k-limited solution for PCP

So there’s following problem, that has been bugging me for the last few days:

A solution of a PCP $ i_{1}.,..,i_{n}$ with the cards $ (x_{1}, y_{1})…(x_{m}, y_{m})$ is considered as k-limited if for all $ j\leq k$ we say that $ \mid\mid x_{i_{1}}.,..,x_{i_{j}}\mid – \mid y_{i_{1}}.,..,y_{i_{j}}\mid\mid \leq k$ .

So after every time we add another card to our sequence, the difference between the upper and lower part of the card should be $ \leq k$ .

Apparently it is decidable for a fixed k, that a PCP-instance has a k-limited solution. We were told that we should try solving this problem with a DFA over the input alphabet {1…m} – where the word $ i_{1}.,..,i_{n}$ will only be accepted if it is k-limited.

Now my problem is that I don’t know how to construct a DFA that can check if a solution is k-limited and a PCP-solution at the same time.

Step by step integration solution that shows where substitutions occur without using Wolfram Alpha

I know there is heaps of people’s code around the forum with code that shows step by step solutions for integration. But they don’t specify where certain things happen like substitution (u sub) or recognition by integration and stuff like that. Is there a code (has to work offline) that can show you what is being made (like u sub, simplify, expand, double angle…) for each step?

How to manipulate solution outputs that appear in a list?

I have a list {E, {p, ϕ}}, and I need values E, p, and ϕ such that I can later on do operations with them (like ArcCos[p], etc.). I will be repeatedly generating the values in the list, as a list, so doing this by-hand is not an option. How do I get them out of the list?

In this case, the values E, p, and ϕ exist such that they appear as p -> 0.097, ϕ -> 0.03, etc. I need them such that calling, for example, for list = {E, {p, ϕ}}, list[[2,1]] yields 0.097 and not p -> 0.097. This is the issue.

Solution to User Initial HTTP Requests Unencrypted Despite HTTPS Redirection?

It is my understanding that requests from a client browser to a webserver will initially follow the specified protocol e.g, HTTPS, and default to HTTP if not specified (Firefox Tested). On the server side it is desired to enforce a strict type HTTPS for all connections for the privacy of request headers and as a result HTTPS redirections are used. The problem is that any initial request where the client does not explicitly request HTTPS will be sent unencrypted. For example, client instructs browser with the below URL command.

google.com/search?q=unencrypted-get

google.com will redirect the client browser to use HTTPS but the initial HTTP request and GET parameters were already sent unencrypted possibly compromising the privacy of the client. Obviously there is nothing full-proof that can be done by the server to mitigate this vulnerability but:

  1. Could this misuse compromise the subsequent TLS security possibly through a known-plaintext
    attack (KPA)?
  2. Are there any less obvious measures that can be done to mitigate this possibly through some DNS protocol solution?
  3. Would it be sensible for a future client standard to always initially attempt with HTTPS as the default?

Is finding solution to a system of 2SAT equations seperated by OR (DNF form) in NP

I want to know if finding solution to a specific number of 2SAT equations sepearted by OR gate (DNF form as below) is in P or NP.

The equation has total n variables and each clause is a 2SAT equation in itself in a subset of variables from 1 to n. Example:

F = [(x1 || x2 ) && ( !x2 || x3) && (x3 || x4)] || [(x2 || x5) && (x3 || x6) && (!x4 || !x6)] || ….

The equation F is a DNF of 2SAT equations, which has say m clauses. Is finding a solution to this in NP? If yes how?

Also, specifically I also want to know if finding False instances of equation F is in P or NP as well.

