On a generalization of the master cyclic group polyhedron

We study the master equality polyhedron (MEP) which generalizes the master cyclic group polyhedron (MCGP) and the master knapsack polyhedron (MKP). We present an explicit characterization of the polar of the nontrivial facet-defining inequalities for MEP. This result generalizes similar results for the MCGP by Gomory (1969) and for the MKP by Araóz (1974). Furthermore, this characterization gives a polynomial time algorithm for separating an arbitrary point from MEP. We describe how facet-defining inequalities for the MCGP can be lifted to obtain facet-defining inequalities for MEP, and also present facet-defining inequalities for MEP that cannot be obtained in such a way. Finally, we study the mixed-integer extension of MEP and present an interpolation theorem that produces valid inequalities for general mixed integer programming problems using facets of MEP.     

Factorizing Kernel Operators

Consider an operator ${T: X(\mu) \rightarrow Y(\mu)}$ between Banach function spaces having adequate order continuity and Fatou properties. Assume that T can be factorized through a Banach space as ${T = S \circ R}$ , where R and the adjoint of S are p-th power and q-th power factorable, respectively. Then a canonical factorization scheme can be given for T. We show that it provides a tool for analyzing T that becomes specially useful for the case of kernel operators. In particular, we show that this square factorization scheme for T is equivalent to some inequalities for the bilinear form defined by T. Kernel operators are studied from this point of view.     

On the Minimization of Dirichlet Eigenvalues of the Laplace Operator

We study variational problems of the form $$\inf\{\lambda_k(\Omega): \Omega\ \mbox{open in}\ \mathbb{R}^m,\ T(\Omega ) \le1 \},$$ where λ _k (Ω) is the k-th eigenvalue of the Dirichlet Laplacian acting in L ²(Ω), and where T is a non-negative set function defined on the open sets in ? ^m , which is invariant under isometries, additive on disjoint families of open sets, and is such that the ball with T(B)=1 is a minimizer for k=1. Upper bounds are obtained for the number of components of any bounded minimizer if T satisfies a scaling relation. For example, we show that if T is Lebesgue measure and if k≤m+1 then any bounded minimizer has at most 7 components. We also consider variational problems over open sets Ω in ? ^m involving the (m?1)-dimensional Hausdorff measure of ?Ω.     

Absolutely continuous operators on function spaces and vector measures

Let (Ω, Σ, μ) be a finite atomless measure space, and let E be an ideal of L ⁰(μ) such that ${L^\infty(\mu) \subset E \subset L^1(\mu)}$ . We study absolutely continuous linear operators from E to a locally convex Hausdorff space ${(X, \xi)}$ . Moreover, we examine the relationships between μ-absolutely continuous vector measures m : Σ → X and the corresponding integration operators T _m : L ^∞(μ) → X. In particular, we characterize relatively compact sets ${\mathcal{M}}$ in ca _μ (Σ, X) (= the space of all μ-absolutely continuous measures m : Σ → X) for the topology ${\mathcal{T}_s}$ of simple convergence in terms of the topological properties of the corresponding set ${\{T_m : m \in \mathcal{M}\}}$ of absolutely continuous operators. We derive a generalized Vitali–Hahn–Saks type theorem for absolutely continuous operators T : L ^∞(μ) → X.     

The size of sums of sets,II

We prove inequalities which give lower bounds for the Lebesgue measures of setsE +K whereK is a certain kind of Cantor set. For example, ifC is the Cantor middle-thirds subset of the circle groupT, then $$m(E)^{1 - log2/log3} \leqq m(E + C)$$ for every BorelE ?T.     

Numerical solution of AXB=C for (R,S)-symmetric matrices

Let $R\in\mathbb{R}^{m\times m}$ and $S\in\mathbb{R}^{n\times n}$ be real nontrivial symmetric involutions; i.e., R=R ^T =R ^?1≠±I _m and S=S ^T =S ^?1≠±I _n . We say that $A\in \mathbb{R}^{m\times n}$ is (R,S)-symmetric ((R,S)-skew symmetric) if RAS=A (RAS=?A). Trench (Linear Algebra Appl. 389:23–31, 2004) has theoretically studied the minimization problems and the related approximation problems of matrix equation AZ=W for (R,S)-symmetric matrices, using their structure properties and Moore-Penrose inverse. In this paper, we extend and develop these research in a totally different way using iterative methods. We propose two algorithms based on the idea of the classical Conjugate Gradient method (CG) and Conjugate Gradient Least Squares method (CGLS), to solve the more general equation AXB=C for (R,S)-symmetric matrices X. Some numerical results confirm the efficiency of these algorithms. More importantly, some numerical stability analysis on the approximation problem is given combining with numerical examples, which is not given in the earlier papers.     

