Sullivan Model Research Articles

A simple step-change method to determine mean residence time in rotary kiln and a predictive model at low inclination

Rotary kilns are widely used for chemical, metallurgical, environmental protection and fossil fuels pyrolysis processes. To react those thermal treatments efficiently for industrial production, it is necessary to study the residence time distribution (RTD) of materials in rotary kilns. In this paper, a simple step-change method is developed to measure the residence time distribution of solid particles in a rotary kiln. Using salts as the tracer and recording the salinity by the conductivity meter, the effluent salinity curves can be converted to E(t) curves, namely the residence time distribution. The influential factors of the mean residence time (MRT) and the residence time distribution such as rotational speed, incline, feed rate and particle size are evaluated and discussed. The experimental results are compared with three different models, namely the Sullivan model, Chatterjee model and Saeman model. The experimental mean residence time results match well with the Sullivan model except at low incline angles. To predict the mean residence time in rotary kilns at low inclination, which has not been studied before, the experiments at low inclination are conducted and predictive model is presented. The calculated values agree well with the experimental results at low inclination.

On Koszul–Tate resolutions and Sullivan models

We report on Koszul-Tate resolutions in Algebra, in Mathematical Physics, in Cohomological Analysis of PDE-s, and in Homotopy Theory. Further, we define an abstract Koszul-Tate resolution in the frame of $\mathcal{D}$-Geometry, i.e., geometry over differential operators. We prove Comparison Theorems for these resolutions, thus providing a dictionary between the different fields. Eventually, we show that all these resolutions are of the new $\mathcal{D}$-geometric type.

A reflection on analytical tornado-like vortex flow field models

Given the difficulties associated with undertaking full-scale measurements in tornadoes, recourse is often made to models. In this field, analytical models have, perhaps surprisingly, stood the test of time, with the Rankine, Burgers-Rott and Sullivan models frequently invoked to model the flow field of a tornado. These mathematical models are by their very nature, a simplification of what is a highly complex phenomenon. However, in many cases they have been represented as the ‘truth’ without the fundamental assumptions governing the model being either explored in detail or even acknowledged. This paper attempts to rectify this by giving detailed information about assumptions and limitations of each vortex model and critically assesses the ability (or otherwise) of the Rankine, Burgers-Rott, Sullivan, and the recently published Baker vortex model to simulate tornado-like flow. Comparisons are made to the flow field of a physically simulated tornado, which by its very nature is also a model, but arguably more realistic.It was found that the vortex models are able to represent certain flow patterns at certain heights but fail, due to their simplifications, in replicating the entire three-dimensional flow structure obtained experimentally.

A new lower bound for LS-category

Let X be a simply connected CW-complex of finite type and {mathbb {K}} an arbitrary field. In this paper, we use the Eilenberg–Moore spectral sequence of C_*(Omega (X), mathbb K) to introduce a new homotopical invariant textsc {r}(X, {mathbb {K}}). If X is a Gorenstein space with nonzero evaluation map, then textsc {r}(X, {mathbb {K}}) turns out to interpolate mathrm {depth}(H_*(Omega (X), {mathbb {K}})) and mathrm {e}_{{mathbb {K}}}(X). We also define for any minimal Sullivan algebra (Lambda V,d) a new spectral sequence and make use of it to associate to any 1-connected commutative differential graded algebra (A, d) a similar invariant textsc {r}(A,d). When (Lambda V,d) is a minimal Sullivan model of X, this invariant fulfills the relation textsc {r}(X, {mathbb {K}}) = textsc {r}(Lambda V,d).

BV structure on the Hochschild cohomology of Sullivan algebras

Abstract Let X be a closed, simply connected manifold of dimension m and LX the space of free loops on X . If (∧ V, d ) is the minimal Sullivan model of X where V is finite dimensional, then there is a Gerstenhaber algebra ( ∧ V ⊗ ∧ s − 1 V # , d 0 ) , where V # is the graded dual of V , and its homology is isomorphic to the loop space homology H * ( L X ) . In this paper we define a BV structure on ( ∧ V ⊗ ∧ s − 1 V # , d 0 ) which extends the Gerstenhaber bracket.

Certain maps preserving self-homotopy equivalences

Let \(\mathcal {E}(X)\) be the group of homotopy classes of self homotopy equivalences for a connected CW complex X. We consider two classes of maps, \(\mathcal {E}\)-maps and co-\(\mathcal {E}\)-maps. They are defined as the maps \(X\rightarrow Y\) that induce homomorphisms \(\mathcal {E}(X)\rightarrow \mathcal {E}( Y)\) and \(\mathcal {E}(Y)\rightarrow \mathcal {E}(X)\), respectively. We give some rationalized examples related to spheres, Lie groups and homogeneous spaces by using Sullivan models. Furthermore, we introduce an \(\mathcal {E}\)-equivalence relation between rationalized spaces \(X_{{\mathbb Q}}\) and \(Y_{{\mathbb Q}}\) as a geometric realization of an isomorphism \(\mathcal {E}(X_{{\mathbb Q}})\cong \mathcal {E}(Y_{{\mathbb Q}})\).

