Cylindric Partitions Research Articles

Burge multipartitions and the AGT correspondence

Burge multipartitions are tuples of partitions that satisfy a cyclic embedding condition. When uncoloured, Burge m-multipartitions give a combinatorial model for characters of the Wm algebras (W 2 is the Virasoro algebra). When n-coloured, they generalise the “cylindric partitions” that provide a combinatorial model (and a crystal graph) for integrable characters of the affine Lie algebra .Here, we show that the n-coloured Burge m-multipartitions yield the characters of the CFT cosets . Having previously shown that the same combinatorial objects give the SU(m) instanton partition functions in 𝒩 = 2 supersymmetric gauge theories on ℂ2/ℤ n , we have thus established a wide-ranging extension of the AGT correspondence.This talk is based on a collaboration with N.Macleod (Melbourne).

A New Approach of Solidification Analysis in Modular Latent Thermal Energy Storage Unit Based on Image Processing

The solidification process of RT18HC in a cylindrical shell and tube storage unit has been studied using a new methodology based on image processing. The main idea of the algorithm is to label the region of solidification and use statistical functions to calculate the dimensions of the solidification front over time. Said analysis includes two methods. The first method is to measure the solid fraction changes during solidification. The novelty of this method, as compared to other literature findings, is that pre-processing and calculation process occurs automatically via a calculation algorithm. This method is used to calculate the solid fraction of RT18HC which is reported to be a bit fast at the beginning that 40 % of its volume solidified in 1000 s while the rest of the process is completed in almost 6500 s. The second method is used to measure and calculate the thickness of the solid front by using image processing. This method’s error is calculated to be less than 7% throughout the entire process. The second method also acts as an experimental database of front thickness to use in a novel, simplified, semi-theoretical model proposed to calculate the solid front thickness as a function of time in this paper. It is also worth presenting solution extended by a general definition of thermal resistance for a cylindrical partition. The above study will enable the development of an enhanced and optimized model for complex geometries based on image processing techniques in the future. It will also allow the investigation of both processes i.e. solidification and melting alongside other influencing parameters such as the geometry of the storage unit in future.

The 𝐴₂ Andrews–Gordon identities and cylindric partitions

Inspired by a number of recent papers by Corteel, Dousse, Foda, Uncu and Welsh on cylindric partitions and Rogers–Ramanujan-type identities, we obtain theA2\mathrm {A}_2(orA2(1)\mathrm {A}_2^{(1)}) analogues of the celebrated Andrews–Gordon identities. We further proveqq-series identities that correspond to the infinite-level limit of the Andrews–Gordon identities forAr−1\mathrm {A}_{r-1}(orAr−1(1)\mathrm {A}_{r-1}^{(1)}) for arbitrary rankrr. Our results forA2\mathrm {A}_2also lead to conjectural, manifestly positive, combinatorial formulas for the22-variable generating function of cylindric partitions of rank33and leveldd, such thatddis not a multiple of33.

Combinatorial Constructions of Generating Functions of Cylindric Partitions with Small Profiles into Unrestricted or Distinct Parts

In this paper, cylindric partitions into profiles $c=(1,1)$ and $c=(2,0)$ are considered. The generating functions into unrestricted cylindric partitions and cylindric partitions into distinct parts with these profiles are constructed. The constructions are combinatorial and they connect the cylindric partitions with ordinary partitions.

Cylindrical and Asymmetrical 3D Convolution Networks for LiDAR-Based Perception.

State-of-the-art methods for driving-scene LiDAR-based perception (including point cloud semantic segmentation, panoptic segmentation and 3D detection, etc.) often project the point clouds to 2D space and then process them via 2D convolution. Although this cooperation shows the competitiveness in the point cloud, it inevitably alters and abandons the 3D topology and geometric relations. A natural remedy is to utilize the 3D voxelization and 3D convolution network. However, we found that in the outdoor point cloud, the improvement obtained in this way is quite limited. An important reason is the property of the outdoor point cloud, namely sparsity and varying density. Motivated by this investigation, we propose a new framework for the outdoor LiDAR segmentation, where cylindrical partition and asymmetrical 3D convolution networks are designed to explore the 3D geometric pattern while maintaining these inherent properties. The proposed model acts as a backbone and the learned features from this model can be used for downstream tasks such as point cloud semantic and panoptic segmentation or 3D detection. In this paper, we benchmark our model on these three tasks. For semantic segmentation, we evaluate the proposed model on several large-scale datasets, i.e., SemanticKITTI, nuScenes and A2D2. Our method achieves the state-of-the-art on the leaderboard of SemanticKITTI (both single-scan and multi-scan challenge), and significantly outperforms existing methods on nuScenes and A2D2 dataset. Furthermore, the proposed 3D framework also shows strong performance and good generalization on LiDAR panoptic segmentation and LiDAR 3D detection.

