Global Solutions to the 3-D Incompressible Anisotropic Navier-Stokes System in the Critical Spaces

In this paper, we consider the global wellposedness of the 3-D incompressible anisotropic Navier-Stokes equations with initial data in the critical Besov-Sobolev type spaces B{\mathcal{B}} and B^{-\frac12,\frac12}₄{\mathcal{B}^{-\frac12,\frac12}_4} (see Definitions 1.1 and 1.2 below). In particular, we proved that there exists a positive constant C such that (ANS _ν ) has a unique global solution with initial data u₀ = (u₀^h, u₀³){u_0 = (u_0^h, u_0^3)} which satisfies ||u₀^h||_B exp(\fracCn⁴ ||u₀³||_B⁴) ￡ c₀n{\|u_0^h\|_{\mathcal{B}} \exp\bigl(\frac{C}{\nu^4} \|u_0^3\|_{\mathcal{B}}^4\bigr) \leq c_0\nu} or ||u₀^h||_{B^{-\frac12,\frac12}4} exp(\fracCn⁴ ||u₀³||_{B^{-\frac12,\frac12}4}⁴) ￡ c₀n{\|u_0^h\|_{\mathcal{B}^{-\frac12,\frac12}_{4}} \exp \bigl(\frac{C}{\nu^4} \|u_0^3\|_{\mathcal{B}^{-\frac12,\frac12}_{4}}^4\bigr)\leq c_0\nu} for some c ₀ sufficiently small. To overcome the difficulty that Gronwall’s inequality can not be applied in the framework of Chemin-Lerner type spaces, [(L^p_t)\tilde](B){\widetilde{L^p_t}(\mathcal{B})}, we introduced here sort of weighted Chemin-Lerner type spaces, [(L²_{t, f})\tilde](B){\widetilde{L^2_{t, f}}(\mathcal{B})} for some apropriate L ¹ function f(t).     

A Frequency Localized Maximum Principle Applied to the 2D Quasi-Geostrophic Equation

In this paper, we prove a maximum principle for a frequency localized transport-diffusion equation. As an application, we prove the local well-posedness of the supercritical quasi-geostrophic equation in the critical Besov spaces \mathringB^1-a_￥,q{\mathring{B}^{1-\alpha}_{\infty,q}}, and global well-posedness of the critical quasi-geostrophic equation in \mathringB⁰_￥,q{\mathring{B}^{0}_{\infty,q}} for all 1 ≤ q < ∞. Here \mathringB^s_￥,q {\mathring{B}^{s}_{\infty,q} } is the closure of the Schwartz functions in the norm of B^s_￥,q{B^{s}_{\infty,q}}.     

A Note on the Extreme Points of Positive Quantum Operations

In this paper, an error in the proof of Theorem 4.9 in Gudder’s paper (Int. J. Theor. Phys. 47(1):268–279, 2008) is pointed out and it is proved that if such that E _i ∈ℂI∖{0} and E _j ∉ℂI for some i,j in {1,2,…,n}, then . This subject is supported by the NNSF of China (No. 10571113, 10871224).     

Stochastic Porous Media Equations and Self-Organized Criticality: Convergence to the Critical State in all Dimensions

If X = X(t, ξ) is the solution to the stochastic porous media equation in O ì R^d, 1 ￡ d ￡ 3,{\mathcal{O}\subset \mathbf{R}^d, 1\le d\le 3,} modelling the self-organized criticality (Barbu et al. in Commun Math Phys 285:901–923, 2009) and X _c is the critical state, then it is proved that ò^￥₀m(O\O^t₀)dt < ￥,\mathbbP-a.s.{\int^{\infty}_0m(\mathcal{O}{\setminus}\mathcal{O}^t_0)dt<{\infty},\mathbb{P}\hbox{-a.s.}} and lim_t?￥ ò_O|X(t)-X_c|dx = l < ￥, \mathbbP-a.s.{\lim_{t\to{\infty}} \int_\mathcal{O}|X(t)-X_c|d\xi=\ell<{\infty},\ \mathbb{P}\hbox{-a.s.}} Here, m is the Lebesgue measure and O^t_c{\mathcal{O}^t_c} is the critical region {x ? O; X(t,x)=X_c(x)}{\{\xi\in\mathcal{O}; X(t,\xi)=X_c(\xi)\}} and X _c (ξ) ≤ X(0, ξ) a.e. x ? O{\xi\in\mathcal{O}}. If the stochastic Gaussian perturbation has only finitely many modes (but is still function-valued), lim_{t ? ￥} ò_K|X(t)-X_c|dx = 0{\lim_{t \to {\infty}} \int_K|X(t)-X_c|d\xi=0} exponentially fast for all compact K ì O{K\subset\mathcal{O}} with probability one, if the noise is sufficiently strong. We also recover that in the deterministic case ℓ = 0.     

