Superfield Quantization in the Sp(2)-Covariant Formalism

We generalize the rules for the superfield Sp(2)-covariant quantization of arbitrary gauge theories to the case of gauge fixing by the generating equations for the gauge functional. We consider possible realizations of the extended antibrackets and show that only one of the realizations is consistent with the extended BRST symmetry transformations in the form of the supertranslations along the Grassmann coordinates of a superspace.     

二分(0,mf-m+1)-图的正交(0,f)-因子分解

设G=(X,Y,E(G))是一个二分图,分别用V(G)=XUY和E(G)表示G的顶点集和边集.设f是定义在V(G)上的整数值函数且对(A)x∈V(G)有f(x)≥k.设H_1,H_2,…,H_k是G的k个顶点不相交的子图,且|E(H_i)|=m,1≤i≤k.本文证明了每个二分(0,mf-m+1)-图G有一个(0,f)-因子分解正交于Hi(i=1,2,…,k).     

Classification of small (0, 1) matrices

Denote by A_n the set of square (0, 1) matrices of order n. The set A_n, n ? 8, is partitioned into row/column permutation equivalence classes enabling derivation of various facts by simple counting. For example, the number of regular (0, 1) matrices of order 8 is 10160459763342013440. Let D_n, S_n denote the set of absolute determinant values and Smith normal forms of matrices from A_n. Denote by a_n the smallest integer not in D_n. The sets D₉ and S₉ are obtained; especially, a₉ = 103. The lower bounds for a_n, 10 ? n ? 19 (exceeding the known lower bound a_n ? 2f_n − 1, where f_n is nth Fibonacci number) are obtained. Row/permutation equivalence classes of A_n correspond to bipartite graphs with n black and n white vertices, and so the other applications of the classification are possible.     

Monotone subsequences in (0, 1)-matrices

A (0, 1)-matrix contains anS ₀(k) if it has 0-cells (i, j ₁), (i + 1,j ₂),..., (i + k – 1,j _k) for somei andj ₁ < ... < j_k, and it contains anS ₁(k) if it has 1-cells (i ₁,j), (i ₂,j + 1),...,(i _k ,j + k – 1) for somej andi ₁ < ... <i _k . We prove that ifM is anm × n rectangular (0, 1)-matrix with 1 m n whose largestk for anS ₀(k) isk ₀ m, thenM must have anS ₁(k) withk m/(k ₀ + 1). Similarly, ifM is anm × m lower-triangular matrix whose largestk for anS ₀(k) (in the cells on or below the main diagonal) isk ₀ m, thenM has anS ₁(k) withk m/(k ₀ + 1). Moreover, these results are best-possible.     

Construction of nonconvertible (0, 1) Matrices

In the present paper, the existence problem for nonconvertible (0, 1) matrices is solved completely. A similar result is obtained for the set of symmetric (0, 1)-matrices.     

Representing (0, 1)-matrices by boolean circuits

A boolean circuit represents an n by n(0,1)-matrix A if it correctly computes the linear transformation over GF(2) on all n unit vectors. If we only allow linear boolean functions as gates, then some matrices cannot be represented using fewer than Ω(n²/lnn) wires. We first show that using non-linear gates one can save a lot of wires: any matrix can be represented by a depth-2 circuit with O(nlnn) wires using multilinear polynomials over GF(2) of relatively small degree as gates. We then show that this cannot be substantially improved: If any two columns of an n by n(0,1)-matrix differ in at least d rows, then the matrix requires Ω(dlnn/lnlnn) wires in any depth-2 circuit, even if arbitrary boolean functions are allowed as gates.     

The length of a (0, 1) matrix

The length of a (0, 1) matrix (a bigraph) is defined and studied.     

Ensembles de classe (0, 1, q,q + 1) de PG (d,q)

We determine all sets Q of points of any finite dimensional protective space P such that each line intersecting Q in more than one point, either is contained in Q or contains exactly one point not on Q. If P is a finite protective space of order q, these sets are the so called sets of class (0, 1, q, q + 1).     

