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Kai Tao

Publications and source records attributed to Kai Tao.

8 recordsLinked to original sources
arXiv2026

Potential Applications of HBF in LLM Serving Systems

LLM serving is increasingly constrained by memory capacity as model weights, KV caches, and the number of served model variants continue to grow. This report examines High-Bandwidth Flash (HBF) as a capacity-oriented extension to HBM-based serving systems. We first discuss how HBF can be integrated into the GPU memory hierarchy without undermining the bandwidth expected by the compute die. We then model the system-level value of added capacity as expanded residency for read-mostly model-state objects. Under this view, HBF can improve MoE serving by enabling more expert replicas and can improve multi-model serving by reducing model loading and supporting hot-model replication. Our simulation results show that these benefits depend on preserving the HBM-resident execution path while using HBF to expand the resident set of model weights.

cs.AR↗
arXiv2025

Discontinuity of Lyapunov exponent in spaces of quasiperiodic cocycles: Smoothness vs Arithmetic

We construct examples of discontinuity of Lyapunov exponent in the spaces of quasiperiodic $\mathrm{SL}(2,\mathbb R)$-cocycles for fixed irrational frequencies. Especially, we prove that the Gevrey space $G^2$ is the transition space of continuity for all strong Diophantine frequencies. We also construct examples of discontinuity for other frequencies in less smooth spaces, which show that the more difficult it is to approximate the frequency with rational numbers, the more likely it is to exhibit discontinuity in smoother spaces.

math.DS↗
arXiv2019

Large Deviation theorems for Dirichlet determinants of analytic quasi-periodic Jacobi operators with Brjuno-Rüssmann frequency

In this paper, we first study the strong Birkhoff Ergodic Theorem for subharmonic functions with the Brjuno-Rüssmann shift on the Torus. Then, we apply it to prove the large deviation theorems for the finite scale Dirichlet determinants of quasi-periodic analytic Jacobi operators with this frequency. It shows that the Brjuno-Rüssmann function, which reflects the irrationality of the frequency, plays the key role in these theorems via the smallest deviation. At last, as an application, we obtain a distribution of the eigenvalues of the Jacobi operators with Dirichlet boundary conditions, which also depends on the smallest deviation, essentially on the irrationality of the frequency.

math.DS↗
arXiv2018

Strong Birkhoff Ergodic Theorem for subharmonic functions with irrational shift and its application to analytic quasi-periodic cocycles

In this paper, we first prove the strong Birkhoff Ergodic Theorem for subharmonic functions with the irrational shift on the Torus. Then, it is applied to the analytic quasi-periodic Jacobi cocycles. We show that if the Lyapunov exponent of these cocycles is positive at one point, then it is positive on an interval centered at this point for suitable frequency and coupling numbers. We also prove that the Lyapunov exponent is Hölder continuous in $E$ on this interval and calculate the expression of its length. What's more, if the coupling number of the potential is large, then the Lyapunov exponent is always positive for all irrational frequencies and Hölder continuous in $E$ for all finite Liouville frequencies. We also study the Lyapunov exponent of the Schrödinger cocycles, a special case of the Jacobi ones, and obtain its Hölder continuity in the frequency.

math.DS↗
arXiv2018

Non-perturbative positive Lyapunov exponent of Schrödinger equations and its applications to skew-shift

We first study the discrete Schrödinger equations with analytic potentials given by a class of transformations. It is shown that if the coupling number is large, then its logarithm equals approximately to the Lyapunov exponents. When the transformation becomes the skew-shift, we prove that the Lyapunov exponent is week Hölder continuous, and the spectrum satisfies Anderson Localization and contains large intervals. Moreover, all of these conclusions are non-perturbative.

math.DS↗
arXiv2015

Hölder continuity of the integrated density of states for quasi-periodic Jacobi operators

We show Hölder continuity for the integrated density of states of a quasi-periodic Jacobi operator with analytic coefficients, in the regime of positive Lyapunov exponent and with a strong Diophantine condition on the frequency. In particular, when the coefficients are trigonometric polynomials we express the Hölder exponent in terms of the degrees of the coefficients.

math.SP↗
arXiv2011

Hölder continuity of Lyapunov exponent for quasi-periodic Jacobi operators

We consider the quasi-periodic Jacobi operator $H_{x,ω}$ in $l^2(\mathbb{Z})$ $(H_{x,ω}ϕ)(n) = -b(x+(n+1)ω)ϕ(n+1) - b(x+nω)ϕ(n-1) + a(x+nω)ϕ(n) = Eϕ(n),\ n\in\mathbb{Z},$ where $a(x),\ b(x)$ are analytic function on $\mathbb{T}$, $b$ is not identically zero, and $ω$ obeys some strong Diophantine condition. We consider the corresponding unimodular cocycle. We prove that if the Lyapunov exponent $L(E)$ of the cocycle is positive for some $E=E_0$, then there exists $ρ_0=ρ_0(a,b,ω,E_0)$, $β=β(a,b,ω)$ such that $|L(E)-L(E')|<|E-E'|^β$ for any $E,E'\in (E_0-ρ_0,E_0+ρ_0)$. If $L(E)>0$ for all $E$ in some compact interval $I$ then $L(E)$ is Hölder continuous on $I$ with a Hölder exponent $β=β(a,b,ω,I)$. In our derivation we follow the refined version of the Goldstein-Schlag method \cite{GS} developed by Bourgain and Jitomirskaya \cite{BJ}.

math.DS↗