Internal approximations of reachable sets of control systems with state constraints

We construct self-intersecting flexible cross-polytopes in the spaces of constant curvature, that is, in Euclidean spaces \(\mathbb{E}^n\) , spheres \(\mathbb{S}^n\) , and Lobachevsky spaces Λ ⁿ of all dimensions n. In dimensions n ≥ 5, these are the first examples of flexible polyhedra. Moreover, we classify all flexible cross-polytopes in each of the spaces \(\mathbb{E}^n\) , \(\mathbb{S}^n\) , and Λ ⁿ . For each type of flexible cross-polytopes, we provide an explicit parametrization of the flexion by either rational or elliptic functions.     

Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster \mathfrak{D}^ \bot-parallel structure Jacobi operator

Regarding the generalized Tanaka-Webster connection, we considered a new notion of \(\mathfrak{D}^ \bot\) -parallel structure Jacobi operator for a real hypersurface in a complex two-plane Grassmannian G ₂(? ^m+2) and proved that a real hypersurface in G ₂(? ^m+2) with generalized Tanaka-Webster \(\mathfrak{D}^ \bot\) -parallel structure Jacobi operator is locally congruent to an open part of a tube around a totally geodesic quaternionic projective space ?P ⁿ in G ₂(? ^m+2), where m = 2n.     

Estimate for index of hypersurfaces in spheres with null higher order mean curvature

In a recent paper (Barros, Sousa in: Kodai Math. J. 2009) the authors proved that closed oriented non-totally geodesic minimal hypersurfaces of the Euclidean unit sphere have index of stability greater than or equal to n + 3 with equality occurring at only Clifford tori provided their second fundamental forms A satisfy the pinching: |A|² ≥ n. The natural generalization for this pinching is ?(r + 2)S _r+2 ≥ (n ? r)S _r  > 0. Under this condition we shall extend such result for closed oriented hypersurface Σ ⁿ of the Euclidean unit sphere ${\mathbb{S}^{n+1}}$ with null S _r+1 mean curvature by showing that the index of r-stability, ${Ind_{\Sigma^n}^{r}}$ , also satisfies ${Ind_{\Sigma^n}^{r}\ge n+3}$ . Instead of the previous hypothesis if we consider ${\frac{S_{r+2}}{{S_r}}}$ constant we have the same conclusion. Moreover, we shall prove that, up to Clifford tori, closed oriented hypersurfaces ${\Sigma^{n}\subset \mathbb{S}^{n+1}}$ with S _r+1 = 0 and S _r+2 < 0 have index of r-stability greater than or equal to 2n + 5.     

Some relations on the ordering of trees by minimal energies between subclasses of trees

The energy of a graph is defined as the sum of the absolute values of all eigenvalues of the graph. A tree is said to be non-starlike if it has at least two vertices with degree more than 2. A caterpillar is a tree in which a removal of all pendent vertices makes a path. Let $\mathcal{T}_{n,d}$ , $\mathbb{T}_{n,p}$ be the set of all trees of order n with diameter d, p pendent vertices respectively. In this paper, we investigate the relations on the ordering of trees and non-starlike trees by minimal energies between $\mathcal{T}_{n,d}$ and $\mathbb{T}_{n,n-d+1}$ . We first show that the first two trees (non-starlike trees, resp.) with minimal energies in $\mathcal{T}_{n,d}$ and $\mathbb{T}_{n,n-d+1}$ are the same for 3≤d≤n?2 (3≤d≤n?3, resp.). Then we obtain that the trees with third-minimal energy in $\mathcal{T}_{n,d}$ and $\mathbb{T}_{n,n-d+1}$ are the same when n≥11, 3≤d≤n?2 and d≠8; and the tree with third-minimal energy in $\mathcal{T}_{n,8}$ is the caterpillar with third-minimal energy in $\mathbb{T}_{n,n-7}$ for n≥11.     

