diff --git a/.eval_results/olmocrbench.yaml b/.eval_results/olmocrbench.yaml new file mode 100644 index 0000000000000000000000000000000000000000..db9be48b80bbef26ddd188ebd001d0be18ba9ab9 --- /dev/null +++ b/.eval_results/olmocrbench.yaml @@ -0,0 +1,108 @@ +- dataset: + id: allenai/olmOCR-bench + task_id: overall + revision: 54a96a6fb6a2bd3b297e59869491db4d3625b711 + value: 86.8214 + date: "2026-09-18" + source: + url: https://huggingface.co/awmg/TrueDoc/blob/main/RESULTS.md + name: "TrueDoc run 97: official scorer, all 1,403 page outputs and the scoring log in this repository" + user: awmg + notes: "Macro-average of the eight sections; 95% bootstrap CI 86.0-87.7. TrueDoc is a conversion pipeline, not a model: the PDF's own text layer is read on CPU for the 1,122 pages that have one (no model); Infinity-Parser2-Flash (2.2B) reads the 281 pages without a text layer, Infinity-Parser2-Pro (35B) the 102 of those the pipeline's own OCR cannot read, and olmOCR-2-7B-1025 92 picture regions on digital pages. Model readings were recorded on a rented GPU on 17 September 2026 and replayed from disk. Official scorer at dataset revision 54a96a6f, all eight sections, no post-processing keyed on the benchmark's folder names. Code: https://github.com/islandtimer/TrueDoc at tag run-97." + +- dataset: + id: allenai/olmOCR-bench + task_id: arxiv_math + revision: 54a96a6fb6a2bd3b297e59869491db4d3625b711 + value: 88.6232 + date: "2026-09-18" + source: + url: https://huggingface.co/awmg/TrueDoc/blob/main/RESULTS.md + name: "TrueDoc run 97: official scorer, all 1,403 page outputs and the scoring log in this repository" + user: awmg + notes: "2594 of 2927 checks. TrueDoc is a conversion pipeline, not a model: the PDF's own text layer is read on CPU for the 1,122 pages that have one (no model); Infinity-Parser2-Flash (2.2B) reads the 281 pages without a text layer, Infinity-Parser2-Pro (35B) the 102 of those the pipeline's own OCR cannot read, and olmOCR-2-7B-1025 92 picture regions on digital pages. Model readings were recorded on a rented GPU on 17 September 2026 and replayed from disk. Official scorer at dataset revision 54a96a6f, all eight sections, no post-processing keyed on the benchmark's folder names. Code: https://github.com/islandtimer/TrueDoc at tag run-97." + +- dataset: + id: allenai/olmOCR-bench + task_id: baseline + revision: 54a96a6fb6a2bd3b297e59869491db4d3625b711 + value: 99.8565 + date: "2026-09-18" + source: + url: https://huggingface.co/awmg/TrueDoc/blob/main/RESULTS.md + name: "TrueDoc run 97: official scorer, all 1,403 page outputs and the scoring log in this repository" + user: awmg + notes: "1392 of 1394 checks. TrueDoc is a conversion pipeline, not a model: the PDF's own text layer is read on CPU for the 1,122 pages that have one (no model); Infinity-Parser2-Flash (2.2B) reads the 281 pages without a text layer, Infinity-Parser2-Pro (35B) the 102 of those the pipeline's own OCR cannot read, and olmOCR-2-7B-1025 92 picture regions on digital pages. Model readings were recorded on a rented GPU on 17 September 2026 and replayed from disk. Official scorer at dataset revision 54a96a6f, all eight sections, no post-processing keyed on the benchmark's folder names. Code: https://github.com/islandtimer/TrueDoc at tag run-97." + +- dataset: + id: allenai/olmOCR-bench + task_id: headers_footers + revision: 54a96a6fb6a2bd3b297e59869491db4d3625b711 + value: 96.7105 + date: "2026-09-18" + source: + url: https://huggingface.co/awmg/TrueDoc/blob/main/RESULTS.md + name: "TrueDoc run 97: official scorer, all 1,403 page outputs and the scoring log in this repository" + user: awmg + notes: "735 of 760 checks. TrueDoc is a conversion pipeline, not a model: the PDF's own text layer is read on CPU for the 1,122 pages that have one (no model); Infinity-Parser2-Flash (2.2B) reads the 281 pages without a text layer, Infinity-Parser2-Pro (35B) the 102 of those the pipeline's own OCR cannot read, and olmOCR-2-7B-1025 92 picture regions on digital pages. Model readings were recorded on a rented GPU on 17 September 2026 and replayed from disk. Official scorer at dataset revision 54a96a6f, all eight sections, no post-processing keyed on the benchmark's folder names. Code: https://github.com/islandtimer/TrueDoc at tag run-97." + +- dataset: + id: allenai/olmOCR-bench + task_id: long_tiny_text + revision: 54a96a6fb6a2bd3b297e59869491db4d3625b711 + value: 91.4027 + date: "2026-09-18" + source: + url: https://huggingface.co/awmg/TrueDoc/blob/main/RESULTS.md + name: "TrueDoc run 97: official scorer, all 1,403 page outputs and the scoring log in this repository" + user: awmg + notes: "404 of 442 checks. TrueDoc is a conversion pipeline, not a model: the PDF's own text layer is read on CPU for the 1,122 pages that have one (no model); Infinity-Parser2-Flash (2.2B) reads the 281 pages without a text layer, Infinity-Parser2-Pro (35B) the 102 of those the pipeline's own OCR cannot read, and olmOCR-2-7B-1025 92 picture regions on digital pages. Model readings were recorded on a rented GPU on 17 September 2026 and replayed from disk. Official scorer at dataset revision 54a96a6f, all eight sections, no post-processing keyed on the benchmark's folder names. Code: https://github.com/islandtimer/TrueDoc at tag run-97." + +- dataset: + id: allenai/olmOCR-bench + task_id: multi_column + revision: 54a96a6fb6a2bd3b297e59869491db4d3625b711 + value: 83.5973 + date: "2026-09-18" + source: + url: https://huggingface.co/awmg/TrueDoc/blob/main/RESULTS.md + name: "TrueDoc run 97: official scorer, all 1,403 page outputs and the scoring log in this repository" + user: awmg + notes: "739 of 884 checks. TrueDoc is a conversion pipeline, not a model: the PDF's own text layer is read on CPU for the 1,122 pages that have one (no model); Infinity-Parser2-Flash (2.2B) reads the 281 pages without a text layer, Infinity-Parser2-Pro (35B) the 102 of those the pipeline's own OCR cannot read, and olmOCR-2-7B-1025 92 picture regions on digital pages. Model readings were recorded on a rented GPU on 17 September 2026 and replayed from disk. Official scorer at dataset revision 54a96a6f, all eight sections, no post-processing keyed on the benchmark's folder names. Code: https://github.com/islandtimer/TrueDoc at tag run-97." + +- dataset: + id: allenai/olmOCR-bench + task_id: old_scans + revision: 54a96a6fb6a2bd3b297e59869491db4d3625b711 + value: 58.365 + date: "2026-09-18" + source: + url: https://huggingface.co/awmg/TrueDoc/blob/main/RESULTS.md + name: "TrueDoc run 97: official scorer, all 1,403 page outputs and the scoring log in this repository" + user: awmg + notes: "307 of 526 checks. TrueDoc is a conversion pipeline, not a model: the PDF's own text layer is read on CPU for the 1,122 pages that have one (no model); Infinity-Parser2-Flash (2.2B) reads the 281 pages without a text layer, Infinity-Parser2-Pro (35B) the 102 of those the pipeline's own OCR cannot read, and olmOCR-2-7B-1025 92 picture regions on digital pages. Model readings were recorded on a rented GPU on 17 September 2026 and replayed from disk. Official scorer at dataset revision 54a96a6f, all eight sections, no post-processing keyed on the benchmark's folder names. Code: https://github.com/islandtimer/TrueDoc at tag run-97." + +- dataset: + id: allenai/olmOCR-bench + task_id: old_scans_math + revision: 54a96a6fb6a2bd3b297e59869491db4d3625b711 + value: 86.6812 + date: "2026-09-18" + source: + url: https://huggingface.co/awmg/TrueDoc/blob/main/RESULTS.md + name: "TrueDoc run 97: official scorer, all 1,403 page outputs and the scoring log in this repository" + user: awmg + notes: "397 of 458 checks. TrueDoc is a conversion pipeline, not a model: the PDF's own text layer is read on CPU for the 1,122 pages that have one (no model); Infinity-Parser2-Flash (2.2B) reads the 281 pages without a text layer, Infinity-Parser2-Pro (35B) the 102 of those the pipeline's own OCR cannot read, and olmOCR-2-7B-1025 92 picture regions on digital pages. Model readings were recorded on a rented GPU on 17 September 2026 and replayed from disk. Official scorer at dataset revision 54a96a6f, all eight sections, no post-processing keyed on the benchmark's folder names. Code: https://github.com/islandtimer/TrueDoc at tag run-97." + +- dataset: + id: allenai/olmOCR-bench + task_id: table_tests + revision: 54a96a6fb6a2bd3b297e59869491db4d3625b711 + value: 89.3346 + date: "2026-09-18" + source: + url: https://huggingface.co/awmg/TrueDoc/blob/main/RESULTS.md + name: "TrueDoc run 97: official scorer, all 1,403 page outputs and the scoring log in this repository" + user: awmg + notes: "913 of 1022 checks. TrueDoc is a conversion pipeline, not a model: the PDF's own text layer is read on CPU for the 1,122 pages that have one (no model); Infinity-Parser2-Flash (2.2B) reads the 281 pages without a text layer, Infinity-Parser2-Pro (35B) the 102 of those the pipeline's own OCR cannot read, and olmOCR-2-7B-1025 92 picture regions on digital pages. Model readings were recorded on a rented GPU on 17 September 2026 and replayed from disk. Official scorer at dataset revision 54a96a6f, all eight sections, no post-processing keyed on the benchmark's folder names. Code: https://github.com/islandtimer/TrueDoc at tag run-97." + diff --git a/outputs/arxiv_math/2502.15977_pg21_pg1_repeat1.md b/outputs/arxiv_math/2502.15977_pg21_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..a030d5a4e6349161e7125d6fd611e09d838860a5 --- /dev/null +++ b/outputs/arxiv_math/2502.15977_pg21_pg1_repeat1.md @@ -0,0 +1,27 @@ +\[V_{\rho_{0}}=0\] + +\[V_{\rho_{1}}=\mathbb{C}\theta_{1}\] + +![](figure) + +Figure 3. The decorated fan of an action of \(Q(1)^{2}\text{on}\mathbb{P}^{2|2}\) + +Example 4.28. Let us use Proposition 4.26 and Corollary 4.27 to classify toric supervarieties with supertorus \(Q(1)^{n}\) and underlying variety \(\mathbb{P}^{\sim^{n}}=X_{\Sigma}\) for the complete fan whose rays are \(\rho_{i}=\mathbb{R}_{+}x_{i}\text{for}i=1,...,n\text{and}\rho_{0}=\mathbb{R}_{+}(-x_{1}-...-x_{n})\). For \(i=1,...,n\), the ray \(\rho_{i}\) must be decorated by a subspace \(V_{\rho_{i}}\text{such that}[V_{\rho_{i}},V_{\rho_{i}}]\subseteq \mathbb{C}x_{i}\). Hence \(V_{\rho_{i}}=\mathbb{C}\theta_{i}\) or 0. Likewise, unless \(V_{\rho_{0}}=0\), we have \(V_{\rho_{0}}=\mathbb{C}(\theta_{1}\pm\theta_{2}\pm...\pm\theta_{n})\) for some \(2^{n-1}\) choices of ±. It is straightforward to verify that condition (b) of the definition of a large-orbit decorated fan holds regardless of which subspaces are chosen. We therefore obtain a collection of \(2^{n}(1+2^{n-1})\) toric supervarieties which are not equivariantly isomorphic. Many, however, are isomorphic via toric morphisms (to be defined in the following section). Notice that if all but one of these decorations is nonzero, then the decorated fan describes a supervariety isomorphic to projective superspace \(\mathbb{P}^{n|n}\). For instance, if \(V_{\rho_{i}}= \mathbb{C}\theta_{i}\text{and}V_{\rho_{0}}=0\), then the coordinate superalgebras of the affine charts can be written as + +\[\mathbb{C}[t_{1},...,t_{n},t_{1}\xi_{1},...,t_{n}\xi_{n}]\] + +and + +\[\mathbb{C}[t_{i}^{-1}t_{1},...,t_{i}^{-1},...,t_{i}^{-1}t_{n}\xi_{1},...,\xi_{i},...,t_{i}^{-1}t_{n}\xi_{n}].\] + +Figure 3 depicts the corresponding decorated fan for \(n=2\), If instead we change the decoration of \(\rho_{0}\) to \(\theta_{1}+...+\theta_{n}\), then the resulting affine charts have coordinate superalgebras + +\[\mathbb{C}[t_{1},...,t_{n},t_{1}\xi_{1},...,t_{n}\xi_{n}]\] + +and + +\[\mathbb{C}[t_{i}^{-1}t_{1}(1+\xi_{i}\xi_{1}),...,t_{i}^{-1},...,t_{i}^{-1}t_{n}(1+\xi_{i}\xi_{n}),t_{i}^{-1}t_{1}(\xi_{i}-\xi_{1}),...,t_{i}^{-1}\xi_{i},...,t_{i}^{-1}t_{n}(\xi_{i}-\xi_{n})],\] + +so the supervariety is decidedly not isomorphic to projective superspace. Figure 4(B) depicts corresponding decorated fan for \(n=1\). In general, when \(T=Q(1)^{n}\), there are finitely many possible decorations for each ray. Namely, for\(\underline{\rho} =\mathbb{R}_{+}(a_{\sqrt{1}} \underline{x_{1}} +...+a_{n}x_{n})\), there are \(2^{d-1}\) many possible “square root subspaces” \(\mathbb{C}( \overline{a_{1}} \theta_{1}\pm...\pm \overline{a_{n}} \theta_{n}),\text{where}d\) is the number of indices \(i=1,...,n\) for which \(a_{i}\neq 0\). The prior example admitted no issues of compatibility between different rays of the same cone. This is not ordinarily the case; if \(\rho_{1}=\mathbb{R}_{+}(a_{1}x_{1}+...+a_{n}x_{n})\text{and}\rho_{2}= \mathbb{R}_{+}(b_{1}x_{1}+...+b_{n}x_{n})\), compatibility is less common if the linear matroid on the \(n\) vectors + +\[\begin{pmatrix} a_{1} \\ b_{1} \end{pmatrix} ,..., \begin{pmatrix} a_{n} \\ b_{n} \end{pmatrix}\] diff --git a/outputs/arxiv_math/2503.02004_pg9_pg1_repeat1.md b/outputs/arxiv_math/2503.02004_pg9_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..78ede2d711a54e98fb2e8b9be49c1146e4cf6bbe --- /dev/null +++ b/outputs/arxiv_math/2503.02004_pg9_pg1_repeat1.md @@ -0,0 +1,62 @@ +The key advance of the MILP model is its global optimality guarantee, since it can explore all possible combinations via branch-and-bound. Furthermore, by employing upper and lower bound pruning strategies, MILP significantly reduces ineffective search efforts, thereby enhancing computational efficiency and guaranteeing the identification of the global optimality. However, despite its global optimality, its computational complexity still grows rapidly as the problem size increases with the time complexity \(\mathcal{O}(2^{M}\cdot K)\). Empirical results show that when \(M>50\text{and}K>100\), the size of the branch-and-bound search tree leads to memory and time costs that exceed practical limits. + +## Algorithm 2 GRSIP: Greedy Row Selection with Isolated Preselection. + +Require: \(\mathbf{G},N_{r},N_{\text{init}},\Delta n\) +Ensure: \(\mathbf{I}_{\boldsymbol{s}}\) +1: Initialize: \(l=0,T_{(0)}=\{\emptyset\}\) +2: while \(l\leq N_{\text{init}}\) do +\(i=\operatorname*{arg\,max}\bar{\mathbf{g}}_{i},\mathrm{s}.\mathrm{t}.D_{1}(\{i\},T_{(l)})\leq\Delta n\) + +3: + +\(i\in T_{(l)}^{c}\) + +4: 5: + +\(l=l+1\) +\(T_{(l)}=T_{(l)}\cup\{i\}\) +6: end while +7: while \(l\leq N_{r}\) do +\(i=\) arg maxmin \(\Vert\mathbf{h}_{k}\Vert_{2},\text{s.t.