45,602 research outputs found
From Quantum to Trigonometric Model: Space-of-Orbits View
A number of affine-Weyl-invariant integrable and exactly-solvable quantum
models with trigonometric potentials is considered in the space of invariants
(the space of orbits). These models are completely-integrable and admit extra
particular integrals. All of them are characterized by (i) a number of
polynomial eigenfunctions and quadratic in quantum numbers eigenvalues for
exactly-solvable cases, (ii) a factorization property for eigenfunctions, (iii)
a rational form of the potential and the polynomial entries of the metric in
the Laplace-Beltrami operator in terms of affine-Weyl (exponential) invariants
(the same holds for rational models when polynomial invariants are used instead
of exponential ones), they admit (iv) an algebraic form of the gauge-rotated
Hamiltonian in the exponential invariants (in the space of orbits) and (v) a
hidden algebraic structure. A hidden algebraic structure for
(A-B-C{-D)-models, both rational and trigonometric, is related to the
universal enveloping algebra . For the exceptional -models,
new, infinite-dimensional, finitely-generated algebras of differential
operators occur. Special attention is given to the one-dimensional model with
symmetry. In particular, the origin
of the so-called TTW model is revealed. This has led to a new quasi-exactly
solvable model on the plane with the hidden algebra .Comment: arXiv admin note: substantial text overlap with arXiv:1106.501
PAC-Bayesian bounds for learning LTI-ss systems with input from empirical loss
In this paper we derive a Probably Approxilmately Correct(PAC)-Bayesian error
bound for linear time-invariant (LTI) stochastic dynamical systems with inputs.
Such bounds are widespread in machine learning, and they are useful for
characterizing the predictive power of models learned from finitely many data
points. In particular, with the bound derived in this paper relates future
average prediction errors with the prediction error generated by the model on
the data used for learning. In turn, this allows us to provide finite-sample
error bounds for a wide class of learning/system identification algorithms.
Furthermore, as LTI systems are a sub-class of recurrent neural networks
(RNNs), these error bounds could be a first step towards PAC-Bayesian bounds
for RNNs.Comment: arXiv admin note: text overlap with arXiv:2212.1483
Unifying W-Algebras
We show that quantum Casimir W-algebras truncate at degenerate values of the
central charge c to a smaller algebra if the rank is high enough: Choosing a
suitable parametrization of the central charge in terms of the rank of the
underlying simple Lie algebra, the field content does not change with the rank
of the Casimir algebra any more. This leads to identifications between the
Casimir algebras themselves but also gives rise to new, `unifying' W-algebras.
For example, the kth unitary minimal model of WA_n has a unifying W-algebra of
type W(2,3,...,k^2 + 3 k + 1). These unifying W-algebras are non-freely
generated on the quantum level and belong to a recently discovered class of
W-algebras with infinitely, non-freely generated classical counterparts. Some
of the identifications are indicated by level-rank-duality leading to a coset
realization of these unifying W-algebras. Other unifying W-algebras are new,
including e.g. algebras of type WD_{-n}. We point out that all unifying quantum
W-algebras are finitely, but non-freely generated.Comment: 13 pages (plain TeX); BONN-TH-94-01, DFTT-15/9
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