Effective Constructions in Plethysms and Weintraub’s Conjecture

We give a short proof of Weintraub’s conjecture (Weintraub J Algebra 129:103–114, 1990), first proved in Bürgisser et al. (J Algebra 328:322–329, 2011), by constructing explicit highest weight vectors in the plethysms S ^p (?∧?^2q W).     

On the theory of fronts on the 2-sphere and the theory of space curves

We apply the singularity theory of caustics and wave fronts on the 2-sphere and the theory of Legendrian curves of the (contact) manifold ST ^{* 2} P ³ (consisting of tangent contact elements of the sphere ²) to obtain new results in the theory of curves in the Euclidean 3-space. In this way, the singularity theory of caustics and wave fronts provides a new insight to the differential geometry of curves in the Euclidean 3-space.Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 7, Suzdal Conference-1, 2003.This revised version was published online in April 2005 with a corrected cover date.     

A note on polar decomposition based Geršgorin-type sets

Let B∈C^n×n denote a finite-dimensional square complex matrix. In [L. Smithies, R.S. Varga, Singular value decomposition Geršgorin sets, J. Linear Algebra Appl. 417 (2004) 370-380; N. Fontes, J. Kover, L. Smithies, R.S. Varga, Singular value decomposition normally estimated Geršgorin sets, Electron. Trans. Numer. Anal. 26 (2007) 320-329], Professor Varga and I introduced Geršgorin-type sets which were developed from singular value decompositions (SVDs) of B. In this note, our work is extended by introducing the polar SV-Geršgorin set, Γ^PSV(B). The set Γ^PSV(B) is a union of n closed discs in C, whose centers and radii are defined in terms of the entries of a polar decomposition B=Q|B|. The set of eigenvalues of B, σ(B), is contained in Γ^PSV(B).     

Linear equations with unknowns from a multiplicative group whose solutions lie in a small number of subspaces

Let K be a field of characteristic 0 and let (K^*)ⁿ denote the n-fold Cartesian product of K^*, endowed with coordinatewise multiplication. Let Γ be a subgroup of (K^*)ⁿ of finite rank. We consider equations (*) a₁x₁ + … + a_nx_n = 1 in x = (x₁x_n)Γ, where a = (a₁,a_n)(K^*)ⁿ. Two tuples a, b(K^*)ⁿ are called Γ-equivalent if there is a uΓ such that b = u · a. Gy?ry and the author [Compositio Math. 66 (1988) 329-354] showed that for all but finitely many Γ-equivalence classes of tuples a(K^*)ⁿ, the set of solutions of (*) is contained in the union of not more than 2⁽ⁿ⁺¹! proper linear subspaces of Kⁿ. Later, this was improved by the author [J. reine angew. Math. 432 (1992) 177-217] to (n!)²ⁿ⁺². In the present paper we will show that for all but finitely many Γ-equivalence classes of tuples of coefficients, the set of non-degenerate solutions of (*) (i.e., with non-vanishing subsums) is contained in the union of not more than 2ⁿ proper linear subspaces of Kⁿ. Further we give an example showing that 2ⁿ cannot be replaced by a quantity smaller than n.     

The asymmetric travelling salesman problem and a reformulation of the Miller–Tucker–Zemlin constraints

In this paper we compare the linear programming (LP) relaxations of several old and new formulations for the asymmetric travelling salesman problem (ATSP). The main result of this paper is the derivation of a compact formulation whose LP relaxation is characterized by a set of circuit inequalities given by Grotschel and Padberg (In: Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A., Shmoys, D.B. (Eds.), The Travelling Salesman Problem: A Guided Tour of Combinatorial Optimization. Wiley, New York, 1985). The new compact model is an improved and disaggregated version of a well-known model for the ATSP based on the subtour elimination constraints (Miller et al., Journal of ACM 7 (1960) 326–329). The circuit inequalities are weaker than the subtour elimination constraints given by Dantzig et al. However, each one of these circuit inequalities can be lifted into several different facet defining inequalities which are not dominated by the subtour elimination inequalities. We show that some of the inequalities involved in the previously mentioned compact formulation can be lifted in such a way that, by projection, we obtain a small subset of the so-called D_k^← and D_k^→ inequalities. This shows that the LP relaxation of our strongest model is not dominated by the LP relaxation of the model presented by Dantzig et al. (Operations Research 2 (1954) 393–410). The new models motivate a new classification of formulations for the ATSP.     

