arXiv subjects
Publications and source records attributed to Ziran Liu.
Let $X_1,X_2,\ldots$ be i.i.d. finitely supported random variables in a torsion-free abelian group, and write $S_k=X_1+\cdots+X_k$, and $H(S_k)$ is the Shannon entropy $S_k$, for all $k \ge 1$. We prove that, for every fixed $n\geq1$, \[ H(S_{n+1})-H(S_n) \geq \frac12\log\frac{n+1}{n} -o_{H(X_1)\to\infty}(1), \] uniformly over the ambient group and the input law. This proves a conjecture of Tao [29] in 2010.
Merging finetuned models combines specialized capabilities without joint training or access to the original data. Most methods operate by linear arithmetic in Euclidean weight space, which cannot carry the geometry of the update. Orthogonal Model Merging (OrthoMerge) uses a single orthogonal transform for each weight matrix, but such a transform cannot change singular values. We propose CORAM, which partitions each target matrix into row slices, represents every expert slice by its singular value decomposition in the corresponding base-model SVD frame, and merges the task-specific factors on their corresponding manifolds. Because manifold averaging contracts the merged update, CORAM applies an amplification coefficient $\lambda=\kappa\hat{c}$. The scale c_hat is estimated from the expert and merged update norms and is approximately $\sqrt{N}$ for $N$ experts with comparable update magnitudes. The restoration strength kappa is selected from the dispersion of expert updates without evaluating candidate merged models. This rule remains within 0.72 points of the best swept value on all evaluated suites. CORAM also includes spread slicing to distribute highly updated rows across slices and a residual pathway for non-target layers. Across four suites covering three model families, 3B to 9B scales, and language and vision-language experts, CORAM improves over OrthoMerge by 0.25 to 1.35 points and matches or exceeds the strongest weight-space baselines.
Atiyah's Minkowski-space version of the configuration-of-points construction assigns to an admissible marked configuration of $n$ worldlines a collection of $n$ binary forms of degree $n-1$, whose roots are the ordered retarded celestial directions. He conjectured that these forms are always linearly independent. We disprove this conjecture for every $n\ge3$. For $n=3$, an explicit planar one-parameter family yields a real coefficient determinant with exactly one simple zero in a specified interval. At this parameter, all six ordered celestial roots are distinct and the coefficient matrix has rank exactly two. A null-translation construction then multiplies the first three forms by a common factor and produces counterexamples for every $n>3$. Consequently, within the class of complete pairwise disjoint timelike affine lines, universal independence holds at $n=2$ and fails for every $n\ge3$; for each of the counterexamples, the normalized Atiyah--Sutcliffe determinant is defined and vanishes.
Parameter-Efficient Fine-Tuning (PEFT) commonly adapts pretrained weights through low-rank updates, and recent methods further exploit the singular value decomposition (SVD) of the base weight for initialization or subspace selection. However, these methods do not explicitly preserve the coupled geometry between the pretrained left and right singular bases. Motivated by recent minimum-perturbation theory, which shows that stable finetuning follows a coherent SVD rotation in which a single orthogonal $Q$ acts on both the left singular basis $U_0$ and the right singular basis $V_0$, we prove a per-slice analogue: each row slice of $W_0$ can be adapted by a shared orthogonal rotation $Q_i$ on its left basis $U_i$ and right basis $V_i$ together with a diagonal spectrum shift. We implement this form as CORA (Coherent Orthogonal Rotation Adaptation), which applies per-slice orthogonal rotations and a per-layer diagonal scale to the rank-$r$ SVD truncation of $W_0$. CORA uses $\tfrac{1}{2}m(r{-}1)$ trainable parameters per linear layer, about $4{\times}$ fewer than LoRA at the same rank. CORA outperforms LoRA, DoRA, PiSSA, and MiLoRA on commonsense reasoning and code generation while using about $8{\times}$ fewer parameters.
This paper studies the problem of discrepancy estimates for pseudorandom vectors constructed by the elliptic curve congruential generator, particularly in the non-translational case. Two families of results are obtained. First, in a full-coset regime characterized by a relative maximal period condition (RMPC) on an induced one-dimensional linear congruential generator, one proves bounds of type $q^{1/2}/t$ for the discrepancy $D$, the serial discrepancy $D_s$, and, under the corresponding derived RMPC, the non-overlapping discrepancy $\widetilde D_s$. Second, in the general sub-period regime, one reduces bounds for $D$, $D_s$, and $\widetilde D_s$ to estimation of Fourier $\ell^1$ masses of admissible index sets attached to one-dimensional linear congruential generators. This isolates the arithmetic bottleneck for further improvement.
