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Liyu Liu

Publications and source records attributed to Liyu Liu.

24 records · Page 2Linked to original sources
arXiv2018

Nakayama automorphisms of Ore extensions over polynomial algebras

Nakayama automorphisms play an important role in several mathematical branches, which are known to be tough to compute in general. We compute the Nakayama automorphism $ν$ of any Ore extension $R[x; σ, δ]$ over a polynomial algebra $R$ in $n$ variables for an arbitrary $n$. The formula of $ν$ is obtained explicitly. When $σ$ is not the identity map, the invariant $E^G$ is also investigated in term of Zhang's twist, where $G$ is a cyclic group sharing the same order with $σ$.

math.RA↗
arXiv2018

Modeling Three-dimensional Invasive Solid Tumor Growth in Heterogeneous Microenvironment under Chemotherapy

A systematic understanding of the evolution and growth dynamics of invasive solid tumors in response to different chemotherapy strategies is crucial for the development of individually optimized oncotherapy. Here, we develop a hybrid three-dimensional (3D) computational model that integrates pharmacokinetic model, continuum diffusion-reaction model and discrete cell automaton model to investigate 3D invasive solid tumor growth in heterogeneous microenvironment under chemotherapy. Specifically, we consider the effects of heterogeneous environment on drug diffusion, tumor growth, invasion and the drug-tumor interaction on individual cell level. We employ the hybrid model to investigate the evolution and growth dynamics of avascular invasive solid tumors under different chemotherapy strategies. Our simulations reproduce the well-established observation that constant dosing is generally more effective in suppressing primary tumor growth than periodic dosing, due to the resulting continuous high drug concentration. In highly heterogeneous microenvironment, the malignancy of the tumor is significantly enhanced, leading to inefficiency of chemotherapies. The effects of geometrically-confined microenvironment and non-uniform drug dosing are also investigated. Our computational model, when supplemented with sufficient clinical data, could eventually lead to the development of efficient in silico tools for prognosis and treatment strategy optimization.

q-bio.TO↗
arXiv2016

Hochschild cohomology of projective hypersurfaces

We compute Hochschild cohomology of projective hypersurfaces starting from the Gerstenhaber-Schack complex of the (restricted) structure sheaf. We are particularly interested in the second cohomology group and its relation with deformations. We show that a projective hypersurface is smooth if and only if the classical HKR decomposition holds for this group. In general, the first Hodge component describing scheme deformations has an interesting inner structure corresponding to the various ways in which first order deformations can be realized: deforming local multiplications, deforming restriction maps, or deforming both. We make our computations precise in the case of quartic hypersurfaces, and compute explicit dimensions in many examples.

math.AG↗
arXiv2015

Non-commutative deformations and quasi-coherent modules

We identify a class of "quasi-compact semi-separated" (qcss) twisted presheaves of algebras A for which well-behaved Grothendieck abelian categories of quasi-coherent modules Qch(A) are defined. This class is stable under algebraic deformation, giving rise to a 1-1 correspondence between algebraic deformations of A and abelian deformations of Qch(A). For a qcss presheaf A, we use the Gerstenhaber-Schack (GS) complex to explicitely parameterize the first order deformations. For a twisted presheaf A with central twists, we descibe an alternative category QPr(A) of quasi-coherent presheaves which is equivalent to Qch(A), leading to an alternative, equivalent association of abelian deformations to GS cocycles of qcss presheaves of commutative algebras. Our construction applies to the restriction O of the structure sheaf of a scheme X to a finite semi-separating open affine cover (for which we have an equivalence between Qch(O) and Qch(X)). Under a natural identification of Gerstenhaber-Schack cohomology of O and Hochschild cohomology of X, our construction is shown to be equivalent to Toda's construction in the smooth case.

math.AG↗
arXiv2014

Homological smoothness and deformations of generalized Weyl algebras

It is an immediate conclusion from Bavula's papers \cite{Bavula:GWA-def}, \cite{Bavula:GWA-tensor-product} that if a generalized Weyl algebra $A=\kk[z;λ,η,φ(z)]$ is homologically smooth, then the polynomial $φ(z)$ has no multiple roots. We prove in this paper that the converse is also true. Moreover, formal deformations of $A$ are studied when $\kk$ is of characteristic zero.

math.RA↗