}\mathbf{h}_{k}\in\text{Col}(\mathbf{H}_{T_{(l)}\cup\{i\}})\) + +8: + +\(i\in T_{(l)}^{k^{c}}\) + +9: 10: + +\(l=l+1\) +\(T_{(l)}=T_{(l)}\cup\{i\}\) +11: end while +12: return \(\mathbf{I}_{\boldsymbol{s}}=T_{(l)}\) + +Hence, in the following we also propose a greedy algorithm in Algorithm 2 with the time complexity \(\mathcal{O}(MKN_{r})\). The algorithm starts by choosing \(N_{\text{init}}\) positions with the top average channel gains for each subcarrier, maintaining a minimum separation of \(\Delta n\) between them, where \(D_{1}(\cdot,\cdot)\) represents the minimal \(\ell_{1}-\)norm distance between two point + +Space Frequency Grid sets. Then it incrementally adds indices to the candidate set, following the principle of maximizing the minimum subcarrier gain, until \(N_{r}\) positions are chosen. Note that \(\mathbf{H}_{T}\) is formed by the row vectors of G associated with row indices in \(T\) . + +![](figure) + +![](figure) + +![](figure) + +![](figure) + +Remark 2 (Applications to Other Models). The theoretical analysis and algorithms of the two-step framework for FAS proposed in this work could be extended to other problems besides FAS. First, the proposed two-step framework could be directly extended to the antenna selection problem with discrete positions, regardless of whether the exact antennas deployment [38]–[41]. The group-sparse recovery formulation and D-GRIP analysis can be directly adapted to delay-Doppler domain channel estimation [42], [43], where structures induce similar group-wise sparsity patterns in reconstruction. The DC-GOMP algorithm employs a correlation-aware selection mechanism to dynamically resolve coherence conflicts, offering a systematic and efficient approach to sparse event detection. Then, MILP-based spatial equalization offers new insights for the resource-constrained optimization in RIS configuration on discrete phase [44]. These potential extensions highlight that our methodology effectively tackles the unified challenge of sparsity-aware optimization under structured constraints, making it applicable to a wide range of domains, including computational sensing, adaptive control, and beyond. + +## V. SIMULATION RESULTS + +In this section, we present the performance of the proposed group-sparsity based frequency-space channel estimation algorithm, i.e., DC-GOMP, in comparison to two traditional algorithms (OMP, GOMP), under FAS-assisted wideband SIMO system. The proposed positions optimization methods, i.e., MILP and GRSIP, are also evaluated through the physical layer simulations and in terms of BER. + +![](figure) + +![](figure) + +![](figure) + +![](figure) + +![](figure) + +Figure 2. (1) The first row has four expressions in frequency-space domain. The first one represents the original SFG and the last three represent the recovered version by three different algorithm (our proposed DC-GOMP, OMP and GOMP). (2) Delay-wavenumber domain expressions corresponding to ones above. Black boxes denote the low power regions and red boxes denote the regions failing to correctly allocate the energy. diff --git a/outputs/arxiv_math/2503.03754_pg10_pg1_repeat1.md b/outputs/arxiv_math/2503.03754_pg10_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..070694cc43f698f1b570cccb6e3a6d07ea7d89f4 --- /dev/null +++ b/outputs/arxiv_math/2503.03754_pg10_pg1_repeat1.md @@ -0,0 +1,39 @@ +[20] P. Skrzypczyk, N. Brunner, and S. Popescu, “Emergence of quantum correlations from nonlocality swapping,” Physical Review Letters, vol. 102, no. 11, p. 110402, 2009. + +[21] A. J. Short, S. Popescu, and N. Gisin, “Entanglement swapping for generalized nonlocal correlations,” Physical Review A—Atomic, Molecular, and Optical Physics, vol. 73, no. 1, p. 012101, 2006. + +[22] J. Barrett, “Information processing in generalized probabilistic theories,” Physical Review A—Atomic, Molecular, and Optical Physics, vol. 75, no. 3, p. 032304, 2007. + +# Proof of Theorem 3 + +# A + +We prove the second part of the theorem first. Take some arbitrary \(\Phi\text{in}\mathscr{F}_{2}\). We can represent a Z-channel source as follows + +\[P_{XY}= \begin{pmatrix} 1-s & 0 \\ sd & s(1-d) \end{pmatrix}\] + +where \(s,d\in[0,1]\). Let \(u=f_{X}(0)\text{and}v=f_{X}(1)\text{forsome}u,v\geq0\). Assume that \(\mathbb{E}[f]=m=(1-s)u+sv\). + +Then \(u= \frac{m-sv}{1-s}\) . One can verify directly that + +\[g(v,m):= \frac{H_{\Phi}(\mathbb{E}[f|Y])}{H_{\Phi}(f)} = \frac{(1-s(1-d))\Phi\left( \tfrac{1-s}{1-s(1-d)} u+ \tfrac{sd}{1-s(1-d)} v\right)+s(1-d)\Phi(v)-\Phi((1-s)u+sv)}{(1-s)\Phi(u)+s\Phi(v)-\Phi((1-s)u+sv)}\] + +\[= \frac{(1-s(1-d))\Phi\left( \tfrac{m-s(1-d)v}{1-s(1-d)} \right)+s(1-d)\Phi(v)-\Phi(m)}{(1-s)\Phi\left( \tfrac{m-sv}{1-s} \right)+s\Phi(v)-\Phi(m)} .\] (14) + +Then, we claim that if (10) holds, then \(g(v,m)\) is decreasing in \(v\) for every fixed \(m\). Therefore, the maximum of \(g(v,m)\) would occur when \(v=0\). This would complete the proof. Taking the partial derivative of \(\log(g(v,m))\) with respect to \(v\), we need to show that + +\[\frac{s\Phi^{\prime}(v)-s\Phi^{\prime}\left( \tfrac{m-sv}{1-s} \right)}{(1-s)\Phi\left( \tfrac{m-sv}{1-s} \right)+s\Phi(v)-\Phi(m)} \geq \frac{s(1-d)\Phi^{\prime}(v)-s(1-d)\Phi^{\prime}\left( \tfrac{m-s(1-d)v}{1-s(1-d)} \right)}{(1-s(1-d))\Phi\left( \tfrac{m-s(1-d)v}{1-s(1-d)} \right)+s(1-d)\Phi(v)-\Phi(m)} .(15)\] + +For any \(t\in[0,1]\), define + +\[k(t)= \frac{st\Phi^{\prime}(v)-st\Phi^{\prime}\left( \tfrac{m-svt}{1-st} \right)}{(1-st)\Phi\left( \tfrac{m-svt}{1-st} \right)+st\Phi(v)-\Phi(m)}\] (16) + +\[= \frac{\Phi^{\prime}(v)-\Phi^{\prime}\left( \tfrac{m-svt}{1-st} \right)}{( \tfrac{1}{st} -1)\Phi\left( \tfrac{m-svt}{1-st} \right)+\Phi(v)- \tfrac{1}{st} \Phi(m)} .\] (17) + +Then, (15) can be written as \(k(1)\geq k(1-d)\). We would be done if we can show that \(k(t)\) is an increasing function. Showing \(k^{\prime}(t)\geq0\) is equivalent with + +\[\begin{aligned} C(t)&=- \frac{1}{st^{2}} \left(\Phi^{\prime}(v)-\Phi^{\prime}\left( \frac{m-svt}{1-st} \right)\right)\left(-\Phi\left( \frac{m-svt}{1-st} \right)+ \frac{(m-v)st}{1-st} \Phi^{\prime}\left( \frac{m-svt}{1-st} \right)+\Phi(m)\right) \\ - \frac{s(m-v)}{(1-st)^{2}} \Phi^{\prime\prime}\left( \frac{m-svt}{1-st} \right)\left(\Phi(v)+ \frac{1-st}{st} \Phi\left( \frac{m-svt}{1-st} \right)- \frac{1}{st} \Phi(m)\right)&\geq0. \end{aligned}\] + +Let \(x_{1}=v,x_{2}= \frac{m-svt}{1-st}\) . Then we can compute \(s\) from \(x_{1}\text{and}x_{2}\) as follows: + +\[s= \frac{m-x_{2}}{t(x_{1}-x_{2})} .\] diff --git a/outputs/arxiv_math/2503.03759_pg9_pg1_repeat1.md b/outputs/arxiv_math/2503.03759_pg9_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..7ac842b6119dcec3899159fd91ae4feb8c97e644 --- /dev/null +++ b/outputs/arxiv_math/2503.03759_pg9_pg1_repeat1.md @@ -0,0 +1,13 @@ +certain types of correlation that cannot be captured using real-valued probabilities alone. Consider a complex random variable \(Z\) whose probability distribution is defined over a discrete set of outcomes. The probability mass function \(P(z)\) is now complex-valued, meaning that \(P(z)\in\mathbb{C}\). This introduces a fundamental shift in how entropy is conceptualized, as the standard Shannon entropy is defined solely for real, nonnegative probabilities. To extend this concept, a suitable framework must account for both the magnitude and phase of complex probabilities. A plausible extension of Shannon entropy to complex probability distributions can be expressed as + +\[H(Z)=\mathbb{E}[I(Z)]=\sum_{z}P(z)I(z)=-\sum_{z}P(z)\log P(z),\] (10) + +where \(P(z)\) is complex. The challenge lies in the interpretation of the logarithm \(\log P(z)\) for complex values, as the logarithm of a complex number is inherently multivalued. To resolve this, \(\log P(z)\) is typically expressed in terms of its polar form + +\[\log P(z)=\log|P(z)|+i\theta(P(z)),\] (11) + +where \(|P(z)|\) is the modulus (or absolute value) of \(P(z),\text{and}\theta(P(z))\) denotes the phase (or argument) of the complex probability. Here, \(\log|P(z)|\) captures the traditional magnitude-based contribution to entropy, while \(i\theta(P(z))\) incorporates the phase information intrinsic to complex probabilities. Substituting the polar form of \(\log P(z)\) into the expression for \(H(Z)\), we obtain the complex form of Shannon entropy: + +\[\begin{aligned} H(Z)&=-\sum_{z}P(z)\log P(z) \\ &=-\sum_{z}P(z)(\log|P(z)|+i\theta(P(z))) \\ &=-\sum_{z}P(z)\log|P(z)|-i\sum_{z}P(z)\theta(P(z)). \end{aligned}\] (12) + +This formulation reveals two distinct components of the complex entropy. The first component, \(-\sum_{z}P(z)\log|P(z)|\), resembles the standard Shannon entropy but now incorporates the magnitudes of the complex probabilities \(|P(z)|\). This term quantifies the uncertainty or information content in the distribution of probability magnitudes, maintaining a familiar structure while adapting to the complex domain. The second component, \(-i\sum_{z}P(z)\theta(P(z))\), introduces a phase-dependent contribution to the entropy, reflecting the coherence, interference, or relative phase relationships among the events. Unlike classical entropy, which focuses solely on magnitude, this phase term highlights the informational significance of relative phases, an essential feature in systems with quantum mechanical or wave-like properties. diff --git a/outputs/arxiv_math/2503.03762_pg1_pg1_repeat1.md b/outputs/arxiv_math/2503.03762_pg1_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..2db4ce8aab29b75d3a106c4f72249c043b82d060 --- /dev/null +++ b/outputs/arxiv_math/2503.03762_pg1_pg1_repeat1.md @@ -0,0 +1,27 @@ +### DISPROVING SOME THEOREMS IN SHARMA AND CHAUHAN et al. \((2018,2021)^{*}\) + +## Ramy Takieldin + +Faculty of Engineering, Ain Shams University, Cairo, Egypt Egypt University of Informatics, New Capital, Cairo, Egypt ramy.farouk@eng.asu.edu.eg + +## Patrick Solé + +I2M (CNRS, University of Aix-Marseille), 13009 Marseilles, France patrick.sole@telecom-paris.fr + +## ABSTRACT + +The main objective of this work is to show, through counterexamples, that some of the theorems presented in the papers of Sharma et al. (2018) and Chauhan et al. ( 2021) are incorrect. Although they used these theorems to establish a sufficient condition for a multi-twisted (MT) code to be linear complementary dual (LCD), we show that this condition itself remains valid. We further improve this condition by removing the restrictions on the shift constants and relaxing the required coprimality condition. We show that compared to the previous condition, the modified condition is able to identify more LCD MT codes. Furthermore, without the need for a normalized set of generators, we develop a formula to determine the dimension of any \(\rho-\)generator MT code. + +Keywords Multi-twisted code · linear complementary dual · Determinantal divisors · Algebraic coding MSC: 94B05, 94B60, 11T71 + +## 1 Introduction + +Multi-twisted (MT) codes over a finite field \(\mathbb{F}_{q}\) constitute a significant and comprehensive class of linear codes. This class contains several well-known subclasses, including cyclic, constacyclic, quasi-cyclic, quasi-twisted, and generalized quasi-cyclic codes. For some integer \(\ell\geq1\), let \(0\neq \lambda_{i}\in\mathbb{F}_{q}\text{and}m_{i}\geq1\text{for}1\leq i\leq\ell.\text{If}\Lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{\ell})\), then a Λ-MT code \(\mathcal{C}\) with block lengths \((m_{1},m_{2},\ldots,m_{\ell})\) is defined in [1, Definition 3.1] as a linear code of length \(n=m_{1}+m_{2}+\cdots +m_{\ell}\) that remains invariant under the Λ-MT linear transformation + +\[\begin{aligned} T_{\Lambda}:(c_{1,0},c_{1,1},\ldots,c_{1,m_{1}-1};c_{2,0},c_{2,1},\ldots,c_{2,m_{2}-1};\ldots;c_{\ell,0},c_{\ell,1},\ldots,c_{\ell,m_{\ell}-1})&\mapsto \\ (\lambda_{1}c_{1,m_{1}-1},c_{1,0},\ldots,c_{1,m_{1}-2};\lambda_{2}c_{2,m_{2}-1},c_{2,0},\ldots,c_{2,m_{2}-2};\ldots;\lambda_{\ell}c_{\ell,m_{\ell}-1},c_{\ell,0},\ldots,c_{\ell,m_{\ell}-2}). \end{aligned}\] + +Throughout this paper, we adopt the same notations as in [1, 2]. Thus, \(\mathcal{C}\) denotes a Λ-MT code over \(\mathbb{F}_{q}\) with block lengths \((m_{1},m_{2},\ldots,m_{\ell})\). The Euclidean dual \(\mathcal{C}^{\perp}\) of \(\mathcal{C}\) is a \(\left(\lambda_{1}^{-1},\lambda_{2}^{-1},\ldots,\lambda_{\ell}^{-1}\right)-\text{MT}\) code with the same block lengths. By using polynomial representation for blocks, \(\mathcal{C}\) can be regarded as an \(\mathbb{F}_{q}[x]-\)submodule of the Λ-MT module + +\[V=\bigoplus_{i=1}^{\ell} \frac{\mathbb{F}_{q}[x]}{\langle x^{m_{i}}-\lambda_{i}\rangle} = \frac{\mathbb{F}_{q}[x]}{\langle x^{m_{1}}-\lambda_{1}\rangle} \oplus \frac{\mathbb{F}_{q}[x]}{\langle x^{m_{2}}-\lambda_{2}\rangle} \oplus\cdots \oplus \frac{\mathbb{F}_{q}[x]}{\langle x^{m_{\ell}}-\lambda_{\ell}\rangle} .