We introduce a unital associative algebra associated with degenerate CP1. We show that is a commutative algebra and whose Poincare' series is given by the number of partitions. Thereby, we can regard as a smooth degeneration limit of the elliptic algebra introduced by Feigin and Odesskii [Int. Math. Res. Notices 11, 531 (1997)]. Then we study the commutative family of the Macdonald difference operators acting on the space of symmetric functions. A canonical basis is proposed for this family by using and the Heisenberg representation of the commutative family studied by Shiraishi [ Commun. Math. Phys. 263, 439 (2006)]. It is found that the Ding-Iohara algebra [Lett. Math. Phys. 41, 183 (1997)] provides us with an algebraic framework for the free field construction. An elliptic deformation of our construction is discussed, showing connections with the Drinfeld quasi-Hopf twisting [Leningrad Math. J. 1, 1419 (1990)] in the sence of Babelon-Bernard-Billey [Phys. Lett. B. 375, 89 (1996)], the Ruijsenaars difference operator [Commun. Math. Phys. 110, 191 (1987)], and the operator M(q,t1,t2) of Okounkov-Pandharipande [e-print arXiv:math-ph/0411210].

We study symmetry properties of quantum systems over 2D graphs or manifolds, with continuous spins, in the spirit

of the Mermin-Wagner' theorem. In the model of quantum rotators the phase space of a single spin is a d-dimensional

torus, and spins (or particles) are attached to sites of a graph satisfying a special bi-dimensional property. We construct

the so-called infinite-volume Gibbs state, with or without boundary conditions. Next, we study the problem of preserving

or spontaneous breaking of continuous symmetry.

We construct a reciprocal transformation for the Geng-Xue equation and show that, with help of this transformation, we relate the first negative flow of the modified Boussinesq hierarchy to the Geng-Xue equation.

For two different natural definitions of Casimir operators for simple Lie algebras we show that their eigenvalues in the adjoint representation can be expressed polyno- mially in the universal Vogel’s parameters α, β, γ and give explicit formulae for the generating functions of these eigenvalues.

We prove the existence for each Hilbert space of the two new quasi hidden variable (qHV) models, statistically noncontextual and context-invariant, reproducing all the von Neumann joint probabilities via non-negative values of real-valued measures and all the quantum product expectations—via the qHV (classical-like) average of the product of the corresponding random variables. In a context-invariant model, a quantum observable X can be represented by a variety of random variables satisfying the functional condition required in quantum foundations but each of these random variables equivalently models X under all joint von Neumann measurements, regardless of their contexts. The proved existence of this model negates the general opinion that, in terms of random variables, the Hilbert space description of all the joint von Neumann measurements for dimH≥3 can be reproduced only contextually. The existence of a statistically noncontextual qHV model, in particular, implies that every N-partite quantum state admits a local quasi hidden variable model introduced in Loubenets [J. Math. Phys. 53, 022201 (2012)]. The new results of the present paper point also to the generality of the quasi-classical probability model proposed in Loubenets [J. Phys. A: Math. Theor. 45, 185306 (2012).

In this paper, we consider the q → 0 limit of the deformed Virasoro algebra and that of the level 1, 2 representation of the Ding-Iohara-Miki algebra. Moreover, 5D AGT correspondence in this limit is discussed. This specialization corresponds to the limit from Macdonalds functions to Hall-Littlewood functions. Using the theory of Hall-Littlewood functions, some problems are solved. For example, the simplest case of 5D AGT conjectures is proven in this limit, and we obtain a formula for the 4-point correlation function of a certain operator.

We show that one-variable reductions of the Pfaff-Toda integrable hierarchy in the dispersionless limit are described by a system of coupled elliptic Löwner (Komatu-Goluzin) equations.

We present a construction of an integrable model as a projective type limit of Calogero-Sutherland models of N fermionic particles, when *N* tends to infinity. Explicit formulas for limits of Dunkl operators and of commuting Hamiltonians by means of vertex operators are given.

