“はじめに ディープラーニングの実装をしているとベクトル微分とかを頻繁に目にしますが、具体的な演算の定義を改めて確認したいなと思い、まとめてみました。”

“The paper generalizes the result to the case of four measurements and deals with the summation form of uncertainty relation for two, three and four observables in a unified way.”

“The paper derives an exact identity for overdamped Langevin dynamics that equates the total EP rate to the mutual-information rate.”

“The paper shows that its restricted holonomy group is SU(2), implying universal single-qubit control by closed loops in shape space.”

“For any objective with log-sum-exp structure over distances or energies, the gradient with respect to each distance is exactly the negative posterior responsibility of the corresponding component: $\partial L / \partial d_j = -r_j$.”

“The paper establishes error bounds for the reconstructed solution and the source term without requiring classical source conditions, and derives an expected convergence rate for the source error in a weaker topology.”

“The dispersive shift and vacuum Rabi splitting emerge from the transcendental eigenvalue equation, with the residues determined by matching to the splitting: $δ_{ge} = 2Lg^2ω_q^2/v^4$, where $g$ is the vacuum Rabi coupling.”

“The paper demonstrates that classical crosstalk effects can significantly alter multi-qubit dynamics, which previous models could not explain.”

“The paper derives a generalized Heisenberg-Robertson uncertainty inequality valid across all spectral regimes.”

“The novelty of this work is two-fold: extending the catalogue of known optimal RMRAs and formulating a sub-optimal RMRA that abides by CFEs.”

“The paper derives least squares estimators for the drift, diffusion, and jump-diffusion coefficients and establishes their asymptotic rate of convergence.”

“The method combines two known field theories into a new composite field theory whose target space is the product of the original target spaces.”

“The paper offers 'deeper insight into the model's development and more detailed implementation guidance, including code examples and mathematical derivations.'”

“By working through the backward pass manually, we gain a deeper intuition for how each operation influences the final output.”

“When the noise acts on the velocity, the rescaled dynamics produce an additional Laplacian damping term, leading to a stochastic derivation of a Westervelt-type acoustic model.”

“The resulting macroscopic model exhibits memory effects and a nonlinear, time-dependent effective current.”

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“Neglecting the PEs in the PDAs leads to an overestimation of the RIS performance gain, explaining the discrepancies between theoretical and measured results.”

“Quantum errors can be modeled as deterministic physical terms rather than purely stochastic perturbations.”

“The paper likely provides a detailed mathematical treatment of the intersection of cylinders.”

“The central proposal of this paper is that qulk-clauses are biclausal structures in which qulk functions a clause-embedding predicate selecting a dull CP complement.”

“The paper establishes a theoretical upper bound on excess risk characterized by a distinct phase transition. In the initial optimization phase, the excess risk decays exponentially relative to the computational cost. However, once a specific resource allocation threshold is crossed, the system enters a statistical phase, where the generalization error follows a power-law decay of Θ(C−1/6).”

“$x(1-t(x + x^{-1})) F(x;t) = x - t F(0;t)$”

“The main contribution is the derivation of unified gradient estimates that connect the near-boundary and far-field regions through a characteristic length scale determined by the Robin parameter, yielding a global description of how boundary effects penetrate into the exterior domain.”

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“We provide a simple derivation — based on Bayes’ rule and conditional expectations — that unifies Gaussian diffusion and flow matching without relying on ODE/SDE…”

“The paper argues that the Gibbs state postulate can be derived from dynamical stability, implying a redundancy of the zeroth law.”

“The article's focus is on the mathematical derivation itself and its subsequent theoretical implications.”

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“Detailed derivations and open-source code to analyze the receptive fields of convnets.”

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