ODE Analysis Research Articles

The Role of Nature in Romantic Poetry—An Educational Analysis of Daffodils and Ode to the West Wind

The Role of Nature in Romantic Poetry—An Educational Analysis of Daffodils and Ode to the West Wind

Nearly Toroidal, Periodic and Quasi-periodic Motions of Fluid Particles Driven by the Gavrilov Solutions of the Euler Equations

We consider the smooth, compactly supported solutions of the steady 3D Euler equations of incompressible fluids constructed by Gavrilov (Geom Funct Anal (GAFA) 29(1):190–197, 2019), and we study the corresponding fluid particle dynamics. This is an ode analysis, which contributes to the description of Gavrilov’s vector field.

The Synesthesia in John Keats' Ode

Poetic synaesthesia is frequently used in poetry, and it can be thought of as a type of poetic metaphor. Synaesthesia can be defined as the human capacity to comprehend the objective world from a cognitive point of view. A tried and true method for generating circles and expressing ideas. Synaesthesia is used by people who create works of literature, particularly poetry, as a means of expressing the feelings and experiences that are unique to them. Synaesthesia is a manifestation of the rich creativity and imagination that people possess. The purpose of this paper is to clarify, from a cognitive point of view, the characteristics of synaesthesia as well as the powerful effect that synaesthesia has on the emotional expression of literary creators through an analysis of Keats' ode. This will allow readers to better understand the emotions that literary creators feel and to feel literary works more profoundly.Keywords: Poetry; Synaesthesia; Cognition; John Keats; Ode

Radial solutions to critical elliptic systems involving p-Laplacian and different Hardy-type terms

We study a system of elliptic equations, which involves p-Laplacian operator, strongly coupled critical nonlinearities and different Hardy-type terms. By ODE analysis methods, the asymptotic properties at the origin and infinity of radially symmetric and strictly decreasing solutions to the system are proved completely. It is found that the properties of solutions depend not only on the coefficients of Hardy terms but also on the exponents of coupled critical terms. The conclusions are new even in the case of p=2.

Stylistic Analysis of Ode to the West Wind by PB Shelley

Stylistics is concerned with the analysis of variation and characteristics of language. Stylistics has gathered imminent role in ascertaining language use, especially poetic language. In this research paper,I have explored the intentional deviation from the conventional rules in the poem Ode to West Wind by P.B Shelly. The poem is rich from a stylistic point of view because the application of deliberate misspellings, creativity in style, the irregularity in the syntax patterns, and the phonological and graphological deviations make the poem a fascinating read. I have noticed that Shelly has freed himself from the structural and traditional norms of language and has presented a new dimension in poetry.This article is the analysis done against the backdrop of deviations at syntactical, morphological, and grammatical levels.

Compactification for asymptotically autonomous dynamical systems: theory, applications and invariant manifolds

We develop a general compactification framework to facilitate analysis of nonautonomous ODEs where nonautonomous terms decay asymptotically. The strategy is to compactify the problem: the phase space is augmented with a bounded but open dimension and then extended at one or both ends by gluing in flow-invariant subspaces that carry autonomous dynamics of the limit systems from infinity. We derive the weakest decay conditions possible for the compactified system to be continuously differentiable on the extended phase space. This enables us to use equilibria and other compact invariant sets of the limit systems from infinity to analyze the original nonautonomous problem in the spirit of dynamical systems theory. Specifically, we prove that solutions of interest are contained in unique invariant manifolds of saddles for the limit systems when embedded in the extended phase space. The uniqueness holds in the general case, that is even if the compactification gives rise to a centre direction and the manifolds become centre or centre-stable manifolds. A wide range of problems including pullback attractors, rate-induced critical transitions (R-tipping) and nonlinear wave solutions fit naturally into our framework, and their analysis can be greatly simplified by the compactification.

