Identities Generalizing the Theorems of Pappus and Desargues

The Theorems of Pappus and Desargues (for the projective plane over a field) are generalized here by two identities involving determinants and cross products. These identities are proved to hold in the three-dimensional vector space over a field. They are closely related to the Arguesian identity in lattice theory and to Cayley-Grassmann identities in invariant theory.

INJECTIVE LINEAR TRANSFORMATIONS WITH EQUAL GAP AND DEFECT

Let V be an infinite-dimensional vector space over a field F and let $I(V)$ be the inverse semigroup of all injective partial linear transformations on V. Given $\alpha \in I(V)$ , we denote the domain and the range of $\alpha $ by ${\mathop {\textrm {dom}}}\,\alpha $ and ${\mathop {\textrm {im}}}\,\alpha $ , and we call the cardinals $g(\alpha )={\mathop {\textrm {codim}}}\,{\mathop {\textrm {dom}}}\,\alpha $ and $d(\alpha )={\mathop {\textrm {codim}}}\,{\mathop {\textrm {im}}}\,\alpha $ the ‘gap’ and the ‘defect’ of $\alpha $ . We study the semigroup $A(V)$ of all injective partial linear transformations with equal gap and defect and characterise Green’s relations and ideals in $A(V)$ . This is analogous to work by Sanwong and Sullivan [‘Injective transformations with equal gap and defect’, Bull. Aust. Math. Soc.79 (2009), 327–336] on a similarly defined semigroup for the set case, but we show that these semigroups are never isomorphic.

Binary Hamming codes and Boolean designs

In this paper we consider a finite-dimensional vector space $${\mathcal {P}}$$ P over the Galois field $${\text {GF}}(2),$$ GF ( 2 ) , and the family $${\mathcal {B}}_k$$ B k (respectively, $${\mathcal {B}}_k^*$$ B k ∗ ) of all the k-sets of elements of $$\mathcal {P}$$ P (respectively, of $${\mathcal {P}}^*= {\mathcal {P}} \setminus \{0\}$$ P ∗ = P \ { 0 } ) summing up to zero. We compute the parameters of the 3-design $$({\mathcal {P}},{\mathcal {B}}_k)$$ ( P , B k ) for any (necessarily even) k, and of the 2-design $$({\mathcal {P}}^{*},{\mathcal {B}}_k^{*})$$ ( P ∗ , B k ∗ ) for any k. Also, we find a new proof for the weight distribution of the binary Hamming code. Moreover, we find the automorphism groups of the above designs by characterizing the permutations of $${\mathcal {P}}$$ P , respectively of $${\mathcal {P}}^*$$ P ∗ , that induce permutations of $${\mathcal {B}}_k$$ B k , respectively of $${\mathcal {B}}_k^*.$$ B k ∗ . In particular, this allows one to relax the definitions of the permutation automorphism groups of the binary Hamming code and of the extended binary Hamming code as the groups of permutations that preserve just the codewords of a given Hamming weight.

Understanding Negative Sampling in Knowledge Graph Embedding

Knowledge graph embedding (KGE) is to project entities and relations of a knowledge graph (KG) into a low-dimensional vector space, which has made steady progress in recent years. Conventional KGE methods, especially translational distance-based models, are trained through discriminating positive samples from negative ones. Most KGs store only positive samples for space efficiency. Negative sampling thus plays a crucial role in encoding triples of a KG. The quality of generated negative samples has a direct impact on the performance of learnt knowledge representation in a myriad of downstream tasks, such as recommendation, link prediction and node classification. We summarize current negative sampling approaches in KGE into three categories, static distribution-based, dynamic distribution-based and custom cluster-based respectively. Based on this categorization we discuss the most prevalent existing approaches and their characteristics. It is a hope that this review can provide some guidelines for new thoughts about negative sampling in KGE.

AI-CTO: Knowledge graph for automated and dependable software stack solution

As the scale of software systems continues expanding, software architecture is receiving more and more attention as the blueprint for the complex software system. An outstanding architecture requires a lot of professional experience and expertise. In current practice, architects try to find solutions manually, which is time-consuming and error-prone because of the knowledge barrier between newcomers and experienced architects. The problem can be solved by easing the process of apply experience from prominent architects. To this end, this paper proposes a novel graph-embedding-based method, AI-CTO, to automatically suggest software stack solutions according to the knowledge and experience of prominent architects. Firstly, AI-CTO converts existing industry experience to knowledge, i.e., knowledge graph. Secondly, the knowledge graph is embedded in a low-dimensional vector space. Then, the entity vectors are used to predict valuable software stack solutions by an SVM model. We evaluate AI-CTO with two case studies and compare its solutions with the software stacks of large companies. The experiment results show that AI-CTO can find effective and correct stack solutions and it outperforms other baseline methods.

With Wronskian through the Looking Glass

In the work of Mukhin and Varchenko from 2002 there was introduced a Wronskian map from the variety of full flags in a finite dimensional vector space into a product of projective spaces. We establish a precise relationship between this map and the Plücker map. This allows us to recover the result of Varchenko and Wright saying that the polynomials appearing in the image of the Wronsky map are the initial values of the tau-functions for the Kadomtsev-Petviashvili hierarchy.

Permutations of zero-sumsets in a finite vector space

In this paper, we consider a finite-dimensional vector space {{\mathcal{P}}} over the Galois field {\operatorname{GF}(p)}, with p being an odd prime, and the family {{\mathcal{B}}_{k}^{x}} of all k-sets of elements of {\mathcal{P}} summing up to a given element x. The main result of the paper is the characterization, for {x=0}, of the permutations of {\mathcal{P}} inducing permutations of {{\mathcal{B}}_{k}^{0}} as the invertible linear mappings of the vector space {\mathcal{P}} if p does not divide k, and as the invertible affinities of the affine space {\mathcal{P}} if p divides k. The same question is answered also in the case where the elements of the k-sets are required to be all nonzero, and, in fact, the two cases prove to be intrinsically inseparable.

Comment on “Quantum key agreement protocol”

The first two-party Quantum Key Agreement (QKA) protocol, based on quantum teleportation, was proposed by Zhou et al. (Electron. Lett. 40(18) (2004) 1149). In this protocol, to obtain the key bit string, one of the parties uses a device to obtain the inner product of two quantum states, one being unknown, and the other one performs Bell measurement. However, in this paper, we show that it is not possible to obtain a device that would output the inner product of two qubits even when only one of the qubits is unknown. This is so because the existence of such a device would imply perfectly distinguishing among four different states in a two-dimensional vector space. This is not permissible in quantum mechanics. Furthermore, we argue that the existence of such a device would also imply a violation of the “No Signaling Theorem” as well.