scholarly journals On positive geometries of quartic interactions: one loop integrands from polytopes

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Mrunmay Jagadale ◽  
Alok Laddha
Keyword(s):  
Large Class ◽  
Scattering Amplitude ◽  
Cluster Algebras ◽  
Seminal Work ◽  
New Class ◽  
Type D ◽  
Kinematic Space ◽  
The One

Abstract Building on the seminal work of Arkani-Hamed, He, Salvatori and Thomas (AHST) [1] we explore the positive geometry encoding one loop scattering amplitude for quartic scalar interactions. We define a new class of combinatorial polytopes that we call pseudo-accordiohedra whose poset structures are associated to singularities of the one loop integrand associated to scalar quartic interactions. Pseudo-accordiohedra parametrize a family of projective forms on the abstract kinematic space defined by AHST and restriction of these forms to the type-D associahedra can be associated to one-loop integrands for quartic interactions. The restriction (of the projective form) can also be thought of as a canonical top form on certain geometric realisations of pseudo-accordiohedra. Our work explores a large class of geometric realisations of the type-D associahedra which include all the AHST realisations. These realisations are based on the pseudo-triangulation model for type-D cluster algebras discovered by Ceballos and Pilaud [2].

scholarly journals Stringy canonical forms and binary geometries from associahedra, cyclohedra and generalized permutohedra

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Song He ◽  
Zhenjie Li ◽  
Prashanth Raman ◽  
Chi Zhang
Keyword(s):  
Large Class ◽  
Configuration Space ◽  
Moduli Space ◽  
Cluster Algebras ◽  
Configuration Spaces ◽  
Minkowski Sum ◽  
Canonical Forms ◽  
Infinite Class ◽  
Special Cases ◽  
Generalized Associahedra

Abstract Stringy canonical forms are a class of integrals that provide α′-deformations of the canonical form of any polytopes. For generalized associahedra of finite-type cluster algebras, there exist completely rigid stringy integrals, whose configuration spaces are the so-called binary geometries, and for classical types are associated with (generalized) scattering of particles and strings. In this paper, we propose a large class of rigid stringy canonical forms for another class of polytopes, generalized permutohedra, which also include associahedra and cyclohedra as special cases (type An and Bn generalized associahedra). Remarkably, we find that the configuration spaces of such integrals are also binary geometries, which were suspected to exist for generalized associahedra only. For any generalized permutohedron that can be written as Minkowski sum of coordinate simplices, we show that its rigid stringy integral factorizes into products of lower integrals for massless poles at finite α′, and the configuration space is binary although the u equations take a more general form than those “perfect” ones for cluster cases. Moreover, we provide an infinite class of examples obtained by degenerations of type An and Bn integrals, which have perfect u equations as well. Our results provide yet another family of generalizations of the usual string integral and moduli space, whose physical interpretations remain to be explored.

Alginate–copper microspheres as efficient and reusable heterogeneous catalysts for the one-pot synthesis of 4-organylselanyl-1H-pyrazoles

2020 ◽  
Vol 10 (12) ◽  
pp. 3918-3930 ◽  
Author(s):  
Jaqueline F. Souza ◽  
Thalita F. B. de Aquino ◽  
José E. R. Nascimento ◽  
Raquel G. Jacob ◽  
André R. Fajardo
Keyword(s):  
Heterogeneous Catalysts ◽  
One Pot ◽  
New Class ◽  
One Pot Synthesis ◽  
Facile Preparation ◽  
The One ◽  
Substituted Pyrazoles

This study demonstrates the facile preparation and use of alginate–Cu2+ microspheres in the catalysis of a new class of organoselenium substituted pyrazoles through one-pot reactions.

