Order statistics of uncertain random variables with application to <Emphasis Type="Italic">k</Emphasis>-out-of-<Emphasis Type="Italic">n</Emphasis> system

Uncertain random variables are tools to deal with a mixture of uncertainty and randomness. A new concept of order statistics associated with uncertain random variables is proposed, and is applied to analyze k-out-of-n systems with uncertain random lifetimes. The chance distributions of order statistics of uncertain random variables are derived from the operational law of uncertain random variables. Finally, the reliability of k-out-of-n systems with uncertain random lifetimes is discussed.     

Optimal (<Emphasis Type="Italic">k</Emphasis>, <Emphasis Type="Italic">n</Emphasis>) visual cryptographic schemes for general <Emphasis Type="Italic">k</Emphasis>

In (k, n) visual cryptographic schemes (VCS), a secret image is encrypted into n pages of cipher text, each printed on a transparency sheet, which are distributed among n participants. The image can be visually decoded if any k(≥2) of these sheets are stacked on top of one another, while this is not possible by stacking any k − 1 or fewer sheets. We employ a Kronecker algebra to obtain necessary and sufficient conditions for the existence of a (k, n) VCS with a prior specification of relative contrasts that quantify the clarity of the recovered image. The connection of these conditions with an L ₁-norm formulation as well as a convenient linear programming formulation is explored. These are employed to settle certain conjectures on contrast optimal VCS for the cases k = 4 and 5. Furthermore, for k = 3, we show how block designs can be used to construct VCS which achieve optimality with respect to the average and minimum relative contrasts but require much smaller pixel expansions than the existing ones.     

Facets of the <Emphasis Type="Italic">r</Emphasis>-Stable (<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">k</Emphasis>)-Hypersimplex

Let k, n, and r be positive integers with k < n and \({r \leq \lfloor \frac{n}{k} \rfloor}\). We determine the facets of the r-stable n, k-hypersimplex. As a result, it turns out that the r-stable n, k-hypersimplex has exactly 2n facets for every \({r < \lfloor \frac{n}{k} \rfloor}\). We then utilize the equations of the facets to study when the r-stable hypersimplex is Gorenstein. For every k > 0 we identify an infinite collection of Gorenstein r-stable hypersimplices, consequently expanding the collection of r-stable hypersimplices known to have unimodal Ehrhart \({\delta}\)-vectors.     

Clique matchings in the <Emphasis Type="Italic">k</Emphasis>-ary <Emphasis Type="Italic">n</Emphasis>-dimensional cube

A clique matching in the k-ary n-dimensional cube (hypercube) is a collection of disjoint one-dimensional faces. A clique matching is called perfect if it covers all vertices of the hypercube. We show that the number of perfect clique matchings in the k-ary n-dimensional cube can be expressed as the k-dimensional permanent of the adjacency array of some hypergraph. We calculate the order of the logarithm of the number of perfect clique matchings in the k-ary n-dimensional cube for an arbitrary positive integer k as n→∞.     

<Emphasis Type="Italic">n</Emphasis>-Flat and <Emphasis Type="Italic">n</Emphasis>-FP-injective modules

In this paper, we study the existence of the n-flat preenvelope and the n-FP-injective cover. We also characterize n-coherent rings in terms of the n-FP-injective and n-flat modules.     

<Emphasis Type="Italic">n</Emphasis>-Abelian and <Emphasis Type="Italic">n</Emphasis>-exact categories

We introduce n-abelian and n-exact categories, these are analogs of abelian and exact categories from the point of view of higher homological algebra. We show that n-cluster-tilting subcategories of abelian (resp. exact) categories are n-abelian (resp. n-exact). These results allow to construct several examples of n-abelian and n-exact categories. Conversely, we prove that n-abelian categories satisfying certain mild assumptions can be realized as n-cluster-tilting subcategories of abelian categories. In analogy with a classical result of Happel, we show that the stable category of a Frobenius n-exact category has a natural \((n+2)\)-angulated structure in the sense of Geiß–Keller–Oppermann. We give several examples of n-abelian and n-exact categories which have appeared in representation theory, commutative algebra, commutative and non-commutative algebraic geometry.     

Eigenvalues of the Manifold Sp（n）/U（n）

In this paper, we give the eigenvalues of the manifold Sp(n)/U(n). We prove that an eigenvalue λ _s (f ₂, f ₂, …, f _n ) of the Lie group Sp(n), corresponding to the representation with label (f ₁, f ₂, ..., f _n ), is an eigenvalue of the manifold Sp(n)/U(n), if and only if f ₁, f ₂, …, f _n are all even.     

