One of the earliest invariants introduced in the study of finite von Neumann algebras is the property Gamma of Murray and von Neumann. The set of separable $\operatorname{II}_1$ factors can be split in two disjoint subsets: those that have the property Gamma and those that do not have it, called full factors by Connes. In this note we prove that it is not possible to classify separable $\operatorname{II}_1$ factors satisfying the property Gamma up to isomorphism by a Borel measurable assignment of countable structures as invariants. We also show that the same holds true for the full $\operatorname{II}_1$ factors.

A fast approximation formula for Euler–Mascheroni's constant based on a certain integral due to Peter Borwein is given. This asymptotic formula for $\gamma$ involves harmonic numbers, $\ln2$ and rational numbers whose denominators are powers of $2$.

The classification of left invariant metrics of neutral signature on the $4$-dimensional nilpotent Lie groups is presented. Their geometry is extensively studied with special emphasis on the holonomy groups and projective equivalence. Additionally, we focus our attention on the Walker metrics. They appear as the underlying structure of metrics on the nilpotent Lie groups with degenerate center. Finally, we give complete description of the isometry groups.

Generalized Hecke groups $H_{p,q}$ are generated by $X(z)=-(z-\lambda _{p})^{-1}$ and $Y(z)=-(z+\lambda _{q})^{-1}$, where $\lambda _{p}=2\cos \frac{\pi }{p}$ , $\lambda _{q}=2\cos \frac{\pi }{q}$, $p,q$ are integers such that $2\leq p\leq q$, $p+q > 4$. Extended generalized Hecke groups $\overline{H}_{p,q}$ are obtained by adding the reflection $R(z)=1/\overline{z}$ to the generators of generalized Hecke groups $H_{p,q}$. We determine the conjugacy classes of the torsion elements in extended generalized Hecke groups $\overline{H}_{p,q}$.

We study the behavior of best simultaneous $(l^q,L^p)$-approximation by rational functions on an interval, when the measure tends to zero. In addition, we consider the case of polynomial approximation on a finite union of intervals. We also get an interpolation result.

We study the null helix and $k$-type null slant helices in $4$-dimensional Minkowski space $\mathbb{E}_{1}^{4}$ and characterize them in terms of the null curvature, null torsion and structure functions. Some null helices and $k$-type null slant helices are presented by solving special differential equations.

We establish the boundedness of singular integral operators, the Fefferman–Stein inequality and the John–Nirenberg inequalities on weighted Lebesgue spaces with variable exponents by using the extrapolation theorem. Moreover, as a consequence of the extrapolation theorem, we have the Fefferman–Stein vector-valued inequalities on weighted Lebesgue spaces, and hence we use it to study the weighted Triebel–Lizorkin spaces with variable exponent.

Our aim is to give a characterization of the BMO norm via Banach function spaces based on the Rubio de Francia algorithm. Our proof is different from the one by Ho [Atomic decomposition of Hardy spaces and characterization of BMO via Banach function spaces, Anal. Math. 38 (2012), 173–185].

An asteroidal triple is a stable set of three vertices such that each pair is connected by a path avoiding the neighborhood of the third vertex. An asteroidal quadruple is a stable set of four vertices such that any three of them is an asteroidal triple.

Two non adjacent vertices are linked by a special connection if either they have a common neighbor or they are the endpoints of two vertex-disjoint chordless paths satisfying certain technical conditions. Cameron, Hoàng, and Lévêque [DIMAP Workshop on Algorithmic Graph Theory, 67–74, Electron. Notes Discrete Math., 32, Elsevier, 2009] proved that if a pair of non adjacent vertices are linked by a special connection then in any directed path model $T$ the subpaths of $T$ corresponding to the vertices forming the special connection have to overlap and they force $T$ to be completely directed in one direction between these vertices. Special connections along with the concept of asteroidal quadruple play an important role to study rooted directed path graphs, which are the intersection graphs of directed paths in a rooted directed tree.

In this work we define other special connections; these special connections along with the ones defined by Cameron, Hoàng, and Lévêque are nine in total, and we prove that every one forces $T$ to be completely directed in one direction between these vertices. Also, we give a characterization of rooted directed path graphs whose rooted models cannot be rooted on a bold maximal clique. As a by-product of our result, we build new forbidden induced subgraphs for rooted directed path graphs.

