We extend the notion of standard pairs to the context of monomial ideals in semigroup rings. Stan... more We extend the notion of standard pairs to the context of monomial ideals in semigroup rings. Standard pairs can be used as a data structure to encode such monomial ideals, providing an alternative to generating sets that is well suited to computing intersections, decompositions, and multiplicities. We give algorithms to compute standard pairs from generating sets and vice versa and make all of our results effective. We assume that the underlying semigroup ring is positively graded, but not necessarily normal. The lack of normality is at the root of most challenges, subtleties, and innovations in this work.
We introduce a linear dimensionality reduction technique preserving topological features via pers... more We introduce a linear dimensionality reduction technique preserving topological features via persistent homology. The method is designed to find linear projection L which preserves the persistent diagram of a point cloud X via simulated annealing. The projection L induces a set of canonical simplicial maps from the Rips (orČech) filtration of X to that of LX. In addition to the distance between persistent diagrams, the projection induces a map between filtrations, called filtration homomorphism. Using the filtration homomorphism, one can measure the difference between shapes of two filtrations directly comparing simplicial complexes with respect to quasi-isomorphism µ quasi-iso or strong homotopy equivalence µ equiv. These µ quasi-iso and µ equiv measures how much portion of corresponding simplicial complexes is quasi-isomorphic or homotopy equivalence respectively. We validate the effectiveness of our framework with simple examples.
We construct a generalize Ishida complex to compute the local cohomology with monomial support of... more We construct a generalize Ishida complex to compute the local cohomology with monomial support of modules over quotients of polynomial rings by cellular binomial ideals. As a consequence, we obtain a combinatorial criterion to determine when such a quotient is Cohen-Macaulay. In particular, this gives a Cohen-Macaulayness criterion for lattice ideals. We also prove a result relating the local cohomology with radical monomial ideal support of an affine semigroup ring to the local cohomology with maximal ideal support of the quotient of the affine semigroup ring by the radical monomial ideal. This requires a combinatorial assumption on the semigroup, which holds for (not necessarily normal) semigroups whose cone is the cone over a simplex.
We give a combinatorial description of local cohomology modules of a graded module over a semigro... more We give a combinatorial description of local cohomology modules of a graded module over a semigroup ring, with support at the graded maximal ideal. This combinatorial framework yields Hochster-type formulas for the Hilbert series of such local cohomology modules in terms of the homology of finitely many polyhedral cell complexes. A Cohen-Macaulay criterion immediately follows. We also provide an alternative proof of a result of [18] characterizing Cohen-Macaulay affine semigroup rings.
Journal of Software for Algebra and Geometry, 2021
We present StdPairs, a SageMath library to compute standard pairs of a monomial ideal over a poin... more We present StdPairs, a SageMath library to compute standard pairs of a monomial ideal over a pointed (nonnormal) affine semigroup ring. Moreover, StdPairs provides the associated prime ideals, the corresponding multiplicities, and an irredundant irreducible primary decomposition of a monomial ideal. The library expands on the standardPairs function on Macaulay2 over polynomial rings, and is based on algorithms from Matusevich and Yu (2020). We also provide methods that allow the outputs from this library to be compatible with the Normaliz package of Macaulay2 and SageMath.
We extend the notion of standard pairs to the context of monomial ideals in semigroup rings. Stan... more We extend the notion of standard pairs to the context of monomial ideals in semigroup rings. Standard pairs can be used as a data structure to encode such monomial ideals, providing an alternative to generating sets that is well suited to computing intersections, decompositions, and multiplicities. We give algorithms to compute standard pairs from generating sets and vice versa and make all of our results effective. We assume that the underlying semigroup ring is positively graded, but not necessarily normal. The lack of normality is at the root of most challenges, subtleties, and innovations in this work.
We introduce a linear dimensionality reduction technique preserving topological features via pers... more We introduce a linear dimensionality reduction technique preserving topological features via persistent homology. The method is designed to find linear projection L which preserves the persistent diagram of a point cloud X via simulated annealing. The projection L induces a set of canonical simplicial maps from the Rips (or Čech) filtration of X to that of LX. In addition to the distance between persistent diagrams, the projection induces a map between filtrations, called filtration homomorphism. Using the filtration homomorphism, one can measure the difference between shapes of two filtrations directly comparing simplicial complexes with respect to quasi-isomorphism μquasi-iso or strong homotopy equivalence μequiv. These μquasi-iso and μequiv measures how much portion of corresponding simplicial complexes is quasi-isomorphic or homotopy equivalence respectively. We validate the effectiveness of our framework with simple examples.
