Infinite Order Research Articles

The strict superlinear order can be faster than the infinite order

The strict superlinear order can be faster than the infinite order

Dynamic asymptotic homogenization for wave propagation in magneto-electro-elastic laminated composite periodic structure

In the present work, anti-plane wave propagation with oblique incidence in a magneto-electro-elastic multi-laminated composite periodic structure with perfect contact is studied. A general asymptotic homogenization method is applied to analyze dispersion phenomena and dynamic process. On assuming the solution having a single-frequency dependency, the higher-order terms for the displacement, electric potential and magnetic potential in asymptotic expansions are studied. Higher order local problems up to a finite order and a generalization of the local problem of infinite order are derived through rigorous calculations satisfying the necessary and sufficient condition for the existence of 1-periodic solutions. The effective equation of motion also known as “Good” Boussinesq equation has been achieved directly which is one of the highlight of the present work. Moreover, closed-form expression of dispersion equation and closed-form solutions of first and second local problems have been obtained. The analytical results have been validated for particular elastic case. Graphical illustrations have been done for tri-laminated magneto-electro-elastic composite periodic structure and the effect of size of unit cell, angle of incidence of wave, volume fraction of the composite and magneto-electric properties on the dispersion curve and also slowness curves have been examined. The numerical results have been validated with previous work and compared with results obtained by other approximation method. The present dynamic homogenization method has been proven to provide a more dispersive system and better approximation.

The distinguishing index of graphs with infinite minimum degree

AbstractThe distinguishing index of a graph is the least number of colors necessary to obtain an edge coloring of that is preserved only by the trivial automorphism. We show that if is a connected ‐regular graph for some infinite cardinal then , proving a conjecture of Lehner, Pilśniak, and Stawiski. We also show that if is a graph with infinite minimum degree and at most vertices of degree for every infinite cardinal , then . In particular, if has infinite minimum degree and order at most .

Factorizing the Rado graph and infinite complete graphs

Let $\mathcal{F}=\{F_{\alpha}: \alpha\in \mathcal{A}\}$ be a family of infinite graphs, together with $\Lambda$. The Factorization Problem $FP(\mathcal{F}, \Lambda)$ asks whether $\mathcal{F}$ can be realized as a factorization of $\Lambda$, namely, whether there is a factorization $\mathcal{G}=\{\Gamma_{\alpha}: \alpha\in \mathcal{A}\}$ of $\Lambda$ such that each $\Gamma_{\alpha}$ is a copy of $F_{\alpha}$.We study this problem when $\Lambda$ is either the Rado graph $R$ or the complete graph $K_\aleph$ of infinite order $\aleph$. When $\mathcal{F}$ is a countable family, we show that $FP(\mathcal{F}, R)$ is solvable if and only if each graph in $\mathcal{F}$ has no finite dominating set. We also prove that $FP(\mathcal{F}, K_\aleph)$ admits a solution whenever the cardinality of $\mathcal{F}$ coincides with the order and the domination numbers of its graphs.For countable complete graphs, we show some non existence results when the domination numbers of the graphs in $\mathcal{F}$ are finite. More precisely, we show that there is no factorization of $K_\N$ into copies of a $k$-star (that is, the vertex disjoint union of $k$ countable stars) when $k=1,2$, whereas it exists when $k\geq 4$, leaving the problem open for $k=3$.Finally, we determine sufficient conditions for the graphs of a decomposition to be arranged into resolution classes.

On k-geodetic graphs and groups

We call a graph k-geodetic, for some [Formula: see text], if it is connected and between any two vertices there are at most k geodesics. It is shown that any hyperbolic group with a k-geodetic Cayley graph is virtually-free. Furthermore, in such a group the centralizer of any infinite order element is an infinite cyclic group. These results were known previously only in the case that [Formula: see text]. A key tool used to develop the theorem is a new graph theoretic result concerning “ladder-like structures” in a k-geodetic graph.

Non-existence theorems on infinite order corks

Suppose that X,X′ are simply-connected closed exotic 4-manifolds. It is well-known that X′ is obtained by an order 2 cork twist of X. We give an infinite family of exotic 4-manifolds not generated by any infinite order cork. This is the first example admitting such a condition. We prove a necessary condition of 4-dimensional Ozsváth-Szabó invariants for a family to be generated by an infinite order cork and as an application give non-contractible relatively exotic 4-manifolds that are never induced by any cork. Furthermore, we give an estimate of the number of Ozsváth-Szabó invariants of 4-manifolds generated by a cork.

