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Θ′exp NℓN(θ⋆ i) +g⋆ i,N⊤θ′+θ′⊤H⋆ Nθ′+O 1√ N p′(θ′)dθ′O1 N , =−R Θ′θ′⊤H⋆ iθ′exp g⋆ i,N⊤θ′+θ′⊤H⋆ Nθ′+O 1√ N p′(θ′)dθ′ R Θ′exp Ng⋆ i,N⊤θ′+θ′⊤H⋆ Nθ′+O 1√ N p′(θ′)dθ′O1 N , =−R Θ′θ′⊤H⋆ iθ′exp g⋆ i,N⊤θ′+θ′⊤H⋆ Nθ′ exp O 1√ N p′(θ′)dθ′ R Θ′exp Ng⋆ i,N⊤θ′+θ′⊤H⋆ Nθ′ exp O 1√ N p′(θ′)dθ′O1 N , =O −... | https://arxiv.org/abs/2505.22442v1 |
in posterior probability as Ngrows large. 30 Lemma 4. Under Assumption 1, EDN∼P⋆ Data P(¯Θ|DN) =O(exp(−N)). Proof. We start by splitting the posterior expectation into integrals over ¯ΘandΘ\¯Θ: P(¯Θ|DN) =R ¯Θexp (NℓN(θ))p(θ)dθR Θexp (NℓN(θ))p(θ)dθ, =R ¯Θexp (NℓN(θ))p(θ)dθR ¯Θexp (NℓN(θ))p(θ)dθ+R Θ\¯Θexp (NℓN(θ))p(θ)d... | https://arxiv.org/abs/2505.22442v1 |
Using Eq. (23), we now re-write each integral in the summation term of Eq. (24) as: Z Θ⋆ i(ℓ⋆−ℓ(θ))p(θ|DN)dθ=R Θ⋆ i(ℓ⋆−ℓ(θ)) exp ( NℓN(θ))p(θ)dθ R Θexp (NℓN(θ))p(θ)dθ, =R Θ⋆ i(ℓ⋆−ℓ(θ)) exp ( NℓN(θ))p(θ)dθ R Θexp (NℓN(θ))p(θ)dθ·R Θ⋆ iexp (NℓN(θ))p(θ)dθ R Θ⋆ iexp (NℓN(θ))p(θ)dθ, =R Θ⋆ i(ℓ⋆−ℓ(θ)) exp ( NℓN(θ))p(θ)dθ R Θ⋆ ... | https://arxiv.org/abs/2505.22442v1 |
required. 34 E Further Results E.1 SOReL Figure 6: SOReL BAMDP hyperparameter sweeps (tuning set ϕIII) for 200,000 randomly sampled transi- tions of the brax datasets. The plots correspond to ϕIII←arg minϕIIIRegretMetric (ϕI, ϕII, ϕIII,DN)in Algorithm 1. SOReL selects the BAMDP hyperparameters that yield the lowest app... | https://arxiv.org/abs/2505.22442v1 |
0.400 0 .134 0 .331 0 .418 Oracle 0.469 0 .000 0 .116 0 .187 TOReL 0.459 0.036 0.339 0.227 d4rl- hopper- medium-v2True 0.411 0 .467 0 .848 0 .595 Oracle 0.375 0 .053 0 .681 0 .183 TOReL 0.428 0.083 0.681 0.327 d4rl- walker2d- medium-replay-v2True 0.450 0 .625 0 .952 1 .000 Oracle 0.339 0 .204 0 .724 1 .000 TOReL 0.358 ... | https://arxiv.org/abs/2505.22442v1 |
The Gaussian reward and state transition models then have the form: PR(s, a, θ ) =N(rθ(s, a), σ2 r,θ(s, a)), P ∆(s, a, θ ) =N(∆θ(s, a), σ2 ∆,θ(s, a)), with mean reward function rθ(s, a)and mean state transition function ∆θ(s, a), as before. Let r(s, a,DN):=Eθ∼PΘ(DN)[rθ(s, a)]and∆(s, a,DN):=Eθ∼PΘ(DN)[∆θ(s, a)]denote the... | https://arxiv.org/abs/2505.22442v1 |
F.3 BAMDP Solver We use the RNN-PPO implementation of Lu et al. [46], which we amend to be compatible with continuous action spaces. We sweep over the hyperparameters given in Table 5. Hyperparameter Value / Sweep Values Learning rate [0.0001, 0.0003] Anneal learning rate True Number of environments [4, 64, 128, 256, 5... | https://arxiv.org/abs/2505.22442v1 |
ratio 0.01 Threshold coefficient [0, 5, 10, 15, 20, 25] Termination penalty offset [-30, -50, -100, -200] Table 7: Hyperparameters and sweep ranges for IQL, ReBRAC, MOPO, and MOReL. For the sample efficiency experiments in Fig. 5, we use the default UCB bandit-based hyperparameter- tuning algorithm hyperparameters. We ... | https://arxiv.org/abs/2505.22442v1 |
NFR: Neural Feature-Guided Non-Rigid Shape Registration Puhua Jiang1,2†, Zhangquan Chen1†, Mingze Sun1†, Ruqi Huang1* 1*Tsinghua Shenzhen International Graduate School, Shenzhen, China. 2Pengcheng Laboratory, Shenzhen, China. *Corresponding author(s). E-mail(s): ruqihuang@sz.tsinghua.edu.cn; Contributing authors: jph21... | https://arxiv.org/abs/2505.22445v1 |
synergistically combines the strengths of clas- sic shape registration and learning-based embed- ding techniques. In a nutshell, we leverage the estimated correspondence from the latter to guide shape deformation via the former iteratively. Our key insight is to enforce similarity between the deformed source and target... | https://arxiv.org/abs/2505.22445v1 |
respect to TpandT. As shown in Prop. 1 in Sec. 3, under certain conditions, we prove that the transformation between the spec- tral embedding of Sand that of Tpcan always be set as the functional map from StoT,regardless of how Tpis generated. The above procedure leads to a simple yet effective training scheme. Namely,... | https://arxiv.org/abs/2505.22445v1 |
tasks. 2 Related Works Non-rigid Shape Registration Non-rigidly aligning shapes presents greater com- plexity compared to the rigid counterpart due to the intricate nature of deformation models. In gen- eral, axiomatic approaches [21] assume the defor- mation of interest can be approximated by local, small-to-moderate,... | https://arxiv.org/abs/2505.22445v1 |