For $\varepsilon>0$, let $\phi^\varepsilon$ be the solution of the ergodic problem \[ \frac12 |D\phi^\varepsilon|^2+F(x)-\varepsilon\Delta\phi^\varepsilon=c(\varepsilon) \qquad \text{on } \mathbb{T}^n, \] normalized by $\phi^\varepsilon(0)=0$. We construct a one-dimensional example with $F\in C^3$ for which the vanishing-viscosity limit $\lim_{\varepsilon\to0}\phi^\varepsilon$ does not exist. This gives a negative answer to a problem proposed by Jauslin, Kreiss, and Moser [10].
Deep residual architectures are modeled as products of near-identity Jacobians. This paper proves deterministic quotient-geometric estimates for singular spectra of Frobenius-normalized layer factors, emphasizing a normalized top-radial Cartan coordinate and fitted power-law chart. Full-rank factors are mapped from $\mathrm{GL}(d)$ to the positive cone by $A\mapsto A^\top A$, then to ordered eigenvalue data. Under Frobenius normalization, exact power-law spectra form a trace-normalized Cartan orbit. This orbit is a Gibbs family on ranks, a Fisher information line, and a Bures--Wasserstein curve with line element $d/4$ times Fisher information. The main rigidity theorem is a slack-aware margin inequality: interface radial amplitude, non-backtracking slack, and signed residual variation control displacement of the fitted Cartan coordinate. In the exact-chart zero-slack case, a depth-$L$ budget gives exponent drift of order $(\log M)/L$; generally, slack and residual increments augment the bound. We separate scalar top-radial from full-Cartan spectral control, which also needs Bures/Hellinger residual variation. We prove approximate-power-law and metric-chart versions, converse lower bounds, Fisher--KL/Bures action estimates, and near-identity expansions for normalized residual chains. Near-identity results verify transport budgets; chart quality remains measurable. Effective rank is a spectral-energy quantile, giving finite-width power-law tail bounds and robust rank-window transition estimates. Empirical static-weight exponent profiles serve as diagnostics; full verification also requires interface budgets, slacks, and residuals for the same operator chain.
This paper develops the angular and static-channel component of Geometric and Spectral Alignment for residual Jacobian chains. Starting from Cartan-coordinate rigidity and fitted effective-rank windows, we study how dominant singular subspaces are transported across adjacent layers and how the resulting finite matrices can be displayed in physical channel coordinates. The main results are deterministic, margin-verified results. We bound the error between full interface transport and its dominant-window truncation, add fitted-tail errors so that empirical spectra can be certified against the Gibbs--Cartan tail model, and distinguish source-mode incidence from fully physical input-output channel incidence. Given row groups and active supports, the Physical Alignment Matrix decomposes orthogonally as core plus overlap plus noise. Active-column gaps, pairwise overlap margins, and noise bounds combine into a static certificate radius under which the full transport and the truncated transport induce the same active supports, pairwise incidence graph, SRS sets, hub columns, and core/overlap/noise masks. The finer SC/SA/ST labels of the Invariant Channel Mapping require additional row-energy and profile-correlation margins, stated as explicit perturbation tests. The empirical section reports the matrices and block-energy heatmaps that measure these certificate quantities across CNNs, language models, and vision/diffusion backbones. The figures are interpreted as finite-dimensional measurements; complete membership in the Physical GSA certificate domain requires checking the numerical margin protocol stated in Section 10.
