15,688 research outputs found
Triangular bases in quantum cluster algebras and monoidal categorification conjectures
We consider the quantum cluster algebras which are injective-reachable and
introduce a triangular basis in every seed. We prove that, under some initial
conditions, there exists a unique common triangular basis with respect to all
seeds. This basis is parametrized by tropical points as expected in the
Fock-Goncharov conjecture.
As an application, we prove the existence of the common triangular bases for
the quantum cluster algebras arising from representations of quantum affine
algebras and partially for those arising from quantum unipotent subgroups. This
result implies monoidal categorification conjectures of Hernandez-Leclerc and
Fomin-Zelevinsky in the corresponding cases: all cluster monomials correspond
to simple modules.Comment: 93 pages; minor corrections, Definition 6.1.1 and Lemma 6.4.2 update
Quantum groups via cyclic quiver varieties I
We construct the quantized enveloping algebra of any simple Lie algebra of
type ADE as the quotient of a Grothendieck ring arising from certain cyclic
quiver varieties. In particular, the dual canonical basis of a one-half quantum
group with respect to Lusztig's bilinear form is contained in the natural basis
of the Grothendieck ring up to rescaling.
This paper expands the categorification established by Hernandez and Leclerc
to the whole quantum groups. It can be viewed as a geometric counterpart of
Bridgeland's recent work for type ADE.Comment: 34 pages, Example 3.2.3 added, clarification and corrections made
following the referee's suggestion
Compare triangular bases of acyclic quantum cluster algebras
Given a quantum cluster algebra, we show that its triangular bases defined by
Berenstein and Zelevinsky and those defined by the author are the same for the
seeds associated with acyclic quivers. This result implies that the
Berenstein-Zelevinsky's basis contains all the quantum cluster monomials.
We also give an easy proof that the two bases are the same for the seeds
associated with bipartite skew-symmetrizable matrices.Comment: 18 pages; minor correction; the proof of Lemma 3.2.1 correcte
Husimi operator and Husimi function for describing electron's probability distribution in uniform magnetic field derived by virtue of the entangled state representation
For the first time we introduce the Husimi operator
Delta_h(gamma,varepsilon;kappa) for studying Husimi distribution in phase
space(gamma,varepsilon) for electron's states in uniform magnetic field, where
kappa is the Gaussian spatial width parameter. Using the Wigner operator in the
entangled state |lambda> representation [Hong-Yi Fan, Phys. Lett. A 301
(2002)153; A 126 (1987) 145) we find that Delta_h(gamma,varepsilon;kappa) is
just a pure squeezed coherent state density operator |gamma,varepsilon>_kappa
kappa<gamma,varepsilon|, which brings convenience for studying and calculating
the Husimi distribution. We in many ways demonstrate that the Husimi
distributions are Gaussian-broadened version of the Wigner distributions.
Throughout our calculation we have fully employed the technique of integration
within an ordered product of operators.Comment: 15page
Further factorization of over a finite field
Let be a finite field with elements and a positive
integer. Mart\'inez, Vergara and Oliveira \cite{MVO} explicitly factorized
over under the condition of . In this
paper, suppose that and , where is a
prime, we explicitly factorize into irreducible factors in and count the number of its irreducible factors
The Highest Dimensional Stochastic Blockmodel with a Regularized Estimator
In the high dimensional Stochastic Blockmodel for a random network, the
number of clusters (or blocks) K grows with the number of nodes N. Two previous
studies have examined the statistical estimation performance of spectral
clustering and the maximum likelihood estimator under the high dimensional
model; neither of these results allow K to grow faster than N^{1/2}. We study a
model where, ignoring log terms, K can grow proportionally to N. Since the
number of clusters must be smaller than the number of nodes, no reasonable
model allows K to grow faster; thus, our asymptotic results are the "highest"
dimensional. To push the asymptotic setting to this extreme, we make additional
assumptions that are motivated by empirical observations in physical
anthropology (Dunbar, 1992), and an in depth study of massive empirical
networks (Leskovec et al 2008). Furthermore, we develop a regularized maximum
likelihood estimator that leverages these insights and we prove that, under
certain conditions, the proportion of nodes that the regularized estimator
misclusters converges to zero. This is the first paper to explicitly introduce
and demonstrate the advantages of statistical regularization in a parametric
form for network analysis
Faster Convergence of a Randomized Coordinate Descent Method for Linearly Constrained Optimization Problems
The problem of minimizing a separable convex function under linearly coupled
constraints arises from various application domains such as economic systems,
distributed control, and network flow. The main challenge for solving this
problem is that the size of data is very large, which makes usual
gradient-based methods infeasible. Recently, Necoara, Nesterov, and Glineur
[Journal of Optimization Theory and Applications, 173 (2017) 227-2254] proposed
an efficient randomized coordinate descent method to solve this type of
optimization problems and presented an appealing convergence analysis. In this
paper, we develop new techniques to analyze the convergence of such algorithms,
which are able to greatly improve the results presented there. This refined
result is achieved by extending Nesterov's second technique developed by
Nesterov [SIAM J. Optim. 22 (2012) 341-362] to the general optimization
problems with linearly coupled constraints. A novel technique in our analysis
is to establish the basis vectors for the subspace of the linearly constraints
Moment estimates and applications for SDEs driven by fractional Brownian motion with irregular drifts
In this paper, high-order moment, even exponential moment, estimates are
established for the H\"older norm of solutions to stochastic differential
equations driven by fractional Brownian motion whose drifts are measurable and
have linear growth. As applications, we first study the weak uniqueness of
solutions to fractional stochastic differential equations. Moreover, combining
our estimates and the Fourier transform, we establish the existence of density
of solutions to equations with irregular drifts.Comment: 25 page
Experimental consequences of -wave spin triplet superconductivity in ACrAs
The experimental observable properties of the triplet -wave pairing
state, proposed by Wu {\em et al.} [arXiv:1503.06707] in quasi-one dimensional
ACrAs materials, are theoretically investigated. This pairing state
is characterized by the line nodes on the plane on the Fermi surfaces.
Based on the three-band tight binding model, we obtain the specific heat,
superfluid density, Knight shift and spin relaxation rate and find that all
these properties at low temperature () show powerlaw behaviors and
are consistent available experiments. Particularly, the superfluid density
determined by the -wave pairing state in this quasi-one dimensional system
is anisotropic: the in-plane superfluid density varies as
but the out-plane one varies as
at low temperature. The anisotropic upper critical
field reported in experiment is consistent with the (i.e.,
) -wave pairing state. We also
suggest the phase-sensitive dc-SQUID measurements to pin down the triplet
-wave pairing state.Comment: 5 pages, 5 figures, + supplemental materials, Fig.3 is update
Finite-time scaling of dynamic quantum criticality
We develop a theory of finite-time scaling for dynamic quantum criticality by
considering the competition among an external time scale, an intrinsic reaction
time scale and an imaginary time scale arising respectively from an external
driving field, the fluctuations of the competing orders and thermal
fluctuations. Through a successful application in determining the critical
properties at zero temperature and the solution of real-time Lindblad master
equation near a quantum critical point at nonzero temperatures, we show that
finite-time scaling offers not only an amenable and systematic approach to
detect the dynamic critical properties, but also a unified framework to
understand and explore nonequilibrium dynamics of quantum criticality, which
shows specificities for open systems.Comment: 5 pages, 4 figure
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