L 2-Harmonic 1-Forms on Submanifolds with Finite Total Curvature

Let $x:M^{m}\to\bar{M}$ , m≥3, be an isometric immersion of a complete noncompact manifold M in a complete simply connected manifold $\bar{M}$ with sectional curvature satisfying $-k^{2}\leq K_{\bar{M}}\leq0$ , for some constant k. Assume that the immersion has finite total curvature in the sense that the traceless second fundamental form has finite L ^m -norm. If $K_{\bar{M}}\not\equiv0$ , assume further that the first eigenvalue of the Laplacian of M is bounded from below by a suitable constant. We prove that the space of the L ² harmonic 1-forms on M has finite dimension. Moreover, there exists a constant Λ>0, explicitly computed, such that if the total curvature is bounded from above by Λ then there are no nontrivial L ²-harmonic 1-forms on M.     

An inequality for a functional on aging distribution functions

We prove an inequality for a functional on aging distribution functions F(t), which makes it possible to obtain inequalities for \(m_r = \int_0^\infty {t^r } dF (t)\) . We show that if \(\left[ {\frac{{m_r }}{{r!}}} \right]^{r + 1} = \left[ {\frac{{m_{r + 1} }}{{(r + 1)!}}} \right]^r \) for some r ≥ 1, then F(t) = 1^?e^?λt; in addition we give upper and lower bounds for the integral \(\int_0^\infty {e^{ - st} } [1 - F(1)] dt,\) expressed in terms of m₁ and m₂.     

Forbidden Configurations and Product Constructions

A simple matrix is a (0,1)-matrix with no repeated columns. For a (0,1)-matrix F, we say that a (0,1)-matrix A has F as a configuration if there is a submatrix of A which is a row and column permutation of F (trace is the set system version of a configuration). Let \({\|A\|}\) denote the number of columns of A. We define \({{\rm forb}(m, F) = {\rm max}\{\|A\| \,:\, A}\) is m-rowed simple matrix and has no configuration F. We extend this to a family \({\mathcal{F} = \{F_1, F_2, \ldots , F_t\}}\) and define \({{\rm forb}(m, \mathcal{F}) = {\rm max}\{\|A\| \,:\, A}\) is m-rowed simple matrix and has no configuration \({F \in \mathcal{F}\}}\) . We consider products of matrices. Given an m ₁ × n ₁ matrix A and an m ₂ × n ₂ matrix B, we define the product A × B as the (m ₁ + m ₂) × n ₁ n ₂ matrix whose columns consist of all possible combinations obtained from placing a column of A on top of a column of B. Let I _k denote the k × k identity matrix, let \({I_k^{c}}\) denote the (0,1)-complement of I _k and let T _k denote the k × k upper triangular (0,1)-matrix with a 1 in position i, j if and only if i ≤ j. We show forb(m, {I ₂ × I ₂, T ₂ × T ₂}) is \({\Theta(m^{3/2})}\) while obtaining a linear bound when forbidding all 2-fold products of all 2 × 2 (0,1)-simple matrices. For two matrices F, P, where P is m-rowed, let \({f(F, P) = {\rm max}_{A} \{\|A\| \,:\,A}\) is m-rowed submatrix of P with no configuration F}. We establish f(I ₂ × I ₂, I _m/2 × I _m/2) is \({\Theta(m^{3/2})}\) whereas f(I ₂ × T ₂, I _m/2 × T _m/2) and f(T ₂ × T ₂, T _m/2 × T _m/2) are both \({\Theta(m)}\) . Additional results are obtained. One of the results requires extensive use of a computer program. We use the results on patterns due to Marcus and Tardos and generalizations due to Klazar and Marcus, Balogh, Bollobás and Morris.     

