Linear Order Research Articles

ON COHESIVE POWERS OF LINEAR ORDERS

AbstractCohesive powersof computable structures are effective analogs of ultrapowers, where cohesive sets play the role of ultrafilters. Let$\omega $,$\zeta $, and$\eta $denote the respective order-types of the natural numbers, the integers, and the rationals when thought of as linear orders. We investigate the cohesive powers of computable linear orders, with special emphasis on computable copies of$\omega $. If$\mathcal {L}$is a computable copy of$\omega $that is computably isomorphic to the usual presentation of$\omega $, then every cohesive power of$\mathcal {L}$has order-type$\omega + \zeta \eta $. However, there are computable copies of$\omega $, necessarily not computably isomorphic to the usual presentation, having cohesive powers not elementarily equivalent to$\omega + \zeta \eta $. For example, we show that there is a computable copy of$\omega $with a cohesive power of order-type$\omega + \eta $. Our most general result is that if$X \subseteq \mathbb {N} \setminus \{0\}$is a Boolean combination of$\Sigma _2$sets, thought of as a set of finite order-types, then there is a computable copy of$\omega $with a cohesive power of order-type$\omega + \boldsymbol {\sigma }(X \cup \{\omega + \zeta \eta + \omega ^*\})$, where$\boldsymbol {\sigma }(X \cup \{\omega + \zeta \eta + \omega ^*\})$denotes the shuffle of the order-types inXand the order-type$\omega + \zeta \eta + \omega ^*$. Furthermore, ifXis finite and non-empty, then there is a computable copy of$\omega $with a cohesive power of order-type$\omega + \boldsymbol {\sigma }(X)$.

Wiener–Hopf Difference Equations and Semi-Cardinal Interpolation with Integrable Convolution Kernels

Let Hsubset {mathbb {Z}}^d be a half-space lattice, defined either relative to a fixed coordinate (e.g. H = {mathbb {Z}}^{d-1}!times !{mathbb {Z}}_+), or relative to a linear order preceq on {mathbb {Z}}^d, i.e. H = {jin {mathbb {Z}}^d: 0preceq j}. We consider the problem of interpolation at the points of H from the space of series expansions in terms of the H-shifts of a decaying kernel phi . Using the Wiener–Hopf factorization of the symbol for cardinal interpolation with phi on {mathbb {Z}}^d, we derive some essential properties of semi-cardinal interpolation on H, such as existence and uniqueness, Lagrange series representation, variational characterization, and convergence to cardinal interpolation. Our main results prove that specific algebraic or exponential decay of the kernel phi is transferred to the Lagrange functions for interpolation on H, as in the case of cardinal interpolation. These results are shown to apply to a variety of examples, including the Gaussian, Matérn, generalized inverse multiquadric, box-spline, and polyharmonic B-spline kernels.

Spinning (A)dS black holes with slow-rotation approximation in dynamical Chern-Simons modified gravity

One of the most crucial areas of gravity research, after the direct observation of gravitational waves, is the possible modification of General Relativity at ultraviolet and infrared scales. In particular, the possibility of parity violation should be considered in strong field regime. The Chern-Simons gravity takes into account parity violation in strong gravity regime. For all conformally flat spacetimes and spacetimes with a maximally symmetric two-dimensional subspace, Chern-Simons gravity is identical to General Relativity. Specifically, the Anti-de Sitter (A)dS-Kerr/Kerr black hole is not a solution for Chern-Simons gravity. The slow-rotating BH and the quadratic order in spin solutions are some of the known solutions to quadratic order in spin and they are rotating solutions in the frame of dynamical Chern-Simons gravity. In the present study, for the (A)dS slow-rotating situation (correct to the first order in spin), we derive the linear perturbation equations controlling the metric and the dynamical Chern-Simons field equation corrected to the linear order in spin and to the second order in the Chern-Simons coupling parameter. We show that the black hole of the (A)dS-Kerr solution is stronger (i.e. more compact and energetic) than the Kerr black hole solution and the reason for this feature comes form contributions at Planck scales. Moreover, we calculate the thermodynamical quantities related to this black hole. Finally, we calculate the geodesic equation and derive the effective potential of the black hole.

