scholarly journals On the mixing time of coordinate Hit-and-Run

2021 ◽  
pp. 1-13
Author(s):  
Hariharan Narayanan ◽  
Piyush Srivastava
Keyword(s):  
Random Walk ◽  
Convex Body ◽  
Uniform Distribution ◽  
Upper Bound ◽  
Mixing Time ◽  
Formula Unit ◽  
Warm Start ◽  
Variation Distance ◽  
Hit And Run ◽  
Open Question

Abstract We obtain a polynomial upper bound on the mixing time $T_{CHR}(\epsilon)$ of the coordinate Hit-and-Run (CHR) random walk on an $n-$ dimensional convex body, where $T_{CHR}(\epsilon)$ is the number of steps needed to reach within $\epsilon$ of the uniform distribution with respect to the total variation distance, starting from a warm start (i.e., a distribution which has a density with respect to the uniform distribution on the convex body that is bounded above by a constant). Our upper bound is polynomial in n, R and $\frac{1}{\epsilon}$ , where we assume that the convex body contains the unit $\Vert\cdot\Vert_\infty$ -unit ball $B_\infty$ and is contained in its R-dilation $R\cdot B_\infty$ . Whether CHR has a polynomial mixing time has been an open question.

scholarly journals Rate of Convergence of the Short Cycle Distribution in Random Regular Graphs Generated by Pegging

10.37236/133 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Pu Gao ◽  
Nicholas Wormald
Keyword(s):  
Rate Of Convergence ◽  
Total Variation ◽  
Upper Bound ◽  
Mixing Time ◽  
Limiting Distribution ◽  
Total Variation Distance ◽  
Regular Graphs ◽  
Short Cycle ◽  
Variation Distance

The pegging algorithm is a method of generating large random regular graphs beginning with small ones. The $\epsilon$-mixing time of the distribution of short cycle counts of these random regular graphs is the time at which the distribution reaches and maintains total variation distance at most $\epsilon$ from its limiting distribution. We show that this $\epsilon$-mixing time is not $o(\epsilon^{-1})$. This demonstrates that the upper bound $O(\epsilon^{-1})$ proved recently by the authors is essentially tight.

scholarly journals On the Determinant Problem for the Relativistic Boltzmann Equation

2021 ◽  
Author(s):  
James Chapman ◽  
Jin Woo Jang ◽  
Robert M. Strain
Keyword(s):  
Boltzmann Equation ◽  
Upper Bound ◽  
Numerical Study ◽  
Jacobian Determinant ◽  
Large Role ◽  
Relativistic Boltzmann Equation ◽  
Relativistic Analog ◽  
Relativistic Collision ◽  
Pointwise Limit ◽  
Open Question

AbstractThis article considers a long-outstanding open question regarding the Jacobian determinant for the relativistic Boltzmann equation in the center-of-momentum coordinates. For the Newtonian Boltzmann equation, the center-of-momentum coordinates have played a large role in the study of the Newtonian non-cutoff Boltzmann equation, in particular we mention the widely used cancellation lemma [1]. In this article we calculate specifically the very complicated Jacobian determinant, in ten variables, for the relativistic collision map from the momentum p to the post collisional momentum $$p'$$ p ′ ; specifically we calculate the determinant for $$p\mapsto u = \theta p'+\left( 1-\theta \right) p$$ p ↦ u = θ p ′ + 1 - θ p for $$\theta \in [0,1]$$ θ ∈ [ 0 , 1 ] . Afterwards we give an upper-bound for this determinant that has no singularity in both p and q variables. Next we give an example where we prove that the Jacobian goes to zero in a specific pointwise limit. We further explain the results of our numerical study which shows that the Jacobian determinant has a very large number of distinct points at which it is machine zero. This generalizes the work of Glassey-Strauss (1991) [8] and Guo-Strain (2012) [12]. These conclusions make it difficult to envision a direct relativistic analog of the Newtonian cancellation lemma in the center-of-momentum coordinates.

