Finite Order Research Articles

Existence of weak solution to initial-boundary value problem for finite order Kelvin–Voigt fluid motion model

Existence of weak solution to initial-boundary value problem for finite order Kelvin–Voigt fluid motion model

Weight modules over Bell–Rogalski algebras

We study a class of Z-graded algebras introduced by Bell and Rogalski. Their construction generalizes in large part that of rank one generalized Weyl algebras (GWAs). We establish certain ring-theoretic properties of these algebras and study their connection to GWAs. We classify the simple weight modules in the infinite orbit case and provide a partial classification in the case of orbits of finite order.

Supersymmetric phases of AdS4/CFT3

We exhibit an infinite family of supersymmetric phases in the three-dimensional ABJM superconformal field theory and the dual asymptotically AdS4 gravity. They are interpreted as partially deconfined phases which generalize the confined/pure AdS phase and deconfined/supersymmetric black hole phase. Our analysis involves finding a family of saddle-points of the superconformal index labelled by rational points (equivalently, roots of unity), separately in the bulk and boundary theories. In the ABJM theory we calculate the free energy of each saddle by the large-N asymptotic expansion of the superconformal index to all orders in perturbation theory near the saddle-point. We find that this expansion terminates at finite order. In the gravitational theory we show that there is a corresponding family of solutions, constructed by orbifolding the eleven-dimensional uplift of the supersymmetric black hole. The on-shell gravitational action of each orbifold agrees with the free energy of the corresponding saddle in the SCFT. We find that there are two saddles in the ABJM theory with the same entropy as the supersymmetric black hole, corresponding to the two primitive fourth-roots of unity, which causes macroscopic oscillations in the microcanonical index.

Decomposition of Admissible Functions in Weighted Coupled Cell Networks

This work makes explicit the degrees of freedom involved in modeling the dynamics of a network, or some other first-order property of a network, such as a measurement function. In previous work, an admissible function in a network was constructed through the evaluation of what we called oracle components. These oracle components are defined through some minimal properties that they are expected to obey. This is a high-level description in the sense that it is not clear how one could design such an object. The goal is to obtain a low-level representation of these objects by unwrapping them into their degrees of freedom. To achieve this, we introduce two decompositions. The first one is the more intuitive one and allows us to define the important concept of coupling order. The second decomposition is built on top of the first one and is valid for the class of coupling components that have finite coupling order. Despite this requirement, we show that this is still a very useful tool for designing coupling components with infinite coupling orders, through a limit approach.

Model structures on finite total orders

We initiate the study of model structures on (categories induced by) lattice posets, a subject we dub homotopical combinatorics. In the case of a finite total order [n], we enumerate all model structures, exhibiting a rich combinatorial structure encoded by Shapiro’s Catalan triangle. This is an application of previous work of the authors on the theory of N_infty -operads for cyclic groups of prime power order, along with new structural insights concerning extending choices of certain model structures on subcategories of [n].

Automorphisms of -free and other Toeplitz shifts

AbstractWe present sufficient conditions for the triviality of the automorphism group of regular Toeplitz subshifts and give a broad class of examples from the class of ${\mathcal B}$ -free subshifts satisfying them, extending the work of Dymek [Automorphisms of Toeplitz ${\mathcal B}$ -free systems. Bull. Pol. Acad. Sci. Math.65(2) (2017), 139–152]. Additionally, we provide an example of a ${\mathcal B}$ -free Toeplitz subshift whose automorphism group has elements of arbitrarily large finite order, answering Question 11 of S. Ferenczi et al [Sarnak’s conjecture: what’s new. Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics (Lecture Notes in Mathematics, 2213). Eds. S. Ferenczi, J. Kułaga-Przymus and M. Lemańczyk. Springer, Cham, 2018, pp. 163–235].

Virtual braid groups, virtual twin groups and crystallographic groups

Let n≥2. Let VBn (resp. VPn) be the virtual braid group (resp. the pure virtual braid group), and let VTn (resp. PVTn) be the virtual twin group (resp. the pure virtual twin group). Let Π be one of the following quotients: VBn/Γ2(VPn) or VTn/Γ2(PVTn) where Γ2(H) is the commutator subgroup of H. In this paper, we show that Π is a crystallographic group and we characterize the elements of finite order and the conjugacy classes of elements in Π. Furthermore, we realize explicitly some Bieberbach groups and infinite virtually cyclic groups in Π. Finally, we also study other braid-like groups (welded, unrestricted, flat virtual, flat welded and Gauss virtual braid group) modulo the respective commutator subgroup in each case.

