Thermodynamic Formalism and Variations of the Hausdorff Dimension of Quadratic Julia Sets

Let d(c) denote the Hausdorff dimension of the Julia set of the polynomial z? z²+cz\mapsto z^2+c. The function d restricted to [0,+X) is real analytic in [0,\frac14)è(\frac14,+￥)[0,\frac{1}{4})\cup (\frac{1}{4},+\infty) ([Ru²]), is left-continuous at ¼ ([Bo,Zi]) but not continuous ([Do,Se,Zi]). We prove that c? d￠(c)c\mapsto d'(c) tends to + X from the left at ¼ as (\frac14-c)^{d(\frac14)-\frac32}(\frac{1}{4}-c)^{d(\frac{1}{4})-\frac{3}{2}}. In particular the graph of d has a vertical tangent on the left at ¼, a result which supports the numerical experiments.     

The Largest Fragment of a Homogeneous Fragmentation Process

We show that in homogeneous fragmentation processes the largest fragment at time t has size

$$\begin{aligned} e^{-t \Phi '(\overline{p})}t^{-\frac{3}{2} (\log \Phi )'(\overline{p})+o(1)}, \end{aligned}$$

where \(\Phi \) is the Lévy exponent of the fragmentation process, and \(\overline{p}\) is the unique solution of the equation \((\log \Phi )'(\bar{p})=\frac{1}{1+\bar{p}}\). We argue that this result is in line with predictions arising from the classification of homogeneous fragmentation processes as logarithmically correlated random fields.

     

A Characterization of Vertex Operator Algebra {L(\frac{1}{2},0)\otimes L(\frac{1}{2},0)}

We study a simple, rational and C ₂-cofinite vertex operator algebra whose weight 1 subspace is zero, the dimension of weight 2 subspace is greater than or equal to 2 and with c = c? = 1. Under some additional conditions it is shown that such a vertex operator algebra is isomorphic to ${L(\frac{1}{2},0)\otimes L(\frac{1}{2},0)}We study a simple, rational and C ₂-cofinite vertex operator algebra whose weight 1 subspace is zero, the dimension of weight 2 subspace is greater than or equal to 2 and with c = c̃ = 1. Under some additional conditions it is shown that such a vertex operator algebra is isomorphic to L(\frac12,0)?L(\frac12,0){L(\frac{1}{2},0)\otimes L(\frac{1}{2},0)}.     

Factorization and Dilation Problems for Completely Positive Maps on von Neumann Algebras

We study factorization and dilation properties of Markov maps between von Neumann algebras equipped with normal faithful states, i.e., completely positive unital maps which preserve the given states and also intertwine their automorphism groups. The starting point for our investigation has been the question of existence of non-factorizable Markov maps, as formulated by C. Anantharaman-Delaroche. We provide simple examples of non-factorizable Markov maps on M_n(\mathbbC){M_n(\mathbb{C})} for all n ≥ 3, as well as an example of a one-parameter semigroup (T(t)) _t≥0 of Markov maps on M₄(\mathbbC){M_4(\mathbb{C})} such that T(t) fails to be factorizable for all small values of t > 0. As applications, we solve in the negative an open problem in quantum information theory concerning an asymptotic version of the quantum Birkhoff conjecture, as well as we sharpen the existing lower bound estimate for the best constant in the noncommutative little Grothendieck inequality.     

Entanglement of the Antisymmetric State

We analyse the entanglement of the antisymmetric state in dimension d × d and present two main results. First, we show that the amount of secrecy that can be extracted from the state is low, more precisely, the distillable key is bounded by O(\frac1d){O(\frac{1}{d})}. Second, we show that the state is highly entangled in the sense that a large number of ebits are needed in order to create the state: entanglement cost is larger than a constant, independent of d. The second result is shown to imply that the regularised relative entropy with respect to separable states is also lower bounded by a constant. Finally, we note that the regularised relative entropy of entanglement is asymptotically continuous in the state.     

