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The purpose of this work is to present a new geometric approach to some problems in differential subordination theory. We also discuss the new results closely related to the generalized Briot-Bouquet differential subordination.

A space *X* is *star-C-Hurewicz* if for each sequence (*U*
_{n} : *n* ∈ *N*) of open covers of X there exists a sequence (*K*
_{n} : *n* ∈ *N*) of countably compact subsets of *X* such that for each *x* ∈ *X*, *x* ∈ *St*(*K*
_{n}, *U*n_{n}) for all but finitely many *n*. In this paper, we investigate the relationship between star-C-Hurewicz spaces and related spaces, and study topological properties of star-C-Hurewicz spaces.

We give estimates for the number of quadratic residue and primitive root values of polynomials in two variables over finite fields.

A space *X* is *weakly linearly Lindelöf* if for any family *U* of non-empty open subsets of *X* of regular uncountable cardinality κ, there exists a point *x* ∈ *X* such that every neighborhood of *x* meets κ-many elements of *U*. We also introduce the concept of *almost discretely Lindelöf spaces* as the ones in which every discrete subspace can be covered by a Lindelöf subspace. We prove that, in addition to linearly Lindelöf spaces, both weakly Lindelöf spaces and almost discretely Lindelöf spaces are weakly linearly Lindelöf.

The main result of the paper is formulated in the title. It implies that every weakly Lindelöf monotonically normal space is Lindelöf, a result obtained earlier in [3].

We show that, under the hypothesis 2^{ω} < *ω*
_{ω}, if the co-diagonal Δ^{c}
_{X} = (*X* × *X*) \Δ_{X} is discretely Lindelöf, then *X* is Lindelöf and has a weaker second countable topology; here Δ_{X} = {(*x*, *x*): *x* ∈ *X*} is the diagonal of the space *X*. Moreover, discrete Lindelöfness of Δ^{c}
_{X} together with the Lindelöf Σ-property of *X* imply that *X* has a countable network.

This paper is concerned with the existence of solutions to a class of *p*(*x*)-Kirchhofftype equations with Robin boundary data as follows: $$-M\left({\displaystyle \underset{\Omega}{\int}\frac{1}{p\left(x\right)}{\left|{\nabla}_{u}\right|}^{p\left(x\right)}dx+{\displaystyle \underset{\partial \Omega}{\int}\frac{\beta \left(x\right)}{p\left(x\right)}{\left|{\nabla}_{u}\right|}^{p\left(x\right)}d\sigma}}\right)div\left({\left|{\nabla}_{u}\right|}^{p\left(x\right)-2}{\nabla}_{u}\right)=f\left(x,u\right)$$ in Ω, $${\left|{\nabla}_{u}\right|}^{p\left(x\right)-2}\frac{\partial u}{\partial v}+\beta \left(x\right){\left|u\right|}^{p\left(x\right)-2}u=0$$ on ∂Ω, where *β* ∈ *L*
^{∞} (∂Ω) and *f* : Ω × $$\mathbb{R}$$ → $$\mathbb{R}$$ satisfies the Carathéodory condition. By means of variational methods and the theory of the variable exponent Sobolev spaces, we establish conditions for the existence of weak solutions.

We classify the extreme 2-homogeneous polynomials on $$\mathbb{R}$$
^{2} with the hexagonal norm of weight ½. As applications, using its extreme points with the Krein-Milman Theorem, we explicitly compute the polarization and unconditional constants of $$P{(}^{2}{\mathbb{R}}_{h(\frac{1}{2})}^{2})$$.

In this article, we define general normal forms for any logic that has propositional part and whose non-propositional connectives distribute over finite disjunctions. We do not require the non-propositional connectives to be closed on the set of formulas, so our normal forms cover logics with partial connectives too. We also show that most of the known normal forms in the literature are in fact particular cases of our general forms. These general normal forms are natural improvement of the distributive normal forms of J. Hintikka [6] and their modal analogues, e.g. [1] and [4].

We compute the second differential osp(*n*|2)-relative cohomology of the Lie superalgebra К(*n*) of contact vector fields with coefficients in the superspace of weighted densities on the (1, *n*)-dimensional real superspace, *n* > 1, where osp(*n*|2) is the orthosymplectic Lie superalgebra. We explicitly give 2-cocycles spanning theses cohomology spaces. This work is the simplest generalization of a result by Basdouri [ On osp(1|2)-Relative Cohomology on *S*
^{1|1}. Communications in Algebra **42:4**, 1698–1710 (2014)].

This paper deals with a class of algebraic hyperstructures called ternary semihypergroups. In this paper, we introduce the notion of generalized quasi (bi)-hyperideals in ternary semihypergroups and study their structure. Some related properties of them are investigated. Several characterizations of ternary semihypergroups in terms of minimal generalized quasi(bi)-hyperideals are provided. Also, the *n*-left simple, *m*-right simple, (*p, q*)-lateral simple and (*m*, (*p, q*), *n*)-quasi-simple ternary semihypergroups are defined and investigated.

We present the algorithms for computing the normal form of unimodular complete intersection surface singularities classified by C. T. C. Wall. He indicated in the list only the *μ*-constant strata and not the complete classification in each case. We give a complete list of surface unimodular singularities. We also give the description of a classifier which is implemented in the computer algebra system Singular.