How to solve a Poisson’s differential equation with a boundary condition at infinity?

Context: This question is relevant to the physical problem of calculating potential for a set of p-n-p junctions. We have to solve a Poisson’s differential equation for a p-n-p junction with potential equal zero outside it on the left and right sides. For simplification and due to symmetry law we analyze only right side from 0 to some delta (from which potential is the same as for Infinity) and do not analyze left side. Boundary conditions are that in point on Infinity function and its 1st derivative is equal 0. Derivative in x=0 is equal 0. Alpha is a random very small number for Fermi step. In code bcd are boundary conditions

α = 0.00001; bcd1 = ϕ'[0] == 0; bcd2 = ϕ'[Infinity] == 0; bcd3 = ϕ[Infinity] == 0; eqn = Div[ Grad  [ϕ[x], x], x] == -((1/(exp ((x - 1)/α) + 1)) - (1/(exp (((-x - 1)/α) + 1))) + exp (-ϕ[x]) - exp (ϕ[x])); DSolve[{eqn, bcd1, bcd2, bcd}, ϕ, {x, 0, Infinity}] 

i have tried to use numbers(some delta from which Phi is 0) instead of Infinity or set boundary conditions like

ϕ'[x == 0] == 0 ϕ[x == -Infinity] == 0 ϕ'[x == -Infinity] == 0 

and put it directly into eqn but it does not seem to work. And I obtain as a result

DSolve[{Div[Grad[ϕ[x],x],x] == 1/(exp (1 + 100000. (-1 - x))) - 1/(1 + 100000. exp (-1 + x)) + 2 exp ϕ[x], Derivative[1][ϕ][0] == 0,    Derivative[1][ϕ][∞] == 0, ϕ[∞] ==  0}, ϕ, {x, 0, ∞}] 

If I try to vary boundary conditions or use more complex version of equation I obtain this

DSolve::dsvar: ∞ (-∞..) cannot be used as a variable. 

Thank you for your time.

Why is $\mathbb{R}^2$ endowed with the taxicab metric isomorphic to the infinity distance model of the cartesian real plane? (Hartshorne exercise 8.9)

My question deals with the following exercise from Hartshorne’s Euclid: Geometry and Beyond:

Following our general principles, we say that two models $ M,M’$ of our geometry are isomorphic if there exists a 1-to-1 mapping $ \phi:M\to M’$ of the set of points of $ M$ onto the set of points $ M’$ , written $ \phi(A) = A’$ , that sends lines to lines, preserves betweenness, i.e., $ A\ast B\ast C$ in M $ \iff$ $ A’\ast B’\ast C’$ in M’, and preserves congruence of line segments, i.e., $ AB\cong CD$ in M$ \iff$ $ A’B’\cong C’D’$ in $ M’$

If $ A=(a_1,a_2),B=(b_1,b_2)\in\mathbb{R}^2$ , the exercise asks you to first show that $ \mathbb{R}^2$ endowed with the taxicab metric: $ $ d(A,B)=|a_1-b_1|+|a_2-b_2|$ $ Is isomorphic to the model of $ \mathbb{R}^2$ endowed with the infinity distance: $ $ d(A,B)=sup\{|a_1-b_1|,|a_2-b_2|\}$ $ How can we proceed in order to prove these models are isomorphic? How should we describe our isomorphism $ \phi:\mathbb{R}^2\to \mathbb{R}^2$ ? I’m definitely having trouble with sending congruent lines to congruent lines.

The second part of the exercise claims that the taxicab metric model is not isomorphic to that of the standard model, i.e., the cartesian plane $ \mathbb{R}^2$ equipped with the usual Euclidean distance. Hartshorne leaves a hint to this exercise: in order to prove two models are not isomorphic, one just has to find a statement which is true in one model but not in the other. I’m thinking, for example, that the line: $ $ y=x+1$ $ Cuts the circle of center the origin $ (0,0)$ , and radius $ 1$ in infinitely many points in the taxicab metric model, a property that never holds in the standard model, which should make it clear that both models are not isomorphic. However, the first part still puzzles me, and I can’t imagine how to proceed.

Thanks in advance for your time, any comments will be appreciated.

Limit of e^-x and 1/e^x as x goes to infinity.

I am pretty sure my understanding may be incorrect so please guide. Wouldn’t the limit of e^-x as x goes to infinity simply be e^-infinity? Looking at the graph of e^-x, e^-infinity would be positive infinity, correct? But I also know that e^-x is also the same as 1/e^x so in this case plugging in infinity would give me 1/infinity which would just be zero?

Why am I getting two different answers? Please shed some light on limits (without using l’hopitals rule). I thought I was getting the concept of limits until this.

Thanks!

Relation between Frobenius norm, infinity norm and sum of maximums

Let $ A$ be a $ n \times n$ matrix so that the Frobenius norm squared $ \|A\|_F^2$ is $ \Theta(n)$ , the infinity norm squared $ \|A\|_{\infty}^2=1$ . Is it true that $ \sum_{i=1}^n\max_{1\leq j\leq n} |A_{ij}|^2$ is $ \Omega(n)$ ? Assume that $ n$ is sufficiently large.

I cannot find a relation between matrix norms that can show this. For the spectral norm it is not true as there is a nice costrunction.

Thanks!

