T. Nakahara proved that the set of minimal indices of bicyclic biquadratic number fields with field index $1$ is unbounded. We strengthen his result by showing that this set just coincides with the set of positive integers, and every positive integer occurs infinitely many times as minimal index of totally complex bicyclic biquadratic number fields. Moreover, we study the analogous problem for all the other possible values of the field index.

We borrow a classical construction from the study of rational billiards in dynamical systems known as the “unfolding construction” and show that it can be used to study the automorphism group of a Platonic surface. More precisely, the monodromy group, or deck group in this case, associated to the cover of a regular polygon or double polygon by the unfolded Platonic surface yields a normal subgroup of the rotation group of the Platonic surface. The quotient of this rotation group by the normal subgroup is always a cyclic group, where explicit bounds on the order of the cyclic group can be given entirely in terms of the Schläfli symbol of the Platonic surface. As a consequence, we provide a new derivation of the rotation groups of the dodecahedron and the Bolza surface.

Fix two points $p$ and $q$ in the plane and a positive number $k\ne 1$. A result credited to Apollonius of Perga states that the set of points $x$ that satisfy $d\left(x,p\right)∕d\left(x,q\right)=k$ forms a circle. In this paper we study the analogous set in taxicab geometry. We find that while Apollonian sets are not taxicab circles, more complicated Apollonian sets can be characterized in terms of simpler ones.

Geometrical aspects of a perfect fluid spacetime are described in terms of different curvature tensors and $\eta$-Ricci and $\eta$-Einstein solitons in a perfect fluid spacetime are determined. Conditions for the Ricci soliton to be steady, expanding or shrinking are also given. In a particular case when the potential vector field $\xi$ of the soliton is of gradient type, $\xi :=grad\left(f\right)$, we derive a Poisson equation from the soliton equation.

We consider a free boundary problem: $u_t=u_{xx}+f\left(t,u\right)$ ($g\left(t\right)<x<h\left(t\right)$) with free boundary conditions $h^{\prime }\left(t\right)=-u_x\left(t,h\left(t\right)\right)-\alpha \left(t\right)$ and $g^{\prime }\left(t\right)=-u_x\left(t,g\left(t\right)\right)+\alpha \left(t\right)$, where $\alpha \left(t\right)$ is a positive $T$-periodic function, $f\left(t,u\right)$ is a Fisher–KPP type of nonlinearity and $T$-periodic in $t$. Such a problem can model the spreading of a biological or chemical species in time-periodic environment, where free boundaries mimic the spreading fronts of the species. We mainly study the convergence of bounded solutions. There is a $T$-periodic function ${\alpha }_0\left(t\right)$ which plays a key role in the dynamics. More precisely:

(i) When $0<\alpha <{\alpha }_0$, we obtain a trichotomy result:

1. Spreading, i.e., $h\left(t\right),-g\left(t\right)\to +\infty$ and $u\left(t,\cdot \right)\to P\left(t\right)$ as $t\to \infty$, where $P\left(t\right)$ is the periodic solution of the ODE $u_t=f\left(t,u\right)$.

2. Vanishing, i.e., $\underset{t\to \mathcal{𝒯}}{lim}h\left(t\right)=\underset{t\to \mathcal{𝒯}}{lim}g\left(t\right)$ and $\underset{t\to \mathcal{𝒯}}{lim}\underset{g\left(t\right)\le x\le h\left(t\right)}{max}u\left(t,x\right)=0$, where $\mathcal{𝒯}$ is some positive constant.

3. Transition, i.e., $0<\underset{t\to \infty }{lim}h\left(t\right),\underset{t\to \infty }{lim}g\left(t\right)<+\infty$, $0<\underset{t\to \infty }{lim}\left[h\left(t\right)-g\left(t\right)\right]<+\infty$ and $u\left(t,\cdot \right)\to V\left(t,\cdot \right)$, where $V$ is a $T$-periodic solution with compact support.

(ii) In the case $\alpha \ge {\alpha }_0$, vanishing happens for any solution.

The purpose of this paper is to introduce and study BiHom-alternative algebras and BiHom-Malcev algebras. It is shown that BiHom-alternative algebras are BiHom-Malcev admissible and BiHom-Jordan admissible. Moreover, BiHom-type generalizations of some well known identities in alternative algebras, including the Moufang identities, are obtained.

