwell–Landau equations

We introduce an extension of the particle-in-cell method that captures the Landau collisional effects in the Vlasov–Maxwell–Landau equations. The method arises from a regularisation of the variational formulation of the Landau equation, leading to a discretisation of the collision operator that conserves mass, charge, momentum and energy, while increasing the (regularised) entropy. The collisional effects appear as a fully deterministic effective force, thus the method does not require any transport–collision splitting. The scheme can be used in arbitrary dimension, and for a general interaction, including the Coulomb case. We validate the scheme on scenarios such as the Landau damping, the two-stream instability and the Weibel instability, demonstrating its effectiveness in the numerical simulation of plasma.