Groups with less than <Emphasis Type="Italic">n</Emphasis> subgroups of index <Emphasis Type="Italic">n</Emphasis>

We characterise (residually-finite) groups which possess less than n subgroups of index n for almost all n ∈ ℕ.     

Universal Cycle Packings and Coverings for <Emphasis Type="Italic">k</Emphasis>-Subsets of an <Emphasis Type="Italic">n</Emphasis>-Set

A cyclic sequence of elements of [n] is an (n, k)-Ucycle packing (respectively, (n, k)-Ucycle covering) if every k-subset of [n] appears in this sequence at most once (resp. at least once) as a subsequence of consecutive terms. Let \(p_{n,k}\) be the length of a longest (n, k)-Ucycle packing and \(c_{n,k}\) the length of a shortest (n, k)-Ucycle covering. We show that, for a fixed \(k,p_{n,k}={n\atopwithdelims ()k}-O(n^{\lfloor k/2\rfloor })\). Moreover, when k is not fixed, we prove that if \(k=k(n)\le n^{\alpha }\), where \(0<\alpha <1/3\), then \(p_{n,k}={n\atopwithdelims ()k}-o({n\atopwithdelims ()k}^\beta )\) and \(c_{n,k}={n\atopwithdelims ()k}+o({n\atopwithdelims ()k}^\beta )\), for some \(\beta <1\). Finally, we show that if \(k=o(n)\), then \(p_{n,k}={n\atopwithdelims ()k}(1-o(1))\).     

The <Emphasis Type="Italic">p</Emphasis>-Adic order of the <Emphasis Type="Italic">k</Emphasis>-Fibonacci and <Emphasis Type="Italic">k</Emphasis>-Lucas numbers

Let (F _k,n ) _n and (L _k,n )n be the k-Fibonacci and k-Lucas sequence, respectively, which satisfies the same recursive relation a _n+1 = ka _n + a _n?1 with initial values F _k,0 = 0, F _k,1 = 1, L _k,0 = 2 and L _k,1 = k. In this paper, we characterize the p-adic orders ν _p (F _k,n ) and ν _p (L _k,n ) for all primes p and all positive integers k.     

On <Emphasis Type="Italic">k</Emphasis>- critical 2<Emphasis Type="Italic">k</Emphasis>- connected graphs

A graph G is called an (n,k)-graph if κ(G-S)=n-|S| for any S ? V(G) with |S| ≤ k, where ?(G) denotes the connectivity of G. Mader conjectured that for k ≥ 3 the graph K_2k+2?(1-factor) is the unique (2k, k)-graph. Kriesell has settled two special cases for k = 3,4. We prove the conjecture for the general case k ≥ 5.     

On a new property of <Emphasis Type="Italic">n</Emphasis>-poised and <Emphasis Type="Italic">G</Emphasis><Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> sets

In this paper we consider n-poised planar node sets, as well as more special ones, called G C _n sets. For the latter sets each n-fundamental polynomial is a product of n linear factors as it always holds in the univariate case. A line ? is called k-node line for a node set \(\mathcal X\) if it passes through exactly k nodes. An (n + 1)-node line is called maximal line. In 1982 M. Gasca and J. I. Maeztu conjectured that every G C _n set possesses necessarily a maximal line. Till now the conjecture is confirmed to be true for n ≤ 5. It is well-known that any maximal line M of \(\mathcal X\) is used by each node in \(\mathcal X\setminus M, \)meaning that it is a factor of the fundamental polynomial. In this paper we prove, in particular, that if the Gasca-Maeztu conjecture is true then any n-node line of G C _n set \(\mathcal {X}\) is used either by exactly \(\binom {n}{2}\) nodes or by exactly \(\binom {n-1}{2}\) nodes. We prove also similar statements concerning n-node or (n ? 1)-node lines in more general n-poised sets. This is a new phenomenon in n-poised and G C _n sets. At the end we present a conjecture concerning any k-node line.     

<Emphasis Type="Italic">k</Emphasis>-invariant nets over an algebraic extension of a field <Emphasis Type="Italic">k</Emphasis>

Let K be an algebraic extension of a field k, let σ = (σ _ij ) be an irreducible full (elementary) net of order n ≥ 2 (respectively, n ≥ 3) over K, while the additive subgroups σ _ij are k-subspaces of K. We prove that all σij coincide with an intermediate subfield P, k ? P ? K, up to conjugation by a diagonal matrix.     

On the <Emphasis Type="Italic">k</Emphasis>-gamma <Emphasis Type="Italic">q</Emphasis>-distribution

We provide combinatorial as well as probabilistic interpretations for the q-analogue of the Pochhammer k-symbol introduced by Díaz and Teruel. We introduce q-analogues of the Mellin transform in order to study the q-analogue of the k-gamma distribution.     

