Virtual lab in Modelica of a cement clinker cooler for operator training

Plant operator training plays a fundamental role in improving the energy efficiency of the cement manufacturing process and in reducing the CO₂ emission. A virtual lab of a clinker grate cooler, intended for the training of cement plant operators, has been developed. The grate cooler model has been derived from first principles, and has been validated by consulting cement industry experts, and comparing the simulated results with published data and available information from the cement industry. The model has been described in the Modelica language. The Interactive Modelica library has been used to develop the interactive user-to-model interface, and the communication between this interface and the model. The virtual lab, which is completely described in Modelica, has been simulated using the Dymola 6.1 modelling environment. The Interactive Modelica library can be freely downloaded from the website http://www.euclides.dia.uned.es/     

A dynamical systems model of unorganized segregation

We consider Schelling’s bounded neighborhood model (BNM) of unorganized segregation, from the perspective of modern dynamical systems theory. We carry out a complete quantitative analysis of the system for linear tolerance schedules. We derive a fully predictive model and associate each term with a social meaning. We recover and generalize Schelling’s qualitative results.

For the case of unlimited population movement, we derive exact formulae for regions in parameter space where stable integrated population mixes can occur, and show how neighborhood tipping can be explained in terms of basins of attraction.

When population movement is limited, we derive exact criteria for the occurrence of new population mixes.

For nonlinear tolerance schedules, we illustrate our approach with numerical simulations.     

Des extensions plus petites que leurs groupes de Galois

In this article we construct some Galois extensions L∕K with finite Galois groups and such that |Gal(L∕K)|>[L:K]. Using an analog of the Noether method, we explain how to obtain, with a fixed center, such a Galois curiosity with a Galois group as large as we want.

Résumé : Dans cet article, nous construisons une extension galoisienne L∕K à groupe de Galois fini et telle que |Gal(L∕K)|>[L:K]. En utilisant un analogue non commutatif de la méthode de Noether, nous expliquons ensuite comment, à centre fixé, l’on peut construire une telle curiosité galoisienne avec un groupe aussi gros que l’on veut.

Mots clés : Corps gauches; théorie de Galois.     

Concrete models and dynamic instruments as early technology tools in classrooms at the dawn of ICMI: from Felix Klein to present applications in mathematics classrooms in different parts of the world

Most national curricula for both primary and secondary grades encourage the active involvement of learners through the manipulation of materials (either concrete models or dynamic instruments). This trend is rooted in the emphasis given, at the dawn of ICMI, to what might be called an experimental approach: the links between mathematics, natural sciences and technology were in the foreground in the early documents of ICMI and also in the papers of its first president, Felix Klein. However, the presence of this perspective in teaching practice is uneven. In this paper, we shall reconstruct first an outline of what happened in three different parts of the world (Europe, USA and Japan) under the direct influence of Klein. Then, we shall report classroom activities realized in the same regions in three different research centres: the Laboratory of Mathematical Machines at the University of Modena and Reggio Emilia, Italy (http://www.mmlab.unimore.it); the pedagogical space of Kinematical Model for Design Digital Library at Cornell, USA (http://kmoddl.library.cornell.edu/); and the Centre for Research on International Cooperation in Educational Development at Tsukuba University, Japan (http://math-info.criced.tsukuba.ac.jp/). They have maintained the reference to concrete materials (either models or instruments), with original interpretations that take advantage of the different cultural conditions. Although in all cases the reference to history is deep and systematic, the synergy with mathematical modelling and with information and communication technologies has been exploited, not to substitute but to complement the advantages of the direct manipulations.     

Preliminary test estimators and phi-divergence measures in generalized linear models with binary data

We consider the problem of estimation of the parameters in Generalized Linear Models (GLM) with binary data when it is suspected that the parameter vector obeys some exact linear restrictions which are linearly independent with some degree of uncertainty. Based on minimum -divergence estimation (ME), we consider some estimators for the parameters of the GLM: Unrestricted ME, restricted ME, Preliminary ME, Shrinkage ME, Shrinkage preliminary ME, James–Stein ME, Positive-part of Stein-Rule ME and Modified preliminary ME. Asymptotic bias as well as risk with a quadratic loss function are studied under contiguous alternative hypotheses. Some discussion about dominance among the estimators studied is presented. Finally, a simulation study is carried out.     

