Sharp estimates for the modulus of a canonical product

2016 ◽  
Vol 207 (2) ◽  
pp. 238-266 ◽  
Author(s):  
S G Merzlyakov
Keyword(s):  
Canonical Product ◽  
Sharp Estimates

scholarly journals Trace operators on fractals, entropy and approximation numbers

2011 ◽  
Vol 18 (3) ◽  
pp. 549-575
Author(s):  
Cornelia Schneider
Keyword(s):  
Besov Spaces ◽  
Trace Operator ◽  
Approximation Numbers ◽  
Atomic Decompositions ◽  
Sharp Estimates ◽  
Compact Embeddings ◽  
Trace Space

Abstract First we compute the trace space of Besov spaces – characterized via atomic decompositions – on fractals Γ, for parameters 0 < p < ∞, 0 < q ≤ min(1, p) and s = (n – d)/p. New Besov spaces on fractals are defined via traces for 0 < p, q ≤ ∞, s ≥ (n – d)/p and some embedding assertions are established. We conclude by studying the compactness of the trace operator TrΓ by giving sharp estimates for entropy and approximation numbers of compact embeddings between Besov spaces. Our results on Besov spaces remain valid considering the classical spaces defined via differences. The trace results are used to study traces in Triebel–Lizorkin spaces as well.

scholarly journals Effect of Energy Degeneracy on the Transition Time for a Series of Metastable States

2021 ◽  
Vol 184 (1) ◽  
Author(s):  
Gianmarco Bet ◽  
Vanessa Jacquier ◽  
Francesca R. Nardi
Keyword(s):  
Metastable State ◽  
Stochastic Dynamics ◽  
Transition Probabilities ◽  
Mixing Time ◽  
Stable State ◽  
Temperature Limit ◽  
Metastable States ◽  
Transition Time ◽  
Sharp Estimates ◽  
Model Independent

AbstractWe consider the problem of metastability for stochastic dynamics with exponentially small transition probabilities in the low temperature limit. We generalize previous model-independent results in several directions. First, we give an estimate of the mixing time of the dynamics in terms of the maximal stability level. Second, assuming the dynamics is reversible, we give an estimate of the associated spectral gap. Third, we give precise asymptotics for the expected transition time from any metastable state to the stable state using potential-theoretic techniques. We do this in a general reversible setting where two or more metastable states are allowed and some of them may even be degenerate. This generalizes previous results that hold for a series of only two metastable states. We then focus on a specific Probabilistic Cellular Automata (PCA) with configuration space $${\mathcal {X}}=\{-1,+1\}^\varLambda $$ X = { - 1 , + 1 } Λ where $$\varLambda \subset {\mathbb {Z}}^2$$ Λ ⊂ Z 2 is a finite box with periodic boundary conditions. We apply our model-independent results to find sharp estimates for the expected transition time from any metastable state in $$\{\underline{-1}, {\underline{c}}^o,{\underline{c}}^e\}$$ { - 1 ̲ , c ̲ o , c ̲ e } to the stable state $$\underline{+1}$$ + 1 ̲ . Here $${\underline{c}}^o,{\underline{c}}^e$$ c ̲ o , c ̲ e denote the odd and the even chessboard respectively. To do this, we identify rigorously the metastable states by giving explicit upper bounds on the stability level of every other configuration. We rely on these estimates to prove a recurrence property of the dynamics, which is a cornerstone of the pathwise approach to metastability.

scholarly journals RETRACTED ARTICLE: An augmented Riesz decomposition method for sharp estimates of certain boundary value problem

2016 ◽  
Vol 2016 (1) ◽  
Author(s):  
Jiaofeng Wang ◽  
Bin Huang ◽  
Nanjundan Yamini
Keyword(s):  
Boundary Value Problem ◽  
Lower Bounds ◽  
Harmonic Functions ◽  
Decomposition Method ◽  
Integral Condition ◽  
Boundary Integral ◽  
Sharp Estimates ◽  
Riesz Decomposition ◽  
Retracted Article ◽  
Riesz Decomposition Method

AbstractIn this paper, by using an augmented Riesz decomposition method, we obtain sharp estimates of harmonic functions with certain boundary integral condition, which provide explicit lower bounds of functions harmonic in a cone. The results given here can be used as tools in the study of integral equations.

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