Non-contemporaneous variations and Hölder’s principle

In the process of deducing the Hölder principle, a key step is to use the concept of non-contemporaneous variations. In this paper, whether starting from analytic method or from graphic solution method, the authors prove that the expression formula of non-contemporaneous variations is incorrect when the variable functions have zero-order nearness degree, and obtain a new expression. From the view of calculus of variations and differential calculus, the non-contemporaneous variations are studied. The study result shows that the concept of non-contemporaneous variations is a combination of the concept of variations and the concept of differentiation. The authors prove that the new expression is correct and obtain an equivalent expression of it. By means of this equivalent expression, this paper proves that the above expression formula of non-contemporaneous variations is correct when the variable functions have one-order nearness degree. Further study shows that, in the process of deducing Hölder’s principle, there is an implicit expression. Whether starting from analytic method or from graphic solution method, the authors discovered that the implicit expression of non-contemporaneous variations is incorrect when the variable functions have zero-order nearness degree and have one-order nearness degree. This paper proves that the implicit expression of non-contemporaneous variations is correct when the variable functions have two-order nearness degree. Further study shows that Hölder’s principle is tenable when the variable functions have two-order nearness degree.     

Projections of Gibbs States for Hölder Potentials

We give a short proof that the projection of a Gibbs state for a Hölder continuous potential on a mixing shift of finite type under a 1-block fiber-wise mixing factor map has a Hölder continuous g function. This improves a number of previous results. The key insight in the proof is to realize the measure of a cylinder set in terms of positive operators and use cone techniques.     

Violation of Bell’s inequality for continuous-variable EPR states

We construct a wide class of bounded continuous variables observables that lead to violations of Bell inequalities for the EPR state and give an intuitive Wigner function explanation how to predetermine which operators wont ever exceed the bounds given by local theories. We show that as examples of such operators, we can use continuous-variable observables that satisfy the commutation relations for the Pauli matrices.     

Equilibrium States of Weakly Hyperbolic One-Dimensional Maps for Hölder Potentials

There is a wealth of results in the literature on the thermodynamic formalism for potentials that are, in some sense, “hyperbolic”. We show that for a sufficiently regular one-dimensional map satisfying a weak hyperbolicity assumption, every Hölder continuous potential is hyperbolic. A sample consequence is the absence of phase transitions: The pressure function is real analytic on the space of Hölder continuous functions. Another consequence is that every Hölder continuous potential has a unique equilibrium state, and that this measure has exponential decay of correlations.     

Regularity Criteria for the Dissipative Quasi-Geostrophic Equations in Hölder Spaces

We study regularity criteria for weak solutions of the dissipative quasi-geostrophic equation (with dissipation (−Δ) ^γ/2, 0 < γ ≤ 1). We show in this paper that if , or with is a weak solution of the 2D quasi-geostrophic equation, then θ is a classical solution in . This result improves our previous result in [18]. Partially supported by a start-up funding from the Division of Applied Mathematics of Brown University and NSF grant number DMS 0800129. Partially supported by a start-up funding from the College of Natural Sciences of the University of Texas at Austin, NSF grant number DMS 0758247 and an Alfred P. Sloan Research Fellowship.     

Hölder Continuity of Absolutely Continuous Spectral Measures for One-Frequency Schrödinger Operators

We establish sharp results on the modulus of continuity of the distribution of the spectral measure for one-frequency Schrödinger operators with Diophantine frequencies in the region of absolutely continuous spectrum. More precisely, we establish 1/2-Hölder continuity near almost reducible energies (an essential support of absolutely continuous spectrum). For non-perturbatively small potentials (and for the almost Mathieu operator with subcritical coupling), our results apply for all energies.     

Generalized Hölder Continuity and Oscillation Functions

We study a notion of generalized Hölder continuity for functions on ?^d. We show that for any bounded function f of bounded support and any r >?0, the r-oscillation of f defined as \(osc_{r} f (x):= \sup _{B_{r}(x)} f - \inf _{B_{r}(x)} f\) is automatically generalized Hölder continuous, and we give an estimate for the appropriate (semi)norm. This is motivated by applications in the theory of dynamical systems.     

Cauchy Problem for Dissipative Hölder Solutions to the Incompressible Euler Equations

We consider solutions to the Cauchy problem for the incompressible Euler equations on the 3-dimensional torus which are continuous or Hölder continuous for any exponent ${\theta < \frac{1}{16}}$ . Using the techniques introduced in De Lellis and Székelyhidi (Inventiones Mathematicae 9:377–407, 2013; Dissipative Euler flows and Onsager’s conjecture, 2012), we prove the existence of infinitely many (Hölder) continuous initial vector fields starting from which there exist infinitely many (Hölder) continuous solutions with preassigned total kinetic energy.     

Bell’s inequality with Dirac particles

We study Bell’s inequality using the Bell states constructed from four component Dirac spinors. Spin operator is related to the Pauli-Lubanski pseudo vector which is relativistic invariant operator. By using Lorentz transformation, in both Bell states and spin operator, we obtain an observer independent Bell’s inequality, so that it is maximally violated as long as it is violated maximally in the rest frame. The article is published in the original.     