The K-moment problem with densities

Given a compact basic semi-algebraic set ${\mathbf{K}} \subset {\mathbb{R}}^n$ , a rational fraction $f:{\mathbb{R}}^n\to{\mathbb{R}}$ , and a sequence of scalars y = (y _α), we investigate when $y_\alpha =\int_{\mathbf{K}} x^\alpha f\,d\mu$ holds for all $\alpha\in{\mathbb{N}}^n$ , i.e., when y is the moment sequence of some measure fdμ, supported on K. This yields a set of (convex) linear matrix inequalities (LMI). We also use semidefinite programming to develop a converging approximation scheme to evaluate the integral ∫ fdμ when the moments of μ are known and f is a given rational fraction. Numerical expreriments are also provided. We finally provide (again LMI) conditions on the moments of two measures $\nu,\mu$ with support contained in K, to have $d\nu=f d\mu$ for some rational fraction f.     

The Lower Weyl Spectrum of a Positive Operator

For the lower Weyl spectrum $$\sigma_{\rm w}^-(T) = \bigcap_{0 \le K \in \mathcal{K}(E) \le T} \sigma(T - K),$$ where T is a positive operator on a Banach lattice E, the conditions for which the equality ${\sigma_{\rm w}^-(T) = \sigma_{\rm w}^-(T^*)}$ holds, are established. In particular, it is true if E has order continuous norm. An example of a weakly compact positive operator T on ? _∞ such that the spectral radius ${r(T) \in \sigma_{\rm w}^-(T) {\setminus} (\sigma_{\rm f}(T) \cup \sigma_{\rm w}^-(T^*))}$ , where σ _f(T) is the Fredholm spectrum, is given. The conditions which guarantee the order continuity of the residue T _?1 of the resolvent R(., T) of an order continuous operator T ≥ 0 at ${r(T) \notin \sigma_{\rm f}(T)}$ , are discussed. For example, it is true if T is o-weakly compact. It follows from the proven results that a Banach lattice E admitting an order continuous operator T ≥ 0, ${r(T) \notin \sigma_{\rm f}(T)}$ , can not have the trivial band ${E_n^\sim}$ of order continuous functionals in general. It is obtained that a non-zero order continuous operator T : E → F can not be approximated in the r-norm by the operators from ${E_\sigma^\sim \otimes F}$ , where F is a Banach lattice, ${E_\sigma^\sim}$ is a disjoint complement of the band ${E_n^\sim}$ of E*.     

Facets with fewest vertices

Forv>d≧3, letm(v, d) be the smallest numberm, such that every convexd-polytope withv vertices has a facet with at mostm vertices. In this paper, bounds form(v, d) are found; in particular, for fixedd≧3, $$\frac{{r - 1}}{r} \leqslant \mathop {\lim \inf }\limits_{\upsilon \to \infty } \frac{{m(\upsilon ,d)}}{\upsilon } \leqslant \mathop {\lim \sup }\limits_{\upsilon \to \infty } \frac{{m(\upsilon ,d)}}{\upsilon } \leqslant \frac{{d - 3}}{{d - 2}}$$ , wherer=[1/3(d+1)].     

Distinguishing-Transversal in Hypergraphs and Identifying Open Codes in Cubic Graphs

The open neighborhood N(v) of a vertex v in a graph G is the set of vertices adjacent to v in G. A graph is twin-free (or open identifiable) if every two distinct vertices have distinct open neighborhoods. A separating open code in G is a set C of vertices such that \({N(u) \cap C \neq N(v) \cap C}\) for all distinct vertices u and v in G. An open dominating set, or total dominating set, in G is a set C of vertices such that \({N(u) \cap C \ne N(v) \cap C}\) for all vertices v in G. An identifying open code of G is a set C that is both a separating open code and an open dominating set. A graph has an identifying open code if and only if it is twin-free. If G is twin-free, we denote by \({\gamma^{\rm IOC}(G)}\) the minimum cardinality of an identifying open code in G. A hypergraph H is identifiable if every two edges in H are distinct. A distinguishing-transversal T in an identifiable hypergraph H is a subset T of vertices in H that has a nonempty intersection with every edge of H (that is, T is a transversal in H) such that T distinguishes the edges, that is, \({e \cap T \neq f \cap T}\) for every two distinct edges e and f in H. The distinguishing-transversal number \({\tau_D(H)}\) of H is the minimum size of a distinguishing-transversal in H. We show that if H is a 3-uniform identifiable hypergraph of order n and size m with maximum degree at most 3, then \({20\tau_D(H) \leq 12n + 3m}\) . Using this result, we then show that if G is a twin-free cubic graph on n vertices, then \({\gamma^{\rm IOC}(G) \leq 3n/4}\) . This bound is achieved, for example, by the hypercube.     

