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Papers by thanh vu

Journal of Algebra

Abstract In this paper, we are interested in the properties of inner and outer projections with a... more Abstract In this paper, we are interested in the properties of inner and outer projections with a view toward the Eisenbud-Goto regularity conjecture or the characterization of varieties satisfying certain extremal conditions. For example, if X is a quadratic scheme, the depth and regularity X and those of its inner projection from a smooth point are equal. In general, the above equalities do not hold for non-quadratic schemes. Therefore it is natural to investigate the algebraic invariants (e.g., depth and regularity) of X and its projected image in general. We develop a framework which provides partial answers and explains their relations using the partial elimination ideal theory. Our main theorems recover several preceding results in the literature. We also give some interesting examples and applications to illustrate our results.

Let k be an algebraically closed field of characteristic 0. Let X be a 2 × e matrix of linear for... more Let k be an algebraically closed field of characteristic 0. Let X be a 2 × e matrix of linear forms over a polynomial ring k[x 1 ,. .. , x n ] (where e, n ≥ 1). We prove that the determinantal ring R = k[x 1 ,. .. , x n ]/I 2 (X) is Koszul if and only if in any Kronecker-Weierstrass normal form of X, the largest length of a nilpotent block is at most twice the smallest length of a scroll block. As an application, we classify rational normal scrolls whose all section rings by natural coordinates are Koszul. This result settles a conjecture due to Conca.

Journal of Physics A: Mathematical and Theoretical, 2018

We address the problem of quantum nonlocality with positive operator valued measures (POVM) in th... more We address the problem of quantum nonlocality with positive operator valued measures (POVM) in the context of Einstein-Podolsky-Rosen quantum steering. We show that, given a candidate for local hidden state (LHS) ensemble, the problem of determining the steerability of a bipartite quantum state of finite dimension with POVMs can be formulated as a nesting problem of two convex objects. One consequence of this is the strengthening of the theorem that justifies choosing the LHS ensemble based on symmetry of the bipartite state. As a more practical application, we study the classic problem of the steerability of two-qubit Werner states with POVMs. We show strong numerical evidence that these states are unsteerable with POVMs up to a mixing probability of 1 2 within an accuracy of 10 −3 .

Acta Mathematica Vietnamica, 2019

Let R and S be polynomial rings of positive dimensions over a field k. Let I ⊆ R, J ⊆ S be non-ze... more Let R and S be polynomial rings of positive dimensions over a field k. Let I ⊆ R, J ⊆ S be non-zero homogeneous ideals none of which contains a linear form. Denote by F the fiber product of I and J in T = R ⊗ k S. We compute homological invariants of the powers of F using the data of I and J. Under the assumption that either char k = 0 or I and J are monomial ideals, we provide explicit formulas for the depth and regularity of powers of F. In particular, we establish for all s ≥ 2 the intriguing formula depth(T /F s) = 0. If moreover each of the ideals I and J is generated in a single degree, we show that for all s ≥ 1, reg F s = max i∈[1,s] {reg I i + s − i, reg J i + s − i}. Finally, we prove that the linearity defect of F is the maximum of the linearity defects of I and J, extending previous work of Conca and Römer. The proofs exploit the so-called Betti splittings of powers of a fiber product.

Proceedings of the American Mathematical Society, 2019

Conca and Herzog proved in [10] that any product of ideals of linear forms in a polynomial ring h... more Conca and Herzog proved in [10] that any product of ideals of linear forms in a polynomial ring has a linear resolution. The goal of this paper is to establish the same result for any quadric hypersurface. The main tool we develop and use is a flexible version of Derksen and Sidman's approximation systems [13].

Nagoya Mathematical Journal, 2017

This work concerns the linearity defect of a module$M$over a Noetherian local ring$R$, introduced... more This work concerns the linearity defect of a module$M$over a Noetherian local ring$R$, introduced by Herzog and Iyengar in 2005, and denoted$\text{ld}_{R}M$. Roughly speaking,$\text{ld}_{R}M$is the homological degree beyond which the minimal free resolution of$M$is linear. It is proved that for any ideal$I$in a regular local ring$R$and for any finitely generated$R$-module$M$, each of the sequences$(\text{ld}_{R}(I^{n}M))_{n}$and$(\text{ld}_{R}(M/I^{n}M))_{n}$is eventually constant. The first statement follows from a more general result about the eventual constancy of the sequence$(\text{ld}_{R}C_{n})_{n}$where$C$is a finitely generated graded module over a standard graded algebra over$R$.

Trends in Mathematics, 2016

We study the free resolution of products of linear forms over a quadratic hypersurface. Our resul... more We study the free resolution of products of linear forms over a quadratic hypersurface. Our results support the conjecture that such free resolutions are linear.

EPL (Europhysics Letters), 2016

Fully characterizing the steerability of a quantum state of a bipartite system has remained an op... more Fully characterizing the steerability of a quantum state of a bipartite system has remained an open problem ever since the concept of steerability was first defined. In this paper, using our recent geometrical approach to steerability, we suggest a necessary and sufficient condition for a two-qubit state to be steerable with respect to projective measurements. To this end, we define the critical radius of local models and show that a state of two qubits is steerable with respect to projective measurements from Alice's side if and only if her critical radius of local models is less than 1. As an example, we calculate the critical radius of local models for the so-called T-states by proving the optimality of a recently-suggested ansatz for Alice's local hidden state model.