We study the periodic homogenization of the viscous Hamilton--Jacobi equation \[ u_t^\varepsilon + \frac{1}{2}|Du^\varepsilon|^2 + V\!\left(\frac{x}{\varepsilon}\right) = \frac{\varepsilon}{2}\Delta u^\varepsilon \qquad \text{in } \mathbb{R}^n \times (0,\infty), \] with initial datum $g \in W^{1,\infty}(\mathbb{R}^n)$, where $V$ is Lipschitz continuous and $\mathbb{Z}^n$-periodic. We prove the sharp global estimate \[ |u^\varepsilon(x,t)-u(x,t)| \leq \varepsilon\!\left(C+\frac{n}{2}\log\!\left(\frac{\max\{t,\varepsilon\}}{\varepsilon}\right)\right) \qquad \text{for all } (x,t)\in \mathbb{R}^n \times [0,\infty), \] where $\varepsilon \in (0,1]$, $u$ solves the limiting (homogenized) equation and $C>0$ is a constant depending only on $\|Dg\|_{L^\infty(\mathbb{R}^n)}$, $\|DV\|_{L^\infty(\mathbb{R}^n)}$, and $n$. We further show that if $g$ is locally semiconcave, then \[|u^\varepsilon(x,t)-u(x,t)| \leq C_{x,t}\varepsilon \qquad \text{for a.e. } (x,t)\in \mathbb{R}^n \times (0,\infty),\] where $C_{x,t}$ depends on $(x,t)$, $\|Dg\|_{L^\infty(\mathbb{R}^n)}$, and $\|DV\|_{L^\infty(\mathbb{R}^n)}$. More precisely, the above improved rate holds at every point $(x,t)$ where $u(\cdot,t)$ is twice differentiable at $x$. In particular, this occurs for a.e. $x\in \mathbb{R}^n$, since $u(\cdot,t)$ is locally semiconcave. We conclude by raising the open problem of whether the same $O(\varepsilon |\log \varepsilon|)$ rate remains valid for general strictly convex Hamiltonians or general periodic diffusions.
Internal noise in deep networks is usually inherited from heuristics such as dropout, hard masking, or additive perturbation. We ask two questions: what correlation geometry should internal noise have, and is the implemented perturbation compatible with the representations it acts on? We answer these questions through Variational Kernel Design (VKD), a framework in which a noise mechanism is specified by a law family, a correlation kernel, and an injection operator, and is derived from learning desiderata. In a solved spatial subfamily, a quadratic maximum-entropy principle over latent log-fields yields a Gaussian optimizer with precision given by the Dirichlet Laplacian, so the induced geometry is the Dirichlet Green kernel. Wick normalization then gives a canonical positive mean-one gate, Gaussian Chaos Noise (GCh). For the sample-wise gate used in practice, we prove exact Gaussian control of pairwise log-ratio deformation, margin-sensitive ranking stability, and an exact expected intrinsic roughness budget; hard binary masks instead induce singular or coherence-amplified distortions on positive coherent representations. On ImageNet and ImageNet-C, GCh consistently improves calibration and under shift also improves NLL at competitive accuracy.
Improving the high-temperature performance and low-temperature plasticity of tantalum (Ta) alloys is a significant scientific challenge. We employed first-principles calculations to study the interaction between screw dislocations and solute atoms in the body centered cubic (BCC) structure of Ta, with a particular focus on solid solution softening and strengthening. We analyzed the impact of various solute elements on the generalized stacking fault energy (GSFE), energy barriers within the single-atom column displacement model, and their interaction with screw dislocations. The results indicate that Hf and Zr, either individually or in combination, exhibit notable solute softening effects in BCC Ta, significantly reducing GSFE, energy barriers, and interaction energies. In contrast, Nb shows relative insensitivity to solute effects, while Mo, W, and Ir demonstrate solute strengthening effects. The calculations suggest that the interaction energy between screw dislocations and solute atoms is a reliable indicator for predicting strengthening and softening effects. Additionally, we extend these predictions to ternary alloys, demonstrating that the strengthening and softening phenomena in these materials can be explained through the electronic work function at the electronic level.
In body-centered cubic (bcc) metals such as molybdenum, screw dislocations often exhibit non-Schmid behavior, moving in directions unpredicted by the Schmid law. The mobility of these dislocations is notably influenced by the presence of solute atoms within the alloy matrix. In this study, employing first-principles calculations, we delve into the electronic origins of these influences.Initially, we construct both single atomic column and triple atomic column models to simulate the formation of screw dislocations with solute atoms. Our investigation reveals that tantalum (Ta) and tungsten (W) increase the formation energy of solute-dislocation complexes, in contrast to osmium (Os), iridium (Ir), and platinum (Pt). Subsequently, employing a comprehensive screw dislocation dipole model under shear deformation, we explore the combined effects of solute atoms and deformation on dislocation core movement. Our findings demonstrate that Ta and W, positioned as first nearest neighbors, reduce the stress required to move dislocation cores away from corresponding dislocation dipoles. Conversely, Os, Ir, and Pt exhibit an attractive effect on dislocation cores, lowering the energy barrier for screw dislocation formation and enticing dislocation cores towards these solute atoms.
