Find a counterexample to the statement 4^n + 1 is divisible by 5. Find the inverse modulo m of each integer n below : a) n = 5, m = 26. b) n = 8, m = 35. c) n = 51, m = 99. What is the remainder of (2x^3 + 4x^2 - 32x + 18) / (x - 3)?
What kind of equation is -2(x + 4) + 1 = -2x - 7? Which is not a counterexample to the formula 1^2 + 3^2 + 5^2 + + (2n - 1)^2 = n(2n + 1)/3? (a) n = 3 (b) n = 2 (c) n = 1 (d) n = 4 Under what situation would one or more solutions of a ration...
This is known as the Toeplitz' conjecture. By "closed curve", we mean you squiggle some line through space in a potentially crazy way and end at the point where you started. We can find four points on this closed loop that form a square. But can we do the same for any loop? If ...
If there are some unwanted linear dependencies in the frequencies, we can do some linear algebra to eliminate one of the frequencies (using some geometry of numbers to keep the quantitative bounds under control) and continue the iteration. If instead the approximation is too inaccurate, then the...
Note that while the Heisenberg group is a counterexample to the complex strong Kakeya conjecture, it is not a counterexample to the complex form of the original Kakeya conjecture, because the complex lines in the Heisenberg counterexample do not point in distinct directions, but instead only point...
Just to add to this - GR is genuinely complicated, but SR isn't really any harder than Euclidean geometry. If you can handle Pythagoras' Theorem then you have the mathematical chops to handle SR, with the caveat that you'd be limited to idealised toy problems unless you also knew the ba...
That may not be the original question, but it helps in understanding and answering the original question. Let's be pedagogic here shall we, so we can make Physicsforum pleasant and informative! GR is based upon differential geometry and the description of space-time as a Lorentzian manifold,...
On that premise, we might surmise that the rise of probabilistic accounts of causation is a natural result of the attempt to understand what we might mean by a 'cause' in a context in which we have dropped the assumption that causes necessitate. And if that were so, we might think, ...
It is not hard to see that not every monotone map is actually a subgradient of a convex function (not even, if we go to “maximally monotone maps”, a notion that we sweep under the rug in this post). A simple counterexample is a (singlevalued) linear monotone map represented by (...
In science, the simplicity of a theory is a hallmark of its elegance. According to Einstein (or Louis Zukofsky or Roger Sessions or William of Ockham…I give up, who knows), “everything should be made as simple as possible, but not simpler.” Hence, the strength of a theory is rela...