Preserving maximal monotonicity with applications in sum and composition rules

In this paper, we study maximal monotonicity preserving mappings on the Banach space X × X ^*. Indeed, for a maximal monotone set ${M \subset X\times X^*}$ and for a multifunction ${T: X \times X^* \multimap Y \times Y^*}$ , under some sufficient conditions on M and T we show that T(M) is maximal monotone. As two consequences of this result we get sum and composition rules for maximal monotone operators.     

On convergence in mixed integer programming

Let ${P \subseteq {\mathbb R}^{m+n}}$ be a rational polyhedron, and let P _I be the convex hull of ${P \cap ({\mathbb Z}^m \times {\mathbb R}^n)}$ . We define the integral lattice-free closure of P as the set obtained from P by adding all inequalities obtained from disjunctions associated with integral lattice-free polyhedra in ${{\mathbb R}^m}$ . We show that the integral lattice-free closure of P is again a polyhedron, and that repeatedly taking the integral lattice-free closure of P gives P _I after a finite number of iterations. Such results can be seen as a mixed integer analogue of theorems by Chvátal and Schrijver for the pure integer case. One ingredient of our proof is an extension of a result by Owen and Mehrotra. In fact, we prove that for each rational polyhedron P, the split closures of P yield in the limit the set P _I .     

Subset currents on free groups

We introduce and study the space ${{\mathcal{S}{\rm Curr} (F_N)}}$ of subset currents on the free group F _N , and, more generally, on a word-hyperbolic group. A subset current on F _N is a positive F _N -invariant locally finite Borel measure on the space ${{\mathfrak{C}_N}}$ of all closed subsets of ?F _N consisting of at least two points. The well-studied space Curr(F _N ) of geodesics currents–positive F _N -invariant locally finite Borel measures defined on pairs of different boundary points–is contained in the space of subset currents as a closed ${{\mathbb{R}}}$ -linear Out(F _N )-invariant subspace. Much of the theory of Curr(F _N ) naturally extends to the ${{\mathcal{S}\;{\rm Curr} (F_N)}}$ context, but new dynamical, geometric and algebraic features also arise there. While geodesic currents generalize conjugacy classes of nontrivial group elements, a subset current is a measure-theoretic generalization of the conjugacy class of a nontrivial finitely generated subgroup in F _N . If a free basis A is fixed in F _N , subset currents may be viewed as F _N -invariant measures on a “branching” analog of the geodesic flow space for F _N , whose elements are infinite subtrees (rather than just geodesic lines) of the Cayley graph of F _N with respect to A. Similarly to the case of geodesics currents, there is a continuous Out(F _N )-invariant “co-volume form” between the Outer space cv _N and the space ${{\mathcal{S}\;{\rm Curr} (F_N)}}$ of subset currents. Given a tree ${{T \in {\rm cv}_N}}$ and the “counting current” ${{\eta_H \in \mathcal{S}\;{\rm Curr} (F_N)}}$ corresponding to a finitely generated nontrivial subgroup H ≤  F _N , the value ${{\langle T, \eta_H \rangle}}$ of this intersection form turns out to be equal to the co-volume of H, that is the volume of the metric graph T _H /H, where ${{T_H \subseteq T}}$ is the unique minimal H-invariant subtree of T. However, unlike in the case of geodesic currents, the co-volume form ${{{\rm cv}_N \times \mathcal{S}\;{\rm Curr}(F_N)\; \to [0,\infty)}}$ does not extend to a continuous map ${{\overline{{\rm cv}}_N \times \mathcal{S}\; {\rm Curr} (F_N) \to [0,\infty)}}$ .     