Isoperimetric inequalities for minimal submanifolds in Riemannian manifolds: a counterexample in higher codimension

For compact Riemannian manifolds with convex boundary, B. White proved the following alternative: either there is an isoperimetric inequality for minimal hypersurfaces or there exists a closed minimal hypersurface, possibly with a small singular set. There is the natural question if a similar result is true for submanifolds of higher codimension. Specifically, B. White asked if the non-existence of an isoperimetric inequality for k-varifolds implies the existence of a nonzero, stationary, integral k-varifold. We present examples showing that this is not true in codimension greater than two. The key step is the construction of a Riemannian metric on the closed four–dimensional ball B ⁴ with the following properties: (i) B ⁴ has strictly convex boundary. (ii) There exists a complete nonconstant geodesic ${c : \mathbb{R} \to B^4}$ . (iii) There does not exist a closed geodesic in B ⁴.     

Geometric aspects of a deformation of the standard addition on integer lattices

Let \(\mathfrak{A}_n\) be the set of all those vectors of the standard lattice ? ⁿ whose coordinates are pairwise incomparable modulo n. In this paper, we analyze the group structure on \(\mathfrak{A}_n\) that arises from the construction of a deformation of multiplication described by V.M. Buchstaber. We present a geometric realization of this group in the ambient space ? ⁿ ? ? ⁿ and find its generators and relations.     

On some integrals over the product of three Legendre functions

The definite integrals \(\int_{-1}^{1}(1-x^{2})^{(\nu-1)/2}[P_{\nu}(x)]^{3}\, \mathrm{d}x\) , \(\int_{-1}^{1}(1-x^{2})^{(\nu-1)/2} [P_{\nu}(x)]^{2}P_{\nu}(-x)\, \mathrm{d}x\) , \(\int_{-1}^{1}x(1-x^{2})^{(\nu-1)/2}[P_{\nu+1}(x)]^{3}\,\mathrm{d}x\) , and \(\int_{-1}^{1}x(1-x^{2})^{(\nu-1)/2} [P_{\nu+1}(x)]^{2}P_{\nu +1}(-x)\,\mathrm{d}x \) are evaluated in closed form, where P _ν is the Legendre function of degree ν, and \(\operatorname{Re}\nu>-1\) . Special cases of these formulae are related to certain integrals over elliptic integrals that have arithmetic interest.     

A weak generalized localization criterion for multiple Walsh-Fourier series with J k -lacunary sequence of rectangular partial sums

We obtain a criterion for the validity of weak generalized localization almost everywhere on an arbitrary set of positive measure \(\mathfrak{A}\) , \(\mathfrak{A} \subset \mathbb{I}^N = \{ x \in \mathbb{R}^N :0 \leqslant x_j < 1,j = 1,2, \ldots ,N\}\) , N ≥ 3 (in terms of the structure and geometry of the set \(\mathfrak{A}\) ), for multiple Walsh-Fourier series (summed over rectangles) of functions f in the classes \(L_p (\mathbb{I}^N )\) , p > 1 (i.e., necessary and sufficient conditions for the convergence almost everywhere of the Fourier series on some subset of positive measure \(\mathfrak{A}_1\) of the set \(\mathfrak{A}\) , when the function expanded in a series equals zero on \(\mathfrak{A}\) ), in the case when the rectangular partial sums S _n (x; f) of this series have indices n = (n ₁, …, n _N ) ∈ ? ^N in which some components are elements of (single) lacunary sequences.     

