Finite Extension Research Articles

Stretched horizon from conformal field theory

Recently, it has been observed that the Hartle-Hawking correlators, a signature of smooth horizon, can emerge from certain heavy excited state correlators in the (manifestly non-smooth) BTZ stretched horizon background, in the limit when the stretched horizon approaches the real horizon. In this note, we develop a framework of quantizing the CFT modular Hamiltonian, that explains the necessity of introducing a stretched horizon and the emergence of thermal features in the AdS-Rindler and (planar) BTZ backgrounds. In more detail, we quantize vacuum modular Hamiltonian on a spatial segment of S1, which can be written as a particular linear combination of sl(2,ℝ) generators. Unlike radial quantization, (Euclidean) time circles emerge naturally here which can be contracted smoothly to the ‘fixed points’(end points of the interval) of this quantization thus providing a direct link to thermal physics. To define a Hilbert space with discrete normalizable states and to construct a Virasoro algebra with finite central extension, a natural regulator (ϵ) is needed around the fixed points. Eventually, in the dual description the fixed points correspond to the horizons of AdS-Rindler patch or (planar) BTZ and the cut-off being the stretched horizon. We construct a (Lorentzian) highest weight representation of that Virasoro algebra where vacuum can be identified with certain boundary states on the cut-off surface. We further demonstrate that two point function in a (vacuum) descendant state of the regulated Hilbert space will reproduce thermal answer in ϵ → 0 limit which is analogous to the recent observation of emergent thermality in (planar) BTZ stretched horizon background. We also argue the thermal entropy of this quantization coincides with entanglement entropy of the subregion. Conversely, the microcanonical entropy corresponding to high energy density of states exactly reproduce the BTZ entropy. Quite remarkably, all these dominant high lying microstates are defined only at finite ϵ in the regulated Hilbert space. We expect that all our observations can be generalized to BTZ in stretched horizon background where the boundary spatial coordinate is compactified.

Unit Groups of a Finite Extension of Galois Rings

Let $R_{o}$ be a Galois ring. It is well known that every element of $R_{o}$ is either a zero divisor or a unit. Galois rings are building blocks of completely primary finite rings which have yielded interesting results towards classification of finite rings. Recent studies have revealed that every finite commutative ring is a direct sum of completely primary finite rings. In fact, extensive account of finite rings have been given in the recent past. However, the classification of finite rings into well known structures still remains an open problem. For instance, the structure of the group of units of $R_{o}$ is known and some results have been obtained on the structure of its zero divisor graphs. In this paper, we construct a finite extension of $R_{o}$ (a special class of completely primary finite rings) and classify its group of units for all the characteristics.

Unit Groups of a Finite Extension of Galois Rings

Let $R_{o}$ be a Galois ring. It is well known that every element of $R_{o}$ is either a zero divisor or a unit. Galois rings are building blocks of completely primary finite rings which have yielded interesting results towards classification of finite rings. Recent studies have revealed that every finite commutative ring is a direct sum of completely primary finite rings. In fact, extensive account of finite rings have been given in the recent past. However, the classification of finite rings into well known structures still remains an open problem. For instance, the structure of the group of units of $R_{o}$ is known and some results have been obtained on the structure of its zero divisor graphs. In this paper, we construct a finite extension of $R_{o}$ (a special class of completely primary finite rings) and classify its group of units for all the characteristics.

Master curves for unidirectional flows of FENE-P fluids in rectilinear and curvilinear geometries

We demonstrate that velocity profiles for steady, unidirectional shear flows of the FENE-P (Finitely-Extensible Nonlinear Elastic, with Peterlin closure) fluid, undergoing canonical rectilinear (pressure-driven flow in a rectangular channel or a circular pipe) or curvilinear (in Taylor–Couette or Dean configurations) flows, obey universal master curves that are a function only of the ratio Wi/L , for a fixed solvent to solution viscosity parameter β. Here, Wi is the Weissenberg number defined as the product of the dumbbell relaxation time and an appropriate shear rate, while L is the ratio of the maximum extension of the polymer to its equilibrium root-mean-square end-to-end distance. The data collapse and the resulting master curves for the velocity profile is a generalization of the recent demonstration of master curves for polymer viscosity and first normal stress coefficient for a FENE-P fluid under steady simple shear flow (Yamani and McKinley, 2023). For pressure-driven channel and pipe flows, we derive simple analytical expressions for the velocity profiles, in the high shear-rate regime of Wi/L≫1, that readily elucidate the role of finite extensibility of the polymer on the velocity profiles. In the Wi/L≫1 regime, for all the flows considered, the limit of zero solvent (β=0) is shown to be singular, owing to the absence of a high-shear plateau in the total solution viscosity, resulting in very different velocity profiles for β=0 and β→0.

