Freiman ideals and the number of generators of powers of monomial ideals

Stanley depth of monomial ideals with small number of generators

Open Mathematics ◽

10.2478/s11533-009-0037-0 ◽

2009 ◽

Vol 7 (4) ◽

Cited By ~ 5

Author(s):

Mircea Cimpoeaş

Keyword(s):

Monomial Ideal ◽

Monomial Ideals ◽

Stanley Depth ◽

Number Of Generators ◽

Stanley Conjecture

AbstractFor a monomial ideal I ⊂ S = K[x 1...,x n], we show that sdepth(S/I) ≥ n − g(I), where g(I) is the number of the minimal monomial generators of I. If I =νI′, where ν ∈ S is a monomial, then we see that sdepth(S/I) = sdepth(S/I′). We prove that if I is a monomial ideal I ⊂ S minimally generated by three monomials, then I and S/I satisfy the Stanley conjecture. Given a saturated monomial ideal I ⊂ K[x 1,x 2,x 3] we show that sdepth(I) = 2. As a consequence, sdepth(I) ≥ sdepth(K[x 1,x 2,x 3]//I) +1 for any monomial ideal in I ⊂ K[x 1,x 2,x 3].

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Hypergraphs with high projective dimension and 1-dimensional hypergraphs

International Journal of Algebra and Computation ◽

10.1142/s0218196717500291 ◽

2017 ◽

Vol 27 (06) ◽

pp. 591-617 ◽

Cited By ~ 1

Author(s):

K.-N. Lin ◽

P. Mantero

Keyword(s):

Large Class ◽

Projective Dimension ◽

Monomial Ideals ◽

Connected Component ◽

Number Of Generators ◽

Ferrers Graph ◽

Explicit Procedure

(Dual) hypergraphs have been used by Kimura, Rinaldo and Terai to characterize squarefree monomial ideals [Formula: see text] with [Formula: see text], i.e. whose projective dimension equals the minimal number of generators of [Formula: see text] minus 1. In this paper, we prove sufficient and necessary combinatorial conditions for [Formula: see text]. The second main result is an effective explicit procedure to compute the projective dimension of a large class of 1-dimensional hypergraphs [Formula: see text] (the ones in which every connected component contains at most one cycle). An algorithm to compute the projective dimension is also provided. Applications of these results are given; they include, for instance, computing the projective dimension of monomial ideals whose associated hypergraph has a spanning Ferrers graph.

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Study on the Modes of Operation of a Multi-Machine Installations with Mechanical Reduction

Alternative Energy and Ecology (ISJAEE) ◽

10.15518/isjaee.2019.10-12.012-022 ◽

2019 ◽

pp. 12-22

Author(s):

A. M. Oleynikov ◽

L. N. Kanov

Keyword(s):

Mathematical Description ◽

Reactive Power ◽

Maximum Efficiency ◽

Wind Flow ◽

System Of Equations ◽

Mechanical Equilibrium ◽

Synchronous Generators ◽

Electromagnetic Excitation ◽

Number Of Generators ◽

Wide Range

The paper gives the description of the original wind electrical installation with mechanical reduction in which the output of vertical axis wind turbine with rather low rotation speed over multiplicator is distributed to a certain number of generators. The number of acting generators is determined by the output of actual operating wind stream at each moment. According to this constructive scheme, it is possible to provide effective and with maximum efficiency installation work in a wide range of wind speeds and under any schedule issued to the consumer of electricity. As there are no any experience in using such complexes, mathematical description of its main elements is given, namely windwheels, generators with electromagnetic excitation of magnetic electrical type, then their interaction with windwheel, and also the results of mathematical modeling of work system regimes under using the offered system of equations. The basis for the mathematical description of the main elements of the installation – synchronous generators – are the system of equations of electrical and mechanical equilibrium in relative units in rotating coordinates without considering saturation of the magnetic circuit. The equation of mechanical equilibrium systems includes torque and brake windwheel electromagnetic moments of generators with taking into account the reduction coefficients and friction. In addition, we specify the alternator rotor dynamics resulting from continuous torque of windwheel fluctuations under the influence of unsteady wind flow and wind speed serving as the original variable is modeled by a set of sinusoids. Model simplification is achieved by equivalization of similar generators and by disregarding these transitions with a small time constant. Calculation the installation with synchronous generators of two types of small and medium capacity taking into account the operational factors allowed us to demonstrate the logic of interactions in the main elements of the reported complex in the process of converting wind flow into the generated active and reactive power. We have shown the possibility of stable system work under changeable wind stream condition by regulating of the plant blade angle and with simultaneous varying of generator number of different types. All these are in great interest for project organizations and power producers.

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