A Generalized Chordal Metric in Control Theory Making Strong Stabilizability a Robust Property

An abstract chordal metric is defined on linear control systems described by their transfer functions. Analogous to a previous result due to Partington (Linear Operators and Linear Systems. An Analytical Approach to Control Theory. London Mathematical Society Student Texts, vol. 60, Cambridge University Press, Cambridge, 2004) for $H^\infty $ , it is shown that strong stabilizability is a robust property in this metric.     

Book Reviews

Numerical Ranges of Operators on Normed spaces and of elements of Normed Algebras, by F. F. Bonsall and J. Duncan. (London Mathematical Society Lecture Note, Series, 2). Cambridge University Press, 1971. ($3.95) Numerical RangesII, by F. F. Bonsall and J. Duncan. (London Mathematical Society Lecture Note, Series, 10). Cambridge University Press, 1973. ($7.50) Latent roots and latent vectors, by S.J. Hammerling ($13.50)

Latent Roots and Latent Vectors, by S.J. Hammerling. ($13.50)     

Book review

Skew Fields, by P. K. Draxl. London Mathematical Society Lecture Notes No. 18, Cambridge University Press, 1983, ix + 182 pp.     

Book review

Skew Fields, by P. K. Draxl. London Mathematical Society Lecture Notes No. 18, Cambridge University Press, 1983, ix + 182 pp.     

Difference sets with few character values

The known families of difference sets can be subdivided into three classes: difference sets with Singer parameters, cyclotomic difference sets, and difference sets with gcd \((v,n)>1\) . It is remarkable that all the known difference sets with gcd \((v,n)>1\) have the so-called character divisibility property. Jungnickel and Schmidt (Difference sets: an update. London Math. Soc. Lecture Note Ser., vol. 245, pp. 89–112, Cambridge University Press, Cambridge 1997) posed the problem of constructing difference sets with gcd \((v,n)>1\) that do not satisfy this property. In an attempt to attack this problem, we use difference sets with three nontrivial character values as candidates, and get some necessary conditions.     

Lyapunov stability for semigroup actions

In this paper we introduce a theory of Lyapunov stability of sets for semigroup actions on Tychonoff spaces. We also present the main properties and the main results relating these new concepts. We generalize several concepts and results of Lyapunov stable sets from Bhatia and Hajek (Local Semi-Dynamical Systems. Lecture Notes in Mathematics, vol. 90. Springer, Berlin, 1969), Bhatia and Szegö (Dynamical Systems: Stability Theory and Applications. Lecture Notes in Mathematics, vol. 35. Springer, Berlin, 1967; and Stability Theory of Dynamical Systems. Springer, Berlin, 1970).     

On the Chvátal–Gomory closure of a compact convex set

In this paper, we show that the Chvátal–Gomory closure of any compact convex set is a rational polytope. This resolves an open question of Schrijver (Ann Discret Math 9:291–296, 1980) for irrational polytopes, and generalizes the same result for the case of rational polytopes (Schrijver in Ann Discret Math 9:291–296, 1980), rational ellipsoids (Dey and Vielma in IPCO XIV, Lecture Notes in Computer Science, vol 6080. Springer, Berlin, pp 327–340, 2010) and strictly convex bodies (Dadush et al. in Math Oper Res 36:227–239, 2011).     

Book reviews

INtroduction to Group Characters, by Lederman, L.A.M.A., Cambridge University Press, 1977.

Matrix Representations in the Theory of Finite Groups, by V. A. Belonogov and A. N. Fomin. Nauka, Moscow, 1976 (in Russian). 48 kopecks.

Transformation Groups, Czes Kosniowski, Editor. London Mathematical Society Lecture Notes 26. Cambridge University Press, New York, 1977.     

Book Reviews

Methods of Intermediate Problems for Eigenvalues: Theory and Ramifications, by Alexander Weinstein and William Stenger. Academic Press, 1971. ($16.00)

Combinatorics, Proceedings of the British Combinatorial Conference 1973, Edited by T.P. McDnough and V.C. Mavron, London Mathematical Society Lecture Note Series 13, Cambridge University Press, London, New York, 1974. 204pp.($9.95)     

Jensen’s inequality on convex spaces

We prove a Jensen’s inequality on $p$ -uniformly convex space in terms of $p$ -barycenters of probability measures with $(p-1)$ -th moment with $p\in ]1,\infty [$ under a geometric condition, which extends the results in Kuwae (Jensen’s inequality over CAT $(\kappa )$ -space with small diameter. In: Proceedings of Potential Theory and Stochastics, Albac Romania, pp. 173–182. Theta Series in Advanced Mathematics, vol. 14. Theta, Bucharest, 2009) , Eells and Fuglede (Harmonic maps between Riemannian polyhedra. In: Cambridge Tracts in Mathematics, vol. 142. Cambridge University Press, Cambridge, 2001) and Sturm (Probability measures on metric spaces of nonpositive curvature. Probability measures on metric spaces of nonpositive curvature. In: Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), pp. 357–390. Contemporary Mathematics, vol. 338. American Mathematical Society, Providence, 2003). As an application, we give a Liouville’s theorem for harmonic maps described by Markov chains into $2$ -uniformly convex space satisfying such a geometric condition. An alternative proof of the Jensen’s inequality over Banach spaces is also presented.     