Lozenge Tilings and the Gaussian Free Field on a Cylinder

We use the periodic Schur process, introduced in arXiv:math/0601019v1, to study the random height function of lozenge tilings (equivalently, dimers) on an infinite cylinder distributed under two variants of the $q^{\operatorname{vol}}$ measure. Under the first variant, corresponding to random cylindric partitions, the height function converges to a deterministic limit shape and fluctuations around it are given by the Gaussian free field in the conformal structure predicted by the Kenyon-Okounkov conjecture. Under the second variant, corresponding to an unrestricted dimer model on the cylinder, the fluctuations are given by the same Gaussian free field with an additional discrete Gaussian shift component. Fluctuations of the latter type have been previously conjectured for dimer models on planar domains with holes.

Completing the A2 Andrews–Schilling–Warnaar Identities

Abstract We study the Andrews–Schilling–Warnaar sum-sides for the principal characters of standard (i.e., integrable, highest weight) modules of ${\operatorname {A}}_2^{(1)}$. These characters have been studied recently by Corteel, Dousse, Foda, Uncu, Warnaar, and Welsh. We prove complete sets of identities for moduli $5$ through $8$ and $10$. The cases of moduli $6$ and $10$ are new. Our proofs use Corteel–Welsh recursions for cylindric partitions and certain relations satisfied by the Andrews–Schilling–Warnaar sum-sides. We further show that at all moduli, a complete set of conjectures may be deduced using certain explicit “seed” conjectures. These seed conjectures are obtained by appropriately truncating our conjectures for the “infinite” level. Additionally, for moduli $3k$, we use an identity of Weierstraß to deduce new sum-product identities starting from the results of Andrews–Schilling–Warnaar.

Weighted cylindric partitions

Recently Corteel and Welsh outlined a technique for finding new sum-product identities by using functional relations between generating functions for cylindric partitions and a theorem of Borodin. Here, we extend this framework to include very general product-sides coming from work of Han and Xiong. In doing so, we are led to consider structures such as weighted cylindric partitions, symmetric cylindric partitions and weighted skew double-shifted plane partitions. We prove some new identities and obtain new proofs of known identities, including the Göllnitz–Gordon and Little Göllnitz identities as well as some beautiful Schmidt-type identities of Andrews and Paule.

LessNet: Lightweight and efficient semantic segmentation for large‐scale point clouds

AbstractWith a wide range of applications in autonomous driving and robotics, semantic segmentation for large‐scale outdoor point clouds is a critical and challenging issue. Due to the large number and irregular arrangement of point clouds, it is difficult to balance the efficiency and effectiveness. In this paper, we propose LessNet, a lightweight and efficient voxel‐based method for LiDAR‐only semantic segmentation, taking advantage of cylindrical partition and intra‐voxel feature fusion. Specifically, we use a cylindrical partition method to distribute the outdoor point clouds more evenly in voxels. To better encode the voxel features, we adopt an intra‐voxel aggregation method without querying neighbours. The voxel features are further input into a lightweight and effective 3D U‐net to aggregate local features and dilate the receptive field. Extensive experiments prove the satisfied semantic segmentation performance and the improvement of each component in our proposed framework. Our method is capable of processing more than one million point clouds at a time while retaining low latency and few parameters. Moreover, our method achieves comparable performance with state‐of‐the‐art approaches and outperforms all projection‐based methods on the SemanticKITTI benchmark.