Classification of the types for which every Hopf–Galois correspondence is bijective

Let L/K be any finite Galois extension with Galois group G. It is known by Chase and Sweedler that the Hopf–Galois correspondence is injective for every Hopf–Galois structure on L/K, but it need not be bijective in general. Hopf–Galois structures are known to be related to skew braces, and recently, the first-named author and Trappeniers proposed a new version of this connection with the property that the intermediate fields of L/K in the image of the Hopf–Galois correspondence are in bijection with the left ideals of the associated skew brace. As an application, they classified the groups G for which the Hopf–Galois correspondence is bijective for every Hopf–Galois structure on any G-Galois extension. In this paper, using a similar approach, we shall classify the groups N for which the Hopf–Galois correspondence is bijective for every Hopf–Galois structure of type N on any Galois extension.

The Failure of Galois Descent for p-Selmer Groups of Elliptic Curves

Abstract We show that if F is $\mathbb{Q}$ or a multiquadratic number field, $p\in\left\{{2,3,5}\right\}$ , and $K/F$ is a Galois extension of degree a power of p, then for elliptic curves $E/\mathbb{Q}$ ordered by height, the average dimension of the p-Selmer groups of $E/K$ is bounded. In particular, this provides a bound for the average K-rank of elliptic curves $E/\mathbb{Q}$ for such K. Additionally, we give bounds for certain representation–theoretic invariants of Mordell–Weil groups over Galois extensions of such F. The central result is that: for each finite Galois extension $K/F$ of number fields and prime number p, as $E/\mathbb{Q}$ varies, the difference in dimension between the Galois fixed space in the p-Selmer group of $E/K$ and the p-Selmer group of $E/F$ has bounded average.

On a Galois property of fields generated by the torsion of an abelian variety

AbstractIn this article, we study a certain Galois property of subextensions of , the minimal field of definition of all torsion points of an abelian variety defined over a number field . Concretely, we show that each subfield of that is Galois over (of possibly infinite degree) and whose Galois group has finite exponent is contained in an abelian extension of some finite extension of . As an immediate corollary of this result and a theorem of Bombieri and Zannier, we deduce that each such field has the Northcott property, that is, does not contain any infinite set of algebraic numbers of bounded height.

Infinite locally finite groups groups with the given properties of the norm of Abelian non-cyclic subgroups

In the paper, the properties of infinite locally finite groups with non-Dedekind locally nil\-potent norms of Abelian non-cyclic subgroups are studied. It is proved that such groups are finite extensions of a quasicyclic subgroup and contain Abelian non-cyclic $p$-subgroups for a unique prime $p$. In particular, in the paper is prove the following assertions: 1) Let $G$ be an infinite locally finite group and contain the locally nilpotent norm $N_{G}^{A}$ with the non-Hamiltonian Sylow $p$-subgroup $(N_{G}^{A})_{p}$. Then $G$ is a finite extension of a quasicyclic $p$-subgroup, all Sylow $p'$-subgroups are finite and do not contain Abelian non-cyclic subgroups. In particular, Sylow $q$-subgroups ($q$ is an odd prime, $q\in \pi(G)$, $q\neq p$) are cyclic, Sylow $2$-subgroups ($p\neq 2$) are either cyclic or finite quaternion $2$-groups (Theorem 1). 2) Let $G$ be a locally finite non-locally nilpotent group with the infinite locally nilpotent non-Dedekind norm $N_{G}^{A}$ of Abelian non-cyclic subgroups. Then $G=G_{p} \leftthreetimes H,$ where $G_{p}$ is an infinite $\overline{HA}_{p}$-group of one of the types (1)--(4) of Proposition~2 in present paper, which coincides with the Sylow $p$-subgroup of the norm $N_{G}^{A}$, $H$ is a finite group, all Abelian subgroups of which are cyclic, and $(|H|,p)=1$. Any element $h\in H$ of odd order that centralizes some Abelian non-cyclic subgroup $M\subset N_{G}^{A}$ is contained in the centralizer of the norm $N_{G}^{A}$. (Theorem 2).3) Let $G$ be an infinite locally finite non-locally nilpotent group with the finite nilpotent non-Dedekind norm $N_{G}^{A}$ of Abelian non-cyclic subgroups. Then$G=H\leftthreetimes K,$ where $H$ is a finite group, all Abelian subgroups of which are cyclic,$\left(\left|H\right|,2\right)=1$, $K$ is an infinite 2-group of one of the types (5)--(6) of Proposition~2 (in present paper). Moreover, the norm $N_{K}^{A}$ of Abelian non-cyclic subgroups of the group $K$ is finite, $K\cap N_{G}^{A}=N_{K}^{A}$ and coincides with the Sylow 2-subgroup $(N_{G}^{A})_2$ of the norm $N_{G}^{A}$ of a group $G$.Moreover, any element $h\in H$ of the centralizer of some Abelian non-cyclic subgroup $M \subset N_{G}^{A}$ is contained in the centralizer of the norm $N_{G}^{A}$. (Theorem 4).