Histopolating splines

Given an integrable function f, we are concerned with the construction of a spline H_n(f) of degree n with prescribed knots that satisfies the histopolation conditions

for some fixed . Additionally, the resulting spline operator should be local and reproduce all polynomials of degree n. Our approach of generating such a histospline is based on a local spline interpolation operator that is exact for all polynomials of degree n.     

A vanishing theorem on Kaehler Finsler manifolds

Let M be a connected compact complex manifold endowed with a strongly pseudoconvex complex Finsler metric F. In this paper, we first define the complex horizontal Laplacian □_h and complex vertical Laplacian □_v on the holomorphic tangent bundle T^1,0M of M, and then we obtain a precise relationship among □_h,□_v and the Hodge–Laplace operator on (T^1,0M,,), where , is the induced Hermitian metric on T^1,0M by F. As an application, we prove a vanishing theorem of holomorphic p-forms on M under the condition that F is a Kaehler Finsler metric on M.     

On multipartite posets

A poset P=(X,) is m-partite if X has a partition X=X₁X_m such that (1) each X_i forms an antichain in P, and (2) xy implies xX_i and yX_j where i<j. In this article we derive a tight asymptotic upper bound on the order dimension of m-partite posets in terms of m and their bipartite sub-posets in a constructive and elementary way.     

The Approximation Functional and Interpolation with Perturbed Continuity

The continuity conditions at the endpoints of interpolation theorems, Ta_BjM_j a_Aj for j=0, 1, can be written with the help of the approximation functional: E(t, Ta; B₁, B₀)_L^∞M₀ a_A0 and E(t, Ta; B₀, B₁)_L^∞M₁ a_A1. As a special case of the results we present here we show that in the hypotheses of the interpolation theorem the L^∞ norms can be replaced by BMO( ₊) norms. This leads to a strong version of the Stein-Weiss theorem on interpolation with change of measure. Another application of our results is that the condition fL⁰, i.e., f*L^∞, where f*(γ)=μ{|f|>γ} is the distribution function of f, can be replaced in interpolation with L(p, q) spaces by the weaker f*BMO( ₊).     

Sharp tridiagonal pairs

Let denote a field and let V denote a vector space over with finite positive dimension. We consider a pair of -linear transformations A:V→V and A^*:V→V that satisfies the following conditions: (i) each of A,A^* is diagonalizable; (ii) there exists an ordering of the eigenspaces of A such that A^*V_iV_i-1+V_i+V_i+1 for 0id, where V_-1=0 and V_d+1=0; (iii) there exists an ordering of the eigenspaces of A^* such that for 0iδ, where and ; (iv) there is no subspace W of V such that AWW, A^*WW, W≠0, W≠V. We call such a pair a tridiagonal pair on V. It is known that d=δ and for 0id the dimensions of coincide. We say the pair A,A^* is sharp whenever dimV₀=1. A conjecture of Tatsuro Ito and the second author states that if is algebraically closed then A,A^* is sharp. In order to better understand and eventually prove the conjecture, in this paper we begin a systematic study of the sharp tridiagonal pairs. Our results are summarized as follows. Assuming A,A^* is sharp and using the data we define a finite sequence of scalars called the parameter array. We display some equations that show the geometric significance of the parameter array. We show how the parameter array is affected if Φ is replaced by or or . We prove that if the isomorphism class of Φ is determined by the parameter array then there exists a nondegenerate symmetric bilinear form , on V such that Au,v=u,Av and A^*u,v=u,A^*v for all u,vV.     

Equivalence relation groupoids associated with certain linearly ordered dimension groups

We consider linearly ordered, Archimedean dimension groups (G,G⁺,u) for which the group G/u is torsion-free. It will be shown that if, in addition, G/u is generated by a single element (i.e., ), then (G,G⁺,u) is isomorphic to for some irrational number τ(0,1). This amounts to an extension of related results where dimension groups for which G/u is torsion were considered. We will prove, in the case of the Fibonacci dimension group, that these results can be used to directly construct an equivalence relation groupoid whose C^*-algebra is the Fibonacci C^*-algebra.     