Some notes on the sectional fibrations

Given a fibration F↪E⟶pB, we study its rational sections from three different points of view.First, we transform the problem to some realization problems under different conditions, and also give some general criterion using the derivations of the Sullivan model of the fibre.In addition, we consider the problem with the base space being a formal space. It turns out that the existence of a section is equivalent to the realization of some commutative differential graded algebra constructed from the minimal model of p.Finally, we construct a Lie model for the space of the sections of p using the localization method, which is also used to study the sections of Lie epimorphisms.

The local structure factor near an interface; beyond extended capillary-wave models

We investigate the local structure factor S (z;q) at a free liquid–gas interface in systems with short-ranged intermolecular forces and determine the corrections to the leading-order, capillary-wave-like, Goldstone mode divergence of S (z;q) known to occur for parallel (i.e. measured along the interface) wavevectors . We show from explicit solution of the inhomogeneous Ornstein–Zernike equation that for distances z far from the interface, where the profile decays exponentially, S (z;q) splits unambiguously into bulk and interfacial contributions. On each side of the interface, the interfacial contributions can be characterised by distinct liquid and gas wavevector dependent surface tensions, and , which are determined solely by the bulk two-body and three-body direct correlation functions. At high temperatures, the wavevector dependence simplifies and is determined almost entirely by the appropriate bulk structure factor, leading to positive rigidity coefficients. Our predictions are confirmed by explicit calculation of S (z;q) within square-gradient theory and the Sullivan model. The results for the latter predict a striking temperature dependence for and , and have implications for fluctuation effects. Our results account quantitatively for the findings of a recent very extensive simulation study by Höfling and Dietrich of the total structure factor in the interfacial region, in a system with a cut-off Lennard-Jones potential, in sharp contrast to extended capillary-wave models which failed completely to describe the simulation results.

Malcev completions, LS category, and depth

Let X be a connected topological spaces with finite rational Betti numbers. We denote by G the radical of its fundamental group and by \((\wedge V,d)\) its minimal Sullivan model. If dim\(\, V^i<\infty \) for each \(i\ge 2\), then the dimension of the rationalization of G is less than or equal to the Lusternik-Schnirelmann category of X.

A fibre-restricted Gottlieb group and its rational realization problem

We define the fibre-restricted Gottlieb group G⁎ξ(X) with respect to a fibration ξ:X→E→Y in CW complexes. It is a subgroup of the Gottlieb group G⁎(X) of X. When X and E are finite simply connected, its rationalized model is given by the arguments of derivations of Sullivan models based on Y. Félix, G. Lupton and S.B. Smith [6]. Then we consider the problem of realizing subgroups of a Gottlieb group as fibre-restricted Gottlieb groups. Especially we define a numerical invariant named as the Gottlieb depth of X over Y, denoted by depthY(X). It gives a measure for the realizations. We illustrate an example of the poset structures of inclusions of realized subgroups in the rational Gottlieb groups, named as the Gottlieb poset of X over Y.

The harmonic field of a Riemannian manifold

Sullivan’s construction of minimal models for topological spaces is refined for the case of a simply connected closed Riemannian manifold, $(M,\lt,\gt)$, to define a unique finitely generated field extension, $\mathbf{k}$ of $\mathbf{Q}$, baptized the harmonic field of $(M,\lt,\gt)$, and a morphism, $m:(\Lambda V,d)\rightarrow A_{DR}(M)$, from a Sullivan model defined over $\mathbf{k}$. The Sullivan model and the morphism are determined up to isomorphism, and the natural extension of $H(m)$ to $H(\Lambda V,d)\otimes_{\mathbf{k}}\mathbf{R}$ is an isomorphism; in particular, $(\Lambda V,d)$ is isomorphic to a rational Sullivan model for $M$ tensored with $\mathbf{k}$. Examples are constructed to show that every finitely generated extension field of $\mathbf{Q}$ occurs as a harmonic field of such a Riemannian manifold.