\] + +∗This research was conducted at Université d’Artois, La Faculté Jean Perrin in Lens, and was funded by the Science, Technology & Innovation Funding Authority (STDF); International Cooperation Grants, project number 49294. Ramy Takieldin would like to express his deepest gratitude to Professor André Leroy for his invaluable guidance throughout this project. diff --git a/outputs/arxiv_math/2503.03765_pg1_pg1_repeat1.md b/outputs/arxiv_math/2503.03765_pg1_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..a60b09807480233ca745102910cf1cc8f4e24d3d --- /dev/null +++ b/outputs/arxiv_math/2503.03765_pg1_pg1_repeat1.md @@ -0,0 +1,19 @@ +# STABLE RECOVERY GUARANTEES FOR BLIND DECONVOLUTION UNDER RANDOM MASK ASSUMPTION + +## SONG LI AND YU XIA + +Abstract. This study addresses the blind deconvolution problem with modulated inputs, focusing on a measurement model where an unknown blurring kernel \(\boldsymbol{h}\) is convolved with multiple random modulations \(\{\boldsymbol{d}_{l}\}_{l=1}^{L}(\)coded masks) of a signal \(\boldsymbol{x}\), subject to \(\ell_{2}-\)bounded noise. We introduce a more generalized framework for coded masks, enhancing the versatility of our approach. Our work begins within a constrained least squares framework, where we establish a robust recovery bound for both \(\boldsymbol{h}\) and \(\boldsymbol{x}\), demonstrating its near-optimality up to a logarithmic factor. Additionally, we present a new recovery scheme that leverages sparsity constraints on \(\boldsymbol{x}\). This approach significantly reduces the sampling complexity to the order of \(L=O(\log n)\) when the non-zero elements of \(\boldsymbol{x}\) are sufficiently separated. Furthermore, we demonstrate that incorporating sparsity constraints yields a refined error bound compared to the traditional constrained least squares model. The proposed method results in more robust and precise signal recovery, as evidenced by both theoretical analysis and numerical simulations. These findings contribute to advancing the field of blind deconvolution and offer potential improvements in various applications requiring signal reconstruction from modulated inputs. + +## 1. Introduction + +1.1. Problem Setup. Blind deconvolution is an inverse problem that aims to reconstruct two unknown signals, \(\boldsymbol{h},\boldsymbol{x}\in\mathbb{C}^{n}\), from their circular convolution \(\boldsymbol{y}\in\mathbb{C}^{n}\), defined as \(\boldsymbol{y}:=\boldsymbol{h}⊛\boldsymbol{x},\text{where}⊛\) denotes the circular convolution operator. This operation can be equivalently expressed in matrix form as \(\boldsymbol{y}=\boldsymbol{h}⊛\boldsymbol{x}=\boldsymbol{C}_{\boldsymbol{h}}\boldsymbol{x},\text{where}\boldsymbol{C}_{\boldsymbol{h}}\) is the circulant matrix generated by \(\boldsymbol{h}=[h_{1},\cdots ,h_{n}]^{T}\) , defined as: + +\[\boldsymbol{C}_{\boldsymbol{h}}= \begin{bmatrix} h_{1} & h_{n} & \cdots & h_{2} \\ h_{2} & h_{1} & \cdots & h_{3} \\ \ddots & \ddots & \ddots & \ddots \\ h_{n} & h_{n-1} & \cdots & h_{1} \end{bmatrix}\] + +| h1 | hn | · · · h2 | +| --- | --- | --- | +| h2 | h1 | · · · h3 | + +This problem arises in numerous fields, including astronomy, optics, image processing, and communications engineering [17, 11, 21, 33]. The blind deconvolution problem is inherently ill-posed due to the presence of scaled-shift symmetry, which implies that there are infinitely many signal pairs that can yield the same convolution result. Consequently, incorporating prior information is crucial to overcoming this ill-posedness. For example, one might impose a subspace condition on \(\boldsymbol{x}[2]\), or enforce a short support condition on \(\boldsymbol{h}\) in combination with a sparsity constraint on \(\boldsymbol{x}[19,31]\). In this work, we examine a related class of blind deconvolution problems, where the blur kernel \(\boldsymbol{h}\in\mathbb{C}^{n}\) is convolved with multiple modulated inputs. Specifically, the observations \(\boldsymbol{y}_{l}\in\mathbb{C}^{n}\), + +2020 Mathematics Subject Classification. Primary 94A15, 46C05; Secondary 94A12, 49N45. Key words and phrases. Self calibration, General random mask, Optimal complexity, Alternating minimization. Song Li is supported by NSFC grant (U21A20426, 12071426). Yu Xia was supported by NSFC grant (12271133, 11901143) and the key project of Zhejiang Provincial Natural Science Foundation grant (LZ23A010002). diff --git a/outputs/arxiv_math/2503.03766_pg12_pg1_repeat1.md b/outputs/arxiv_math/2503.03766_pg12_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..93b74f2039a9cb80a1e0759113b61c6fb2540d58 --- /dev/null +++ b/outputs/arxiv_math/2503.03766_pg12_pg1_repeat1.md @@ -0,0 +1,39 @@ +Now, we prove that \(\Psi_{c}\setminus\{(0,m):m\geq c\}\subset\Psi_{c}^{*}\). For any \(00\), every ordered + +pair in \(\Psi_{c}\) except for a region with Lebesgue measure 0 (namely the region \(\{(0,m):m\geq c\})\) is achievable by some random variable T. Specifically: + +- When \(\Pr\{T\geq c\}>0\), Markov’s inequality, namely + +\[E[T]\geq c\cdot\Pr\{T\geq c\},\] (10) + +which gives a lower bound on E[T], is the only constraint on E[T] in terms of \(\Pr\{T\geq c\}\). + +- When \(\Pr\{T\geq c\}=0\), Markov’s inequality as in (10), which becomes \(E[T]\geq0\), continues to + +be valid. However, we also have + +Combining (10) and (11), we have + +\[E[T]0\) are given by \(\alpha_{j}(v_{i})=0\text{for all}j\neq i,0\text{and}\alpha_{0}(v_{i})=1\). We now expand \(\alpha_{0}\) in the basis given by \(\Pi\) and find that + +\[\alpha_{0}=\sum_{i=1}^{d}c_{i}\alpha_{i}\] + +for some positive integers \(c_{i}\). We define \(\omega_{i}:=c_{i}v_{i}\) and then we note that \(\alpha_{i}(\omega_{j})= \delta_{ij}\). Thus the \(\omega_{i}\) form a basis for \(V\) dual to \(\Pi\) and they are called the fundamental coweights. Associated to \(C\) there is a fundamental Weyl chamber \(C^{+}:=\mathbb{R}_{\geq0}\cdot C\). Because \(C\) is a fundamental domain for the action of \(N\text{on}\mathbb{A}\), every vertex \(x\text{in}\mathbb{A}\) is \(G-\) conjugate to exactly one of the \(v_{i}\), which gives rise to a map \(\lambda:\mathbb{A}_{0}\rightarrow\{0,\ldots,d\}\) diff --git a/outputs/arxiv_math/2503.03861_pg30_pg1_repeat1.md b/outputs/arxiv_math/2503.03861_pg30_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..c5345815723910377927c256bacd4293af26c867 --- /dev/null +++ b/outputs/arxiv_math/2503.03861_pg30_pg1_repeat1.md @@ -0,0 +1,21 @@ +Since \(H^{2}(X^{\text{an}},\mathscr{O}_{X^{\text{an}}})\) is torsion free, any torsion element of \(H^{2}(X;Z)\) vanishes under \(\beta\). Therefore, to conclude the proof, it suffices to show \(\alpha\) is an injection. Since \(H^{1}(X,\mathscr{O}_{X^{\text{an}}}^{\times})\) is identified with the Picard group, to prove the desired injection, we only need to show \(H^{1}(X^{\text{an}},\mathscr{O}_{X^{\text{an}}})=0\). Using GAGA for Deligne-Mumford stacks, [Hal11, Proposition A.4], we have \(H^{1}(X^{\text{an}},\mathscr{O}_{X^{\text{an}}})=H^{1}(X,\mathscr{O}_{X})\). Since \(H^{1}(X;Q)=0\), we also have \(H^{1}(X;C)=0\), and hence we conclude \(H^{1}(X,\mathscr{O}_{X})=0\) using [Sat12, Corollary 1.7], which says that the Hodge de Rham spectral sequence degenerates for smooth proper Deligne-Mumford stacks. + +- + +7. 3. Proving the stable Picard rank conjecture. We now aim to prove Theorem 7.1.1. To do this, we next compute the first two stable cohomology groups of \([[\text{CHur}_{\in^{1},n}^{G,c}\) /G]/ PGL2]. To do so, we need a basic lemma about the number of connected components of Hurwitz spaces. + +Lemma 7.3.1. Let G be a group and \(c\subset\)Ga conjugacy class generating G. For n sufficiently large, the set of connected components of CHurcn with boundary monodromy \(g\in G\) is either empty or forms a torsor under H2(G, c); it is nonempty if and only if the image of n in Gab (under the map \(Z\rightarrow G^{\text{ab}}\) sending the positive generator to the image of any element of c) agrees with the image of g in Gab. + +Rephrasing the statement above, there are H2(G, c) many components if the image of n in Gab agrees with the image of g, and 0 components otherwise. Proof. This essentially follows from [Woo21] as we now explain. Indeed, using [Woo21, Theorem 2.5 and Theorem 3.1] we can identify the number of components of CHurcn for n sufficiently large with the set of elements in a certain reduced Schur cover \(S_{c}\rightarrow G\) having the same image in Gab as n. Moreover, the boundary monodromy of these components is the same as their image in G under the map \(S_{c}\rightarrow G\). The kernel of \(S_{c}\rightarrow G\) is identified with H2(G, c) and so connected components with boundary monodromy g either form a torsor under H2(G, c) when the image of n in Gab agrees with the image of g, or else there are no such connected components. For the next lemma and its proof, the reader may wish to recall notation from Notation 2.4.1. + +- + +Lemma 7.3.2. Let G be a group, \(c\subset G\) be a conjugacy class generating G, and \(R:=Z[1/2|G|]\). For n sufficiently large depending on c and for each component \(\tilde{Z}\subset[\text{CHur}_{\in^{1},n}^{G,c}\) /G], with corresponding component \(Z\subset[[\text{CHur}_{\in^{1},n}^{G,c}\) /G]/ PGL2], we have + +\[\begin{aligned} H^{1}(\tilde{Z};R)&=H^{1}(Z;R)=0, \\ H^{2}(\tilde{Z};R)&=H^{2}(Z;R)=((Z/(2n-2)Z)\otimes R). \end{aligned}\] + +Proof. Taking g = id in Lemma 7.3.1, we obtain that for n sufficiently large, both \([\text{CHur}_{n}^{G,c,\partial\in\text{id}}\) /G] and \([\text{CHur}_{\in^{1},n}^{G,c}\) /G] have \(|H_{2}(G,c)|\) many connected components. Indeed, the statement for \([\text{CHur}^{G,c,\partial\in\text{id}}\) trivially on H2(G, c) as it is identified with the central kernel of \(S_{c}\rightarrow G\) by definition. Since \([\text{CHur}^{G,c,\partial\in\text{id}}\) + +/G] follows from Lemma 7.3.1 and the fact that G conjugation acts + +/G] is dense open in \([\text{CHur}_{\in^{1},n}^{G,c}\) /G], we obtain \([\text{CHur}_{\in^{1},n}^{G,c}\) /G] also has diff --git a/outputs/arxiv_math/2503.03873_pg5_pg1_repeat1.md b/outputs/arxiv_math/2503.03873_pg5_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..319e44a0600a5770ab3b37aba430de3dcc0597e4 --- /dev/null +++ b/outputs/arxiv_math/2503.03873_pg5_pg1_repeat1.md @@ -0,0 +1,37 @@ +We want to study the relations between the dual group \(\widehat{A}\) of \(A\) and the dual groups \(\widehat{H}\) and \(A/H^{\cup}\) with the subgroup \(H^{\perp}\), called the annihilator of \(H\), defined as + +[ of the subgroup \(H\) and the quotient group \(A/H\). In fact, one can identify \(A/H\) + +\[H^{\perp}:=\left\{\chi\in\widehat{A}:\chi(x)=1\forall x\in H\right\},\] + +so that if we take \(\phi\in L^{1}(A)\) and define \(\phi^{H}\in L^{1}(A/H)\text{as}\phi^{H}(xH)=\int_{H}\phi(xh)dh\), then by + +the above identification, we get \(\mathscr{F}_{A/H}(\phi^{H})=\mathscr{F}_{A}(\phi)|_{H^{\perp}}\), see the proof of the theorem below, which explains this technique. + +Theorem 2.1 (General Poisson summation formula). Let \(H\) be a closed subgroup of the + +locally compact Abelian group \(A.\text{For}\phi\in L^{1}(A),if\mathscr{F}_{A}(\phi)|_{H^{\perp}}\in L^{1}(H^{\perp})\), then + +\[\int_{H}\phi(xh)dh=\int_{H^{\perp}}\mathscr{F}_{A}(\phi)(\chi)\chi(x)d\chi,\] (2.3) + +[ is the Plancherel measure with respect to + +for all \(x\in A\), where Haar measure on \(H^{\sim^{\perp}}=A/H\) the chosen Haar measure on \(A/H\). + +Proof. For \(\chi\in H^{\perp}\) we have \(\chi(xh)=\chi(x)\text{for every}x\in A\text{and}h\in H\). We therefore get from the quotient integral formula (2.2) that + +\[\begin{aligned} \mathscr{F}_{A/H}(\phi^{H})(\chi)&=\phi^{H}(xH) \overline{\chi(x)} d(xH) \\ &=\int_{A_{A}}^{A/H}\int_{/HH}\phi(xh) \overline{\chi(xh)} dhd(xH) \\ &=\phi(x) \overline{\chi(x)} dx=\mathscr{F}_{A}(\phi)(\chi) \end{aligned}\] + +\(A/H\) , then the Fourier inversion + +for every \(\chi\in H^{\perp}\). Moreover, if \(\mathscr{F}_{A}(\phi)|_{H^{\perp}}\in L^{1}(H^{\perp})=L^{1}\left(^{\cup}\right)\) formula implies that for all \(x\in A\) + +\[\begin{aligned} \int_{H}\phi(xh)dh&=\phi^{H}(xH)=\mathscr{F}_{H^{\perp}}\mathscr{F}_{A/H}(\phi^{H})(-xH) \\ &=\mathscr{F}_{H^{\perp}}\mathscr{F}_{A}(\phi)(-xH)=\int_{H^{\perp}}\mathscr{F}_{A}(\phi)(\chi) \overline{-xH(\chi)} d\chi \\ &=\int_{H^{\perp}}\mathscr{F}_{A}(\phi)(\chi) \overline{\chi(-x)} d\chi=\int_{H^{\perp}}\mathscr{F}_{A}(\phi)(\chi)\chi(x)d\chi. \\ □ \end{aligned}\] + +In all our applications, the group \(A\) will be of the form \(A=(\mathbb{Z}/N\mathbb{Z})^{d_{1}}\times\mathbb{R}^{d_{2}}\text{for}N\in\mathbb{N}_{+}\), + +beginning of §1, and for any \(N\in\mathbb{N}_{+}\), we still denote the symmetric bilinear form on \((\mathbb{Z}/N\mathbb{Z})^{d_{1}}\) + +\(d_{1},d_{2}\in\mathbb{N}\). Recall we have defined the standard symmetric bilinear form on \(\mathbb{R}^{d_{2}}\) as in the by \(Q\) as defined in the standard way. We then identify \(A\) with its dual group \(\widehat{A}\) through the pairing + +\[\begin{aligned} A&\cong \widehat{A} \\ (s,t)&\mapsto\chi_{s,t}(l,\xi)=e^{(^{2\pi i} \frac{Q(s,l)}{N} +Q(t,\xi))}, \end{aligned}\] diff --git a/outputs/arxiv_math/2503.03879_pg4_pg1_repeat1.md b/outputs/arxiv_math/2503.03879_pg4_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..3d1328a9afb85b10172c4d3404d438d84262a9d7 --- /dev/null +++ b/outputs/arxiv_math/2503.03879_pg4_pg1_repeat1.md @@ -0,0 +1,55 @@ +![A diagram showing a timeline labeled t from t_{i-2} to t_{i+2}, with markers at each time step. Green stars represent points in the past, red crosses represent points in the future, and blue stars represent points near the current time \( t_i \). A blue arrow labeled 'Move Finite Element' points from \( t_i \) to the right.](figure)[^inferred] + +![A diagram showing a timeline labeled t from t_{i-2} to t_{i+2}, with markers at each time step. Green stars represent points in the past, red crosses represent points in the future, and blue stars represent points near the current time \( t_i \). A blue arrow labeled 'Move Finite Element' points from \( t_i \) to the right.](figure)[^inferred] + +Fig. 2. Location of switch point at the end of finite element + +The proposed step-equilibration approach is based on the principle that only the finite element(s) with the switch point(s) have non-uniform discretization \([t_{l-1},t_{l}]\). The approach uses an indicator variable \((\eta)\) for the switch point(s), which is determined by calculating and multiplying the sum of the complementarity variables in two consecutive finite elements. The auxiliary variables for the sum of the complementarity variables at each finite element are defined as: + +\[\hat{\lambda}_{l}=\sum_{k=1}^{K}\lambda_{l,k},\hat{\nu}_{l}=\sum_{k=1}^{K}\nu_{l,k}\] (10a) + +Then, the Hadamard product of the forward and backward sum of the complementarity variables determine if they have switched from positive to zero (or vice-versa). + +\[\pi_{l}^{\lambda}=\hat{\lambda}_{l-1}\odot\hat{\lambda}_{l},\pi_{l}^{\nu}=\hat{\nu}_{l-1}\odot\hat{\nu}_{l}\] (10b) + +(Here, ⊙ represents pointwise or elementwise product of vectors.) Since at least one of the vectors \(\pi_{l}^{\lambda}\text{or}\pi_{l}^{\nu}\) is zero at each element, and they are exactly equal to zero at the element corresponding to the switching point, the sum of the two vectors is a good candidate for the indicator function + +\[\tau_{l}=\pi_{l}^{\lambda}+\pi_{l}^{\nu},\eta_{l}=\prod_{j=1}^{n_{f}}\tau_{l,j}\] (10c) + +Since the indicator variable \(\eta_{l}\) is non-negative and only zero at the switching element, the relation between step size and indicator variable can be represented by the following complementarity constraints. + +\[\begin{aligned} \text{For}l&=1,\ldots,N-1 \\ 0&\leq(\Delta h_{l}^{+}+\Delta h_{l}^{-})\perp\eta_{l}\geq0 \end{aligned}\] (10d) + +where \(h_{l-1}-h_{l}=\Delta h_{l}^{+}-\Delta h_{l}^{-},\Delta h_{l}^{+},\Delta h_{l}^{-}\geq0\), The finite element with switch detection (FESD) algorithm was implemented as a package NOSNOC in [19]. As mentioned in (10), the Nurkanovic formulation augments an additional \([2N\{\hat{\lambda},\hat{\nu}\}+(2N-2)\{\pi^{\lambda},\pi^{\nu}\}+(N- 1)\{\tau\}+(N-1)\{\eta\}]\) variables and \([2N\{(10\mathrm{a})\}+(2N- 2)\{(10\mathrm{b})\}+2(N-1)\{(10\mathrm{c})\}+(N-1)\{(10d)\}]\) constraints for each complementarity constraint. This