A Liouville type result and quantization effects on the system $-\Delta u = u J'(1-|u|^{2})$ for a potential convex near zero

We consider a Ginzburg-Landau type equation in $\mathbb R^2$ of the form $-\Delta u = u J'(1-|u|^{2})$ with a potential function $J$ satisfying weak conditions allowing for example a zero of infinite order in the origin. We extend in this context the results concerning quantization of finite potential solutions of H. Brezis, F. Merle, T. Rivière from [10] who treat the case when $J$ behaves polinomially near $0,$ as well as a result of Th. Cazenave, found in the same reference, and concerning the form of finite energy solutions.

On an interpolation sequence for a weighted Bergman space on a Hilbert unit ball

The purpose is to provide a generalization of Carleson's Theorem on interpolating sequences when dealing with a sequence in the open unit ball of a Hilbert space. Precisely, we interpolate a sequence by a function belonging to a weighted Bergman space of infinite order on a unit Hilbert ball and we furnish explicitly the upper bound corresponding to the interpolation constant.

Full view on the dynamics of an impurity coupled to two one-dimensional baths

We consider a model for the motion of an impurity interacting with two parallel, one-dimensional baths, described as two Tomonaga-Luttinger liquid systems. The impurity is able to move along the baths, and to jump from one to the other. We provide a perturbative expression for the evolution of the system when the impurity is injected in one of the baths, with a given wave packet. We obtain an approximation formally of infinite order in the impurity-bath coupling, which allows us to reproduce the orthogonality catastrophe. We monitor and discuss the dynamics of the impurity and its effect on the baths, in particular for a Gaussian wave packet. Besides the motion of the impurity, we also analyze the dynamics of the bath density and momentum density (i.e., the particle current), and show that it fits an intuitive semiclassical interpretation. We also quantify the correlation that is established between the baths by calculating the interbath, equal-time spatial correlation functions of both bath density and momentum, finding a complex pattern. We show that this pattern contains information on both the impurity motion and on the baths, and that these can be unveiled by taking appropriate ``slices'' of the time evolution.

Conjugator lengths in hierarchically hyperbolic groups

In this paper, we establish upper bounds on the length of the shortest conjugator between pairs of infinite order elements in a wide class of groups. We obtain a general result which applies to all hierarchically hyperbolic groups, a class which includes mapping class groups, right-angled Artin groups, Burger–Mozes-type groups, most 3-manifold groups, and many others. In this setting, we establish a linear bound on the length of the shortest conjugator for any pair of conjugate Morse elements. For a subclass of these groups, including, in particular, all virtually compact special groups, we prove a sharper result by obtaining a linear bound on the length of the shortest conjugator between a suitable power of any pair of conjugate infinite order elements.

Bijections, generalizations, and other properties of sequentially congruent partitions

Recently, Schneider and Schneider defined a new class of partitions called sequentially congruent partitions, in which each part is congruent to the next part modulo its index, and they proved two partition bijections involving these partitions. We introduce a new partition notation specific to sequentially congruent partitions which allows us to more easily study these bijections and their compositions, and we reinterpret them in terms of Young diagram transformations. We also define a generalization of sequentially congruent partitions, and we provide several new partition bijections for these generalized sequentially congruent partitions. Finally, we investigate a question of Schneider and Schneider regarding how sequentially congruent partitions fit into Andrews’ theory of partition ideals. We prove that the maximal partition ideal of sequentially congruent partitions has infinite order and is therefore not linked, and we identify its order 1 subideals.

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]].

Blown-up toric surfaces with non-polyhedral effective cone

Abstract We construct projective toric surfaces whose blow-up at a general point has a non-polyhedral pseudo-effective cone. As a consequence, we prove that the pseudo-effective cone of the Grothendieck–Knudsen moduli space M ¯ 0 , n \overline{M}_{0,n} of stable rational curves is not polyhedral for n ≥ 10 n\geq 10 . These results hold both in characteristic 0 and in characteristic 𝑝, for all primes 𝑝. Many of these toric surfaces are related to an interesting class of arithmetic threefolds that we call arithmetic elliptic pairs of infinite order. Our analysis relies on tools of arithmetic geometry and Galois representations in the spirit of the Lang–Trotter conjecture, producing toric surfaces whose blow-up at a general point has a non-polyhedral pseudo-effective cone in characteristic 0 and in characteristic 𝑝, for an infinite set of primes 𝑝 of positive density.