Regularity in the growth of the loop space homology of a finite CW complex

To any path connected topological space X X , such that rk ⁡ H i ( X ) &gt; ∞ \operatorname {rk} H_i(X) &gt;\infty for all i ≥ 0 i\geq 0 , are associated the following two sequences of integers: b i = rk ⁡ H i ( Ω X ) b_i= \operatorname {rk} H_i(\Omega X) and r i = rk ⁡ π i + 1 ( X ) r_i= \operatorname {rk} \pi _{i+1}(X) . If X X is simply connected, the Milnor-Moore theorem together with the Poincaré-Birkoff-Witt theorem provides an explicit relation between these two sequences. If we assume moreover that H i ( X ; Q ) = 0 H_i(X;\mathbb Q)=0 , for all i ≫ 0 i\gg 0 , it is a classical result that the sequence of Betti numbers ( b i ) (b_i) grows polynomially or exponentially, depending on whether the sequence ( r i ) (r_i) is eventually zero or not. The purpose of this note is to prove, in both cases, that the r t h r^{\mathrm {th}} Betti number b r b_r is controlled by the immediately preceding ones. The proof of this result is based on a careful analysis of the Sullivan model of the free loop space X S 1 X^{S^1} .

Of Sullivan models, Massey products, and twisted Pontrjagin products

Associated to every connected, topological space $X$ there is a Hopf algebra - the Pontrjagin ring of the based loop space of the configuration space of two points in X. We prove that this Hopf algebra is not a homotopy invariant of the space. We also exhibit interesting examples of H-spaces, which are homotopy equivalent as spaces, which either lead to isomorphic rational Hopf algebras or not, depending crucially on the existence of Whitehead products. Moreover, we investigate a (naturally motivated) twisted version of these Pontrjagin rings in the various aforementioned contexts. In all of these examples, Massey products abound and play a key role.

Reply to Sullivan and Kelly and Skowronski et al

To the Editor—We appreciate the insightful comments of Sullivan and Kelly and Skowronski et al concerning our article on influenza vaccine effectiveness in households during the 2010–2011 season. The comments of Sullivan and Kelly overall confirm, in another setting and influenza season (2012), that past vaccination may have a negative effect on current season vaccine effectiveness (VE). They also bring to light important issues of what might be the appropriate comparison groups for determination of VE when considering prior year vaccination. In doing the initial exploratory analyses to explain the low effectiveness observed in our study, we identified that there was significant interaction between current and prior year vaccination. For this reason, current year VE could not be estimated without taking this into account, which we did in a standard stratified analysis. We realized that, from a biologic basis, there were 4 groups to consider: vaccinated both years, vaccinated in the current year and not the previous year, vaccinated in the previous year but not the current year, and not vaccinated in either year. Instead of running another model, similar to Sullivan and Kelly's model B, we opted, in our final table, to present risk of influenza infection, divided by these 4 groups. Examination of risk was the chosen option because we also had to deal with the unusual issue of previous vaccination with the pandemic vaccine, which was also available in the prior year. This approach also allowed making comparisons without having to choose a single or combined referent group. Given the limitation of small numbers, it is still possible to order the risks from the lowest to highest in those ≥9 years of age, where the need for 2 vaccinations for protection is not recommended [1]. For past receipt of seasonal vaccine, the lowest risk was seen in persons vaccinated in the current year and not the prior year, and the highest in those vaccinated in both years; risk was intermediate for those unvaccinated in both years. It is impossible to draw definitive conclusions from these data, in light of the small numbers, but it does suggest that the effect of repeated annual vaccination needs further examination. This is the overall conclusion drawn by Sullivan and Kelly as well. Skowronski and colleagues, citing observations from their network in Canada in 2010–2011, raise another explanation for the low VE against influenza A(H3N2) demonstrated in our study, namely, antigenic drift of the circulating strains away from the vaccine strains [2]. The World Health Organization (WHO) strain selection process of updating the virus composition in the vaccine is based on antigenic similarities and differences, informed by amino acid sequences. Based on strains circulating in 2010–2011, the WHO recommendation for the 2011–2012 vaccine did not result in a change in the influenza A(H3N2) virus strain selected; this decision was based on lack of significant differences in antigenicity in representative strains analyzed [3]. The amino acid substitutions pointed out by Skowronski et al were known at the time, and as she acknowledges, did not result in changes in antigenicity. It should be noted that this year (2012–2013), we have a somewhat similar situation, relatively low VE against influenza A(H3N2) in preliminary estimates from the US Influenza Vaccine Effectiveness Network [4], but little evidence of antigenic changes in the circulating viruses as an explanation [5]. In final analyses of these network data, the effect of past vaccination will be examined, but that is not possible at this time. Overall, these questions raised by current examinations of VE indicate the value of such studies, but also indicate the need to understand better the biologic mechanisms that bring about lower than expected VE.

