## Does ‘Distance Spell’ apply to ‘Sword Burst’

If this has been asked, I could not find it. Sorcerer has Sword burst cantrip, and is thinking of taking ‘Distance Spell’ metamagic. Will it change the range to 10′?

Similar to Metamagic Distant Spell and AoE spells but not quite answered, though Crawford’s tweet seems to imply it would work.

Although the spell has a range of 5′ it seems like distance spell would be increasing the area of affect in this case. So RAW it works, but likely shouldn’t??

## Thorup : What is the meaning of super distance?

While reading Thorup’s Algorithm to solve SSSP problem, I have one point that I can’t understand: super distance.

It says: “For each vertex we have a super distance $$D(v)\geq d(v)$$

$$d(v)$$ must refer the shortest distance from origin to $$v$$, but what is $$D$$ ?

Is it just a distance value from origin while calculation until $$D(v)=d(v)$$ ?

## Lower and upper bounds of the distance between two Frobenius numbers

I consider two sequences of numbers: $$A=\{a_1,…,a_{m-1},n\}$$ and $$B=\{n-a_{m-1},…,n-a_1,n\}$$, where $$a_1 < a_2 < … < a_{m-1} < n$$ and $$\gcd(A) = \gcd(B) = 1$$.

I investigate the lower and upper bounds of the distance between two Frobenius Numbers:

$$Dist Lower Bound \le \left| F(A) – F(B) \right| \le Dist Upper Bound$$

I found only one trivial estimate for lower bound: It is known that $$F(a_1,…,a_m) \ge a_1 – 1$$ if $$a_1 \ge 2$$. Using this property, we obtain that:

$$\left| F(A) – F(B) \right| \ge \left| n-a_{m-1}-a_1 \right|$$.

Also, I investigate the lower and upper bounds of the sum of two Frobenius Numbers:

$$Sum Lower Bound \le F(A) + F(B) \le Sum Upper Bound$$

I found only one trivial estimate for upper bound: I am using simple way. I took well-known estimates and combine it because I having knowledge about $$a_{m-1}$$.

$$1. F(a_1,…,a_m) \le 2a_{m-1} \lfloor{\frac{a_m}{m}}\rfloor – a_m = 2a_{m-1} \lfloor{\frac{n}{m}}\rfloor – n$$.

$$2. F(b_1,…,b_m) \le 2b_m \lfloor{\frac{b_1}{m}}\rfloor – b_1 = 2n \lfloor{\frac{n-a_{m-1}}{m}}\rfloor – n + a_{m-1}$$.

I am convinced that there are other nearer solutions. I will be grateful for any help in search $$Sum Lower Bound$$ and $$Dist Upper Bound$$ as well as improving my estimates.

## Flight search engine offering results ordered by distance flown

Put simply: I’d like to fly more, I’d like to be more in the air than on the ground.

I have asked similar questions before, and yes, they sound weird, but please hear me out.

Ref: Flight search for 3+/4+ stops

I know that actual distance flown differs from flight to flight depending on weather(winds,rain,snow), airline (fuel/cost decisions) and aircraft,

Is there a flight search engine that displays tentative/approximate total distance flown? Also, i’m looking such a search engine that allows filtering by distance such as total > 10000 miles. Obviously, i expect such a system to allow sorting too.

On the other hand, this can be seen as a person trying to reduce distance flown or Co2 emission etc.

I really hope one such thing exists, please help me find it. For example, Tokyo to Mumbai: Direct Flight : NH829, 4200 miles or 6700 km via Hong Kong Flight : 2800 km + 4300 km = 7100 km via Singapore Flight : 5300 km + 3900 km = 9200 km via Bangkok Flight : 4600 km + 3000 km = 7600 km

Thanks.

## Expected values of packing distance between vectors with Bernoulli trials?

Pick set $$\mathcal T$$ of $$2^k$$ vectors in $$\{-1,+1\}^n$$ with Bernoulli trials.

What are exact expected values or at least tight bounds of $$\min_{\substack {u,v\in\mathcal T\u\neq v}}\|u-v\|_2^2$$ $$\min_{\substack {u,v\in\mathcal T\u\neq v}}\|u-v\|_1$$ and their distributions?

## Comparison of Wasserstein Distance And Push-Forward By Lipschitz Map

Suppose that $$f,g :(X,d_X)\rightarrow (Y,d_Y)$$ are contiuous (we may assume Lipschitz if its necessary) and let $$\mu,\delta_{x}$$ be probability measure on $$X$$ (the latter is the point-mass at some point $$x \in X$$) lying in the Wasserstein 1 space $$W_1(X)$$ over $$X$$.

If I know that $$\sup_{x \in X} d(f(x),g(x))\leq \epsilon$$ for some positive number $$\epsilon$$ then how can I use this to obtain a (non-trivial) upper bound on $$d_{W_1(Y)}(f_{\star}(\mu),g_{\star}(\delta_x)) ?$$

So Far All I can do is: $$d_{W_1(Y)}(f_{\star}(\mu),g_{\star}(\delta_x))\leq \int_{z \in X} d_{Y}(f(z),g(x)) d\mu(z) ?$$ But this seems rather useless…

## Speeding Up Excel Distance Calculation Using Bing API Calls

I am writing VB code in Excel to calculate the distance between an employee’s home address and work address using Bing Maps API calls. The process follows this general flow:

1) Convert the employee’s address to Lat-Long values using the GetLatLong function

2) Convert the employee’s work address to Lat-Long values using the GetLatLong function

3) Calculate the distance between these two points using the GetDistance function

4) Calculate the drive time between these two points using the GetTime function

The process is working, but it is excruciatingly slow. The employee population is approximately 2300, and it takes almost an hour to execute.