tion between the respective spectral spaces. This approach effectively transforms the problem of finding point-wise correspondences into a simpler task of aligning spectral coefficients. Originating from the foundational work on functional maps, along with a series of follow-ups [32–36], spec- tral methods have made si... | https://arxiv.org/abs/2505.22445v1 |
shape registration between Sand the target point clouds, and finally compute the map by composition T12=Ts2◦T1s. Specifically, the non-rigid registration approach developed to tackle the problem of full-2-full is named DFR , and the methodology designed for partial-2-full problem is named Partial-DFR . 3.1 DFR Our DFR ... | https://arxiv.org/abs/2505.22445v1 |
shown in Alg. 1 3.1.4 Two-stage Registration Finally, we observe that solely depending on learned features to infer correspondence is sub- optimal. At the converging point, the deformed source shape is often at the right pose but has defi- ciencies in shape style. To this end, we perform a second-stage registration, ba... | https://arxiv.org/abs/2505.22445v1 |
contrast, we propose to consider the func- tional maps encoded in the spatially truncated spectral embedding forTp. Specifically, we set ΦTp= ΠTpTΦT (4) Proposition 1. LetS,Tbe a pair of shapes each having non-repeating Laplacian eigenvalues, which are the same ( i.e., ∆ S= ∆T), and ΠT S be an isometry between TandS.ΦS... | https://arxiv.org/abs/2505.22445v1 |
Φ S, regardless of how Tpis generated. While it seems natural to formulate the following losspa L(F) = arg min F∥ΦTpCST−˜ΠTpSΦS∥2 Fro, we notice that optimizing the above loss requires to compute Frobenius norm of a matrix of vary- ing dimension, which depends on the number of points in Tp. To alleviate such implementa... | https://arxiv.org/abs/2505.22445v1 |
state-of-the-art techniques for estimating corre- spondences between deformable shapes on an array of benchmarks as follows: FAUST r:The remeshed version of FAUST dataset [51], which consists of 100 human shapes (10 individuals performing the same 10 actions). We split the first 80 as training shapes and the rest as te... | https://arxiv.org/abs/2505.22445v1 |
associat- ing them via a common source mesh S. In Tab. 1 and Tab. 2, we indicate our choice of source mesh by Ours- name , where name indicates the origin of the source mesh. For simplicity, we fix the source mesh from each dataset and visualize them in the appendix. On the other hand, when imple- menting axiomatic sha... | https://arxiv.org/abs/2505.22445v1 |
performs allof the baselines, including the state- of-the-art methods that take meshes as input. Regarding point-based methods, SSMSM [8] per- forms well in the standard case and outper- forms ours in FASUT r/FAUST r, but general- izes poorly to unseen shapes. Another important observation is that our method manifests ... | https://arxiv.org/abs/2505.22445v1 |
Especially, among the point-based methods, our method outperforms the latest SOTA method by a large margin, namely, over 40% relative error reduction ( 7.1vs. 12 .3). We refer readers to the appendix for a qualitative comparison, which agrees with the quantitative results above. 4.1.2 Ablation Study We report ablation ... | https://arxiv.org/abs/2505.22445v1 |
r and test on SHREC’07. w/o Registration w/o Stage I w/o Stage II w/o updating corre. w/o cons. filter Full 11.5 10.6 10.1 8.1 7.2 5.9 Table 5 Mean geodesic errors ( ×100) of Ours, NDP, AMM based on the same initial maps. SCAPE rSHREC19 rSHREC07-H Ini. 5.5 8.1 11.5 NDP 5.4 11.4 8.9 AMM 11.4 10.7 8.8 Ours 2.6 5.1 5.9 Ta... | https://arxiv.org/abs/2505.22445v1 |
on PFARM dataset. The best is highlighted. Method Geo. error( ×100) PFM(+ZO) [39] 42.34 FSP(+ZO) [64] 53.15 DOC(+ZO) [65] 51.78 GeomFMaps(+ZO) [59] 22.22 DPFM(+ZO) [17] 10.53 SSMSM [8] 12.32 Ours-S&F 7.35 Ours-M&P 8.30 Table 8 Mean geodesic errors ( ×100) on SURREAL partial view dataset from SSMSM. The best is highligh... | https://arxiv.org/abs/2505.22445v1 |
0.951 In the second set of experiment illustrated in Fig 8, apart from HCLV2S, we also compare with non-rigid shape registration methods. Similar to the last example, we keep using the template in Fig. 7 as source shape. Visual inspection clearly shows that our texture transfer outperforms all the baselines by a signif... | https://arxiv.org/abs/2505.22445v1 |
best is highlighted. CD (mm) PN-AE [67] DG-AE [13] CPAE [68] ISR [69] DPC [5] Point2SSM [70] Ours Spleen 43.7 43.5 61.3 17.6 10.6 3.4 1.7 Pancreas 22.0 21.0 18.8 7.4 6.1 2.7 1.3 Table 11 Mean geodesic errors ( ×100) on SCAPE-FAUST dataset. The best is highlighted. Method Geo. error( ×100) Ours full 2.33 Ours feature 4.... | https://arxiv.org/abs/2505.22445v1 |
