Functional differential equations and nonlinear semigroups in Lp-spaces

The autonomous nonlinear functional differential equation x(t) = F(x_t), t ? 0, x₀ = φ is studied as a semigroup of nonlinear operators in L_p function spaces. The method employed is to construct a semigroup of nonlinear operators which may be associated with the solutions of this equation. New existence and stability results are obtained for this equation by means of the semigroup approach.     

Torsion-free G3-connections that are not analytic

Up to now all the known torsion-free G₃-connections have been analytic. We provide an explicit PDE-solving approach to construct a family of smooth torsion-free G₃-connections that are not equivalent to any analytic ones, although each of these connections is locally homogeneous away from a hypersurface of the base manifold.     

Three-dimensional modelling of NOx and particulate traps using CFD: A porous medium approach

Lean burn after-treatment systems are the current focus for reducing emissions from diesel exhaust. The trend is for commercial CFD packages to use a single channel modelling approach. Due to computational demands, this necessitates specification of representative channels for modelling, implying prior knowledge of the flow field. This paper investigates a methodology for applying the porous medium approach to lean burn after-treatment systems. This approach has proved successful for three-way catalysis modelling and has the advantage that the flow field is predicted. Chemical kinetic rates for NO_x trapping and regeneration in the model are based on information available in the open literature. Similarly, filtration information based on mass accumulation and soot combustion kinetics are also readily available. Modification of the source terms in a commercial CFD package enables prediction of trapping and release of NO_x. This is an effective way to model a NO_x trap after-treatment system and provides simultaneous 3D modelling of the flow field. With diesel, particulate filtration is required. In the case of particulate traps, however, because of channel geometry, some assumptions are necessary for use of the porous medium approach and these are discussed in this paper. Both models produce qualitatively correct output and have parameters that can be tuned to conform to experimental data. Data to validate the NO_x trap model is to be measured. The particulate trap model, on the other hand, is a feasibility study for modelling the complete diesel after-treatment system using the porous medium approach.     

The bifurcation of limit cycles in Zn-equivariant vector fields

In this paper, we study the bifurcation of limit cycles from fine focus in Z_n-equivariant vector fields. An approach for investigating bifurcation was obtained. In order to show our work is efficacious, an example on bifurcations behavior is given, namely five order singular points values are given in the seventh degree Z₈-equivariant systems. We discuss their bifurcation behavior of limit cycles, and show that there are eight fine focuses of five order and five small amplitude limit cycles can bifurcate from each. So 40 small amplitude limit cycles can bifurcate from eight fine focuses under a certain condition. In terms of the number of limit cycles for seventh degree Z₈-equivariant systems, our results are good and interesting.     

Robust finite-time H∞ control for a class of uncertain switched neutral systems

This paper investigates the robust finite-time H_∞ control problem for a class of uncertain switched neutral systems with unknown time-varying disturbance. The uncertainties under consideration are norm bounded. By using the average dwell time approach, a sufficient condition for finite-time boundedness of switched neutral systems is derived. Then, finite-time H_∞ performance analysis for switched neutral systems is developed, and a robust finite-time H_∞ state feedback controller is proposed to guarantee that the closed-loop system is finite-time bounded with H_∞ disturbance attenuation level γ. All the results are given in terms of linear matrix inequalities (LMIs). Finally, two numerical examples are provided to show the effectiveness of the proposed method.     

Minimum w-entropy interpolants for matricial Carathéodory functions and maximum determinant completions of associated block Pick matrix

The so-called modified block Toeplitz vector approach is used to connect a class of particular solutions G_w for w∈D of a nondegenerate interpolation problem of the Nevanlinna-Pick type with a class of particular solutions F_w of a certain matricial Carathéodory coefficient problem in a transparent way. This will suggest a simple approach to the minimum w-entropy interpolants and the maximum determinant completions of the associated block Pick matrix within the framework of that Nevanlinna-Pick type interpolation problem by using the known assertions corresponding to F_w. It turns out that G_w(w∈D) is exactly or provides us with the unique solution to these two extremal problems in a manner.     

Block sparse recovery via mixed l 2/l 1 minimization

We consider efficient methods for the recovery of block sparse signals from underdetermined system of linear equations. We show that if the measurement matrix satisfies the block RIP with δ2s 0.4931, then every block s-sparse signal can be recovered through the proposed mixed l2 /l1 -minimization approach in the noiseless case and is stably recovered in the presence of noise and mismodeling error. This improves the result of Eldar and Mishali (in IEEE Trans. Inform. Theory 55: 5302-5316, 2009). We also give another sufficient condition on block RIP for such recovery method: δs 0.307.     

