The existence of (2 × c, λ) grid-block designs with {c\in \{3, 4, 5\}} and λ ≥ 1

Let X be a v-set and ${\mathcal{B}}$ a collection of r × c arrays with elements in X. Two elements of X are collinear if they are on the same grid line (row or column). A pair ${(X, \mathcal{B})}$ is called an (r × c, λ) grid-block design if every two distinct elements in X are collinear exactly λ times in the arrays of ${\mathcal{B}}$ . This design has absorbed much attention due to its use in DNA library screening. In this paper, we prove that the necessary conditions for the existence of (2 × c, λ) grid-block designs of order v with ${c\in \{3, 4, 5\}}$ and any integer λ ≥ 1 are also sufficient.     

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.     

Higher Order Transverse Bundles and Riemannian Foliations

The purpose of this paper is to prove that each of the following conditions is equivalent to that the foliation ${{\mathcal{F}}}$ is riemannian: (1) the lifted foliation ${{\mathcal{F}}^{r}}$ on the r-transverse bundle ${\nu^{r}{\mathcal{F}}}$ is riemannian for an r ≥  1; (2) the foliation ${{\mathcal{F}}_{0}^{r}}$ on a slashed ${\nu _{\ast }^{r}{\mathcal{F}}}$ is riemannian and vertically exact for an r ≥  1; (3) there is a positively admissible transverse lagrangian on a ${\nu _{\ast }^{r}{\mathcal{F}}}$ , for an r ≥  1. Analogous results have been proved previously for normal jet vector bundles.     

Discrepancy, separation and Riesz energy of finite point sets on the unit sphere

For $d \geqslant 2,$ we consider asymptotically equidistributed sequences of $\mathbb S^d$ codes, with an upper bound $\operatorname{\boldsymbol{\delta}}$ on spherical cap discrepancy, and a lower bound Δ on separation. For such sequences, if 0?<?s?<?d, then the difference between the normalized Riesz s energy of each code, and the normalized s-energy double integral on the sphere is bounded above by $\operatorname{O}\big(\operatorname{\boldsymbol{\delta}}^{1-s/d}\,\Delta^{-s}\,N^{-s/d}\big),$ where N is the number of code points. For well separated sequences of spherical codes, this bound becomes $\operatorname{O}\big(\operatorname{\boldsymbol{\delta}}^{1-s/d}\big).$ We apply these bounds to minimum energy sequences, sequences of well separated spherical designs, sequences of extremal fundamental systems, and sequences of equal area points.     

Infinite systems of noncolliding generalized meanders and Riemann–Liouville differintegrals

Yor’s generalized meander is a temporally inhomogeneous modification of the 2(ν + 1)-dimensional Bessel process with ν  >   ? 1, in which the inhomogeneity is indexed by $\kappa \in [0, 2(\nu+1))$ . We introduce the noncolliding particle systems of the generalized meanders and prove that they are Pfaffian processes, in the sense that any multitime correlation function is given by a Pfaffian. In the infinite particle limit, we show that the elements of matrix kernels of the obtained infinite Pfaffian processes are generally expressed by the Riemann–Liouville differintegrals of functions comprising the Bessel functions J _ν used in the fractional calculus, where orders of differintegration are determined by ν ? κ. As special cases of the two parameters (ν, κ), the present infinite systems include the quaternion determinantal processes studied by Forrester, Nagao and Honner and by Nagao, which exhibit the temporal transitions between the universality classes of random matrix theory.     

Classification of radial solutions to the Emden–Fowler equation on the hyperbolic space

We study the Emden–Fowler equation ?Δu = |u| ^p?1 u on the hyperbolic space ${{\mathbb H}^n}$ . We are interested in radial solutions, namely solutions depending only on the geodesic distance from a given point. The critical exponent for such equation is p = (n + 2)/(n ? 2) as in the Euclidean setting, but the properties of the solutions show striking differences with the Euclidean case. While the papers (Bhakta and Sandeep, Poincaré Sobolev equations in the hyperbolic space, 2011; Mancini and Sandeep, Ann Sci Norm Sup Pisa Cl Sci 7(5):635–671, 2008) consider finite energy solutions, we shall deal here with infinite energy solutions and we determine the exact asymptotic behavior of wide classes of finite and infinite energy solutions.     