The second main theorem of hypersurfaces in the projective space

We deal with a holomorphic map from the complex plane ${\mathbb{C}}$ to the n-dimensional complex projective space ${\mathbb{P}^{n}(\mathbb{C})}$ and prove the Nevanlinna Second Main Theorem for some families of non-linear hypersurfaces in ${\mathbb{P}^{n}(\mathbb{C})}$ . This Second Main Theorem implies the defect relation. If the degree of the hypersurfaces are sufficiently large, the defect of the map is smaller than one. This means that holomorphic maps which omit the irreducible hypersurface of large degree is algebraically degenerate. To prove the Second Main Theorem, we use a meromorphic partial projective connection which is totally geodesic with respect to these hypersurfaces. A meromorphic partial projective connection is a family of locally defined meromorphic connections such which work as an entirely defined meromorphic connection under the Wronskian operator. By resolving the singularity and pulling back a meromorphic partial projective connection, we also prove the Second Main Theorem for singular hypersurfaces in ${\mathbb{P}^{n}(\mathbb{C})}$ , and prove the Second Main Theorem for smooth hypersurfaces in ${\mathbb{P}^{2}(\mathbb{C})}$ which are not normal crossing.     

Rigidity theorems for hypersurfaces with constant mean curvature

Let M ⁿ be a compact oriented hypersurface of a unit sphere \(\mathbb{S}^{n + 1} \) (1) with constant mean curvature H. Given an integer k between 2 and n ? 1, we introduce a tensor ? related to H and to the second fundamental form A of M, and show that if |?|² ≤ B _H,k and tr(? ³) ≤ C _n,k |?|³, where B _H,k and C _n,k are numbers depending only on H, n and k, then either |?|² ≡ 0 or |?|² ≡ B _H,k . We characterize all M ⁿ with |?|² ≡ B _H,k . We also prove that if \(\left| A \right|^2 \leqslant 2\sqrt {k(n - k)}\) and tr(? ³) ≤ C _n,k |?|³ then |A|² is constant and characterize all M ⁿ with |A|² in the interval \(\left[ {0,2\sqrt {k\left( {n - k} \right)} } \right] \) . We also study the behavior of |?|², with the condition additional tr(? ³) ≤ C _n,k |?|³, for complete hypersurfaces with constant mean curvature immersed in space forms and show that if sup _M |?|² = B _H,k and this supremum is attained in M ⁿ then M ⁿ is an isoparametric hypersurface with two distinct principal curvatures of multiplicities k y n ? k. Finally, we use rotation hypersurfaces to show that the condition on the trace of ? ³ is necessary in our results; more precisely, for each integer k with 2 ≤ k ≤ n ? 1 and \(H \geqslant 1/\sqrt {2n - 1} \) there is a complete hypersurface M ⁿ in \(\mathbb{S}^{n + 1} \) (1) with constant mean curvature H such that sup _M |?|² = B _H,k , and this supremum is attained in M ⁿ , and which is not a product of spheres.     

The Minimal Size of Infinite Maximal Antichains in Direct Products of Partial Orders

For a partial order \(\mathbb {P}\) having infinite antichains by \(\mathfrak {a}(\mathbb {P})\) we denote the minimal cardinality of an infinite maximal antichain in \(\mathbb {P}\) and investigate how does this cardinal invariant of posets behave in finite products. In particular we show that \(\min \{ \mathfrak {a}(\mathbb {P}),\mathfrak {p} (\text {sq} \mathbb {P}) \} \leq \mathfrak {a} (\mathbb {P}^{n} ) \leq \mathfrak {a} (\mathbb {P})\), for all \(n\in \mathbb {N}\), where \(\mathfrak {p} (\text {sq} \mathbb {P})\) is the minimal size of a centered family without a lower bound in the separative quotient of the poset \(\mathbb {P}\), or \(\mathfrak {p} (\text {sq} \mathbb {P})=\infty \), if there is no such family. So we have \(\mathfrak {a} (\mathbb {P} \times \mathbb {P})=\mathfrak {a} (\mathbb {P})\) whenever \(\mathfrak {p} (\text {sq} \mathbb {P})\geq \mathfrak {a} (\mathbb {P})\) and we show that, in addition, this equality holds for all posets obtained from infinite Boolean algebras of size ≤ø ₁ by removing zero, all reversed trees, all atomic posets and, in particular, for all posets of the form \(\langle \mathcal {C} ,\subset \rangle \), where \(\mathcal {C}\) is a family of nonempty closed sets in a compact T ₁-space containing all singletons. As a by-product we obtain the following combinatorial statement: If X is an infinite set and {A _i ×B _i :i∈I} an infinite partition of the square X ², then at least one of the families {A _i :i∈I} and {B _i :i∈I} contains an infinite partition of X.     

