Minimal ellipsoids and their duals

It is proved that “most” convex bodies in IE ^d touch the boundaries of their minimal circumscribed and their maximal inscribed ellipsoids in preciselyd(d+3)/2 points. A version of the former result shows that for “most” compact sets in IE ^d the corresponding optimal designs, i.e. probability measures with a certain extremal property, are concentrated ond(d+1)/2 points.     

Sufficient Condition to Resolve Costa's First Conjecture

In this work, we give a sufficient condition to resolve Costa's first conjecture for each positive integer n and d with n ≥ 4. Precisely, we show that if there exists a local ring (A, M) such that λ _A (M) = n, and if there exists an (n + 2)-presented A-submodule of M ^m , where m is a positive integer (for instance, if M contains a regular element), then we may construct an example of (n + 4, d)-ring which is neither an (n + 3, d)-ring nor an (n + 4, d ? 1)-ring. Finally, we construct a local ring (B, M) such that λ _B (M) = 0 (resp., λ _B (M) = 1) and so we exhibit for each positive integer d, an example of a (4, d)-ring (resp., (5, d)-ring) which is neither a (4, d ? 1)-ring (resp., neither a (5, d ? 1)-ring) nor a (2, d′)-ring (resp., nor a (3, d′)-ring) for each positive integer d′.     

On the minimum length of ternary linear codes

From the geometrical point of view, we prove that [g ₃(6, d) + 1, 6, d]₃ codes exist for d = 118–123, 283–297 and that [g ₃(6, d), 6, d]₃ codes for d = 100, 341, 342 and [g ₃(6, d) + 1, 6, d]₃ codes for d = 130, 131, 132 do not exist, where ${g_3(k,\,d)=\sum_{i=0}^{k-1}\left\lceil d/3^i \right\rceil}$ . These determine the exact value of n ₃(6, d) for d = 100, 118–123, 130, 131, 132, 283–297, 341, 342, where n _q (k, d) is the minimum length n for which an [n, k, d] _q code exists.     

Small point sets of PG(n, q 3) intersecting each k-subspace in 1 mod q points

The main result of this paper is that point sets of PG(n, q ³), q = p ^h , p ≥ 7 prime, of size less than 3(q ^3(n?k) + 1)/2 intersecting each k-space in 1 modulo q points (these are always small minimal blocking sets with respect to k-spaces) are linear blocking sets. As a consequence, we get that minimal blocking sets of PG(n, p ³), p ≥ 7 prime, of size less than 3(p ^3(n?k) + 1)/2 with respect to k-spaces are linear. We also give a classification of small linear blocking sets of PG(n, q ³) which meet every (n ? 2)-space in 1 modulo q points.     

Circuits of Each Length in Tournaments

A tournament is a directed graph whose underlying graph is a complete graph. A circuit is an alternating sequence of vertices and arcs of the form v ₁, a ₁, v ₂, a ₂, v ₃, . . . , v _n-1, a _n-1, v _n in which vertex v _n  = v ₁, arc a _i  = v _i v _i+1 for i = 1, 2, . . . , n?1, and \({a_i \neq a_j}\) if \({i \neq j}\) . In this paper, we shall show that every tournament T _n in a subclass of tournaments has a circuit of each length k for \({3 \leqslant k \leqslant \theta(T_n)}\) , where \({\theta(T_n) = \frac{n(n-1)}{2}-3}\) if n is odd and \({\theta(T_n) = \frac{n(n-1)}{2}-\frac{n}{2}}\) otherwise. Note that a graph having θ(G) > n can be used as a host graph on embedding cycles with lengths larger than n to it if congestions are allowed only on vertices.     

Small point sets of PG(n, p 3h ) intersecting each line in 1 mod p h points

The main result of this paper is that point sets of PG(n, q), q = p ^3h , p ≥ 7 prime, of size < 3(q ^n-1 + 1)/2 intersecting each line in 1 modulo ${\sqrt[3] q}$ points (these are always small minimal blocking sets with respect to lines) are linear blocking sets. As a consequence, we get that minimal blocking sets of PG(n, p ³), p ≥ 7 prime, of size < 3(p ^3(n-1) + 1)/2 with respect to lines are always linear.     