effectively decreases the degrees of freedom by \(N-1\) to that of the original problem and avoids non-unique solutions for the step size variables \(h_{i}\). + +### 2.4 Proposed Formulation + +Although the Nurkanovic formulation makes the problem consistent with respect to the degrees of freedom and ensures uniform grid discretization away from the switch point(s), the formulation may be numerically unstable (i.e. the derivatives have large condition number) and increases the size of the problem, making it difficult to implement on larger optimal control problems. Inspired by the Nurkanovic [21] formulation, we propose a modification of the approach in [1], in order to keep the degrees of the problem consistent. In our proposed approach, we first apply the formulation in [1] with cross complementarities (9f). This locates the switching point(s) \((t_{s})\) at the end of the finite element(s). We define the set of finite elements which have the switching point at the end (i.e. right) as: + +\[\lambda_{l-1,K}+\nu_{l-}\chi_{1^{s},K}=_{>}\{l|_{0}[t_{\mathrm{o}^{l-}\mathrm{r}^{1}},_{\lambda}t_{l^{l}+}],_{1,K}\lambda_{+^{l,K}\nu}=_{l+}\nu_{>_{1,K}^{l,K}}=_{0}0_{\}},_{(11\mathrm{a})}\] + +In the next step, we add additional constraints to the formulation which forces the finite elements to be equally spaced away from the switching points. + +\[h_{l}-h_{l+1}=0\forall l\in\{1,\ldots,N-1\}\setminus\chi_{s}\] (11b) + +Also, we add constraints to force the switching to happen at the boundary of finite elements found in the first step. + +\[\lambda_{l,K}+\nu_{l,K}=0\forall l\in\chi_{s}\] (11c) + +This formulation adds the necessary \(N-1\) linear constraints without any additional variables making the formulation much more adaptable and applicable for large optimal control problems. The main assumption in our approach is that the location of the switching points in the optimal solution is independent of the step-size variables and the formulations. Thus, the switching points in the Baumrucker formulation would be the same as in the Nurkanovic formulation. The only difference between their solutions is in the value of the step size variables away from the switching elements. Therefore, we implement uniform discretization between the switch points, start time and the final time using (11) instead. + +## 3. SOLUTION METHODS FOR MPCCS + +To develop the solution strategy for the MPCC derived in the previous section, we discretize and rewrite (7) in the more general form: + +\[\begin{aligned} \min\varphi(x) \\ \text{s.t.}g_{I}(x)&\leq0;g_{E}(x)=0, \\ 0&\leq G(x)\perp H(x)\geq0 \end{aligned}\] (12a) (12b) (12c) + +Here the complementarity constraints (12d) represent the cross-complementarity constraints (9f) and step equilibration constraints (10e). The NLP equivalent formulation of the complementarity constraints is + +\[G(x)\geq0,H(x)\geq0,G_{i}(x)H_{i}(x)=0,\forall i=1\ldots n_{c}\] + +### 3.1 MPCC Basics and Stationary Points + +The following index sets are defined at every feasible point \(\bar{x}\) of the MPCC (12): + +[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter. diff --git a/outputs/arxiv_math/2503.03899_pg9_pg1_repeat1.md b/outputs/arxiv_math/2503.03899_pg9_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..ff4bdd7c0dc5dae144860fce837de0cbb1d10112 --- /dev/null +++ b/outputs/arxiv_math/2503.03899_pg9_pg1_repeat1.md @@ -0,0 +1,27 @@ +Proposition A.1. For fixed \(0<\delta< \frac{1}{4}\), we have + +\[\limsup_{n\rightarrow\infty}n^{-\delta}\log P_{n}\left(\inf_{t\geq0}\Bigg| \frac{1}{\sqrt{n}} \sum_{k\leq t \sqrt{n}}X_{k}-L(t)\Bigg|>n^{- \frac{1}{4} +\delta}\right)<0.\] + +Remark. In particular, we have + +\[\lim_{n\rightarrow\infty}P_{n}\left(\sup_{t\geq0}\Bigg| \frac{1}{\sqrt{n}} \sum_{k\leq t \sqrt{n}}X_{k}-L(t)\Bigg|\leq n^{- \frac{1}{4} +\delta}\right)=1.\] + +Proof. Let \(d(n)\) be the number of distinct parts partitions of \(n\). We require only a weak form of the well-known asymptotic expansion of \(d(n)[11]\), + +\[d(n)=e^{\frac{2 \sqrt{n}}{A} +O(\log n)}.\] + +Let \(a_{n},b_{n}\in\mathbb{N}_{0}\) and define \(\alpha_{n},\beta_{n}\) by √ + +\[a_{n}=\alpha_{n} \overline{n} ,b_{n}=\beta_{n} \overline{n} .\] + +Assume that \(\alpha_{n},\beta_{n}\geq n^{- \frac{1}{4} +\delta}\). We use the saddle point bound to write, for any \(x_{n}\in\mathbb{R}\), + +\[\begin{aligned} P_{n}\left( \frac{1}{\sqrt{n}} \sum_{k\leq a_{n}}X_{k}=b_{n}\right) \\ &= \frac{1}{d(n)} [q^{n}][\zeta^{b_{n}}]\prod_{k\leq a}(1+\zeta q^{k})\prod_{k>a_{n}}(1+q^{k}) \\ &\leq \frac{1}{d(n)} q_{n}^{-n}e^{-b_{n}x_{n}}\prod_{k\leq^{n}a_{n}}(1+e^{x_{n}}q_{n}^{k})\prod_{k>a_{n}}(1+q_{n}^{k}) \\ &=\exp\left( \frac{\sqrt{n}}{A} -\log d(n)-\beta_{n}x_{n} \sqrt{n} +\sum_{k\leq a_{n}}\log\left(1+e^{x_{n}- \frac{k}{A \sqrt{n}}}\right)+\sum_{k>a_{n}}\log\left(1+e^{- \frac{k}{A \sqrt{n}}}\right)\right). \end{aligned}\] + +We will take \(x_{n}\in\{\pm n^{- \frac{1}{4}}\}\), where the sign will depend on \(b_{n}\). By Taylor’s Theorem, + +\[\begin{aligned} \Bigg|\log\left(1+e^{x_{n}- \frac{k}{A \sqrt{n}}}\right)-\log\left(1+e^{- \frac{k}{A \sqrt{n}}}\right)-x_{n} \frac{e^{- \frac{k}{A \sqrt{n}}}}{1+e^{- \frac{k}{A \sqrt{n}}}} \Bigg|&\leq \frac{x_{n}^{2}}{2}_{|x} \sup_{|\leq n^{\frac{1}{4}}} \frac{e^{x- \frac{k}{A \sqrt{n}}}}{\left(1+e^{x- \frac{k}{A \sqrt{n}}}\right)^{2}} \\ &\leq x_{n}^{2} \frac{e^{- \frac{k}{A \sqrt{n}}}}{\left(1+ \tfrac{1}{2} e^{- \frac{k}{A \sqrt{n}}}\right)^{2}} . \end{aligned}\] + +Thus, + +\[\begin{aligned} P_{n}\left( \frac{1}{\sqrt{n}} \sum_{k\leq a_{n}}X_{k}=b_{n}\right)&\leq\exp\big( \frac{\sqrt{n}}{A} -\log d(n)+\sum_{k\geq1}\log\left(1+e^{- \frac{k}{A \sqrt{n}}}\right)-\beta_{n}x_{n} \sqrt{n} \\ +x_{n}\sum_{k\leq a_{n}} \frac{e^{- \frac{k}{A \sqrt{n}}}}{1+e^{- \frac{k}{A \sqrt{n}}}} +O\left(x_{n}^{2}\sum_{k\leq a_{n}} \frac{e^{- \frac{k}{A \sqrt{n}}}}{\left(1+ \tfrac{1}{2} e^{- \frac{k}{A \sqrt{n}}}\right)^{2}} \right)\big) \end{aligned}\] diff --git a/outputs/arxiv_math/2503.03903_pg9_pg1_repeat1.md b/outputs/arxiv_math/2503.03903_pg9_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..7892c6982ea0dd985dc07541d77a20cb47befa0d --- /dev/null +++ b/outputs/arxiv_math/2503.03903_pg9_pg1_repeat1.md @@ -0,0 +1,13 @@ +![](figure) + +![](figure) + +Figure 5. Above is the bottom pipe dream for 1427356 (cross signs denote crossings, and empty boxes denote non-crossings). If we draw a diagonal line out from the first crossing in row 3, then this diagonal line never intersects or is directly to the right of another crossing. However, the diagonal line emitting from the first box in row 5 enters the square directly to the right of the last crossing of row 3. Indeed, the subsequence 473 is a 231 pattern. + +- a square containing a crossing in \(P\) , nor does any square immediately to the left of this line contain a crossing in \(P\) . See Figure 3.2 for an example. Proof. For contradiction, let \(i\) the largest row index such that the line emanating from the leftmost crossing of \(\text{row}i\) intersects or is directly to the right of a crossing in \(\text{row}j\). Since \(L(i)>0\text{and}i\) was the largest such row, we must have \(L(i+1)=0\), and thus, since \(L(i)>L(i+1)\), we have \(w(i)>w(i+1)\). But then, we claim that \(w(j)>w(i+1)\). Either \(w(j)>w(i)\), in which this follows by transitivity, or \(w(j)0\). Either way, we have either a 231 or a 321 pattern in \(w\) given by the indices \(j,i,i+1\). So, we have proved the claim. Lemma 3.6 allows us to prove that 321 and 231-avoidance are sufficient: Lemma 3.7. If \(w\) avoids 231 and 321, then \(S_{w}\) is a single CHM. In particular, \(S_{w}=h_{L(w)}\). Proof. We use again the fact that all pipe dreams are obtained from the bottom pipe dream by ladder moves. Here, again all of the ladder moves we can perform are simple ladder moves that just move crossings along their diagonals. Two crossings in the same row of the bottom pipe dream can never slide past each other, and any two rows can slide independently by Lemma 3.6. Thus, \(\text{row}i\) contributes a factor \(h_{L(i)}^{i}\), and multiplying these factors gives us \(S_{w}=\prod_{i}h_{L(i)}^{i}\). Notice that, unlike the case of SEMs and analogously to the case for usual monomials, the maximal monomial in the CHM expansion of \(S_{w}\) is always \(h_{L(w)}\). Finally, we prove that 321 and 231 avoidance are necessary conditions in order for \(S_{w}\) to be a CHM. + +- + +## Lemma 3.8. If \(w\) contains a 321 pattern, then \(S_{w}\) is not a CHM. + +Proof. We induct on the length of \(w\). First, suppose that \(w\) contains a 321 pattern and \(S_{w}\) is a single CHM. Let \(iw_{j}>w_{k}\). First, we show how to reduce to the case where \(i,j,k=i,i+1,i+2\). diff --git a/outputs/arxiv_math/2503.03905_pg7_pg1_repeat1.md b/outputs/arxiv_math/2503.03905_pg7_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..c02d5763586e0dd2d5d3c3d8432dde8db67d6ec7 --- /dev/null +++ b/outputs/arxiv_math/2503.03905_pg7_pg1_repeat1.md @@ -0,0 +1,47 @@ +We show that \(\sigma^{\prime}\) is bijective on \(\text{GF}(2^{m})\). Let us fix \(u,y_{1},y_{2}\in\text{GF}(2^{m}),y_{1}\neq y_{2}\). There is an + +element \(v\in\text{GF}(2^{t})\) such that + +\[\text{Tr}_{\text{GF}(2^{t})/\text{GF}(2)}(uv)\neq h^{\prime}(y_{1})+h^{\prime}(y_{2}).\] + +As \(\tau\) is surjective, there is a \(z\in\text{GF}(2^{m})\text{with}\tau(z)=v\). By Lemma 7, there is \(x\in\text{GF}(2^{m})\) with \(y_{1}\star x+y_{2}\star x=z\). Then + +\[\begin{aligned} \alpha_{y_{1}}(x)+\alpha_{y_{2}}(x)&=\text{Tr}_{\text{GF}(2^{t})/\text{GF}(2)}(u\tau(y_{1}\star x+y_{2}\star x)+uh(y_{1})+uh(y_{2})) \\ &=\text{Tr}_{\text{GF}(2^{t})/\text{GF}(2)}(u\tau(z))+h^{\prime}(y_{1})+h^{\prime}(y_{2}) \\ &=\text{Tr}_{\text{GF}(2^{t})/\text{GF}(2)}(uv)+h^{\prime}(y_{1})+h^{\prime}(y_{2}) \\ &\neq 0. \end{aligned}\] + +This implies \(\alpha_{y_{1}}(x)\neq \alpha_{y_{2}}(x)\text{and}\sigma^{\prime}(y_{1})\neq \sigma^{\prime}(y_{2})\). It follows that all component functions of \(f\) are of Maiorana-McFarland type bent function. This finishes the proof of the theorem. + +- + +In this paper, we do not study the question of EA-equivalence of the \((2m,t)-\)bent functions defined above. In general, this is a very difficult question. We only remark that Weng, Feng and Qui [23] proved that most of the \(\mathcal{P}\mathcal{S}\) type bent functions, obtained from a Desarguesian spread are not EA-equivalent to any Maiorana–McFarland bent function. This leads us to conclude that, typically, for \(t>1\), the \((2m,t)-\)bent functions described by (9) are generally not EA-equivalent to the other two classes, as specified in (6) and (8). + +3. 4. Proof of Theorem 1. Let us recall the Carlet-Ding-Yuan bound (2) for the distance between affine and \((n,m)-\)bent functions: + +\[\left(1- \frac{1}{2^{m}} \right)\left(2^{n}-2^{n/2}\right)\leq d_{H}(f,\mathcal{A})\leq\left(1- \frac{1}{2^{m}} \right)\left(2^{n}+2^{n/2}\right).\] + +For an \((n,m)-\)bent function \(f\), the Walsh coefficients are + +\[W_{f}(a,b)= \begin{cases} \pm2^{n/2} & \text{if}b\neq 0, \\ 0 & \text{if}a\neq 0,b=0, \\ 2^{n} & \text{if}a=0,b=0. \end{cases}\] + +Hence, the Carlet-Ding-Yuan bound follows from Lemma 3 easily. The Liu-Mesnager-Chen + +Conjecture implies that the true value of \(d_{H}(f,\mathcal{A})\) is \(\left(1- \frac{1}{2^{m}} \right)\left(2^{n}-2^{n/2}\right)\). Theorem 1 claims that this holds for two classes of \((n,m)-\)bent functions. + +Proof of Theorem 1. Let \(E_{i}\) be the set of pairs \((x,y)\in\text{GF}(2^{m})^{2}\text{such that}f_{i}(x,y)=f_{i}(0,0)\), + +\(i=1,2\). We show that \(|E_{i}|=2^{2m-t}+2^{m}-2^{m-t}\), which implies that \(f_{i}\) has Hamming distance \(2^{m})\), and the theorem follows from the Carlet-Ding-Yuan bound. + +\((1-2^{-t})(2^{2m}-2^{m})\) from the constant function \(f_{i}(0,0)\). Therefore, \(d_{H}(f_{i},\mathcal{A})\leq(1-2^{-t})(2^{2m}-\) + +Let \(T\) be the set of elements \(z\in\text{GF}(q^{m})\text{with}\gamma(z)=f_{1}(0,0)\). Since \(\gamma\) is balanced, \(|T|=2^{m-t}\), + +and \(f_{1}(x,y)=f_{1}(0,0)\) if and only if \(\star \frac{x}{y} \in T\) . Moreover, since \(\star \frac{0}{0} =0\), we have \(0\in T\) . The + +number of solutions of \(\star \frac{x}{y} =0\) is \(2\cdot2^{m}-1\), and the number of solutions of \(\star \frac{x}{y} =t\in T\setminus\{0\}\) is + +\(2^{m}-1\). This implies + +\[|E_{1}|=|f_{1}^{-1}(f_{1}(0,0))|=2\cdot2^{m}-1+(2^{m-t}-1)(2^{m}-1).\] + +The same argument applies to \(f_{2}\). In this case, \(f_{2}(0,0)=h(0)\), and the number of solutions of □ + +\(\sigma(y)\star x=t\) is \(2\cdot2^{m}-1\text{or}2^{m}-1\), depending on \(t=0\text{or}t\neq 0\). diff --git a/outputs/arxiv_math/2503.03909_pg14_pg1_repeat1.md b/outputs/arxiv_math/2503.03909_pg14_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..3f976446422b9275def0155fe419dedc714e0d4c --- /dev/null +++ b/outputs/arxiv_math/2503.03909_pg14_pg1_repeat1.md @@ -0,0 +1,62 @@ +## 4.2.2. Bratu problem. We use lrAA to solve the non-linear Bratu problem + +\[u_{xx}+u_{yy}+\lambda e^{u}=0,(x,y)\in[0,1]\times[0,1],\] + +with \(\lambda=1\) and homogeneous Dirichlet boundary conditions. To find the approximate solution we discretize this equation using standard second order finite difference approximations for the \(x\text{and}y\) derivatives. Given an approximation \(X(i,j)\approx u(x_{i},y_{j})\) this results in a function \(G_{\mathrm{B}}(i,j;X)\) describing the equation + +\[G_{+^{\mathrm{B}}} \frac{(_{1}i,}{h_{x}^{2}} j(^{;}X^{X}(^{)}i,^{=}j \frac{1}{+^{h_{x}^{2}}1} )(X_{-}(_{2}i_{X}+_{(i}1_{,},_{j}j_{)})_{+}-_{X}2X_{(i}(_{,}i_{j},j_{-})_{1}+_{))}X_{+}(i_{\lambda}-_{e^{X}}1_{(},_{i,}j_{j})_{)})_{.}\] (4.2) + +\[\frac{1}{m+1}\] + +Near the boundaries some of the terms in this expression will be set to zero to account for the homogeneous Dirichlet boundary conditions. In the numerical examples below we take the mesh to be \(x_{i}=ih_{x},h_{x}=\) and \(y_{j}=jh_{y},h_{y}= \frac{1}{n+1} \text{with}m=n=200\), making the setup the same as in [52]. The fixed point function \(G(i,j)\) is obtained by applying the preconditioned Richardson iteration. We have + +\[X^{k+1}(i,j)=G(i,j;X^{k},\alpha)\equiv X^{k}(i,j)+\alpha M(G_{\mathrm{B}}(i,j;X^{k})).