Definable nilpotent and soluble envelopes in groups without the independence property

We prove that in a group without the independence property a nilpotent subgroup is always contained in a definable nilpotent subgroup of the same nilpotency class. The analogue for the soluble case is also shown when the subgroup is normal in the ambient group.     

Moduli spaces of arrangements of 10 projective lines with quadruple points

We classify moduli spaces of arrangements of 10 lines with quadruple points. We show that moduli spaces of arrangements of 10 lines with quadruple points may consist of more than 2 disconnected components, namely 3 or 4 distinct points. We also present defining equations to those arrangements whose moduli spaces are still reducible after taking quotients of complex conjugations.     

A BARRIER BOUNDARY VALUE PROBLEM FOR PARABOLIC AND ELLIPTIC EQUATIONS

Barrier type boundary conditions are modeled for describing the substance diffusion in a medium when obstacles in the medium are considered. The existence of solutions of parabolic and elliptic differential equations with barrier boundary condition is presented in this paper.     

How algebra spoiled recreational problems: A case study in the cross-cultural dissemination of mathematics

This paper deals with a sub-class of recreational problems which are solved by a simple memorized rule resulting from an elementary arithmetical or algebraic solution, called proto-algebraic rules. Their recreational aspect is derived from a surprise or trick solution which is not immediately obvious to the subjects involved. Around 1560 many such problems wane from arithmetic and algebra textbooks to reappear in the eighteenth century. Several hypotheses are investigated why popular Renaissance recreational problems lost their appeal. We arrive at the conclusion that the emergence of algebra as a general problem solving method changed the scope of what is considered recreational in mathematics.     

On solitary wave solutions to the nonlinear Schrödinger equations with two parameters

In this paper, we discuss the existence and multiplicity of positive solitary wave solutions for nonlinear Schrödinger equations with two parameters. The proof is based on the method of upper and lower solutions and the fixed point index.     

Indecomposable involutive set-theoretic solutions of the Yang–Baxter equation

We describe the indecomposable involutive non-degenerate set-theoretic solutions of the Yang–Baxter equation as dynamical extensions of non-degenerate left cycle sets. Moreover we characterize the indecomposable dynamical extensions and we produce several examples. As an application we construct a family of finite indecomposable solutions whose structure groups have not the unique product property.     

Embedding Banach spaces into the space of bounded functions with countable support

We prove that a WLD subspace of the space 201800308:mana201800308-math-0001" class="section_image" src="/cms/asset/71777a6e-fdf1-40ca-9538-0a7a42d9a48e/mana201800308-math-0001.png"> consisting of all bounded, countably supported functions on a set Γ embeds isomorphically into 201800308:mana201800308-math-0002" class="section_image" src="/cms/asset/eb0c08eb-9208-4e36-959b-0416665b92f2/mana201800308-math-0002.png"> if and only if it does not contain isometric copies of 201800308:mana201800308-math-0003" class="section_image" src="/cms/asset/27533b6f-fe13-4878-8d52-8f452519bcd5/mana201800308-math-0003.png">. Moreover, a subspace of 201800308:mana201800308-math-0004" class="section_image" src="/cms/asset/941943a9-9b8d-4252-ac6a-a03a05981684/mana201800308-math-0004.png"> is constructed that has an unconditional basis, does not embed into 201800308:mana201800308-math-0005" class="section_image" src="/cms/asset/6bfe9958-c29d-4bdb-984d-fc35620aa6e0/mana201800308-math-0005.png">, and whose every weakly compact subset is separable (in particular, it cannot contain any isomorphic copies of 201800308:mana201800308-math-0006" class="section_image" src="/cms/asset/48be3f92-0893-4a7a-8c9a-50b0de42c7bf/mana201800308-math-0006.png">).