Highest Weight Modules Over Quantum Queer Superalgebra {U_q(\mathfrak {q}(n))}

In this paper, we investigate the structure of highest weight modules over the quantum queer superalgebra U_q(\mathfrak q(n)){U_q(\mathfrak {q}(n))}. The key ingredients are the triangular decomposition of U_q(\mathfrak q(n)){U_q(\mathfrak {q}(n))} and the classification of finite dimensional irreducible modules over quantum Clifford superalgebras. The main results we prove are the classical limit theorem and the complete reducibility theorem for U_q(\mathfrak q(n)){U_q(\mathfrak {q}(n))}-modules in the category O_q ^{3 0}{\mathcal {O}_{q}^{\geq 0}}.     

Poincaré-Dulac Normal Form Reduction for Unconditional Well-Posedness of the Periodic Cubic NLS

We implement an infinite iteration scheme of Poincaré-Dulac normal form reductions to establish an energy estimate on the one-dimensional cubic nonlinear Schrödinger equation (NLS) in ${C_tL^2(\mathbb{T})}$ C t L 2 ( T ) , without using any auxiliary function space. This allows us to construct weak solutions of NLS in ${C_tL^2(\mathbb{T})}$ C t L 2 ( T ) with initial data in ${L^2(\mathbb{T})}$ L 2 ( T ) as limits of classical solutions. As a consequence of our construction, we also prove unconditional well-posedness of NLS in ${H^s(\mathbb{T})}$ H s ( T ) for ${s \geq \frac{1}{6}}$ s ≥ 1 6 .     

Chiral properties of baryon interpolating fields

We study the chiral transformation properties of all possible local (non-derivative) interpolating field operators for baryons consisting of three quarks with two flavors, assuming good isospin symmetry. We derive and use the relations/identities among the baryon operators with identical quantum numbers that follow from the combined color, Dirac and isospin Fierz transformations. These relations reduce the number of independent baryon operators with any given spin and isospin. The Fierz identities also effectively restrict the allowed baryon chiral multiplets. It turns out that the non-derivative baryons’ chiral multiplets have the same dimensionality as their Lorentz representations. For the two independent nucleon operators the only permissible chiral multiplet is the fundamental one, . For the Δ, admissible Lorentz representations are and . In the case of the chiral multiplet, the Δ field has one chiral partner; otherwise it has none. We also consider the Abelian (U _A(1)) chiral transformation properties of the fields and show that each baryon comes in two varieties: (1) with Abelian axial charge +3; and (2) with Abelian axial charge −1. In case of the nucleon these are the two Ioffe fields; in case of the Δ, the multiplet has an Abelian axial charge −1 and the multiplet has an Abelian axial charge +3.     

Wave Decay on Convex Co-Compact Hyperbolic Manifolds

For convex co-compact hyperbolic quotients , we analyze the long-time asymptotic of the solution of the wave equation u(t) with smooth compactly supported initial data f = (f ₀, f ₁). We show that, if the Hausdorff dimension δ of the limit set is less than n/2, then where and . We explain, in terms of conformal theory of the conformal infinity of X, the special cases , where the leading asymptotic term vanishes. In a second part, we show for all the existence of an infinite number of resonances (and thus zeros of Selberg zeta function) in the strip . As a byproduct we obtain a lower bound on the remainder R(t) for generic initial data f.     

Parabolic Presentations of the Super Yangian Y(\mathfrakglM|N){Y(\mathfrak{gl}_{M|N})}

Associated to a composition of M and a composition of N, a new presentation of the super Yangian of the general linear Lie superalgebra Y(\mathfrakgl_M|N){Y(\mathfrak{gl}_{M|N})} is obtained.     