Cylindric partitions and some new $A_2$ Rogers–Ramanujan identities

We study the generating functions for cylindric partitions with profile $(c_1,c_2,c_3)$ for all $c_1,c_2,c_3$ such that $c_1+c_2+c_3=5$. This allows us to discover and prove seven new $A_2$ Rogers-Ramanujan identities modulo $8$ with quadruple sums, related with work of Andrews, Schilling, and Warnaar.

QFunctions – A Mathematica package for q-series and partition theory applications

We describe the qFunctions Mathematica package for q-series and partition theory applications. This package includes both experimental and symbolic tools. The experimental set of elements includes guessers for q-difference equations and recurrences for given q-difference and fitting/finding explicit expressions for sequences of polynomials. This package can symbolically handle formal manipulations on q-differential, q-difference equations and recurrences, such as switching between these forms, finding the greatest common divisor of recurrences, and formal substitutions. The package qFunctions has implementations that automate the recurrence system creation of the weighted words approach as well as a scheme on cylindric partitions. We also discuss a slight extension of the classical method of weighed words and present some new sum-product identities.

Skew doubled shifted plane partitions: Calculus and asymptotics

<p style='text-indent:20px;'>In this paper, we establish a new summation formula for Schur processes, called the <i>complete summation formula</i>. As an application, we obtain the generating function and the asymptotic formula for the number of <i>doubled shifted plane partitions</i>, which can be viewed as plane partitions "shifted at the two sides". We prove that the order of the asymptotic formula depends only on the diagonal width of the doubled shifted plane partition, not on the profile (the skew zone) itself. By using similar methods, the generating function and the asymptotic formula for the number of <i>symmetric cylindric partitions</i> are also derived.</p>

SWIRL STAGES FOR HEAT AND MASS EXCHANGE EQUIPMENT

The results of experimental studies of hydrodynamics and mass exchange at vortex stages in bubbling mode are presented, as well as data of numerical simulation by finite element method using models of k-ε This made it possible to establish the presence of circulation movement of liquid at the stage, to determine profiles of phase speeds, to obtain the value of initial speed of gas phase movement, at which coordination of experimental and design values of liquid speed is observed. The design method shows the effect of the cylindrical partition installed between the swirler and the stage wall on the fluid velocity profile and its distribution in the working volume. It is proposed to use the obtained results of numerical modeling when designing multi-element stages to provide the required hydrodynamics on it.
 New vortex contact devices with annular channels, having lower hydraulic resistance compared to tangential devices and providing a developed interfacial surface, stable without fluctuations, operation of the stage at high gas and liquid flow rates, have been investigated.
 Data are presented for determination of hydraulic resistance of vortex stage, value of interfacial surface, coefficients of mass output and efficiency in investigated range of loads of liquid and gas L/G = 10-150.
 Designs of vortex contact stages for industrial brew column have been developed. Contact devices with ring channels allow to bring dissipation of energy in a gas phase in the bubbling and ring mode to 5 W/kg, to increase an interphase surface to 2000 m-1 and to reach the size of superficial coefficient of a massotdacha about (0.5-0.8)·10-3 of m/s.

The $$\varvec{A}_2$$ Rogers–Ramanujan Identities Revisited

In this note, we show how to use cylindric partitions to rederive the four $$A_2$$ Rogers–Ramanujan identities originally proven by Andrews, Schilling and Warnaar, and provide a proof of a similar fifth identity.

Some useful theorems for asymptotic formulas and their applications to skew plane partitions and cylindric partitions

Inspired by the works of Dewar, Murty and Kotěšovec, we establish some useful theorems for asymptotic formulas. As an application, we obtain asymptotic formulas for the numbers of skew plane partitions and cylindric partitions. We prove that the order of the asymptotic formula for the number of skew plane partitions of fixed width depends only on the width of the region, not on the profile (the skew zone) itself, while this is not true for cylindric partitions.