I am not a coder, but I can functionally modify found code to my purposes. This is an amalgamation of a couple different process found through Google searching. The code pieces in use are:

Public Function GetDistance(start As String, dest As String)     Dim firstVal As String, secondVal As String, lastVal As String     firstVal = "https://dev.virtualearth.net/REST/v1/Routes/DistanceMatrix?origins="     secondVal = "&destinations="     lastVal = "&travelMode=driving&o=xml&key=<My Key>&distanceUnit=mi"     Set objHTTP = CreateObject("MSXML2.ServerXMLHTTP")     Url = firstVal & start & secondVal & dest & lastVal     objHTTP.Open "GET", Url, False     objHTTP.setRequestHeader "User-Agent", "Mozilla/4.0 (compatible; MSIE 6.0; Windows NT 5.0)"     objHTTP.send ("")     GetDistance = Round(WorksheetFunction.FilterXML(objHTTP.responseText, "//TravelDistance"), 0) & " miles" End Function  Public Function GetTime(start As String, dest As String)     Dim firstVal As String, secondVal As String, lastVal As String     firstVal = "https://dev.virtualearth.net/REST/v1/Routes/DistanceMatrix?origins="     secondVal = "&destinations="     lastVal = "&travelMode=driving&o=xml&key=<My Key>&distanceUnit=mi"     Set objHTTP = CreateObject("MSXML2.ServerXMLHTTP")     Url = firstVal & start & secondVal & dest & lastVal     objHTTP.Open "GET", Url, False     objHTTP.setRequestHeader "User-Agent", "Mozilla/4.0 (compatible; MSIE 6.0; Windows NT 5.0)"     objHTTP.send ("")     GetTime = Round(WorksheetFunction.FilterXML(objHTTP.responseText, "//TravelDuration"), 0) & " minutes" End Function  Public Function GetLatLong(address As String, city As String, state As String, zip As String)     Dim firstVal As String, secondVal As String, thirdVal As String, fourthVal As String, lastVal As String     firstVal = "https://dev.virtualearth.net/REST/v1/Locations?countryRegion=United States of America&adminDistrict="     secondVal = "&locality="     thirdVal = "&postalCode="     fourthVal = "&addressLine="     lastVal = "&maxResults=1&o=xml&key=<My Key>"     Url = firstVal & state & secondVal & city & thirdVal & zip & fourthVal & address & lastVal     Set objHTTP = CreateObject("MSXML2.ServerXMLHTTP")     objHTTP.Open "GET", Url, False     objHTTP.setRequestHeader "User-Agent", "Mozilla/4.0 (compatible; MSIE 6.0; Windows NT 5.0)"     objHTTP.send ("")     GetLatLong = WorksheetFunction.FilterXML(objHTTP.responseText, "//Point//Latitude") & "," & WorksheetFunction.FilterXML(objHTTP.responseText, "//Point//Longitude") End Function 

To be clear, the process works well, just extremely slowly. Any thoughts on speeding this up?

Thanks, Lee

## Fuel consumption and distance

Robin has owned her car for 2 years. In that time, she has driven a total of 68, 000 km. She has had the following maintenance costs over the past 2 years.

• 13 lube, oil and filter services at $$31.95/service – 6 tire rotations at$$24.95/service
• 1 cooling system service at 95.00 – 2 wheel alignments at71.95/service

Robin has a car and has a fuel consumption of 9.8L/100km. Her license plate costs $$74.00 a year and her insurance is$$1475.00 a year. The average cost of gas over the past two years has been $1.28/L. a) Calculate Robins maintenance costs over the past 2 years b) Calculate Robin’s fuel costs over the past 2 years c) Calculate Robin’s operating costs over the past 2 years (maintenance, fuel, license, insurance costs) d) Calculate the average monthly cost over the past 2 years e) Calculate her costs per 1 km. ## Distance function to the boundary and Harnack inequality Suppose $$\Omega \subset \mathbb{R}^d$$ be a domain, and let $$\rho(x) = \mathrm{dist} (x, \partial \Omega)$$ be the distance function to the boundary of $$\Omega$$. I want to know for which domains $$\rho$$ satisfies a Harnack type inequality. Harnack inequality says that $$\sup _{x \in B} \rho (x) \leq C \inf _{x \in B} \rho(x)$$ on a ball $$B= B(a,r), a \in \Omega$$, and $$C$$ is a constant depend on $$B$$. It is known that harmonic functions satisfy Harnack inequality. Is it enough if $$\Omega$$ satisfy regularity property (e.g.,if it is a Lipschitz or a NTA-domain)? What about boundary Harnack inequality? ## Isoperimetry on$[0, 1]^n$w.r.t$\ell_p\$ distance

Let $$A$$ be a measurable subset of the metric space $$\mathcal X = ([0, 1]^n,\ell_p)$$, and define its $$\epsilon$$-blowup by $$A^\varepsilon:=\{x \in \mathcal X \mid \|x-a\|_p \le \epsilon\text{ for some }a \in A\}$$.

# Question

• If $$\operatorname{vol}(A) > 0$$, what is a good lower bound on $$\operatorname{vol}(A^\epsilon)$$ ?

• Same question with $$\operatorname{vol}(A) \ge 1/2$$.