把好奇心写进星光里https://verapyuyi.github.ioenhttps://verapyuyi.github.io/en/papers/horizon-uniform-sensitivitypaper:horizon-uniform-sensitivityWe study local stationary solutions of finite-horizon discrete-time Pontryagin systems near a steady extremal. Under regular stationarity, hyperbolicity of the reduced state-costate map, and scaled transversality of the endpoint conditions, the linearized boundary-value problem has a horizon-uniform Green estimate. A weighted-norm contraction argument gives existence, uniqueness, uniform Lipschitz estimates, and pointwise quadratic remainders. For graph boundary conditions, terminal reward perturbations have exponentially small effects on the initial control and the stationary objective gradient; the linear-quadratic case yields exponential convergence of Riccati matrices and initial feedback gains.Mon, 17 Aug 2026 00:00:00 GMT<p>We study local stationary solutions of finite-horizon discrete-time Pontryagin systems near a steady extremal. Under regular stationarity, hyperbolicity of the reduced state-costate map, and scaled transversality of the endpoint conditions, the linearized boundary-value problem has a horizon-uniform Green estimate. A weighted-norm contraction argument gives existence, uniqueness, uniform Lipschitz estimates, and pointwise quadratic remainders. For graph boundary conditions, terminal reward perturbations have exponentially small effects on the initial control and the stationary objective gradient; the linear-quadratic case yields exponential convergence of Riccati matrices and initial feedback gains.</p><p><a href="/en/papers/horizon-uniform-sensitivity">Read HTML and download PDF</a></p>https://verapyuyi.github.io/en/papers/cycle-decorated-ribbon-complexespaper:cycle-decorated-ribbon-complexesWe define a two-variable specialization of the ribbon basis of noncommutative symmetric functions from cycle enumerators of an ordinary permutation and a rooted permutation, with reflection length recorded by the second variable. Its factorial multiple is realized as the shifted bigraded Euler characteristic of an equivariant ordered-set-partition complex. Total decorations determine simultaneous factorization cuts and classical ribbon-complex fibers, yielding explicit nonnegative ribbon expansions. A compatible cut coproduct is constructed, and the alternating-fence specialization is shown to have nonnegative coefficients together with controlled homological support and explicit defect-zero formulas.Sat, 15 Aug 2026 00:00:00 GMT<p>We define a two-variable specialization of the ribbon basis of noncommutative symmetric functions from cycle enumerators of an ordinary permutation and a rooted permutation, with reflection length recorded by the second variable. Its factorial multiple is realized as the shifted bigraded Euler characteristic of an equivariant ordered-set-partition complex. Total decorations determine simultaneous factorization cuts and classical ribbon-complex fibers, yielding explicit nonnegative ribbon expansions. A compatible cut coproduct is constructed, and the alternating-fence specialization is shown to have nonnegative coefficients together with controlled homological support and explicit defect-zero formulas.</p><p><a href="/en/papers/cycle-decorated-ribbon-complexes">Read HTML and download PDF</a></p>https://verapyuyi.github.io/en/papers/bernstein-transfers-greedy-recordspaper:bernstein-transfers-greedy-recordsFor the fence poset associated with an orientation of a path, we define a greedy right-to-left record statistic on the symmetric group and prove that its generating function equals the factorial-scaled order polynomial. The proof uses a Bernstein-basis transfer between a continuous threshold recurrence and endpoint-refined order-preserving maps. A finite transfer gives a direct recursive bijection, while refinements by record set, direction, and terminal value identify fibers with decorated endpoint paths and pointed linear extensions of record posets. A cyclic record statistic gives the corresponding formula for every nonconstant orientation of a cycle and resolves the circular-fence conjecture discussed in the paper.Fri, 31 Jul 2026 00:00:00 GMT<p>For the fence poset associated with an orientation of a path, we define a greedy right-to-left record statistic on the symmetric group and prove that its generating function equals the factorial-scaled order polynomial. The proof uses a Bernstein-basis transfer between a continuous threshold recurrence and endpoint-refined order-preserving maps. A finite transfer gives a direct recursive bijection, while refinements by record set, direction, and terminal value identify fibers with decorated endpoint paths and pointed linear extensions of record posets. A cyclic record statistic gives the corresponding formula for every nonconstant orientation of a cycle and resolves the circular-fence conjecture discussed in the paper.</p><p><a href="/en/papers/bernstein-transfers-greedy-records">Read HTML and download PDF</a></p>https://verapyuyi.github.io/en/blogs/monte-carlo-control-natural-startsblog:monte-carlo-control-natural-startsA route from Basic MC, Exploring Starts, and ε-greedy control to initial distributions, biased random walks, hitting probabilities, and natural starts.Sun, 30 Aug 2026 07:48:15 GMT<p>A route from Basic MC, Exploring Starts, and ε-greedy control to initial distributions, biased random walks, hitting probabilities, and natural starts.</p><p><a href="/en/blogs/monte-carlo-control-natural-starts">Read the essay</a></p>Inspirationshttps://verapyuyi.github.io/en/blogs/attention-as-spiking-dynamicsblog:attention-as-spiking-dynamicsSeparating attention modules from attention functions, then exploring a constrained route through excitatory–inhibitory competition, spike measures, and causal operators.Fri, 28 Aug 2026 05:07:34 GMT<p>Separating attention modules from attention functions, then exploring a constrained route through excitatory–inhibitory competition, spike measures, and causal operators.</p><p><a href="/en/blogs/attention-as-spiking-dynamics">Read the essay</a></p>Inspirationshttps://verapyuyi.github.io/en/blogs/llm-sampling-energy-viewblog:llm-sampling-energy-viewSeparating data, token, and reasoning-trajectory sampling in large models, and asking what a Gibbs energy view can genuinely contribute.Tue, 25 Aug 2026 17:58:33 GMT<p>Separating data, token, and reasoning-trajectory sampling in large models, and asking what a Gibbs energy view can genuinely contribute.</p><p><a href="/en/blogs/llm-sampling-energy-view">Read the essay</a></p>Inspirationshttps://verapyuyi.github.io/en/blogs/sgd-zigzag-posetsblog:sgd-zigzag-posetsProjecting stochastic-gradient trajectories into ordinal patterns, and examining what fence posets may reveal—and fail to reveal—about local optimization dynamics.Sun, 23 Aug 2026 18:37:52 GMT<p>Projecting stochastic-gradient trajectories into ordinal patterns, and examining what fence posets may reveal—and fail to reveal—about local optimization dynamics.</p><p><a href="/en/blogs/sgd-zigzag-posets">Read the essay</a></p>Inspirationshttps://verapyuyi.github.io/en/blogs/workflow-to-agent-harnessblog:workflow-to-agent-harnessAs models learn to plan and use tools dynamically, workflows do not disappear; they become executable, recoverable, and verifiable control structures inside a harness.Thu, 20 Aug 2026 17:35:36 GMT<p>As models learn to plan and use tools dynamically, workflows do not disappear; they become executable, recoverable, and verifiable control structures inside a harness.</p><p><a href="/en/blogs/workflow-to-agent-harness">Read the essay</a></p>Inspirations