Convergence of Discrete Green Functions with Neumann Boundary Conditions

In this article we prove convergence of Green functions with Neumann boundary conditions for the random walk to their continuous counterparts. Our methods rely on local central limit theorems for convergence of random walks on discretizations of smooth domains to Reflected Brownian motion.     

New Coins from Old, Smoothly

Given a (known) function f:[0,1]→(0,1), we consider the problem of simulating a coin with probability of heads f(p) by tossing a coin with unknown heads probability p, as well as a fair coin, N times each, where N may be random. The work of Keane and O’Brien (ACM Trans. Model. Comput. Simul. 4(2):213–219, 1994) implies that such a simulation scheme with the probability ℙ _p (N<∞) equal to 1 exists if and only if f is continuous. Nacu and Peres (Ann. Appl. Probab. 15(1A):93–115, 2005) proved that f is real analytic in an open set S⊂(0,1) if and only if such a simulation scheme exists with the probability ℙ _p (N>n) decaying exponentially in n for every p∈S. We prove that for α>0 noninteger, f is in the space C ^α [0,1] if and only if a simulation scheme as above exists with ℙ _p (N>n)≤C(Δ _n (p)) ^α , where \varDelta _n(x):=max{?{x(1-x)/n},1/n}\varDelta _{n}(x):=\max\{\sqrt{x(1-x)/n},1/n\}. The key to the proof is a new result in approximation theory: Let B⁺_n\mathcal{B}^{+}_{n} be the cone of univariate polynomials with nonnegative Bernstein coefficients of degree n. We show that a function f:[0,1]→(0,1) is in C ^α [0,1] if and only if f has a series representation ?_n=1^￥F_n\sum_{n=1}^{\infty}F_{n} with F_n ? B⁺_nF_{n}\in \mathcal{B}^{+}_{n} and ∑ _k>n F _k (x)≤C(Δ _n (x)) ^α for all x∈[0,1] and n≥1. We also provide a counterexample to a theorem stated without proof by Lorentz (Math. Ann. 151:239–251, 1963), who claimed that if some j_n ? B⁺_n\varphi_{n}\in\mathcal{B}^{+}_{n} satisfy |f(x)−φ _n (x)|≤C(Δ _n (x)) ^α for all x∈[0,1] and n≥1, then f∈C ^α [0,1].     

Mixing of the upper triangular matrix walk

We study a natural random walk over the upper triangular matrices, with entries in the field ${\mathbb{Z}_2}$ , generated by steps which add row i + 1 to row i. We show that the mixing time of the lazy random walk is O(n ²) which is optimal up to constants. Our proof makes key use of the linear structure of the group and extends to walks on the upper triangular matrices over the fields ${\mathbb{Z}_q}$ for q prime.     

Markov type and threshold embeddings

For two metric spaces X and Y, say that X threshold-embeds into Y if there exist a number K > 0 and a family of Lipschitz maps ${\{\varphi_{\tau} : X \to Y : \tau > 0\}}$ such that for every ${x,y \in X}$ , $$d_X(x, y) \geq \tau \implies d_Y(\varphi_\tau (x),\varphi_\tau (y)) \geq \|{\varphi}_\tau\|_{\rm Lip}\tau/K,$$ where ${\|{\varphi}_{\tau}\|_{\rm Lip}}$ denotes the Lipschitz constant of ${\varphi_{\tau}}$ . We show that if a metric space X threshold-embeds into a Hilbert space, then X has Markov type 2. As a consequence, planar graph metrics and doubling metrics have Markov type 2, answering questions of Naor, Peres, Schramm, and Sheffield. More generally, if a metric space X threshold-embeds into a p-uniformly smooth Banach space, then X has Markov type p. Our results suggest some non-linear analogs of Kwapien’s theorem. For instance, a subset ${X \subseteq L_1}$ threshold-embeds into Hilbert space if and only if X has Markov type 2.     

Glauber Dynamics for the Mean-Field Potts Model

We study Glauber dynamics for the mean-field (Curie-Weiss) Potts model with q??3 states and show that it undergoes a critical slowdown at an inverse-temperature ?? _s (q) strictly lower than the critical ?? _c (q) for uniqueness of the thermodynamic limit. The dynamical critical ?? _s (q) is the spinodal point marking the onset of metastability. We prove that when ??<?? _s (q) the mixing time is asymptotically C(??,q)nlogn and the dynamics exhibits the cutoff phenomena, a sharp transition in mixing, with a window of order n. At ??=?? _s (q) the dynamics no longer exhibits cutoff and its mixing obeys a power-law of order n ^4/3. For ??>?? _s (q) the mixing time is exponentially large in n. Furthermore, as ?????? _s with n, the mixing time interpolates smoothly from subcritical to critical behavior, with the latter reached at a scaling window of O(n ^?2/3) around ?? _s . These results form the first complete analysis of mixing around the critical dynamical temperature??including the critical power law??for a model with a first order phase transition.     

How do dimethyl oxalate ions CH3O-C(=O)-C(=O)-OCH 3 ·+ break in half? Loss of CH 3 · + CO2 versus CH3O-C-O·

The interesting unimolecular dissociation chemistry of dimethyl oxalate (DMO) ions, CH₃O-C(=O)-C(=O)-OCH ₃ ^·+ , has been studied by vacuum ultraviolet photoionization and tandem mass spectrometry based experiments. The measured appearance energy (AE) for the generation of CH₃O-C=O⁺ (10. 5 eV) is not compatible with a simple bond cleavage involving the cogeneration of the radical CH₃O-C=O^· whose calculated AE is 11 kcal/mol higher. However, because the CH₃O-C=O^· radical is thermodynamically less stable than its dissociation products CH₃ ^· and CO₂, by 19 kcal/mol, a two-step dissociation of ionized DMO into CH₃O-C=O⁺ + CH ₃ ^· + CO₂ is energetically feasible. Collision induced dissociative ionization experiments clearly show that low energy DMO ions dissociate into CH₃ ^· + CO₂ without the intermediacy of CH₃O-C=O^·. Experiments using a charged collision chamber further indicate that CO₂ is released first, followed by loss of CH₃ ^· and not vice versa and a mechanism is proposed. The measured AE, which we assign to the two-step process, is 8 kcal/mol higher than the calculated value. This could be due to a competitive shift caused by a prominent low energy decarbonylation reaction yielding the hydrogen bridged radical cation CH₂=O … H … O=C-OCH₃ ^·+. However, from metastable ion observations and AE measurements on deuterium labeled DMO ions, it follows that there is no competitive shift and that the elevated AE for the two-step process corresponds to the barrier for the first step, loss of CO₂. Finally, neutralization-reionization experiments on ionized DMO and CH₃O-C=O⁺ provide evidence for the existence of CH₃O-C=O^· as a kinetically stable radical.     

Mössbauer effect study of the pseudo-binary system (Ti1−x Nb x )Fe2

Mössbauer spectroscopy and X-ray diffraction measurements have been done on (Ti_1–x Nb _x )Fe₂ compounds in order to investigate the effect of Nb on the magnetic properties of TiFe₂. The experimental results show that Nb enters the lattice by filling Ti sites, thereby forming a continuous phase over the whole range of Nb concentrations. The Mössbauer spectra at 80 K fitted with a magnetic hyperfine field distribution show a continuous decrease of the average magnetic hyperfine field with increasing Nb concentration, as well as several different magnetic configurations forx0.3.