Z., Sarma, S.E., Bronstein, M.M., Solomon, J.M.: Dynamic graph cnn for learning on point clouds. ACM Transactions on Graphics (TOG) (2019) [14] Li, Y., Harada, T.: Non-rigid point cloud reg- istration with neural deformation pyramid. In: NeurIPS (2022) [15] Yao, Y., Deng, B., Xu, W., Zhang, J.: Fast and robust non-rigi... | https://arxiv.org/abs/2505.22445v1 |
functional maps. ACM Trans. Graph. 37(6), 248–124816 (2018) [35] Melzi, S., Ren, J., Rodol` a, E., Wonka, P., Ovsjanikov, M.: Zoomout: Spectral upsam- pling for efficient shape correspondence. Proc. SIGGRAPH Asia (2019) [36] Huang, R., Ren, J., Wonka, P., Ovsjanikov, M.: Consistent zoomout: Efficient spectral map synch... | https://arxiv.org/abs/2505.22445v1 |
dirichlet energy opti- mization. In: 3DV (2022) [55] L¨ ahner, Z., Rodol` a, E., Bronstein, M.M., Cremers, D., Burghard, O., Cosmo, L., Dieck- mann, A., Klein, R., Sahillioglu, Y.: Shrec’16: Matching of deformable shapes with topolog- ical noise. Proc. 3DOR 2(10.2312) (2016) [56] Eisenberger, M., Lahner, Z., Cremers, D... | https://arxiv.org/abs/2505.22445v1 |
arXiv:2505.22451v1 [cs.AI] 28 May 2025AI Mathematician: Towards Fully Automated Frontier Mathematical Research Yuanhang Liu4∗, Yanxing Huang3∗, Yanqiao Wang4∗, Peng Li2†, Yang Liu1,2† 1Dept. of Comp. Sci. & Tech., Institute for AI, Tsinghua University, Beijing, China 2Institute for AI Industry Research (AIR), Tsinghua ... | https://arxiv.org/abs/2505.22451v1 |
them to undertake frontier mathematical research. In this work, we take initial steps in this promising and underexplored direction, reporting preliminary yet encouraging results to inspire future exploration. We propose an LRM-based mathematical research agent framework named AIM (AI Mathematician), with a special foc... | https://arxiv.org/abs/2505.22451v1 |
information of this research topic. This could include the definitions of terminologies, and some preliminary conclusions required for this problem. These contents will then be formatted and treated as a system prompt, which is visible to all three agents. After this, we can directly pass the research problem to the ag... | https://arxiv.org/abs/2505.22451v1 |
module consistently incorporates certain fixed essential requirements when refining proofs. The refined proof is then returned to the verifier for re-evaluation. This creates an iterative refinement- verification loop that continuously enhances proof quality. Once a proof passes verification, it is accepted as a lemma ... | https://arxiv.org/abs/2505.22451v1 |
The output syntax of OpenAI o4-mini is encoded in Unicode, which we subsequently transcribed into standard L ATEXwith DeepSeek-V3 [DeepSeek-AI et al., 2025] for better readability. The following provides a brief overview of the targeted mathematical research problems and the current progress of the proofs produced by A... | https://arxiv.org/abs/2505.22451v1 |
model by transform- ing the PDE into a heat equation and leveraging quantum algorithms. The core steps are: •PDE Transformation and Spatial Discretization : Use variable substitution to convert the BSM PDE into a standard heat equation. Then discretize the spatial variable (Lemma 1). •Operator Decomposition and Integra... | https://arxiv.org/abs/2505.22451v1 |
representation in the LCHS lemma for the BSM model can be approximated with precision ϵ using O(1/ϵ2)terms through an adaptive discretization of the k-integral, leveraging the rapid decay of the Cauchy kernel η(k) =1 π(1+k2). Proof. **Truncation Error Analysis**: The integral I=R∞ −∞η(k)u(t, k)dkis truncated to [−K, K ... | https://arxiv.org/abs/2505.22451v1 |
˜O∥L∥T ϵ3 . **Step 5: Ancilla and Gate Complexity** The LCU framework requires logM=O(log(1 /ϵ))an- cilla qubits. Gate complexity inherits ˜O(M)-scaling from the superposition state, augmented by O(poly(log(∥L∥T/ϵ)))factors from simulation subroutines. The dominant terms remain polynomial in∥L∥,T, and1/ϵ, with polylo... | https://arxiv.org/abs/2505.22451v1 |
process. 4.2.3 Detailed Analysis of Selected Proofs We explain some specific results of AIM. 9 **A priori energy estimate for the coupled system** Let uandφk(k= 1, . . . , m ) satisfy the system (2.10) withu0∈L2(Ω)andφk(0) = 0 . Then, there exists a constant C > 0depending on T, β, α k, dk, but independent of u0, such ... | https://arxiv.org/abs/2505.22451v1 |