Entropy optimization of a FENE-P viscoelastic model: a numerically guided comprehensive analysis

AbstractThe influence of polymers on entropy generation processes is substantial, particularly in the fields of fluid dynamics and rheology. The FENE-P (Finitely Extensible Nonlinear Elastic-Peterlin) model describes the polymer’s dynamics as a result of the interaction between the stretching caused by the velocity gradient and the elastic force that restores the polymer to its equilibrium position. Models such as FENE-P aid in understanding and predicting polymer flow behaviour allowing for the reduction of entropy generation by optimizing system designs. A continuum approach is employed to express the heat flux vector and polymer confirmation tensor of the model. The study investigates the complex relationship between polymer conformation, flow dynamics, and heat transfer taking into account the thermophoresis (Soret effect) and mass diffusion-thermal diffusion coupling (Dufour effect) phenomena to optimize processes by reducing entropy. This study illuminates polymer’s significance in entropy minimization, improving engineering design methodologies and applications in materials science, chemical engineering, and fluid dynamics. As result, the presence of polymers leads to a substantial decrease in the total entropy of the system. This understanding provides opportunities for enhancing heat transfer systems, thereby facilitating the development of more efficient and sustainable technology.

Transverse Oscillations and Kelvin–Helmholtz Instability in Curved Arcade Loops with Siphon Flows

The effect of plasma flow in curved arcade loops on transverse waves and oscillations is examined analytically. The model under study is a semicircular magnetic slab with finite transverse extensions and a mass flow inside, in the zero-β plasma approximation. It is found that in the quasi-perpendicular propagation limit, the model supports two fast surface modes: one with higher (FSW+) and another with lower (FSW−) frequency. For a weak flow, the frequency of the FSW+ (FSW−) increases (decreases) as the flow speed grows in both propagating and quasi-standing wave regimes. We show that the FSW+ and FSW− are subjected to the Kelvin–Helmholtz (KH) instability, and the threshold flow is greater (less than) the internal Alfvén speed for the FSW+ (FSW−). The presence of plasma flow results in modifying the period ratio P 1/2P 2 of the fundamental harmonic to the first overtone with P 1/2P 2 less (more) than 1 for the FSW+ (FSW−), and this effect degenerates in the straight waveguide limit. The sub-Alfvénic flow can prohibit resonant absorption of kink modes when the frequencies of the FSW+ and FSW− become out of the Alfvén continuum. It is also shown that in the static case and for a weak flow case, the FSW+ (FSW−) is interpreted as a vertically (horizontally) polarized kink mode, while for moderate flow, both modes have oblique polarization. We apply the developed theory to interpret the observational cases of kink oscillations in coronal loops with signatures of a siphon flow and the onset of KH instability induced by the blowout jet along a loop-shaped magnetic structure.

The probability of non-isomorphic group structures of isogenous elliptic curves in finite field extensions, II

Let E and E′ be 2-isogenous elliptic curves over Q. Following [6], we call a prime of good reduction p anomalous if E(Fp)≃E′(Fp) but E(Fp2)≄E′(Fp2). Our main result is an explicit formula for the proportion of anomalous primes for any such pair of elliptic curves. We consider both the CM case and the non-CM case.

When the Poisson Ratio of Polymer Networks and Gels Is Larger Than 0.5?

We use coarse-grained molecular dynamics simulations to study deformation of networks and gels of linear and brush strands in both linear and nonlinear deformation regimes under constant pressure conditions. The simulations show that the Poisson ratio of networks and gels could exceed 0.5 in the nonlinear deformation regime. This behavior is due to the ability of the network and gel strands to sustain large reversible deformation, which, in combination with the finite strand extensibility results in strand alignment and monomer density, increases with increasing strand elongation. We developed a nonlinear network and gel deformation model which defines conditions for the Poisson ratio to exceed 0.5. The model predictions are in good agreement with the simulation results.