On the Identity of Some von Neumann Algebras and Some Consequences

Among von Neumann algebras, the Weyl algebra W{\mathcal{W}} generated by two unitary groups {U(α)} and {V(β)}, the algebra U{\mathcal{U}} generated by a completely non-unitary semigroup of isometries {U ₊(α)} and the Weyl algebra W₊^h{\mathcal{W}_{+}^{h}} pertaining to a semi-bounded space with homogeneous spectrum of the generator of {V(β)}, all share the property that their representations are completely reducible and the irreducible representations are equivalent. We trace this fact to the identity of these algebras, in the sense that any of them contains a representation of any of the remaining two algebras, which in turn contains the original algebra. We prove this statement by explicit construction. The aforementioned results about the representations of the algebras follow immediately from the proof for any of them. Also, by the above construction we prove for W^h₊{\mathcal{W}^{h}_{+}} the analog of a theorem by Sinai for W{\mathcal{W}} : given {V(β)} with semi-bounded homogeneous spectrum, there exists a completely non-unitary semigroup {U ₊(α)} such that {V(β)} and {U ₊(α)} generate W₊^h{\mathcal{W}_{+}^{h}}.     

Thermal States in Conformal QFT. I

We analyze the set of locally normal KMS states w.r.t. the translation group for a local conformal net A{{\mathcal A}} of von Neumann algebras on \mathbb R{\mathbb R} . In this first part, we focus on the completely rational net A{{\mathcal A}} . Our main result here states that, if A{{\mathcal{A}}} is completely rational, there exists exactly one locally normal KMS state j{\varphi} . Moreover, j{\varphi} is canonically constructed by a geometric procedure. A crucial r?le is played by the analysis of the “thermal completion net” associated with a locally normal KMS state. A similar uniqueness result holds for KMS states of two-dimensional local conformal nets w.r.t. the time-translation one-parameter group.     

The Orthogonal Weingarten Formula in Compact Form

We present a compact formulation of the orthogonal Weingarten formula, with the traditional quantity \({I(i_1,\ldots,i_{2k}:j_1,\ldots,j_{2k})=\int_{O_n}u_{i_1j_1}\cdots u_{i_{2k}j_{2k}}\,{\rm d}u}\) replaced by the more advanced quantity \({I(a)=\int_{O_n}\Pi u_{ij}^{a_{ij}}\,{\rm d}u}\), depending on a matrix of exponents \({a\in M_n(\mathbb{N})}\). Among consequences, we establish a number of basic facts regarding the integrals I(a): vanishing conditions, possible poles, asymptotic behavior.     

Twistor Actions for Self-Dual Supergravities

We give holomorphic Chern-Simons-like action functionals on supertwistor space for self-dual supergravity theories in four dimensions, dealing with supersymmetries, the cases where different parts of the R-symmetry are gauged, and with or without a cosmological constant. The gauge group is formally the group of holomorphic Poisson transformations of supertwistor space where the form of the Poisson structure determines the amount of R-symmetry gauged and the value of the cosmological constant. We give a formulation in terms of a finite deformation of an integrable -operator on a supertwistor space, i.e., on regions in . For , we also give a formulation that does not require the choice of a background.     

Exact Solution for a Class of Random Walk on the Hypercube

A class of families of Markov chains defined on the vertices of the n-dimensional hypercube, Ω _n ={0,1} ⁿ , is studied. The single-step transition probabilities P _n,ij , with i,j∈Ω _n , are given by P_n,ij=\frac(1-a)^d_ij(2-a)ⁿP_{n,ij}=\frac{(1-{\alpha})^{d_{ij}}}{(2-{\alpha})^{n}}, where α∈(0,1) and d _ij is the Hamming distance between i and j. This corresponds to flip independently each component of the vertex with probability \frac1-a2-a\frac{1-{\alpha}}{2-{\alpha}}. The m-step transition matrix P_n,ij^mP_{n,ij}^{m} is explicitly computed in a close form. The class is proved to exhibit cutoff. A model-independent result about the vanishing of the first m terms of the expansion in α of P_n,ij^mP_{n,ij}^{m} is also proved.     