Poisson approximation for a sum of dependent indicators: an alternative approach

2002 ◽  
Vol 34 (03) ◽  
pp. 609-625 ◽  
Author(s):  
N. Papadatos ◽  
V. Papathanasiou
Keyword(s):  
Total Variation ◽  
Upper Bound ◽  
Random Variables ◽  
Random Variable ◽  
Poisson Approximation ◽  
Total Variation Distance ◽  
Poisson Random Variable ◽  
Birthday Problem ◽  
Alternative Approach ◽  
Variation Distance

The random variablesX1,X2, …,Xnare said to be totally negatively dependent (TND) if and only if the random variablesXiand ∑j≠iXjare negatively quadrant dependent for alli. Our main result provides, for TND 0-1 indicatorsX1,x2, …,Xnwith P[Xi= 1] =pi= 1 - P[Xi= 0], an upper bound for the total variation distance between ∑ni=1Xiand a Poisson random variable with mean λ ≥ ∑ni=1pi. An application to a generalized birthday problem is considered and, moreover, some related results concerning the existence of monotone couplings are discussed.

scholarly journals Polynomial Mixing of the Edge-Flip Markov Chain for Unbiased Dyadic Tilings

2018 ◽  
Vol 28 (3) ◽  
pp. 365-387
Author(s):  
S. CANNON ◽  
D. A. LEVIN ◽  
A. STAUFFER
Keyword(s):  
Markov Chain ◽  
Relaxation Time ◽  
Lower Bound ◽  
Upper Bound ◽  
Open Problem ◽  
Mixing Time ◽  
Perpendicular Bisector ◽  
Unit Square

We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall and Spencer in 2002 [14]. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2−s, (a + 1)2−s] × [b2−t, (b + 1)2−t] for a, b, s, t ∈ ℤ⩾ 0. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n4.09), which implies that the mixing time is at most O(n5.09). We complement this by showing that the relaxation time is at least Ω(n1.38), improving upon the previously best lower bound of Ω(n log n) coming from the diameter of the chain.

scholarly journals On the relative distances of six points in a plane convex body

2005 ◽  
Vol 42 (3) ◽  
pp. 253-264
Author(s):  
Károly Böröczky ◽  
Zsolt Lángi
Keyword(s):  
Convex Body ◽  
Upper Bound ◽  
Euclidean Distance ◽  
Euclidean Plane ◽  
Relative Distance ◽  
Plane Convex ◽  
Euclidean Length ◽  
Plane Convex Body

Let C be a convex body in the Euclidean plane. By the relative distance of points p and q we mean the ratio of the Euclidean distance of p and q to the half of the Euclidean length of a longest chord of C parallel to pq. In this note we find the least upper bound of the minimum pairwise relative distance of six points in a plane convex body.

scholarly journals Effect of Disintegration Times of the Homogeneity of Soil prior to Treatment

2019 ◽  
Vol 9 (22) ◽  
pp. 4791
Author(s):  
Wathiq Al-Jabban ◽  
Jan Laue ◽  
Sven Knutsson ◽  
Nadhir Al-Ansari
Keyword(s):  
Water Content ◽  
Uniform Distribution ◽  
Construction Projects ◽  
Mixing Time ◽  
Soil Improvement ◽  
Natural Soil ◽  
High Plasticity ◽  
Negative Effects ◽  
Disintegration Time ◽  
Soil Particles

