Let $$N=2n^2-1$$ or $$N=n^2+n-1$$ , for any $$n\ge 2$$ . $$M=\frac{N-1}{2}$$ We construct families of prime knots with Jones polynomials $$(-1)^M\sum _{k=-M}^{M} (-1)^kt^k$$ Such have Mahler measure equal to 1. If N is prime, these are cyclotomic $$\Phi _{2N}(t)$$ up some shift in the powers t. Otherwise, they products such polynomials, including In particular, all roots unity $$\zeta _{2N}$$ o...

The notion of the square submodule of a module M over an arbitrary commutative ring R, which is denoted by RM, was introduced by Aghdam and Najafizadeh in [3]. In fact, RM is the R−submodule of M generated by the images of all bilinear maps on M. Furthermore, given a submodule N of an R−module M, we say that M is nil modulo N if μ(M×M) ≤ N for all bilinear maps μ on M. The main question about t...

Suppose that is an innnite set and k is a natural number. Let ] k denote the set of all k-subsets of and let F be a eld. In this paper we study the FSym(()-submodule structure of the permutation module F] k. Using the representation theory of nite symmetric groups, we show that every submodule of F] k can be written as an intersection of kernels of certain FSym(()-homomorphisms F] k ?! F] l for...

We theoretically show that in the presence of conventional or higher order van Hove singularities(VHS), bare finite momentum pairing, also known as pair density wave (PDW), susceptibility can be promoted to same most divergent BCS through a valley-contrasting flux 3$\phi$ each triangular plaquette at $\phi=\frac{\pi}{3}$ and $\phi=\frac{\pi}{6}$ moir\'e systems. This makes PDW possible leading ...

Searches for new low-mass matter and mediator particles have actively been pursued at fixed target experiments $e^+e^-$ colliders. It is challenging the CERN LHC, but they searched in Higgs boson decays $B$ meson by ATLAS CMS Collaborations, as well a low transverse momentum phenomena from forward scattering processes (e.g., FASER). We propose search scalar particle association with heavy vecto...

Submodule construction is the problem of finding a new submodule which, together with a given submodule, provides a behavior that conforms to a given desired global behavior. A new formulation of this problem and its solution in first-order logic is presented, and it is shown how the known solutions to this problem in the context of various communication paradigms and specification formalisms c...

An R-module A is called GF-regular if, for each a ∈ A and r ∈ R, there exist t ∈ R and a positive integer n such that r(n)tr(n)a = r(n)a. We proved that each unitary R-module A contains a unique maximal GF-regular submodule, which we denoted by M GF(A). Furthermore, the radical properties of A are investigated; we proved that if A is an R-module and K is a submodule of A, then MGF(K) = K∩M GF(A...

Top-down design methodology is one of the widely used approaches to the design of complex concurrent systems. In this approach, the speciication of a system is decomposed into a set of submodules whose concurrent behavior is equivalent to that of the system speciication. The following problem is of particular importance when using this methodology: given the speciication of a system and some of...

In a finite-dimensional vector space, every subspace is finite-dimensional and the dimension of a subspace is at most the dimension of the whole space. Unfortunately, the naive analogue of this for modules and submodules is wrong: (1) A submodule of a finitely generated module need not be finitely generated. (2) Even if a submodule of a finitely generated module is finitely generated, the minim...

This study was carried out to determine population structure, growth and reproduction properties of barbel. A total of 198 individuals were sampled. Ages of samples were found between I and VI years, fork lengths between 4.3 and 16.6 cm and total weights between 1.2 and 65.8 g. Length-weight relationship was calculated as W=0.0146×L2.934. Munro’s phi prime index was estimated as 1.9...