Hulls of codes from incidence matrices of connected regular graphs

The hulls of codes from the row span over ${\mathbb{F}_p}$ , for any prime p, of incidence matrices of connected k-regular graphs are examined, and the dimension of the hull is given in terms of the dimension of the row span of A + kI over ${\mathbb{F}_p}$ , where A is an adjacency matrix for the graph. If p = 2, for most classes of connected regular graphs with some further form of symmetry, it was shown by Dankelmann et al. (Des. Codes Cryptogr. 2012) that the hull is either {0} or has minimum weight at least 2k?2. Here we show that if the graph is strongly regular with parameter set (n, k, λ, μ), then, unless k is even and μ is odd, the binary hull is non-trivial, of minimum weight generally greater than 2k ? 2, and we construct words of low weight in the hull; if k is even and μ is odd, we show that the binary hull is zero. Further, if a graph is the line graph of a k-regular graph, k ≥ 3, that has an ?-cycle for some ? ≥ 3, the binary hull is shown to be non-trivial with minimum weight at most 2?(k?2). Properties of the p-ary hulls are also established.     

About the stability of the tangent bundle of {\mathbb{P}^n} restricted to a surface

Let X be a smooth projective surface over ${\mathbb{C}}$ and let L be a line bundle on X generated by its global sections. Let ${\phi _L:X\longrightarrow{{\mathbb{P}}^r}}$ be the morphism associated to L; we investigate the μ?stability of ${\phi _L^*T_{{\mathbb{P}}^r}}$ with respect to L when X is either a regular surface with p _g  = 0, a K3 surface or an abelian surface. In particular, we show that ${\phi_L^*T_{{\mathbb{P}}^r}}$ is μ? stable when X is K3 and L is ample and when X is abelian and L ² ≥ 14.     

On the Schröder equation and iterative sequences of C r diffeomorphisms in {\mathbb{R}^{N}} space

Let ${U \subset \mathbb{R}^{N}}$ be a neighbourhood of the origin and a function ${F:U\rightarrow U}$ be of class C ^r , r ≥ 2, F(0) = 0. Denote by F ⁿ the n-th iterate of F and let ${0<|s_1|\leq \cdots \leq|s_N| <1 }$ , where ${s_1, \ldots , s_N}$ are the eigenvalues of dF(0). Assume that the Schröder equation ${\varphi(F(x))=S\varphi(x)}$ , where S: = dF(0) has a C ² solution φ such that dφ(0) = id. If ${\frac{log|s_1|}{log|s_N|} <2 }$ then the sequence {S ^?n F ⁿ (x)} converges for every point x from the basin of attraction of F to a C ² solution φ of (1). If ${2\leq\frac{log|s_1|}{log|s_N|} }$ then this sequence can be diverging. In this case we give some sufficient conditions for the convergence and divergence of the sequence {S ^?n F ⁿ (x)}. Moreover, we show that if F is of class C ^r and ${r>\big[\frac{log|s_1|}{log|s_N|} \big ]:=p \geq 2}$ then every C ^r solution of the Schröder equation such that dφ(0) = id is given by the formula $$\begin{array}{ll}\varphi (x)={\lim\limits_{n \rightarrow \infty}} (S^{-n}F^n(x) + {\sum\limits _{k=2}^{p}} S^{-n}L_k (F^n(x))),\end{array}$$ where ${L_k:\mathbb{R}^{N} \rightarrow \mathbb{R}^{N}}$ are some homogeneous polynomials of degree k, which are determined by the differentials d ^(j) F(0) for 1 < j ≤  p.     

On the chromatic numbers of spheres in ℝ n

Let χ(S _r ^n?1 )) be the minimum number of colours needed to colour the points of a sphere S _r ^n?1 of radius $r \geqslant \tfrac{1} {2}$ in ? ⁿ so that any two points at the distance 1 apart receive different colours. In 1981 P. Erd?s conjectured that χ(S _r ^n?1 )→∞ for all $r \geqslant \tfrac{1} {2}$ . This conjecture was proved in 1983 by L. Lovász who showed in [11] that χ(S _r ^n?1 ) ≥ n. In the same paper, Lovász claimed that if $r < \sqrt {\frac{n} {{2n + 2}}}$ , then χ(S _r ^n?1 ) ≤ n+1, and he conjectured that χ(S _r ^n?1 ) grows exponentially, provided $r \geqslant \sqrt {\frac{n} {{2n + 2}}}$ . In this paper, we show that Lovász’ claim is wrong and his conjecture is true: actually we prove that the quantity χ(S _r ^n?1 ) grows exponentially for any $r > \tfrac{1} {2}$ .     

