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In this paper, we consider statistics on PERMUTATIONS of lengthn represented geometrically as bargraphs having the same number of horizontal steps. More precisely, we nd the joint distribution of the descent and up step statistics on the bargraph representations, thereby obtaining a new rened count of PERMUTATIONS of a given length. To do so, we consider the distribution of the parameters on PERMUTATIONS of a more general multiset of which Sn is a subset. In addition to nding an explicit formula for the joint distribution on this multiset, we provide counts for the total number of descents and up steps of all its members, supplying both algebraic and combinatorial proofs. Finally, we derive explicit EXPRESSIONS for the sign balance of these statistics, from which the comparable results on PERMUTATIONS follow as special cases.

Let Sn be the symmetric group on the set [n] = {1, 2, . . ., n}. For g ∈ Sn let fix(g) denote the number of fixed points of g. A subset S of Sn is called t-transitive if for any two t-tuples (x1, x2, . . ., xt) and (y1, y2, . . ., yt) of distinct elements of [n], there exists g ∈ S such that xg i = yi for any 1 ≤ i ≤ t and additionally S is called sharply t-transitive if for any given pair of t-tuples, exactly one element of S carries the first to the second. In addition, a subset S of Sn is called t-intersecting if fix(h− 1g) ≥ t for any two distinct PERMUTATIONS h and g of S. In this paper, we prove that there are only two sharply (n− 2)-transitive subsets of Sn and finally we establish some relations between sharply k-transitive subsets and t-intersecting subsets of Sn where k, t ∈ Z and 0 ≤ t ≤ k ≤ n.

In this paper, by using the notion of fuzzy subsets, the concept of F-permutation is introduced. Then by applying this notion the concepts of presentation of an F-polygroup, graph of an F-permutation and F-permutation matrices are investigated.

3D is a 512-bit block cipher whose design is inspired from the Advanced Encryption Standard (AES). Like the AES, each round of 3D is composed of 4 transformations including a round-key addition, a byte-wise substitution, a byte-wise shuffle and an MDS matrix multiplication. In 3D, two distinct byte-wise PERMUTATIONS are employed for odd and even rounds. In this paper, using concepts from graph theory, we design a unified byte permutation for both odd and even rounds with the same diffusion property as the original cipher. The main advantage of this new transformation is in hardware implementation of the cipher where with less resources we can speed up the encryption/decryption process.

Fixed-point-free PERMUTATIONS, also known as derangements, have been studied for centuries. In particular, depending on their applications, derangements of prime-power order and of prime order have always played a crucial role in a variety of different branches of mathematics: from number theory to algebraic graph theory. Substantial progress has been made on the study of derangements, many long-standing open problems have been solved, and many new research problems have arisen. The results obtained and the methods developed in this area have also effectively been used to solve other problems regarding  nite vertex-transitive graphs. The methods used in this area range from deep group theory, including the classi cation of the  nite simple groups, to combinatorial techniques. This article is devoted to surveying results, open problems and methods in this area.

The objective of this study was to determine the allowed and desired limits for heavy soils of texture (clay loom), as it has operational capability (friability), These limits were 6.34 mm for planting and 10.62 mm for harvesting in each raining or irrigation episode. We determined some factors that affect soil wetness and water content as follow: the amount and severity of rain, the condition of soil’s drainage, the percent of air moisture, evapotranspiration, the amount of runoff, the amount and severity of sunlight, the ambient temperature, the condition of vegetation and, the soil texture. The time needed for dryness and removal of soil moisture as evapotranspiration in each over- desired and allowed capacity rains are 5-7 days. Also, because of higher evaporation in planting time (late April and early May) than in harvesting time (November), and also higher surface temperature in spring, both result in moving the vapor from deeper layers of soil to surface. Therefore, as our result showed, the time needed for readiness of soil for machinery operation at time of planting are 5 days for planting (late April and early May) and 7 days for harvesting time (November). According to our calculation, 19.5% of rains at planting time and 7.5% at harvest time changed to run- off; and with taking this amount of run- off in account at allowed- limit calculation, and also because of texture of the studied soil, the maximum allowed rain in which the soil remains in proper condition for machinery planting and harvesting activities exceeded 7.6 and 11.41 mm in each rainy day. As a result, according to limiting factor of well doing of operations and also factors affecting these restricting factors, the number of working days for mechanized planting operation was found to be about 19.69 days and also 21.98 days for harvesting operations in farms with heavy soil with 98% of probability.

Principle of Contradiction has of high importance in philosophy. Aristotle calls this principle “First of the First” and has discussed it in some of his works, particularly in the Metaphysics. In the beginning of his discussions on this principle, he has attempted to define it. The present article shows that Firstly, the definitions of this principle have not the same components. Secondly, they have prerequisites which are inconsistent with Aristotle’s other opinions on the problem of “negation” and the theory of the “number of the parts of a proposition” and so on. Thirdly, Aristotle’s expression of contradiction principle is not a “statement” indicating a definition, because it consist neither of all unities nor of all differences logically valid in contradictory propositions; in fact, his expression is a “pseudo-statement” which may not be considered as a proper literal definition.