On nominal tense

2020 ◽  
Vol 24 (2) ◽  
pp. 311-352
Author(s):  
Pier Marco Bertinetto
Keyword(s):  
Side Effect ◽  
Ongoing Debate ◽  
Seminal Work ◽  
Semantic Component ◽  
Aspect Markers ◽  
Paraguayan Guaraní ◽  
The One ◽  
The Individual ◽  
Intense Debate

AbstractNordlinger & Sadler’s (2004. Nominal tense in crosslinguistic perspective. Language 80. 776–806) seminal work fostered an intense debate on the semantics of nominal tense systems, with the side effect of widening the typological coverage of this grammatical feature. This paper aims at contributing to the ongoing debate. In contrast with work by Tonhauser, who excluded ‘tense’ as a semantic component of the Paraguayan Guaraní nominal tense system, the paper claims that all TAM dimensions are involved – temporality, aspect, modality – with different proportions in the individual markers. Most importantly, it claims that nominal tense does not presuppose a semantics of its own, other than the one needed for verbal tenses. Moreover, the paper presents evidence that the semantic component of aspect, besides being necessarily activated in any nominal tense marker, is also directly conveyed by some of them, which can legitimately be called ‘nominal aspect’ markers. This integrates Nordlinger & Sadler’s (2004) survey, in which aspect was notably absent. In addition, the paper points out possible cases of nominal actionality (a.k.a. Aktionsart). Finally, the paper suggests that the pervasive presence of aspect (and also, but rarely, actionality) among nominal tense markers finds interesting parallels in some types of deverbal nominalizations, although these belong in another grammatical drawer.

scholarly journals SKEW CATEGORY ALGEBRAS ASSOCIATED WITH PARTIALLY DEFINED DYNAMICAL SYSTEMS

2012 ◽  
Vol 23 (04) ◽  
pp. 1250040 ◽  
Author(s):  
PATRIK LUNDSTRÖM ◽  
JOHAN ÖINERT
Keyword(s):  
Dynamical Systems ◽  
Topological Space ◽  
Large Class ◽  
Additive Group ◽  
Nonzero Ideal ◽  
Intersection Property ◽  
The Other ◽  
The One ◽  
Category Algebras ◽  
The Ideal

We introduce partially defined dynamical systems defined on a topological space. To each such system we associate a functor s from a category G to Topop and show that it defines what we call a skew category algebra A ⋊σ G. We study the connection between topological freeness of s and, on the one hand, ideal properties of A ⋊σ G and, on the other hand, maximal commutativity of A in A ⋊σ G. In particular, we show that if G is a groupoid and for each e ∈ ob (G) the group of all morphisms e → e is countable and the topological space s(e) is Tychonoff and Baire. Then the following assertions are equivalent: (i) s is topologically free; (ii) A has the ideal intersection property, i.e. if I is a nonzero ideal of A ⋊σ G, then I ∩ A ≠ {0}; (iii) the ring A is a maximal abelian complex subalgebra of A ⋊σ G. Thereby, we generalize a result by Svensson, Silvestrov and de Jeu from the additive group of integers to a large class of groupoids.

A CLASS OF CLOSED SOLUTIONS FOR THE BOGOLIUBOV EXCITATIONS ON SMOOTH GROUND STATE OF A TRAPPED BOSE–EINSTEIN CONDENSATE

2005 ◽  
Vol 19 (15) ◽  
pp. 713-720
Author(s):  
YONG-LI MA ◽  
HAICHEN ZHU
Keyword(s):  
Ground State ◽  
Einstein Condensate ◽  
Zero Energy ◽  
Ground State Wavefunction ◽  
New Class ◽  
Bose Einstein Condensate ◽  
Energy Mode ◽  
The One ◽  
Bose Einstein ◽  
Bogoliubov Excitations

Bogoliubov–de Gennes equations (BdGEs) for collective excitations from a trapped Bose–Einstein condensate described by a spatially smooth ground-state wavefunction can be treated analytically. A new class of closed solutions for the BdGEs is obtained for the one-dimensional (1D) and 3D spherically harmonic traps. The solutions of zero-energy mode of the BdGEs are also provided. The eigenfunctions of the excitations consist of zero-energy mode, zero-quantum-number mode and entire excitation modes when the approximate ground state is a background Bose gas sea.

scholarly journals Cluster structures for 2-Calabi–Yau categories and unipotent groups

2009 ◽  
Vol 145 (4) ◽  
pp. 1035-1079 ◽  
Author(s):  
A. B. Buan ◽  
O. Iyama ◽  
I. Reiten ◽  
J. Scott
Keyword(s):  
Cluster Structure ◽  
Cluster Algebras ◽  
New Class ◽  
Preprojective Algebras ◽  
Unipotent Groups ◽  
Special Cases ◽  
Dynkin Quivers ◽  
Cluster Tilting ◽  
Stable Categories ◽  
Tilting Objects