of the solution to the equation. The agent con- siders the basis functions in the functional space of the overall space and boundary space respectively, and constructs the existence space of approximated solutions. After that, the agent constructs the form of the approximated solutions and the weak form of the satisfyi... | https://arxiv.org/abs/2505.22451v1 |
conclusions by applying the variational method. The agent first establishes energy control mechanisms through rigorous mathematical derivation and analysis for both the solution uϵand pressure term pϵof the governing equations. Building on this theoretical foundation, the analysis further develops an error control fram... | https://arxiv.org/abs/2505.22451v1 |
of the problem. However, the critical steps lack detailed derivations to substantiate the reasoning, which diminishes the transparency of the logical progression. Under the given assumptions, the difference wε=ulim−uεbetween the solutions of the limit problem (10) and the original problem (9) satisfies the homogenized ... | https://arxiv.org/abs/2505.22451v1 |
proof steps and precise key conclusions. Although certain derivational details are omitted, the underlying reasoning framework and final conclusions remain rigorously accurate and align with the problem’s requirements. And we can also find that the agent verifies the condition of theorems. This demonstrates that the ag... | https://arxiv.org/abs/2505.22451v1 |
Yf. By the scaling x=εyone deduces that on each ε-cellε(Yf+k) for all v∈H1(ε(Yf+k)),∥v∥H1(ε(Yf+k))≤C0 ∥v∥L2(ε(Yf+k))+∥sym∇v∥L2(ε(Yf+k)) . [Vague] Here the agent considers the Korn inequality but the derivation process is not detailed enough. 2. (Partition of unity subordinate to the periodic tiling.) Let {ψk}k∈Kεbe a... | https://arxiv.org/abs/2505.22451v1 |
ambiguities in microscale definitions may propagate into approximations at the macroscale. 4. Homogenized tensor and macroscopic equation By choosing in the cell system the test-function θ=χE, one derives the energy representation ChomE:E=Z Ye(E+DyχE e) :C(E+DyχE e)dy+Z Yi2µDyχE i:DyχE idy. On the other hand, testing t... | https://arxiv.org/abs/2505.22451v1 |
proper exploration and verification mechanisms, AIM already achieved impressive results in some open problems, and the same framework can be easily adopted for various research topics. However, our system still has many shortcomings in terms of reliability, that the proofs generated by AIM can not be directly accepted ... | https://arxiv.org/abs/2505.22451v1 |
Our agent can directly attempt to solve these open problems and provide core conclusions, key processes, and reasonable inferences. Mathematicians can analyze the output results obtained, try more methods and ideas, and continue to use our agents to further explore the conclusions of this open problem. 6 Future Works O... | https://arxiv.org/abs/2505.22451v1 |
D. Yang, H. Zhang, J. Song, R. Zhang, R. Xu, Q. Zhu, S. Ma, P. Wang, X. Bi, X. Zhang, X. Yu, Y . Wu, Z. F. Wu, Z. Gou, Z. Shao, Z. Li, Z. Gao, A. Liu, B. Xue, B. Wang, B. Wu, B. Feng, C. Lu, C. Zhao, C. Deng, C. Zhang, C. Ruan, D. Dai, D. Chen, D. Ji, E. Li, F. Lin, F. Dai, F. Luo, G. Hao, G. Chen, G. Li, H. Zhang, H. ... | https://arxiv.org/abs/2505.22451v1 |
H. Xu, H. Wang, H. Zhang, H. Ding, H. Xin, H. Gao, H. Li, H. Qu, J. L. Cai, J. Liang, J. Guo, J. Ni, J. Li, J. Wang, J. Chen, J. Chen, J. Yuan, J. Qiu, J. Li, J. Song, K. Dong, K. Hu, K. Gao, K. Guan, K. Huang, K. Yu, L. Wang, L. Zhang, L. Xu, L. Xia, L. Zhao, L. Wang, L. Zhang, M. Li, M. Wang, M. Zhang, M. Zhang, M. T... | https://arxiv.org/abs/2505.22451v1 |
Empirical explorations with the logic theory machine: A case study in heuristics. In Proceedings of the Western Joint Computer Conference , volume 15, pages 218–239. ACM, 1957. doi: 10.1145/1455567.1455605. OpenAI. OpenAI o3 and o4-mini system card. https://cdn.openai.com/pdf/ 2221c875-02dc-4789-800b-e7758f3722c1/o3-an... | https://arxiv.org/abs/2505.22451v1 |
T. Wang, T. Gordon, T. Sanders, T. Patwardhan, T. Sottiaux, T. Degry, T. Dimson, T. Zheng, T. Garipov, T. Stasi, T. Bansal, T. Creech, T. Peterson, T. Eloundou, V . Qi, V . Kosaraju, V . Monaco, V . Pong, V . Fomenko, W. Zheng, W. Zhou, W. McCabe, W. Zaremba, Y . Dubois, Y . Lu, Y . Chen, Y . Cha, Y . Bai, Y . He, Y . ... | https://arxiv.org/abs/2505.22451v1 |
. Shen. Reinforcement learning for reasoning in large language models with one training example, 2025. URL https://arxiv.org/abs/2504.20571 . Y . Zuo, K. Zhang, L. Sheng, S. Qu, G. Cui, X. Zhu, H. Li, Y . Zhang, X. Long, E. Hua, B. Qi, Y . Sun, Z. Ma, L. Yuan, N. Ding, and B. Zhou. TTRL: Test-time reinforcement learnin... | https://arxiv.org/abs/2505.22451v1 |