The parabolic map f1/e(z)=(1/e)e

We consider the exponential maps ?_λ : ? → ? defined by the formula ?_λ (z) = λe^z, λ(0,1/e]. Let J_r(?_λ) be the subset of the Julia set consisting of points that do not escape to infinity under forward iterates of ?. Our main result is that the function λh_λ :=HD(J_r(?_λ),)), λ(0, 1/e], is continuons at the point 1/e. As a preparation for this result we deal with the map ?₁/e itself. We prove that the h₁/e-dimensional Hausdorff measure of J_r(?₁/e) is positive and finite on each horizontal strip, and that the h₁/e-dimensional packing measure of J_r(?_λ) is locally infinite at each point of J_r(?_λ). Our main technical devices are formed by the, associated with ?_λ, maps F_λ defined on some strip P of height 2π and also associated with them tonformal measures.     

On the sum ∑k=13 | 2πk − αk |, αk algebraic numbers

If α₁, α₂, α₃ are algebraic numbers satisfying (i) the height of α₁, α₂, α₃ do not exceed H (ii) the degree of the field generated by α₁, α₂, α₃ over the field of rational numbers do not exceed D, then a positive lower bound for

$\text{∑}\text{k=1}\text{3}\text{|2π}{}^{\mathrm{k}}\text{?αk|}$

is determined explicitly (except for an absolute constant) in terms of D and H.     

Interpolation orbits in couples of Lp spacesOrbites d'interpolation pour les couples d'espaces Lp

We consider linear operators T mapping a couple of weighted L_p spaces {L_p0(U₀), L_p1(U₁)} into {L_q0(V₀),L_q1(V₁)} for any 1?p₀, p₁, q₀, q₁?∞, and describe the interpolation orbit of any a∈L_p0(U₀)+L_p1(U₁) that is we describe a space of all {Ta}, where T runs over all linear bounded mappings from {L_p0(U₀),L_p1(U₁)} into {L_q0(V₀),L_q1(V₁)}. We show that interpolation orbit is obtained by the Lions–Peetre method of means with functional parameter as well as by the K-method with a weighted Orlicz space as a parameter. To cite this article: V.I. Ovchinnikov, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 881–884.     

On some special codes over Fp + uFp + uFp

We define a new map between codes over F_p + uF_p + u²F_p and F_p which is different to that defined in [2]. It is proved that the image of the linear cyclic code over the commutative ring F_p + uF_p + u²F_p with length n under this map is a distance-invariant quasi-cyclic code of index p² with length p²n over F_p. Moreover, it is proved that, if (n, p) = 1, then every code with length p²n over F_p which is the image of a linear (1 − u²)-cyclic code with length n over F_p + uF_p + u²F_p under this map is permutation equivalent to a quasi-cyclic code of index p².     

On a continuous analogue of the stochastic difference equation Xn=ρXn-1+Bn

Let B₁, B₂, ... be a sequence of independent, identically distributed random variables, letX₀ be a random variable that is independent ofB_n forn?1, let ρ be a constant such that 0<ρ<1 and letX₁,X₂, ... be another sequence of random variables that are defined recursively by the relationshipsX_n=ρX_n-1+B_n. It can be shown that the sequence of random variablesX₁,X₂, ... converges in law to a random variableX if and only ifE[log⁺¦B₁¦]<∞. In this paper we let {B(t):0≦t<∞} be a stochastic process with independent, homogeneous increments and define another stochastic process {X(t):0?t<∞} that stands in the same relationship to the stochastic process {B(t):0?t<∞} as the sequence of random variablesX₁,X₂,...stands toB₁,B₂,.... It is shown thatX(t) converges in law to a random variableX ast →+∞ if and only ifE[log⁺¦B(1)¦]<∞ in which caseX has a distribution function of class L. Several other related results are obtained. The main analytical tool used to obtain these results is a theorem of Lukacs concerning characteristic functions of certain stochastic integrals.     

Minimal free resolutions of monomial curves in P3k

Minimal free resolutions for prime ideals with generic zero (tn3,tn3?n10tn11,tn3?n20tn2, tn31), n1<n2<n3 positive integers, (n1,n2,n3)=1, are determined.     