To a $N \times N$ real symmetric matrix Kerov assigns a piecewise linear function whose local minima are the eigenvalues of this matrix and whose local maxima are the eigenvalues of its $(N-1) \times (N-1)$ submatrix. We study the scaling limit of Kerov's piecewise linear functions for Wigner and Wishart matrices. For Wigner matrices the scaling limit is given by the Verhik-Kerov-Logan-Shepp curve which is known from asymptotic representation theory. For Wishart matrices the scaling limit is also explicitly found, and we explain its relation to the Marchenko-Pastur limit spectral law.

We consider a class of deterministic local collisional dynamics, showing how to approximate them by means of stochastic models and then studying the fluctuations of the current of energy. We show first that the variance of the time-integrated current is finite and related to the conductivity by the Green-Kubo relation. Next we show that the law of the empirical average current satisfies a large deviations principle and compute explicitly the rate functional in a suitable scaling limit. We observe that this functional is not strictly convex.

We introduce for a general correlation scenario a new simulation model, a local quasi hidden variable(LqHV) model, where locality and the measure-theoretic construction inherent to a local hidden variable (LHV) model are preserved but positivity of a simulation measure is dropped. We specify a necessary and sufficient condition for LqHV modelling and, based on this, prove that every quantum correlation scenario admits a LqHV simulation. Via the LqHV approach, we construct analogs of Bell-type inequalities for an N-partite quantum state and find a new analytical upper bound on the maximal violation by an N-partite quantum state of S 1 × ⋯ × S N -setting Bell-type inequalities – either on correlation functions or on joint probabilities and for outcomes of an arbitrary spectral type, discrete or continuous. This general analytical upper bound is expressed in terms of the new state dilation characteristics introduced in the present paper and not only traces quantum states admitting an S 1 × ⋯ × S N -setting LHV description but also leads to the new exact numerical upper estimates on the maximal Bell violations for concrete N-partite quantum states used in quantum information processing and for an arbitrary N-partite quantum state. We, in particular, prove that violation by an N-partite quantum state of an arbitrary Bell-type inequality (either oncorrelation functions or on joint probabilities) for S settings per site cannot exceed (2S − 1) N − 1 even in case of an infinite dimensional quantum state and infinitely many outcomes.

Last years, bounds on the maximal quantum violation of general Bell inequalities were intensively discussed in the literature via different mathematical tools. In the present paper, we analyze quantum violation of general Bell inequalities via the LqHV (local quasi hidden variable) modelling framework, correctly reproducing the probabilistic description of every quantum correlation scenario. The LqHV mathematical framework allows us to derive for all d and N a new upper bound (2d-1)^{N-1} on the maximal violation by an N-qudit state of all general Bell inequalities, also, new upper bounds on the maximal violation by an N-qudit state of general Bell inequalities for S settings per site. These new upper bounds essentially improve all the known precise upper bounds on quantum violation of general multipartite Bell inequalities. For some S, d and N, the new upper bounds are attainable.

In this paper, we explore the set of linear maps sending the set of quantum Gaussian states into itself. These maps are in general not positive, a feature which can be exploited as a test to check whether a given quantum state belongs to the convex hull of Gaussian states (if one of the considered maps sends it into a non-positive operator, the above state is certified not to belong to the set). Gener-alizing a result known to be valid under the assumption of complete positivity, we provide a characterization of these Gaussian-to-Gaussian (not necessarily positive) superoperators in terms of their action on the characteristic function of the inputs. For the special case of one-mode mappings, we also show that any Gaussian-to-Gaussian superoperator can be expressed as a concatenation of a phase-space dilatation, followed by the action of a completely positive Gaussian channel, possibly composed with a transposition. While a similar decomposition is shown to fail in the multi-mode scenario, we prove that it still holds at least under the further hypoth-esis of homogeneous action on the covariance matrix. © 2015 AIP Publishing LLC.

We introduce an odd supersymmetric version of the Kronecker elliptic function. It satisfies the genus one Fay identity and supersymmetric version of the heat equation. As an application, we construct odd supersymmetric extensions of the elliptic *R*-matrices, which satisfy the classical and the associative Yang–Baxter equations.