Non-Hermitian Hamiltonians for linear and nonlinear optical response: A model for plexcitons.

In polaritons, the properties of matter are modified by mixing the molecular transitions with light modes inside a cavity. Resultant hybrid light-matter states exhibit energy level shifts, are delocalized over many molecular units, and have a different excited-state potential energy landscape, which leads to modified exciton dynamics. Previously, non-Hermitian Hamiltonians have been derived to describe the excited states of molecules coupled to surface plasmons (i.e., plexcitons), and these operators have been successfully used in the description of linear and third order optical response. In this article, we rigorously derive non-Hermitian Hamiltonians in the response function formalism of nonlinear spectroscopy by means of Feshbach operators and apply them to explore spectroscopic signatures of plexcitons. In particular, we analyze the optical response below and above the exceptional point that arises for matching transition energies for plasmon and molecular components and study their decomposition using double-sided Feynman diagrams. We find a clear distinction between interference and Rabi splitting in linear spectroscopy and a qualitative change in the symmetry of the line shape of the nonlinear signal when crossing the exceptional point. This change corresponds to one in the symmetry of the eigenvalues of the Hamiltonian. Our work presents an approach for simulating the optical response of sublevels within an electronic system and opens new applications of nonlinear spectroscopy to examine the different regimes of the spectrum of non-Hermitian Hamiltonians.

Scalable growth models for time‐series multispectral images

AbstractVegetation indices (VIs) are produced as a combination of different reflectance bands that are captured by multispectral images (MSIs). These indices, such as normalized difference vegetation index (NDVI), are reported to be proxy indicators of photosynthetic activity, plant canopy biomass, and leaf area index. To determine the utility of using VI derived from MSI to model plant growth, random regression (RR) models with linear splines and different orders of Legendre polynomials were applied to data collected (years 2019 and 2020) as part of the Genome‐to‐Fields initiative. Growth curves of maize (Zea mays L.) hybrids were modeled using both NDVI and cumulative NDVI (cNDVI) phenotypes. Due to the difference in MSI recording dates, and sparse overlap in hybrids between years, all the analyses were nested within a year. Results indicate that RR models using Legendre polynomials provide a robust and scalable method for modeling growth curves using phenotypes extracted from MSI; however, RR models using linear splines showed inconsistent convergence. Growth curves estimated using NDVI and cNDVI showed low‐to‐moderate heritability (0.11–0.44) and a range of genetic correlations (−0.15 to 0.97) with grain yield. This study demonstrates the utility of MSI for modeling genetic growth trends, with the best modeling results obtained when using Legendre polynomials and cNDVI.

Spin-Electric Coupling in Lead Halide Perovskites.

Lead halide perovskites enjoy a number of remarkable optoelectronic properties. To explain their origin, it is necessary to study how electromagnetic fields interact with these systems. We address this problem here by studying two classical quantities: Faraday rotation and the complex refractive index in a paradigmatic perovskite CH_{3}NH_{3}PbBr_{3} in a broad wavelength range. We find that the minimal coupling of electromagnetic fields to the k·p Hamiltonian is insufficient to describe the observed data even on the qualitative level. To amend this, we demonstrate that there exists a relevant atomic-level coupling between electromagnetic fields and the spin degree of freedom. This spin-electric coupling allows for quantitative description of a number of previous as well as present experimental data. In particular, we use it here to show that the Faraday effect in lead halide perovskites is dominated by the Zeeman splitting of the energy levels and has a substantial beyond-Becquerel contribution. Finally, we present general symmetry-based phenomenological arguments that in the low-energy limit our effective model includes all basis coupling terms to the electromagnetic field in the linear order.