Classification and nonexistence results for linear codes with prescribed minimum distances

Starting from a linear [n, k, d] _q code with dual distance ${d^{\bot}}$ , we may construct an ${[n - d^\bot, k - d^\bot +1,\geq d]_q}$ code with dual distance at least ${\left\lceil\frac{d^\bot}{q}\right\rceil}$ using construction Y ₁. The inverse construction gives a rule for the classification of all [n, k, d] _q codes with dual distance ${d^{\bot}}$ by adding ${d^\bot}$ further columns to the parity check matrices of the smaller codes. Isomorph rejection is applied to guarantee a small search space for this iterative approach. Performing a complete search based on this observation, we are able to prove the nonexistence of linear codes for 16 open parameter sets [n, k, d] _q , q =  2, 3, 4, 5, 7, 8. These results imply 217 new upper bounds in the known tables for the minimum distance of linear codes and establish the exact value in 109 cases.     

Rainbow Connection Number, Bridges and Radius

Let G be a connected graph. The notion of rainbow connection number rc(G) of a graph G was introduced by Chartrand et al. (Math Bohem 133:85–98, 2008). Basavaraju et al. (arXiv:1011.0620v1 [math.CO], 2010) proved that for every bridgeless graph G with radius r, ${rc(G)\leq r(r+2)}$ and the bound is tight. In this paper, we show that for a connected graph G with radius r and center vertex u, if we let D ^r  = {u}, then G has r?1 connected dominating sets ${ D^{r-1}, D^{r-2},\ldots, D^{1}}$ such that ${D^{r} \subset D^{r-1} \subset D^{r-2} \cdots\subset D^{1} \subset D^{0}=V(G)}$ and ${rc(G)\leq \sum_{i=1}^{r} \max \{2i+1,b_i\}}$ , where b _i is the number of bridges in E[D ⁱ , N(D ⁱ )] for ${1\leq i \leq r}$ . From the result, we can get that if ${b_i\leq 2i+1}$ for all ${1\leq i\leq r}$ , then ${rc(G)\leq \sum_{i=1}^{r}(2i+1)= r(r+2)}$ ; if b _i  > 2i + 1 for all ${1\leq i\leq r}$ , then ${rc(G)= \sum_{i=1}^{r}b_i}$ , the number of bridges of G. This generalizes the result of Basavaraju et al. In addition, an example is given to show that there exist infinitely graphs with bridges whose rc(G) is only dependent on the radius of G, and another example is given to show that there exist infinitely graphs with bridges whose rc(G) is only dependent on the number of bridges in G.     

A Liouville comparison principle for sub- and super-solutions of the equation w t −Δ p (w) = |w| q-1 w

We establish a Liouville comparison principle for entire weak sub- and super-solutions of the equation (*) w _t ?Δ _p (w) =  |w| ^q-1 w in the half-space ${\mathbb {S}= \mathbb {R}^1_+\times \mathbb {R}^n}$ , where n ≥ 1, q > 0, and ${ \Delta_p(w) := {\rm div}_x \left(|\nabla_x w|^{p-2}\nabla_x w \right)}$ , 1 < p ≤ 2. In our study we impose neither restrictions on the behaviour of entire weak sub- and super-solutions of (*) on the hyper-plane t = 0, nor any growth conditions on their behaviour and on that of any of their partial derivatives at infinity. We prove that if ${1< q \leq p-1+\frac {p}{n}}$ and u and v are, respectively, an entire weak super-solution and an entire weak sub-solution of (*) in ${\mathbb {S}}$ which belong, only locally in ${\mathbb {S}}$ , to the corresponding Sobolev space and are such that u ≥  v, then u ≡ v. The result is sharp. As direct corollaries we obtain known Fujita-type and Liouville-type theorems.     

On the Hilbert scheme of the moduli space of vector bundles over an algebraic curve

Let M(n, ξ) be the moduli space of stable vector bundles of rank n ≥ 3 and fixed determinant ξ over a complex smooth projective algebraic curve X of genus g ≥ 4. We use the gonality of the curve and r-Hecke morphisms to describe a smooth open set of an irreducible component of the Hilbert scheme of M(n, ξ), and to compute its dimension. We prove similar results for the scheme of morphisms ${M or_P (\mathbb{G}, M(n, \xi))}$ and the moduli space of stable bundles over ${X \times \mathbb{G}}$ , where ${\mathbb{G}}$ is the Grassmannian ${\mathbb{G}(n - r, \mathbb{C}^n)}$ . Moreover, we give sufficient conditions for ${M or_{2ns}(\mathbb{P}^1, M(n, \xi))}$ to be non-empty, when s ≥ 1.     

