Càlcul vectorial · Teorema de Stokes Integral de línia sobre una corba d'intersecció Aplicació del teorema de Stokes i càlcul de l'àrea d'un disc el·líptic Enunciat Sigui \(C\) la corba intersecció entre la superfície \[ z=2x^2+y^2 \] i el pla \[ y+3z=1. \] La corba és recorreguda en sentit antihorari respecte del vector \[ u=(0,1,3). \] Calculeu \[ \oint_{C^+} \langle F,d\ell\rangle,

Rotacional i camp conservatiu | Matemàtiques Matemàtiques · Càlcul vectorial Rotacional i camps conservatius Circulació d'un camp vectorial en una circumferència 0 Enunciat Sigui el camp vectorial \[ \boxed{ \mathbf F(x,y,z)= \left( \frac{-y}{x^2+y^2}, \frac{x}{x^2+y^2}, 0 \right) } \] Definit en el seu domini de definició. Es demana: Calculeu el seu rotacional. Calculeu \[ \oint_C \langle \mathbf F,d\boldsymbol\ell\rangle, \] on \(C\) és la circumferència de centre l'origen i radi…

**Authors:** Fangqi Dong, Alex Lombardi We prove near-optimal time-space lower bounds for breaking quantum cryptography in the random oracle model. Specifically, we show that a $T$-query adversary with $S$ qubits of non-uniform advice can recover a random key $k$ from the $n$-qubit binary phase state $|ψ_k\rangle \propto \sum_{x} R(k,x) |x\rangle$ with probability at most $O(\frac{T^2 + \sqrt{ST}}{N})$ for $N=2^n$. In contrast, the best known bound for post-quantum one-way functions is $O(\frac{T^2 + ST}{N})$, with a trivial attack at $S = N$. This demonstrates a new advantage of quantum cryptography over classical cryptography: $n$ qubits of communication suffice for security against preprocessing attacks with space up to $N^2$ rather than $N$. Our methodology is simple: express the optimal preprocessing attack as the operator norm of a random matrix, and bound this value in expectation over the random oracle via the trace-moment method. These trace moments have a natural interpretation using compressed oracles [Zhandry, Crypto 2019], which we then analyze. This can be viewed as a simplification and generalization of the approach of Liu [Eurocrypt 2023] for proving time-space tradeoffs for breaking post-quantum cryptography. We also prove the following results: (1) We tighten Liu's analysis of post-quantum PRGs in QROM, achieving a distinguishing advantage bound of $O(\frac{T^2}N + \sqrt{\frac{ST}N})$. (2) For unitary synthesis, we extend the one-query lower bound of Lombardi-Ma-Wright [STOC 2024] to hold against adversaries that can make one arbitrary function query along with polynomially many (adaptive) queries to the random oracle, either before or after the function query. This also interprets the original LMW24 result in terms of compressed oracles. (3) Finally, we prove a tight $O(\frac{\sqrt{S}}N)$ bound for the pseudorandomness of random binary phase states against space $S$ distinguishers.

The Bulletproofs ZKP chapter states that The prove_commitments_log subroutine will prove that $t_u\stackrel{?}=\langle\mathbf{l}_u,\mathbf{r}_u\rangle$ and $C\stackrel{?}{=} \langle \mathbf{l}_u, \...

**Authors:** Aleksandr Storozhenko For an arbitrary isotropic log-concave distribution $P$ on $\mathbb{R}^d$, we prove that the polynomial $(Cm)^m\|v\|_2^m - \mathbb{E}_{X\sim P}\langle X,v\rangle^m$ is a sum of squares for every even $m\ge2$, where $C>0$ is a universal constant. This removes the dependence on the Poincaré constant in the theorem of Kothari and Steinhardt (arXiv:1711.07465), recovering the optimal moment bounds for log-concave distributions. As an immediate corollary, we obtain computationally efficient algorithms with dimension-free error guarantees for a wide range of high-dimensional statistical estimation problems. Our proof uses stochastic localization to decompose $P$ as an average of random strongly log-concave measures, whose centered moments admit the subgaussian certificates of Diakonikolas, Hopkins, Pensia, and Tiegel (STOC 2025; arXiv:2410.21194). With a covariance-adapted choice of localization, we show that a fourth-moment certificate derived from Letwin's variance inequality for quadratic forms (arXiv:2607.24164) suffices to control this averaging at every even degree.

Building on proton light-front wave functions in the $|qqq\rangle$ and $|qqqg\rangle$ Fock sectors, obtained by diagonalizing the proton light-front Hamiltonian within the Basis Light-Front Quantizati...

All optical clock systems use some form of averaging, in which the frequencies of two or more transitions are combined such that the average frequency is insensitive to perturbations from external fields. Most commonly, this is an average over one or more pairs of Zeeman lines, which cancels the linear Zeeman shift34. In the case of 176Lu+, we use averaging over the transitions from \(|g\rangle \equiv {|}^{1}{S}_{0},7,0\rangle \) to \(|F\rangle \equiv {|}^{3}{D}_{1},F,0\rangle \) for F = 6, 7 and 8 at the optical frequencies fF. Conceptually, this hyperfine averaging technique35 leads to an effective J = 0 level for which the quadratic Zeeman and quadrupole shifts of the (J = 1) \({|}^{3}{D}_{1},F,0\rangle \) states are largely eliminated in the hyperfine average (HA) frequency, \(f\equiv \frac{1}{3}({f}_{6}+{f}_{7}+{f}_{8})\). Rather than sequentially interrogate multiple optical transitions, the average can be conveniently realized by Ramsey spectroscopy on a single optical transition by using a hybrid microwave and optical pulse sequence16,36. As shown in Fig. 1b,c, the interrogation sequence consists of a Ramsey interrogation on the |g⟩ to |8⟩ transition and microwave transitions within the interrogation time to transfer population between the hyperfine states. Timing is such that the effective time in each of the hyperfine...