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### Multilinear polynomials and Frankl-Ray-Chaudhuri-Wilson type intersection theorems

N Alon, L Babai, H Suzuki
1991 Journal of combinatorial theory. Series A
Let us arrange all subsets of [n] in a linear order, denoted -c, such that J< I implies (JI < 111.  ...  The q-analogue of this example is the set of partial linear functions mapping a subspace of a linear space A over F, into a linear space B over F,, again ordered by restriction.  ...

### On a conjecture of Marcus and de Oliveira

Alexander Kovačec
1994 Linear Algebra and its Applications
First the problem is reduced to the question of nonnegative solvability of a certain system of linear equations.  ...  In other words, we shall show the solvability of the following system (I) of 2n + n2 linear equations in reals Sij > 0: In order to state our second lemma we introduce for i, j E [n] the quantities wl(  ...  there t,, CT E S,, such that for all equicardinal pairs ( I, K) for Clearly, instead of (1,2) one can choose any fried two-element su[n].  ...

### Splitting necklaces, with constraints [article]

Duško Jojić, Gaiane Panina, Rade Živaljević
2020 arXiv   pre-print
the number of cuts, one guarantees the existence of a fair splitting such that each thief is allocated (approximately) one and the same number of pieces of the necklace (including "degenerate pieces"  ...  Alon's "necklace-splitting theorem", subject to additional constraints, as illustrated by the following results. (1) The "almost equicardinal necklace-splitting theorem" claims that, without increasing  ...  Note that the cut points (all N = (r − 1)n of them) are linearly ordered, which induces a linear ordering on all N + 1 intervals (including degenerate).  ...

### On Ascertaining Inductively the Dimension of the Joint Kernel of Certain Commuting Linear Operators, II

Carl de Boor, Amos Ron, Zuowei Shen
1996 Advances in Mathematics
Given an index set X, a collection IB of subsets of X, and a collection ('x : x 2 X) of commuting linear maps on some linear space, the family of linear operators whose joint kernel K = K(IB) is sought  ...  It is shown that certain conditions on IB and', used in BRS] to obtain the inequality or the corresponding equality, can be weakened.  ...  Let S be a real linear space, let`: X ! L(S) : x 7 !`x be some map into the space of linear maps on S, and assume that its images commute. Then, the association X 7 !  ...

### Non-Count Symmetries in Boolean & Multi-Valued Prob. Graphical Models [article]

Ankit Anand, Ritesh Noothigattu, Parag Singla, Mausam
2017 arXiv   pre-print
However, existing algorithms for Boolean-valued domains can identify only those pairs of states as symmetric, in which the number of ones and zeros match exactly (count symmetries).  ...  Given a value v ∈ D i , we choose a representative value from its equivalence class based on some canonical ordering. We denote this value by rep i (v).  ...  Orbital MCMC suffices for downstream inference over most kinds of symmetries except non-equicardinal ones, for which a Metropolis Hastings extension is needed.  ...

### Nullity and Loop Complementation for Delta-Matroids

Robert Brijder, Hendrik Jan Hoogeboom
2013 SIAM Journal on Discrete Mathematics
We characterize delta-matroids in terms of equicardinality of minimal sets with respect to inclusion (in addition we obtain similar characterizations for matroids).  ...  ., the delta-matroids obtained after loop complementation and after pivot on a single element together with the original delta-matroid fulfill the property that two of them have equal "null space" while  ...  We are much indebted to the anonymous reviewers for valuable comments on earlier versions of this paper.  ...

### Dominating cliques in chordal graphs

Dieter Kratsch, Peter Damaschke, Anna Lubiw
1994 Discrete Mathematics
A strongly chordal graph which has a dominating clique has one as small as the smallest dominating set-and, furthermore, there is a linear-time algorithm to find such a small dominating clique.  ...  Farber's algorithm, given a strongly chordal graph G with a strong elimination ordering of the vertices, finds in linear time a dominating set D and an equicardinal set A of vertices whose neighbourhoods  ...  C and an equicardinal set A of vertices whose neighbourhoods are disjoint.  ...