Non-integrability of strings in AdS_{6}times S^{2}times Sigma background and its 5D holographic duals

In this manuscript we study Liouvillian non-integrability of strings in AdS_{6}times S^{2}times Sigma background. We consider soliton strings and look for simple solutions in order to reduce the equations to only one linear second order differential equation called Normal Variation Equation (NVE). We study truncations in eta and sigma variables showing their applicability or not to catch (non) integrability of models. With this technique we are able to study many recent cases considered in the literature: the abelian and non-abelian T-duals, the (p, q)-5-brane system, the T_{N}, +_{MN} theories and the {tilde{T}}_{N,P} and +_{P,N} quivers. We show that all of them are not integrable. Finally, we consider the general case at the boundary sigma =sigma _0 for large sigma _0 and show that we can get general conclusions about integrability. For example, beyond the above quivers, we show generically that long quivers are not integrable. In order to establish the results, we numerically study the string dynamical system seeking by chaotic behaviour. Such a characteristic gives one more piece of evidence for non-integrability of the background studied.

Numerical simulation of fluid flow with a fast Boundary‐Domain Integral Method

AbstractFor the simulation of fluid flow the velocity‐vorticity formulation of the Navier‐Stokes equations can be used. Approximating the first order time derivative in these equations by finite differences, essentially, the Yukawa equation for the velocity has to be solved with an inhomogeneity governed by the vorticity. Applying the Boundary Element Method to such problems is not straightforward. However, the Boundary‐Domain Integral Method (BDIM) can be used, which requires to solve beside the boundary integrals as well a domain integral due to the inhomogeneity. Classical BEM formulations have a quadratic complexity with respect to the unknowns on the boundary and, for the BDIM additionally, for the unknowns in the domain. This prevents the BDIM from being applied to real‐world problems.To reduce the quadratic complexity in BEM, approximation methods like the fast multipole method, ℋ‐matrices in combination with the adaptive cross approximation or the ℋ2 ‐matrix technique can be used. All these methods reduce the complexity to an almost linear order. Here, the ℋ2 ‐matrix technique is applied to reduce storage and computing time for the right hand side, i.e., for the matrix of the domain integrals. Based on interpolatory kernel expansions and an heuristic criterium for too small matrix entries an almost linear complexity can be achieved. The numerical studies focus on a lid‐driven cavity. The complexity reduction is demonstrated.

Numerical Solution of Linear and Nonlinear Second Order Initial Value Problems Using Three-Step Generalized Off-Step Hybrid Block Method

Numerical Solution of Linear and Nonlinear Second Order Initial Value Problems Using Three-Step Generalized Off-Step Hybrid Block Method

Analyzing and Modeling the Spatial-Temporal Changes and the Impact of GLOTI Index on Precipitation in the Marmara Region of Türkiye

Precipitation is a particularly important part of the Earth’s hydrological cycle and, therefore, is a necessary variable for maintaining natural balance. This study investigated past, present, and future changes in precipitation in the Marmara region, and examined the effects of global warming on this variable. The study period was from 1960 to 2020, and the climate data of 15 synoptic stations in the Marmara region were used for this purpose. To achieve the objectives of the study, linear and 6th order polynomial regression, ombrothermic and hythergraph diagrams, geostatistical models, Mann-Kendall test, Pearson correlation, standard Z-scores, and multi-layer perceptron artificial neural network models (MLP-ANN) were used to model and predict precipitation. The results of the linear regression analysis showed that of the 15 stations, 6 stations had an increasing trend, 6 stations had a trendless pattern, and 3 stations had a decreasing trend. In terms of periodic analysis, the main downward trend started in 1964 and continued until 1992, while the main periodic upward trend started in 1992 and continued until 2016. The synoptic stations in the Marmara region showed a lack of precipitation over six to seven months of the year, and the precipitation changes in the region were stronger than the temperature changes. In addition, the highest precipitation was observed on the southeast coast of the Black Sea, and the lowest precipitation was observed in the eastern parts of the region. Moreover, except for the Bilecik and Kocaeli stations, the changes in the long-term trend of precipitation at the other stations were significant. Among the 15 stations, only the Kocaeli and Sarıyer stations showed a positive correlation with global temperature during the annual period. In addition, the developed ANN model was accurate in simulating and predicting precipitation and showed an upward trend over the next seven years.

