Related papers: Cluster ensembles, quantization and the dilogarith…
Recent advances in image clustering typically focus on learning better deep representations. In contrast, we present an orthogonal approach that does not rely on abstract features but instead learns to predict image transformations and…
This is the third part in a series of papers in which we introduce and develop a natural, general tensor category theory for suitable module categories for a vertex (operator) algebra. In this paper (Part III), we introduce and study…
We apply the abelianization technique to obtain an explicit ring presentation for the quasimap quantum cohomology of GIT quotients. As an application, for quiver varieties associated with oriented-acyclic quivers, we establish a cluster…
We study the cluster category of a canonical algebra A in terms of the hereditary category of coherent sheaves over the corresponding weighted projective line X. As an application we determine the automorphism group of the cluster category…
These are expanded notes from three survey lectures given at the 14th International Conference on Representations of Algebras (ICRA XIV) held in Tokyo in August 2010. We first study identities between products of quantum dilogarithm series…
Cluster states are key resources for measurement-based quantum information processing. Photonic cluster and graph states, in particular, play indispensable roles in quantum network and quantum metrology. We demonstrate a semiconductor…
Collective spins of large atomic samples trapped inside optical resonators can carry quantum information that can be processed in a way similar to quantum computation with continuous variables. It is shown here that by combining the…
We extend the notion of the quantization of the coefficients of the ordinary cluster algebras to the generalized cluster algebras by Chekhov and Shapiro. In parallel to the ordinary case, it is tightly integrated with certain…
In the loop quantum gravity framework, spin network states carry entanglement between quantum excitations of the geometry at different space points. This intertwiner entanglement is gauge-invariant and comes from quantum superposition of…
Quantum signal processing is a framework for implementing polynomial functions on quantum computers. To implement a given polynomial $P$, one must first construct a corresponding complementary polynomial $Q$. Existing approaches to this…
The paper is devoted to integral quantization, a procedure based on operator-valued measure and resolution of the identity. We insist on covariance properties in the important case where group representation theory is involved. We also…
In a recent paper the notion of {\em quantum perceptron} has been introduced in connection with projection operators. Here we extend this idea, using these kind of operators to produce a {\em clustering machine}, i.e. a framework which…
Clustering is grouping of data by the proximity of some properties. We report on the possibility of increasing the efficiency of clustering of points in a plane using artificial quantum neural networks after the replacement of the two-level…
We mostly survey results concerning the $L^2$ boundedness of oscillatory and Fourier integral operators. This article does not intend to give a broad overview; it mainly focusses on a few topics directly related to the work of the authors.
Intertwiner is a homomorphism between two existing dynamical R matrices, first introduced by Baxter in eight vertex-SOS correspondence, we develop certain equivalence relations among R matrices using intertwiners. Twist is a homomorphism…
We develop theoretical methods for the implementation of creation and destruction operators in separate registers of a quantum computer, allowing for a transparent and dynamical creation and destruction of particle modes in second…
We study the root of unity degeneration of cluster algebras and quantum dilogarithm identities. We prove identities for the cyclic dilogarithm associated with a mutation sequence of a quiver, and as a consequence new identities for the…
The preceding paper constructed tangle machines as diagrammatic models, and illustrated their utility with a number of examples. The information content of a tangle machine is contained in characteristic quantities associated to equivalence…
In a cluster algebra, a subset of initial cluster variables can be specialised in such a way that all elements of the resulting algebra become polynomial in the remaining variables.
Clustering is one of the main tasks in exploratory data analysis and descriptive statistics where the main objective is partitioning observations in groups. Clustering has a broad range of application in varied domains like climate,…