On formulas and some combinatorial properties of Schubert Polynomials

By applying a Gr\"{o}bner-Shirshov basis of the symmetric group $S_{n}$, we give two formulas for Schubert polynomials, either of which involves only nonnegative monomials. We also prove some combinatorial properties of Schubert polynomials. As applications, we give two algorithms to calculate the structure constants for Schubert polynomials, one of which depends on Monk's formula.Comment: 32 page

Free Products of digroups

We construct the free products of arbitrary digroups, and thus we solve an open problem of Zhuchok.Comment: 13 page

No dialgebra has Gelfand-Kirillov dimension strictly between 1 and 2

The Gelfand-Kirillov dimension measures the asymptotic growth rate of algebras. For every associative dialgebra $\mathcal{D}$, the quotient $\mathcal{A}_\mathcal{D}:=\mathcal{D}/\mathsf{Id}(S)$, where $\mathsf{Id}(S)$ is the ideal of $\mathcal{D}$ generated by the set $S:=\{x \vdash y-x\dashv y \mid x,y\in \mathcal{D}\}$, is called the associative algebra associated to $\mathcal{D}$. Here we show that the Gelfand--Kirillov dimension of $\mathcal{D}$ is bounded above by twice the Gelfand--Kirillov dimension of $\mathcal{A}_\mathcal{D}$. Moreover, we prove that no associative dialgebra has Gelfand-Kirillov dimension strictly between 1 and 2.Comment: 12 page

Gr\"{o}bner-Shirshov bases method for Gelfand-Dorfman-Novikov algebras

We establish Gr\"{o}bner-Shirshov bases theory for Gelfand-Dorfman-Novikov algebras over a field of characteristic $0$. As applications, a PBW type theorem in Shirshov form is given and we provide an algorithm for solving the word problem of Gelfand-Dorfman-Novikov algebras with finite homogeneous relations. We also construct a subalgebra of one generated free Gelfand-Dorfman-Novikov algebra which is not free.Comment: 22 page

Composition-Diamond lemma for associative nn-conformal algebras

In this paper, we study the concept of associative $n$-conformal algebra over a field of characteristic 0 and establish Composition-Diamond lemma for a free associative $n$-conformal algebra. As an application, we construct Gr\"{o}bner-Shirshov bases for Lie $n$-conformal algebras presented by generators and defining relations.Comment: 47 page

On free Gelfand-Dorfman-Novikov-Poisson algebras and a PBW theorem

In 1997, X. Xu \cite{Xiaoping Xu Poisson} invented a concept of Novikov-Poisson algebras (we call them Gelfand-Dorfman-Novikov-Poisson (GDN-Poisson) algebras). We construct a linear basis of a free GDN-Poisson algebra. We define a notion of a special GDN-Poisson admissible algebra, based on X. Xu's definition and an S.I. Gelfand's observation (see \cite{Gelfand}). It is a differential algebra with two commutative associative products and some extra identities. We prove that any GDN-Poisson algebra is embeddable into its universal enveloping special GDN-Poisson admissible algebra. Also we prove that any GDN-Poisson algebra with the identity $x\circ(y\cdot z)=(x\circ y )\cdot z +(x\circ z) \cdot y$ is isomorphic to a commutative associative differential algebra.Comment: 23 page

Characterizing and Detecting CUDA Program Bugs

While CUDA has become a major parallel computing platform and programming model for general-purpose GPU computing, CUDA-induced bug patterns have not yet been well explored. In this paper, we conduct the first empirical study to reveal important categories of CUDA program bug patterns based on 319 bugs identified within 5 popular CUDA projects in GitHub. Our findings demonstrate that CUDA-specific characteristics may cause program bugs such as synchronization bugs that are rather difficult to detect. To efficiently detect such synchronization bugs, we establish the first lightweight general CUDA bug detection framework, namely Simulee, to simulate CUDA program execution by interpreting the corresponding llvm bytecode and collecting the memory-access information to automatically detect CUDA synchronization bugs. To evaluate the effectiveness and efficiency of Simulee, we conduct a set of experiments and the experimental results suggest that Simulee can detect 20 out of the 27 studied synchronization bugs and successfully detects 26 previously unknown synchronization bugs, 10 of which have been confirmed by the developers.Comment: 12 pages, 8 figure

A construction of free digroup

We give a construction of a free digroup $F(X)$ on a set $X$ and formulate the halo and the group parts of $F(X)$. We prove that $F(X)$ is isomorphic to $F(Y)$ if and only if $card(X)=card(Y)$.Comment: 12 page

Gr\"obner--Shirshov bases for commutative dialgebras

We establish Gr\"obner--Shirshov bases theory for commutative dialgebras. We show that for any ideal $I$ of $Di[X]$, $I$ has a unique reduced Gr\"obner--Shirshov basis, where $Di[X]$ is the free commutative dialgebra generated by a set $X$, in particular, $I$ has a finite Gr\"obner--Shirshov basis if $X$ is finite. As applications, we give normal forms of elements of an arbitrary commutative disemigroup, prove that the word problem for finitely presented commutative dialgebras (disemigroups) is solvable, and show that if $X$ is finite, then the problem whether two ideals of $Di[X]$ are identical is solvable. We construct a Gr\"obner--Shirshov basis in associative dialgebra $Di\langle X\rangle$ by lifting a Gr\"obner--Shirshov basis in $Di[X]$

Gelfand-Kirillov dimension of bicommutative algebras

We first offer a fast method for calculating the Gelfand-Kirillov dimension of a finitely presented commutative algebra by investigating certain finite set. Then we establish a Groebner-Shirshov bases theory for bicommutative algebras, and show that every finitely generated bicommutative algebra has a finite Groebner-Shirshov basis. As an application, we show that the Gelfand-Kirillov dimension of a finitely generated bicommutative algebra is a nonnegative integer