We study some K3 surfaces obtained as minimal resolutions of quotients of subgroups of special reflection groups. Some of these were already studied in a previous paper by W. Barth and the second author. We give here an easy proof that these are K3 surfaces, give equations in weighted projective space and describe their geometry.

The 4-dimensional Sklyanin algebras are a well-studied 2-parameter family of non-commutative graded algebras, often denoted A(E, τ), that depend on a quartic elliptic curve E ⊆ ℙ³ and a translation automorphism τ of E. They are graded algebras generated by four degree-one elements subject to six quadratic relations and in many important ways they behave like the polynomial ring on four indeterminates except that they are not commutative. They can be seen as "elliptic analogues" of the enveloping algebra of 𝖌𝖑(2, ℂ) and the quantized enveloping algebras U_q(𝖌𝖑₂). Recently, Cho, Hong, and Lau conjectured that a certain 2-parameter family of algebras arising in their work on homological mirror symmetry consists of 4-dimensional Sklyanin algebras. This paper shows their conjecture is false in the generality they make it. On the positive side, we show their algebras exhibit features that are similar to, and differ from, analogous features of the 4-dimensional Sklyanin algebras in interesting ways. We show that most of the Cho-Hong-Lau algebras determine, and are determined by, the graph of a bijection between two 20-point subsets of the projective space ℙ³. The paper also examines a 3-parameter family of 4-generator 6-relator algebras admitting presentations analogous to those of the 4-dimensional Sklyanin algebras. This class includes the 4-dimensional Sklyanin algebras and most of the Cho-Hong-Lau algebras.

We compute explicitly traces of one-dimensional diffusion processes. The obtained trace forms can be regarded as Dirichlet forms on graphs. Then we discuss conditions ensuring the trace forms to be conservative. Finally, the obtained results are applied to the Bessel process of order ν.

Let 𝔽_q be a finite field with q elements. A function f : 𝔽_q^d × 𝔽_q^d → 𝔽_q is called a Mattila-Sjölin type function of index γ ∈ ℝ if γ is the smallest real number such that whenever |E| ≥ Cq^γ for a sufficiently large constant C, the set f(E, E) := {f(x, y) : x, y ∈ E} is equal to 𝔽_q. In this article, we construct an example of a Mattila-Sjölin type function f and provide its index, generalizing the result of Cheong, Koh, Pham and Shen [1].

In this paper, we define cup product on relative bounded cohomology, and study its basic properties. Then, by extending it to a more generalized formula, we prove that all cup products of bounded cohomology classes of an amalgamated free product G₁ *_A G₂ are zero for every positive degree, assuming that free factors G_i are amenable and amalgamated subgroup A is normal in both of them. As its consequences, we show that all cup products of bounded cohomology classes of the groups ℤ * ℤ and ${\mathbb{Z}}_n\;{\ast}_{{\mathbb{Z}}_d}\;{\mathbb{Z}}_m$, where d is the greatest common divisor of n and m, are zero for every positive degree.

We give a classification of real solvable Lie algebras whose non-trivial coadjoint orbits of corresponding to simply connected Lie groups are all of codimension 2. These Lie algebras belong to a well-known class, called the class of MD-algebras.

Let f(z) be a meromorphic function in several variables satisfying $$\lim_{r\rightarrow\infty}sup\frac{log T(r, f)}{r}=o.$$ We mainly investigate the uniqueness problem on f in ℂ^m sharing polynomial or periodic small function with its difference polynomials from a new perspective. Our main theorems can be seen as the improvement and extension of previous results.

In this paper, we introduce and study preresolving subcategories in an extriangulated category ${\mathfrak{C}}$. Let ${\mathcal{Y}}$ be a ${\mathcal{Z}}$-preresolving subcategory of ${\mathfrak{C}}$ admitting a ${\mathcal{Z}}$-proper ξ-generator ${\mathcal{X}}$. We give the characterization of ${\mathcal{Z}}$-proper ${\mathcal{Y}}$-resolution dimension of an object in ${\mathfrak{C}}$. Next, for an object A in ${\mathfrak{C}}$, if the ${\mathcal{Z}}$-proper ${\mathcal{Y}}$-resolution dimension of A is at most n, then all "n-${\mathcal{X}}$-syzygies" of A are objects in ${\mathcal{Y}}$. Finally, we prove that A has a ${\mathcal{Z}}$-proper ${\mathcal{X}}$-resolution if and only if A has a ${\mathcal{Z}}$-proper ${\mathcal{Y}}$-resolution. As an application, we introduce (${\mathcal{X}},\;{\mathcal{Z}}$)-Gorenstein subcategory ${\mathcal{G}{\mathcal{X}}_{\mathcal{Z}}({\xi})$ of ${\mathfrak{C}}$ and prove that ${\mathcal{G}{\mathcal{X}}_{\mathcal{Z}}({\xi})$ is both ${\mathcal{Z}}$-resolving subcategory and ${\mathcal{Z}}$-coresolving subcategory of ${\mathfrak{C}}$.

We consider thick generalized hexagons fully embedded in metasymplectic spaces, and we show that such an embedding either happens in a point residue (giving rise to a full embedding inside a dual polar space of rank 3), or happens inside a symplecton (giving rise to a full embedding in a polar space of rank 3), or is isometric (that is, point pairs of the hexagon have the same mutual position whether viewed in the hexagon or in the metasymplectic space-these mutual positions are equality, collinearity, being special, opposition). In the isometric case, we show that the hexagon is always a Moufang hexagon, its little projective group is induced by the collineation group of the metasymplectic space, and the metasymplectic space itself admits central collineations (hence, in symbols, it is of type F_4,1). We allow non-thick metasymplectic spaces without non-thick lines and obtain a full classification of the isometric embeddings in this case.