scholarly journals Tropical ideals do not realise all Bergman fans

2021 ◽  
Vol 8 (3) ◽  
Author(s):  
Jan Draisma ◽  
Felipe Rincón
Keyword(s):  
Tropical Geometry ◽  
Tensor Products ◽  
Tropical Variety ◽  
Polyhedral Complex ◽  
Direct Sum ◽  
Weight One ◽  
Polyhedral Complexes

AbstractEvery tropical ideal in the sense of Maclagan–Rincón has an associated tropical variety, a finite polyhedral complex equipped with positive integral weights on its maximal cells. This leads to the realisability question, ubiquitous in tropical geometry, of which weighted polyhedral complexes arise in this manner. Using work of Las Vergnas on the non-existence of tensor products of matroids, we prove that there is no tropical ideal whose variety is the Bergman fan of the direct sum of the Vámos matroid and the uniform matroid of rank two on three elements and in which all maximal cones have weight one.

scholarly journals COMPLETIONS, REVERSALS, AND DUALITY FOR TROPICAL VARIETIES

2011 ◽  
Vol 10 (06) ◽  
pp. 1141-1163
Author(s):  
ZUR IZHAKIAN ◽  
LOUIS ROWEN
Keyword(s):  
Algebraic Structure ◽  
Tropical Geometry ◽  
Tropical Variety ◽  
Single Face ◽  
Polyhedral Complexes ◽  
Tropical Varieties

The object of this paper is to present two algebraic results with straightforward proofs, which have interesting consequences in tropical geometry. We start with an identity for polynomials over the max-plus algebra, which shows that any polynomial divides a product of binomials. Interpreted in tropical geometry, any tropical variety W can be completed to a union of tropical primitives, i.e. single-face polyhedral complexes. In certain situations, a tropical variety W has a "reversal" variety, which together with W already yields the union of primitives; this phenomenon is explained in terms of a map defined on the algebraic structure, and yields a duality on tropical hypersurfaces.

scholarly journals On Properties of ClassA(n)andn-Paranormal Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Xiaochun Li ◽  
Fugen Gao
Keyword(s):  
Positive Integer ◽  
Tensor Products ◽  
Unit Vector ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Direct Sum ◽  
Paranormal Operator ◽  
Necessary And Sufficient

Letnbe a positive integer, and an operatorT∈B(ℋ)is called a classA(n)operator ifT1+n2/1+n≥|T|2andn-paranormal operator ifT1+nx1/1+n≥||Tx||for every unit vectorx∈ℋ, which are common generalizations of classAand paranormal, respectively. In this paper, firstly we consider the tensor products for classA(n)operators, giving a necessary and sufficient condition forT⊗Sto be a classA(n)operator whenTandSare both non-zero operators; secondly we consider the properties forn-paranormal operators, showing that an-paranormal contraction is the direct sum of a unitary and aC.0completely non-unitary contraction.

scholarly journals Tropical geometry and Newton–Okounkov cones for Grassmannian of planes from compactifications

2020 ◽  
pp. 1-33
Author(s):  
Christopher Manon ◽  
Jihyeon Jessie Yang
Keyword(s):  
Tropical Geometry ◽  
Flag Varieties ◽  
Tropical Variety ◽  
Affine Cone ◽  
Toric Degenerations

Abstract We construct a family of compactifications of the affine cone of the Grassmannian variety of $2$ -planes. We show that both the tropical variety of the Plücker ideal and familiar valuations associated to the construction of Newton–Okounkov bodies for the Grassmannian variety can be recovered from these compactifications. In this way, we unite various perspectives for constructing toric degenerations of flag varieties.

scholarly journals Tropical ideals

2018 ◽  
Vol 154 (3) ◽  
pp. 640-670 ◽  
Author(s):  
Diane Maclagan ◽  
Felipe Rincón
Keyword(s):  
Special Class ◽  
Tropical Geometry ◽  
Basic Structure ◽  
Chain Condition ◽  
Duke Math ◽  
Polyhedral Complex ◽  
Algebraic Foundation ◽  
Tropical Varieties ◽  
Associated Variety ◽  
Ascending Chain Condition

We introduce and study a special class of ideals, called tropical ideals, in the semiring of tropical polynomials, with the goal of developing a useful and solid algebraic foundation for tropical geometry. The class of tropical ideals strictly includes the tropicalizations of classical ideals, and allows us to define subschemes of tropical toric varieties, generalizing Giansiracusa and Giansiracusa [Equations of tropical varieties, Duke Math. J. 165 (2016), 3379–3433]. We investigate some of the basic structure of tropical ideals, and show that they satisfy many desirable properties that mimic the classical setup. In particular, every tropical ideal has an associated variety, which we prove is always a finite polyhedral complex. In addition we show that tropical ideals satisfy the ascending chain condition, even though they are typically not finitely generated, and also the weak Nullstellensatz.