Task and Motion Planning (TAMP) algorithms can generate plans that combine logic and motion aspects for robots. However, these plans are sensitive to interference and control errors. To make TAMP more applicable in real-world, we propose the modular multi-level replanning TAMP framework(MMRF), blending the probabilistic completeness of sampling-based TAMP algorithm with the robustness of reactive replanning. MMRF generates an nominal plan from the initial state, then dynamically reconstructs this nominal plan in real-time, reorders robot manipulations. Following the logic-level adjustment, GMRF will try to replan a new motion path to ensure the updated plan is feasible at the motion level. Finally, we conducted real-world experiments involving stack and rearrange task domains. The result demonstrate MMRF's ability to swiftly complete tasks in scenarios with varying degrees of interference.
In flat-band materials, the strong Coulomb interaction between electrons can lead to exotic physical phenomena. Recently, $α$-In$_2$Se$_3$ thin films were found to possess ferroelectricity and flat bands. In this work, using first-principles calculations, we find that for the monolayer, there is a Weyl point at $Γ$ in the flat band, where the inclusion of the spin-orbit coupling opens a gap. Shifting the Fermi level into the spin-orbit gap gives rise to nontrivial band topology, which is preserved for the bilayer regardless of the interlayer polarization couplings. We further calculate the Chern number and edge states for both the monolayer and bilayer, for which the results suggest that they become quantum anomalous Hall insulators under appropriate dopings. Moreover, we find that the doping-induced magnetism for In$_2$Se$_3$ bilayer is strongly dependent on the interlayer polarization coupling. Therefore, doping the flat bands in In$_2$Se$_3$ bilayer can also yield multiferroicity, where the magnetism is electrically tunable as the system transforms between different polarization states. Our study thus reveals that multiferroicity and nontrivial band topology can be unified into one material for designing multifunctional electronic devices.
We propose a discrete approach for approximating solutions to the prescribed Gaussian curvature problem in two-dimensional manifolds, based on the notion of discrete conformality. Our approach provides an efficient numerical method to compute the solution by minimizing a convex functional.
We study reflecting Brownian motion with drift constrained to a wedge in the plane. Our first set of results provide necessary and sufficient conditions for existence and uniqueness of a solution to the corresponding submartingale problem with drift, and show that its solution possesses the Markov and Feller properties. Next, we study a version of the problem with absorption at the vertex of the wedge. In this case, we provide a condition for existence and uniqueness of a solution to the problem and some results on the probability of the vertex being reached.
Two-dimensional (2D) Janus transition metal dichalcogenides (TMDs) are a new class of materials with unique physical properties. However, the carrier mobility of most Janus TMDs calculated by deformation potential theory (DPT) is not reliable due to the unconsidered part of lattice scattering. In this work, we propose a new method of Born effective charge (BEC) to calculate the carrier mobility of Janus TMDs by including the important factors that neglected in the DPT. The BEC could be used in the calculation of both pure and defective Janus TMDs by employing density functional perturbation theory. We have figured out the relationship between the carrier mobility and the value of BEC, which is the lower the absolute BEC, the higher the electron or hole mobility. Using the new method, we have calculated the carrier mobility of commonly studied Janus TMDs with and without defect. The method may shed light on the high-throughout calculation of selecting high carrier mobility 2D materials.
Active control and manipulation of electromagnetic waves are highly desirable for advanced photonic device technology, such as optical cloaking, active camouflage and information processing. Designing optical resonators with high ease-of-control and reconfigurability remains a open challenge thus far. Here we propose a novel mechanism to continuously reconfigure an optical resonator between one-port and two-port configurations via \emph{phase-change material} for efficient optical modulation. By incorporating a phase-change material VO$_2$ substrate into a photonic crystal optical resonator, we computationally show that the system behaves as a one-port device with near-perfect absorption and two-port device with high transmission up to 92% when VO$_2$ is in the metallic rutile phase and insulating monoclinic phase, respectively. The optical response can be continuously and reversibly modulated between various intermediate states. More importantly, the proposed device is compatible with wide-angle operation and is robust against structural distortion. Our findings reveal a novel device architecture of \emph{port reconfigurable} optical resonator uniquely enabled by switchable optical properties of phase change material.