Physical Review A, 2016

When two qubits A and B are in an appropriate state, Alice can remotely steer Bob's system B into... more When two qubits A and B are in an appropriate state, Alice can remotely steer Bob's system B into different ensembles by making different measurements on A. This famous phenomenon is known as quantum steering, or Einstein-Podolsky-Rosen steering. Importantly, quantum steering establishes the correspondence not only between a measurement on A (made by Alice) and an ensemble of B (owned by Bob) but also between each of Alice's measurement outcomes and an unnormalized conditional state of Bob's system. The unnormalized conditional states of B corresponding to all possible measurement outcomes of Alice are called Alice's steering outcomes. We show that, surprisingly, the 4-dimensional geometry of Alice's steering outcomes completely determines both the non-separability of the two-qubit state and its steerability from her side. Consequently, the problem of classifying two-qubit states into non-separable and steerable classes is equivalent to geometrically classifying certain 4-dimensional skewed double-cones.

Journal of Algebraic Combinatorics, 2016

Fröberg's classical theorem about edge ideals with 2-linear resolution can be regarded as a class... more Fröberg's classical theorem about edge ideals with 2-linear resolution can be regarded as a classification of graphs whose edge ideals have linearity defect zero. Extending his theorem, we classify all graphs whose edge ideals have linearity defect at most 1. Our characterization is independent of the characteristic of the base field: the graphs in question are exactly weakly chordal graphs with induced matching number at most 2. The proof uses the theory of Betti splittings of monomial ideals due to Francisco, Hà, and Van Tuyl and the structure of weakly chordal graphs. Along the way, we compute the linearity defect of edge ideals of cycles and weakly chordal graphs. We are also able to recover and generalize previous results due to Dochtermann-Engström, Kimura and Woodroofe on the projective dimension and Castelnuovo-Mumford regularity of edge ideals.

This work concerns the linearity defect of a module M over a noetherian local ring R, introduced ... more This work concerns the linearity defect of a module M over a noetherian local ring R, introduced by Herzog and Iyengar in 2005, and denoted by ld R M. Roughly speaking, ld R M is the homological degree beyond which the minimal free resolution of M is linear. In the paper, it is proved that for any ideal I in a regular local ring R and for any finitely generated Rmodule M , each of the sequences (ld R (I n M)) n and (ld R (M/I n M)) n is eventually constant. The first statement follows from a more general result about the eventual constancy of the sequence (ld R C n) n where C is a finitely generated graded module over a standard graded algebra over R. The second statement follows from the first together with a result of Avramov on small homomorphisms.

The Michigan Mathematical Journal, 2015

Journal of Algebra, 2015

Let R be a standard graded algebra over an F-finite field of characteristic p > 0. Let φ : R → R ... more Let R be a standard graded algebra over an F-finite field of characteristic p > 0. Let φ : R → R be the Frobenius endomorphism. For each finitely generated graded R-module M , let φ M be the abelian group M with the R-module structure induced by the Frobenius endomorphism. The R-module φ M has a natural grading given by deg x = j if x ∈ M jp+i for some 0 ≤ i ≤ p − 1. In this paper, we prove that R is Koszul if and only if there exists a non-zero finitely generated graded R-module M such that reg R φ M < ∞. This result supplies another instance for the ability of the Frobenius in detecting homological properties, as exemplified by Kunz's famous regularity criterion. The main technical tool is the notion of Castelnuovo-Mumford regularity over certain homomorphisms between N-graded algebras. The latter notion is a common generalization of the relative and absolute Castelnuovo-Mumford regularity of modules.

Advances in Intelligent Systems and Computing, 2015

With the huge number of available images on the web, an effective image retrieval system has been... more With the huge number of available images on the web, an effective image retrieval system has been more and more needed. Improving the performance is one of crucial tasks in modern text-based image retrieval systems such as Google Image Search, Frickr, etc. In this paper, we propose a unified framework to cluster and re-rank returned images with respect to an input query. However, owning to a difference to previous methods of using only either textual or visual features of an image, we combine the textual and visual features to improve search performance. The experimental results show that our proposed model can significantly improve the performance of a text-based image search system (i.e. Flickr). Moreover, the performance of the system with the combination of textual and visual features outperforms the performance of both the textual-based system and the visual-based system.

Companion Proceedings of the Web Conference 2022

We propose a simple yet effective embedding model to learn quaternion embeddings for entities and... more We propose a simple yet effective embedding model to learn quaternion embeddings for entities and relations in knowledge graphs. Our model aims to enhance correlations between head and tail entities given a relation within the Quaternion space with Hamilton product. The model achieves this goal by further associating each relation with two relation-aware rotations, which are used to rotate quaternion embeddings of the head and tail entities, respectively. Experimental results show that our proposed model produces state-of-the-art performances on well-known benchmark datasets for knowledge graph completion. Our code is available at: https://github.com/daiquocnguyen/QuatRE. CCS CONCEPTS • Computing methodologies → Natural language processing; Neural networks.