Lawrence–Sullivan models for the interval

Two constructions of a Lie model of the interval were performed by R. Lawrence and D. Sullivan. The first model uses an inductive process and the second one comes directly from solving a differential equation. They conjectured that these two models are the same. We prove this conjecture here.

CARDINALITY OF RATIONAL HOMOTOPY TYPES OF SIMPLY CONNECTED CW COMPLEXES

The aim of this paper is to consider the following problem: given a simply connected CW complex X of finite type, find the cardinal of the set of rational homotopy types of CW complexes Y such that H*(Y, ℚ) is isomorphic, as a commutative graded algebra, to H*(X, ℚ). For this purpose we introduce a long exact sequence, called the Whitehead exact sequence, which we associated with the minimal Sullivan model of X and which allows us to define the notion of the adapted couple associated with X. Thus we show that the rational homotopy type of X is completely determined by the equivalence class of its adapted couple.

Cohomology of configuration spaces of complex projective spaces

In this paper we compute topological invariants for some configuration spaces of complex projective spaces. We shall describe Sullivan models for these configuration spaces.

(n)-pairing with axes in rational homotopy

Let $f:X\to Z$ and $g:Y\to Z$ be maps between connected pointed CW-complexes. Recall the definition of pairing with axes $f$ and $g$ due to N.Oda. In this paper, we introduce {\it (n)-pairing}, which is a generalization of {\it H(n)}-space due to Y.Félix and D.Tanré and define a family of subsets of the homotopy set of maps. We give some rational characterizations of it and illustrate some examples in Sullivan models. Also we consider about the $G(n)$-sequence of a fibration which is a generalization of $G$-sequence.

What Will Happen If We Do Nothing To Control Trachoma: Health Expectancies for Blinding Trachoma in Southern Sudan

BackgroundUncontrolled trachoma is a leading cause of blindness. Current global trachoma burden summary measures are presented as disability adjusted life years but have limitations due to inconsistent methods and inadequate population-based data on trachomatous low vision and blindness. We aimed to describe more completely the burden of blinding trachoma in Southern Sudan using health expectancies.Methodology/Principal FindingsAge and gender specific trachomatous trichiasis (TT) prevalence was estimated from 11 districts in Southern Sudan. The distribution of visual acuity (VA) in persons with TT was recorded in one district. Sudan life tables, TT prevalence, and VA were used to calculate Trichiasis Free Life Expectancy (TTFLE) and Trichiasis Life Expectancy (TTLE) using the Sullivan method. TTLE was broken down by VA to derive TTLE with normal vision, TTLE with low vision, and TTLE with blindness. Total life expectancy at birth in 2001 was 54.2 years for males and 58.1 for females. From our Sullivan models, trichiasis life expectancy at the age of 5 years was estimated to be 7.0 (95% confidence interval [CI] = 6.2–7.8) years (12% [95% CI = 11–14] of remaining life) for males and 10.9 (95% CI = 9.9–11.9 ) years (18% [95% CI = 16–20] of remaining life) for females. Trichiasis life expectancy with low vision or blindness was 5.1 (95% CI = 3.9–6.4) years (9% [95% CI = 7–11] of remaining life) and 7.6 (95% CI = 6.0–9.1) years (12% [95% CI = 10–15] of remaining life) for males and females, respectively. Women were predicted to live longer and spend a greater proportion of their lives with disabling trichiasis, low vision, and blindness compared to men.ConclusionsThe study shows the future burden associated with doing nothing to control trachoma in Southern Sudan, that is, a substantial proportion of remaining life expectancy spent with trichiasis and low vision or blindness for both men and women, with a disproportionate burden falling on women.

Rational homotopy of the polyhedral product functor

Let (X, *) be a pointed CW-complex, K be a simplicial complex on n vertices and X-K be the associated polyhedral power. In this paper, we construct a Sullivan model of XK from K and from a model of X. Let F(K, X) be the homotopy fiber of the inclusion X-K -> X-n. Recent results of Grbic and Theriault, on one side, and of Denham and Suciu, on the other side, show the diversity of the possible homotopy types for F( K, X). Here, we prove that the corresponding map between Sullivan models is Golod attached, generalizing a result of J. Backelin. This property is deduced from the existence of a succession of vibrations whose fibers are suspensions. We consider also the Lusternik-Schnirelmann category of XK. In the case that cat X-n = n cat X, we prove that cat X-K = (cat X)(1 + dimK). Finally, we mention that this work is written in the case of a sequence of pairs, (X) under bar = (X-i, A(i)) 1 <= i <= n, as in a recent work of Bahri, Bendersky, Cohen and Gitler.