### Latroids and their representation by codes over modules

Dirk Vertigan
2003 Transactions of the American Mathematical Society
In fact a generator matrix for a linear code over a field is also a representation of a matroid over that field.  ...  It has been known for some time that there is a connection between linear codes over fields and matroids represented over fields.  ...  We arbitrarily order these k simple modules as S 1 , S 2 , . . . , S k , and then fix this ordering.  ...

### Page 4332 of Mathematical Reviews Vol. , Issue 98G [page]

1998 Mathematical Reviews
Summary: “Given an index set X, a collection B of subsets of X, and a collection (/,: x € X) of commuting linear maps on some linear space, the family of linear operators whose joint kernel K = K(B) is  ...  He gives a characterization of the approximation order provided by a refinable function in terms of the order of the sum rules satisfied by the refinable mask.  ...

### A general model for matroids and the greedy algorithm

Ulrich Faigle, Satoru Fujishige
2008 Mathematical programming
Whether a similar representation is possible for matroids on convex geometries is an open question. *  ...  We discuss the relationship between these general matroids and classical matroids and provide a Dilworth embedding that allows us to represent matroids with underlying partial order structures within classical  ...  In order to establish (E, M) as an H * -matroid, we need an additional assumption on H * .  ...

### Pure pairs. I. Trees and linear anticomplete pairs [article]

Maria Chudnovsky, Alex Scott, Paul Seymour, Sophie Spirkl
2020 arXiv   pre-print
Using concavity An ordered graph is a graph J together with a linear order of its vertex set. Isomorphism of ordered graphs is defined in the natural way. Let B = (B i : i ∈ I) be a blockade in G.  ...  Let v ∈ Z, and take a linear order of C; and let i ∈ C be the first member of C (under this order) such that v meets E i ∪ F i (there is such a member i from the definition of Z).  ...

### Page 1789 of Mathematical Reviews Vol. , Issue 92c [page]

1992 Mathematical Reviews
L. (1-SIL); Jackson, Bill (4-LNDG); Lou, Ding Jun (PRC-ZHO-C); Saito, Akira (J-NIHO) Partitioning regular graphs into equicardinal linear forests. Discrete Math. 88 (1991), no. 1, 1-9.  ...  The authors have previously introduced a partial ordering of finite graphs in order to study embeddings on surfaces.  ...

### Page 1789 of Mathematical Reviews Vol. , Issue 92d [page]

1992 Mathematical Reviews
L. (1-SIL); Jackson, Bill (4-LNDG); Lou, Ding Jun (PRC-ZHO-C); Saito, Akira (J-NIHO) Partitioning regular graphs into equicardinal linear forests. Discrete Math. 88 (1991), no. 1, 1-9.  ...  The authors have previously introduced a partial ordering of finite graphs in order to study embeddings on surfaces.  ...

### Dependence systems with the operator–image exchange property

Marcin J. Schroeder
1994 Discrete Mathematics
This is one of several equivalent ways to define a dependence system.  ...  An operator on a set S, i.e. an extensive and monotone (but not necessarily idempotent) function on the power set of S, generalizes the familiar notion of closure operator (transitive operator).  ...  A different kind of exchange property, well known from the early works on linear dependence and matroids, which played an essential role in the development of the theory, is exchange property for finite  ...

### Partitioning regular graphs into equicardinal linear forests

R.E.L. Aldred, Bill Jackson, Dingjun Lou, Akira Saito
1991 Discrete Mathematics
On the other hand, in  there is a conjecture that if the number of edges of a 3-regular graph is even (i.e. its order is divisible by 4), then E(G) can be partitioned into two isomorphic linear forests  ...  In order to prove Theorem 1 and Theorem 2, we first consider connected graphs G with A(G) s 3 and 6(G) = 2. Definition.  ...
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