통계학적 관점에서 본 집단적 유해물질 사건에서의 인과관계 법리에 관한 연구 -"담배소송"(대법원 2014. 4. 10. 선고 2011다22092 판결)을 중심으로
본고에서는 고엽제 사건에서 정립되어 담배소송에서도 적용된 집단적 유해물질 사건에서의 인과관계 요건의 합리적 증명책임 정도에 관하여 고찰한다. 특히 원고가 제출하는 역학조사 결... more 본고에서는 고엽제 사건에서 정립되어 담배소송에서도 적용된 집단적 유해물질 사건에서의 인과관계 요건의 합리적 증명책임 정도에 관하여 고찰한다. 특히 원고가 제출하는 역학조사 결과자료(역학적 상관관계)의 (집단적) 역학적 인과관계로의 인정기준 및 역학적 인과관계의 (개별적) 법적 인과관계로의 전환기준을 각 제시한다. 그중 후자에 관해서는 원고가 집단인지, 개인인지에 따라 다음과 같은 세부기준이 도출된다. 원고가 집단인 경우에는 대수의 강법칙에 따라 곧바로 집단적 인과관계를 인정할 수 있고, 개인인 경우에는 베이즈정리에 따라 역학적 인과관계를 계량화된 개별적 인과관계의 확률로 전환시킬 수 있다. 이때 관련된 여러 요인들이 경합할 경우에도, 베이즈정리는 판사가 이들을 비교하여 그중 가장 참일 가능성이 높은 요인을 사건의 원인으로써 판단할 수 있게 한다. 위 기준을 실제 사례인 국민건강보험공단의 담배소송과 개인의 담배소송 사례에 각 적용함으로써 이에 대한 시사점을 검토해본다.
We extend the notion of standard pairs to the context of monomial ideals in semigroup rings. Stan... more We extend the notion of standard pairs to the context of monomial ideals in semigroup rings. Standard pairs can be used as a data structure to encode such monomial ideals, providing an alternative to generating sets that is well suited to computing intersections, decompositions, and multiplicities. We give algorithms to compute standard pairs from generating sets and vice versa and make all of our results effective. We assume that the underlying semigroup ring is positively graded, but not necessarily normal. The lack of normality is at the root of most challenges, subtleties, and innovations in this work.
We introduce a linear dimensionality reduction technique preserving topological features via pers... more We introduce a linear dimensionality reduction technique preserving topological features via persistent homology. The method is designed to find linear projection L which preserves the persistent diagram of a point cloud X via simulated annealing. The projection L induces a set of canonical simplicial maps from the Rips (orČech) filtration of X to that of LX. In addition to the distance between persistent diagrams, the projection induces a map between filtrations, called filtration homomorphism. Using the filtration homomorphism, one can measure the difference between shapes of two filtrations directly comparing simplicial complexes with respect to quasi-isomorphism µ quasi-iso or strong homotopy equivalence µ equiv. These µ quasi-iso and µ equiv measures how much portion of corresponding simplicial complexes is quasi-isomorphic or homotopy equivalence respectively. We validate the effectiveness of our framework with simple examples.
We construct a generalize Ishida complex to compute the local cohomology with monomial support of... more We construct a generalize Ishida complex to compute the local cohomology with monomial support of modules over quotients of polynomial rings by cellular binomial ideals. As a consequence, we obtain a combinatorial criterion to determine when such a quotient is Cohen-Macaulay. In particular, this gives a Cohen-Macaulayness criterion for lattice ideals. We also prove a result relating the local cohomology with radical monomial ideal support of an affine semigroup ring to the local cohomology with maximal ideal support of the quotient of the affine semigroup ring by the radical monomial ideal. This requires a combinatorial assumption on the semigroup, which holds for (not necessarily normal) semigroups whose cone is the cone over a simplex.
