Geometric nilpotent Lie algebras and zero-dimensional simple complete intersection singularities

The Levi theorem tells us that every finite-dimensional Lie algebra is the semi-direct product of a semi-simple Lie algebra and a solvable Lie algebra. Brieskorn gave the connection between simple Lie algebras and simple singularities. Simple Lie algebras have been well understood, but not the solvable (nilpotent) Lie algebras. Therefore, it is important to establish connections between singularities and solvable (nilpotent) Lie algebras. In this paper, we give a new connection between nilpotent Lie algebras and nilradicals of derivation Lie algebras of isolated complete intersection singularities. As an application, we obtain the correspondence between the nilpotent Lie algebras of dimension less than or equal to 7 and the nilradicals of derivation Lie algebras of isolated complete intersection singularities with modality less than or equal to 1. Moreover, we give a new characterization theorem for zero-dimensional simple complete intersection singularities.

Hybrid structures applied to modules over semirings

The concept of a hybrid structure in X -semimodules, where X is a semiring, is introduced in this paper. The notions of hybrid subsemimodule and hybrid right (resp., left) ideals are defined and discussed in semirings. We investigate the representations of hybrid subsemimodules and hybrid ideals using hybrid products. We also get some interesting results on t-pure hybrid ideals in X . Furthermore, we show how hybrid products and hybrid intersections are linked. Finally, the characterization theorem is proved in terms of hybrid structures for fully idempotent semirings.

Global Dynamics of Sixth-Order Fuzzy Difference Equation

Across many fields, such as engineering, ecology, and social science, fuzzy differences are becoming more widely used; there is a wide variety of applications for difference equations in real-life problems. Our study shows that the fuzzy difference equation of sixth order has a nonnegative solution, an equilibrium point and asymptotic behavior. y i + 1 = D y i − 1 y i − 2 / E + F y i − 3 + G y i − 4 + H y i − 5 , i = 0,1,2 , … , where y i is the sequence of fuzzy numbers and the parameter D , E , F , G , H and the initial condition y − 5 , y − 4 , y − 3 , y − 2 , y − 1 , y 0 are nonnegative fuzzy number. The characterization theorem is used to convert each single fuzzy difference equation into a set of two crisp difference equations in a fuzzy environment. So, a pair of crisp difference equations is formed by converting the difference equation. The stability of the equilibrium point of a fuzzy system has been evaluated. By using variational iteration techniques and inequality skills as well as a theory of comparison for fuzzy difference equations, we investigated the governing equation dynamics, such as its boundedness, existence, and local and global stability analysis. In addition, we provide some numerical solutions for the equation describing the system to verify our results.

Homotopy Hubbard trees for post-singularly finite exponential maps

We extend the concept of a Hubbard tree, well established and useful in the theory of polynomial dynamics, to the dynamics of transcendental entire functions. We show that Hubbard trees in the strict traditional sense, as invariant compact trees embedded in $\mathbb {C}$ , do not exist even for post-singularly finite exponential maps; the difficulty lies in the existence of asymptotic values. We therefore introduce the concept of a homotopy Hubbard tree that takes care of these difficulties. Specifically for the family of exponential maps, we show that every post-singularly finite map has a homotopy Hubbard tree that is unique up to homotopy, and that post-singularly finite exponential maps can be classified in terms of homotopy Hubbard trees, using a transcendental analogue of Thurston’s topological characterization theorem of rational maps.

Laplacian Fractional Revival on Graphs

We develop the theory of fractional revival in the quantum walk on a graph using its Laplacian matrix as the Hamiltonian. We first give a spectral characterization of Laplacian fractional revival, which leads to a polynomial time algorithm to check this phenomenon and find the earliest time when it occurs. We then apply the characterization theorem to special families of graphs. In particular, we show that no tree admits Laplacian fractional revival except for the paths on two and three vertices, and the only graphs on a prime number of vertices that admit Laplacian fractional revival are double cones. Finally, we construct, through Cartesian products and joins, several infinite families of graphs that admit Laplacian fractional revival; some of these graphs exhibit polygamous fractional revival.

Ideals of Residuated Lattices

In this article, we study ideals in residuated lattice and present a characterization theorem for them. We investigate some related results between the obstinate ideals and other types of ideals of a residuated lattice, likeness Boolean, primary, prime, implicative, maximal and ʘ-prime ideals. Characterization theorems and extension property for obstinate ideal are stated and proved. For the class of ʘ-residuated lattices, by using the ʘ-prime ideals we propose a characterization, and prove that an ideal is an ʘ-prime ideal iff its quotient algebra is an ʘ-residuated lattice. Finally, by using ideals, the class of Noetherian (Artinian) residuated lattices is introduced and Cohen’s theorem is proved.

On numerical approximation of Atangana-Baleanu-Caputo fractional integro-differential equations under uncertainty in Hilbert Space

The paper uses the Atangana-Baleanu-Caputo(ABC) fractional operator for an effective advanced numerical-analysis approach to apply in handling various classes of fuzzy integro-differential equations of fractional order along with uncertain constraints conditions. We adopt the fractional derivative of ABC under generalized H-differentiability(g-HD) that uses the Mittag-Leffler function as a nonlocal kernel to better describe the timescale in fuzzy models and reduce complicity of numerical computations. Towards this end, the applications of reproducing kernel algorithm are extended for solving classes of linear and non-linear fuzzy fractional ABC Volterra-Fredholm integro-differential equations. The interval parametric solutions are provided in term of rapidly convergent series in Sobolev spaces. Based on the characterization theorem, preconditions are established to characterize the fuzzy solution in a coupled equivalent system of crisp ABC integro-differential equations. The viability and efficiency of the putative algorithm are tested by solving several fuzzy ABC Volterra-Fredholm types examples under the g-HD. The achieved numerical results are given for both classical Caputo and ABC fractional derivatives to show the effect of the ABC derivative on the interval parametric solutions of the fuzzy models, which reveal that the present method is systematic and suitable for dealing with fuzzy fractional problems arising in physics, technology, and engineering.

Generalized Minkowski type inequality for pseudo-integral

In this paper some properties of the pseudo-integral are summarized and a characterization theorem for this integral is proposed. Using the characterization theorem, we obtain that the pseudo-integral with respect to the pseudo-product of two σ-⊕-measures can be reduced to repeated pseudo-integrals. As a consequence of that claim and the Hölder type inequality for the pseudo-integral, we get the generalized Minkowski inequality for the pseudo-integral.