Finite Loop Research Articles

Terwilliger algebras constructed from Cayley tables of finite Bol loops

Terwilliger algebras constructed from Cayley tables of finite Bol loops

Amenability of finite energy path and loop groups

It is shown that the groups of finite energy (i.e., Sobolev class H^{1} ) paths and loops with values in a compact Lie group are amenable in the sense of Pierre de la Harpe, that is, every continuous action of such a group on a compact space admits an invariant regular Borel probability measure. To our knowledge, the strongest previously known result concerned the amenability of groups of continuous paths and loops (Malliavin and Malliavin 1992).

Thermal effects on sound velocity peak and conformality in isospin QCD

We study thermal effects on equations of state (EOS) in isospin QCD, utilizing a quark-meson model coupled to a Polyakov loop. The quark-meson model is analyzed at one-loop that is the minimal order to include quark substructure constraints on pions which condense at finite isospin density. In the previous study we showed that the quark-meson model at zero temperature produces the sound velocity peak and the negative trace anomaly in the domain between the chiral effective theory regime at low density and the perturbative QCD regime at high density, in reasonable agreement with lattice simulations. We now include thermal effects from quarks in the Polyakov loop background and examine EOS, especially the sound velocity and trace anomaly along isentropic trajectories. At large isospin density, there are three temperature windows; (i) the pion condensed region with almost vanishing Polyakov loops, (ii) the pion condensed region with finite Polyakov loops, and (iii) the quark gas without pion condensates. In the domain (i), the gap associated with the pion condensate strongly quenches thermal excitations. As the system approaches the domain (ii), thermal quarks, which behave as nonrelativistic particles, add energy density but little pressure, substantially reducing the sound velocity to the value less than the conformal value while increasing the trace anomaly toward the positive value. Approaching the domain (iii), thermal quarks become more relativistic as pion condensates melt, increasing sound velocity toward the conformal limit. Corrections from thermal pions are also briefly discussed. Published by the American Physical Society 2024

The significance of fuzzy boundaries of the barrier regions in single-molecule measurements of failed barrier crossing attempts.

A recent ground-breaking experimental study [Lyons et al., Phys. Rev. X 14(1), 011017 (2024)] reports on measuring the temporal duration and the spatial extent of failed attempts to cross an activation barrier (i.e., "loops") for a folding transition in a single molecule and for a Brownian particle trapped within a bistable potential. Within the model of diffusive dynamics, however, both of these quantities are, on average, exactly zero because of the recrossings of the barrier region boundary. That is, an observer endowed with infinite spatial and temporal resolution would find that finite loops do not exist (or, more precisely, form a set of measure zero). Here we develop a description of the experiment that takes the "fuzziness" of the boundaries caused by finite experimental resolution into account and show how the experimental uncertainty of localizing the point, in time and space, where the barrier is crossed leads to observable distributions of loop times and sizes. Although these distributions generally depend on the experimental resolution, this dependence, in certain cases, may amount to a simple resolution-dependent factor and, therefore, the experiments do probe inherent properties of barrier crossing dynamics.

On neutral Tannakian subcategories of loop quiver representations

We explored the categories of (twisted) representations of a loop quiver. These representation categories have two choices of tensor structures: Kronecker tensor and Simpson tensor. By studying the rigidity properties, we have provided several examples of (semi-) Tannakian categories using the category of (twisted) representations of a loop quiver for both tensors.We have introduced the concept of essentially finite loop quiver bundles based on the work of Nori, Borne, and Vistoli. As an application, we have given some examples of (semi-) Tannakian categories of equivariant bundles and Hitchin pairs. Additionally, we have defined the notion of H-nflat twisted loop quiver bundles and have established Tannakian category structures for certain classes of varieties.

Accretion onto oscillating cosmic string loops

Cosmic string loops are non-linear density fluctuations which form in the early universe and could play an important role in explaining many phenomena which are in tension with the standard ΛCDM model. Hence, the accretion process onto cosmic string loops should be studied in detail. Most previous works view loops as point masses and ignore the impact of a finite loop size. In this work, we utilize the Zel'dovich approximation to calculate the non-linear mass sourced by a static extended loop with a time-averaged density profile derived from the trajectory of the loop oscillation, and compare the result with what is obtained for a point-mass source. We find that the finite size of a loop mainly affects the evolution of turnaround shells during the early stages of accretion, converging to the point mass result after a critical redshift, zc ( II)/( III). For zc ( II)/( III), the total accreted mass surrounding a loop is suppressed relative to the point mass case and has a growth rate proportional to (1+z)-3/2.As an immediate extension, we also qualitatively analyse the accretion onto moving point masses and onto moving extended loops. In addition to the reduction in the nonlinear mass, the loop finite size also changes the shape of the turnaround surface at early stages of accretion.

