112,946 research outputs found
Factorization in generalized Calogero-Moser spaces
No full textUsing a recent construction of Bezrukavnikov and Etingof we prove that there is a factorization of the Etingof-Ginzburg sheaf on the generalized Calogero-Moser space associated to a complex reflection group. In the case W = S_n, this confirms a conjecture of Etingof and Ginzburg
On singular Calogero-Moser spaces
No full textUsing combinatorial properties of complex reflection groups, we show that the generalised Calogero-Moser space associated to the centre of the corresponding rational Cherednik algebra is singular for all values of its deformation parameter c if and only if the group is different from the wreath product and the binary tetrahedral group. This result and a theorem of Ginzburg and Kaledin imply that there does not exist a symplectic resolution of the singular symplectic variety h+h*/W outside of these cases; conversely we show that there exists a symplectic resolution for the binary tetrahedral group (Hilbert schemes provide resolutions for the wreath product case)
Periodic Solutions of a System of Complex ODEs. II. Higher Periods
No full textIn a previous paper the real evolution of the system of ODEs ¨zn + zn = N m=1, m=n gnm(zn - zm) -3 , zn zn(t), zn dzn(t) dt , n = 1, . . . , N is discussed in CN , namely the N dependent variables zn, as well as the N(N - 1) (arbitrary!) "coupling constants" gnm, are considered to be complex numbers, while the independent variable t ("time") is real. In that context it was proven that there exists, in the phase space of the initial data zn(0), zn(0), an open domain having infinite measure, such that all trajectories emerging from it are completely periodic with period 2, zn(t + 2) = zn(t). In this paper we investigate, both by analytcal techniques and via the display of numerical simulations, the remaining solutions, and in particular we show that there exist many -- emerging out of sets of initial data having nonvanishing measures in the phase space of such data -- that are also completely periodic but with periods which are integer multiples of 2. We also elcidate the mechanism that yields nonperiodic solutions, including those characterized by a "chaotic" behavior, namely those associated, in the context of the initial-value problem, with a sensitive dependence on the initial data
Per i busti ritratto in marmo di Alfonso Lombardi (con una proposta per il perduto Carlo V e una lettera del cardinale Innocenzo Cibo
No full textAmong the many marble portrait busts executed by the Ferrarese sculptor Alfonso Lombardi (c. 1497 - 1537), Giorgio Vasari mentions only four in his biography of the artist: the portraits of Giuliano de’ Medici, Duke of Nemours and Pope Clement VII, now both in the Palazzo Vecchio in Florence, and two versions of the bust of Charles V, considered lost. The article seeks to retrace the physical history of these four objects, from their genesis to how they were collected. Thanks to a new photographic campaign, a thorough analysis of the Palazzo Vecchio busts can now be made. Viewed from the side, the bust of Giuliano is revealed as a faithful derivation from a medal produced in Rome in 1513. The bust of Clement VII shows that it was conceived in stylistic dialogue with portraits of the pontiff by other artists at the Papal Court: Sebastiano del Piombo, Benvenuto Cellini and Giovanni Bernardi. The second part of the article hypothesises that the lost bust of Charles V carved by Lombardi in 1533 was a portrait in armour. The sculptor’s correspondence reveals that a second version of this bust was made in the same year for Alessandro de’ Medici, Duke of Florence: once owned by Cardinal Innocenzo Cibo, this can be identified as the fragmentary sculpture now housed in the fortress in Massa
Calogero–Moser models V: supersymmetry and quantum
No full textIt is shown that the Calogero-Moser models based on all root systems of the finite reflection groups (both the crystallographic and non-crystallographic cases) with the rational (with/without a harmonic confining potential), trigonometric and hyperbolic potentials can be simply supersymmetrised in terms of superpotentials. There is a universal formula for the supersymmetric ground state wavefunction. Since the bosonic part of each supersymmetric model is the usual quantum Calogero-Moser model, this gives a universal formula for its ground state wavefunction and energy, which is determined purely algebraically. Quantum Lax pair operators and conserved quantities for all the above Calogero-Moser models are established. The supersymmetric generalisation of quantum Calogero-Moser models in terms of superpotentials is presented. It applies to all of the Calogero-Moser models based on the crystallographic and non-crystallographic root systems and with the degen-erate potentials, i.e. the rational, hyperbolic and trigonometric potentials. The su
Symmetries of Spin Calogero Models
No full textWe investigate the symmetry algebras of integrable spin Calogero systems constructed from Dunkl operators associated to finite Coxeter groups. Based on two explicit examples, we show that the common view of associating one symmetry algebra to a given Coxeter group W is wrong. More precisely, the symmetry algebra heavily depends on the representation of W on the spins. We prove this by identifying two different symmetry algebras for a B-L spin Calogero model and three for G(2) spin Calogero model. They are all related to the half-loop algebra and its twisted versions. Some of the result are extended to any finite Coxeter group
Symmetries of Spin Calogero Models
No full textWe investigate the symmetry algebras of integrable spin Calogero systems constructed from Dunkl operators associated to finite Coxeter groups. Based on two explicit examples, we show that the common view of associating one symmetry algebra to a given Coxeter group W is wrong. More precisely, the symmetry algebra heavily depends on the representation of W on the spins. We prove this by identifying two different symmetry algebras for a BL spin Calogero model and three for G₂ spin Calogero model. They are all related to the half-loop algebra and its twisted versions. Some of the result are extended to any finite Coxeter group.This paper is a contribution to the Special Issue on Dunkl Operators and Related Topics. N.C. would like to thank the hospitality of the Centre for Mathematical Science, City University, where this work was initiated
Symmetries of Spin Calogero Models
No full textWe investigate the symmetry algebras of integrable spin Calogero systems constructed from Dunkl operators associated to finite Coxeter groups. Based on two explicit examples, we show that the common view of associating one symmetry algebra to a given Coxeter group W is wrong. More precisely, the symmetry algebra heavily depends on the representation of W on the spins. We prove this by identifying two different symmetry algebras for a BL spin Calogero model and three for G₂ spin Calogero model. They are all related to the half-loop algebra and its twisted versions. Some of the result are extended to any finite Coxeter group.This paper is a contribution to the Special Issue on Dunkl Operators and Related Topics. N.C. would like to thank the hospitality of the Centre for Mathematical Science, City University, where this work was initiated
Symmetries of Spin Calogero Models
No full textWe investigate the symmetry algebras of integrable spin Calogero systems constructed from Dunkl operators associated to finite Coxeter groups. Based on two explicit examples, we show that the common view of associating one symmetry algebra to a given Coxeter group W is wrong. More precisely, the symmetry algebra heavily depends on the representation of W on the spins. We prove this by identifying two different symmetry algebras for a BL spin Calogero model and three for G₂ spin Calogero model. They are all related to the half-loop algebra and its twisted versions. Some of the result are extended to any finite Coxeter group.This paper is a contribution to the Special Issue on Dunkl Operators and Related Topics. N.C. would like to thank the hospitality of the Centre for Mathematical Science, City University, where this work was initiated
SEX STEROID METABOLISM AND ACTION, INFLAMMATION AND PROSTATE CANCER: HYPOTHETICAL MECHANISMS AND NETWORKS
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