One-way quantum state generators (OWSGs), which serve as the quantum analog of one-way functions (OWFs), have attracted significant interest due to their potential applications and the reduced assumption requir
We will prove Theorem 2 Periodic states that have an anchor lie in cycles in the state diagram of \chi . These cycles have a length that is a power of two and this length ranges from 1 to the largest power of two not larger than n. Recall that \chi operates as multiplication of ...
Thus, eliminating x(s), the state vector, by combining the output equation and equation (5.50), the output vector y(s) is given by (5.51)y(s)=[C(sI−A)−1B+D]u(s)=G(s)u(s) where G(s) is called the transfer function matrix. In general, the transfer function matrix has...
1. We prove that given ϵ = Θ(1), Here, the notation f(x) = Θ(g(x)) denotes that f(x)=O(g(x)) and f(x) = Ω(g(x)) both hold. Hence, f(x) is asymptotically equal to g(x) up to constant factors. the improved ML algorithm can use a dataset size ...
Nash–Moser inverse function theorem35Q3537C2537K6546T0558C15It is well known that the incompressible Euler equations can be formulated in a very geometric language. The geometric structures provide very valuable insights into the properties of the solutions. Analogies with the finite-dimensional model...
We can also directly prove [6.36] by noting that according to the equivalences [6.34], we have Π13 = Π23 = Π33 = 0 through the thickness and, thus, Π.G3 = Πi3Gi = 0. • In fact, in certain cases, the conditions σ13 = σ23 = 0 are not really new, but are equival...
We prove the Theorem in the Supplementary Material, where we also show that it cannot be formulated as an if and only if statement. We will now apply the methods we developed for studying the irreversibility of coherence theory. For any quantum resource theory, the conversion rate \(R\) ful...
Whether the theorem statement is true or not, I don’t know; but at the moment I am unable to prove it. I have to accept that there is a bug. As painful as it is I realize that I am writing this post from a relatively comfortable position. Who knows if I would have been able ...
∗represents the convolution between two functions. We assume that the potential functionV(x)satisfies general periodic condition. Moreover, by using variational tools from the Nehari manifold method developed by Szulkin and Weth, we obtain the existence results of ground state solutions and ...
In mathematics, a function is a relationship between two variables, an input variable and an output variable, such that each output variable relates to exactly one input variable. The range of a function is the set of all of the outputs of a given function....