We give a combinatorial description of local cohomology modules of a graded module over a semigro... more We give a combinatorial description of local cohomology modules of a graded module over a semigroup ring, with support at the graded maximal ideal. This combinatorial framework yields Hochster-type formulas for the Hilbert series of such local cohomology modules in terms of the homology of finitely many polyhedral cell complexes. A Cohen-Macaulay criterion immediately follows. We also provide an alternative proof of a result of [18] characterizing Cohen-Macaulay affine semigroup rings.
Journal of Software for Algebra and Geometry, 2021
We present StdPairs, a SageMath library to compute standard pairs of a monomial ideal over a poin... more We present StdPairs, a SageMath library to compute standard pairs of a monomial ideal over a pointed (nonnormal) affine semigroup ring. Moreover, StdPairs provides the associated prime ideals, the corresponding multiplicities, and an irredundant irreducible primary decomposition of a monomial ideal. The library expands on the standardPairs function on Macaulay2 over polynomial rings, and is based on algorithms from Matusevich and Yu (2020). We also provide methods that allow the outputs from this library to be compatible with the Normaliz package of Macaulay2 and SageMath.
We extend the notion of standard pairs to the context of monomial ideals in semigroup rings. Stan... more We extend the notion of standard pairs to the context of monomial ideals in semigroup rings. Standard pairs can be used as a data structure to encode such monomial ideals, providing an alternative to generating sets that is well suited to computing intersections, decompositions, and multiplicities. We give algorithms to compute standard pairs from generating sets and vice versa and make all of our results effective. We assume that the underlying semigroup ring is positively graded, but not necessarily normal. The lack of normality is at the root of most challenges, subtleties, and innovations in this work.
We introduce a linear dimensionality reduction technique preserving topological features via pers... more We introduce a linear dimensionality reduction technique preserving topological features via persistent homology. The method is designed to find linear projection L which preserves the persistent diagram of a point cloud X via simulated annealing. The projection L induces a set of canonical simplicial maps from the Rips (or Čech) filtration of X to that of LX. In addition to the distance between persistent diagrams, the projection induces a map between filtrations, called filtration homomorphism. Using the filtration homomorphism, one can measure the difference between shapes of two filtrations directly comparing simplicial complexes with respect to quasi-isomorphism μquasi-iso or strong homotopy equivalence μequiv. These μquasi-iso and μequiv measures how much portion of corresponding simplicial complexes is quasi-isomorphic or homotopy equivalence respectively. We validate the effectiveness of our framework with simple examples.
통계학적 관점에서 본 집단적 유해물질 사건에서의 인과관계 법리에 관한 연구 -"담배소송"(대법원 2014. 4. 10. 선고 2011다22092 판결)을 중심으로
본고에서는 고엽제 사건에서 정립되어 담배소송에서도 적용된 집단적 유해물질 사건에서의 인과관계 요건의 합리적 증명책임 정도에 관하여 고찰한다. 특히 원고가 제출하는 역학조사 결... more 본고에서는 고엽제 사건에서 정립되어 담배소송에서도 적용된 집단적 유해물질 사건에서의 인과관계 요건의 합리적 증명책임 정도에 관하여 고찰한다. 특히 원고가 제출하는 역학조사 결과자료(역학적 상관관계)의 (집단적) 역학적 인과관계로의 인정기준 및 역학적 인과관계의 (개별적) 법적 인과관계로의 전환기준을 각 제시한다. 그중 후자에 관해서는 원고가 집단인지, 개인인지에 따라 다음과 같은 세부기준이 도출된다. 원고가 집단인 경우에는 대수의 강법칙에 따라 곧바로 집단적 인과관계를 인정할 수 있고, 개인인 경우에는 베이즈정리에 따라 역학적 인과관계를 계량화된 개별적 인과관계의 확률로 전환시킬 수 있다. 이때 관련된 여러 요인들이 경합할 경우에도, 베이즈정리는 판사가 이들을 비교하여 그중 가장 참일 가능성이 높은 요인을 사건의 원인으로써 판단할 수 있게 한다. 위 기준을 실제 사례인 국민건강보험공단의 담배소송과 개인의 담배소송 사례에 각 적용함으로써 이에 대한 시사점을 검토해본다.
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