An Improvement for Error-Correcting Pairs of Some Special MDS Codes

The error-correcting pair is a general algebraic decoding method for linear codes. Since every linear code is contained in an MDS linear code with the same minimum distance over some finite field extensions, we focus on MDS linear codes. Recently, He and Liao showed that for an MDS linear code [Formula: see text] with minimum distance [Formula: see text], if it has an [Formula: see text]-error-correcting pair, then the parameters of the pair have three possibilities. Moreover, for the first case, they gave a necessary condition for an MDS linear code [Formula: see text] with minimum distance [Formula: see text] to have an [Formula: see text]-error-correcting pair, and for the other two cases, they only gave some counterexamples. For the second case, in this paper, we give a necessary condition for an MDS linear code [Formula: see text] with minimum distance [Formula: see text] to have an [Formula: see text]-error-correcting pair, and then basing on the Product Singleton Bound, we prove that there are two cases for such pairs, and then give some counterexamples basing on twisted generalized Reed–Solomon codes for these cases.

Mechanical properties of composite materials with low-covered polyrotaxane and plasma-surface-modified hexagonal boron nitride

An effective method for fabricating flexible materials containing thermally conductive fillers, such as hexagonal boron nitride (hBN), utilizes plasma surface modification and polyrotaxane (PR) with movable cross-linking points. This study demonstrated that lowering the coverage rates of PR composites increased the mobility of the cross-linking points. Plasma surface modification improved the dispersion of hBN and enabled homogeneous deformation of the PR composites owing to movable cross-linking points. Studying the deformation with a model including a finite extension effect indicated that the sliding range in the low-covered PR composites was extended compared to that in the high-covered PR composites, resulting in a smaller Young’s modulus, longer extension, and higher toughness.

Torsion Groups with the Norm of pd-Subgroup of Finite Index

The authors study the relations between the properties of torsion groups and their norms of $pd$-subgroups. The norm $N_G^{pdI}$ of $pd$-subgroups of a group $G$ is the intersection of the normalizers of all its $pd$-subgroups or a group itself, if the set of such subgroups is empty in a group. The structure of the norm of $pd$-subgroups in torsion groups is described and the conditions of Dedekindness of this norm is proved (Dedekind group is a group in which all subgroups are normal). It is proved that a torsion group is a finite extension of its norm of $pd$-subgroups if and only if it is a finite extension of its center. By this fact and the structure of the norm of $pd$-subgroups, we get that any torsion group that is a finite extension of this norm is locally finite.

Numerical simulation of droplet characterized by Rolie–Poly model with finite extensibility passing through cylinder obstacles

The deformation and rupture of viscoelastic droplet passing through cylinder obstacles in a microchannel are investigated using OpenFOAM. The constitute relationship of droplet is modeled by the Rolie–Poly model with finite extensibility, and the two-phase interface is tracked by the volume of fluid method. The effects of capillary number (Ca), the distance between cylinders (l1), relaxation time ratio (ξ), Weissenberg number (Wi), etc., on droplet deformation and rupture are mainly explored. When Ca decreases, the symmetry of droplet rupture changes and three behaviors of the droplet, i.e., symmetrical rupture, asymmetrical rupture, and non-rupture, can be captured. Further research shows that the stagnation area formed between cylinders is broken with the increase in l1, where the two sub-droplets merge again. Viscoelastic droplet with a smaller relaxation time ratio ξ is more likely to extend into thin and durable filament. Especially, when ξ=0.2, the filament will break many times during the stretching process. During above-mentioned two kinds of development, the normal stress difference develops obviously at the places, where the filament breaks or the sub-droplets combine together. This may imply that the normal stress difference facilitates the rupture and coalescence of droplets. In addition, with the increase in elasticity, the normal stress difference tends to develop at the phase interface.