Doubly-Iterative Sparsified MMSE Turbo Equalization for OTFS Modulation

Currently, orthogonal time frequency space (OTFS) modulation has drawn much attention to reliable communications in high-mobility scenarios. This paper proposes a doubly-iterative sparsified minimum mean square error (DI-S-MMSE) turbo equalizer, which iteratively exchanges the extrinsic information between a soft-input-soft-input (SISO) MMSE estimator and a SISO decoder. Our proposed equalizer does not suffer from short loops and approaches the performance of the near-optimal symbol-wise maximum a posteriori (MAP) algorithm. To exploit the inherent sparsity of OTFS system, we resort to graph theory to investigate the sparsity pattern of the channel matrix, and propose two sparsification guidelines to reduce the complexity of calculating the matrix inverse at the MMSE estimator. Then, we apply two iterative algorithms to MMSE estimation, i.e., the Generalized Minimal Residual (GMRES) and Factorized Sparse Approximate Inverse (FSPAI) algorithms. The former is used at the initial turbo iteration, whose global convergence is proven in our equalizer, while the latter is used at the subsequent turbo iterations with the help of our proposed guidelines. Simulation results demonstrate that our equalizer can achieve a good bit-error-rate (BER) performance regardless of the fractional Doppler, and only has a linear order of complexity.

Conjunction Agreement and the Coordinate Structure Constraint

A long-standing question in syntax is what role linear order plays in a hierarchical grammar. Phenomena that on the surface show sensitivity to linear order have been particularly illuminating. When agreeing with coordinated subjects with different gender features, participles in Bosnian-Croatian-Serbian (BCS), Slovenian, and other languages allow multiple options including agreeing with the linearly closest conjunct. This pattern motivated an analysis where linear order can play a role in a syntactic operation such as agreement (Marušič et al. 2015; Willer-Gold et al. 2016 among others). On the other hand, Murphy & Puškar (2018) show that the pattern can be accounted for without resorting to linear order. This paper provides novel evidence from Coordinate Structure Constraint violating movement in BCS to argue for the non-linear approach. If the argument is on the right track, agreement can be kept within syntax without resorting to PF conditions.

Normal functions and maximal order types

AbstractTransformations of well partial orders induce functions on the ordinals, via the notion of maximal order type. In most examples from the literature, these functions are not normal, in marked contrast with the central role that normal functions play in ordinal analysis and related work from computability theory. The present paper aims to explain this phenomenon. In order to do so, we investigate a rich class of order transformations that are known as $\textsf {WPO}$-dilators. According to a first main result of this paper, $\textsf {WPO}$-dilators induce normal functions when they satisfy a rather restrictive condition, which we call strong normality. Moreover, the reverse implication holds as well, for reasonably well-behaved $\textsf {WPO}$-dilators. Strong normality also allows us to explain another phenomenon: by previous work of Freund, Rathjen and Weiermann, a uniform Kruskal theorem for $\textsf {WPO}$-dilators is as strong as $\varPi ^1_1$-comprehension, while the corresponding result for normal dilators on linear orders is equivalent to the much weaker principle of $\varPi ^1_1$-induction. As our second main result, we show that $\varPi ^1_1$-induction is equivalent to the uniform Kruskal theorem for $\textsf {WPO}$-dilators that are strongly normal.

Sufficient Condition for the Existence of an Additional Zone in Singularly Perturbated Second-Order Boundary Problem

Studies the Dirichlet, Neman and Robin boundary value problems for a singularly perturbed linear inhomogeneous second order ordinary differential equation. The considered boundary value problems have three features: the singular presence of a small parameter; the solution of the corresponding unperturbed equation has a k order pole and an additional boundary layer. The singular presence of a small parameter generates the classical boundary layer, and the singular point of the corresponding unperturbed equation generates the second boundary layer. As a result, we get a double boundary layer. A sufficient condition for the existence of an additional boundary layer is found.

On the Application of the Block Hybrid Methods to Solve Linear and Non-Linear First Order Differential Equations

Block hybrid methods with intra-step points are considered in this study. These methods are implemented to solve linear and nonlinear single and systems of first order differential equations. The stability, convergence, and accuracy of the proposed methods are qualitatively investigated through the absolute and residual error analysis in some selected cases. A number of different numerical examples are tested to demonstrate the efficiency and applicability of the proposed methods. In this study we also implement the proposed methods to solve chaotic systems such as the Glukhvsky–Dolzhansky system, producing very comparable results to those already in the literature.