$\def\lieg{\mathfrak{g}} \def\QEA{U_{\mathbf{q}}(\lieg_{A})} \def\U{{\mathcal U}} \def\D{{\mathcal D}} \def\red{\mathrm{red}}$The multiparameter quantized enveloping algebras $\QEA$ constructed by Pei, Hu and Rosso [Quantum affine algebras, extended affine Lie algebras, and their applications, 145–171, Amer. Math. Soc., Providence, 2010] are presented as the pointed Hopf algebras $\widetilde{\U}(\D_{\red},\ell)$ defined by Andruskiewitsch and Schneider [Ann. of Math. (2) 171 (2010), 375–417]. The result is applied to show that under a certain assumption $\QEA$ depends, up to cocycle deformation, on only one parameter in each connected component of the associated Dynkin diagram. In the special case that $\mathfrak{g}_{A}$ is simple, this was already shown by Pei, Hu and Rosso in an alternative way.

We study the geometry of the moduli space of principal Spin bundles through the study of the subvarieties of fixed points of automorphisms of finite order coming from outer automorphisms of the structure group. We describe the nonstable and the singular locus of the moduli space, prove that nonstable bundles are singular points, and identify the fixed points, in the case of $M(\operatorname{Spin}(8,\mathbb{C}))$, in the stable or strictly polystable locus of the moduli space.

We study two sharp inequalities involving the instrinsic scalar curvature and extrinsic generalized normalized $\delta$-Casorati curvature of submanifolds of real space forms endowed with a semi-symmetric metric connection, which are the generalization of some recent results related to the Casorati curvature for submanifolds in a real space form with a semi-symmetric metric connection, obtained by Lee et al. [J. Inequal. Appl. 2014, 2014:327].

$\def\implica{\mathbin{\rightarrow}} \def\cur{\mathcal}$Wajsberg hoops are the $\{\odot,\implica, 1\}$-subreducts (hoop-subreducts) of Wajsberg algebras, which are term equivalent to MV-algebras and are the algebraic models of Łukasiewicz infinite-valued logic. Monadic MV-algebras were introduced by Rutledge [Ph.D. thesis, Cornell University, 1959] as an algebraic model for the monadic predicate calculus of Łukasiewicz infinite-valued logic, in which only a single individual variable occurs. In this paper we study the class of $\{\odot, \implica, \forall, 1\}$-subreducts (monadic hoop-subreducts) of monadic MV-algebras. We prove that this class, denoted by $\cur{MWH}$, is an equational class and we give the identities that define it. An algebra in $\cur{MWH}$ is called a monadic Wajsberg hoop. We characterize the subdirectly irreducible members in $\cur{MWH}$ and the congruences by monadic filters. We prove that $\cur{MWH}$ is generated by its finite members. Then, we introduce the notion of width of a monadic Wajsberg hoop and study some of the subvarieties of monadic Wajsberg hoops of finite width $k$. Finally, we describe a monadic Wajsberg hoop as a monadic maximal filter within a certain monadic MV-algebra such that the quotient is the two element chain.

In this paper the Fourier transform is studied using the Henstock–Kurzweil integral on $\mathbb{R}$. We obtain that the classical Fourier transform $\mathcal{F}_{p}: L^{p}(\mathbb{R})\rightarrow L^{q}(\mathbb{R})$, $1/p+1/q=1$ and $1 < p\leq 2$, is represented by the integral on a subspace of $L^{p}(\mathbb{R})$, which strictly contains $L^{1}(\mathbb{R})\cap L^{p}(\mathbb{R})$. Moreover, for any function $f$ in that subspace, $\mathcal{F}_{p} (f)$ obeys a generalized Riemann–Lebesgue lemma.

Although computing the Khovanov homology of links is common in literature, no general formulae have been given for all families of knots and links. We give the general formulae of the Khovanov homology of the family of $2$-strand braid links $\widehat{x_{1}^{n}}$ and the family of $3$-strand braid links $\widehat{\Delta^{2k}}$, where $\Delta=x_{1}x_{2}x_{1}$.

We study a special subclass of real solvable Lie algebras having small dimensional or small codimensional derived ideals. It is well-known that the derived ideal of any Heisenberg Lie algebra is 1-dimensional and the derived ideal of the 4-dimensional real Diamond algebra is 1-codimensional. Moreover, all the coadjoint orbits of any Heisenberg Lie group as well as 4-dimensional real Diamond group are orbits of dimension zero or maximal dimension. In general, a (finite dimensional) real solvable Lie group is called an MD-group if its coadjoint orbits are zero-dimensional or maximal dimensional. The Lie algebra of an MD-group is called an MD-algebra and the class of all MD-algebras is called MD-class. Simulating the mentioned above characteristic of Heisenberg Lie algebras and 4-dimensional real Diamond algebra, we give a complete classification of MD-algebras having 1-dimensional or 1-codimensional derived ideals.