A New Proof for the Convergence of an Individual Based Model to the Trait Substitution Sequence

We consider a continuous time stochastic individual based model for a population structured only by an inherited vector trait and with logistic interactions. We consider its limit in a context from adaptive dynamics: the population is large, the mutations are rare and the process is viewed in the timescale of mutations. Using averaging techniques due to Kurtz (in Lecture Notes in Control and Inform. Sci., vol. 177, pp. 186–209, 1992), we give a new proof of the convergence of the individual based model to the trait substitution sequence of Metz et al. (in Trends in Ecology and Evolution 7(6), 198–202, 1992), first worked out by Dieckman and Law (in Journal of Mathematical Biology 34(5–6), 579–612, 1996) and rigorously proved by Champagnat (in Theoretical Population Biology 69, 297–321, 2006): rigging the model such that “invasion implies substitution”, we obtain in the limit a process that jumps from one population equilibrium to another when mutations occur and invade the population.     

Proper and adjoint exhausters in nonsmooth analysis: optimality conditions

The notions of upper and lower exhausters represent generalizations of the notions of exhaustive families of upper convex and lower concave approximations (u.c.a., l.c.a.). The notions of u.c.a.’s and l.c.a.’s were introduced by Pshenichnyi (Convex Analysis and Extremal Problems, Series in Nonlinear Analysis and its Applications, 1980), while the notions of exhaustive families of u.c.a.’s and l.c.a.’s were described by Demyanov and Rubinov in Nonsmooth Problems of Optimization Theory and Control, Leningrad University Press, Leningrad, 1982. These notions allow one to solve the problem of optimization of an arbitrary function by means of Convex Analysis thus essentially extending the area of application of Convex Analysis. In terms of exhausters it is possible to describe extremality conditions, and it turns out that conditions for a minimum are expressed via an upper exhauster while conditions for a maximum are formulated in terms of a lower exhauster (Abbasov and Demyanov (2010), Demyanov and Roshchina (Appl Comput Math 4(2): 114–124, 2005), Demyanov and Roshchina (2007), Demyanov and Roshchina (Optimization 55(5–6): 525–540, 2006)). This is why an upper exhauster is called a proper exhauster for minimization problems while a lower exhauster is called a proper one for maximization problems. The results obtained provide a simple geometric interpretation and allow one to construct steepest descent and ascent directions. Until recently, the problem of expressing extremality conditions in terms of adjoint exhausters remained open. Demyanov and Roshchina (Appl Comput Math 4(2): 114–124, 2005), Demyanov and Roshchina (Optimization 55(5–6): 525–540, 2006) was the first to derive such conditions. However, using the conditions obtained (unlike the conditions expressed in terms of proper exhausters) it was not possible to find directions of descent and ascent. In Abbasov (2011) new extremality conditions in terms of adjoint exhausters were discovered. In the present paper, a different proof of these conditions is given and it is shown how to find steepest descent and ascent conditions in terms of adjoint exhausters. The results obtained open the way to constructing numerical methods based on the usage of adjoint exhausters thus avoiding the necessity of converting the adjoint exhauster into a proper one.     

Strong NP-hardness of scheduling problems with learning or aging effect

Proofs of strong NP-hardness of single machine and two-machine flowshop scheduling problems with learning or aging effect given in Rudek (Computers & Industrial Engineering 61:20–31, 2011; Annals of Operations Research 196(1):491–516, 2012a; International Journal of Advanced Manufacturing Technology 59:299–309, 2012b; Applied Mathematics and Computations 218:6498–6510, 2012c; Applied Mathematical Modelling 37:1523–1536, 2013) contain a common mistake that make them incomplete. We reveal the mistake and provide necessary corrections for the problems in Rudek (Computers & Industrial Engineering 61:20–31, 2011; Annals of Operations Research 196(1):491–516, 2012a; Applied Mathematical Modelling 37:1523–1536, 2013). NP-hardness of problems in Rudek (International Journal of Advanced Manufacturing Technology 59:299–309, 2012b; Applied Mathematics and Computations 218:6498–6510, 2012c) remains unknown because of another mistake which we are unable to correct.     