Passing NaN and Infinity through the JXBrowser Javascript bridge

I can’t figure out how to pass special number values NaN and positive/negative infinity between Java and Javascript with JXBrowser.

Running e.g.

browser.executeJavaScriptAndReturnValue("0/0"); 

results in a NumberFormatException in the conversion step back to a Java number:

java.lang.NumberFormatException: For input string: "nan"     at java.base/jdk.internal.math.FloatingDecimal.readJavaFormatString(FloatingDecimal.java:2054)     at java.base/jdk.internal.math.FloatingDecimal.parseDouble(FloatingDecimal.java:110)     at java.base/java.lang.Double.parseDouble(Double.java:543)     at java.base/java.lang.Double.valueOf(Double.java:506)     at com.teamdev.jxbrowser.chromium.as.a(SourceFile:1117)     at com.teamdev.jxbrowser.chromium.as.b(SourceFile:70)     at com.teamdev.jxbrowser.chromium.Browser.executeJavaScriptAndReturnValue(SourceFile:2191)     at com.teamdev.jxbrowser.chromium.Browser.executeJavaScriptAndReturnValue(SourceFile:2120) 

Same for 1/0 and “inf”. I would expect to get back a Double.NaN or Double.POSITIVE_INFINITY respectively.

And the other way around, if I do

JSObject obj = browser.getJSContext().createObject(); obj.setProperty("special", Double.NaN); String json = obj.toJSONString(); Double resultValue = tmpObj.getProperty("special").asNumber().getDouble; 

the resultValue is 0.0 for both NaN and POSITIVE_INFINITY, while the json string evaluates to:

{"special":1.0982741961637e-311} 

(again, the same for either case.)

Is it possible to get JXBrowser to convert these values correctly somehow, or is this just not possible?

As a workaround, I could actually live with catching the NumberFormatException and handling these values on Java side, but the API I need to feed the return value into expects a JSValue object, and I can’t figure out how to create a JSNumber that properly wraps Double.NaN or Double.POSITIVE/NEGATIVE_INFINITY (as per the the second case above).

How does infinity affect things that are impossible, and include things that are very rare?

Basic question but complex answer. My question is that how does an infinite amount of time affect things that are impossible or rare. For instance, and I am not a scientist so this may be wrong, but how many times would a person die of burning in water or when 0 can be divisible. Theoretically, having infinite amount of time anything can happen infinitely, but my question is how many time could impossible things happen from case to case.

i want reference about the method(below) of describing Infinity…

let think x is real number, then

x=x

x=(x/2)+(x/2),

x=(x/3)+(x/3)+(x/3),

x=(x/4)+(x/4)+(x/4)+(x/4), . . ., x=(x/n)+(x/n)+(x/n)+………………………+(x/n),

if n is too large and equal to infinity, then

x=(x/inf)+(x/inf)+(x/inf)+(x/inf)+……………..+(x/inf)

then

(x/inf)= epsilon and this epsilon is equal ant to zero. because in goes below planck scale and we can’t observe it. my question is that who described it?

Hankel norm and H infinity norm model reduction exam question.

Shown below is a question from a model reduction exam. I’m not sure how to answer the questions and I’m wondering if my approach is correct.

A continuous time system relates the inputs $ u_1$ and $ u_2$ to the output $ y$ according to the differential equation.

$ $ \dot{y}+ \rho y= u_1 + 2u_2$ $

Where $ \rho$ is a real parameter.

a) $ \quad$ Determine for arbitrary $ \rho > 0$ the Hankel norm of this system.

b) $ \quad$ Determine for arbitrary $ \rho > 0$ the $ H_{\infty}$ norm of this system.

For the hankel norm we first must determine the state space representation. We assume $ \dot{y} = \dot{x}$ . Which leads to: $ $ \dot{x}=-\rho x+u_1+2u_2, \quad y=1$ $ So the state space form becomes: $ $ \dot{x} = \begin{bmatrix} -\rho \end{bmatrix}x + \begin{bmatrix} 1&2 \end{bmatrix} \begin{bmatrix} u_1\u_2 \end{bmatrix}, \quad y = \begin{bmatrix} 1 \end{bmatrix} x $ $ So $ A = \begin{bmatrix} -\rho \end{bmatrix}, \quad B = \begin{bmatrix} 1&2 \end{bmatrix}, \quad C = \begin{bmatrix} 1 \end{bmatrix}$ and $ D = 0$

Next, we need to determine the continuous time $ \infty$ horizon reachability and observability gramians using. $ $ 0 = AP+PA^{\top}+BB^{\top}$ $ $ $ 0 = A^{\top}Q + QA +C^{\top}C$ $ This leads to $ P = \frac{5}{2 \rho}$ and $ Q = \frac{1}{2 \rho}$

The Hankel norm can then be determined using: $ ||\Sigma||_H=\sqrt{\lambda_{max}(PQ)}= \sqrt{\lambda_{max}(\frac{5}{4 \rho^2})}$

The $ H_{\infty}$ norm can be determined using $ ||\Sigma||_{H_{\infty}}=sup \ \sigma_{max}(G(i \omega))$

In which $ G(i \omega)=C(SI-A)^{-1}B+D$ But the matrix dimensions are incorrect to perform this calculation. So I don’t have an idea on how to calculate the $ ||\Sigma||_{H_{\infty}}$ norm.

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