\] + +We test lrAA with no preconditioner and with the ES preconditioner (corresponding to Rel1_x_n10.1E10) described above. The lrAA parameters used are the following \(\text{TOL}=10^{-6},\hat{m}=5,\theta=0.9,\alpha=0.125h_{x}^{2}\) (un-preconditioned case) and \(\alpha=0.1\) (preconditioned case, no scheduling is used). In all experiments we use a rank 1 matrix with Frobenius norm around 1. The results in Figure 11 display the numerical solutions obtained by lrAA methods with and without the ES preconditioner. In particular, both methods obtain visually similar numerical results in terms of solution contours and column and row index section for the final iterates. The un-preconditioned lrAA gives a montonically increasing intermediate ranks. The ES-preconditioned lrAA converges very rapidly in 8 iterations. + +0 + +102 + +no preconditioner Preconditioned + +-0.01 + +0.8 + +-0.02 + +100 + +0.6 + +-0.03 k + +10-2 + +0.4 + +| | | -0.04 | | | [icon] | | | | | +| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | +| 0.2 | | 10-4 | | | | [icon] | [icon] | | | +| | | -0.05 | | | | | | | | +| | | -0.06 10-6 | | | | | | [icon] | ↑ | +| 0.2 | 0.4 0.6 0.8 | | 0 | 2 | 4 | | 6 | 8 | 10 | +| | x | | | | | k | | | | +| | | 100 | | | | | | | | +| 10 → | | | | | | | | no preconditioner | | + +preconditioned + +8 + +6 + +| 4 → | | | 10-5 | | +| --- | --- | --- | --- | --- | +| 2 ● | | no preconditioner preconditioned ● | | | +| 0 | | | | | +| 0 | 100 | 200 300 400 500 | 0 | 100 200 300 400 500 | +| | | # iteration | | # iteration | + +Fig. 11. Bratu problem solved by lrAA. The top left figure displays the contour levels of the converged solution along with markers at the intersection points of the final index sets \(\mathcal{I}\text{and}\mathcal{J}\) (for both lrAA with and without ES preconditioner). The top right figure displays the singular values from lrAA solutions with and without the ES preconditioner. The bottom left figure and right figures display the rank evolution and the decay of the residual throughout the lrAA iterations for lrAA with and without the ES preconditioner. diff --git a/outputs/arxiv_math/2503.03948_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.03948_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..d12b1f3c15b0f5c383907871bb2962bafba5c8b9 --- /dev/null +++ b/outputs/arxiv_math/2503.03948_pg3_pg1_repeat1.md @@ -0,0 +1,39 @@ +2 + +−1 + +−2 + +−3 + +![](figure) + +\(\Gamma\) + +Egap + +K + +M Brillouin Zone + +−xm + +![](figure) + +FIG. 1. + +![](figure) + +0 Position x xm + +![](figure) + +Closing + +Closing + +0.3 Local Gap \(\mu_{(0,0,E)}\) + +(a) Bulk band structure for the \(C_{2}\mathcal{T}\) -symmetric TBG model. Inset diagrams this lattice model. (b) The local gap \(\mu_{(x,y,E)}\text{for}x\text{and}E\) at \(y=0\). Position \(x\) is scaled in terms of the lattice constant \(a\text{and}x_{m}\) denotes the length of the edge from the origin. (c) \(\mu_{(x,y,E)}\text{for}E\) at the center of rotation denoted by blue dotted line in panel (b). The energy-resolved invariant \(\zeta_{E}\) is shown by the green dots, where \(\kappa=0.1t/a\) for all calculations. + +Specifically, when perturbing the system \(H\rightarrow H+\delta H\), \(\zeta_{E}\) is guaranteed by Weyl’s inequality to be preserved so long as \(\Vert\delta H\Vert<\mu_{(0,0,E)}(X,Y,H)[60,61]\). In addition, as locations where \(\mu_{(x,y,E)}=0\) are associated with the locations of a system’s states [62], changes in a system’s topological marker necessarily imply changes in the structure of its states. A detailed mathematical discussion of Eq. (9), its essential properties, and its relation to atomic limits is given in Supplementary Secs. SI.E-G. In particular, Examples SI.11 and SI.12 show the form of the two different classes of atomic limits distinguished by \(\zeta_{E}\). Summarizing our derivation, we first used a system’s physical symmetries to define a transpose-like matrix operation that translates the physical symmetries to matrix symmetries of the system’s \(H,X,\text{and}Y\) . Then, we found a change of basis that transformed this matrix operation into the standard matrix transpose, such that in this atypical basis the system’s operators were either symmetric or skew-symmetric. Finally, by tensor-ing these operators using the Pauli matrices, we formed a single skew-symmetric matrix whose Pfaffian’s sign discriminates between which atomic limits a given system can be connected to without closing the system’s local gap. Altogether, by using results from matrix homotopy, this argument yields an energy-resolved invariant for classifying fragile topology as well as a quantitative measure of topological protection. Having derived a classification framework applicable to finite systems, we demonstrate its use in a four-band model that is a low-energy approximation of TBG and exhibits fragile topology [18, 21]. This model consists of a bilayer honeycomb lattice, as schematically shown in Fig. 1(a), where \(t_{1}\text{and}t_{2}\) represent the intra- and inter-layer hopping amplitudes, respectively. The blue lines spirally connecting inter-layer sites represent next-nearest neighbor (NNN) hoppings with the hopping phase \(\pm\phi\), which can induce a nontrivial fragile band gap. We have provided the expressions of the Hamiltonian in both position and momentum space the End Matter and further information in Supplementary Sec. SII. Comparison of the bulk band structure of the infinite four-band TBG model with the local gap of a finite system confirms that the locations in \((x,y,E)-\)space with \(\mu_{(x,y,E)}\approx0\) indicate the presence of states at the specified energies and positions, see Figs. 1(a),(b). For choices of \(E\) residing within the spectral extent of the bulk bands, extended Bloch states are distributed throughout the system, whereas within the bulk band gap, only localized states exist at the system’s boundaries. Note that the fluctuation pattern of \(\mu_{(x,0,E)}\) only intermittently touching zero near the band gap inherently suggests the weak topological nature of this fragile system; strong topological phases instead exhibit a spheroid of appropriate dimension where \(\mu_{(\mathbf{x},E)}=0\) (see Supplementary Sec. SI.G). Within the bulk band gap, the energy-resolved marker \(\zeta_{E}\) confirms the finite system’s fragile topology, while the large local gap at the rotation center indicates this phase’s strong topological protection, see Fig. 1(c). For energies above and below the bulk gap, \(\zeta_{E}\) maintains a nontrivial value of −1 until the first closing points of \(\mu_{(0,0,E)}\), beyond which it switches to +1. Although the exact energy where \(\mu_{(0,0,E)}=0\) may vary with the parameter \(\kappa\) (see Supplementary Sec. SIII), these spectral regions with small local gaps are not topologically robust, as very small system perturbations are able to change the topology. To confirm that the local fragile marker \(\zeta_{E}\) captures phase transitions, we uniformly vary \(\phi\) between all NNN sites from \(-\pi\) to \(\pi\). As can be seen in Fig. 2(a), the width of the bulk spectral gap under this alteration is symmetric about \(\phi=0\) and touches zero twice at \(\phi=\pi/3\) and \(\phi=2\pi/3\). Similarly, the local gap closes at precisely the same points where \(E_{\text{gap}}=0\text{and}\zeta_{E}\) changes across these values of \(\phi\), indicating a change in the material’s fragile topological phase. As our framework works directly with a finite system expressed in position-space, it can inherently be applied to disordered and aperiodic systems. To illustrate this capability, we consider an ensemble of disordered variants of the four-band TBG model where the hopping phases \(\phi_{jk}\) between each pair of NNN sites \(j\text{and}k\) are randomly assigned a value within an angle range of \(2S\) from \(\pm i\) from a uniform distribution while preserving \(C_{2}\mathcal{T}\) -symmetry. Therefore, \(S\) represents the median value of the disorder strength, allowing us to investigate the phase diagram of the disordered system based on this variable. In Figs. 2(d), (e), and (f), we present \(E_{\text{gap}},\mu_{(0,0,E)},\text{and}\zeta_{E}\) as functions of \(S\) for 10 different disorder realizations. We numerically observe that the disordered samples exhibit diff --git a/outputs/arxiv_math/2503.03949_pg1_pg1_repeat1.md b/outputs/arxiv_math/2503.03949_pg1_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..aabd624a0519ef431aaba8676a93d0e637e761d3 --- /dev/null +++ b/outputs/arxiv_math/2503.03949_pg1_pg1_repeat1.md @@ -0,0 +1,33 @@ +# Volumes in Calabi-Yau Complete Intersection of Products of Projective Space + +##### Yi-Heng Tsai + +Abstract. We prove that the birational automorphism group of a general Calabi-yau complete + +intersection \(X\) given by ample divisors in \(\mathbb{P}^{n_{1}}\times\cdots \times\mathbb{P}^{n_{l}}\) is always Lorentzain. Applying the Kawamata-Morrison cone theorem on such \(X\), we compute \(\text{vol}_{X}(D+sA)\) for any divisor \(D\in \partial \overline{\text{Eff}} (X)\) and ample divisor \(A\text{when}s\) is small. We also provide examples of volumes of certain Cartier divisors that involve the digamma function. + +## 1 Introduction + +It is conjectured in [Mor93] and [Kaw97] that the movable effective cone of a Calabi-Yau manifold has a rational fundamental domain under the action of the birational automorphism group. + +Conjecture 1.0.1 (Kawamata-Morrison cone conjecture) Let \(X\) be a Calabi-Yau manifold. Then there exists a rational polyhedral fundamental domain \(\Pi\) for the action of the birational automorphism group \(\text{Bir}(X)\) on the movable effective cone Mov\((^{e}X)\) in the sense that + +\[(\mathrm{i}) \overline{\text{Mov}}^{e} (X)=\bigcup_{g\in\mathrm{B}\text{ir}(X)}g^{*}\Pi,\] + +\[(\text{ii})\text{int}(\Pi)\cap\text{int}(g^{*}\Pi)=\emptyset ifg^{*}\neq id.\] + +The conjecture has been proven in the case when \(X\) is a general Wehler \(N-\)fold i.e. a general + +hypersurface of multidegree \((2,\cdots ,2)\text{in}(\mathbb{P}^{1})^{N+1}\text{for}N\geq3\) ([CO15, Theorem 1.3]). In [FLT23], the structure of the boundary of the pesudoeffective cone \(\overline{\text{Eff}} (X)\) has been studied. The divergent–recurrent decomposition of \(\partial \overline{\text{Eff}} (X)\) is proven in [FLT23, Theorem 2.4.2], and the following result on the asymptotic behavior of volume function near \(\partial \overline{\text{Eff}} (X)\) is derived: + +Theorem 1.0.2 ([FLT23, Theorem 1.3.7]) Suppose \(X\) is a general Wehler Calabi-Yau \(N-\text{fold}\). Then, for every pseudoeffective \(\mathbb{R}-divisorDin\partial \overline{\text{Eff}} (X)\) and sufficiently ample divisor \(A\), there + +exists an integer \(s(D)\in\{0,\cdots ,N-2\}\) and a real number \(\delta(D)\in[1, \frac{1}{2} (N-s(D))]\) such that + +\[\begin{aligned} \liminf_{s0} \frac{\log\text{vol}(D+sA)}{\log s} &=\delta(D), \\ \limsup_{s\downarrow0}^{\downarrow} \frac{\log\text{vol}(D+sA)}{\log s} &= \frac{N-s(D)}{2} . \end{aligned}\] + +The fact that the birational automorphism group of a general Wehler \(N-\)fold is Lorentzian provides a hyperbolic subspace in \(\mathbb{P}N^{1}(X)\), and plays an important role when investigating the stucture + +of \(\partial \overline{\text{Eff}} (X)\) (see [FLT23, §2.3]). Let \(\mathbf{n}=(n_{1},\cdots ,n_{l})\in\mathbb{N}\text{such that}|\mathbf{n}|\geq4\text{and}\mathbf{n}\neq (2,2)\). In [Yá22], the author generalized the results in [CO15], and proved the Kawamata-Morrison + +cone conjecture when \(X\) is a general Calabi-Yau complete intersection in \(\mathbb{P}^{\mathbf{n}}:=\mathbb{P}^{n_{1}}\times\cdots \times\mathbb{P}^{n_{l}}\) given by the intersection of \(n\) ample divisors, with \(n\leq\min\{n_{i}\}\). He also conjectured that the diff --git a/outputs/arxiv_math/2503.03952_pg5_pg1_repeat1.md b/outputs/arxiv_math/2503.03952_pg5_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..57d7a1d70b4e24d823d7d4308f88970fa4a9ca86 --- /dev/null +++ b/outputs/arxiv_math/2503.03952_pg5_pg1_repeat1.md @@ -0,0 +1,27 @@ +Also taking the limit \(u\rightarrow0\), the single letter index is given by + +\[f(t,0;q)= \frac{t-q}{1-q} .\] (2.8) + +In this case, the corresponding index is known as the “half-index” (of Neumann boundary condition) [14]. Due to an obvious symmetry between \(t\text{and}u\), the limit \(t\rightarrow0\) is essentially same as \(u\rightarrow0\). However, in the analysis in the next subsection, these two limits look different, and lead to the equivalent result non-trivially. Finally, if we set \(u=q/t\), the resulting index is known as the flavored Schur index. The single letter index is now given by + +\[f(t,q/t;q)= \frac{t+q/t-2q}{1-q} .\] (2.9) + +For the further specialization to \(t=q^{1/2}\) (i.e., \(u=q^{1/2})\), the index \(I_{N}(q^{1/2},q^{1/2};q)\) is nothing but the original Schur index [3, 4]. We stress that all of them are obtained from the index (2.10) as special limits. The reduced index \(I_{N}(t,u;q)\) is regarded as a two-parameter deformation of the Schur index \(I_{N}(q^{1/2},q^{1/2};q)\). As mentioned in the introductory section, we refer to \(I_{N}(t,u;q)\) as the deformed Schur index. + +### 2.2 Exact evaluation of deformed Schur indices + +In this subsection, we evaluate the matrix integral of the deformed Schur index exactly. When \(v=p=0\), we can rewrite the integral representation (2.4) as a more convenient form in terms of the \(q-\)Pochhammer symbol: + +\[I_{N}(t,u;q)= \frac{1}{N!} \frac{(q;q)_{\infty}^{N}(tu;q)_{\infty}^{N}}{(t;q)_{\infty}^{N}(u;q)_{\infty}^{N}} \oint_{\mathbb{T}}\prod_{i^{N}=1}^{N} \frac{dx_{i}}{2\pi ix_{i}} \prod_{1\leq i\neq j\leq N} \frac{(x_{i}/x_{j};q)_{\infty}(tux_{i}/x_{j};q)_{\infty}}{(tx_{i}/x_{j};q)_{\infty}(ux_{i}/x_{j};q)_{\infty}} .(2.10)\] + +This is a starting point of our analysis. The \(q-\)Pochhammer symbol is defined by + +\[(x;q)_{\infty}=\prod_{k=0}^{\infty}(1-xq^{k}),(x;q)_{n}=\prod_{k=0}^{n-1}(1-xq^{k}),(x;q)_{0}=1,\] (2.11) + +and we have used an identity, + +\[(qx;q)_{\infty}= \frac{(x;q)_{\infty}}{1-x} ,\] (2.12) + +to derive (2.10). A method to perform the integral (2.10) is simple. The computation consists of three steps. In the first step, we recognize that the integrand of (2.10) includes a weight function of Macdonald polynomials of type \(A_{N-1}\). In Appendix A, we review basics on the Macdonald polynomials of type A, based on [15, 16], for the reader’s convenience. The weight function of the Macdonald polynomials of type \(A_{N-1}\) is given by + +\[w(x)=\prod_{1\leq i\neq j\leq N} \frac{(x_{i}/x_{j};q)_{\infty}}{(tx_{i}/x_{j};q)_{\infty}} .