Given a binary matroid M and a subset T  E(M), Luis A. Goddyn posed a problem that the dual of the splitting of M, i. e., ((MT ) ∗ ) is not always equal to the splitting of the dual of M, ((M ∗ )T ). This persuade us to ask if we can characterize those binary matroids for which (MT ) ∗ = (M ∗ )T. Santosh B. Dhotre answered this question for a two-element subset T. In this paper, we generalize his result for any subset T  E(M) and exhibit a criterion for a binary matroid M and subsets T for which (MT ) ∗ and (M ∗ )T are the equal. We also show that there is no subset T  E(M) for which, the dual of element splitting of M, i. e., ((M ′ T ) ∗ ) equals to the element splitting of the dual of M, ((M ∗ ) ′ T ).

In this paper, we combine two binary operations ℾ,-Extension and element split-ting under special conditions, to extend binary matroids. For a given binary matroid M, we call a matroid obtained in this way a Λ,-Extension of M. We note some attractive properties of this matroid operation, particularly constructing a chordal matroid from a chordal binary matroid.

Varchenko introduced in 1993 a distance function on the chambers of a hyperplane arrangement that gave rise to a determinant whose entry in position $(C, D)$ is the distance between the chambers $C$ and $D$, and computed that determinant. In 2017, Aguiar and Mahajan provided a generalization of that distance function, and computed the corresponding determinant. This article extends their distance function to the topes of an oriented matroid, and computes the determinant thus defined. Oriented matroids have the nice property to be abstractions of some mathematical structures including hyperplane and sphere arrangements, polytopes, directed graphs, and even chirality in molecular chemistry. Independently and with another method, Hochst\"{a}ttler and Welker also computed in 2019 the same determinant.

The aim of this paper is to study the categorical relations between matroids, Goetschel-Voxman’s fuzzy matroids and Shi’s fuzzifying matroids. It is shown that the category of fuzzifying matroids is isomorphic to that of closed fuzzy matroids and the latter is concretely coreflective in the category of fuzzy matroids. The category of matroids can be embedded in that of fuzzifying matroids as a simultaneously concretely reflective and coreflective subcategory.

To extract some more information from the constructions of matroids that arise from new operations, computing the Tutte polynomial, plays an important role. In this paper, we consider applying three operations of splitting, element splitting and splitting o to a binary matroid and then introduce the Tutte polynomial of resulting matroids by these operations in terms of that of original matroids.

This article shows that any type of binary data can be defined as a collection from codewords of variable length. This feature helps us to define an Injective and surjective function from the suggested codewords to the required codewords. Therefore, by replacing the new codewords, the binary data becomes another binary data regarding the intended goals. One of these goals is to reduce data size. It means that instead of the original codewords of each binary data, it replaced the Huffman codewords to reduce the data size. One of the features of this method is the result of positive compression for any type of binary data, that is, regardless of the size of the code table, the difference between the original data size and the data size after compression will be greater than or equal to zero. Another important and practical feature of this method is the use of symmetric codewords instead of the suggested codewords in order to create symmetry, reversibility and error resistance properties with two-way decoding.

In this article, first we define a Kripke semantics for normal modal Logic with a binary operator and we introduce a system K^2 which is sound and complete for this semantics. Then, we will introduce two translations and show that binary normal modal logic K^2, and unary normal modal logic K, i.e. modal logic with one binary operator, are very closely related by these two translations. We call a translation a faithful interpretation if provability is preserved in both directions. So, with this terminology we will show that these two translations are faithful interpretation of K into K^2 and vice versa. A logic extending K will be a set of formulas containing K closed under its rules and uniform substitution. A logic extending K^2 is similarly defined. Finally, we will prove that the classes of logics extending K and K^2 are closely related as well and there is a 1-1-correspondence between the logics extending K and extending K^2.

Elimination of redundancies in the memory representation is necessary for fast and efficient analysis of large sets of fuzzy data. In this work, we use MTBDDs as the underlying data-structure to represent fuzzy sets and binary fuzzy relations. This leads to elimination of redundancies in the representation, less computations, and faster analyses. We also extended a BDD package (BuDDy) to support MTBDDs in general and fuzzy sets and relations in particular. Representation and manipulation of MTBDD based fuzzy sets and binary fuzzy relations are described in this paper. These include design and implementation of different fuzzy operations such as max, min and max-min composition. In particular, an efficient algorithm for computing max-min composition is presented. Effectiveness of our MTBDD based implementation is shown by applying it on fuzzy connectedness and image segmentation problem. Compared to a base implementation, the running time of the MTBDD based implementation was faster (in our test cases) by a factor ranging from 2 to 27. Also, when the MTBDD based data-structure was employed, the memory needed to represent the final results was improved by a factor ranging from 37.9 to 265.5. We also describe our base implementation which is based on matrices.