A (hyper)graph G is called k-critical if it has chromatic number k, but every proper sub(hyper)graph of it is (k− 1)-colourable. We prove that for sufficiently large k, every k-critical triangle-free graph on n vertices has at least (k−o(k))n edges. Furthermore, we show that every (k+1)-critical hypergraph on n vertices and without graph edges has at least (k−3/ 3 √ k)n edges. Both bounds diffe...

We consider a residue form for a singular hypersurface K with isolated singularities. Suppose there are neighbourhoods of the singular points with coordinates in which hypersurface is described by quasihomogeneous polynomials. We find a condition on the weights under which the norm of the Leray residue form is square integrable. For dim K ≥ 2 all simple singularities satisfy this condition. The...

We consider the fuzzification of the notion of a positive implicative ordered filter in implicative semigroups. We show that every fuzzy positive implicative ordered filter is both a fuzzy ordered filter and a fuzzy implicative ordered filter. We give examples that a fuzzy (implicative) ordered filter may not be a fuzzy positive implicative ordered filter. We also state equivalent conditions of...

a heyting algebra is a distributive lattice with implication and a dual $bck$-algebra is an algebraic system having as models logical systems equipped with implication. the aim of this paper is to investigate the relation of heyting algebras between dual $bck$-algebras. we define notions of $i$-invariant and $m$-invariant on dual $bck$-semilattices and prove that a heyting semilattice is equiva...

In this paper, we introduce the hyper-star graph HS(n, k) as a new interconnection network, and discuss its properties such as faulttolerance, scalability, isomorphism, routing algorithm, and diameter. A hyper-star graph has merits when degree × diameter is used as a desirable quality measure of an interconnection network because it has a small degree and diameter. We also introduce a variation...

In this paper we study a 3-fold X ⊂ P that is a complete intersection of two hypersurfaces Fn and Gk of degree n and k respectively such that n ≥ k and the only singularities of X are ordinary double points. The 3-fold X is called factorial if every surface in X is cut by a hypersurface in P. The factoriality of the 3-fold X is equivalent to a global topological condition rkH4(X,Z) = 1, which f...

In this paper we introduce the concepts of hyper P -semisimple Ialgebra and Abelian hyper I-algebra and we state and prove some related results. The hyper p-semi simple I-algebras is a generalization of p-semisimple BCI-algebras and a subclass of hyper I-algebras. Specially we show that every Abelian hyper I-algebra is a hyper psemisimple I-algebra and a commutative canonical hypergroup. In fol...

Let K ⊂ G be a compact subgroup of a real Lie group G. Denote by D(X) thealgebra of G-invariant differential operators on the homogeneous space X = G/K. ThenX is called commutative or the pair (G,K) is called a Gelfand pair if the algebra D(X)is commutative. Symmetric Riemannian homogeneous spaces introduced by Élie Cartanand weakly symmetric homogeneous spaces introduced by Sel...

The concept of residuated relational systems ordered under a quasi-order relation was introduced in 2018 by S. Bonzio and I. Chajda as structure $\mathfrak{A} = \langle A, \cdot, \rightarrow, 1, \preccurlyeq \rangle$, where $(A,\cdot)$ is commutative semigroup with the identity $1$ top element this monoid $\preccurlyeq$. In 2020, author analyzed concepts filters type algebraic structures. addit...