arXiv subjects
Publications and source records attributed to Xianzu Lin.
In this paper, using $p$-adic analysis and $p$-adic L-functions, we show how to extend classical congruences (due to Wilson, Gauss, Dirichlet, Jacobi, Wolstenholme, Glaisher, Morley, Lemher and other people) to modulo $p^k$ for any $k>0$.
In this paper, we develop Furstenberg's proof of infinity of primes, and prove several results about prime divisors of sequences of integers, including the celebrated Schur's theorem. In particular, we give a simple proof of a classical result which says that a non-degenerate linear recurrence sequence of integers of order k>1 has infinitely many prime divisors.
In this paper we prove the existence of Hall's ray for the quadratic Lagrange spectrums of all real quadratic numbers. For a large class of real quadratic numbers, we compute the Hurwitz constants of their quadratic Lagrange spectrums
In this paper we establish properties of independence for the continued fraction expansions of two algebraic numbers. Roughly speaking, if the continued fraction expansions of two irrational algebraic numbers have the same long sub-word, then the two continued fraction expansions have the same tails. If the two expansions have mirror symmetry long sub-words, then both the two algebraic numbers are quadratic. Applying the above results, we prove a theorem analogous to the Roth's theorem about approximation by algebraic numbers.
In this paper we give a generalization of the main results in \cite{ab,ab1} about $b$-ary expansions of algebraic numbers. As a byproduct we get a large class of new transcendence criteria. One of our corollaries implies that $b$-ary expansions of linearly independent irrational algebraic numbers are quite independent. Motivated by this result, we propose a generalized Borel conjecture.
We consider $AD$-type orbifolds of the triplet vertex algebras $\mathcal{W}(p)$ extending the well-known $c=1$ orbifolds of lattice vertex algebras. We study the structure of Zhu's algebras $A(\mathcal{W}(p)^{A_m})$ and $A(\mathcal{W}(p)^{D_m})$, where $A_m$ and $D_m$ are cyclic and dihedral groups, respectively. A combinatorial algorithm for classification of irreducible $\mathcal{W}(p)^Γ$-modules is developed, which relies on a family of constant term identities and properties of certain polynomials based on constant terms. All these properties can be checked for small values of $m$ and $p$ with a computer software. As a result, we argue that if certain constant term properties hold, the irreducible modules constructed in [Commun. Contemp. Math. 15 (2013), 1350028, 30 pages, arXiv:1212.5453; Internat. J. Math. 25 (2014), 1450001, 34 pages, arXiv:1304.5711] provide a complete list of irreducible $\mathcal{W}(p)^{A_m}$ and $\mathcal{W}(p)^{D_m}$-modules. This paper is a continuation of our previous work on the $ADE$ subalgebras of the triplet vertex algebra $\mathcal{W}(p)$.
Let $c:SU(n)\rightarrow PSU(n)=SU(n)/\mathbb{Z}_{n}$ be the quotient map of the special unitary group $SU(n)$ by its center subgroup $\mathbb{Z}_{n}$. We determine the induced homomorphism $c^{\ast}:$ $H^{\ast}(PSU(n))\rightarrow H^{\ast}(SU(n))$ on cohomologies by computing with the prime orders of binomial coefficients
Let $\mathcal {W}(p)$ be the triplet vertex algebra of central charge $c_{p}=1-\frac{6(p-1)^{2}}{p}$, $p\geq2$. As a Virasoro module, we have $$\mathcal {W}(p)=\bigoplus_{n=0} ^{\infty}(2n+1) L(c_{p}, n^{2}p+np-n).$$ It was pointed out in \cite{am1} that $\mathcal {W}(p)$ admits an action of $\mathfrak{sl}_{2}$. In this paper we give a combinatorics description of $\mathcal {W}(p)$, from which the action of $\mathfrak{sl}_{2}$ follows quite directly. In the end of this paper we give similar descriptions of the invariant subalgebra $\mathcal {W}(p)^Γ$, these will be useful for the characterizations of the exceptional vertex operator algebras of central charge $1$ in forthcoming papers. We also hope to extend the method of this paper to subalgebra of lattice vertex operator algebras of higher rank.
In this paper, we give a characterization of the rational vertex operator algebra VTL, where L is the root lattice of type A1 and T is the tetrahedral group.
It is the second paper in a series devoted to the investigation of characterizations of the exceptional vertex operator algebras of central charge 1. In this paper, we give a characterization of the rational vertex operator algebra VOL, where L is the root lattice of type A1 and O is the octahedral group.
Motivated by \cite{am1}, for every finite subgroup $Γ\subset PSL(2,\mathbb{C})$ we investigate the fixed point subalgebra $\triplet^Γ$ of the triplet vertex $\mathcal {W}(p)$, of central charge $1-\frac{6(p-1)^{2}}{p}$, $p\geq2$. This part deals with the $A$-series in the ADE classification of finite subgroups of $PSL(2,\mathbb{C})$. First, we prove the $C_2$-cofiniteness of the $A_m$-fixed subalgebra $\triplet^{A_m}$. Then we construct a family of $\am$-modules, which are expected to form a complete set of irreps. As a strong support to our conjecture, we prove modular invariance of (generalized) characters of the relevant (logarithmic) modules. Further evidence is provided by calculations in Zhu's algebra for $m=2$. We also present a rigorous proof of the fact that the full automorphism group of $\triplet$ is $PSL(2,\mathbb{C})$.
We are continuing our study of ADE-orbifold subalgebras of the triplet vertex algebra W(p). This part deals with the dihedral series. First, subject to a certain constant term identity, we classify all irreducible modules for the vertex algebra $\bar{M(1)} ^+$, the $\Z_2$--orbifold of the singlet vertex algebra $\bar{M(1)}$. Then we classify irreducible modules and determine Zhu's and $C_2$--algebra for the vertex algebra $\triplet ^{D_2}$. A general method for construction of twisted $\triplet$--modules is also introduced. We also discuss classification of twisted $\bar{M(1)}$--modules including the twisted Zhu's algebra $A_Ψ (\bar{M(1)})$, which is of independent interest. The category of admissible $Ψ$-twisted $\bar{M(1)}$-modules is expected to be semisimple. We also prove $C_2$-cofiniteness of $\triplet^{D_m}$ for all $m$, and give a conjectural list of irreducible $\triplet^{D_m}$-modules. Finally, we compute characters of the relevant irreducible modules and describe their modular closure.