Weak solvability of the initial-boundary value problem for inhomogeneous incompressible Kelvin–Voigt fluid motion model of arbitrary finite order

Weak solvability of the initial-boundary value problem for inhomogeneous incompressible Kelvin–Voigt fluid motion model of arbitrary finite order

Plaquette valence bond state in the spin- 12 J1−J2 XY model on the square lattice

We studied the ground-state phase diagram of spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2} XY$ model on the square lattice with first-neighbor ${J}_{1}$ and second-neighbor ${J}_{2}$ antiferromagnetic interactions using both infinite density matrix renormalization group (iDMRG) and DMRG approaches. We show that a plaquette valence bond phase is realized in an intermediate region $0.50\ensuremath{\le}{J}_{2}/{J}_{1}<0.54$ between a N\'eel magnetic ordered phase at ${J}_{2}/{J}_{1}<0.50$ and a stripy magnetic ordered phase at ${J}_{2}/{J}_{1}\ensuremath{\ge}0.54$. The plaquette valence bond phase is characterized by finite dimer orders in both the horizontal and vertical directions. Contrary to the spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model, we do not find numerical evidence for a quantum spin liquid phase in the ${J}_{1}\text{\ensuremath{-}}{J}_{2} XY$ model.

On the type of the meromorphic function of finite order

Let $f(z)$ be a meromorphic function on the complex plane of finite order $\rho>0$. Let $\rho(r)$ be a proximate order in the sense of Boutroux such that $\limsup\limits_{r\to\infty}\rho(r)=\rho$, $\liminf\limits_{r\to\infty}\rho(r)=\alpha>0$. If $[\alpha]<\alpha\leqslant\rho<[\alpha]+1$ then the types of $T(r,f)$ and $|N|(r,f)$ coincide with respect to $\rho(r)$. If there are integers between $\alpha$ and $\rho$, then the resulting criterion is formulated in terms of the upper density of zeros and poles of the function $f$ and their argument symmetry.

Critical and injective modules over skew polynomial rings

Let R be a commutative local k-algebra of Krull dimension one, where k is a field. Let α be a k-algebra automorphism of R, and define S to be the skew polynomial algebra R[θ;α]. We offer, under some additional assumptions on R, a criterion for S to have injective hulls of all simple S-modules locally Artinian - that is, for S to satisfy property (⋄). It is easy and well known that if α is of finite order, then S has this property, but in order to get the criterion when α has infinite order we found it necessary to classify all cyclic (Krull) critical S-modules in this case, a result which may be of independent interest. With the help of the above we show that Sˆ=k[[X]][θ,α] satisfies (⋄) for all k-algebra automorphisms α of k[[X]].

∇-prime rings and their commutativity

Consider a ring with an (anti)-automorphism ∇ of finite order. The fundamental aim of this manuscript is to introduce the notions of ∇-(semi)prime ideal and ∇-(semi)prime ring as a generalization of the notions of (semi)prime ideal, ∗ -(semi)prime ideal, (semi)prime ring and ∗ -(semi)prime ring. Furthermore, we will investigate their basic properties and study the commutative property of ∇-prime rings which satisfy certain central differential identities on ∇-ideals.

On the trace of the wave group and regularity of potentials

For the wave equation $\partial_t^2-\Delta+V$ on $\mathbb{R}^d$ with compactly supported, real valued potential $V$, we establish a sharp relation between Sobolev regularity of $V$ and the existence of finite order expansions as $t\rightarrow 0$ for the relative trace of the wave group.

Fisher information of correlated stochastic processes

Many real-world tasks include some kind of parameter estimation, i.e. the determination of a parameter encoded in a probability distribution. Often, such probability distributions arise from stochastic processes. For a stationary stochastic process with temporal correlations, the random variables that constitute it are identically distributed but not independent. This is the case, for instance, for quantum continuous measurements. In this article, we derive the asymptotic Fisher information rate for a stationary process with finite Markov order. We give a precise expression for this rate which is determined by the process’ conditional distribution up to its Markov order. Second, we demonstrate with suitable examples that correlations may both enhance or hamper the metrological precision. Indeed, unlike for entropic information quantities, in general nothing can be said about the sub- or super-additivity of the joint Fisher information in the presence of correlations. To illustrate our results, we apply them to thermometry on an Ising spin chain, considering nearest-neighbour and next-to-nearest neighbour coupling. In this case, the asymptotic Fisher information rate is directly connected to the specific heat capacity of the spin chain. We observe that the presence of correlations strongly enhances the estimation precision in an anti-ferromagnetic chain, while in a ferromagnetic chain this is not the case.

