scholarly journals A note on the local weak limit of a sequence of expander graphs

2021 ◽  
Vol 26 (none) ◽  
Author(s):  
Sourav Sarkar
Keyword(s):  
Weak Limit ◽  
Expander Graphs

scholarly journals Combined Mean-Field and Semiclassical Limits of Large Fermionic Systems

2021 ◽  
Vol 182 (2) ◽  
Author(s):  
Li Chen ◽  
Jinyeop Lee ◽  
Matthew Liew
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Mean Field ◽  
Weak Limit ◽  
Weak Compactness ◽  
Three Dimensions ◽  
Uniform Estimates ◽  
Fermionic Systems ◽  
Semiclassical Limits ◽  
Semiclassical Regime

AbstractWe study the time dependent Schrödinger equation for large spinless fermions with the semiclassical scale $$\hbar = N^{-1/3}$$ ħ = N - 1 / 3 in three dimensions. By using the Husimi measure defined by coherent states, we rewrite the Schrödinger equation into a BBGKY type of hierarchy for the k particle Husimi measure. Further estimates are derived to obtain the weak compactness of the Husimi measure, and in addition uniform estimates for the remainder terms in the hierarchy are derived in order to show that in the semiclassical regime the weak limit of the Husimi measure is exactly the solution of the Vlasov equation.

scholarly journals On a stochastic Burgers equation with Dirichlet boundary conditions

2003 ◽  
Vol 2003 (43) ◽  
pp. 2735-2746 ◽  
Author(s):  
Ekaterina T. Kolkovska
Keyword(s):  
Weak Solution ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Burgers Equation ◽  
Martingale Problem ◽  
Weak Limit ◽  
Dirichlet Boundary Conditions ◽  
One Dimensional ◽  
Stochastic Burgers Equation ◽  
The One

We consider the one-dimensional Burgers equation perturbed by a white noise term with Dirichlet boundary conditions and a non-Lipschitz coefficient. We obtain existence of a weak solution proving tightness for a sequence of polygonal approximations for the equation and solving a martingale problem for the weak limit.

scholarly journals ON QUANTUM MARKOV CHAINS ON CAYLEY TREE I: UNIQUENESS OF THE ASSOCIATED CHAIN WITH XY-MODEL ON THE CAYLEY TREE OF ORDER TWO

2011 ◽  
Vol 14 (03) ◽  
pp. 443-463 ◽  
Author(s):  
LUIGI ACCARDI ◽  
FARRUKH MUKHAMEDOV ◽  
MANSOOR SABUROV
Keyword(s):  
Markov Chains ◽  
Boundary Conditions ◽  
Cayley Tree ◽  
Weak Limit ◽  
Finite Volumes ◽  
Xy Model ◽  
Quantum Markov Chains

In this paper we study forward quantum Markov chains (QMC) defined on Cayley tree. A construction of such QMC is provided, namely we construct states on finite volumes with boundary conditions, and define QMC as a weak limit of those states which depends on the boundary conditions. Using the provided construction, we investigate QMC associated with XY-model on a Cayley tree of order two. We prove uniqueness of QMC associated with such a model, this means the QMC does not depend on the boundary conditions.

On the age distribution of a Markov chain

1978 ◽  
Vol 15 (1) ◽  
pp. 65-77 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Limit Theorems ◽  
Age Distribution ◽  
Weak Limit ◽  
Absorbing State ◽  
Absorbing Markov Chain ◽  
Continuous Random Walk ◽  
A Chain ◽  
Watson Process

This paper develops the notion of the limiting age of an absorbing Markov chain, conditional on the present state. Chains with a single absorbing state {0} are considered and with such a chain can be associated a return chain, obtained by restarting the original chain at a fixed state after each absorption. The limiting age, A(j), is the weak limit of the time given Xn = j (n → ∞).A criterion for the existence of this limit is given and this is shown to be fulfilled in the case of the return chains constructed from the Galton–Watson process and the left-continuous random walk. Limit theorems for A (J) (J → ∞) are given for these examples.

scholarly journals The weak limit of Ising models on locally tree-like graphs

2010 ◽  
Vol 152 (1-2) ◽  
pp. 31-51 ◽  
Author(s):  
Andrea Montanari ◽  
Elchanan Mossel ◽  
Allan Sly
Keyword(s):  
Ising Models ◽  
Weak Limit

scholarly journals On $$\alpha $$-Firmly Nonexpansive Operators in r-Uniformly Convex Spaces