Levi-Flat Hypersurfaces Tangent to Projective Foliations

Let M?? ⁿ be a singular real-analytic Levi-flat hypersurface tangent to a codimension-one holomorphic foliation \(\mathcal{F}\) on ? ⁿ . For n≥3, we give sufficient conditions to guarantee the existence of degenerate singularities in M, (in the sense of Segre varieties) and as a consequence we prove that \(\mathcal{F}\) can be defined by a global closed meromorphic 1-form.     

On the rigidity of hypersurfaces into space forms

Our purpose is to study the rigidity of complete hypersurfaces immersed into a Riemannian space form. In this setting, first we use a classical characterization of the Euclidean sphere \(\mathbb S ^{n+1}\) due to Obata (J Math Soc Jpn 14:333–340, 1962) in order to prove that a closed orientable hypersurface \(\Sigma ^n\) immersed with null second-order mean curvature in \(\mathbb S ^{n+1}\) must be isometric to a totally geodesic sphere \(\mathbb S ^{n}\) , provided that its Gauss mapping is contained in a closed hemisphere. Furthermore, as suitable applications of a maximum principle at the infinity for complete noncompact Riemannian manifolds due to Yau (Indiana Univ Math J 25:659–670, 1976), we establish new characterizations of totally geodesic hypersurfaces in the Euclidean and hyperbolic spaces. We also obtain a lower estimate of the index of minimum relative nullity concerning complete noncompact hypersurfaces immersed in such ambient spaces.     

On some new identities for the Fibonomial coefficients

Let F _n be the nth Fibonacci number. The Fibonomial coefficients \(\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F\) are defined for n ≥ k > 0 as follows $$\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F = \frac{{F_n F_{n - 1} \cdots F_{n - k + 1} }} {{F_1 F_2 \cdots F_k }},$$ with \(\left[ {\begin{array}{*{20}c} n \\ 0 \\ \end{array} } \right]_F = 1\) and \(\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F = 0\) . In this paper, we shall provide several identities among Fibonomial coefficients. In particular, we prove that $$\sum\limits_{j = 0}^{4l + 1} {\operatorname{sgn} (2l - j)\left[ {\begin{array}{*{20}c} {4l + 1} \\ j \\ \end{array} } \right]_F F_{n - j} = \frac{{F_{2l - 1} }} {{F_{4l + 1} }}\left[ {\begin{array}{*{20}c} {4l + 1} \\ {2l} \\ \end{array} } \right]_F F_{n - 4l - 1} ,}$$ holds for all non-negative integers n and l.     

Sur un problème de capitulation du corps ℚ\sqrt {p_1 p_2 } , i dont le 2-groupe de classes est élémentaire

Let p ₁ ≡ p ₂ ≡ 1 (mod 8) be primes such that \(\left( {\tfrac{{p_1 }} {{p_2 }}} \right) = - 1\) and \(\left( {\tfrac{2} {{a + b}}} \right) = - 1\) , where p ₁ p ₂ = a ²+b ². Let \(i = \sqrt { - 1} \) , d = p ₁ p ₂, \(\Bbbk = \mathbb{Q}(\sqrt {d,} i),\Bbbk _2^{(1)} \) be the Hilbert 2-class field and \(\Bbbk ^{(*)} = \mathbb{Q}(\sqrt {p_1 } ,\sqrt {p_2 } ,i)\) be the genus field of \(\Bbbk \) . The 2-part \(C_{\Bbbk ,2} \) of the class group of \(\Bbbk \) is of type (2, 2, 2), so \(\Bbbk _2^{(1)} \) contains seven unramified quadratic extensions \(\mathbb{K}_j /\Bbbk \) and seven unramified biquadratic extensions \(\mathbb{L}_j /\Bbbk \) . Our goal is to determine the fourteen extensions, the group \(C_{\Bbbk ,2} \) and to study the capitulation problem of the 2-classes of \(\Bbbk \) .     