Analysis**: The integral I=R∞ −∞η(k)u(t, k)dkis truncated to [−K, K ]. The tail error is bounded by: Z |k|>Kη(k)dk=2 πZ∞ K1 1 +k2dk=2 ππ 2−arctan( K) ≈1 πKforK≫1. Setting1 πK≤ϵ/2gives K≥2 πϵ. Thus, K=O(1/ϵ). **Adaptive Discretization**: 1. **Central Interval [−K, K ]**: The entire truncated domain [−K, K ] must be di... | https://arxiv.org/abs/2505.22451v1 |
And the calculation process lacks detail. comment : None type: lemma Lemma 4. The solution operator e−τBfor the heat equation derived from the BSM model can be approximated with error ϵusing a quantum algorithm that implements a discretized version of the LCHS lemma. This algorithm requires M=O 1 ϵ2 terms in the quadr... | https://arxiv.org/abs/2505.22451v1 |
by ϵ, each term requires precision δ=O(ϵ). Using high-order Trotter or QSP, the per-term query complexity is ˜O ∥Hj∥τ δ =˜O |kj|∥B∥τ ϵ . **Step 5: Total Query Complexity** Summing over Mterms and leveragingP j|kj|=O(M): ˜O ∥B∥τ ϵ·X j|kj| =˜O∥B∥τ ϵ·M . 27 Substituting M=O(1/ϵ2)andτ=σ2T/2: ˜O∥B∥T ϵ3 . **Step ... | https://arxiv.org/abs/2505.22451v1 |
complexity scales as poly ∥B∥, T,log1 ϵ,1 ϵ , with ancilla qubit count O log1 ϵ . correctness : True Proof. **Step 1: Heat Equation Reduction** From Lemma-0, the BSM PDE transforms into the heat equation ∂τU=BU, where B=−∂2 x. The solution is U(τ) =e−BτU(0). The LCHS lemma provides the integral representation e−Bτ=... | https://arxiv.org/abs/2505.22451v1 |
Unitaries (LCU) method: 1. **Ancilla Preparation**: Prepare an m-qubit ancilla register in the state1√βPM j=1√wj|j⟩, where β=P jwj=O(1). This requires O(logM) =O(log1 ϵ)ancillas and poly(M)gates. 2. **Controlled Hamiltonian Simulation**: For each ancilla state |j⟩, apply the controlled unitary |j⟩⟨j| ⊗e−ikjBτ. Each e−i... | https://arxiv.org/abs/2505.22451v1 |
Finally, by integral over time, the agent proves the energy estimate conclusion. comment : None type: conjecture Lemma 9. **Uniqueness of solutions** The system (2.10) admits at most one solution (u,{φk})in the spaces specified. correctness : True Proof. **Proof of uniqueness:** Suppose two solutions (u1,{φk1})and(u2,{... | https://arxiv.org/abs/2505.22451v1 |
L∞(0, T;L2(Ω)) and weakly in L2(0, T;H1(Ω)), -∂tuN⇀ ∂ tuweakly in L2(0, T;H−1(Ω)), -φN k⇀ φ kanalogously in L∞(0, T;L2(S))∩L2(0, T;H1(S)), - By the compact trace embedding H1(Ω) ,→L2(S), uN|S→u|Sstrongly in L2(0, T;L2(S)). [Correct] Here the agents considers the weak convergence to get the limited function. **Step 6: P... | https://arxiv.org/abs/2505.22451v1 |
(u,{φk})in the specified spaces. comment : None 34 C Proof of High Contrast Limit Generated by AIM Based on DeepSeek-R1 type: lemma Lemma 13. The pressure field plimin the Stokes inclusion satisfies the uniform bound: ∥plim∥L2(Dϵ)≤C∥g∥H−1 2(∂Ω), where Cdepends only on λ0, the domain Ω, and the reference cell Y, but is ... | https://arxiv.org/abs/2505.22451v1 |
eλand taking the square root: ∥divuϵ∥L2(Dϵ)≤Cp eλ∥g∥H−1/2. [Correct] The deprivation here is accurate. 36 comment : None type: lemma Lemma 15. The pressure field pϵin the Stokes inclusion of the original problem (9) satisfies the estimate: ∥pϵ+ (eλ+µ)divuϵ∥L2(Dϵ)≤Cp eλ∥g∥H−1 2(∂Ω), where Cis independent of eλ. correctn... | https://arxiv.org/abs/2505.22451v1 |
=µχΩ\Dϵ+eµχDϵ Suppose the solution of equation (9) above is uϵ. The existence and uniqueness of the solution uϵare guaranteed by the ellipticity condition (1) and the Lax-Milgram theorem. Finally, the homogenization needed to be solved is the following. When eλ→ ∞ andeµis fixed, the coupled system in the incompressible... | https://arxiv.org/abs/2505.22451v1 |
vinto the variational equation: −eλZ Dϵ(divuϵ)2dx+ 2Z Ωµ(x)Duϵ:D(∇ϕ)dx=Z ∂Ωg· ∇ϕ ds. (2) Rearranging and taking absolute values: eλ∥divuϵ∥2 L2(Dϵ)≤ 2Z ΩµDuϵ:D(∇ϕ)dx + Z ∂Ωg· ∇ϕ ds . Using Hölder inequalities and trace duality: 2Z ΩµDuϵ:D(∇ϕ)dx ≤C∥g∥H−1/2∥divuϵ∥L2(Dϵ), Z ∂Ωg· ∇ϕ ds ≤C∥g∥H−1/2∥divuϵ∥L2(Dϵ). **Step 4: Fin... | https://arxiv.org/abs/2505.22451v1 |
Ω[λdivulimdivv+ 2µD(ulim) :D(v)]dx+Z Dϵplimdivvdx=Z ∂Ωg·vds. 2. **Subtraction**: Taking v=wϵ=ulim−uϵ: Z Ω λ|divwϵ|2+ 2µ|D(wϵ)|2 dx=Z Dϵ(plim+eλdivuϵ) divwϵdx. 42 3. **Pressure Analysis**: By Lemma 2, ∥eλdivuϵ∥L2(Dϵ)≤C∥g∥H−1/2. From Stokes regularity, ∥plim∥L2(Dϵ)≤C∥g∥H−1/2. Thus: ∥plim+eλdivuϵ∥L2(Dϵ)≤C∥g∥H−1/2. 4. **... | https://arxiv.org/abs/2505.22451v1 |
(10) yield solutions uεandulim, respectively. By Lemma 2, the divergence in Dεsatisfies: ∥divuε∥L2(Dε)≤C eλ∥g∥H−1/2(∂Ω). The pressure pε=−eλdivuεis bounded in L2(Dε)by Lemma 2. **Step 2: Energy Norm Decomposition** Define wε=ulim−uε. Let λ(x) =eλinDεandλ(x) =λ0(fixed) in Ω\Dε. The energy norm splits as:Z Ωλ(x)|divwε|2d... | https://arxiv.org/abs/2505.22451v1 |
