Author Archives: anuj - Page 11
The Dirichlet problem and quantum entanglement
Dirichlet Problem and Quantum Entanglement The Dirichlet problem and quantum entanglement are concepts from different branches of mathematics and physics, respectively, but there are indirect connections through the underlying…
Verify that a boost in the x-direction that any object traveling at speed c in an inertial frame S travels at speed c in the boosted frame.
Verify directly from the form of the Lorentz transformation representing a boost in the x-direction that any object traveling at speed c in an inertial frame S travels at speed…
Find those subsets S ⊂ Z+ such that all but finitely many sums of elements from S (possibly with repetitions) are composite numbers.
Find those subsets S ⊂ Z+ such that all but finitely many sums of elements from S (possibly with repetitions) are composite numbers. To find the subsets S⊂Z+S \subset \mathbb{Z}^+S⊂Z+…
Qubit Exercises
Part (i) Find the scalar product ⟨Ψ−(θ)∣Ψ+(θ)⟩\langle \Psi_{-}(\theta) | \Psi_{+}(\theta) \rangle⟨Ψ−(θ)∣Ψ+(θ)⟩ and discuss. Given: ∣Ψ+(θ)⟩=cos(θ)∣0⟩+sin(θ)∣1⟩|\Psi_{+}(\theta)\rangle = \cos(\theta) |0\rangle + \sin(\theta) |1\rangle∣Ψ+(θ)⟩=cos(θ)∣0⟩+sin(θ)∣1⟩ ∣Ψ−(θ)⟩=cos(θ)∣0⟩−sin(θ)∣1⟩|\Psi_{-}(\theta)\rangle = \cos(\theta) |0\rangle - \sin(\theta) |1\rangle∣Ψ−(θ)⟩=cos(θ)∣0⟩−sin(θ)∣1⟩ To find…
what is the langlands program?
The Langlands program is a set of far-reaching and deep conjectures proposed by Robert Langlands in 1967, which aims to relate and unify various areas of mathematics, including number theory,…
Let R > 1 and let f be analytic on IzI < R except at z = 1, where f has a simple pole. If M n=O is the Maclaurin series for f, show that hn,+,M alL exists.
To show that the limit limn→∞nan\lim_{n \to \infty} n a_nlimn→∞nan exists for the Maclaurin series f(z)=∑n=0∞anznf(z) = \sum_{n=0}^{\infty} a_n z^nf(z)=∑n=0∞anzn of the function fff, which is analytic in ∣z∣<R|z| <…
An accurate map of California is spread out flat on a table in Evans Hall, in Berkeley. Prove that there is exactly one point on the map lying directly over the point it represents.
To prove that there is exactly one point on the map of California that lies directly over the point it represents, we can use a combination of the Brouwer Fixed-Point…
Let f : R + R be continuous, with 00 s_, If(x)l dx < o. Show that there is a sequence (x,,) such that x, -+ 00, x, f (x,) 4 0, and x, f(-xc,) 4 0 as n -+ o.
Let f : R + R be continuous, with 00 s_, If(x)l dx < o. Show that there is a sequence (x,,) such that x, -+ 00, x, f (x,)…
Given conditions: lim 𝑡 → 𝑎 𝑔 ( 𝑡 ) = 𝑏 lim t→a g(t)=b lim 𝑥 → 𝑏 𝑓 ( 𝑥 ) = 𝑐 lim x→b f(x)=c To prove: lim 𝑡 → 𝑎 𝑓 ( 𝑔 ( 𝑡 ) ) = 𝑐 lim t→a f(g(t))=c
Given conditions: limt→ag(t)=b\lim_{t \to a} g(t) = blimt→ag(t)=b limx→bf(x)=c\lim_{x \to b} f(x) = climx→bf(x)=c To prove: limt→af(g(t))=c\lim_{t \to a} f(g(t)) = climt→af(g(t))=c Using the definition of the limit: limt→ag(t)=b\lim_{t \to…
Contour Integration and Feynman Propagator
Contour integration is a powerful technique in complex analysis, used extensively in physics for evaluating integrals that arise in quantum field theory, especially when calculating Feynman propagators. Here's an explanation…