\] (2.13) diff --git a/outputs/arxiv_math/2503.03994_pg108_pg1_repeat1.md b/outputs/arxiv_math/2503.03994_pg108_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..18015199311f213765c7611a12d2af5ecde54cba --- /dev/null +++ b/outputs/arxiv_math/2503.03994_pg108_pg1_repeat1.md @@ -0,0 +1,37 @@ +\[\max\{-1,-1-v_{p}(x_{1})\}\leq v_{p}(\Theta_{0})=1-v_{p}(\Theta_{1})\leq\min\{0,v_{p}(x_{0})\}.\] + +We will divide \(R(\vec{r};\vec{k}^{\prime})\) into following four areas, and in each area we will determine the \(\mod-p\) reduction: + +(i) \(\vec{t}\in R_{int}(\vec{r};\vec{k}^{\prime})\): Note that we have either \(v_{p}(\Theta_{0})\not\in \{0,v_{p}(x_{0})\}\text{or}v_{p}(\Theta_{0})\not\in \{-1,-1-v_{p}(x_{1})\}\), in this case. Consider two \(\overline{S}_{\mathbf{F}} -\text{submodules}\mathcal{M}^{\prime}= \overline{S}_{\mathbf{F}} ( \overline{E}_{1}^{(0)} , \overline{E}_{2}^{(1)} )\) and \(\mathcal{M}^{\prime\prime}= \overline{S}_{\mathbf{F}} ( \overline{E}_{2}^{(0)} , \overline{E}_{1}^{(1)} )\) of \(\mathcal{M}\). Then one can observe that + + \(\circ\text{if}v_{p}(\Theta_{0})\not\in \{0,v_{p}(x_{0})\},\text{then}\mathcal{M}^{\prime}\) is a Breuil submodule of \(\mathcal{M}\) such that \(\mathcal{M}^{\sim^{\prime}}=\mathcal{M}(1,1;-\beta_{0},\alpha_{1})\text{and}\mathcal{M}/\mathcal{M}^{\sim^{\prime}}=\mathcal{M}(1,1;\alpha_{0},-\beta_{1})\); + + \(\circ\text{if}v_{p}(\Theta_{0})\not\in \{-1,-1-v_{p}(x_{1})\},\text{then}\mathcal{M}^{\prime\prime}\) is a Breuil submodule of \(\mathcal{M}\) such that \(\mathcal{M}^{\sim^{\prime\prime}}=\mathcal{M}(1,1;\alpha_{0},-\beta_{1})\text{and}\mathcal{M}/\mathcal{M}^{\sim^{\prime\prime}}=\mathcal{M}(1,1;-\beta_{0},\alpha_{1})\). + +In particular, if \(v_{p}(\Theta_{0})\not\in \{-1,0,v_{p}(x_{0}),-1-v_{p}(x_{1})\},\text{then}\mathcal{M}=\mathcal{M}^{\prime}\oplus\mathcal{M}^{\prime\prime}\) as Breuil modules. Otherwise, It is not difficult to show that the short exact sequence determined by \(\mathcal{M}^{\prime}\hookrightarrow\mathcal{M}\text{or}\mathcal{M}^{\prime\prime}\hookrightarrow\mathcal{M}\) is non-split. So, \(\overline{\rho}^{ss} |_{I_{\mathbf{Q}_{p2}}}\cong \omega_{2}^{p+1}\oplus\omega_{2}^{p+1}\), by Lemma 2.3.6 (i). Furthermore, the monodromy type is given as follow. + + (a) If \(v_{p}(\Theta_{0})=0,\text{then}t_{0}\geq0\), and the monodromy type is \((1,0)\). + + (b) If \(v_{p}(\Theta_{0})=-1,\text{then}t_{1}\geq0\), and the monodromy type is \((0,1)\). + + (c) Otherwise, the monodromy type is \((0,0)\). + +(ii) \(t_{1}=-1,t_{0}\geq0\): We have \(v_{p}(\Theta_{0})=0=-1-v_{p}(x_{1})\), and so \(\frac{1}{\Theta_{0}} , \frac{x_{1}}{\Theta_{1}} \in\mathbf{F}^{\times}\text{and} \frac{1}{\Theta_{1}} =0\) in F. In particular, the monodromy type is \((1,0)\). Put \(\mathcal{M}^{\prime}= \overline{S}_{\mathbf{F}} ( \overline{E}_{2}^{(0)} , \overline{E}_{1}^{(1)} )\). It is easy to see that \(\mathcal{M}^{\sim^{\prime}}=\mathcal{M}(1,2;\alpha_{0},-px_{1}\beta_{1})\text{and}\mathcal{M}/\mathcal{M}^{\sim^{\prime}}=\mathcal{M}(1,0;-\beta_{0}, \frac{\alpha_{1}}{px_{1}} )\), and so we conclude that \(\overline{\rho}^{ss} |_{I_{\mathbf{Q}_{p2}}}\cong \omega_{2}^{p}\oplus\omega_{2}^{p+2}\), by Lemma 2.3.6 (i), and \(\overline{\rho}\) is non-split as the monodromy type is nonzero. + +(iii) \(t_{0}+t_{1}=-1,-10\) is the diffusion coefficient, while \(\boldsymbol{c}\in H_{\text{div}}^{1}(\Omega)=\{\boldsymbol{c}\in[L^{2}(\Omega)]^{2}|\nabla\cdot\boldsymbol{c}\in L^{2}(\Omega)\}\) is the velocity field. We will use (2.2) as a canonical example for the numerical results in Section 4. + +### 2.1 Variational form and symmetrization + +The variational formulation of (2.1) is given by: find \(u\in\mathcal{V}\) such that + +\[a(u,w;\boldsymbol{g})=(f,w)+(\eta,w)_{\Gamma_{\eta}}\forall w\in\mathcal{V},\] (2.3) + +where \((.,.)\) is the \(L^{2}(\Omega)\) inner-product, \((.,.)_{\Gamma_{\eta}}\) is the \(L^{2}(\Gamma_{\eta})\) inner-product, while \(a(u,w;\boldsymbol{g})\) is the associated bilinear form parameterized by \(\boldsymbol{g}\). We also assume that the bilinear form is coercive, which requires additional conditions on \(\boldsymbol{g}\). With this assumption, a unique solution of (2.3) exists, as guaranteed by the Lax-Milgram theorem [10, 9]. + +For the particular case of the advection-diffusion equation, we have + +\[a(u,w;\kappa,\boldsymbol{c}):=(\kappa\nabla u,\nabla w)+\left(\boldsymbol{c}\cdot\nabla u,w\right)\] (2.4) diff --git a/outputs/arxiv_math/2503.04026_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.04026_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..ad639b517a28b1a9fc0d8f34a26be09095d36bfe --- /dev/null +++ b/outputs/arxiv_math/2503.04026_pg2_pg1_repeat1.md @@ -0,0 +1,7 @@ +Euclidean and non-Euclidean 3-dimensional spaces is a fundamental area of research in geometry and physics. Over time, researchers have developed various results, theorems, and insights concerning these surfaces. In [8], the authors investigated the Bertrand offset for ruled surfaces and showed that, similarly to planar curves, a ruled surface can have an infinite number of Bertrand offsets. The relationship between the Bertrand offsets of trajectory ruled surfaces and their projections on spherical areas has been explored in [9, 10], along with their corresponding invariants. The Bertrand offsets of ruled surfaces in Minkowski 3-space were considered in [11, 12]. Additionally, the study of Mannheim offsets for timelike ruled and developable surfaces, particularly their invariants, was addressed in [13, 14]. In [15], Senturk and Yuce analyzed involute-evolute offsets of ruled surfaces and developed associated invariants using the geodesic Frenet frame. More recently, research on evolute offsets of ruled surfaces, especially those with constant Gaussian and mean curvatures, in both Euclidean and Minkowski 3-spaces has been presented in [16, 17]. The characteristics of ruled surfaces and their offset surfaces have been extensively studied in both Euclidean and non-Euclidean spaces (see [15-21]). However, existing literature lacks a detailed approach to constructing evolute offsets of slant \(TL-\)ruled surfaces in terms of the striction curve. In this study, we explore the geometric properties of slant \(TL-\)ruled surfaces and their evolute offsets within three-dimensional Minkowski space \(\mathbb{E}_{1}^{3}\). By establishing a bijective correspondence through their rulings, we derive conditions under which an evolute offset \(M^{*}\) maintains a coaxial relationship with the central normal of \(M\). Furthermore, we formulate expressions governing curvature behavior and classify \(M\text{and}M^{*}\) based on specific functional parameters. Special cases, including \(TL-\text{developable},TL-\text{binormal},\text{and}TL-\)cone surfaces, are also examined, with graphical visualizations provided to support the analysis. This paper investigates slant timelike \((TL)\) ruled surfaces and their evolute offsets in Minkowski 3-space \(\mathbb{E}_{1}^{3}\). We present a parametric formulation of skew \(TL-\)ruled surfaces and establish conditions ensuring the coaxial alignment of the central normal with the ruling direction of the corresponding offset surface. The study explores the geometric properties using the Blaschke and Darboux frames, deriving curvature characteristics and fundamental invariants. Special cases, including \(TL-\text{developable and}TL-\)binormal surfaces, are analyzed with illustrative examples. The findings provide a deeper understanding of the differential geometry of \(TL-\)ruled surfaces and their evolute offsets in Lorentzian space. + +## 2 Basic concepts + +Let \(\mathbb{E}_{1}^{3}\) denote Minkowski 3-space [22, 23]. For vectors \(e=(e_{1},e_{2},e_{3})\) and \(\nu=(\nu_{1},\nu_{2},\nu_{3})\text{in}\mathbb{E}_{1}^{3}\), the inner product is defined as: + +\[=e_{1}\nu_{1}+e_{2}\nu_{2}-e_{3}\nu_{3}.\] diff --git a/outputs/arxiv_math/2503.04033_pg9_pg1_repeat1.md b/outputs/arxiv_math/2503.04033_pg9_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..e41633301085b9cbab463111ff14c53b9ea2b1c3 --- /dev/null +++ b/outputs/arxiv_math/2503.04033_pg9_pg1_repeat1.md @@ -0,0 +1,15 @@ +\[\mathbf{Au}=\mathbf{f}\equiv \begin{bmatrix} \mathbf{A}_{11} & \mathbf{A}_{15} \\ \mathbf{A}_{22} & \mathbf{A}_{25} \\ \mathbf{A}_{51} & \mathbf{A}_{52} & \mathbf{A}_{55} \end{bmatrix} \begin{bmatrix} \mathbf{u}_{1} \\ \mathbf{u}_{2} \\ \mathbf{u}_{5} \end{bmatrix} = \begin{bmatrix} \mathbf{f}_{1} \\ \mathbf{f}_{2} \\ \mathbf{f}_{5} \end{bmatrix} .\] (8) + +Here u is the solution and f is the body load on the interior of \(\Omega\), split into the two subdomains and shared boundary. Because an elliptic PDO is a local operator there is no interaction through A on f1 by u2 or f2 by u1, so A = 0 in those submatrices. However, both u1 and u2, are local to the shared boundary where u5 lies, so they both interact with the body load there, f5, through A. To interpret the first two block rows, note that + +\[\mathbf{A}_{11}\mathbf{u}_{1}+\mathbf{A}_{15}\mathbf{u}_{5}=\mathbf{f}_{1}\Longleftrightarrow \mathbf{u}_{1}=-\mathbf{A}_{11}^{-1}\mathbf{A}_{15}\mathbf{u}_{5}+\mathbf{A}_{11}^{-1}\mathbf{f}_{1}\] (9) + +This is similar to our solution operator formulation shown in (4), with the part of the subdomain boundary on \(\partial\Omega\) now folded into u1. Next, let \(\mathbf{ũ}_{1}=\mathbf{A}_{11}^{-1}\mathbf{f}_{1}\text{and}\mathbf{ũ}_{2}=\mathbf{A}_{22}^{-1}\mathbf{f}_{2}\). If we supply these vectors in place of \(\mathbf{u}_{1},\mathbf{u}_{2}\) in + +(8) and set u5 = 0, then the equation still holds with the same body loads f1 and f2. Thus ũ1 and ũ2 are the particular solutions to our PDE on the interiors of \(\Omega^{(1)},\text{and}\Omega^{(2)}\) with u5 = 0. Now consider an upper triangular matrix U that satisfies + +\[\begin{bmatrix} \mathbf{I} & \mathbf{A}_{11}^{-1}\mathbf{A}_{15} \\ \mathbf{I} & \mathbf{A}_{22}^{-1}\mathbf{A}_{25} \\ \mathbf{I} \end{bmatrix} \overline{\mathbf{U}} \begin{bmatrix} \mathbf{u}_{1} \\ \mathbf{u}_{2} \\ \mathbf{u}_{5} \end{bmatrix} = \begin{bmatrix} \tilde{\mathbf{ũ}}_{1} \\ \tilde{\mathbf{ũ}}_{2} \\ \mathbf{u}_{5} \end{bmatrix} .\] (10) + +This linear system can be confirmed with (9). U decouples our solutions \(\mathbf{u}_{1},\mathbf{u}_{2}\) into the particular solutions \(\mathbf{ũ}_{1},\mathbf{ũ}_{2}\) which are derived from our global Dirichlet BC without the subdomain-only (shared face) Dirichlet BC, and u5 which is that subdomain-only boundary condition. Thus \(\mathbf{U}^{-1}\) collects both components of our solutions - in effect it represents the end of solving the merged system, where we use u3 and u5 to get u1, and u4 and u5 to get u2. Next, let L be a lower triangular matrix defined such that + +\[\begin{bmatrix} \mathbf{I} \\ \mathbf{I} \\ \mathbf{A}_{51}\mathbf{A}_{11}^{-1} & \mathbf{A}_{52}\mathbf{A}_{22}^{-1} & \mathbf{I} \end{bmatrix} \overline{\mathbf{L}} \begin{bmatrix} \mathbf{f}_{1} \\ \mathbf{f}_{2} \\ \frac{\mathbf{f}_{5}-\mathbf{A}_{51}\mathbf{A}_{11}^{-1}\mathbf{f}_{1}-\mathbf{A}_{52}\mathbf{A}_{22}^{-1}\mathbf{f}_{2}}{\tilde{\mathbf{f}}_{5}} \end{bmatrix} = \begin{bmatrix} \mathbf{f}_{1} \\ \mathbf{f}_{2} \\ \mathbf{f}_{5} \end{bmatrix} .(11)\] diff --git a/outputs/arxiv_math/2503.04040_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.04040_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..e4ebf571ce450ccaa7d21b9e5bf23f3333eeb0ce --- /dev/null +++ b/outputs/arxiv_math/2503.04040_pg2_pg1_repeat1.md @@ -0,0 +1,19 @@ +proach cannot be directly applied to this joint optimization problem. To address this challenge, the authors of [25] studied a secure rate maximization problem with joint beamforming, FA position control, and artificial noise optimization problem. Then, they proposed the minimum mean square error (MMSE) method to decouple the optimization variables, demonstrating that the joint optimization significantly improves performance. The FAS was further extended to multiuser systems by [12], where the authors considered the transmit power minimization problem in an FA-assisted multiuser system with joint beamforming and FA position optimization. They decoupled the variables using zero-forcing (ZF) and MMSE methods. Then, they optimized FA positions via a multi-directional descent (MDD) framework. It was demonstrated that FAS can effectively cancel MUI, thereby significantly reducing transmit power while ensuring the data rate. More recently, [26] evaluated the system capacity gain of the FAS by solving the weighted sum rate (WSR) maximization problem in an FA-assisted downlink MU-multiple-input single-output (MISO) system. The authors first applied the weighted minimum mean square error (WMMSE) algorithm to decouple beamforming vectors and FA positions. Then, they leveraged the majorization maximization (MM) framework to solve the resulting nonconcave FA position optimization problem. Simulation results demonstrated that FAS enhances overall system capacity in MU-MISO networks. However, the MM framework in [26] has unaffordable complexity when the number of FAs at the BS is large. Specifically, the MM framework optimizes one FA position at a time while keeping the others fixed in each iteration, leading to the increased computational time as the number of FAs grows. + +## B. Contributions + +In this paper, we propose a novel block coordinate ascent (BCA)-based algorithm for joint beamforming and FA position optimization in the FA-assisted downlink MU-MIMO system, as well as provide a decentralized implementation of the proposed algorithm. Our contributions are summarized as follows. + +- Different from [26] that focuses on MU-MISO networks, we consider the more general FA-assisted MU-MIMO networks. We formulate the joint beamforming and antenna position optimization as a WSR maximization problem, where the objective function is non-concave and the optimization variables are highly coupled. To decouple the beamforming matrices and FA positions, we utilize two matrix fractional programming (FP) techniques, i.e., the quadratic transform and the Lagrangian dual transform [27], [28]. These FP techniques enable us to solve the decoupled subproblems by a BCA-based algorithm. + +- Unlike existing FA position optimization algorithms that sequentially update FA positions, we propose a novel optimization algorithm based on the MM framework that optimizes all FA positions simultaneously. The proposed algorithm reduces computational time by exploiting parallelism and can be easily extended to a decentralized implementation. Additionally, we