Search results for "13a"

A numerical property of Hilbert functions and lex segment ideals

2017

We introduce the fractal expansions, sequences of integers associated to a number. We show that these sequences characterize the Osequences and encode some information about lex segment ideals. Moreover, we introduce numerical functions called fractal functions, and we use them to solve the open problem of the classification of the Hilbert functions of any bigraded algebra.

On the Betti numbers of three fat points in P1 × P1

2019

In these notes we introduce a numerical function which allows us to describe explicitly (and nonrecursively) the Betti numbers, and hence, the Hilbert function of a set Z of three fat points whose support is an almost complete intersection (ACI) in P1 × P1 . A nonrecursively formula for the Betti numbers and the Hilbert function of these configurations is hard to give even for the corresponding set of five points on a special support in P2 and we did not find any kind of this result in the literature. Moreover, we also give a criterion that allows us to characterize the Hilbert functions of these special set of fat points.

Locally nilpotent derivations of rings with roots adjoined

2013

Computing with Rational Symmetric Functions and Applications to Invariant Theory and PI-algebras

2012

The research of the first named author was partially supported by INdAM. The research of the second, third, and fourth named authors was partially supported by Grant for Bilateral Scientific Cooperation between Bulgaria and Ukraine. The research of the fifth named author was partially supported by NSF Grant DMS-1016086.

Invariant deformation theory of affine schemes with reductive group action

2015

We develop an invariant deformation theory, in a form accessible to practice, for affine schemes $W$ equipped with an action of a reductive algebraic group $G$. Given the defining equations of a $G$-invariant subscheme $X \subset W$, we device an algorithm to compute the universal deformation of $X$ in terms of generators and relations up to a given order. In many situations, our algorithm even computes an algebraization of the universal deformation. As an application, we determine new families of examples of the invariant Hilbert scheme of Alexeev and Brion, where $G$ is a classical group acting on a classical representation, and describe their singularities.

Zvaigžņotā Debess: 2011, Vasara (212)

2011

Contents: “ZVAIGŽŅOTĀ DEBESS” FORTY YEARS AGO: A.Balklavs. Extremely Distant New Objects of Metagalaxy (abridged) ; Newspaper „Literaturnaja Rossija” 9 April 1971. Astronautics: Yesterday, Today, Tomorrow (abridged) ; CONFERENCE “The VIEW from SPACE. FIRST MANNED SPACE FLIGHT – 50”: M.Gills. Conference on the Eve of Human Space Flight Anniversary ; J.Ekmanis, S.Negrejeva. Mstislav Keldysh and Golden Age of Soviet Science ; T.Millers. Contribution of Scientists of the Institute of Inorganic Chemistry to the Development of Space Technologies and Materials ; G.Vilka. Remembrance of Friedrich Zander in Rīga ; O.Plepis. My Contribution to the Development of Space Medicine ; J.Stradiņš. Latvia’s …

F-signature of pairs: Continuity, p-fractals and minimal log discrepancies

2011

This paper contains a number of observations on the {$F$-signature} of triples $(R,\Delta,\ba^t)$ introduced in our previous joint work. We first show that the $F$-signature $s(R,\Delta,\ba^t)$ is continuous as a function of $t$, and for principal ideals $\ba$ even convex. We then further deduce, for fixed $t$, that the $F$-signature is lower semi-continuous as a function on $\Spec R$ when $R$ is regular and $\ba$ is principal. We also point out the close relationship of the signature function in this setting to the works of Monsky and Teixeira on Hilbert-Kunz multiplicity and $p$-fractals. Finally, we conclude by showing that the minimal log discrepancy of an arbitrary triple $(R,\Delta,\b…

F-singularities via alterations

2011

For a normal F-finite variety $X$ and a boundary divisor $\Delta$ we give a uniform description of an ideal which in characteristic zero yields the multiplier ideal, and in positive characteristic the test ideal of the pair $(X,\Delta)$. Our description is in terms of regular alterations over $X$, and one consequence of it is a common characterization of rational singularities (in characteristic zero) and F-rational singularities (in characteristic $p$) by the surjectivity of the trace map $\pi_* \omega_Y \to \omega_X$ for every such alteration $\pi \: Y \to X$. Furthermore, building on work of B. Bhatt, we establish up-to-finite-map versions of Grauert-Riemenscheneider and Nadel/Kawamata-V…

Mutations in ATP13A2 (PARK9) are associated with an amyotrophic lateral sclerosis-like phenotype, implicating this locus in further phenotypic expans…

2019

Background Amyotrophic lateral sclerosis [1] is a genetically heterogeneous neurodegenerative disorder, characterized by late-onset degeneration of motor neurons leading to progressive limb and bulbar weakness, as well as of the respiratory muscles, which is the primary cause of disease fatality. To date, over 25 genes have been implicated as causative in ALS with C9orf72, SOD1, FUS, and TARDBP accounting for the majority of genetically positive cases. Results We identified two patients of Italian and French ancestry with a clinical diagnosis of juvenile-onset ALS who were mutation-negative in any of the known ALS causative genes. Starting with the index case, a consanguineous family of Ita…

Detection and clinical implications of a novel BCR-ABL1 E12A2 insertion/deletion in a CML patient expressing the E13A2 isoform

2019

Background/Aim: The Philadelphia chromosome is the most frequent cytogenetic abnormality in chronic myelogenous (CML). More than 95% of CML patients are diagnosed with the e13a2 or e14a2 BCR-ABL1 fusion transcripts while, in about 1% of these individuals, the break generates the e1a2 rearrangement. Furthermore, about 5% of CML patients are diagnosed with rare BCR-ABL1 fusion transcripts, such as e19a2, e8a2, e13a3, e14a3, e1a3 and e6a2. However, there is limited evidence concerning the clinical and prognostic implications of these infrequent oncogenic variants for CML patients receiving tyrosine kinase inhibitors (TKIs). Case Report: We describe a novel atypical e12a2 insertion/deletion (In…