2,977 research outputs found
Variations on a result of Bressoud
The well-known Rogers-Ramanujan identities have been a rich source of mathematical study over the last fifty years. In particular, Gordon’s generalization in the early 1960s led to additional work by Andrews and Bressoud in subsequent years. Unfortunately, these results lacked a certain amount of uniformity in terms of combinatorial interpretation. In this work, we provide a single combinatorial interpretation of the series sides of
these generating function results by using the concept of cluster parities. This unifies the aforementioned results of Andrews and Bressoud and also allows for a strikingly broader family of q–series results to be obtained. We close the paper by proving congruences for a “degenerate case” of Bressoud’s theorem
Andrews Style Partition Identities
We propose a method to construct a variety of partition identities at once.
The main application is an all-moduli generalization of some of Andrews'
results in [5]. The novelty is that the method constructs solutions to
functional equations which are satisfied by the generating functions. In
contrast, the conventional approach is to show that a variant of well-known
series satisfies the system of functional equations, thus reconciling two
separate lines of computations
Proving two partition identities
In this paper we give combinatorial proofs for two partition identities. The first one solves a recent open question formulated by G. E. Andrews.Neste artigo fornecemos provas combinatórias para duas identidades de partições. A primeira resolve uma questão recentemente formulada por G. E. Andrews.13314
Modulo extension of parity results in Rogers-Ramanujan-Gordon type overpartition identities
Sang, Shi and Yee, in 2020, found overpartition analogs of Andrews' results
involving parity in Rogers-Ramanujan-Gordon identities. Their result partially
answered an open question of Andrews'. The open question was to involve parity
in overpartition identities. We extend Sang, Shi, and Yee's work to arbitrary
moduli, and also provide a missing case in their identities. We also unify
proofs of Rogers-Ramanujan-Gordon identities for overpartitions due to Lovejoy
and Chen et.al.; Sang, Shi, and Yee's results; and ours. Although verification
type proofs are given for brevity, a construction of series as solutions of
functional equations between partition generating functions is sketched.Comment: 18 page
Chiral discrimination in optical trapping and manipulation
When circularly polarized light interacts with chiral molecules or nanoscale particles powerful symmetry principles determine the possibility of achieving chiral discrimination, and the detailed form of electrodynamic mechanisms dictate the types of interaction that can be involved. The optical trapping of molecules and nanoscale particles can be described in terms of a forward-Rayleigh scattering mechanism, with trapping forces being dependent on the positioning within the commonly non-uniform intensity beam profile. In such a scheme, nanoparticles are commonly attracted to local potential energy minima, ordinarily towards the centre of the beam. For achiral particles the pertinent material response property usually entails an electronic polarizability involving transition electric dipole moments. However, in the case of chiral molecules, additional effects arise through the engagement of magnetic counterpart transition dipoles. It emerges that, when circularly polarized light is used for the trapping, a discriminatory response can be identified between left- and right-handed polarizations. Developing a quantum framework to accurately describe this phenomenon, with a tensor formulation to correctly represent the relevant molecular properties, the theory leads to exact analytical expressions for the associated energy landscape contributions. Specific results are identified for liquids and solutions, both for isotropic media and also where partial alignment arises due to a static electric field. The paper concludes with a pragmatic analysis of the scope for achieving enantiomer separation by such methods
Quasi-Particles, Conformal Field Theory, and -Series
We review recent results concerning the representation of conformal field theory characters in terms of fermionic quasi-particle excitations, and describe in detail their construction in the case of the integrable three-state Potts chain. These fermionic representations are q-series which are generalizations of the sums occurring in the Rogers-Ramanujan identities
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