If a differential operator $D$ on a smooth Hermitian vector bundle $S$ over a compact manifold $M$ is symmetric, it is essentially self-adjoint and so admits the use of functional calculus. If $D$ is also elliptic, then the Hilbert space of square integrable sections of $S$ with the canonical left $C\left(M\right)$-action and the operator $\chi \left(D\right)$ for $\chi$ a normalizing function is a Fredholm module, and its $K$-homology class is independent of $\chi$. In this expository article, we provide a detailed proof of this fact following the outline in the book “Analytic K-homology” by Higson and Roe.

Two subanalytic subsets of ${ℝ}^n$ are called $s$-equivalent at a common point $P$ if the Hausdorff distance between their intersections with the sphere centered at $P$ of radius $r$ vanishes to order $>s$ as $r$ tends to $0$. We strengthen this notion in the case of real subanalytic subsets of ${ℝ}^n$ with isolated singular points, introducing the notion of tangential $s$-equivalence at a common singular point, which considers also the distance between the tangent planes to the sets near the point. We prove that, if $V\left(f\right)$ is the zero set of an analytic map $f$ and if we assume that $V\left(f\right)$ has an isolated singularity, say at the origin $O$, then for any $s\ge 1$ the truncation of the Taylor series of $f$ of sufficiently high order defines an algebraic set with isolated singularity at $O$ which is tangentially $s$-equivalent to $V\left(f\right)$.

With an eye towards studying curve systems on low-complexity surfaces, we introduce and analyze the $k$-Farey graphs ${\mathcal{ℱ}}_k$ and ${\mathcal{ℱ}}_{\le k}$, two natural variants of the Farey graph $\mathcal{ℱ}$ in which we relax the edge condition to indicate intersection number $=k$ or $\le k$, respectively.

The former, ${\mathcal{ℱ}}_k$, is disconnected when $k>1$. In fact, we find that the number of connected components is infinite if and only if $k$ is not a prime power. Moreover, we find that each component of ${\mathcal{ℱ}}_k$ is a quasitree (in fact, a tree when $k$ is even) and $Aut\left({\mathcal{ℱ}}_k\right)$ is uncountable for $k>1$.

As for ${\mathcal{ℱ}}_{\le k}$, Agol obtained an upper bound of $1+min\left\{p:p\text{is a prime}>k\right\}$ for both chromatic and clique numbers, and observed that this is an equality when $k$ is either one or two less than a prime. We add to this list the values of $k$ that are three less than a prime equivalent to $11\left(mod12\right)$, and we show computer-assisted computations of many values of $k$ for which equality fails.

A group is called metahamiltonian if all its non-abelian subgroups are normal. It is known that any infinite locally graded group whose proper subgroups are metahamiltonian is likewise metahamiltonian, and the aim of this paper is to describe the structure of locally graded groups whose proper subgroups contain a metahamiltonian subgroup of finite index.

Let $P$ be an arbitrary partially ordered set, $R$ be a commutative ring with identity and $FI\left(P,R\right)$ be the finitary incidence algebra of $P$ over $R$. Under some natural assumption on $R$, we prove that each Lie-type derivation of $FI\left(P,R\right)$ is proper, which partially generalizes the main results of Zhang and Khrypchenko (2017), Wang and Xiao (2019), and Xiao and Yang (2019).

Recently in 2018, four open questions were raised by Oman and Salminen (2018). We answer three of them in this article.

We survey the Hirzebruch signature theorem as a special case of the Atiyah–Singer index theorem. The family version of the Atiyah–Singer index theorem in the form of the Riemann–Roch–Grothendieck–Quillen (RRGQ) formula is then applied to the complexified signature operators varying along the universal family of elliptic curves. The RRGQ formula allows us to determine a generalized cohomology class on the base of the elliptic fibration that is known in physics as (a measure of) the local and global anomaly. Combining several anomalous operators allows us to cancel the local anomaly on a Jacobian elliptic surface, a construction that is based on the construction of the Poincaré line bundle over an elliptic surface.

We prove that von Neumann’s inequality holds for circulant contractions. We show that every complex polynomial $f\left(z_1,\dots ,z_n\right)$ over ${\mathbb{𝔻}}^n$ is associated to a constant $d\left(f\right)$ such that von Neumann’s inequality can hold up to $d\left(f\right)$, for $n$-tuples of commuting contractions on a Hilbert space. We characterise complex polynomials over ${\mathbb{𝔻}}^n$ in which $d\left(f\right)=2$. We introduce the properties of upper (or lower) complex triangular Toeplitz matrices. We show that von Neumann’s inequality holds for $n$-tuples of upper (or lower) complex triangular Toeplitz contractions. We construct contractive homomorphisms.

In this paper we prove new sum formulas for Mordell–Tornheim zeta values in the case of depth 2 and 3, expressing the sums as single multiples of Riemann zeta values. Also, we obtain weighted sum formulas for double Mordell–Tornheim zeta values. Moreover, we present a sum formula for the Mordell–Tornheim series of even arguments.