Cylindric partitions, characters and the Andrews–Gordon–Bressoud identities

We study the Andrews–Gordon–Bressoud (AGB) generalisations of the Rogers–Ramanujan q-series identities in the context of cylindric partitions. We recall the definition of r-cylindric partitions, and provide a simple proof of Borodin’s product expression for their generating functions, that can be regarded as a limiting case of an unpublished proof by Krattenthaler. We also recall the relationships between the r-cylindric partition generating functions, the principal characters of algebras, the minimal model characters of algebras, and the r-string abaci generating functions, providing simple proofs for each. We then set r = 2, and use two-cylindric partitions to re-derive the AGB identities as follows. Firstly, we use Borodin’s product expression for the generating functions of the two-cylindric partitions with infinitely long parts, to obtain the product sides of the AGB identities, times a factor , which is the generating function of ordinary partitions. Next, we obtain a bijection from the two-cylindric partitions, via two-string abaci, into decorated versions of Bressoud’s restricted lattice paths. Extending Bressoud’s method of transforming between restricted paths that obey different restrictions, we obtain sum expressions with manifestly non-negative coefficients for the generating functions of the two-cylindric partitions which contains a factor . Equating the product and sum expressions of the same two-cylindric partitions, and canceling a factor of on each side, we obtain the AGB identities.

Brane webs and random processes

We study (p, q) 5-brane webs dual to certain N M5-brane configurations and show that the partition function of these brane webs gives rise to cylindric Schur process with period N. This generalizes the previously studied case of period 1. We also show that open string amplitudes corresponding to these brane webs are captured by the generating function of cylindric plane partitions with profile determined by the boundary conditions imposed on the open string amplitudes.

Modular Invariants and Twisted EquivariantK-theory II: Dynkin diagram symmetries

AbstractThe most basic structure ofchiralconformal field theory (CFT) is the Verlinde ring. Freed-Hopkins-Teleman have expressed the Verlinde ring for the CFT's associated to loop groups, as twisted equivariant K-theory. We build on their work to express K-theoretically the structures offullCFT. In particular, the modular invariant partition functions (which essentially parametrise the possible full CFTs) have a rich interpretation within von Neumann algebras (subfactors), which has led to the developments of structures of full CFT such as the full system (fusion ring of defect lines), nimrep (cylindrical partition function), alpha-induction etc. Modular categorical interpretations for these have followed. For the generic families of modular invariants (i.e. those associated to Dynkin diagram symmetries), we provide aK-theoretic framework for these other CFT structures, and show how they relate to D-brane charges and charge-groups. We also study conformal embeddings and themodular invariant of SU(2), as well as some families of finite group doubles. This newK-theoretic framework allows us to simplify and extend the less transparent, more ad hoc descriptions of these structures obtained previously within CFT.

Enumeration of Cylindric Plane Partitions

Cylindric plane partitions may be thought of as a natural generalization of reverse plane partitions. A generating series for the enumeration of cylindric plane partitions was recently given by Borodin. As in the reverse plane partition case, the right hand side of this identity admits a simple factorization form in terms of the "hook lengths'' of the individual boxes in the underlying shape. The first result of this paper is a new bijective proof of Borodin's identity which makes use of Fomin's growth diagram framework for generalized RSK correspondences. The second result of this paper is a $(q,t)$-analog of Borodin's identity which extends previous work by Okada in the reverse plane partition case. The third result of this paper is an explicit combinatorial interpretation of the Macdonald weight occurring in the $(q,t)$-analog in terms of the non-intersecting lattice path model for cylindric plane partitions. Les partitions planes cylindriques sont une généralisation naturelle des partitions planes renversées. Une série génératrice pour énumération des partitions planes cylindriques a été donnée récemment par Borodin. Comme dans le cas des partitions planes renversées, la partie droite de cette identité peut être factoriser en terme de "longueur d’équerres'' des carrés dans la forme sous-jacente. Le premier résultat de cet article est une nouvelle preuve bijective de l'identité de Borodin qui utilise le cadre de "diagramme de croissance'' de Fomin pour la correspondance de RSK généralisée. Le deuxième résultat de cette article est une $(q,t)$-déformation d'identité de Borodin qui généralise un résultat de Okada dans le cas des partitions planes renversées. Le troisième résultat de cet article est une formule combinatoire explicite pour le poids de Macdonald qui utilise le modèle des chemins non-intersectant pour les partitions planes cylindriques.

Plane overpartitions and cylindric partitions

Generating functions for plane overpartitions are obtained using various methods such as nonintersecting paths, RSK type algorithms and symmetric functions. We extend some of the generating functions to cylindric partitions. Also, we show that plane overpartitions correspond to certain domino tilings and we give some basic properties of this correspondence.