On the outputs of linear control systems

This paper studies autonomous, single-input, single-output linear control systems on finite time intervals. The object of interest is the output operatorO, which associates to each input function and initial state vector the corresponding system output. Main result: If the system has relative degree r<∞, then for any “admissible” Banach space U of inputs, O is a bounded operator taking U×Cn onto the “Sobolev space” of complex functions f∈C(r−1)([0,T]) for which the (r−1)-order derivative f(r−1) is absolutely continuous, with f(r)∈U. This completes recent results of Jönsson and Martin [Ulf Jönsson, Clyde Martin, Approximation with the output of linear control systems, J. Math. Anal. Appl. 329 (2007) 798-821] who showed that if the system is minimal and U is either L²([0,T]) or C([0,T]), then has dense range.     

Regularity of the free boundary in two-phase problems for linear elliptic operators

In this paper we complete the study of the regularity of the free boundary in two-phase problems for linear elliptic operators started in [M.C. Cerutti, F. Ferrari, S. Salsa, Two-phase problems for linear elliptic operators with variable coefficients: Lipschitz free boundaries are C1,γ, Arch. Ration. Mech. Anal. 171 (2004) 329-348]. In particular we prove that Lipschitz and flat free boundaries (in a suitable sense) are smooth. As byproduct, we prove that Lipschitz free boundaries are smooth in the case of quasilinear operators of the form div(A(x,u)∇u) with Lipschitz coefficients.     

A Laplace integral on a Kähler manifold and Calabi's diastasis function

In this paper we give a different proof of Engliš's result [J. Reine Angew. Math. 528 (2000) 1-39] about the asymptotic expansion of a Laplace integral on a real analytic Kähler manifold (M,g) by using the link between the metric g and the associated Calabi's diastasis function D. We also make explicit the connection between the coefficients of Engliš' expansion and Gray's invariants [Michigan Math. J. (1973) 329-344].     

Planar 3DM is NP-complete

A restriction of the three-dimensional matching problem 3DM, in which the associated bipartite graph is planar, is shown to remain NP-complete. The restriction is inspired by that of Lichtenstein's planar 3SAT (Planar formulae and their uses, SIAM J. Comput. 11 (1982), 329–343). Like Planar 3SAT, Planar 3DM is principally a tool for use in NP-completeness proofs for planar restrictions of other problems. Several examples of its applications in this respect are given.     

Circular distributions and a result of Solomon

In his paper (Invent. Math. 109 (1992) 329-350), Solomon finds an information on the prime factorization of an element coming from a circular unit 1-ζ over the ideal class group of a real abelian number field L, where ζ denotes a root of unity. Using this he obtains an annihilator of the p-Sylow subgroup of the subgroup of the ideal class group of L generated by the classes of prime ideals lying above p. We generalize this result to the circular distributions which has the axiomatic definition of Euler systems as its defining property.     

Estimating the diameter of the space of plane convex figures with respect to an affine-invariant metric

A convex figure K ⊂ ℝ² is a compact convex set with nonempty interior, and αK is a homothetic image of K with coefficient α ∈ ℝ. It is proved that for any two convex figures K₁, K₂ ⊂ ℝ² there is an affine transformation T of the plane such that K₁ ⊂ T(K₂) ⊂ 2.7K₁. Bibliography: 2 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 58–66.     

Stable complete surfaces with constant mean curvature

Letx:M ²N ³ be a stable immersion with constant mean curvatureH of a complete orientable surfaceM ² into a complete oriented three dimensional Riemannian manifoldN ³. In this paper we prove that, ifM ² is compact andH ²> –1/2 inf _M Ricc _N , thenM ² has genusg3, here Ricc _N is the Ricci curvature ofN ³. We also prove that, ifM ² is complete non compact andN ² has bounded geometry, the area ofM ² is infinite in the metric induced byx. In this case, ifH ²–1/2 inf _M Ricc _N thenx is umbilic and the equality holds.     

Homogenization and concentrated capacity in reticular almost disconnected structuresHomogénéisation et capacité concentrée dans les structures réticulaires presque déconnectées

We compute the homogenized-concentrated limit for a pair of non-linearly coupled diffusion equations in a perforated cylindric domain with coaxial cylindric holes periodically distributed along its axis. This problem arises from visual transduction. To cite this article: D. Andreucci et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 329–332.     