This article documents my journey down the rabbit hole, chasing what I have come to know as a particularly unyielding problem in Ramsey theory on the integers: the $2$-large conjecture. This conjecture states that if $D\subseteq {\mathbb{Z}}^{+}$ has the property that every $2$-coloring of ${\mathbb{Z}}^{+}$ admits arbitrarily long monochromatic arithmetic progressions with common difference from $D$, then the same property holds for any finite number of colors. We hope to provide a roadmap for future researchers and also provide some new results related to the $2$-large conjecture.

The category of torus orbifolds is a generalization of the category of toric orbifolds which contains projective toric varieties associated to complete simplicial fans. We introduce the concept of “divisive” torus orbifolds following divisive weighted projective spaces. The divisive condition may ensure an invariant cell structure on a locally standard torus orbifold. We give a combinatorial description of equivariant $K$-theory, equivariant cobordism theory and equivariant cohomology theory of divisive torus orbifolds. In particular, we get a combinatorial description of these generalize cohomology theories for torus manifolds over acyclic polytopes.

We show that there exist infinitely many quartic number fields $K$ with large class numbers such that $K∕\mathbf{Q}$ is a Galois extension whose Galois group is isomorphic to a given finite group. Cho and Kim proved that there are infinitely many totally real cyclic extensions over $\mathbf{Q}$ of degree $4$ with large class numbers. We consider all the other cases of quartic Galois extensions.

We compare the invariants of flat vector bundles defined by Atiyah et al. and Jones et al. and prove that, up to weak homotopy, they induce the same map from the $0$-connective algebraic $K$-theory space of the complex numbers to the homotopy fiber of the Chern character. We examine homotopy properties of this map and its relation with other known invariants. In addition, using the formula for $\tilde{\xi }$-invariants of lens spaces derived from Donnelly’s fixed point theorem and the $4$-dimensional cobordisms constructed via relative Kirby diagrams, we recover the formula of the $e$-invariants of Seifert homology spheres given by Jones and Westbury, up to sign.

With the Hirota bilinear method and symbolic computation, explicit forms of $N$-soliton of the $\left(2+1\right)$-dimensional Hirota–Satsuma–Ito equation are derived. General high-order breather solutions are constructed through appropriate parameter restrictions. By performing an appropriate limiting procedures on soliton solutions and then making further parameter constraints, general lump solutions to the $\left(2+1\right)$-dimensional Hirota–Satsuma–Ito equation would be succinctly constructed. Furthermore, we provide the hybrid solutions which means different types of combinations in breathers, lumps and line solitons. In order to better understand the dynamical behaviors of the equation, the novel interaction and propagation characteristics are discussed graphically.

In this paper, we define and study irreducible convergence and irreducible order-convergence in $T_0$ spaces, which can be seen as topological counterparts of lim-inf-convergence and order-convergence in posets, respectively. Especially, we give sufficient and necessary conditions for irreducible convergence and irreducible order-convergence in $T_0$ spaces to be topological.

Let $\mathcal{ℛ}\left(n\right)$ denote the number of representations of the positive integer $n$ as the sum of two squares and three biquadrates of primes and we write $\mathcal{ℰ}\left(N\right)$ for the number of positive integers $n$ satisfying $n\le N$, $n\equiv 5,53,101\left(mod120\right)$ and

$$\left|\mathcal{ℛ}\left(n\right)-\frac{{\mathrm{\Gamma }}^2\left(\frac{1}{2}\right){\mathrm{\Gamma }}^3\left(\frac{1}{4}\right)}{\mathrm{\Gamma }\left(\frac{7}{4}\right)}\frac{\mathfrak{𝔖}\left(n\right)n^{\frac{3}{4}}}{{log}^{5}n}\right|\ge \frac{n^{\frac{3}{4}}}{{log}^{\frac{11}{2}}n},$$

where $0<\mathfrak{𝔖}\left(n\right)\ll 1$ is the singular series. In this paper, we prove

$$\mathcal{ℰ}\left(N\right)\ll N^{\frac{15}{32}+𝜀}$$

for any $𝜀>0$. This result constitutes a refinement upon that of Friedlander and Wooley (2014).

A general fourth-order regular ordinary differential equation with eigenvalue dependent boundary conditions and transmission conditions are considered. We prove that the eigenvalues depend continuously and smoothly on the coefficients of the differential equation and on the boundary and transmission matrices. We provide as well formulas for the derivatives with respect to each of these parameters.