Cotauberian Operators on L 1(0, 1) Obtained by Lifting

We show that the set \({\mathcal{T}^d(L_1(0, 1))}\) of cotauberian operators acting on L ₁(0, 1) is not open, and \({T \in \mathcal{T}^d(L_1(0, 1))}\) does not imply T** cotauberian. As a consequence, we derive that the set \({\mathcal{T}(L_\infty(0, 1))}\) of tauberian operators acting on L _∞(0, 1) is not open, and that \({T \in \mathcal{T}(L_\infty(0, 1))}\) does not imply T** tauberian.     

Permanental compounds and permanents of (0, 1)-circulants

It was shown by the author in a recent paper that a recurrence relation for permanents of (0, 1)-circulants can be generated from the product of the characteristic polynomials of permanental compounds of the companion matrix of a polynomial associated with (0, 1)-circulants of the given type. In the present paper general properties of permanental compounds of companion matrices are studied, and in particular of convertible companion matrices, i.e., matrices whose permanental compounds are equal to the determinantal compounds after changing the signs of some of their entries. These results are used to obtain formulas for the limit of the nth root of the permanent of the n × n (0, 1)-circulant of a given type, as n tends to infinity. The root-squaring method is then used to evaluate this limit for a wide range of circulant types whose associated polynomials have convertible companion matrices.     

Properties of (0, 1)-matrices without certain configurations

We generalize results of Ryser on (0, 1)-matrices without triangles, 3 × 3 submatrices with row and column sums 2. The extremal case of matrices without triangles was previously studied by the author. Let the row intersection of row i and row j (i ≠ j) of some matrix, when regarded as a vector, have a 1 in a given column if both row i and row j do not 0 otherwise. For matrices satisfying some conditions on forbidden configurations and column sums ? 2, we find that the number of linearly independent row intersections is equal to the number of distinct columns. The extremal matrices with m rows and $\text{(}\text{m}\text{2}\text{)}$ distinct columns have a unique SDR of pairs of rows with 1's. A triangle bordered with a column of 0's and its (0, 1)-complement are also considered as forbidden configurations. Similar results are obtained and the extremal matrices are closely related to the extremal matrices without triangles.     

A Hardy type inequality for W m,1(0, 1) functions

In this paper, we consider functions ${u\in W^{m,1}(0,1)}$ where m ≥ 2 and u(0) = Du(0) = · · · = D ^m-1 u(0) = 0. Although it is not true in general that ${\frac{D^ju(x)}{x^{m-j}} \in L^1(0,1)}$ for ${j\in \{0,1,\ldots,m-1\}}$ , we prove that ${\frac{D^ju(x)}{x^{m-j-k}} \in W^{k,1}(0,1)}$ if k ≥ 1 and 1 ≤ j + k ≤ m, with j, k integers. Furthermore, we have the following Hardy type inequality, $$\left\|{D^k\left({\frac{D^ju(x)}{x^{m-j-k}}}\right)}\right\|_{L^1(0,1)} \leq \frac {(k-1)!}{(m-j-1)!} \|{D^mu}\|_{L^1(0,1)},$$ where the constant is optimal.     

Properties of (0, 1)-matrices with no triangles

We study (0, 1)-matrices which contain no triangles (submatrices of order 3 with row and column sums 2) previously studied by Ryser. Let the row intersection of row i and row j of some matrix, when regarded as a vector, have a 1 in a given column if both row i and row j do and a zero otherwise. For matrices with no triangles, columns sums ?2, we find that the number of linearly independent row intersections is equal to the number of distinct columns. We then study the extremal (0, 1)-matrices with no triangles, column sums ?2, distinct columns, i.e., those of size mx(^m₂). The number of columns of column sum l is m ? l + 1 and they form a (l ? 1)-tree. The ((^m₂)) columns have a unique SDR of pairs of rows with 1's. Also, these matrices have a fascinating inductive buildup. We finish with an algorithm for constructing these matrices.