A weighted Hardy inequality and nonexistence of positive solutions

In this article, we prove that the following weighted Hardy inequality $$\begin{array}{ll}\big(\frac{|{d-p}|}{p}\big)^{p}\, \int\limits_{\Omega}\, \frac{|{u}|^{p}}{|{x}|^{p}}\;d\mu \\ \quad \quad \le \int\limits_{\Omega}\,|{\nabla u}|^{p}\;d\mu+ \big(\frac{|{d-p}|}{p}\big)^{p-1}\,\textrm{sgn}(d-p)\,\int\limits_{\Omega}|{u}|^{p}\,\frac{(x^{t}Ax)^{p/2}}{|{x}|^{p}}\; d\mu \quad \quad \quad (1) \end{array}$$ holds with optimal Hardy constant ${\big(\frac{|d-p|}{p}\big)^{p}}$ for all ${u \in W^{1,p}_{\mu,0}(\Omega)}$ if the dimension d ≥ 2, 1 < p < d, and for all ${u \in W^{1,p}_{\mu,0}(\Omega{\setminus}\{0\})}$ if p > d ≥ 1. Here we assume that Ω is an open subset of ${\mathbb{R}^{d}}$ with ${0 \in \Omega}$ , A is a real d × d-symmetric positive definite matrix, c > 0, and $$ d \mu: = \rho(x) \,dx \qquad \textrm{with} \quad \rho(x) = c \cdot \exp(-\frac{1}{p}(x^{t}Ax)^{p/2}), \quad x \in\Omega.\quad \quad (2) $$ If p > d ≥ 1, then we can deduce from (1) a weighted Poincaré inequality on ${W^{1,p}_{\mu,0}(\Omega \setminus\{0\})}$ . Due to the optimality of the Hardy constant in (1), we can establish the nonexistence (locally in time) of positive weak solutions of a p-Kolmogorov parabolic equation perturbed by a singular potential in dimension d = 1, for 1 < p <  + ∞, and when Ω =  ]0, + ∞[.     

Gelfand numbers and metric entropy of convex hulls in Hilbert spaces

We establish optimal estimates of Gelfand numbers or Gelfand widths of absolutely convex hulls cov(K) of precompact subsets ${K\subset H}$ of a Hilbert space H by the metric entropy of the set K where the covering numbers ${N(K, \varepsilon)}$ of K by ${\varepsilon}$ -balls of H satisfy the Lorentz condition $$ \int\limits_{0}^{\infty} \left(\log N(K,\varepsilon) \right)^{r/s}\, d\varepsilon^{s} < \infty $$ for some fixed ${0 < r, s \le \infty}$ with the usual modifications in the cases r = ∞, 0 < s < ∞ and 0 < r < ∞, s = ∞. The integral here is an improper Stieltjes integral. Moreover, we obtain optimal estimates of Gelfand numbers of absolutely convex hulls if the metric entropy satisfies the entropy condition $$\sup_{\varepsilon >0 }\varepsilon \left(\log N(K,\varepsilon) \right)^{1/r}\left(\log(2+\log N(K,\varepsilon))\right)^\beta < \infty$$ for some fixed 0 < r < ∞, ?∞ < β < ∞. Using inequalities between Gelfand and entropy numbers we also get optimal estimates of the metric entropy of the absolutely convex hull cov(K). As an interesting feature of the estimates, a sudden jump of the asymptotic behavior of Gelfand numbers as well as of the metric entropy of absolutely convex hulls occurs for fixed s if the parameter r crosses the point r = 2 and, if r = 2 is fixed, if the parameter β crosses the point β = 1. The results established in Hilbert spaces extend and recover corresponding results of several authors.     