Positivstellens?tze for real function algebras

We look for algebraic certificates of positivity for functions which are not necessarily polynomial functions. Similar questions were examined earlier by Lasserre and Putinar [Positivity and optimization for semi-algebraic functions (to appear), Proposition 1] and by Putinar [A Striktpositivestellensatz for measurable functions (corrected version) (to appear), Theorem 2.1]. We explain how these results can be understood as results on hidden positivity: The required positivity of the functions implies their positivity when considered as polynomials on the real variety of the respective algebra of functions. This variety is however not directly visible in general. We show how algebras and quadratic modules with this hidden positivity property can be constructed. We can then use known results, for example Jacobi’s representation theorem (Jacobi in Math Z 237:259–273, 2001, Theorem 4), or the Krivine-Stengle Positivstellensatz (Marshall in Positive polynomials and sums of squares. Mathematical Surveys and Monographs 146, 2008, page 25), to obtain certificates of positivity relative to a quadratic module of an algebra of real-valued functions. Our results go beyond the results of Lasserre and Putinar, for example when dealing with non-continuous functions. The conditions are also easier to check. We explain the application of our result to various sorts of real finitely generated algebras of semialgebraic functions. The emphasis is on the case where the quadratic module is also finitely generated. Our results also have application to optimization of real-valued functions, using the semidefinite programming relaxation methods pioneered by Lasserre [SIAM J Optim 11(3): 796–817, 2001; Lasserre in Moments, positive polynomials and their applications. Imperial College Press, London, 2009; Lasserre and Putinar in Positivity and optimization for semi-algebraic functions (to appear); Marshall in Positive polynomials and sums of squares. Mathematical Surveys and Monographs 146, 2008, page 25].     

Pollaczek polynomials and hypergeometric representation

This paper gives a solution, without the use of the three-term recurrence relation, of the problem posed in Ismail (Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, Cambridge, 2005) (Problem 24.8.2, p. 658): that the hypergeometric representation of the general Pollaczek polynomials is a polynomial in cos(θ) of degree n. Chu solved in (Ramanujan J. 13(1–3): 221–225, 2007) the problem in a particular case. We use elementary properties of functions of complex variables and Pfaff’s transformation on hypergeometric ₂ F ₁-series.     

Further variations on the theme of FC-groups

Groups that are FC, or more generally satisfy any of the weakenings of the FC-condition considered in de Giovanni (Serdica Math. J. 28:241?C254, 2002) and Robinson et?al. (J. Algebra 326:218?C226, 2011), have local systems consisting of normal finite-by-nilpotent subgroups. Apart from generalizing results from de Giovanni (Serdica Math. J. 28:241?C254, 2002) and Robinson et al. (J. Algebra 326:218?C226, 2011) to the more general context of locally (normal and finite-by-nilpotent) groups, we partially settle an open problem raised in Robinson et?al. (J. Algebra 326:218?C226, 2011) concerning the isomorphism of maximal p-subgroups, but in this more general setting of locally (normal and finite-by-nilpotent) groups.     

Un lemme matriciel effectif

We show an almost optimal effective version of Masser’s matrix lemma (Lecture Notes Math. 1290:109–148, 1987), giving a lower bound of the height of an abelian variety in terms of its period lattices.     

Beamlets are densely embedded in H −1

The insufficiency of using ordinary measurable functions to model complex natural images was first emphasized by David Mumford (Q Appl Math 59:85–111, 2001). The idea was later rediscovered by Yves Meyer (Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, vol. 22, University Lecture Series, AMS, Providence, 2001) who introduced proper texture models based on generalized functions or distributions. The simpler but effective Sobolev texture model of H ^???1 was subsequently explored by Osher et al. (Multiscale Model Simul 1:349–370, 2003) to facilitate practical computation. H ^???1 textures have also been further employed in the recent works of Daubechies and Teschke (Appl Comput Harmon Anal 19(1):1–16, 2005), Lieu and Vese (UCLA CAM Tech Report, 05–33, 2005), Shen (Appl Math Res Express 4:143–167, 2005), and many others, leading to a new generation of models for image processing and analysis. On the other hand, beamlets are the unconventional class of geometric wavelets invented by Donoho and Huo (Multiscale and Multiresolution Methods, Lect Notes Comput Sci Eng, vol. 20, pp. 149–196. Springer, Berlin, 2002) to efficiently represent and detect lower dimensional singular image features. In the current work, we make an intriguing connection between the above two realms by demonstrating that H ^???1 is the natural space (of generalized functions) that hosts beamlets, and in return can be completely described by them. Computational evidences existing in the literature also help confirm this newly discovered bond.