Interval estimation on hyper-order of meromorphic solutions of complex linear differential equations with uncertain coefficients

The authors address the complex oscillation problems of all solutions of homogenous linear differential equations with meromorphic coefficients. Sufficient conditions for estimating the growth of meromorphic solution with infinite order have been proposed based on Nevanlinna value distribution theory. Compared with the existing results, the proposed hyper-order of all meromorphic solutions with infinite order can be estimated in terms of a bounded interval which includes information of order of growth of meromorphic functions and meromorphic polynomial coefficients.

Quantum modeling, beyond secularity, of the collisional dissipation of molecular alignment using the energy-corrected sudden approximation.

We propose a Markovian quantum model for the time dependence of the pressure-induced decoherence of rotational wave packets of gas-phase molecules beyond the secular approximation. It is based on a collisional relaxation matrix constructed using the energy-corrected sudden approximation, which improves the previously proposed infinite order sudden one by taking the molecule rotation during collisions into account. The model is tested by comparisons with time-domain measurements of the pressure-induced decays of molecular-axis alignment features (revivals and echoes) for HCl and CO2 gases, pure and diluted in He. For the Markovian systems HCl-He and CO2-He, the comparisons between computed and measured data demonstrate the robustness of our approach, even when the secular approximation largely breaks down. In contrast, significant differences are obtained in the cases of pure HCl and CO2, for which the model underestimates the decay rate of the alignment at short times. This result is attributed to the non-Markovianity of HCl-HCl and CO2-CO2 interactions and the important contribution of those collisions that are ongoing at the time when the system is excited by the aligning laser pulse.

Adaptive Self-Organizing Map Using Optimal Control

The self-organizing map (SOM), which is a type of artificial neural network (ANN), was formulated as an optimal control problem. Its objective function is to minimize the mean quantization error, and the state equation is the weight updating equation of SOM. Based on the objective function and the state equations, the Hamiltonian equation based on Pontryagin’s minimum principle (PMP) was formed. This study presents two models of SOM formulated as an optimal control problem. In the first model, called SOMOC1, the design is based on the state equation representing the weight updating equation of the best matching units of the SOM nodes in each iteration, whereas in the second model, called SOMOC2, it considers the weight updating equation of all the nodes in the SOM as the state updating equation. The learning rate is treated as the control variable. Based on the solution of the switching function, a bang-bang control was applied with a high and low learning rate. The proposed SOMOC2 model performs better than the SOMOC1 model and conventional SOM as it considers all the nodes in the Hamiltonian equation, and the switching function obtained from it is influenced by all the states, which provides one costate variable for each. The costate determines the marginal cost of violating the constraint by the state equations, and the switching function is influenced by this, hence producing a greater improvement in terms of the mean quantization error at the final iteration. It was found that the solution leads to an infinite order singular arc. The possible solutions for the suitable learning rates during the singular arc period are discussed in this study.

Profinite groups with few conjugacy classes of elements of infinite order

It is proved that every finitely generated profinite group with fewer than 2^{aleph _0} conjugacy classes of elements of infinite order is finite.

Evolution of superoscillations for spinning particles

Superoscillating functions are band-limited functions that can oscillate faster than their fastest Fourier component. These functions appear in various fields of science and technology, in particular they were discovered in quantum mechanics in the context of weak values introduced by Y. Aharonov and collaborators. The evolution problem of superoscillatory functions as initial conditions for the Schrödinger equation is intensively studied nowadays and the supershift property of the solution of Schrödinger equation encodes the persistence of superoscillatory phenomenon during the evolution. In this paper, we prove that the evolution of a superoscillatory initial datum for spinning particles in a magnetic field has the supershift property. Our techniques are based on the exact propagator of spinning particles, the associated infinite order differential operators and their continuity on suitable spaces of entire functions with growth conditions.

Modern perspectives in Proof Theory.

Modern perspectives in Proof Theory.

Recursive relations and quantum eigensolver algorithms within modified Schrieffer-Wolff transformations for the Hubbard dimer

We derive recursive relations for the Schrieffer--Wolff (SW) transformation applied to the half-filled Hubbard dimer. While the standard SW transformation is set to block-diagonalize the transformed Hamiltonian solely at the first order of perturbation, we infer from recursive relations two types of modifications, variational or iterative, that approach, or even enforce for the homogeneous case, the desired block-diagonalization at infinite order of perturbation. The modified SW unitary transformations are then used to design an test quantum algorithms adapted to the noisy and fault-tolerant era. This work paves the way toward the design of alternative quantum algorithms for the general Hubbard Hamiltonian.