On the Classification of Automorphic Lie Algebras

The problem of reduction of integrable equations can be formulated in a uniform way using the theory of invariants. This provides a powerful tool of analysis and it opens the road to new applications of Automorphic Lie Algebras, beyond the context of integrable systems. In this paper it is shown that \mathfraksl₂(\mathbbC){\mathfrak{sl}_{2}(\mathbb{C})}–based Automorphic Lie Algebras associated to the icosahedral group \mathbb I{{\mathbb I}}, the octahedral group \mathbb O{{\mathbb O}}, the tetrahedral group \mathbb T{{\mathbb T}}, and the dihedral group \mathbb D_n{{\mathbb D}_n} are isomorphic. The proof is based on techniques from classical invariant theory and makes use of Clebsch-Gordan decomposition and transvectants, Molien functions and the trace-form. This result provides a complete classification of \mathfraksl₂(\mathbbC){\mathfrak{sl}_{2}(\mathbb{C})}–based Automorphic Lie Algebras associated to finite groups when the group representations are chosen to be the same and it is a crucial step towards the complete classification of Automorphic Lie Algebras.     

Super-sensitivity measurement of tiny Doppler frequency shifts based on parametric amplification and squeezed vacuum state

The precision measurement of Doppler frequency shifts is of great significance for improving the precision of speed measurement. This paper proposes a precision measurement scheme of tiny Doppler shifts by a parametric amplification process and squeezed vacuum state. This scheme takes a parametric amplification process and squeezed vacuum state into a detection system, so that the measurement precision of tiny Doppler shifts can exceed the Cram′er–Rao bound of coherent light. Simultaneously, a simulation study is carried out on the theoretical basis, and the following results are obtained: for the signal light of Gaussian mode, when the amplification factor g = 1 and the squeezed factor r = 0.5, the measurement error of Doppler frequency shifts is 14.4% of the Cramer–Rao bound of the coherent light in our system. At the same time,when the local light mode and squeezed vacuum state mode are optimized, the measurement precision of this scheme can be further improved by ■ times, where n is the mode-order of the signal light.     

Generalized parton distributions in impact parameter space

The kinematical factor in the positivity bound (36) is incorrect. The bound correctly reads Our corrected result agrees with inequality (25) in [1], taking into account the different normalization conventions here and there.Published online: 9 October 2003Erratum published online: 10 October 2003     

SBV Regularity for Genuinely Nonlinear, Strictly Hyperbolic Systems of Conservation Laws in one Space Dimension

We prove that if t ? u(t) ? BV(\mathbbR){t \mapsto u(t) \in BV(\mathbb{R})} is the entropy solution to a N × N strictly hyperbolic system of conservation laws with genuinely nonlinear characteristic fields     

Marginally Trapped Tubes Generated from Nonlinear Scalar Field Initial Data

We show that the maximal future development of asymptotically flat spherically symmetric black hole initial data for a self-gravitating nonlinear scalar field, also called a Higgs field, contains a connected, achronal, spherically symmetric marginally trapped tube which is asymptotic to the event horizon of the black hole, provided the initial data is sufficiently small and decays like O(r^-\frac12){O(r^{-\frac{1}{2}})}, and the potential function V is nonnegative with bounded second derivative. This result can be loosely interpreted as a statement about the stability of ‘nice’ asymptotic behavior of marginally trapped tubes under certain small perturbations of Schwarzschild.     

Higher Dimensional Cosmological Model with Quark and Strange Quark Matter

We study higher dimensional homogeneous cosmological model in the presence of quark and strange quark matter. The dynamical behavior of the model for the strange quark matter equation of state of the form p = \frac13 (r- 4 B_c)p= \frac{1}{3} (\rho- 4 B_{c}) are studied.     