scholarly journals Tropical Linear Spaces and Tropical Convexity

10.37236/5271 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Simon Hampe
Keyword(s):  
Convex Set ◽  
Tropical Geometry ◽  
Tropical Variety ◽  
Linear Combinations ◽  
Linear Spaces ◽  
Classical Geometry ◽  
Tropical Convexity ◽  
Valuated Matroid ◽  
Tropical Varieties ◽  
Global Principle

In classical geometry, a linear space is a space that is closed under linear combinations. In tropical geometry, it has long been a consensus that tropical varieties defined by valuated matroids are the tropical analogue of linear spaces. It is not difficult to see that each such space is tropically convex, i.e. closed under tropical linear combinations. However, we will also show that the converse is true: Each tropical variety that is also tropically convex is supported on the complex of a valuated matroid. We also prove a tropical local-to-global principle: Any closed, connected, locally tropically convex set is tropically convex.

scholarly journals Fusion frames and g-frames in tensor product and direct sum of Hilbert spaces

2012 ◽  
Vol 6 (2) ◽  
pp. 287-303
Author(s):  
Amir Khosravi ◽  
Azandaryani Mirzaee
Keyword(s):  
Tensor Product ◽  
Finite Number ◽  
Hilbert Spaces ◽  
Product Space ◽  
Tensor Products ◽  
Riesz Bases ◽  
Direct Sums ◽  
Fusion Frames ◽  
Direct Sum ◽  
Tensor Product Space

In this paper we study fusion frames and g-frames for the tensor products and direct sums of Hilbert spaces. We show that the tensor product of a finite number of g-frames (resp. fusion frames, g-Riesz bases) is a g-frame (resp. fusion frame, g-Riesz basis) for the tensor product space and vice versa. Moreover we obtain some important results in tensor products and direct sums of g-frames, fusion frames, resolutions of the identity and duals.

A class of non-weight modules of 𝑈𝑝(𝖘𝖑2) and Clebsch–Gordan type formulas

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Yan-an Cai ◽  
Hongjia Chen ◽  
Xiangqian Guo ◽  
Yao Ma ◽  
Mianmian Zhu
Keyword(s):  
Quantum Group ◽  
Tensor Products ◽  
Simple Modules ◽  
Isomorphism Classes ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Direct Sum Decomposition ◽  
Free Modules ◽  
Weight Modules

Abstract In this paper, we construct a class of new modules for the quantum group U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) which are free of rank 1 when restricted to C ⁢ [ K ± 1 ] \mathbb{C}[K^{\pm 1}] . The irreducibility of these modules and submodule structure for reducible ones are determined. It is proved that any C ⁢ [ K ± 1 ] \mathbb{C}[K^{\pm 1}] -free U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) -module of rank 1 is isomorphic to one of the modules we constructed, and their isomorphism classes are obtained. We also investigate the tensor products of the C ⁢ [ K ± 1 ] \mathbb{C}[K^{\pm 1}] -free modules with finite-dimensional simple modules over U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) , and for the generic cases, we obtain direct sum decomposition formulas for them, which are similar to the well-known Clebsch–Gordan formula for tensor products between finite-dimensional weight modules over U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) .

Central Histamine, the H3-Receptor and Obesity Therapy

2019 ◽  
Vol 18 (7) ◽  
pp. 516-522
Author(s):  
Néstor F. Díaz ◽  
Héctor Flores-Herrera ◽  
Guadalupe García-López ◽  
Anayansi Molina-Hernández
Keyword(s):  
Body Weight ◽  
Food Intake ◽  
Animal Studies ◽  
Energy Homeostasis ◽  
Receptor Activation ◽  
Receptor Ligands ◽  
Histaminergic System ◽  
H3 Receptor ◽  
Weight One ◽  
The Central Nervous System

The brain histaminergic system plays a pivotal role in energy homeostasis, through H1- receptor activation, it increases the hypothalamic release of histamine that decreases food intake and reduces body weight. One way to increase the release of hypothalamic histamine is through the use of antagonist/inverse agonist for the H3-receptor. Histamine H3-receptors are auto-receptors and heteroreceptors located on the presynaptic membranes and cell soma of neurons, where they negatively regulate the synthesis and release of histamine and other neurotransmitters in the central nervous system. Although several compounds acting as H3-receptor antagonist/inverse agonists have been developed, conflicting results have been reported and only one has been tested as anti-obesity in humans. Animal studies revealed the opposite effect in food intake, energy expeditor, and body weight, depending on the drug, spice, and route of administration, among others. The present review will explore the state of art on the effects of H3-receptor ligands on appetite and body-weight, going through the following: a brief overview of the circuit involved in the control of food intake and energy homeostasis, the participation of the histaminergic system in food intake and body weight, and the H3-receptor as a potential therapeutic target for obesity.

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