open domain with a connected Lipschitz boundary ∂Ω. 2.Dis an open domain with a finite number of connected components, each having a Lipschitz boundary ∂D. 3.Ω\Dis connected and has a Lipschitz boundary ∂Ω∪∂D. The connected components of Dare denoted as Di,i= 1, . . . , N , where Nis finite. type: assumption content : ... | https://arxiv.org/abs/2505.22451v1 |
of small inclusions arranged in an ε-periodic pattern. The construction steps are as follows: 1. Take the unit cell Y= (−1 2,1 2)d. 2. ω⊂Yis a simply connected open subset with a Lipschitz boundary and satisfies dist(ω, ∂Y )>0. Here, Yf=Y\ωrepresents the model environment at the unit scale. 3. For a given ε >0and n∈Zd,... | https://arxiv.org/abs/2505.22451v1 |
the form is uniformly coercive on Vε. [Vague] The conclusion here is basically correct, but the process requires more proof details, which are not given here. comment : None type: lemma Lemma 26. (Uniform Korn Inequality on Periodically Perforated Domains, corrected) Let Ω⊂Rd (d≥2) be a bounded connected Lipschitz doma... | https://arxiv.org/abs/2505.22451v1 |
D∇w:∇w dx = −˜µR D|∇w|2dx. 2. Again integrating by parts,R D(˜λ+ ˜µ)∇φ·w dx =−(˜λ+ ˜µ)R Dφdivw dx =−(˜λ+ ˜µ)R Dφ·φ dx =−(˜λ+ ˜µ)R D|φ|2dx, since divw= div ue−divulim=φ−0. 3. Finally,R D∇plim·w dx =−R Dplimdivw dx =−R Dplimφ dx . Putting these together, the left-hand side becomes −˜µR D|∇w|2−(˜λ+ ˜µ)R D|φ|2and the right... | https://arxiv.org/abs/2505.22451v1 |
·σ(u)∈H−1(D)with ∥∇ ·σ(u)∥H−1(D)≤C2[λ∥divu∥L2+µ∥∇u∥L2], by the definition of σ(u). Hence Z D∇ ·σ(u)·Φdx ≤ ∥∇ · σ(u)∥H−1(D)∥Φ∥H1(D) ≤C2[λ∥divu∥+µ∥∇u∥]·C1∥φ∥H1/2. (ii) The second termR Dσ(u) :∇Φdx: since σ(u)∈L2(D),∇Φ∈L2(D), Z Dσ(u) :∇Φdx ≤ ∥σ(u)∥L2(D)∥∇Φ∥L2(D). But ∥σ(u)∥L2 ≤λ∥divu∥L2+ 2µ∥D(u)∥L2 ≤λ∥divu∥+µ∥∇u∥, and∥∇Φ∥... | https://arxiv.org/abs/2505.22451v1 |
the local inf–sup constant on each inclusion is scale-invariant and the global constant γ(the minimum over finitely many identical blocks) is independent of ε. Step 5. Conclusion by Babuška–Brezzi. By Steps 2–4 the pair of forms (A, B)onW×Qsatisfies the continuity, coercivity on KerB, and the inf–sup condition with con... | https://arxiv.org/abs/2505.22451v1 |
variational formulation plus the uniform ellipticity (Lemma 4) and Korn’s inequality (Lemma 5) give a(w, w) =Z ∂DϵJϵ·w≤ ∥Jϵ∥H−1/2∥w∥H1/2(∂Dϵ)≤C∥Jϵ∥H−1/2∥w∥H1(Ωϵ). 3. Coercivity a(w, w)≳∥w∥2 H1then yields the stated bound. comment : None type: lemma Lemma 34. LetD⊂Rd(d≥2) be a bounded Lipschitz domain with outward unit ... | https://arxiv.org/abs/2505.22451v1 |
of the Neumann trace. Using Cauchy–Schwarz and the two energy bounds above, |⟨τ, g⟩| ≤ ∥∇ u∥L2(D)∥∇G∥L2(D)+∥f∥H−1∥G∥H1(D)≤(C1C2+C2)∥f∥H−1(D)∥g∥H1/2(∂D). Therefore τ∈H−1/2(∂D)and ∥τ∥H−1/2(∂D)≤C∥f∥H−1(D), withCdepending only on the domain constants C1, C2. 4. Identification of τwith the classical normal derivative. By st... | https://arxiv.org/abs/2505.22451v1 |
traction-jump. On ∂D, σ(˜λ,˜µ)(uε) =˜λφI+ 2˜µD(uε), and (−plimI+ 2˜µD(ulim)) =−plimI+ 2˜µD(ulim). 61 Hence Rε=σ(uε)N−(−plimI+ 2˜µD(ulim))N= (˜λφ+plim)N+ 2˜µD(w)N. By the Neumann-trace continuity on a Lipschitz domain (Lemma 8), ∥t∥H−1/2(∂D)≤C1h ˜λ∥divv∥L2(D)+ ˜µ∥∇v∥L2(D)i for any v. Applying this with v=uε−ulimfor the ... | https://arxiv.org/abs/2505.22451v1 |
+ inf–sup) the unique solution (ulim, plim)satisfies ∥ulim∥H1(Ω)+∥plim∥L2(De)≤˜Csup v,qL(v) + 0 coercivity +inf–sup=C∥g∥H−1 2(∂Ω). Thus the pressure in the inclusions is uniformly bounded by the boundary data, with no dependence onε. [Vague] Here, more detailed explanation is required for the application of boundary pr... | https://arxiv.org/abs/2505.22451v1 |
aλ1(w, w)≥α∥w∥2 H1(D)for all w∈V. Take v=w:=u1−u2. Then α∥u1−u2∥2 H1(D)≤aλ1(u1−u2, u1−u2) = ( λ2−λ1)R D(divu2)(div( u1−u2))dx. 4.Cauchy-Schwarz. R D(divu2)(div( u1−u2)) ≤ ∥divu2∥L2(D)∥div(u1−u2)∥L2(D) ≤C∥u2∥H1(D)∥u1−u2∥H1(D), where we used the continuous embedding H1→H(div) and Korn’s inequality to bound ∥div·∥by∥ · ∥H... | https://arxiv.org/abs/2505.22451v1 |
remainders RK+1(˜λ)∈H1(D;Rd),QK(˜λ)∈L2 0(D)such that u˜λ=u0+˜λ−1v1+···+˜λ−(K+1)vK+1+RK+1(˜λ), p˜λ=p0+˜λ−1p1+···+˜λ−KpK+QK(˜λ), and one has the uniform remainder estimate ∥RK+1(˜λ)∥H1(D)+∥QK(˜λ)∥L2(D)=O(˜λ−(K+1)) as˜λ→ ∞ . In particular the first-order corrector (v1, p1)is the unique solution of a∞(v1,w)−b(w, p1) = 0 , ... | https://arxiv.org/abs/2505.22451v1 |