Probability distribution and Bell’s inequality for the quantum qubit state

The problem of Bell’s inequality violation for a particle with spin 1/2 is studied within the tomographic approach. Two possible methods for constructing the distribution functions associated with the qubit quantum state are presented. The Bell parameter maximum is studied for each proposed distribution.     

Hölder Continuity of the Integrated Density of States for the Fibonacci Hamiltonian

We obtain the asymptotics of the optimal global Hölder exponent of the integrated density of states of the Fibonacci Hamiltonian for large and small couplings.     

The H?lder Inequality for KMS States

We prove a H?lder inequality for KMS States, which generalise a well-known trace-inequality. Our results are based on the theory of non-commutative L _p -spaces.     

Hörmander’s method for the characteristic Cauchy problem and conformal scattering for a nonlinear wave equation

Letters in Mathematical Physics - The purpose of this note is to prove the existence of a conformal scattering operator for the cubic defocusing wave equation on a non-stationary background. The...     

A Fast Algorithm for the First-Passage Times of Gauss-Markov Processes with Hölder Continuous Boundaries

Even for simple diffusion processes, treating first-passage problems analytically proves intractable for generic barriers and existing numerical methods are inaccurate and computationally costly. Here, we present a novel numerical method that is faster and has more tightly controlled accuracy. Our algorithm is a probabilistic variant of dichotomic search for the computation of first passage times through non-negative homogeneously Hölder continuous boundaries by Gauss-Markov processes. These include the Ornstein-Uhlenbeck process underlying the ubiquitous “leaky integrate-and-fire” model of neuronal excitation. Our method evaluates discrete points in a sample path exactly, and refines this representation recursively only in regions where a passage is rigorously estimated to be probable (e.g. when close to the boundary).As a result, for a given temporal accuracy in the location of the first passage time, our method is orders of magnitude faster than direct forward integration such as Euler or stochastic Runge-Kutta schemata. Moreover, our algorithm rigorously bounds the probability that such crossings are not true first-passage times.     

Hölder continuity of sample paths in Euclidean field theory

Hölder continuity of sample paths of the stochastic process _t (f)=( _t f) (f Y(R ^d–1)) in Euclidean field theory is proved under some assumptions on correlation functions. These assumptions are fulfilled inP()₂ and in theories in which the GHS inequality holds. The continuity index is determined by the condition d(p)|p ₀|²<, whered(p) is the Fourier transform of the two-point function.On leave of absence from Institute of Theoretical Physics, University of Wrocaw, Wrocaw, Poland     

Hölder Regularity of the Solutions of the Cohomological Equation for Roth Type Interval Exchange Maps

There is a general method for constructing a soliton hierarchy from a splitting \({{L_{\pm}}}\) of a loop group as positive and negative sub-groups together with a commuting linearly independent sequence in the positive Lie algebra \({\mathcal{L}_{+}}\). Many known soliton hierarchies can be constructed this way. The formal inverse scattering associates to each f in the negative subgroup \({L_-}\) a solution \({u_{f}}\) of the hierarchy. When there is a 2 co-cycle of the Lie algebra that vanishes on both sub-algebras, Wilson constructed a tau function \({\tau_{f}}\) for each element \({f \in L_-}\). In this paper, we give integral formulas for variations of \({\ln\tau_{f}}\) and second partials of \({\ln\tau_{f}}\), discuss whether we can recover solutions \({u_{f}}\) from \({\tau_{f}}\), and give a general construction of actions of the positive half of the Virasoro algebra on tau functions. We write down formulas relating tau functions and formal inverse scattering solutions and the Virasoro vector fields for the \({GL(n,\mathbb{C})}\)-hierarchy.     

An explicit schrödinger picture for aharonov’s modular-variable concept

We propose to address in a natural manner the modular-variable concept explicitly in a Schrödinger picture. The idea of modular variables was introduced in 1969 by Aharonov, Pendleton, and Petersen to explain certain nonlocal properties of quantum mechanics. Our approach to this subject is based on Schwinger’s finite quantum kinematics and its continuous limit.     

Test of bell’s inequality using the one-atom micromaser

We examine a local realist bound in the case of a one-atom micromaser. It is shown that such a bound is violated using a simplified treatment of the micromaser. We consider the effect of dissipation in a proposed experiment with the real micromaser. It is seen that the magnitude of violation of a Bell-type inequality depends significantly on the cavity parameters.     

Hölder Regularity of Integrated Density of States for the Almost Mathieu Operator in a Perturbative Regime

Consider the Almost Mathieu operator H = cos 2(k +)+ on the lattice. It is shown that for large , the integrated density of states is Hölder continuous of exponent < . This result gives a precise version in the perturbative regime of recent work by M. Goldstein and W. Schlag on Hölder regularity of the integrated density of states for 1D quasi-periodic lattice Schrödinger operators, assuming positivity of the Lyapunov exponent (and proven by different means). Our approach provides also a new way to control Green's functions, in the spirit of the author's work in KAM theory. It is by no means restricted to the cosine-potential and extends to band operators.