Differences between Mathematica versions 11 and 12 regarding ODE solution

Solving the ODE

$ $ (\lambda +y(x)) y”(x)-y'(x)^2-1=0 $ $

with Version 11 I got the solution

yx = 1/2 (Exp[-Exp[C[1]] (C[2] + x) - 2 C[1]] + Exp[Exp[C[1]] (C[2] + x)] - 2 lambda) 

while in Version 12 for the same ODE I got the solution

yx = -lambda - Tanh[E^C[1] (x + C[2])]^2/Sqrt[-E^(2 C[1]) Sech[E^C[1] (x + C[2])]^2 Tanh[E^C[1] (x + C[2])]^2] 

This last result isn’t ever real: see the denominator. My question is regarding how to ask the solver in Version 12 to obtain the Version 11 answer. Thanks.

Approximate solution of a nonlinear ODE in the form of a Fourier series containing the coefficients of the initial ODE

In this topic we considering nonlinear ODE:

$ \frac{dx}{dt}= (x^4) \cdot a_1 \cdot sin(\omega_1 \cdot t)-a_1 \cdot sin(\omega_1 \cdot t + \frac{\pi}{2})$ – Chini ODE

And system of nonlinears ODE:

$ \frac{dx}{dt}= (x^4+y^4) \cdot a_1 \cdot sin(\omega_1 \cdot t)-a_1 \cdot sin(\omega_1 \cdot t + \frac{\pi}{2})$

$ \frac{dy}{dt}= (x^4+y^4) \cdot a_2 \cdot sin(\omega_2 \cdot t)-a_2 \cdot sin(\omega_2 \cdot t + \frac{\pi}{2})$

Chini ODE’s NDSolve in Mathematica:

pars = {a1 = 0.25, \[Omega]1 = 1} sol1 = NDSolve[{x'[t] == (x[t]^4) a1 Sin[\[Omega]1 t] - a1 Cos[\[Omega]1 t], x[0] == 1}, {x}, {t, 0, 200}] Plot[Evaluate[x[t] /. sol1], {t, 0, 200}, PlotRange -> Full] 

System of Chini ODE’s NDSolve in Mathematica:

pars = {a1 = 0.25, \[Omega]1 = 3, a2 = 0.2, \[Omega]2 = 4} sol2 = NDSolve[{x'[t] == (x[t]^4 + y[t]^4) a1 Sin[\[Omega]1 t] - a1 Cos[\[Omega]1 t], y'[t] == (x[t]^4 + y[t]^4) a2 Sin[\[Omega]2 t] - a2 Cos[\[Omega]2 t], x[0] == 1, y[0] == -1}, {x, y}, {t, 0, 250}] Plot[Evaluate[{x[t], y[t]} /. sol2], {t, 0, 250}, PlotRange -> Full] 

There is no exact solution to these equations, therefore, the task is to obtain an approximate solution.

Using AsymptoticDSolveValue was ineffective, because the solution is not expanded anywhere except point 0.

The numerical solution contains a strong periodic component; moreover, it is necessary to evaluate the oscillation parameters. Earlier, we solved this problem with some users as numerically: Estimation of parameters of limit cycles for systems of high-order differential equations (n> = 3)

How to approximate the solution of the equation by the Fourier series so that it contains the parameters of the original differential equation in symbolic form, namely $ a_1$ , $ \omega_1$ , $ a_2$ and $ \omega_2$ .

Forcing NIntegrate not to store full solution, in order to save memory

When using NIntegrate to integrate a time-varying dynamical system, I often only care about the final value of the dependent variables at the end of integration. However, NIntegrate returns interpolation functions, implying that it is storing all intermediate values of the dynamical variables in memory.

Is there a way to save memory by forcing NIntegrate to only return the final values of the dependent variables?

Is using a firewall debit card a good solution to protect my main credit card? [closed]

I’m using Curve, a physical debit card that connects all your credit and debit cards in one and let you decide which one to use at each transaction by using its own app.

I’ve used it with my lower limit credit cards and I really enjoy its features (like instant notifications etc). I recently managed to get a premium credit card from my bank and I wanted to ask you if you think it’s a good idea to link it to this service (or similar ones) in a way to protect my main premium credit card data to be exposed online and in real life. I would store my premium credit card in a safe place and spend with this curve card.

Is this a good idea from a security standpoint?