Lemma 22. Then there is a constant C, depending only on D,d,˜µ, and the C1,1–character of ∂D, such that for all ˜λ≫1, ∥u˜λ−u0∥H1 2(∂D)≤C˜λ−1∥t∥H−1 2(∂D). In other words, the boundary–trace error between the compressible–elastic solution and its Stokes– limit vanishes at rate O(1/˜λ). 69 correctness : True Proof. Proof.... | https://arxiv.org/abs/2505.22451v1 |
(˜λ+ ˜µ)∇(divu) = 0 inD0, σ(˜λ,˜µ)(u)N=ton∂D0, normalized to kill rigid motions. By combining the asymptotic-expansion of the resolvent (Lemma 28 and Lemma 27) with the Lipschitz-dependence estimate (Lemma 23) one shows that Λ(˜λ)admits a convergent expansion in powers of 1/˜λ: Λ(˜λ) = Λ( ∞) +˜λ−1Λ1+R(˜λ),∥R(˜λ)∥H−1/2→... | https://arxiv.org/abs/2505.22451v1 |
plim)be the coupled Stokes–elastic limit solution (Stokes in Dε, elasticity in Ωε) with prescribed boundary traction g∈H−1/2(∂Ω). Define the global first-order corrector U(1)∈ H1(Ω;Rd), unique up to rigid motions, as the solution of the homogeneous Lamé transmission- Neumann problem eµ∆U(1)+ (eλ+eµ)∇(div... | https://arxiv.org/abs/2505.22451v1 |
in Ωewith zero Neumann data on ∂Ωand with traction-jump J(1) eon∂De. By the uniform exterior-energy estimate (Lemmas 9 and 18), ∥w∥H1(Ωe)≤C∥J(1) e∥H−1/2(∂De)≤C˜λ−1∥g∥H−1/2(∂Ω). 4. Finally, since ue−ulim= (ue−˜ue) + ˜ue−ulim =w+˜λ−1U(1), and∥U(1)∥H1=O(∥plim∥) =O(∥g∥), the same O(˜λ−1)–rate holds for ∥ue−ulim∥H1. This ... | https://arxiv.org/abs/2505.22451v1 |
∥τ∥H1 2(∂De)=∥ue|∂De−ulim|∂De−˜λ−1U(1)|∂De∥H1 2(∂De)≤C˜λ−1∥g∥H−1 2(∂Ω), by Lemma 30 (Boundary-Trace Rate-Improvement via First-Order Asymptotics). Step 3. Interior H1-estimate from boundary H1 2-error. Define w:=ue−˜ueonΩe. Then wsolves the homogeneous Lamé system in Ωe, with zero traction on ∂Ωand Dirichlet data w|∂De... | https://arxiv.org/abs/2505.22451v1 |
full high-contrast Lamé solution and ulimthe incompressible-limit (Stokes–elastic) solution, both driven by the same boundary traction gon∂Ω. Then there exists C >0, independent ofεand of the high-contrast parameter ˜λ, such that ∥wext∥H1(Ωε)≤C∥wext∥H1/2(∂Dε). Equivalently, ∥uε−ulim∥H1(Ωε)≤C∥uε−ulim∥H1/2(∂Dε). correctn... | https://arxiv.org/abs/2505.22451v1 |
r d)T:D(r) = 0 inRd (6) The dimension of Risd(d+1) 2, spanned by the following basis vectors: e1, . . . ,ed, xjei−xiej,for1≤i < j≤d where eidenotes the standard basis vector. These basis vectors are denoted as rj, j= 1, . . . ,d(d+1) 2. Define the space orthogonal to rigid body motions: H−1 2 R(∂Dϵ) :=n ϕ∈H−1(∂Dϵ) : (ϕ... | https://arxiv.org/abs/2505.22451v1 |
all ε-cells in Ω, and Kεas the buffer region: Kε= Ω\ [ n∈ΠεYnε! , Y ε= Ω\Kε ForDεconstructed according to Assumption 2, it can be verified that all conditions of Assumption 1 are satisfied. type: hint content : You can use the two-scale expansion method to obtain the cell problem and subsequently utilize the cell probl... | https://arxiv.org/abs/2505.22451v1 |
ε-cellε(Yf+k) for all v∈H1(ε(Yf+k)),∥v∥H1(ε(Yf+k))≤C0 ∥v∥L2(ε(Yf+k))+∥sym∇v∥L2(ε(Yf+k)) . [Vague] Here the agent considers the Korn inequality but the derivation process is not detailed enough. 2. (Partition of unity subordinate to the periodic tiling.) Let {ψk}k∈Kεbe a smooth partition of unity onΩ: • each ψk∈C∞ c(ε... | https://arxiv.org/abs/2505.22451v1 |
two-scale expansion here is not detailed enough. The proof of the convergence of the function lacks details. 4. Identification of the oscillating part as a y-gradient. We set W(x, y) = lim ε→0∇yTεuεinL2(Ω×Yf). On one hand, by commutation of partial derivatives and the smoothness of the unfolding map, one shows in distr... | https://arxiv.org/abs/2505.22451v1 |
[Vague] This conclusion needs more detailed process. 87 2. Two-scale compactness Extend uεby zero into Dεandpεby zero into Ωε; still denote the extensions by ˆuε∈H1(Ω)and˙pε∈L2(Ω). From the uniform bound and the periodic unfolding or classical two-scale compactness theorems (Lemma A.3, A.4) we extract a subsequence and... | https://arxiv.org/abs/2505.22451v1 |
property sup |E|≤M∥∂EχE∥H1per(Ye∪Yi)≤C(M). Define the two-scale approximation on Ωεby Uε(x) =u0(x) +εχDxu0(x)x ε , extended by zero in each inclusion. Then there exist α∈(0,1)andC >0, independent of εandg, such that for all sufficiently small ε >0, ∥uε−Uε∥H1(Ωε)≤Cεα∥g∥ H−1 2 R(∂Ω). correctness : True Proof. We split ... | https://arxiv.org/abs/2505.22451v1 |
the proof of the εα–rate in H1–norm. comment : None type: lemma Lemma 58. Conjecture 5. (Well–posedness of the periodic transmission cell–problem and strict ellipticity of the homogenized tensor) Let Y= (−1 2,1 2)dbe the reference cell, decomposed into two disjoint Lipschitz–subdomains Ye(elastic) and Yi(incompressible... | https://arxiv.org/abs/2505.22451v1 |