On the supremum of the pseudocompact group topologies

P is the class of pseudocompact Hausdorff topological groups, and P^′ is the class of groups which admit a topology T such that (G,T)∈P. It is known that every G=(G,T)∈P is totally bounded, so for G∈P^′ the supremum T^∨(G) of all pseudocompact group topologies on G and the supremum T#(G) of all totally bounded group topologies on G satisfy T^∨⊆T#.The authors conjecture for abelian G∈P^′ that T^∨=T#. That equality is established here for abelian G∈P^′ with any of these (overlapping) properties. (a) G is a torsion group; (b) |G|?c²; (c) r₀(G)=|G|=ω|G|; (d) |G| is a strong limit cardinal, and r₀(G)=|G|; (e) some topology T with (G,T)∈P satisfies w(G,T)?c; (f) some pseudocompact group topology on G is metrizable; (g) G admits a compact group topology, and r₀(G)=|G|. Furthermore, the product of finitely many abelian G∈P^′, each with the property T^∨(G)=T#(G), has the same property.     

On the BIB design having the minimum p-rank

It is shown that (i) among BIB designs with parameters (2^t+1 ? 1, 2^t+1 ? 1, 2^t ? 1, 2^t ? 1, 2^t?1 ? 1), the incidence matrix of the BIB design PG(t, 2):t ? 1 derived from a finite projective geometry PG(t, 2) has the minimum 2-rank and (ii) among BIB designs with parameters (2^t, 2^t+1 ? 2, 2^t ? 1, 2^?1, 2^t?1 ? 1), the incidence matrix of the BIB design EG(t, 2):t ? 1 derived from an affine geometry EG(t, 2) has the minimum 2-rank.     

Further results on the reverse-order law

An explicit expression is obtained for a pair of generalized inverses (B^?,A^?) such that B^?A^?=(AB)⁺_MN, and a class of pairs (B^?,A^? of this property is shown. A necessary and sufficient condition for (AB)^? to have the expression B^?A^? is also given.     

On the covariance of the Moore-Penrose inverse

Given a square complex matrix A with Moore-Penrose inverse A², we describe the class of invertible matrices T such that (TAT^-1)²=TA²T^-1.     

Upper bounds for the autocorrelation coefficients of the Rudin-Shapiro polynomials

Givena _m to be them ^th correlation coefficient of the Rudin-Shapiro polynomials of degrees 2 ⁿ ? 1, ¦a _m¦ ≤ C(2 ⁿ )^3/4 and there existsk ≠ 0 such that ¦a _k¦ >D(2 ⁿ )^0.73 (C andD are universal constants). Here we show that the 0.73 is optimal in the upper bound case.     

Classifying simple closed curve pairs in the 2-sphere and a generalization of the Schoenflies theorem

A classification theory is developed for pairs of simple closed curves (A,B) in the sphere S², assuming that A∩B has finitely many components. Such a pair of simple closed curves is called an SCC-pair, and two SCC-pairs (A,B) and (A^′,B^′) are equivalent if there is a homeomorphism from S² to itself sending A to A^′ and B to B^′. The simple cases where A and B coincide or A and B are disjoint are easily handled. The component code is defined to provide a classification of all of the other possibilities. The component code is not uniquely determined for a given SCC-pair, but it is straightforward that it is an invariant; i.e., that if (A,B) and (A^′,B^′) are equivalent and C is a component code for (A,B), then C is a component code for (A^′,B^′) as well. It is proved that the component code is a classifying invariant in the sense that if two SCC-pairs have a component code in common, then the SCC-pairs are equivalent. Furthermore code transformations on component codes are defined so that if one component code is known for a particular SCC-pair, then all other component codes for the SCC-pair can be determined via code transformations. This provides a notion of equivalence for component codes; specifically, two component codes are equivalent if there is a code transformation mapping one to the other. The main result of the paper asserts that if C and C^′ are component codes for SCC-pairs (A,B) and (A^′,B^′), respectively, then (A,B) and (A^′,B^′) are equivalent if and only if C and C^′ are equivalent. Finally, a generalization of the Schoenflies theorem to SCC-pairs is presented.