Classification of smooth structures on a homotopy complex projective space

We classify, up to diffeomorphism, all closed smooth manifolds homeomorphic to the complex projective n-space \(\mathbb {C}\textbf {P}^{n}\), where n=3 and 4. Let M²ⁿ be a closed smooth 2n-manifold homotopy equivalent to \(\mathbb {C}\textbf {P}^{n}\). We show that, up to diffeomorphism, M⁶ has a unique differentiable structure and M⁸ has at most two distinct differentiable structures. We also show that, up to concordance, there exist at least two distinct differentiable structures on a finite sheeted cover N²ⁿ of \(\mathbb {C}\textbf {P}^{n}\) for n=4,7 or 8 and six distinct differentiable structures on N¹⁰.     

An isometrical ℂP<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>-theorem

Let M ⁿ (n ? 3) be a complete Riemannian manifold with sec _M ? 1, and let \(M_i^{n_i }\) (i = 1, 2) be two complete totally geodesic submanifolds in M. We prove that if n₁ + n₂ = n ? 2 and if the distance |M₁M₂| ? π/2, then M _i is isometric to \(\mathbb{S}^{n_i } /\mathbb{Z}_h\), \(\mathbb{C}P^{n_i /2}\), or \(\mathbb{C}P^{n_i /2} /\mathbb{Z}_2 \) with the canonical metric when n _i > 0; and thus, M is isometric to S ⁿ /? _h , ?P^n/2, or ?P^n/2/?₂ except possibly when n = 3 and \(M_1 (or M_2 )\mathop \cong \limits^{iso} \mathbb{S}^1 /\mathbb{Z}_h \) with h ? 2 or n = 4 and \(M_1 (or M_2 )\mathop \cong \limits^{iso} \mathbb{R}P^2 \).     

Pluriharmonic mappings and linearly connected domains in ℂ n

In this paper we obtain certain sufficient conditions for the univalence of pluriharmonic mappings defined in the unit ball \(\mathbb{B}^n \) of ? ⁿ . The results are generalizations of conditions of Chuaqui and Hernández that relate the univalence of planar harmonic mappings with linearly connected domains, and show how such domains can play a role in questions regarding injectivity in higher dimensions. In addition, we extend recent work of Hernández and Martín on a shear type construction for planar harmonic mappings, by adapting the concept of stable univalence to pluriharmonic mappings of the unit ball \(\mathbb{B}^n \) into ? ⁿ .     

Euclidean hypersurfaces with genuine deformations in codimension two

We classify hypersurfaces of rank two of Euclidean space ${\mathbb{R}^{n+1}}$ that admit genuine isometric deformations in ${\mathbb{R}^{n+2}}$ . That an isometric immersion ${\hat{f}\colon M^n \to \mathbb{R}^{n+2}}$ is a genuine isometric deformation of a hypersurface ${f\colon M^n\to\mathbb{R}^{n+1}}$ means that ${\hat f}$ is nowhere a composition ${\hat f=\hat F\circ f}$ , where ${\hat{F} \colon V\subset \mathbb{R}^{n+1} \to\mathbb{R}^{n+2}}$ is an isometric immersion of an open subset V containing the hypersurface.     

The Liouville theorem for conformal mappings on Carnot groups with Goursat-Darboux distribution

Under minimal assumptions on smoothness we prove the Liouville theorem on conformal mappings for one infinite series of Carnot groups \(\mathbb{J}^k\) with sub-Riemannian metric with Goursat-Darboux distribution, k ≥ 2: each mapping with 1-bounded distortion of a connected domain U on \(\mathbb{J}^k\) is equal to the restriction to U of the action of an element of the finite-dimensional group of 1-quasiconformal smooth mappings.