On Jumping Densities of Hypergraphs

A number \({\alpha\in [0, 1)}\) is a jump for an integer r ≥ 2 if there exists a constant c > 0 such that for any family \({{\mathcal F}}\) of r-uniform graphs, if the Turán density of \({{\mathcal F}}\) is greater than α, then the Turán density of \({{\mathcal F}}\) is at least α + c. A fundamental result in extremal graph theory due to Erd?s and Stone implies that every number in [0, 1) is a jump for r = 2. Erd?s also showed that every number in [0, r!/r ^r ) is a jump for r ≥ 3. However, not every number in [0, 1) is a jump for r ≥ 3. In fact, Frankl and Rödl showed the existence of non-jumps for r ≥ 3. By a similar approach, more non-jumps were found for some r ≥ 3 recently. But there are still a lot of unknowns regarding jumps for hypergraphs. In this note, we show that if \({c\cdot{\frac{r!}{r^r}}}\) is a non-jump for r ≥ 3, then for every p ≥ r, \({c\cdot{\frac{p!}{p^p}}}\) is a non-jump for p.     

A Comment on Ryser’s Conjecture for Intersecting Hypergraphs

Let \(\tau({\mathcal{H}})\) be the cover number and \(\nu({\mathcal{H}})\) be the matching number of a hypergraph \({\mathcal{H}}\). Ryser conjectured that every r-partite hypergraph \({\mathcal{H}}\) satisfies the inequality \(\tau({\mathcal{H}}) \leq (r-1) \nu ({\mathcal{H}})\). This conjecture is open for all r ≥ 4. For intersecting hypergraphs, namely those with \(\nu({\mathcal{H}}) = 1\), Ryser’s conjecture reduces to \(\tau({\mathcal{H}}) \leq r-1\). Even this conjecture is extremely difficult and is open for all r ≥ 6. For infinitely many r there are examples of intersecting r-partite hypergraphs with \(\tau({\mathcal{H}}) = r-1\), demonstrating the tightness of the conjecture for such r. However, all previously known constructions are not optimal as they use far too many edges. How sparse can an intersecting r-partite hypergraph be, given that its cover number is as large as possible, namely \(\tau({\mathcal{H}}) \ge r-1\)? In this paper we solve this question for r ≤ 5, give an almost optimal construction for r = 6, prove that any r-partite intersecting hypergraph with τ(H) ≥ r ? 1 must have at least \((3-\frac{1}{\sqrt{18}})r(1-o(1)) \approx 2.764r(1-o(1))\) edges, and conjecture that there exist constructions with Θ(r) edges.     

Further study of entire radial solutions of a biharmonic equation with exponential nonlinearity

We study the structure of entire radial solutions of a biharmonic equation with exponential nonlinearity: $$\begin{array}{ll}\Delta^2 u = \lambda {\rm e}^u \;\; {\rm in}\; \mathbb{R}^N, N \geq 5 \quad\quad\quad (0.1)\end{array}$$ with λ = 8(N ? 2)(N ? 4). It is known from a recent interesting paper by Arioli et al. that (0.1) admits a singular solution U _s (r) = ln r ^?4. We show that for 5 ≤ N ≤ 12, any regular entire radial solution u with u(r) ? ln r ^?4 → 0 as r → ∞ of (0.1) intersects with U _s (r) infinitely many times. On the other hand, if N ≥ 13, then u(r) < U _s (r) for all r > 0, and the solutions are strictly ordered with respect to the initial value a = u(0). Moreover, the asymptotic expansions of the entire radial solutions near ∞ are also obtained. Our main results give a positive answer to a conjecture in Arioli et al. (J Differ Equ 230:743–770, 2006) [see lines ?11 to ?9, p. 747 of Arioli et al. (J Differ Equ 230:743–770, 2006)].     

Matrix Representations of the Low Order Real Clifford Algebras

In this paper we construct the matrix subalgebras ${L_{r,s}(\mathbb{R})}$ of the real matrix algebra ${M_{2^{r+s}} (\mathbb{R})}$ when 2 ≤ r + s ≤ 3 and we show that each ${L_{r,s}(\mathbb{R})}$ is isomorphic to the real Clifford algebra ${\mathcal{C} \ell_{r,s}}$ . In particular, we prove that the algebras ${L_{r,s}(\mathbb{R})}$ can be induced from ${L_{0,n}(\mathbb{R})}$ when 2 ≤ r + s = n ≤ 3 by deforming vector generators of ${L_{0,n}(\mathbb{R})}$ to multiply the specific diagonal matrices. Also, we construct two subalgebras ${T_4(\mathbb{C})}$ and ${T_2(\mathbb{H})}$ of matrix algebras ${M_4(\mathbb{C})}$ and ${M_2(\mathbb{H})}$ , respectively, which are both isomorphic to the Clifford algebra ${\mathcal{C} \ell_{0,3}}$ , and apply them to obtain the properties related to the Clifford group Γ_0,3.