Analyzing Schoenberg’s War Compositions as Satire and Sincerity: A Comparative Analysis of Ode to Napoleon Buonaparte and A Survivor from Warsaw

This paper presents a comparative analysis of Ode to Napoleon Buonaparte op. 41 (1942) and A Survivor from Warsaw op. 46 (1947) to show that Schoenberg’s compositional decisions for each piece reflect the public’s (and his) change of attitude regarding the historical events surrounding World War II. In particular, I argue that Ode frequently features humorous depiction of the Nazi regime through various types of musical satire whereas Survivor’s overall compositional planning and execution are more in line with his other compositions without a satirical tone, for example, the String Trio op. 45. The analyses in this paper show humorous elements in Schoenberg’s Ode to Napoleon and demonstrate how these elements are avoided in Survivor from Warsaw to make space for the sincere criticism of the Nazi party and to raise awareness for the survivors. I begin by interpreting satirical passages in Ode as instances of incongruent juxtapositions and then examine the different ways the incongruence manifests. I then discuss the leitmotivic partitioning of various twelve-tone row forms in Survivor to show the compositional congruence between the text, musical textures, and partitioning schemes. Given the change of compositional mood between Ode and Survivor (in addition to the historical evidence), this paper speculates that Schoenberg’s motivation for composing Survivor was to rectify the narrative tone he adopted in Ode.

An improved failure mode and effects analysis method based on uncertainty measure in the evidence theory

Abstractmode and effects analysis (FMEA) is an effective tool to assess the risk of a system or process under uncertain environment. However, how to handle the uncertainty in the subjective assessment is an open issue. In this paper, a novel method to deal with the uncertainty coming from subjective assessments of FMEA experts is proposed in the framework of Dempster–Shafer evidence theory. First, the uncertain degree of the assessment is measured by the ambiguity measure. Then, the uncertainty is transformed to the reliability of each FMEA expert and the relative importance of each risk factor. After that, the assessments from FMEA team will be fused with a discounting‐based combination rule to address the potential conflict. Moreover, to avoid the situation that different risk priorities of failure modes may have the same ranking based on classical risk priority number method, the gray relational projection method (GRPM) is adopted for ranking risk priorities of failure modes. Finally, an application of the improved FMEA model in sheet steel production process verifies the reliability and validity of the proposed method.

The Stability Principle and global weak solutions of the free surface semi-geostrophic equations in geostrophic coordinates

The semi-geostrophic equations are used widely in the modelling of large-scale atmospheric flows. In this note, we prove the global existence of weak solutions of the incompressible semi-geostrophic equations, in geostrophic coordinates, in a three-dimensional domain with a free upper boundary. The proof, based on an energy minimization argument originally inspired by the Stability Principle as studied by Cullen, Purser and others, uses optimal transport techniques as well as the analysis of Hamiltonian ODEs in spaces of probability measures as studied by Ambrosio and Gangbo. We also give a general formulation of the Stability Principle in a rigorous mathematical framework.

Stability analysis in some strongly prestressed rectangular plates

We consider an evolution plate equation aiming to model the motion of the deck of a periodically forced strongly prestressed suspension bridge. Using the prestress assumption, we show the appearance of multiple time-periodic uni-modal longitudinal solutions and we discuss their stability. Then, we investigate how these solutions exchange energy with a torsional mode. Although the problem is forced, we find a portrait where stability and instability regions alternate. The techniques used rely on ODE analysis of stability and are complemented with numerical simulations.

A note on the uniqueness and the non-degeneracy of positive radial solutions for semilinear elliptic problems and its application

In this paper, we are concerned with the uniqueness and the non-degeneracy of positive radial solutions for a class of semilinear elliptic equations. Using detailed ODE analysis, we extend previous results to cases where nonlinear terms may have sublinear growth. As an application, we obtain the uniqueness and the non-degeneracy of ground states for modified Schrödinger equations.