Prime and composite integers close to powers of a number

We prove that, for any real numbers ξ ≠ 0 and ν, the sequence of integer parts [ξ2 ⁿ  + ν], n = 0, 1, 2, . . . , contains infinitely many composite numbers. Moreover, if the number ξ is irrational, then the above sequence contains infinitely many elements divisible by 2 or 3. The same holds for the sequence [ξ( ? 2) ⁿ  + ν _n ], n = 0, 1, 2, . . . , where ν ₀, ν ₁, ν ₂, . . . all lie in a half open real interval of length 1/3. For this, we show that if a sequence of integers x ₁, x ₂, x ₃, . . . satisfies the recurrence relation x _n+d  = cx _n  + F(x _n+1, . . . , x _n+d-1) for each n  ≥  1, where c ≠ 0 is an integer, \({F(z_1,\dots,z_{d-1}) \in \mathbb {Z}[z_1,\dots,z_{d-1}],}\) and lim _{n→ ∞}|x _n | = ∞, then the number |x _n | is composite for infinitely many positive integers n. The proofs involve techniques from number theory, linear algebra, combinatorics on words and some kind of symbolic computation modulo 3.     

Ideals of varieties parameterized by certain symmetric tensors

The ideal of a Segre variety P^n₁×?×P^n_t?P^{(n₁+1)?(n_t+1)−1} is generated by the 2-minors of a generic hypermatrix of indeterminates (see [H.T. Hà, Box-shaped matrices and the defining ideal of certain blowup surface, J. Pure Appl. Algebra 167 (2-3) (2002) 203-224. MR1874542 (2002h:13020)] and [R. Grone, Decomposable tensors as a quadratic variety, Proc. Amer. Math. 43 (2) (1977) 227-230. MR0472853 (57 #12542)]). We extend this result to the case of Segre-Veronese varieties. The main tool is the concept of “weak generic hypermatrix” which allows us to treat also the case of projection of Veronese surfaces from a set of general points and of Veronese varieties from a Cohen-Macaulay subvariety of codimension 2.     

On sets of type (q + 1, n)2 in finite three-dimensional projective spaces

It is proved that a k–set of type (q + 1, n)₂ in PG(3, q) either is a plane or it has size k ≥ (q + 1)² and a characterization of some sets of size (q + 1)² is given.     

Principal Functions for Bi-free Central Limit Distributions

We find the principal function of the completely non-normal operator l(v₁) + l(v₁)* + i(r(v₂) + r(v₂)*) on a subspace of the full Fock space \({\mathcal{F}}({\mathcal{H}})\) which arises from a bi-free central limit distribution. As an application, we find the essential spectrum of this operator.     

Minimal Blocking Sets in PG(n, 2) and Covering Groups by Subgroups

In this article we prove that a set of points B of PG(n, 2) is a minimal blocking set if and only if ?B? = PG(d, 2) with d odd and B is a set of d + 2 points of PG(d, 2) no d + 1 of them in the same hyperplane. As a corollary to the latter result we show that if G is a finite 2-group and n is a positive integer, then G admits a ? _n+1-cover if and only if n is even and G? (C ₂) ⁿ , where by a ? _m -cover for a group H we mean a set 𝒞 of size m of maximal subgroups of H whose set-theoretic union is the whole H and no proper subset of 𝒞 has the latter property and the intersection of the maximal subgroups is core-free. Also for all n < 10 we find all pairs (m,p) (m > 0 an integer and p a prime number) for which there is a blocking set B of size n in PG(m,p) such that ?B? = PG(m,p).     

Decomposition of complete multigraphs into crown graphs

Let λ K _v be the complete multigraph, G a finite simple graph. A G-design of λ K _v is denoted by GD(v,G,λ). The crown graph Q _n is obtained by joining single pendant edge to each vertex of an n-cycle. We give new constructions for Q _n -designs. Let v and λ be two positive integers. For n=4, 6, 8 and λ≥1, there exists a GD(v,Q _n ,λ) if and only if either (1) v>2n and λ v(v?1)≡0 (mod 4n), or (2) v=2n and λ≡0 (mod 4). Let n≥4 be even. Then (1) there exists a GD(2n,Q _n ,λ) if and only if λ≡0 (mod 4). (2) There exists a GD(2n+1,Q _n ,λ) when λ≡0 (mod 4).     

The Frölicher spectral sequence can be arbitrarily non-degenerate

The Frölicher spectral sequence of a compact complex manifold X measures the difference between Dolbeault cohomology and de Rham cohomology. If X is Kähler then the spectral sequence collapses at the E ₁term and no example with d _n  ≠  0 for n > 3 has been described in the literature.We construct for n ≥  2 nilmanifolds with left-invariant complex structure X _n such that the n-th differential d _n does not vanish. This answers a question mentioned in the book of Griffiths and Harris.     

