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The concept of mixed multiset topology was introduced and investigated by different researchers from different aspects. In this paper, we introduce the notion of mixed multiset ideal topological space. Further, we define the concepts of τ₁(τ₂)-pre-𝓘-open mset, τ₁(τ₂)-semi-𝓘-open mset, τ₁(τ₂)-α-𝓘-open mset and τ₁(τ₂)-δ-𝓘-open mset in mixed multiset ideal topological space. We investigate on these generalized open multisets.

Fuzzy multiset finite automata with output represent fuzzy version of finite automata (with output) working over multisets. This paper introduces Mealy-like, Moore-like, and compact fuzzy multiset finite automata. Their mutual transformations are described to prove their equivalent behaviours. Furthermore, various variants of reduced fuzzy multiset finite automata are studied where the reductions are directed to decrease the number of fuzzy components (like fuzzy initial distribution, fuzzy transition relation, or fuzzy output relation) of the fuzzy automata. The research confirmed that all fuzzy multiset finite automata with output can be reduced without change of their behaviours.

We prove a theorem giving the asymptotic number of binary quartic forms having bounded invariants; this extends, to the quartic case, the classical results of Gauss and Davenport in the quadratic and cubic cases, respectively. Our techniques are quite general and may be applied to counting integral orbits in other representations of algebraic groups. We use these counting results to prove that the average rank of elliptic curves over ℚ, when ordered by their heights, is bounded. In particular, we show that when elliptic curves are ordered by height, the mean size of the 2-Selmer group is 3. This implies that the limsup of the average rank of elliptic curves is at most 1.5.

We prove an asymptotic formula for the number of SL3(ℤ)-equivalence classes of integral ternary cubic forms having bounded invariants. We use this result to show that the average size of the 3-Selmer group of all elliptic curves, when ordered by height, is equal to 4. This implies that the average rank of all elliptic curves, when ordered by height, is less than 1.17. Combining our counting techniques with a recent result of Dokchitser and Dokchitser, we prove that a positive proportion of all elliptic curves have rank 0. Assuming the finiteness of the Tate–Shafarevich group, we also show that a positive proportion of elliptic curves have rank 1. Finally, combining our counting results with the recent work of Skinner and Urban, we show that a positive proportion of elliptic curves have analytic rank 0; i.e., a positive proportion of elliptic curves have nonvanishing L-function at s = 1. It follows that a positive proportion of all elliptic curves satisfy BSD.

Inspired by the generalizations from grammars and finite automata to fuzzy grammars and fuzzy finite automata, respectively, we introduce the concepts of fuzzy multiset grammars and fuzzy multiset finite automata (FMFAs), as the generalizations of multiset grammars and multiset finite automata, respectively. The relationship between fuzzy multiset regular grammars and FMFAs is discussed. Furthermore, we define some operations on fuzzy multiset languages, and prove that the family of FMFA languages is closed under the operations.

Let V be a finite dimensional complex vector space and W ⊆ GL(V) be a finite complex reflection group. Let Vreg be the complement in V of the reflecting hyperplanes. We prove that Vreg is a K(π, 1) space. This was predicted by a classical conjecture, originally stated by Brieskorn for complexified real reflection groups. The complexified real case follows from a theorem of Deligne and, after contributions by Nakamura and Orlik-Solomon, only six exceptional cases remained open. In addition to solving these six cases, our approach is applicable to most previously known cases, including complexified real groups for which we obtain a new proof, based on new geometric objects. We also address a number of questions about π1(W\\Vreg), the braid group of W. This includes a description of periodic elements in terms of a braid analog of Springer's theory of regular elements.

The root open msets and root closed msets associated with an mset topology are introduced and their basic properties are discussed.

The entropy of rough neutrosophic multisets is introduced to measure the fuzziness degree of rough multisets information. The entropy is defined in two ways, which is the entropy of rough neutrosophic multisets generalize from existing entropy of single value neutrosophic set and the rough neutrosophic multisets entropy based on roughness approximation. The definition is derived from being satisfied in the following conditions required for rough neutrosophic multisets entropy. Note that the entropy will be null when the set is crisp, while maximum if the set is a completely rough neutrosophic multiset. Moreover, the rough neutrosophic multisets entropy and its complement are equal. Also, if the degree of lower and upper approximation for truth membership, indeterminacy membership, and falsity membership of each element decrease, then the sum will decrease. Therefore, this set becomes fuzzier, causing the entropy to increase.

Since its original formulation, the theory of fuzzy sets has spawned a number of extensions where the role of membership values in the real unit interval $[0, 1]$ is handed over to more complex mathematical entities. Amongst the many existing extensions, two similar ones, the fuzzy multisets and the hesitant fuzzy sets, rely on collections of several distinct values to represent fuzzy membership, the key difference being that the fuzzy multisets allow for repeated membership values whereas the hesitant fuzzy sets do not. But in neither case are these collections of values ordered, as they are simply represented through multisets or sets. In this paper, we study ordered fuzzy multisets, where the membership value can be an ordered $n$-tuple of values, thus accounting for both order and repetition. We present some basic definitions and results and explore the relation between these ordered fuzzy multisets and the fuzzy multisets and hesitant fuzzy sets.

In this paper, notions of multi and soft multi LA-Γ-semigroup are defined. Some generalizations of certain operations on the families of the soft and fuzzy soft multisets are introduced. A concept of fuzzy multi LA-Γ-semigroup is presented. The concept of fuzzy soft multisets is applied to LA-Γ-semigroups and various characteristics of these all structures defined on LA-Γ-semigroups are investigated. Some properties of generalized operations on multi soft and fuzzy multi soft LA-Γ-semigroups are studied.