Begin with a question. Explore equations, diagrams, and experiments.从一个问题开始，在公式、图形与实验之间找到答案。
For a dense strided tensor, strides and storage_offset map logical indices to storage locations. Transpose shares the original storage by swapping sizes and strides; making a noncontiguous result contiguous requires copying its data.对稠密步幅张量，strides 与 storage_offset 决定逻辑索引对应的存储位置。转置通过交换形状和步幅共享原存储；将非连续结果转为连续布局则需要复制数据。
LIF describes membrane integration, threshold firing and reset. BPTT propagates gradients through time, STDP updates synapses from spike timing, and EventProp differentiates continuous event trajectories. UltraLIF modifies the forward model to support differentiable training.LIF 描述膜电位积分、阈值发放与重置。BPTT 沿时间展开图传播梯度，STDP 根据脉冲时序更新突触，EventProp 对连续时间的事件轨迹求导。UltraLIF 则调整前向模型以支持可微训练。
Safe Rust uses ownership and borrowing rules to restrict invalid references and unsynchronized shared mutation. C++ uses RAII to manage resource lifetimes, while reference validity depends on API contracts and program design. Performance in either language depends on data layout, algorithms and generated code.Safe Rust 通过所有权和借用规则约束无效引用与未同步的共享修改。C++ 用 RAII 管理资源生命周期，引用有效性仍依赖接口约定和程序设计。两者的性能都取决于数据布局、算法与生成的机器码。
SIMT kernels assign scalar work to individual threads; tile kernels express operations over data blocks and leave thread mapping to the compiler. Correct output partitioning and boundary handling prevent duplicate writes and omissions, while memory access and data reuse shape performance.SIMT 内核为单个线程分配标量计算，Tile 内核则描述数据块上的运算，由编译器组织线程映射。正确的输出分区与边界处理避免重复写入和遗漏，访存方式与数据复用影响性能表现。
For dense 2-D inputs in eager mode, torch.matmul calls mm and dispatches to a CPU or CUDA implementation. With gradient recording enabled and inputs requiring gradients, autograd records dependencies for backward to apply the chain rule and accumulate leaf gradients.在即时执行模式下，稠密二维张量的 torch.matmul 调用 mm，并分派到 CPU 或 CUDA 实现。开启梯度记录且输入需要梯度时，autograd 记录依赖关系，供反向传播应用链式法则并累积叶子张量的梯度。
Warm-up reduces initial JIT compilation effects, batched timing amortizes timer overhead, and repeated samples reveal runtime variability. Performance comparisons require consistent inputs and equivalent results; medians and quartiles describe the measured runtime distribution.预热减轻首次 JIT 编译的影响，批量计时摊薄计时开销，重复采样揭示耗时波动。性能比较以输入一致、结果等价为前提，再用中位数和四分位数描述实测耗时的分布。
In a linear model, waves superpose by adding their displacements. For two sinusoidal waves with equal amplitude, frequency and wave number, phase difference sets the resultant amplitude: zero phase difference doubles it, while π produces complete cancellation.在线性模型中，波的叠加等于位移相加。两列振幅、频率和波数相同的正弦波，其合成振幅由相位差决定：同相时振幅加倍，相差 π 时完全相消。
Band storage keeps only diagonals within the lower and upper bandwidths, allowing GAXPY to skip entries outside the band. For an n×n narrow-band matrix with stored band width w, storage and arithmetic work scale as O(nw) rather than O(n²).带状存储只保留上下带宽以内的对角线，使 GAXPY 跳过带外元素。对于 n×n 窄带矩阵，若存储带宽为 w，存储量与运算量均由 O(n²) 降为 O(nw)。
A real symmetric matrix can store its lower triangle in n(n+1)/2 entries. In GAXPY, each off-diagonal entry contributes to two output components, so storage is nearly halved while arithmetic work remains O(n²).实对称矩阵只需保存下三角的 n(n+1)/2 个元素。GAXPY 中，每个非对角元素都要贡献到两个输出分量，因此存储量接近减半，运算量仍为 O(n²)。
Radix-2 FFT recursively splits a DFT into transforms over even- and odd-indexed samples. Butterfly operations reuse these results, reducing arithmetic from O(n²) to O(n log n) while computing the same discrete Fourier transform.基 2 FFT 将 DFT 递归分解为偶数位置与奇数位置样本的变换，再通过蝶形运算复用子问题的结果。它计算的仍是同一个离散傅里叶变换，运算量却从 O(n²) 降为 O(n log n)。
Loop order determines memory strides and opportunities for data reuse. Contiguous accesses use cache lines efficiently, while blocking reuses submatrices in cache to reduce data movement without changing the arithmetic work.矩阵乘法的循环顺序决定访存步长与数据复用机会。连续访问有利于利用缓存行，分块则反复使用缓存中的子矩阵，在算术量不变的情况下减少数据搬运。
Frank–Wolfe minimizes a linear approximation over a convex feasible set, then updates the iterate by a convex combination, preserving feasibility without projection. For a differentiable convex objective, the Frank–Wolfe gap bounds the remaining objective error.Frank–Wolfe 在凸可行域上求解线性化子问题，再以凸组合更新迭代点，从而无需投影即可保持可行性。对可微凸目标，Frank–Wolfe 间隙给出剩余目标误差的上界。
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