Quantum Circuits with Multiterminal Josephson-Andreev Junctions

We explore superconducting quantum circuits where several leads are simultaneously connected beyond the tunneling regime, such that the fermionic structure of Andreev bound states in the resulting multiterminal Josephson junction influences the states of the full circuit. Using a simple model of single-channel contacts and a single level in the middle region, we discuss different circuit configurations where the leads are islands with finite capacitance and/or form loops with finite inductance. We find situations of practical interest where the circuits can be used to define noise-protected qubits, which map to the bifluxon and 0−π qubits in the tunneling regime. We also point out the subtleties of the gauge choice for a proper description of these quantum circuit dynamics. Published by the American Physical Society 2024

Graphs Connected to Isotopes of Inverse Property Quasigroups: A Few Applications

Many real‐world applications can be modelled as graphs or networks, including social networks and biological networks. The theory of algebraic combinatorics provides tools to analyze the functioning of these networks, and it also contributes to the understanding of complex systems and their dynamics. Algebraic methods help uncover hidden patterns and properties that may not be immediately apparent in a visual representation of a graph. In this paper, we introduce left and right inverse graphs associated with finite loop structures and two mappings, P‐edge labeling and V‐edge labeling, of Latin squares. Moreover, this work includes some structural and graphical results of the commutator subloop, nucleus, and loop isotopes of inverse property quasigroups.

Emergence of interlacements from the finite volume Bose soup

We show that conditioned on the (empirical) particle density exceeding the critical value, the finite volume Bose loop soup converges to the superposition of the Bosonic loop soup (on the whole space) and the Poisson point process of random interlacements. The intensity of the latter is given by the excess density above the critical point. We consider both the free case and the mean-field case.

Josephson diode effect in monolithic dc-SQUIDs based on 3D Dayem nanobridges

It was recently experimentally proved that the superconducting counterpart of a diode, i.e., a device that realizes nonreciprocal Cooper pairs transport, can be realized by breaking the spatial and time-reversal symmetry of a system simultaneously. Here, we report the theory, fabrication, and operation of a monolithic dc superconducting quantum interference device that embedding three-dimensional Dayem nanobridges as weak links realizes an efficient and magnetic flux-tunable supercurrent diode. The device is entirely realized in Al and achieves a maximum rectification efficiency of ∼20%, which stems from the high harmonic content of its current-to-phase relation only without the need of any sizable screening current caused by a finite loop inductance. Our interferometer can be easily integrated with state-of-the-art superconducting electronics and since it does not require a finite loop inductance to provide large rectification, its downsizing is not limited by the geometrical constraints of the superconducting ring.

Reversible Networks Made of Star Polymers: Mean-Field Treatment with Consideration of Finite Loops

Reversible Networks Made of Star Polymers: Mean-Field Treatment with Consideration of Finite Loops

A short proof for the central nilpotency of Moufang loops of prime power order

The main result is an elementary proof of the nilpotency of Moufang loops which are of prime power order. Besides basic properties of Moufang loops and Sylow theorems, the proof relies on the fact that a relative multiplication group of a (centrally nilpotent) subloop of order pk is a p-group in any finite Moufang loop Q, and that Lφ(x)−1φLxφ−1 and Rφ(x)−1φRxφ−1 are pseudoautomorphisms of Q whenever x∈Q and φ is a pseudoautomorphism of Q.

A Virtual-Flux State Observer-Based Inductance Identification Method for Model Predictive Control of Grid-Tied Inverters With a Finite Phase Angle Set-Based PLL

To reduce the calculation burden of the conventional finite phase angle set-based phase-locked loop (PLL) and enhance the parameter robustness of the model predictive control for grid-tied inverters (GTIs), an improved finite phase angle set-based PLL and a new inductanceidentification method basedonvirtual-flux state observer (VFSO) are proposed in this paper. First, three improved PLLs are proposed based on the principles of the conventional finite phase angle set-based PLL. Second, the proposed PLLs are further improved by redesigning the cost function, which can search the optimal phase angle within the region of [0, π] instead of [0, 2π], reducing the iteration times. Third, a new VFSO with only one observer parameter is proposed, based on which a new inductance identification method is presented. Fourth, by analyzing the characteristics of the proposed VFSO, it is found that a larger observer gain can improve the dynamic convergence speed of the proposed VFSO, while a smaller one can enhance its robustness against grid frequency deviation. So, an observer gain adaptive method is proposed, aiming to improve both the dynamic and steady state control performance of the proposed inductance identification method. Finally, comparative experimental studies verify the effectiveness of the proposed strategies.

Three Concepts of Nilpotence in Loops

We introduce the abstract concept of supernilpotence in loop theory, and relate it to existing concepts, namely, central nilpotence and nilpotence of the multiplication group. We prove that the class of supernilpotence is greater or equal than the class of nilpotence of the multiplication group, and combining existing results, we show that a finite loop is supernilpotent if and only if its multiplication group is nilpotent. We also provide a new exposition of a classical result and crucial ingredient, that loops with a nilpotent multiplication group are centrally nilpotent and admit a prime decomposition.