An alternative justification for the stationary assumption made by many reduced models for nonlocal electron heat flow in plasmas

Nonlocal models are widely used for approximating kinetic effects on electron heat flow in fusion-relevant plasmas. Almost universally, such models have no explicit time dependence and are designed to make heat flow predictions based directly on instantaneous profiles of macroscopic plasma parameters. While this is usually justified by the claim that transient effects fade before temperature profiles evolve appreciably, a more rigorous justification of the stationarity assumption in terms of kinetic theory is desirable. In this Letter, such a justification is provided by demonstrating that nonstationary effects related to the time dependence of the isotropic part of the electron distribution function vanish up to third order in Chapman–Enskog theory (irrespective of ion charge state or presence of magnetic fields). However, it is found that the electron inertia term (whose appearance in Ohm's law stems from the time derivative of the anisotropic part of the electron distribution function) does have a small but finite third order effect that is most prominent for plasmas with low average ion charges. This Letter additionally provides a convenient analytic inverse for the isotropic part of the Landau electron–electron collision operator.

Extending the predictive power of perturbative QCD using the principle of maximum conformality and the Bayesian analysis

In addition to the evaluation of high-order loop contributions, the precision and predictive power of perturbative QCD (pQCD) predictions depends on two important issues: (1) how to achieve a reliable, convergent fixed-order series, and (2) how to reliably estimate the contributions of unknown higher-order terms. The recursive use of renormalization group equation, together with the Principle of Maximum Conformality (PMC), eliminates the renormalization scheme-and-scale ambiguities of the conventional pQCD series. The result is a conformal, scale-invariant series of finite order which also satisfies all of the principles of the renormalization group. In this paper we propose a novel Bayesian-based approach to estimate the size of the unknown higher order contributions based on an optimized analysis of probability distributions. We show that by using the PMC conformal series, in combination with the Bayesian analysis, one can consistently achieve high degree of reliability estimates for the unknown high order terms. Thus the predictive power of pQCD can be greatly improved. We illustrate this procedure for two pQCD observables: R_{e^+e^-} and R_tau , which are each known up to four loops in pQCD. Numerical analyses confirm that by using the scale-independent and more convergent PMC conformal series, one can achieve reliable Bayesian probability estimates for the unknown higher-order contributions.

Nonlinear dynamics of a vertical pipe subjected to a two-phase, solid-liquid internal flow

In this paper, a numerical study is carried out on a simply supported pipe model which conveys a two-phase, solid-liquid flow. First, a system of nonlinear, partial differential equations of dimensionless, two-phase, solid-liquid flow pipeline is established. Then, the above mentioned partial differential equation system is discretized by the Galerkin method of finite order to obtain a system of ordinary differential equations. Finally, the natural frequency characteristics of the corresponding linear system are solved to approximate the stability range of the nonlinear system, the structural response characteristics of which are solved by the Newton-Raphson method. The dimensionless aspect ratio, the dimensionless liquid-phase volume, and the dimensionless gravity coefficient are chosen to analyze the impact on the critical velocity of buckling and structural nonlinear response characteristic in this research. The research results indicate that a change to the above parameters will cause the critical speed to change in a linear or non-linear form. Moreover, at subcritical speeds, the vibrations are damped; at critical speeds, steady-state vibrations with extremely small amplitudes with a single frequency are seen; and at supercritical speeds, the pipeline model will produce buckling in different directions according to initial values.

Value Sharing Results for Generalized Shifts of Entire Functions

In this paper, we investigate the uniqueness problem of generalized shift polynomials of finite order entire functions sharing a small function under the notion of weakly and relaxed weighted sharing environment. Some examples have been exhibited which are relevant to the content of the paper.

Unicity on Entire Functions Concerning Their Difference Operators and Derivatives

In this paper we study the uniqueness of entire functions concerning their difference operator and derivatives. The idea of entire and meromorphic functions relies heavily on this direction. Rubel and Yang considered the uniqueness of entire function and its derivative and proved that if $f(z)$ and $f'(z)$ share two values $a,b$ counting multilicities then $f(z)\equiv f'(z)$. Later, Li Ping and Yang improved the result given by Rubel and Yang and proved that if $f(z)$ is a non-constant entire function and $a,b$ are two finite distinct complex values and if $f(z)$ and $f^{(k)}(z)$ share $a$ counting multiplicities and $b$ ignoring multiplicities then $f(z)\equiv f^{(k)}(z)$. In recent years, the value distribution of meromorphic functions of finite order with respect to difference analogue has become a subject of interest. By replacing finite distinct complex values by polynomials, we prove the following result: Let $\Delta f(z)$ be trancendental entire functions of finite order, $ k \geq 0$ be integer and $P_{1}$ and $P_{2}$ be two polynomials. If $\Delta f(z)$ and $f^{(k)}$ share $P_{1}$ CM and share $P_{2}$ IM, then $\Delta f \equiv f^{(k)}$. A non-trivial proof of this result uses Nevanlinna's value distribution theory.

Riesz summability on boundary lines of holomorphic functions of finite order generated by Dirichlet series

Riesz summability on boundary lines of holomorphic functions of finite order generated by Dirichlet series