Rayleigh-schrödinger series for defective spectral elements of compact operators in banach spaces

We prove a posteriori error bounds and convergence of Rayleigh-Schrödinger Series for bases of maximal invariant subspaces of compact operators in Banach spaces. These results generalize those for eigenvectors associated with simple eigenvalues. Recursive formulae are given for the numerical computation of the series' coefficients.

This work has been partially supported by Grant 89–0879 Fondecyt Chile.     

Relative rearrangement and Lebesgue spaces with variable exponent

We apply the techniques of monotone and relative rearrangements to the nonrearrangement invariant spaces L^p()(Ω) with variable exponent. In particular, we show that the maps uL^p()(Ω)→k(t)u_*L^p_*()(0,measΩ) and uL^p()(Ω)→u_*L^p*()(0,measΩ) are locally -Hölderian (u_* (resp. p^*) is the decreasing (resp. increasing) rearrangement of u (resp. p)). The pointwise relations for the relative rearrangement are applied to derive the Sobolev embedding with eventually discontinuous exponents.     

Widths and shape-preserving widths of Sobolev-type classes of s-monotone functions

Let I be a finite interval, , and 1p∞. Given a set M, of functions defined on I, denote by the subset of all functions yM such that the s-difference is nonnegative on I, τ>0. Further, denote by the Sobolev class of functions x on I with the seminorm x^(r)_Lp1. We obtain the exact orders of the Kolmogorov and the linear widths, and of the shape-preserving widths of the classes in L_q for s>r+1 and (r,p,q)≠(1,1,∞). We show that while the widths of the classes depend in an essential way on the parameter s, which characterizes the shape of functions, the shape-preserving widths of these classes remain asymptotically ≈n^-2.     

On groups with conjugacy classes of distinct sizes

A finite group G is called an ah-group if any two distinct conjugacy classes of G have distinct cardinality. We show that if G is an ah-group, then the non-abelian socle of G is isomorphic to one of the following:

1. , for 1a5, a≠2.

2. A₈.

3. PSL(3,4)^e, for 1e10.

4. A₅×PSL(3,4)^e, for 1e10.

Based on this result, we virtually show that if G is an ah-group with π(G) 2,3,5,7 , then F(G)≠1, or equivalently, that G has an abelian normal subgroup.In addition, we show that if G is an ah-group of minimal size which is not isomorphic to S₃, then the non-abelian socle of G is either trivial or isomorphic to one of the following:

1. , for 3a5.

2. PSL(3,4)^e, for 1e10.

Our research lead us to interesting results related to transitivity and homogeneousity in permutation groups, and to subgroups of wreath products of form Z₂S_n. These results are of independent interest and are located in appendices for greater autonomy.     

Maximal Inequalities and Lebesgue's Differentiation Theorem for Best Approximant by Constant over Balls

For fL_p( ⁿ), with 1p<∞, >0 and x ⁿ we denote by T(f)(x) the set of every best constant approximant to f in the ball B(x, ). In this paper we extend the operators T_p to the space L_p−1( ⁿ)+L_∞( ⁿ), where L₀ is the set of every measurable functions finite almost everywhere. Moreover we consider the maximal operators associated to the operators T_p and we prove maximal inequalities for them. As a consequence of these inequalities we obtain a generalization of Lebesgue's Differentiation Theorem.     

A polynomial time approximation scheme for the two-source minimum routing cost spanning trees

Let G be an undirected graph with nonnegative edge lengths. Given two vertices as sources and all vertices as destinations, we investigated the problem how to construct a spanning tree of G such that the sum of distances from sources to destinations is minimum. In the paper, we show the NP-hardness of the problem and present a polynomial time approximation scheme. For any >0, the approximation scheme finds a (1+)-approximation solution in O(n^1/+1) time. We also generalize the approximation algorithm to the weighted case for distances that form a metric space.     

Lattices generated by subspaces in d-bounded distance-regular graphs

Let Γ denote a d-bounded distance-regular graph with diameter d2. A regular strongly closed subgraph of Γ is said to be a subspace of Γ. Define the empty set to be the subspace with diameter -1 in Γ. For 0ii+sd-1, let denote the set of all subspaces in Γ with diameters i,i+1,…,i+s including Γ and . If we define the partial order on by ordinary inclusion (resp. reverse inclusion), then is a poset, denoted by (resp. ). In the present paper we show that both and are atomic lattices, and classify their geometricity.