Normed Amenability and Bounded Cohomology over Non-Archimedean Fields

We study continuous bounded cohomology of totally disconnected locally compact groups with coefficients in a non-Archimedean valued field K \mathbb {K} . To capture the features of classical amenability that induce the vanishing of bounded cohomology with real coefficients, we start by introducing the notion of normed K \mathbb {K} -amenability, of which we prove an algebraic characterization. It implies that normed K \mathbb {K} -amenable groups are locally elliptic, and it relates an invariant, the norm of a K \mathbb {K} -amenable group, to the order of its discrete finite p p -subquotients, where p p is the characteristic of the residue field of K \mathbb {K} . Moreover, we prove a characterization of discrete normed K \mathbb {K} -amenable groups in terms of vanishing of bounded cohomology with coefficients in K \mathbb {K} . The algebraic characterization shows that normed K \mathbb {K} -amenability is a very restrictive condition, so the bounded cohomological one suggests that there should be plenty of groups with rich bounded cohomology with trivial K \mathbb {K} coefficients. We explore this intuition by studying the injectivity and surjectivity of the comparison map, for which surprisingly general statements are available. Among these, we show that if either K \mathbb {K} has positive characteristic or its residue field has characteristic 0 0 , then the comparison map is injective in all degrees. If K \mathbb {K} is a finite extension of Q p \mathbb {Q}_p , we classify unbounded and non-trivial quasimorphisms of a group and relate them to its subgroup structure. For discrete groups, we show that suitable finiteness conditions imply that the comparison map is an isomorphism; this applies in particular to finitely presented groups in degree 2 2 . A motivation as to why the comparison map is often an isomorphism, in stark contrast with the real case, is given by moving to topological spaces. We show that over a non-Archimedean field, bounded cohomology is a cohomology theory in the sense of Eilenberg–Steenrod, except for a weaker version of the additivity axiom which is however equivalent for finite disjoint unions. In particular there exists a Mayer–Vietoris sequence, the main missing piece for computing real bounded cohomology.

Descent and Étale-Brauer Obstructions for 0-Cycles

Abstract For 0-cycles on a variety over a number field, we define an analogue of the classical descent set for rational points. This leads to, among other things, a definition of the étale-Brauer obstruction set for 0-cycles. We show that all these constructions are compatible with Suslin’s singular homology of degree 0. We then transfer some tools and techniques used to study the arithmetic of rational points into the setting of 0-cycles. For example, we extend the strategy developed by Y. Liang, relating the arithmetic of rational points over finite extensions of the base field to that of 0-cycles, to torsors. We give applications of our results to study the arithmetic behaviour of 0-cycles for Enriques surfaces, torsors given by (twisted) Kummer varieties, universal torsors, and torsors under tori.

Perspective on the description of viscoelastic flows via continuum elastic dumbbell models

Non-Newtonian fluid mechanics and computational rheology widely exploit elastic dumbbell models such as Oldroyd-B and FENE-P for a continuum description of viscoelastic fluid flows. However, these constitutive equations fail to accurately capture some characteristics of realistic polymers, such as the steady extension in simple shear and extensional flows, thus questioning the ability of continuum-level modeling to predict the hydrodynamic behavior of viscoelastic fluids in more complex flows. Here, we present seven elastic dumbbell models, which include different microstructurally inspired terms, i.e., (i) the finite polymer extensibility, (ii) the conformation-dependent friction coefficient, and (iii) the conformation-dependent non-affine deformation. We provide the expressions for the steady dumbbell extension in shear and extensional flows and the corresponding viscosities for various elastic dumbbell models incorporating different microscopic features. We show the necessity of including these microscopic features in a constitutive equation to reproduce the experimentally observed polymer extension in shear and extensional flows, highlighting their potential significance in accurately modeling viscoelastic channel flow with mixed kinematics.

Reductive covers of klt varieties

In this article, we study G -covers of klt varieties, where G is a reductive group. First, we exhibit an example of a klt singularity admitting a \mathbb{P}{\mathrm{GL}}_{n}(\mathbb{K}) -cover that is not of klt type. Then, we restrict ourselves to G -quasi-torsors, a special class of G -covers that behave like G -torsors outside closed subsets of codimension two. Given a G -quasi-torsor X\rightarrow Y , where G is a finite extension of a torus \mathbb{T} , we show that X is of klt type if and only if Y is of klt type. We prove a structural theorem for \mathbb{T} -quasi-torsors over normal varieties in terms of Cox rings. As an application, we show that every sequence of \mathbb{T} -quasi-torsors over a variety with klt type singularities is eventually a sequence of \mathbb{T} -torsors. This is the torus version of a result due to Greb–Kebekus–Peternell regarding finite quasi-torsors of varieties with klt type singularities. On the contrary, we show that in any dimension there exists a sequence of finite quasi-torsors and \mathbb{T} -quasi-torsors over a klt type variety, such that infinitely many of them are not torsors. We show that every variety with klt type singularities is a quotient of a variety with canonical factorial singularities. We prove that a variety with Zariski locally toric singularities is indeed the quotient of a smooth variety by a solvable group. Finally, motivated by the work of Stibitz, we study the optimal class of singularities for which the previous results hold.