Solving Ck/Cm/1/N queues by using characteristic roots in matrix analytic methods

In this paper, we study a C_k/C_m/1/N open queueing system with finite capacity. We investigate the property which shows that a product of the Laplace Stieltjes Transforms of interarrival and service times distributions satisfies an equation of a simple form. According to this equation, we present that the stationary probabilities on the unboundary states can be written as a linear combination of vector product-forms. Each component of these products is expressed in terms of roots of an associated characteristic polynomial. As a result, we carry out an algorithm for solving stationary probabilities in C_k/C_m/1/N systems, which is independent of N, hence greatly reducing the computational complexity.     

From U-bounds to isoperimetry with applications to H-type groups

In this paper we study U-bounds in relation to L₁-type coercive inequalities and isoperimetric problems for a class of probability measures on a general metric space (RN,d). We prove the equivalence of an isoperimetric inequality with several other coercive inequalities in this general framework. The usefulness of our approach is illustrated by an application to the setting of H-type groups, and an extension to infinite dimensions.     

The Eulerian generating function of q-derangements

Let F_q denote the finite field with q elements. For nonnegative integers n,k, let d_q(n,k) denote the number of n×nF_q-matrices having k as the sum of the dimensions of the eigenspaces (of the eigenvalues lying in F_q). Let d_q(n)=d_q(n,0), i.e., d_q(n) denotes the number of n×nF_q-matrices having no eigenvalues in F_q. The Eulerian generating function of d_q(n) has been well studied in the last 20 years [Kung, The cycle structure of a linear transformation over a finite field, Linear Algebra Appl. 36 (1981) 141-155, Neumann and Praeger, Derangements and eigenvalue-free elements in finite classical groups, J. London Math. Soc. (2) 58 (1998) 564-586 and Stong, Some asymptotic results on finite vector spaces, Adv. Appl. Math. 9(2) (1988) 167-199]. The main tools have been the rational canonical form, nilpotent matrices, and a q-series identity of Euler. In this paper we take an elementary approach to this problem, based on Möbius inversion, and find the following bivariate generating function:

     

Nonlinear analysis of the orbital motions of immersed rotors using a spectral/Galerkin approach

We recently developed a symbolic-numerical formulation for the nonlinear planar motion of rotors under fluid confinement, based on a spectral/Galerkin approach, for gap geometries of about δ=H/R≈0.1––where H is the average annular gap and R is the rotor radius. Results showed a quite good agreement between the class of approximate models generated, the corresponding analytical exact planar model and experiments. This methodology can be almost entirely automated on a symbolic computing environment.In the present paper this symbolic-numerical spectral/Galerkin procedure is extended in order to deal with nonlinear orbital motions––X(t) and Y(t) taking place in orthogonal directions.Numerical simulations performed over a centered rotor configuration maintained by nonisotropic supports (K^st_Y/K^st_X=0.4, where K^st_X and K^st_Y stand for the structural stiffnesses), which exhibit interesting dynamics, show a quite good agreement between this type of approximate models and the corresponding analytical exact (but quite involved) model, developed in the past by the authors.With the proposed symbolic-numerical approach one can obtain accurate nonlinear dynamical formulations enabling the study, understanding and prediction of nonlinear orbital rotor dynamics.     

Generalized comparison theorems for disfocality types of the equation Lny + py = 0

Comparison theorems for disfocality types on [a, ∞) of a pair of equations L_n(r_n, r_{n ? 1},…, r₀) y + py = 0 and L_v(?_v, ?_{v ? 1},…, ?₀) y + qy = 0 are given, where L_n and L_v are disconjugate linear differential operators, not necessarily of the same order, and p and q are continuous and of constant sign.     

Constraint programming for computing non-stationary (R, S) inventory policies

This paper proposes a constraint programming model for computing the finite horizon single-item inventory problem with stochastic demands in discrete time periods with service-level constraints under the non-stationary version of the “periodic review, order-up-to-level” policy (i.e., non-stationary (R, S) or, simply (R_n, S_n)). It is observed that the modeling process is more natural and the required number of variables is smaller compared to the MIP formulation of the same problem. The computational tests show that the CP approach is more tractable than the conventional MIP formulation. Two different domain reduction methods are proposed to improve the computational performance of solution algorithms. The numerical experiments confirmed the effectiveness of these methods.