W-Algebra W(2, 2) and the Vertex Operator Algebra $${L(\frac{1}{2},\,0)\,\otimes\, L(\frac{1}{2},\,0)}$$

In this paper the W-algebra W(2, 2) and its representation theory are studied. It is proved that a simple vertex operator algebra generated by two weight 2 vectors is either a vertex operator algebra associated to an irreducible highest weight W(2, 2)- module or a tensor product of two simple Virasoro vertex operator algebras. Furthermore, we show that any rational, C ₂-cofinite and simple vertex operator algebra whose weight 1 subspace is zero, weight 2 subspace is 2-dimensional and with central charge c = 1 is isomorphic to . Supported by NSF grants and a research grant from the Committee on Research, UC Santa Cruz.     

On variational expressions for quantum relative entropies

Distance measures between quantum states like the trace distance and the fidelity can naturally be defined by optimizing a classical distance measure over all measurement statistics that can be obtained from the respective quantum states. In contrast, Petz showed that the measured relative entropy, defined as a maximization of the Kullback–Leibler divergence over projective measurement statistics, is strictly smaller than Umegaki’s quantum relative entropy whenever the states do not commute. We extend this result in two ways. First, we show that Petz’ conclusion remains true if we allow general positive operator-valued measures. Second, we extend the result to Rényi relative entropies and show that for non-commuting states the sandwiched Rényi relative entropy is strictly larger than the measured Rényi relative entropy for \(\alpha \in (\frac{1}{2}, \infty )\) and strictly smaller for \(\alpha \in [0,\frac{1}{2})\). The latter statement provides counterexamples for the data processing inequality of the sandwiched Rényi relative entropy for \(\alpha < \frac{1}{2}\). Our main tool is a new variational expression for the measured Rényi relative entropy, which we further exploit to show that certain lower bounds on quantum conditional mutual information are superadditive.     

On the Best Constant in the Moser-Onofri-Aubin Inequality

Let S ² be the 2-dimensional unit sphere and let J _α denote the nonlinear functional on the Sobolev space H ¹(S ²) defined by

A frustrated quantum spin-s model on the Union Jack lattice with spins s > 1/2

The zero-temperature phase diagrams of a two-dimensional (2D) frustrated quantum antiferromagnetic system, namely the Union Jack model, are studied using the coupled cluster method (CCM) for the two cases when the lattice spins have spin quantum number s = 1 and s = \frac32\frac{3}{2}. The system is defined on a square lattice and the spins interact via isotropic Heisenberg interactions such that all nearest-neighbour (NN) exchange bonds are present with identical strength J ₁ > 0, and only half of the next-nearest-neighbour (NNN) exchange bonds are present with identical strength J ₂ ≡ κ J ₁ > 0. The bonds are arranged such that on the 2×2 unit cell they form the pattern of the Union Jack flag. Clearly, the NN bonds by themselves (viz., with J ₂ = 0) produce an antiferromagnetic Néel-ordered phase, but as the relative strength κ of the frustrating NNN bonds is increased a phase transition occurs in the classical case (s→ ∞) at κ _c ^cl = 0.5 to a canted ferrimagnetic phase. In the quantum cases considered here we also find strong evidence for a corresponding phase transition between a Néel-ordered phase and a quantum canted ferrimagnetic phase at a critical coupling κ _c1 = 0.580 ± 0.015 for s = 1 and κ _c1 = 0.545 ± 0.015 for s = \frac32\frac{3}{2}. In both cases the ground-state energy E and its first derivative dE/dκ seem continuous, thus providing a typical scenario of a second-order phase transition at κ = κ _c1. However, the order parameter for the transition (viz., the average ground-state on-site magnetization) does not go to zero there on either side of the transition. Thus, the phase transition at κ = κ _c1 between the Néel antiferromagnetic phase and the canted ferrimagnetic phase for both the s = 1 and s = \frac32\frac{3}{2} Union Jack models is similar in nature to that found previously for the s = \frac12\frac{1}{2} Union Jack model. It is thus also completely comparable to the transition in the s = \frac12\frac{1}{2} XXZ model on the 2D square lattice between two Néel antiferromagnetic phases, one aligned along the z-axis and the other along some perpendicular direction in the xy-plane.