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}}.     

On the Global Wellposedness to the 3-D Incompressible Anisotropic Navier-Stokes Equations

Corresponding to the wellposedness result [2] for the classical 3-D Navier-Stokes equations (NS _ν) with initial data in the scaling invariant Besov space, here we consider a similar problem for the 3-D anisotropic Navier-Stokes equations (ANS _ν), where the vertical viscosity is zero. In order to do so, we first introduce the Besov-Sobolev type spaces, and Then with initial data in the scaling invariant space we prove the global wellposedness for (ANS _ν) provided the norm of initial data is small enough compared to the horizontal viscosity. In particular, this result implies the global wellposedness of (ANS _ν) with high oscillatory initial data (1.2).     

Universality of the Local Regime for the Block Band Matrices with a Finite Number of Blocks

We consider the block band matrices, i.e. the Hermitian matrices $H_N$ , $N=|\Lambda |W$ with elements $H_{jk,\alpha \beta }$ , where $j,k \in \Lambda =[1,m]^d\cap \mathbb {Z}^d$ (they parameterize the lattice sites) and $\alpha , \beta = 1,\ldots , W$ (they parameterize the orbitals on each site). The entries $H_{jk,\alpha \beta }$ are random Gaussian variables with mean zero such that $\langle H_{j_1k_1,\alpha _1\beta _1}H_{j_2k_2,\alpha _2\beta _2}\rangle =\delta _{j_1k_2}\delta _{j_2k_1} \delta _{\alpha _1\beta _2}\delta _{\beta _1\alpha _2} J_{j_1k_1},$ where $J=1/W+\alpha \Delta /W$ , $\alpha < 1/4d$ . This matrices are the special case of Wegner’s $W$ -orbital models. Assuming that the number of sites $|\Lambda |$ is finite, we prove universality of the local eigenvalue statistics of $H_N$ for the energies $|\lambda _0|< \sqrt{2}$ .     

Weakly Non-Ergodic Statistical Physics

For weakly non ergodic systems, the probability density function of a time average observable is where is the value of the observable when the system is in state j=1,…L. p _j ^eq is the probability that a member of an ensemble of systems occupies state j in equilibrium. For a particle undergoing a fractional diffusion process in a binding force field, with thermal detailed balance conditions, p _j ^eq is Boltzmann’s canonical probability. Within the unbiased sub-diffusive continuous time random walk model, the exponent 0<α<1 is the anomalous diffusion exponent 〈x ²〉∼t ^α found for free boundary conditions. When α→1 ergodic statistical mechanics is recovered . We briefly discuss possible physical applications in single particle experiments.     

On the Product of Real Spectral Triples

The product of two real spectral triples and , the first of which is necessarily even, was defined by A.Connes as given by and, in the even-even case, by . Generically it is assumed that the real structure obeys the relations , , , where the -sign table depends on the dimension n modulo 8 of the spectral triple. If both spectral triples obey Connes' >-sign table, it is seen that their product, defined in the straightforward way above, does not necessarily obey this -sign table. In this Letter, we propose an alternative definition of the product real structure such that the -sign table is also satisfied by the product.     