Wave propagation in a strongly nonlinear locally resonant granular crystal

In this work, we study the wave propagation in a recently proposed acoustic structure, the locally resonant granular crystal. This structure is composed of a one-dimensional granular crystal of hollow spherical particles in contact, containing linear resonators. The relevant model is presented and examined through a combination of analytical approximations (based on ODE and nonlinear map analysis) and of numerical results. The generic dynamics of the system involves a degradation of the well-known traveling pulse of the standard Hertzian chain of elastic beads. Nevertheless, the present system is richer, in that as the primary pulse decays, secondary ones emerge and eventually interfere with it creating modulated wavetrains. Remarkably, upon suitable choices of parameters, this interference “distills” a weakly nonlocal solitary wave (a “nanopteron”). This motivates the consideration of such nonlinear structures through a separate Fourier space technique, whose results suggest the existence of such entities not only with a single-side tail, but also with periodic tails on both ends. These tails are found to oscillate with the intrinsic oscillation frequency of the out-of-phase motion between the outer hollow bead and its internal linear attachment.

Bayesian Analysis of ODEs: Solver Optimal Accuracy and Bayes Factors

In most cases in the Bayesian analysis of ODE inverse problems, a numerical solver needs to be used. Therefore, we cannot work with the exact theoretical posterior distribution but only with an approximate posterior derived from the error in the numerical solver. To compare an approximate posterior distribution with the theoretical one, we propose using Bayes factors (BFs), considering both of them as models for the data at hand. From a theoretical point of view, we prove that the theoretical vs. numerical posterior BF tends to 1, in the same order as the numerical solver used. In practice, we illustrate the fact that for higher order solvers (e.g., Runge--Kutta) the BF is already nearly 1 for step sizes that would take far less computational effort. Considerable CPU time may be saved by using coarser solvers that nevertheless produce practically error-free posteriors. Two examples are presented where nearly 90% CPU time is saved, with all inference results being identical to those obtained using a solver with a much finer time step.

Existence of periodic orbits in nonlinear oscillators of Emden–Fowler form

The nonlinear pseudo-oscillator recently tackled by Gadella and Lara is mapped to an Emden–Fowler (EF) equation that is written as an autonomous two-dimensional ODE system for which we provide the phase-space analysis and the parametric solution. Through an invariant transformation we find periodic solutions to a certain class of EF equations that pass an integrability condition. We show that this condition is necessary to have periodic solutions and via the ODE analysis we also find the sufficient condition for periodic orbits. EF equations that do not pass integrability conditions can be made integrable via an invariant transformation which also allows us to construct periodic solutions to them. Two other nonlinear equations, a zero-frequency Ermakov equation and a positive power Emden–Fowler equation, are discussed in the same context.

Uniform upper bounds for the cyclicity of the zero solution of the Abel differential equation

Given two polynomials P,q we consider the following question: “how large can the index of the first non-zero moment m˜k=∫abPkq be, assuming the sequence is not identically zero?” The answer K to this question is known as the moment Bautin index, and we provide the first general upper bound: K⩽2+deg⁡q+3(deg⁡P−1)2. The proof is based on qualitative analysis of linear ODEs, applied to Cauchy-type integrals of certain algebraic functions.The moment Bautin index plays an important role in the study of bifurcations of periodic solution in the polynomial Abel equation y′=py2+εqy3 for p,q polynomials and ε≪1. In particular, our result implies that for p satisfying a well-known generic condition, the number of periodic solutions near the zero solution does not exceed 5+deg⁡q+3deg2⁡p. This is the first such bound depending solely on the degrees of the Abel equation.