This paper presents an experimental study to investigate the effect of various disintegration times on the homogeneity of pre-treated natural soil before mixing with cementitious binders. Various disintegration times were applied, ranging from 10 s to 120 s. Four different soils were used with different characteristics from high, medium and low plasticity properties. Visual and sieving assessment were used to evaluate the best disintegration times to allow for a uniform distribution of water content and small-sized particles that would produce a uniform distribution of the binder around the soil particles. Results showed that a proper mixing time to homogenize and disintegrate the soil prior to treatment depended on several factors: soil type, water content and plasticity properties. For high plasticity soil, the disintegration time should be kept as short as possible. Increasing the disintegration time ha negative effects on the uniformity of distribution of the binder around soil particles. The homogenizing and disintegration time were less important for low plasticity soils with low water content than for medium to high plasticity soils. The findings could assist various construction projects that deal with soil improvement through preparation of soil before adding a cementitious binder to ensure uniformity of distribution of the binder around soil particles and obtain uniform soil–binder mixtures.

scholarly journals Uniform mixing time for random walk on lamplighter graphs

2014 ◽  
Vol 50 (4) ◽  
pp. 1140-1160 ◽  
Author(s):  
Júlia Komjáthy ◽  
Jason Miller ◽  
Yuval Peres
Keyword(s):  
Random Walk ◽  
Mixing Time

scholarly journals The Maximum of a Random Walk and Its Application to Rectangle Packing

1998 ◽  
Vol 12 (3) ◽  
pp. 373-386 ◽  
Author(s):  
E. G. Coffman ◽  
Philippe Flajolet ◽  
Leopold Flatto ◽  
Micha Hofri
Keyword(s):  
Random Walk ◽  
Upper Bound ◽  
Symmetric Random Walk ◽  
Rectangle Packing ◽  
Unit Width ◽  
Minimum Height ◽  
Image Position

Let S0,…,Sn be a symmetric random walk that starts at the origin (S0 = 0) and takes steps uniformly distributed on [— 1,+1]. We study the large-n behavior of the expected maximum excursion and prove the estimate,where c = 0.297952.... This estimate applies to the problem of packing n rectangles into a unit-width strip; in particular, it makes much more precise the known upper bound on the expected minimum height, O(n½), when the rectangle sides are 2n independent uniform random draws from [0,1].

Nilpotent Ideals in Alternative Rings

1980 ◽  
Vol 23 (3) ◽  
pp. 299-303 ◽  
Author(s):  
Michael Rich
Keyword(s):  
Upper Bound ◽  
Associative Ring ◽  
Analogous Result ◽  
Alternative Ring ◽  
Locally Nilpotent ◽  
Open Question ◽  
The Ideal

It is well known and immediate that in an associative ring a nilpotent one-sided ideal generates a nilpotent two-sided ideal. The corresponding open question for alternative rings was raised by M. Slater [6, p. 476]. Hitherto the question has been answered only in the case of a trivial one-sided ideal J (i.e., in case J2 = 0) [5]. In this note we solve the question in its entirety by showing that a nilpotent one-sided ideal K of an alternative ring generates a nilpotent two-sided ideal. In the process we find an upper bound for the index of nilpotency of the ideal generated. The main theorem provides another proof of the fact that a semiprime alternative ring contains no nilpotent one-sided ideals. Finally we note the analogous result for locally nilpotent one-sided ideals.

WELFARE LOSS IN LINEAR PRICE-PREFERENCE PROCUREMENT AUCTIONS

2007 ◽  
Vol 09 (04) ◽  
pp. 719-730
Author(s):  
WINSTON T. H. KOH
Keyword(s):  
Uniform Distribution ◽  
Upper Bound ◽  
Government Procurement ◽  
Welfare Loss ◽  
Procurement Auctions ◽  
Domestic Firms ◽  
Sealed Bid ◽  
Non Linear ◽  
Price Auction

In government procurement auctions, discrimination in favor of one group of participants (e.g. domestic firms, minority bidders) over another group is a common practice. The optimal discriminatory rules for these auctions are typically non-linear and could be administratively complex and costly to implement. In practice, procurement auctions are usually organized as sealed-bid first-price auction with a simple percentage price-preference policy. In this paper, we analyze a model with two bidders that draw their costs from a common uniform distribution, and derive an upper bound to the welfare loss resulting from the use of linear-price preference auctions.

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