Asymptotic behavior of Mayer cluster sums for the one-dimensional Ising model

The properties of the high-field polynomialsL _n (u) for the one-dimensional spin 1/2 Ising model are investigated. [The polynomialsL _n (u) are essentially lattice gas analogues of the Mayer cluster integralsb _n (T) for a continuum gas.] It is shown thatu ^?1 L _n (u) can be expressed in terms of a shifted Jacobi polynomial of degreen?1. From this result it follows thatu ^?1 L _n (u); n=1, 2,... is a set of orthogonal polynomials in the interval (0, 1) with a weight functionω(u)=u, andu ^?1 L _n (u) hasn?1 simple zerosu _n (v); v=1, 2,...,n?1 which all lie in the interval 0<u<1. Next the detailed behavior ofL _n (u) asn→∞ is studied. In particular, various asymptotic expansions forL _n (u) are derived which areuniformly valid in the intervalsu<0, 0<u<1, andu>1. These expansions are then used to analyze the asymptotic properties of the zeros {u _n (v); v=1, 2,...,n?1}. It is found that $$\begin{array}{*{20}c} {u_n (v) \sim \tfrac{1}{4}({{j_{1,v} } \mathord{\left/ {\vphantom {{j_{1,v} } n}} \right. \kern-\nulldelimiterspace} n})^2 [1 - ({{j_{1,v}^2 } \mathord{\left/ {\vphantom {{j_{1,v}^2 } {12}}} \right. \kern-\nulldelimiterspace} {12}})n^{ - 1} + ({{j_{1,v}^2 } \mathord{\left/ {\vphantom {{j_{1,v}^2 } {700)( - 3 + 2j_{1,v}^2 )n^{ - 4} }}} \right. \kern-\nulldelimiterspace} {700)( - 3 + 2j_{1,v}^2 )n^{ - 4} }}} \\ { + ({{j_{1,v}^2 } \mathord{\left/ {\vphantom {{j_{1,v}^2 } {20160)(40 + 4j_{1,v}^2 - j_{1,v}^4 }}} \right. \kern-\nulldelimiterspace} {20160)(40 + 4j_{1,v}^2 - j_{1,v}^4 }})n^{ - 6} + \cdot \cdot \cdot ]} \\ {u_n (n - v) \sim 1 - ({{j_{0,v}^2 } \mathord{\left/ {\vphantom {{j_{0,v}^2 } 4}} \right. \kern-\nulldelimiterspace} 4})n^{ - 2} + ({{j_{0,v}^2 } \mathord{\left/ {\vphantom {{j_{0,v}^2 } {48)( - 2 + j_{0,v}^2 )n^{ - 4} }}} \right. \kern-\nulldelimiterspace} {48)( - 2 + j_{0,v}^2 )n^{ - 4} }}} \\ { + ({{j_{0,v}^2 } \mathord{\left/ {\vphantom {{j_{0,v}^2 } {2880)(2 + 9j_{0,v}^2 - 2j_{0,v}^4 )n^{ - 6} + \cdot \cdot \cdot }}} \right. \kern-\nulldelimiterspace} {2880)(2 + 9j_{0,v}^2 - 2j_{0,v}^4 )n^{ - 6} + \cdot \cdot \cdot }}} \\ \end{array} $$ asn→∞v fixed, wherej _k,v denotes thevth zero of the Bessel functionJ _k(^z)     

The Green-Kubo Formula for the Spin-Fermion System

The spin-fermion model describes a two level quantum system (spin 1/2) coupled to finitely many free Fermi gas reservoirs which are in thermal equilibrium at inverse temperatures β _j . We consider non-equilibrium initial conditions where not all β _j are the same. It is known that, at small coupling, the combined system has a unique non-equilibrium steady state (NESS) characterized by strictly sitive entropy production. In this paper we study linear response in this NESS and prove the Green-Kubo formula and the Onsager reciprocity relations for heat fluxes generated by temperature differentials. Dedicated to Jean Michel Combes on the occasion of his sixtyfifth birthday     

Non-Abelian Vortices,Super Yang–Mills Theory and Spin(7)-Instantons

We consider a complex vector bundle E{\mathcal{E}} endowed with a connection A{\mathcal{A}} over the eight-dimensional manifold \mathbbR²×G/H{\mathbb{R}^2\times G/H}, where G/H = SU(3)/U(1) × U(1) is a homogeneous space provided with a never-integrable almost-complex structure and a family of SU(3)-structures. We establish an equivalence between G-invariant solutions A{\mathcal{A}} of the Spin(7)-instanton equations on \mathbbR²×G/H{\mathbb{R}^2\times G/H} and general solutions of non-Abelian coupled vortex equations on \mathbbR²{\mathbb{R}^2}. These vortices are BPS solitons in a d = 4 gauge theory obtained from N = 1{\mathcal{N} =1} supersymmetric Yang–Mills theory in ten dimensions compactified on the coset space G/H with an SU(3)-structure. The novelty of the obtained vortex equations lies in the fact that Higgs fields, defining morphisms of vector bundles over \mathbbR²{\mathbb{R}^2}, are not holomorphic in the generic case. Finally, we introduce BPS vortex equations in N = 4{\mathcal{N} =4} super Yang–Mills theory and show that they have the same feature.