Numerical Analysis of Steady State Patterns in Cell-Based Auxin Transport Models

Cell-based models that describe the pattern formation and the flow of chemicals in plant organs are important building blocks in a multiscale simulation of a whole plant. An example of an important mechanism is the transport of the hormone auxin throughout the plant's organs because it is closely related to the growth characteristics of roots, shoots and leaves. Based on experimental evidence, a number of cell-based auxin transport models were developed. Due to the intercellular transport of chemicals, these models are complex dynamical systems with a large set of endogenous and exogenous parameters. The models share underlying mathematicalprinciples w.r.t. steady state pattern formation which plays a central role in the growth and development of plant organs. This calls for a uniform computational approach. In our research we focus on computer simulations of general cell-based transport models and more specically we use numerical bifurcation analysis to study the steady state patterns. It indicates how the stability ofpatterns is lost or gained as the system parameters change. Bifurcation analysis of ODEs is widespread in biology and various numerical tools produce bifurcation diagrams. However, these automatic tools do not work for large scale problems, such as biological patterning. Indeed, realistic simulations of large tissues that take multiple interacting chemicals into account give rise to very large and sparse systems of coupled ODEs. We analyze recent large scale transport models with new mathematical and computational tools that enable quantitative prediction of the bifurcations that appear at the macroscopic level in these models. This allows us to predict the patterns and self-similar solutions that appear during organ growth and to see how their stability changes as endogenous parameters are modified or as externally applied changes are enforced. We use these methods to compare the model output with observed data such as the auxin distribution and venation patterns in leaves in order to get a better understanding of the processes that regulate organ development in plants.

On a variational approach for the analysis and numerical simulation of ODEs

This paper is devoted to the study and approximation of systems ofordinary differential equations based on an analysis of a certainerror functional associated, in a natural way, with the originalproblem. We prove that in seeking to minimize the error by usingstandard descent schemes, the procedure can never get stuck inlocal minima, but will always and steadily decrease the erroruntil getting to the original solution. One main step in theprocedure relies on a very particular linearization of theproblem: in some sense, it is like a globally convergent Newtontype method. Although our objective here is not to perform a rigorousnumerical study of the method, we illustrate its potential forapproximation by considering some stiff systemsof equations. The performance is astonishingly very good due to the fact that we can use very robust methods to approximatelinear stiff problems like implicit collocation schemes. We also include a couple of typicaltest models for the Lorentz system and the Kepler problem, again confirming a very goodperformance. Webelieve that this approach can be used in a systematic way toexamine other situations and other types of equations due to itsflexibility and its simplicity.

A structured approach to the study of metabolic control principles in intact and impaired mitochondria

We devised an approach to extract control principles of cellular bioenergetics for intact and impaired mitochondria from ODE-based models and applied it to a recently established bioenergetic model of cancer cells. The approach used two methods for varying ODE model parameters to determine those model components that, either alone or in combination with other components, most decisively regulated bioenergetic state variables. We found that, while polarisation of the mitochondrial membrane potential (ΔΨ(m)) and, therefore, the protomotive force were critically determined by respiratory complex I activity in healthy mitochondria, complex III activity was dominant for ΔΨ(m) during conditions of cytochrome-c deficiency. As a further important result, cellular bioenergetics in healthy, ATP-producing mitochondria was regulated by three parameter clusters that describe (1) mitochondrial respiration, (2) ATP production and consumption and (3) coupling of ATP-production and respiration. These parameter clusters resembled metabolic blocks and their intermediaries from top-down control analyses. However, parameter clusters changed significantly when cells changed from low to high ATP levels or when mitochondria were considered to be impaired by loss of cytochrome-c. This change suggests that the assumption of static metabolic blocks by conventional top-down control analyses is not valid under these conditions. Our approach is complementary to both ODE and top-down control analysis approaches and allows a better insight into cellular bioenergetics and its pathological alterations.

Impurity of Blood: Defining Race in Spain, 1870–1930 (review)