2021 ◽  
Vol 76 (4) ◽  
Author(s):  
Arian Bërdëllima ◽  
Gabriele Steidl
Keyword(s):  
Banach Spaces ◽  
Semigroup Theory ◽  
General Setting ◽  
Weak Limit ◽  
Uniformly Convex ◽  
Fixed Point Set ◽  
Uniformly Convex Banach Spaces ◽  
Differentiable Norm ◽  
Firmly Nonexpansive ◽  
Nonexpansive Operators

AbstractWe introduce the class of $$\alpha $$ α -firmly nonexpansive and quasi $$\alpha $$ α -firmly nonexpansive operators on r-uniformly convex Banach spaces. This extends the existing notion from Hilbert spaces, where $$\alpha $$ α -firmly nonexpansive operators coincide with so-called $$\alpha $$ α -averaged operators. For our more general setting, we show that $$\alpha $$ α -averaged operators form a subset of $$\alpha $$ α -firmly nonexpansive operators. We develop some basic calculus rules for (quasi) $$\alpha $$ α -firmly nonexpansive operators. In particular, we show that their compositions and convex combinations are again (quasi) $$\alpha $$ α -firmly nonexpansive. Moreover, we will see that quasi $$\alpha $$ α -firmly nonexpansive operators enjoy the asymptotic regularity property. Then, based on Browder’s demiclosedness principle, we prove for r-uniformly convex Banach spaces that the weak cluster points of the iterates $$x_{n+1}:=Tx_{n}$$ x n + 1 : = T x n belong to the fixed point set $${{\,\mathrm{Fix}\,}}T$$ Fix T whenever the operator T is nonexpansive and quasi $$\alpha $$ α -firmly. If additionally the space has a Fréchet differentiable norm or satisfies Opial’s property, then these iterates converge weakly to some element in $${{\,\mathrm{Fix}\,}}T$$ Fix T . Further, the projections $$P_{{{\,\mathrm{Fix}\,}}T}x_n$$ P Fix T x n converge strongly to this weak limit point. Finally, we give three illustrative examples, where our theory can be applied, namely from infinite dimensional neural networks, semigroup theory, and contractive projections in $$L_p$$ L p , $$p \in (1,\infty ) \backslash \{2\}$$ p ∈ ( 1 , ∞ ) \ { 2 } spaces on probability measure spaces.

scholarly journals Pathways for intramolecular relaxation in S 1 benzene

1990 ◽  
Vol 332 (1625) ◽  
pp. 189-201 ◽  
Keyword(s):  
Energy Coupling ◽  
Weak Limit ◽  
Excess Energy ◽  
Matrix Elements ◽  
Strong Limit ◽  
Decay Channels ◽  
Two Photon ◽  
Decay Behaviour ◽  
Quantum Numbers ◽  
Fast Electronic

Sub-Doppler spectra of various one- and two-photon vibronic bands of benzene are discussed and analysed to determine the pathways of intramolecular relaxation for Sx benzene. New results are presented for the 14J1q16| band of C 6 H 6 and the 14^1^ band of 13 C 6 H 6 . The decay behaviour depends strongly on the excess energy and the rotational quantum numbers rather than on the vibrational character and symmetry of the excited state. At low vibrational excess energy the pathway for intramolecular relaxation is a coupling in the strong limit between pairs of states in Sx leading to shifts of lines, whereas at intermediate excess energy coupling in the weak limit to background states in Sx is present. These background states are strongly broadened due to a fast electronic non-radiative process. The intramolecular relaxation is found to be initiated by the coupling to the broadened S 1 background states and energy can flow via these states to the T 1 or S 0 state. The rotationally selective disappearance of lines is believed to be due to an intricate interplay of the rotational dependence of the coupling matrix elements and accidental resonances, which lead to interference of possible decay channels.

scholarly journals A note on H-convergence

2021 ◽  
Vol 17 ◽  
pp. 150
Author(s):  
O.P. Kogut ◽  
P.I. Kogut ◽  
T.N. Rudyanova
Keyword(s):  
Convergence Property ◽  
Sufficient Conditions ◽  
Weak Limit ◽  
Uniformly Bounded ◽  
Bounded Sequences

In this paper we study the H-convergence property for the uniformly bounded sequences of square matrices $\left\{ A_{\varepsilon} \in L^{\infty} (D; \mathbb{R}^{n \times n}) \right\}_{\varepsilon > 0}$. We derive the sufficient conditions, which guarantee the coincidence of $H$-limit with the weak-* limit of such sequences in $L^{\infty} (D; \mathbb{R}^{n \times n})$.

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