## co.combinatorics – counting self-avoiding walks in a strip

Look at the streak $${0,1, ldots n } times {0,1,2 }$$ in the $$mathbb {N} ^ 2.$$ A formula is known for the total number of self-avoiding walks in this strip $$(0.0)$$ in terms of the parameter $$n$$?

To edit: I mean all the walks, the steps of $$( pm 1,0), (0, pm1)$$ as long as they are limited to this strip.

Note: Asked on math.stackexchange (see here) a week ago with no specific answer.

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## Reference request – Random Walks: How often does the largest component change?

In my understanding, this continues for an unbiased, random walk (starting at the origin) $$mathbb R$$ With $$N$$ Steps, which is the expected number of sign changes $$O ( sqrt N)$$, For a biased walk, I think the expected number is $$O (1)$$, In both cases $$O ( sqrt N)$$,

Now let's say we take a random walk $$(X_n, Y_n)$$ on $$mathbb R ^ 2$$ and are interested in how often the largest component changes. Define formally $$M (n) = 1$$ if $$X_n ge Y_n$$ and $$M (n) = 2$$ if $$X_n , We want to bind $$mathbb E | {n le N: M (n) ne M (n + 1) } |$$,

To reach a limit, we just have to consider the random walk $$X_n-Y_n$$ on $$mathbb R$$, The sign changes for the previous result $$O ( sqrt N)$$ sometimes on expectation. But the sign change corresponds to the largest component change and we are done.

For dimensions $$d> 2$$ This trick no longer works. In this case, is it known how often the largest component is likely to change? Are there any known order limits depending on $$d$$ and $$N$$? I guess $$O ( log (d) sqrt N)$$ Limits may be possible.

I'm trying to find answers online, but apparently I can't even find a reference for them $$O (1)$$ Result I mentioned.

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## random walks – strategy to find yourself in a crowd

Is it better if a person moves and a person stays, or if both people move?

Suppose we have one $$n$$ by $$n$$ Grid of squares. Each person starts on a random field. When a person decides to move, he or she moves 1 field in a random direction every second, except for the direction in which the person comes to the field he was previously on. If at one point both people are in the same field, we are done. We want to compare the expected meeting time of both strategies.

This is like a casual walk $$mathbb {Z} ^ 2$$but the limitedness prevents us from taking the difference in their positions and considering this as another random walk. This is also like a random walk on a graph, but the random direction depends not only on the current but also on the previous square.

## Probability – Martingale and crossing of random walks

To let $$G = (V, E)$$ be a graph with $$n$$ Corners. Consider a few simple random walks $$(X, Y)$$ in the graph the length $$L$$ starting from a node $$v in V$$, We refer to a length$$L$$ aimless walk $$X$$ as a tuple in $$V ^ L$$, as $$(X_1, ldots, X_L)$$, Now consider an estimate of the number of crossings in such a pair of random walks given by
begin {align} T (X, Y) = sum_ {j = 1} ^ L sum_ {k = 1} ^ L mathbb {I} _ { {X_j = Y_k }} end
from where $$mathbb {I} _ { { cdot }}$$ is the display function of the event $${ cdot }$$, My question is, can the random variable be $$T (X, Y)$$ shown as martingale (plus some memory terms)?

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## Reference Request – Common drunkard walks

The drunkard walk is a game in which two players play $$a$$ and $$b$$ Dollars, and they play a series of fair games (both risking a dollar in each game) until one of them goes bankrupt.

My question is: was this taken into account in a graph? More specifically, we get a (finite, connected) simple graph, and each vertex is assigned a positive integer. In each round, we randomly choose an edge randomly and throw a fair coin to decide which vertex gives the other a dollar. The game is played until someone goes broke.

(I have reasonably good estimates for the expected runtime if the graph is a cycle.)

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## App Development Cost Calculator

Mobile App Development Company is extremely competitive with several million mobile apps on the market. The question of how to reach your target users and how to achieve your ROI goals is really difficult. Businesses are looking to find a way to calculate the average cost of app development in advance. To meet this demand, several online calculators have been developed, of which we will highlight only the most popular ones.

Otreva machines

This calculator was developed by Scranton, an Otreva mobile apps development company in Pennsylvania. The calc provides statistics that cover multiple platforms, app and admin functions, average cost of building these platforms, and average costs. It also allows estimating the cost of developing iOS, iOS, Windows Phone or even web app. It takes users through two steps, when users need to choose a platform and then create certain features that they want in mobile apps, and then create a quote in USD.

Imason machines

, Just like Otreva calc, it is built on a three-level structure. As a first step, users can choose between app type for iPhone, iPad, phone, tablet, Windows Phone or Windows tablet and, in a second step, select certain features they need for a development offer such as login, social login, second language support and more ,

Kinvey machines

Unlike two previous calculators, Kinvey has gone through more steps for the user and allows for a more detailed app development cost estimate. The steps are – choosing platforms, choosing internal / external team, using or not using any cloud tech, size, user management, type of data to store, selection of data sources, using / not using location data, third-party cloud APIs, Selection of user contact channels and more. After all this information has been provided, users can provide their e-mail address for a quote as well.

Enterprise Mobile App Calculator

It takes company users through many steps to provide an estimate. The steps are – Platform, B2C / B2B / B2E, requires authentic or non-type of app (access to corporate data / re-launch of a business process / mobilization of a new business process), UA complexity, features using push notifications or not, number of corporate networks to which a connection can be made and more.

,

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## Survival probability for casual walks

The probability of survival for a hiker starting at the origin is defined as the probability that the hiker remains positive by n steps. Thanks to the Sparre-Andersen theorem, I know that this PDF is from

``````plot[Binomial[2 n, n]* 2 ^ {- 2 n}, {n, 0, 100}]
``````

However, I would like to confirm this empirically.

My attempt to confirm this for n = 100:

``````FoldList[
If[#2 < 0, 0, #1 + #2] &,
prefixing[Accumulate[RandomVariate[NormalDistribution[0, 1], 100]]0]]
``````

I want`FoldList` to stop if `# 2 <0` not just replace in 0.

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## Probability or Statistics – Modeling Snowfall with Random Walks with Drift

I try to simulate a (very) simple model of snowfall / accumulation with random walks in the following way:

``````sf = accumulate[RandomVariate[BernoulliDistribution[0.2], 100]*
RandomVariate[GammaDistribution[1, 2], 100]/. {0th -> -0,4}]ListLinePlot[sf]
``````

I generate Bernoulli trials with a probability of 0.2 to simulate days when it snows. On a day when it does not snow, instead of a simple 0 entry, I insert a negative drift duration of -0.4 to simulate the melting of the snow.
Where I have problems is that you can never have negative snowfall. I want the Walker to always be greater than or equal to 0. However, I can not send all negative entries to 0 as this would eliminate the data of the days it is snowing, but the drift term is greater than the snowfall.

Many Thanks.

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