GOODE, JOSHUA. Impurity of Blood: Defining Race in Spain, 1870-1930. Baton Rouge: Louisiana State UP, 2009. xii + 295 pp.One sometimes hears rumblings in certain academic circles that race is an American concern, wholly foreign to modern Spain. Joshua Goode's meticulous study of the multiple attitudes towards racial discourse during the nineteenth and twentieth centuries, and their expression in historiography, imperial (and postimperial) strategy, and policies regarding the descendents of Iberia's Sephardic exiles will quiet those voices: Spain is decidedly not devoid of autochthonous racial thought.Impurities of Blood is a lengthy, comprehensive study of race, racism, and racializing thought in Spain. After a discussion of the medieval and early modern periods, the primary focus is on the decades that coincided with Romanticism and nationalism, the twinned concepts that defined membership among its citizenry by means of cultural and ethnic homogeneity. With careful attention to the subtleties of shifting political winds over the course of the nineteenth century, the author persuasively argues that race was permanently central to Spain's conception of itself, even as its meaning constantly shifted along ideological and political lines. Not only the centralizing nation-state and imperial army, but also Basque nationalism, Sephardic repatriation, and Catholicism were all inflected by the reigning racial theories. In fact, as Goode reveals, race offers a privileged view of the seemingly chaotic social and political history of Spain between 1870 and 1920. Instead of a clear opposition between conservative and liberal visions of Spanish identity, race demonstrated that a far more unified discourse existed within Spanish political and intellectual life at the turn of the twentieth century. There were many more than two Spains (82). The final sections of the book offer an incisive analysis of racial policies in the decades following the Spanish Civil War and World War II.Modern Spanish race developed at the turn of the century as a means of explaining Spanish imperial decline in terms of inherited biological factors and morphological degeneration. Unlike the case for other imperial powers, in Spain race was not a tool to distinguish inferior colonized people from their colonizers, but rather to understand the failure of empire. Anthropology and race theories developed, in part, in an attempt to pinpoint a racial and physical deficiency among Spanish soldiers that could explain their perpetual colonial defeats.The common distinction between race theories in Northern and Southern Europe posits that in the former, race was celebrated as purity and exclusion, while in the latter it was forged through assimilation, fusion, and inclusion. As a result, Spanish race has been seen as somehow positive or welcoming, whereas, for example, German racial thought was negative (leaving aside Nazi policies, for the moment). However, Goode departs from standard historiographical comparisons of European racisms and instead explores how racialized thought was debated and implemented in specific policies in Spain. This methodological approach allows him to illuminate the means by which various racial traditions produce distinct (or similar) racial policies. Go ode's analysis of the ideas and practices that preceded Spanish fascism is complex and nuanced, and truly uncovers how and why particular ideas took hold. Ultimately, the author succeeds in setting aside the idea that race and genocide in Europe must always be defined in relation to the policies of Nazi Germany. He brilliantly dismisses the all too common exculpatory view that because race, eugenics, and population control in Spain under Franco did not reach the murderous heights of the Third Reich, or because race was constructed around a discourse of fusion rather than purity, Spain was and is somehow free from racism.Grounding his arguments on specific political, social, and academic debates, and following the birth and growth of the field of anthropology, Goode details the uses of the conceptualization of Spain as a nation of assimilation and fusion (the racial alloy) by both the left and the right, and its interpretation by writers as diverse as Pio Baroja, Benito Perez Galdos, and Miguel de Unamuno. …

Uniqueness and multiplicity results for N-Laplace equation with critical and singular nonlinearity in a ball

Let B1 be the unit open ball with center at the origin in IRN , N ≥ 2, . We consider the following quasilinear problem depending on a real parameter λ > 0:   −∆N u = λf (u) in Ω, u > 0 (Pλ )  u = 0 on ∂Ω, N/N −1 where f (t) is a nonlinearity that grows like et as t → ∞ and behaves like tα , + . More precisely, we require f to satisfy assump- for some α ∈ (−∞, 0), as t → 0 tions (A1)-(A2) listed in Section 1. For such a general nonlinearity we show that if λ > 0 is small enough, (Pλ ) admits at least one weak solution (in the sense of dis- tributions). We further study the question of uniqueness and multiplicity of solutions to (Pλ ) when Ω = B1 under additional structural conditions on f (see assumptions (A3)-(A8) in Section 2). Using shooting methods and asymptotic analysis of ODEs, under the additional assumptions (A3)-(A5), we prove uniqueness of solution to (Pλ ) for all λ > 0 small whereas under (A6)-(A7) or (A8), we show multiplicity of so- lutions to (Pλ ) for all λ > 0 in a maximal interval. These results clearly show that the borderline between uniqueness and multiplicity is given by the growth condition 1 lim inf h(t)te t N −1 t→∞ = ∞ ∀ > 0.