Tambara-Yamagami, loop groups, bundles and KK-theory

This paper is part of a sequence interpreting quantities of conformal field theories K-theoretically. Here we give geometric constructions of the associated module categories (modular invariants, nimreps, etc). In particular, we give a KK-theory interpretation of all modular invariants for the loop groups of tori, as well as most known modular invariants of loop groups. In addition, we find unexpectedly that the Tambara-Yamagami fusion category has an elegant description as bundles over a groupoid, and use that to interpret its module categories as KK-elements. We establish reconstruction for the doubles of all Tambara-Yamagami categories, generalising work of Bischoff to even-order groups. We conclude by relating the modular group representations coming from finite groups and loop groups to the Chern character and to the Fourier-Mukai transform respectively.

Stochastic Geometric Iterative Method for Loop Subdivision Surface Fitting

In this paper, we propose a stochastic geometric iterative method (S-GIM) to approximate the high-resolution 3D models by finite loop subdivision surfaces. Given an input mesh as the fitting target, the initial control mesh is generated using the mesh simplification algorithm. Then, our method adjusts the control mesh iteratively to make its finite loop subdivision surface approximate the input mesh. In each geometric iteration, we randomly select part of points on the subdivision surface to calculate the difference vectors and distribute the vectors to the control points. Finally, the control points are updated by adding the weighted average of these difference vectors. We prove the convergence of S-GIM and verify it by demonstrating error curves in the experiment. In addition, compared with existing geometric iterative methods, S-GIM has a shorter running time under the same number of iteration steps.

Corrosion behavior of Ni62Nb33Zr5 bulk metallic glasses after annealing and cryogenic treatments

The microstructural evolution and corrosion behavior of Ni62Nb33Zr5 bulk metallic glasses (BMGs) after annealing treatment (AT) at different crystallization temperatures and cryogenic treatment (CT) at −100 ℃ are experimentally investigated. Appropriate AT and CT can both improve the thermal stability and comprehensive corrosion resistance of as-cast BMG in 3.5 wt.% NaCl solution. The annealed and cryo-treated BMGs exhibit one more finite diffusion layer loop in the electrical equivalent circuit than the as-cast BMG, indicating the complexity of the corrosion behavior. Superior corrosion resistance is obtained in the cryo-treated BMG because the high degree of amorphization caused by CT reduces the structural inhomogeneity. Lower Cd and higher Rd values are obtained for the cryo-treated BMG, revealing the formation of a more stable passive film. Among the annealed BMGs, the fully crystallized sample exhibits a higher anti-corrosion performance owing to the existence of Nb-rich oxides in the crystallization products. The passive film is found to be composed mainly of Nb2O5 and ZrO2, demonstrating that Nb and Zr are conducive to reacting with oxygen to form a passive film. Based on the goal of maintaining a fully amorphous phase, appropriate CT causing structural homogeneity of the BMG is a simple and effective means to improve the comprehensive corrosion resistance.

On Doro’s conjecture for finite Moufang loops

In 1978, Doro, in his paper [S. Doro, Simple moufang loops, Math. Proc. Camb. Philos. Soc. 83 (1978) 377–392] published the following conjecture: If the nucleus of a Moufang loop is trivial, then the commutant is a normal subloop. By working in the multiplication group of the loop we prove that in case of finite Moufang loops with trivial nucleus, the commutant is normal if and only if it is trivial.

Intrinsic Anomalous Hall Effect in a Bosonic Chiral Superfluid.

The anomalous Hall effect has had a profound influence on the understanding of many electronic topological materials but is much less studied in their bosonic counterparts. We predict that an intrinsic anomalous Hall effect exists in a recently realized bosonic chiral superfluid, a p-orbital Bose-Einstein condensate in a 2D hexagonal boron nitride optical lattice [Wang etal., Nature (London) 596, 227 (2021)NATUAS0028-083610.1038/s41586-021-03702-0]. We evaluate the frequency-dependent Hall conductivity within a multi-orbital Bose-Hubbard model that accurately captures the real experimental system. We find that in the high frequency limit, the Hall conductivity is determined by finite loop current correlations on the s-orbital residing sublattice, the latter a defining feature of the system's chirality. In the opposite limit, the dc Hall conductivity can trace its origin back to the noninteracting band Berry curvature at the condensation momentum, although the contribution from atomic interactions can be significant. We discuss available experimental probes to observe this intrinsic anomalous Hall effect at both zero and finite frequencies.

Half-automorphism group of Chein loops

A bijection f of a loop L is a half-automorphism if $$f(xy) \in \{f(x)f(y), f(y)f(x)\}$$ , for any $$x, y \in L$$ . A half-automorphism is nontrivial when it is neither an automorphism nor an anti-automorphism. A Chein loop $$L = G \cup Gu$$ is a Moufang loop constructed from a group G and an element u of order 2 outside G. In this paper, the half-automorphism group of finite Chein loops is described.