Quantum complexity of parametric integration

We study parametric integration of functions from the class C^r([0,1]^d₁+d₂) to C([0,1]^d₁) in the quantum model of computation. We analyze the convergence rate of parametric integration in this model and show that it is always faster than the optimal deterministic rate and in some cases faster than the rate of optimal randomized classical algorithms.     

Empty Monochromatic Simplices

Let S be a k-colored (finite) set of n points in $\mathbb{R}^{d}$ , d≥3, in general position, that is, no (d+1) points of S lie in a common (d?1)-dimensional hyperplane. We count the number of empty monochromatic d-simplices determined by S, that is, simplices which have only points from one color class of S as vertices and no points of S in their interior. For 3≤k≤d we provide a lower bound of $\varOmega(n^{d-k+1+2^{-d}})$ and strengthen this to Ω(n ^d?2/3) for k=2. On the way we provide various results on triangulations of point sets in  $\mathbb{R}^{d}$ . In particular, for any constant dimension d≥3, we prove that every set of n points (n sufficiently large), in general position in $\mathbb{R}^{d}$ , admits a triangulation with at least dn+Ω(logn) simplices.     

On the Extendability of Linear Codes

R. Hill and P. Lizak (1995, in “Proc. IEEE Int. Symposium on Inform. Theory, Whistler, Canada,” pp. 345) proved that every [n, k, d]_q code with gcd(d, q)=1 and with all weights congruent to 0 or d (modulo q) is extendable to an [n+1, k, d+1]_q code with all weights congruent to 0 or d+1 (modulo q). We give another elementary geometrical proof of this theorem, which also yields the uniqueness of the extension.     

PATHS AND CYCLES EMBEDDING ON FAULTY ENHANCED HYPERCUBE NETWORKS

Let Qn,k(n≥3,1≤k≤n-1) be an n-dimensional enhanced hypercube which is an attractive variant of the hypercube and can be obtained by adding some complementary edges,fv and fe be the numbers of faulty vertices and faulty edges,respectively.In this paper,we give three main results.First,a fault-free path P [u,v] of length at least 2n-2fv-1(respectively,2n-2fv-2) can be embedded on Qn,k with fv+fe≤n-1 when d Qn,k(u,v) is odd(respectively,d Qn,k(u,v) is even).Secondly,an Qn,k is(n-2) edgefault-free hyper Hamiltonian-laceable when n(≥3) and k have the same parity.Lastly,a fault-free cycle of length at least 2n-2fv can be embedded on Qn,k with fe≤n-1 and fv+fe≤2n-4.     

Coisotropic displacement and small subsets of a symplectic manifold

We prove a coisotropic intersection result and deduce the following: (a) Lower bounds on the displacement energy of a subset of a symplectic manifold, in particular a sharp stable energy-Gromov-width inequality. (b) A stable non-squeezing result for neighborhoods of products of unit spheres. (c) Existence of a “badly squeezable” set in ${\mathbb R^{2n}}$ of Hausdorff dimension at most d, for every n ≥ 2 and d ≥ n. (d) Existence of a stably exotic symplectic form on ${\mathbb R^{2n}}$ , for every n ≥ 2. (e) Non-triviality of a new capacity, which is based on the minimal action of a regular coisotropic submanifold of dimension d.     

Sets of even type in PG(3, 4), alias the binary (85, 24) projective geometry code

To characterize Hermitian varieties in projective space PG(d, q) of d dimensions over the Galois field GF(q), it is necessary to find those subsets K for which there exists a fixed integer n satisfying (i) 3 ? n ? q ? 1, (ii) every line meets K in 1, n or q + 1 points. K is called singular or non-singular as there does or does not exist a point P for which every line through P meets K in 1 or q + 1 points. For q odd, a non-singular K is a non-singular Hermitian variety (M. Tallini Scafati “Caratterizzazione grafica delle forme hermitiane di un S_{r, q}” Rend. Mat. Appl.26 (1967), 273–303). For q even, q > 4 and d = 3, a non-singular K is a Hermitian surface or “looks like” the projection of a non-singular quadric in PG(4, q) (J.W.P. Hirschfeld and J.A. Thas “Sets of type (1, n, q + 1) in PG(d, q)” to appear). The case q = 4 is quite exceptional, since the complements of these sets K form a projective geometry code, a (21, 11) code for d = 2 and an (85, 24) code for d = 3. The full list of these sets is given.