Cyclotomic Hecke Algebras of <Emphasis Type="Italic">G</Emphasis>(<Emphasis Type="Italic">r</Emphasis>, <Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">n</Emphasis>)

In this note, we find a monomial basis of the cyclotomic Hecke algebra \({\mathcal{H}_{r,p,n}}\) of G(r,p,n) and show that the Ariki-Koike algebra \({\mathcal{H}_{r,n}}\) is a free module over \({\mathcal{H}_{r,p,n}}\), using the Gröbner-Shirshov basis theory. For each irreducible representation of \({\mathcal{H}_{r,p,n}}\), we give a polynomial basis consisting of linear combinations of the monomials corresponding to cozy tableaux of a given shape.     

<Emphasis Type="Italic">k</Emphasis>-Free-like groups

The following results are proved. In Theorem 1, it is stated that there exist both finitely presented and not finitely presented 2-generated nonfree groups which are k-free-like for any k ⩾ 2. In Theorem 2, it is claimed that every nonvirtually cyclic (resp., noncyclic and torsion-free) hyperbolic m-generated group is k-free-like for every k ⩾ m + 1 (resp., k ⩾ m). Finally, Theorem 3 asserts that there exists a 2-generated periodic group G which is k-free-like for every k ⩾ 3. Supported by NSF (grant Nos. DMS 0455881 and DMS-0700811). (A. Yu. Olshanskii, M. V. Sapir) Supported by RFBR project No. 08-01-00573. (A. Yu. Olshanskii) Supported by BSF grant (USA–Israel). (M. V. Sapir) Translated from Algebra i Logika, Vol. 48, No. 2, pp. 245–257, March–April, 2009.     

Remarks on the <Emphasis Type="Italic">k</Emphasis>-error linear complexity of <Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>-periodic sequences

Recently the first author presented exact formulas for the number of 2 ⁿ -periodic binary sequences with given 1-error linear complexity, and an exact formula for the expected 1-error linear complexity and upper and lower bounds for the expected k-error linear complexity, k ≥ 2, of a random 2 ⁿ -periodic binary sequence. A crucial role for the analysis played the Chan–Games algorithm. We use a more sophisticated generalization of the Chan–Games algorithm by Ding et al. to obtain exact formulas for the counting function and the expected value for the 1-error linear complexity for p ⁿ -periodic sequences over prime. Additionally we discuss the calculation of lower and upper bounds on the k-error linear complexity of p ⁿ -periodic sequences over .      

Removing vertices from a <Emphasis Type="Italic">k</Emphasis>-connected graph without losing its <Emphasis Type="Italic">k</Emphasis>-connectivity

The paper studies the problem of removing vertices from a k-connected graph without losing its k-connectivity. It is proved that certain inner vertices can be removed from k-blocks, provided that the interior of each block is sufficiently large with respect to its boundary, and the degree of every vertex of the graph is no less than or . Bibliography: 3 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 340, 2006, pp. 103–116.     

EKR Sets for Large <Emphasis Type="Italic">n</Emphasis> and <Emphasis Type="Italic">r</Emphasis>

Let \(\mathcal {A}\subset \left( {\begin{array}{c}[n]\\ r\end{array}}\right) \) be a compressed, intersecting family and let \(X\subset [n]\). Let \(\mathcal {A}(X)=\{A\in \mathcal {A}:A\cap X\ne \emptyset \}\) and \(\mathcal {S}_{n,r}=\left( {\begin{array}{c}[n]\\ r\end{array}}\right) (\{1\})\). Motivated by the Erd?s–Ko–Rado theorem, Borg asked for which \(X\subset [2,n]\) do we have \(|\mathcal {A}(X)|\le |\mathcal {S}_{n,r}(X)|\) for all compressed, intersecting families \(\mathcal {A}\)? We call X that satisfy this property EKR. Borg classified EKR sets X such that \(|X|\ge r\). Barber classified X, with \(|X|\le r\), such that X is EKR for sufficiently large n, and asked how large n must be. We prove n is sufficiently large when n grows quadratically in r. In the case where \(\mathcal {A}\) has a maximal element, we sharpen this bound to \(n>\varphi ^{2}r\) implies \(|\mathcal {A}(X)|\le |\mathcal {S}_{n,r}(X)|\). We conclude by giving a generating function that speeds up computation of \(|\mathcal {A}(X)|\) in comparison with the naïve methods.     

On the directions problem in <Emphasis Type="Italic">AG</Emphasis>(<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)

We prove that if q = p ^h , p a prime, do not exist sets U í AG(n,q){U {\subseteq} AG(n,q)}, with |U| = q ^k and 1 < k < n, determining N directions where