Optimal selling time in stock market over a finite time horizon

In this paper, we examine the best time to sell a stock at a price being as close as possible to its highest price over a finite time horizon [0, T ], where the stock price is modelled by a geometric Brownian motion and the ’closeness’ is measured by the relative error of the stock price to its highest price over [0, T ]. More precisely, we want to optimize the expression: where (V t ) t≥0 is a geometric Brownian motion with constant drift α and constant volatility σ > 0, M t = max Vs is the running maximum of the stock price, and the supremum is taken over all possible stopping times 0 ≤τ≤ T adapted to the natural filtration (F t ) t≥0 of the stock price. The above problem has been considered by Shiryaev, Xu and Zhou (2008) and Du Toit and Peskir (2009). In this paper we provide an independent proof that when α = 1 2 σ 2 , a selling strategy is optimal if and only if it sells the stock either at the terminal time T or at the moment when the stock price hits its maximum price so far. Besides, when α > 1 2 σ 2 , selling the stock at the terminal time T is the unique optimal selling strategy. Our approach to the problem is purely probabilistic and has been inspired by relating the notion of dominant stopping ρτ of a stopping time τ to the optimal stopping strategy arisen in the classical "Secretary Problem".     

On the regular convergence of multiple series of numbers and multiple integrals of locally integrable functions over ℝ̄ + m

We investigate the regular convergence of the m-multiple series (*) $$\sum\limits_{j_1 = 0}^\infty {\sum\limits_{j_2 = 0}^\infty \cdots \sum\limits_{j_m = 0}^\infty {c_{j_1 ,j_2 } , \ldots j_m } }$$ of complex numbers, where m ≥ 2 is a fixed integer. We prove Fubini’s theorem in the discrete setting as follows. If the multiple series (*) converges regularly, then its sum in Pringsheim’s sense can also be computed by successive summation. We introduce and investigate the regular convergence of the m-multiple integral (**) $$\int_0^\infty {\int_0^\infty { \cdots \int_0^\infty {f\left( {t_1 ,t_2 , \ldots ,t_m } \right)dt_1 } } } dt_2 \cdots dt_m ,$$ where f : ?? ₊ ^m → ? is a locally integrable function in Lebesgue’s sense over the closed nonnegative octant ?? ₊ ^m := [0,∞) ^m . Our main result is a generalized version of Fubini’s theorem on successive integration formulated in Theorem 4.1 as follows. If f ∈ L _loc ¹ (?? ₊ ^m ), the multiple integral (**) converges regularly, and m = p + q, where p and q are positive integers, then the finite limit $$\mathop {\lim }\limits_{v_{_{p + 1} } , \cdots ,v_m \to \infty } \int_{u_1 }^{v_1 } {\int_{u_2 }^{v_2 } { \cdots \int_0^{v_{p + 1} } { \cdots \int_0^{v_m } {f\left( {t_1 ,t_2 , \ldots t_m } \right)dt_1 dt_2 } \cdots dt_m = :J\left( {u_1 ,v_1 ;u_2 v_2 ; \ldots ;u_p ,v_p } \right)} , 0 \leqslant u_k \leqslant v_k < \infty } ,k = 1,2, \ldots p,}$$ exists uniformly in each of its variables, and the finite limit $$\mathop {\lim }\limits_{v_1 ,v_2 \cdots ,v_p \to \infty } J\left( {0,v_1 ;0,v_2 ; \ldots ;0,v_p } \right) = I$$ also exists, where I is the limit of the multiple integral (**) in Pringsheim’s sense. The main results of this paper were announced without proofs in the Comptes Rendus Sci. Paris (see [8] in the References).     

Inequalities and properties of some generalized Orlicz classes and spaces

We discuss and complement the knowledge about generalized Orlicz classes $ \tilde X_\Phi $ and Orlicz spaces X _Φ obtained by replacing the space L ¹ in the classical construction by an arbitrary Banach function space X. Our main aim is to focus on the task to study inequalities in such spaces. We prove a number of new inequalities and also natural generalizations of some classical ones (e.g., Minkowski’s, Hölder’s and Young’s inequalities). Moreover, a number of other basic facts for further study of inequalities and function spaces are included.