Lexicographic preference representation: Intrinsic length of linear orders on infinite sets

The intrinsic length of a linear order is the minimum of all ordinals δ such that there is a binary-criteria lexicographic representation of the linear order in {0,1}δ. Assuming the Generalized Continuum Hypothesis, we show that, for each ordinal γ and infinite set X with cardinality κ, there exist a linear order on X such that γ is the intrinsic length of that linear order if and only if logκ≤γ≤κ. This intrinsic-length partition imposes a structure on the profusion of linear orders on an infinite set.

Impact resistance-compressive strength relationship of steel fiber-reinforced concrete at ambient and high temperatures

Despite the great scientific developments in fire resistance technologies and construction materials industry, fires threaten human lives and negatively impact the economy every day. One of the possible scenarios when a structure is subjected to accidental fire is the fragmenting of structural elements or falling of heavy objects from higher stories on fired concrete elements, which imposes a dual accidental loading. The aim of this research is to spotlight the relationship between the compressive strength and impact energy of concrete subjected to the dual action of high temperatures and repeated impacts. Cube and disk specimens from three mixtures with 0 %, 0.5 %, and 1.0 % of steel fibers were cast to investigate the compressive strength and repeated impact strength, respectively. Groups of specimens from each mixture were subjected to temperatures of 200 °C, 400 °C, and 600 °C in addition to the reference unheated specimens. The results showed that compressive strength and impact energy behave differently under the effect of high temperatures. However, regardless of the exposure temperature, the impact energy increases with the increase of compressive strength. This increase is more evident for failure impact energy compared to cracking impact energy. Simple linear and second order formulas were found suitable to describe these relations with R2values ranging from 0.71 to 0.96.

Post-Syntactic Linearization of Yes/No-Question Particle in Turkish

This study focuses on the linear order of the yes/no-question particle mI in Turkish. The distribution of mI is intriguing, since it must reverse the linear order with the immediately adjacent morpheme unless the morpheme immediately follows the verb root. To account for such a distribution, the linear order of mI is attributed to the interaction between two post-syntactic concatenation statements under adjacency (Local Dislocation in Embick 2007). Assuming that the question particle spells out the C[+Q] head, two ordered Local Dislocation statements account for the distribution of mI: the first statement holds between the verb root and its immediately adjacent inflectional morpheme; and the second statement is particular to T and C[+Q].

Explicitness and implicitness of discourse relations in a multilingual discourse bank

Abstract Proposals such as continuity and causality-by-default relate the level of expectedness of a relation to its linguistic marking as an explicit or implicit relation. We investigate these two proposals with regard to the English transcripts of six TED Talks and their Lithuanian, Portuguese and Turkish translations in the TED-Multilingual Discourse Bank (TED-MDB), annotated for discourse relations, following the Penn Discourse Treebank style of annotation. Our data shows that the discontinuous relations contrast and concession are indeed frequently explicit in all languages. But continuous relations show differences per relation and language. For instance, cause is frequently conveyed implicitly in English and Portuguese, but not in Lithuanian and Turkish. We explore temporal continuity by analysing whether the forward-order sense result is more frequently implicit than the backward-order reason. The hypothesis is confirmed by English and Portuguese, but not Lithuanian and Turkish. However, in Turkish, the arguments of the backward-order relation reason are frequently presented by the reversed order of arguments, retaining the linear order of events even in the presence of the connective. The causality-by-default hypothesis is not confirmed, as cause is not the most frequent implicit relation in the four languages.

Asymptotic Average Solutions to Linear Second Order Semi-Elliptic PDEs: A Pizzetti-Type Theorem

By exploiting an old idea first used by Pizzetti for the classical Laplacian, we introduce a notion of asymptotic average solutions making pointwise solvable every Poisson equation ${\mathcal {L}} u(x)=-f(x)$ with continuous data f, where ${\mathcal {L}}$ is a hypoelliptic linear partial differential operator with positive semidefinite characteristic form.