AbstractWe investigate cluster-tilting objects (and subcategories) in triangulated 2-Calabi–Yau and related categories. In particular, we construct a new class of such categories related to preprojective algebras of non-Dynkin quivers associated with elements in the Coxeter group. This class of 2-Calabi–Yau categories contains, as special cases, the cluster categories and the stable categories of preprojective algebras of Dynkin graphs. For these 2-Calabi–Yau categories, we construct cluster-tilting objects associated with each reduced expression. The associated quiver is described in terms of the reduced expression. Motivated by the theory of cluster algebras, we formulate the notions of (weak) cluster structure and substructure, and give several illustrations of these concepts. We discuss connections with cluster algebras and subcluster algebras related to unipotent groups, in both the Dynkin and non-Dynkin cases.

scholarly journals MORE ABOUT THE WILSONIAN ANALYSIS ON THE PIONLESS NEFT

2009 ◽  
Vol 24 (16n17) ◽  
pp. 3191-3225 ◽  
Author(s):  
KOJI HARADA ◽  
HIROFUMI KUBO ◽  
ATSUSHI NINOMIYA
Keyword(s):  
Fixed Points ◽  
Scattering Amplitude ◽  
Flow Equation ◽  
Effective Field ◽  
P Wave ◽  
P Waves ◽  
S Waves ◽  
Partial Waves ◽  
The One ◽  
Power Divergence

We extend our Wilsonian renormalization group (RG) analysis on the pionless nuclear effective field theory in the two-nucleon sector in two ways; on the one hand, (1) we enlarge the space of operators up to including those of [Formula: see text] in the S waves, and, on the other hand, (2) we consider the RG flows in higher partial waves (P and D waves). In the larger space calculations, we find, in addition to nontrivial fixed points, two "fixed lines" and a "fixed surface" which are related to marginal operators. In the higher partial wave calculations, we find similar phase structures to that of the S waves, but there are two relevant directions in the P waves at the nontrivial fixed points and three in the D waves. We explain the physical meaning of the P-wave phase structure by explicitly calculating the low-energy scattering amplitude. We also discuss the relation between the Legendre flow equation which we employ and the RG equation by Birse, McGovern and Richardson, and possible implementation of power divergence subtraction in higher partial waves.

scholarly journals A THEORY OF ALGEBRAIC INTEGRATION

2000 ◽  
Vol 14 (22n23) ◽  
pp. 2293-2297
Author(s):  
R. CASALBUONI
Keyword(s):  
Large Class ◽  
Configuration Space ◽  
Path Integral ◽  
Physical Principle ◽  
Integral Approach ◽  
Path Integral Approach ◽  
The One ◽  
Definition Of

In this paper we study the problem of quantizing theories defined over a nonclassical configuration space. If one follows the path-integral approach, the first problem one is faced with is the one of definition of the integral over such spaces. We consider this problem and we show how to define an integration which respects the physical principle of composition of the probability amplitudes for a very large class of algebras.

scholarly journals Crossings and Nestings in Set Partitions of Classical Types

10.37236/392 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Martin Rubey ◽  
Christian Stump
Keyword(s):  
Type A ◽  
The Other ◽  
Type B ◽  
Set Partition ◽  
Set Partitions ◽  
Type D ◽  
Other Hand ◽  
The One

In this article, we investigate bijections on various classes of set partitions of classical types that preserve openers and closers. On the one hand we present bijections for types $B$ and $C$ that interchange crossings and nestings, which generalize a construction by Kasraoui and Zeng for type $A$. On the other hand we generalize a bijection to type $B$ and $C$ that interchanges the cardinality of a maximal crossing with the cardinality of a maximal nesting, as given by Chen, Deng, Du, Stanley and Yan for type $A$. For type $D$, we were only able to construct a bijection between non-crossing and non-nesting set partitions. For all classical types we show that the set of openers and the set of closers determine a non-crossing or non-nesting set partition essentially uniquely.

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