(χe, χi)into the energy identity ChomE:E=a((E y+χe, E y +χi),(E y+χe, E y +χi)), which is manifestly symmetric and gives Chom∈Rd×d sym⊗Rd×d sym. [Vague] The derivation process here is not detailed enough, and the above conclusions used are also incorrect. 4. Strict ellipticity. Coercivity on Kplus the zero–mean and per... | https://arxiv.org/abs/2505.22451v1 |
is: σ(u) :=λ(∇ ·u)Id+ 2µD(u) (3) D(u) =1 2(∇+∇T)u=1 2(∂iuj+∂jui)ij (4) Here, Idis the identity matrix of order d, andDrepresents the symmetrized differential operator. The normal derivative (boundary traction) on the boundary of a region Eis defined as: ∂u ∂νλ,µ ∂E:=σ(u)N=λ(divu)N+ 2µD(u)N on∂E (5) where Nis the unit o... | https://arxiv.org/abs/2505.22451v1 |
1, one obtains a uniform H1–bound ∥uε∥H1(Ωε)≤C∥g∥H−1/2 R(∂Ω). 94 By standard extension–by–zero and the two–scale compactness of Lemma 2, up to a subsequence uε⇀ u 0inH1(Ω), ∇uε→ ∇ xu0+∇yu1two–scale in Ω×Yf, for some u0∈H1(Ω;Rd)and corrector u1(x, y). A matching argument in the inclusions shows thatu0satisfies the homog... | https://arxiv.org/abs/2505.22451v1 |
Unsupervised Post-Training for Multi-Modal LLM Reasoning via GRPO Lai Wei1,3Yuting Li1Chen Wang3Yue Wang3Linghe Kong1 Weiran Huang1,2∗Lichao Sun4 1School of Computer Science, Shanghai Jiao Tong University 2Shanghai Innovation Institute 3Zhongguancun Academy4Lehigh University Abstract Improving Multi-modal Large Languag... | https://arxiv.org/abs/2505.22453v1 |
studies demonstrate notable success using online reinforcement learning (e.g. GRPO [ 34]) with verifiable rewards [ 6,10, 28,66] to enhance the reasoning capabilities of LLMs and MLLMs. A concurrent work, TTRL [ 71], further extends this line by applying GRPO on test-time scaling of LLMs. Thus, it is promising that onl... | https://arxiv.org/abs/2505.22453v1 |
Improvement. High-quality data obtained from human annotations has been shown to sig- nificantly boost the performance of LLMs across a wide range of tasks [ 10,16,29]. However, such high-quality annotated data may be exhausted in the future. This presents a substantial obstacle to the continual learning of advanced mo... | https://arxiv.org/abs/2505.22453v1 |
human preference data (Ii, qi, y+ i, y− i), where yidenotes the answer of qiand(y+ i, y− i)denotes the preference pair of qi. In contrast, we only allow to operate in a fully unsupervised manner for this setting, leveraging only the model’s own responses to generate training signals. This presents significant challenge... | https://arxiv.org/abs/2505.22453v1 |
riis then determined based on the y∗: ri=1,ifˆyi=y∗, 0,otherwise .(2) 4 Table 1: Main results on four multi-modal mathematical reasoning benchmarks. We report accuracy (%) for each method on MathVision, MathVerse, MathVista, and We-Math. All methods are conducted on the Qwen2.5-VL-7B backbone. MM-UPT outperforms other... | https://arxiv.org/abs/2505.22453v1 |
MLLM to first self-generate an answer and then self-check it. STIC applies DPO where original images and good prompts are used to generate preferred answers, and corrupted images and bad prompts to produce rejected answers. Additionally, we also compare with GRPO [ 34] and rejection sampling-based SFT [ 39], which are ... | https://arxiv.org/abs/2505.22453v1 |
LMSI, Genixer, and STIC. Notably, MM-UPT is able to improve the average score from 49.47 (base model) to 53.17 (with MMR1 dataset), demonstrating its effectiveness in leveraging unlabeled data for self-improvement. In comparison, previous baselines provide only marginal gains or even degrade performance on certain benc... | https://arxiv.org/abs/2505.22453v1 |
data rephrasing. In contrast, direct synthesizing generates more diverse and novel questions. While some of the directly synthesized questions still contain hallucinations, many are of high quality and beneficial for unsupervised post-training. This underscores the potential of the direct synthesizing approach as a sim... | https://arxiv.org/abs/2505.22453v1 |
better understand the behavior of MM-UPT during training, we monitor several diagnostic metrics, including the majority voting reward and entropy, both of which are label-free and provide insights in the absence of ground-truth supervision. In particular, majority voting reward is calculated following Equation 2. Entro... | https://arxiv.org/abs/2505.22453v1 |
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