introduce a box-constrained movement mode to simplify both the formulated problem and the engineering implementation of FAS. This movement mode provides closed-form solutions to reduce complexity and enables decentralized implementation by allowing the independent movement of each FA. + +- To further reduce computational, storage, and interconnection costs, we propose a decentralized implementation of our algorithm by utilizing the decentralized baseband processing (DBP) architecture [29], which partitions the transmit FA array into several clusters. The DBP architecture can decompose the optimization problem into smaller subproblems, and enable decentralized units (DUs) to solve them in parallel. To enable the decentralized implementation of the proposed BCA-based algorithm, we employ the non-homogeneous transform and Nesterov’s extrapolation [30], [31] to avoid the matrix inversion in the optimization of beamforming matrices. The decentralized implementation can significantly alleviate computational, storage, and interconnection costs with negligible performance loss. + +- We conduct various numerical experiments to verify the performance of our proposed algorithm. It is shown that the proposed algorithm can improve the WSR of FA-assisted MU-MIMO networks by 47% compared with conventional MIMO networks using FPAs. Moreover, the decentralized implementation reduces computation time by approximately 70% and has similar performance compared with the centralized implementation. + +The remainder of this paper is organized as follows. Section II presents the channel model of the FA-assisted MU-MIMO system and formulates the WSR problem. Section III reformulates the problem using FP techniques and solves it using BCA and MM. The decentralized implementation of the proposed algorithm is introduced in Section IV. Simulation results are provided in Section V, and conclusions are drawn in Section VI. Notation: Italic letters, boldface lowercase letters, and boldface uppercase letters denote scalars, vectors, and matrices, respectively. The imaginary unit is denoted by \(ȷ\). For a complex number \(a\), its amplitude and phase are given by \(|a|\text{and}\angle a\), respectively. The \(\ell_{2}\) norm of a vector a is \(\Vert\mathbf{a}\Vert_{2}.[\mathbf{A}]_{m},[\mathbf{A}]_{mn}\), \(\mathbf{A}^{\mathrm{T}},\mathbf{A}^{\mathrm{H}},\det(\mathbf{A}),\text{tr}(\mathbf{A}),\text{vec}(\mathbf{A}),\Vert\mathbf{A}\Vert_{\infty},\text{and}\Vert\mathbf{A}\Vert_{\mathrm{F}}\) denote the \(m-\)th row, the \((m,n)-\)th element, transpose, conjugate transpose, determinant, trace, vectorization, the infinity norm, and the Frobenius norm of matrix A, respectively. \(\mathbf{A}\succeq\mathbf{0}\) and \(\mathbf{A}\succ\mathbf{0}\) indicate that A is positive semi-definite and positive definite, respectively. \(\mathbb{C}^{M\times N},\mathbb{R}^{M\times N},\text{and}\mathbb{R}_{+}^{M\times N}\) denote the sets of \(M\times N\) complex, real, and non-negative real matrices, respectively. The circularly symmetric complex Gaussian (CSCG) distribution with zero mean and covariance \(\sigma^{2}\mathbf{I}\) is represented as \(\mathcal{C}\mathcal{N}(\mathbf{0},\sigma^{2}\mathbf{I})\), and the uniform distribution over \([a,b]\) is denoted by \(\mathcal{U}[a,b]\). Operator \(\partial(\cdot)\) denotes the partial differential. \(\nabla_{\mathbf{x}}f(\mathbf{x})\text{and}\nabla_{\mathbf{x}}^{2}f(\mathbf{x})\) denote the gradient vector and Hessian matrix of \(f(\mathbf{x})\) with respect to (w.r.t.) x respectively. + +## II. SYSTEM MODEL AND PROBLEM FORMULATION + +As shown in Fig. 1, we consider a downlink MU-MIMO system where a BS with \(M\) FAs serves \(K\) users, each equipped with \(N\) FAs. A three-dimensional (3D) Cartesian coordinate system is established to describe the positions of the transmit FAs at the BS and receive FAs at the users. diff --git a/outputs/arxiv_math/2503.04041_pg20_pg1_repeat1.md b/outputs/arxiv_math/2503.04041_pg20_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..5dcb0290b4f8714551279fc5998c61d8e82d42f0 --- /dev/null +++ b/outputs/arxiv_math/2503.04041_pg20_pg1_repeat1.md @@ -0,0 +1,23 @@ +due to the loss of orthogonality of basis vectors. Thus we set the maximum iteration steps to \(10n\). We have observed that for the test problems in our experiments, \(10n\) is enough to ensure convergence. We perform full reorthogonalization on \(V_{k}^{\prime},U_{k}\) and \(\widehat{U}_{k}\) to ensure that they are numerically orthonormal to the working precision \(\epsilon\). By default, the convergence of line 2 in Algorithm 6.1 is judged by (3.18) with the stopping tolerance \(tol=10^{-8}\). The starting vector \(u_{1}\) is randomly generated by the standard normal distribution function randn in MATLAB and then normalized. + +| Parameters | Default values | Description | +| --- | --- | --- | +| target kmax | 5 | \|target\| = l is the number of desired GSVD components, where target > 0 or target < 0 means that the largest or smallest GSVD components are required. Maximum subspace dimension | +| adjust | 3 | Integer added to l to speed up convergence | +| tol | 10−8 | Stopping tolerance | +| maxit | 1000 | Maximum number of restarts | +| lsqrtol | 10ǫ | Stopping tolerance of lsqr | +| lsqrmaxit | 10n | Maximum number of iterations of lsqr | +| u1 | generated by randn | The unit length starting vector | + +and then normalized + +Table 1 Parameters of IRJBD + +7. Numerical experiments. We now present a number of numerical experiments to demonstrate the performance of Algorithm 6.1, written in MATLAB language, and compare it with the public C language code of TRJBD in [1]. The experiments will illustrate that Algorithm 6.1 is at least competitive with and can outperform TRJBD considerably in terms of restarts. Since section 4.4 has shown that each restart of IRJBD and TRJBD basically costs the same for the same \(l\text{and}k_{\text{max}}\), fewer restarts mean higher efficiency. The experiments were performed on an AMD Ryzen 7 5800X CPU with 78 GB RAM and 8 cores using the MATLAB R2024a with the machine precision \(\epsilon=2.22\times10^{-16}\) under the Ubuntu 20.04.3 LTS 64-bit system. + +We use some matrices from the SuiteSparse Matrix Collection [6] as \(A,\text{and}L\) is taken as (cf. [11]): + +\[L= \begin{pmatrix} 3 & 1 \\ 1 & \ddots & \ddots \\ \ddots & \ddots & 1 \\ 1 & 3 \end{pmatrix} ,\] + +which is well conditioned. Table 2 lists the test matrices \(A\) and some of their properties, where \(nnz\) is the total number of nonzero entries of \(\{A,L\}\text{and}\kappa(A)= \frac{\sigma_{\text{max}}(A)}{\sigma_{\text{min}}(A)}\) , available from the SuiteSparse Matrix Collection, is the condition number of \(A\) and indicates the level of ill-conditioning of \(A\) and reflects magnitudes of generalized singular values of \(\{A,L\}\). In order to ensure that all the \(A\) are flat or square, i.e., \(m\leqslant n\), we transpose some matrices, denoted by superscript T. Note that the smallest singular value of kneser 8 3 1T is tiny, which causes that the matrix is numerically row diff --git a/outputs/arxiv_math/2503.04045_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.04045_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..69e8952c2971231545a733e35902666185618930 --- /dev/null +++ b/outputs/arxiv_math/2503.04045_pg2_pg1_repeat1.md @@ -0,0 +1,33 @@ +Theorem 1.2 (Chen). Every sufficiently large even number \(N\) can be expressed as + +\[N=p+\eta,\] (1.2) + +where \(p\) is prime and \(\eta>0\) has at most 2 prime factors. Theorem 1.3 (Li). Every sufficiently large odd number \(N\) can be expressed as + +\[N=p+2\eta,\] (1.3) + +where \(p\) is prime and \(\eta>0\) has at most 2 prime factors. Here, the factor of 2 in (1.3) is to force \(p+2\eta\) to be odd; provided of course \(p\neq 2\). Combining Theorems 1.2 and 1.3, we have that every large \(N\geq1\) can be represented as the sum of a prime and a number with at most 3 prime factors. Any further lowering of the number of prime factors in Chen or Li’s results appears out of reach, so we do not attempt this here. Now, representing \(N\) of the form (1.1) when \(k\geq2\) is naturally more difficult, owing to sparseness of the set of \(k\)th prime powers, and additional “forced” prime factors of \(N-p^{k}\). Thus in this setting, much less is known. However, the following classical result, due to Erdős [7] and Rao [24], can be viewed as partial progress towards representing large \(N\) in the form (1.1). Theorem 1.4 (Erdős–Rao). Let \(k\geq2\). Then every sufficiently large integer \(N\) such that + +\[N\not\equiv 1 \begin{cases} \pmod{16}, & \text{when}k=4, \\ \pmod{4}, & \text{when}k=2, \end{cases}\] (1.4) + +can be expressed as + +\[N=p^{k}+\eta\] + +where \(p\) is prime and \(\eta>0\) is \(k\)th power-free. Here, the condition that \(\eta\) is \(k\)th power-free is much simpler than \(\eta\) having a bounded number of prime factors. In particular, most integers (> 60%) are square-free (and thus \(k\)th power-free), whereas the set of integers with a bounded number of prime factors has a natural density of 0. + +### 1.2 Statement of results + +Our main result is as follows. Theorem 1.5. Let \(k\geq2\). There exists an integer \(M(k)\) such that every sufficiently large integer \(N\) can be expressed as + +\[N=p^{k}+\eta\] (1.5) + +where \(p\) is prime and \(\eta>0\) has at most \(M(k)\) prime factors. Here, \(M(k)=6k\) is admissible for all even \(k\geq2,\text{and}M(k)=4k\) is admissible for all odd \(k\geq3\). In addition, for sufficiently large \(k\geq k_{\varepsilon}\) one can set + +\[M(k)=(2+\varepsilon)k,\] (1.6) + +for any \(\varepsilon>0\). Or, under assumption of the Elliott–Halberstam conjecture1, + +\[M(k)=(1+\varepsilon)k.\] (1.7) + +1More precisely, a standard variant of the Elliott–Halberstam conjecture (see Conjecture 2.7). diff --git a/outputs/arxiv_math/2503.04047_pg8_pg1_repeat1.md b/outputs/arxiv_math/2503.04047_pg8_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..6176cb788fb324367b9ccd6093bbeb126f0f1840 --- /dev/null +++ b/outputs/arxiv_math/2503.04047_pg8_pg1_repeat1.md @@ -0,0 +1,31 @@ +The efficiency of simulated annealing around \(T_{c}\) has been reported in numerous studies. Kirkpatrick et al. (1983) demonstrated that during phase transition, the search becomes more efficient. Strobl & Barker (2016) showed that when using simulated annealing to solve phylogeny reconstruction, the search is constrained to a small valley of the search space when the temperature is below \(T_{c}\). The search efficiency around \(T_{c}\) can also be seen in Figure 2 of (Cai & Ma, 2010b). + +While some algorithms (Basu & Frazer, 1990; Cai & Ma, 2010a) have used critical temperature \(T_{c}\) to design the initial and final temperatures, its use for reheating is less common. Abramson et al. (1999) proposes a reheating strategy tied to function cost that implicitly involves \(T_{c}\). However, this method lacks theoretical support and is sensitive to hyperparameters. Instead, our approach directly reheats to the critical temperature, which injects just the right amount of stochastic energy back into the system, enabling it to escape “wandering in contours” without incurring excessive randomness of a high-temperature regime. + +#### 5.2.2 Specific Heat + +Determining the critical temperature in simulated annealing requires identifying the phase transition point during optimization, which is characterized by peaks in specific heat (Kirkpatrick et al., 1983; Strobl & Barker, 2016). In statistical physics, the system’s energy at temperature \(T\) , denoted as \(E(T)\), adheres to the Boltzmann distribution. Thus, the expected energy of a system can be seen as a function of the temperature, \(\mathbb{E}[E(T)]\). The specific heat at temperature \(T\) , denoted as \(C_{T}\) , is traditionally defined in thermodynamics as the rate of change of the expected energy \(\mathbb{E}[E(T)]\) to temperature (Aarts et al., 1987), given by \(C_{T}= \frac{\partial\mathbb{E}[E(T)]}{\partial T}\) . Integrating over the Boltzmann distribution allows deriving specific heat in terms of energy variance: + +\[C_{T}= \frac{\sigma^{2}(E(T))}{T^{2}}\] (9) + +where \(\sigma^{2}(E(T))\) is the variance of the system energy at \(T\) . + +#### 5.2.3 Determination of Critical Temperature + +To determine the critical temperature based on specific heat, we first denote \(T(t)\) as the temperature at step \(t,C(t)\) as the specific heat at temperature \(T(t),\text{and}x_{t}\) as the solution sampled at step \(t\), then by selecting an appropriate sample size \(M\), we define the approximation of \(C(t)\) as : + +\[\hat{C}(t)= \frac{\sigma^{2}(\{f(x_{t-M+1}),\cdots ,f(x_{t})\})}{T(t)^{2}} ,t\geq M\] (10) + +\[t\geq M\] + +where \(\sigma^{2}(\{f(x_{t-M+1}),\cdots ,f(x_{t})\})\) represents the variance in objective values over the \(M\) most recent steps. The critical temperature \(T_{c}\) can be determined as \(T(t^{*}),\text{where}t^{*}=\text{argmax}\hat{C}(t)\). However, SA with gradient-based discrete samplers convergences rapidly in the initial stage, resulting in an abnormal initial peak in specific heat (due to the high variance). As the annealing progresses, the specific heat quickly decreases, eventually stabilizing at a level more typical of a critical temperature. This behavior is shown in Figure 4a&4b. + +To address the abnormal peak, we introduce a “skip step” threshold, denoted as \(t_{\text{skip}}\), ensuring that the initial transient behavior is excluded from the analysis. Thus, the critical temperature is identified as \(T(\tilde{t}^{*})\), where + +\[\begin{aligned} \tilde{t}&=\operatorname*{arg\,max}^{*}\hat{C}(t). \\ t&\geq t_{\text{skip}} \end{aligned}\] (11) + +Our method diverges from traditional SA reheat strategies (Abramson et al., 1999) by accounting for inhomogeneous chains and addressing gradient-based methods’ unique abnormal peaks. + +### 5.3 Reheated Gradient-based Discrete Sampling for Combinatorial Optimization + +Combining results in Section 5.1 and 5.2, we obtain the Reheated Sampling for Combinatorial Optimization algorithm (ReSCO), which is summarized in Algorithm 1. ReSCO is compatible with any gradient-based diff --git a/outputs/arxiv_math/2503.04048_pg46_pg1_repeat1.md b/outputs/arxiv_math/2503.04048_pg46_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..0577e98300e50bec272ef10349d83bd56e67cb84 --- /dev/null +++ b/outputs/arxiv_math/2503.04048_pg46_pg1_repeat1.md @@ -0,0 +1,17 @@ +![](figure) + +![](figure) + +which implies that: + +Figure 9: + +\[x_{m}= \frac{r_{1}\tan\theta_{1}-r_{2}\tan\theta_{2}-(u_{1}-u_{2})}{\tan\theta_{1}+\tan\theta_{2}}\] (130) + +Then we use \(x_{m}\text{and}\psi_{m}\) to express the total volume. By using geometric computation, we have: + +\[\begin{aligned} \mathcal{V}(\psi_{m})&=(r_{1}u_{1}-\int_{\psi_{1}}\int_{\psi}^{\psi_{m}\psi_{m}} \frac{\cos\gamma}{g \sqrt{\tfrac{2\sigma}{g} (\cos\psi_{m}-\cos\gamma)}} d\gamma \frac{-\sin\psi}{g \sqrt{\tfrac{2\sigma}{g} (\cos\psi_{m}-\cos\psi)}} d\psi) \\ +\int_{\psi_{2}}\int_{\psi}^{\psi_{m}\psi_{m}} \frac{\cos\gamma}{g \sqrt{\tfrac{2\sigma}{g} (\cos\psi_{m}-\cos\gamma)}} d\gamma \frac{-\sin\psi}{g \sqrt{\tfrac{2\sigma}{g} (\cos\psi_{m}-\cos\psi)}} d\psi \\ + \frac{1}{2} (u_{1}+u_{2})^{2} \frac{1}{-\tan\theta_{1}} +(-u_{2})r_{1}+ \frac{1}{2} (u_{2}-u_{1}-x_{1}\tan\theta_{1})^{2}( \frac{1}{\tan\theta_{1}} + \frac{1}{\tan\theta_{2}} ) \end{aligned}\] + +Then we need to prove the following theorem about the total volume: Theorem 3.12. When \(\psi_{m}\rightarrow0\), we have \(\mathcal{V}(\psi_{m})\rightarrow+\infty\). Proof. Using the computation in Theorem 3.7, we know that: + +\[\int_{\psi}^{\psi_{m}} \frac{\cos\gamma}{g \sqrt{\tfrac{2\sigma}{g} (\cos\psi_{m}-\cos\gamma)}} d\gamma\rightarrow+\infty\] diff --git a/outputs/arxiv_math/2503.04056_pg25_pg1_repeat1.md b/outputs/arxiv_math/2503.04056_pg25_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..f34d910ef7738629a5606fd7bc0b9bbb8f360790 --- /dev/null +++ b/outputs/arxiv_math/2503.04056_pg25_pg1_repeat1.md @@ -0,0 +1,57 @@ +![Six surface plots showing the evolution of a function u over time t and space x at different y values: y = -0.72, y = 1.08, y = 2.88. Each plot includes a color bar indicating the value range. Below each surface plot is a corresponding contour plot with a color bar.](figure)[^inferred] + +![Six surface plots showing the evolution of a function u over time t and space x at different y values: y = -0.72, y = 1.08, y = 2.88. Each plot includes a color bar indicating the value range. Below each surface plot is a corresponding contour plot with a color bar.](figure)[^inferred] + +Figure 17: (Color online) Data-driven discovery of the nonlinear coefficient v(t) = cos(t) in the variable coefficient KP equation: the upper part of the figure displays the 3D predicted solution at these three cross-sections. The red dashed lines represent the predicted solution, while the blue lines represent the true solution; the lower part displays the errors between the predicted and true solutions at the same cross-sections. + +✗ 10-3 2.5 + +2 + +1.5 + +2 + +1 + +0.5 + +1.5 + +0 + +1 + +-0.5 + +-1 + +0.5 + +-1.5 + +-2 + +0 + +-6 + +-4 + +-2 + +0 + +2 + +4 + +6 + +t + +![](figure) + +Figure 18: (Color online) The data-driven variable coefficient V(t) discovery of the variable coefficient KP equation by R-gPINN related to solution (21): the comparison of the error between the learning and exact variable coefficient V(t) over the interval \([-6,6]\). + +[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter. diff --git a/outputs/arxiv_math/2503.04068_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.04068_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..db0a6677bf4bc14f505437f0166f311fa254a250 --- /dev/null +++ b/outputs/arxiv_math/2503.04068_pg2_pg1_repeat1.md @@ -0,0 +1,9 @@ +The time-one map \(\Psi\) is defined by setting \(\Psi(x_{0})=x(1)\) for each \(x_{0}\in\mathbb{R}^{d}\). Classical results on approximation properties of shallow neural networks do not immediately apply since \(A(t)\) is fixed to be a square matrix of input dimension. For this reason, the neural ODE (1) is referred to as narrow following the definition in [2]. In [11], it has been shown that flows of this ODE can be used to approximate continuous maps on \(\mathbb{R}^{d}\) in the uniform norm by considering a NODEs on \(\mathbb{R}^{2d+1}\), provided the activation function satisfies a quadratic equation. In [9], it has been shown that any \(L_{2}\) map can be approximated using a narrow NODE. The capability NODEs for transporting probability densities have been explored in [5, 9, 2]. For \(L_{p}\) maps, universal approximations have been shown in [4]. The strategy used in this paper departs from the techniques of the other cited works, where either the arguments are constructive or control theoretic arguments are used. Firstly, we note that it can be shown that flows of (1) can approximate flows of the wide NODE + +\[\dot{x}(t)=\sum_{i=1}^{m}A_{i}\Sigma(W_{i}x+b_{i})x(0)=x_{0}\] (3) + +This, in turn, can be used to show that flow maps of (1) can be used to approximate flows of any dynamical system of the form + +\[\dot{x}(t)=V(x,t)x(0)=x_{0}\] (4) + +where \(V(x,t)\) is a time dependent vector field and hence, \(V(x,t)\) can be approximated by an arbitrarily wide network \(\sum_{i=1}^{m}A_{i}\Sigma(W_{i}x+b_{i})\) for any \(t\). Therefore, narrow neural ODEs inherit the approximation properties of their shallow but wide counterparts. This strategy is used in [5] and in [6] to study flow approximation properties using narrow NODEs. In [6] it was in fact shown that the dimension of the weight parameters used to approximate maps can be taken to be \(m\times d,\text{with}m\) less than the input dimension \(d\), by additionally exploiting geometric non-commutative properties of some chosen \(m\) basis vector fields, owing to diffeormophism controllability results due to [1]. Our goal in this paper is to derive quantitative rates of approximation of (1). Existing quantitative rates are either established to approximate homoemorphism in the \(L_{2}\) norm [9] or for the purposes of generative modeling [9, 2]. In contrast, we are interested in quantitative rates for approximation of flow maps in the uniform norm. The problem we address is how many switches are needed in \(A(t),W(t),b(t)\), to approximate the flow of (3). In this way, our result complements [9, 2]. While we restrict to reference flows of the differential equations of the form (3), an extension to general flows is straightforward since shallow but wide neural networks are known to be dense in the set of continuous functions. Hence, quantitative approximation properties of shallow neural networks would immediately translate to quantitative approximation properties of narrow NODEs. Before we present our analysis, we give some brief intuition behind our proof strategy. The idea behind our analysis is the following. Given a differential equation of the diff --git a/outputs/arxiv_math/2503.04077_pg1_pg1_repeat1.md b/outputs/arxiv_math/2503.04077_pg1_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..cc72c65b3f6d79a607afdf3abf1fb1559f090c22 --- /dev/null +++ b/outputs/arxiv_math/2503.04077_pg1_pg1_repeat1.md @@ -0,0 +1,27 @@ +# Periodic Inscription of Isosceles Trapezoids + +# Ali Naseri Sadr + +###### Abstract + +We prove that a pair of continuous disjoint periodic curves in \(\mathbb{C}\) inscribes an isosceles trapezoid with any similarity type. The case of smooth curves can be identified with a Lagrangian intersection problem for a pair of Lagrangian cylinders in \(\mathbb{R}\times S^{1}\times\mathbb{C}\), and the continuous case follows from the smooth one by a standard convergence argument. + +## 1 Introduction + +Let \(\gamma_{1},\gamma_{2}:\mathbb{R}\rightarrow\mathbb{C}\) be two continuous embeddings of the real line into \(\mathbb{C}\) that satisfy the periodicity condition + +\[\gamma_{i}(t+1)=\gamma_{i}(t)+ \sqrt{-1}\] + +for every \(t\text{and}i=1,2\). Furthermore, assume the images of \(\gamma_{1}\text{and}\gamma_{2}\) are disjoint. Tao conjectured in [10] that there exist four points in \(\gamma_{1}(\mathbb{R})\cup\gamma_{2}(\mathbb{R})\) which are vertices of a square; this is a variation of the Toeplitz square peg problem for periodic curves, and Hugelmeyer proved it in [5]. For any given isosceles trapezoid \(Q\), we show there are four points in \(\gamma_{1}(\mathbb{R})\cup\gamma_{2}(\mathbb{R})\) that are vertices of a quadrilateral similar to \(Q\). The approach of [5] does not directly generalize even to the case of rectangles. By contrast, in this article, we use a different approach to prove not only that every pair of periodic curves inscribes every similarity type of rectangles, but also every similarity type of isosceles trapezoids. + +Definition 1.1. Assume \(Q\) is an isosceles trapezoid. We say that the pair \((\gamma_{1},\gamma_{2})\) admits a balanced inscription of \(Q\) if there exist \(p_{1},p_{2}\in\gamma_{1} (\mathbb{R}) \text{and}p ,_{3}p_{4} \in\gamma_{2}(\mathbb{R})\) such that the quadrilateral formed by \(p_{1},p_{2},p_{3},p_{4}\) is similar to \(Q\), the line segments \(\overline{p_{1}p_{2}} \text{and} \overline{p_{3}p_{4}}\) are parallel, and \(| \overline{p_{1}p_{2}} |\leq| \overline{p_{3}p_{4}} |\). Note that our definition depends on the order of the pair \((\gamma_{1},\gamma_{2})\) unless \(Q\) is a rectangle. + +Theorem 1.2. Suppose \(\gamma_{1}\text{and}\gamma_{2}\) are two continuous disjoint periodic embeddings of the real line into the plane, and suppose \(Q\) is an isosceles trapezoid. Then \((\gamma_{1},\gamma_{2})\) admits a balanced inscription of \(Q\). Furthermore, there is a generic subset of smooth disjoint periodic pairs such that each pair in this set admits + +at least two balanced inscriptions of \(Q\) that are not related under translation by \(\sqrt{-1}\). + +Corollary 1.3. Let \(\theta\in(0, \frac{\pi}{2} ]\); then every pair of continuous disjoint periodic curves in the plane inscribes a rectangle with angle \(\theta\) between its two diagonals. + +We conjecture that Theorem 1.2 is optimal, in the following sense. + +Conjecture 1.4. Let \(Q\) be a quadrilateral that admits an inscription in any pair of disjoint periodic curves in \(\mathbb{C}.\text{Then}Q\) is an isosceles trapezoid. diff --git a/outputs/arxiv_math/2503.04086_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.04086_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..bdc8e6ad2d1dec98d5c6b466067299cb1ee11d18 --- /dev/null +++ b/outputs/arxiv_math/2503.04086_pg3_pg1_repeat1.md @@ -0,0 +1,19 @@ +recall that a positive integer n and two integers a, b, gcd(a, n) = gcd(b, n) if and only if a and b generate the same ideal in the ring \(Z/n\). This is also equivalent to the condition that \(a\equiv\) ub (mod n) for some \(u\in(Z/n)^{\times}\). This property generalizes well for Artinian rings, and in particular, to finite rings. More precisely + +Lemma 2.2. (See [5, Lemma 2.1] Let R be an Artinian ring. If Ra = Rb, then there exists \(u\in R^{\times}\) such that b = ua. + +From this perspective, a crucial observation here is that a generating set S in a finite ring R such as \(Z/n\text{or}F_{q}[x]/f\) gives rise to a gcd-graph if and only if S is stable under the action of \(R^{\times}\); that is, if \(s\in S\), then us \(\in S\text{for all}u\in R^{\times}\). Motivated by this observation, we introduce the following definition. + +Definition 2.3. We say that \(\Gamma(R,S)\) is a gcd-graph if S is stable under the action of \(R^{\times}\). + +Remark 2.4. When \(S=R^{\times}\), the graph \(\Gamma(R,R^{\times})\) is known as a unitary Cayley graph (see [1, 6]). This shows that the unitary Cayley graph over R is a special case of gcd-graphs. Other examples, as explained previously, include the gcd-graphs defined in [6] for the ring \(Z/n\) and the gcd-graphs defined in [9] for the ring \(F_{q}[x]/f\text{where}F_{q}\) is a finite field with q elements and f is a non-zero polynomial in \(F_{q}[x]\). + +We provide below the necessary and sufficient conditions for a Cayley graph over R to be a gcd-graph. These conditions are somewhat more explicit than Definition 2.3. Furthermore, together with Corollary 2.7, this description will be important later on when we study the spectrum of these gcd-graphs. + +Proposition 2.5. Let R be a finite commutative ring and S a subset of R. Then the following are equivalent. + +\((1)\Gamma(R,S)\) is a gcd-graph. + +2. There exist distinct nonzero principal ideals \(\mathcal{I}_{1},\mathcal{I}_{2},\ldots,\mathcal{I}_{k}\) such that for each \(r\in R\), \(r\in S\) if and only if there exists \(1\leq i\leq\text{k such that}\mathcal{I}_{i}=\) Rr. + +Proof. First, we claim that \((2)\Longrightarrow (1)\). Clearly, S is symmetric and \(0\not\in S\). By definition, we need to show that if \(s\in S\text{and}u\in R^{\times}\) then us \(\in S\). Since \(s\in S\), there exists an ideal \(\mathcal{I}_{i}\text{with}1\leq i\leq k\) such that Rs \(=\mathcal{I}_{k}\). Since \(u\in R^{\times},\mathcal{I}_{k}=\) Rs = Rus. By (2), this shows that us \(\in S\). Let us show that (1) implies (2) as well. For each \(s\in S\), Let \(\mathcal{I}_{s}=\) Rs be the ideal generated by s. Let \(\{\mathcal{I}_{1},\mathcal{I}_{2},\ldots,\mathcal{I}_{k}\}\) be the set of all \(\mathcal{I}_{s}\). We conclude that for each \(s\in S\), \(\mathcal{I}_{s}=\mathcal{I}_{i}\text{for some}1\leq i\leq k\). Conversely, if \(r\in R\) such that Rr \(\in\{\mathcal{I}_{1},\mathcal{I}_{2},\ldots,\mathcal{I}_{k}\}\), then Rr = Rs for some \(s\in S\). By Lemma 2.2, we know that r = us for some \(u\in R^{\times}\). Since \(\Gamma(R,S)\) is a gcd-graph, S is stable under the action of \(R^{\times}\). This shows that \(r\in S\) as well. diff --git a/outputs/arxiv_math/2503.04092_pg10_pg1_repeat1.md b/outputs/arxiv_math/2503.04092_pg10_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..1ba3822f0b8b2cbf3686b6d8a0254631167e34e8 --- /dev/null +++ b/outputs/arxiv_math/2503.04092_pg10_pg1_repeat1.md @@ -0,0 +1,28 @@ +with \(T_{c}=0.8\text{and}T=0.36\). The physical parameters are set as seen in Table 1. The forward problem is solved using + +| Parameter | Value | +| --- | --- | +| ρ (gr · cm3) | 1.2 | +| µ (P ) | 0.035 | +| U (cm · s−1) | 75 | +| Tc (s) | 0.80 | +| T (s) | 0.36 | +| κ (s−1) | 70 | + +| | Γ1 | Γ2 | Γ3 | Γ4 | +| --- | --- | --- | --- | --- | +| Rp (dyn · s · cm−5) | 480 | 520 | 520 | 200 | +| Rd (dyn · s · cm−5) | 7200 | 11520 | 11520 | 4800 | +| C (dyn−1 · cm5) | 4 · 10−4 | 3 · 10−4 | 3 · 10−4 | 4 · 10−4 | + +Table 1: Physical parameters and numerical values of the three-element Windkessel parameters for every outlet. + +a semi-implicit 3D-0D coupling scheme as seen in [9]. The full algorithm is detailed in the appendix. + +#### 3.1.2 Synthetic measurements + +The forward solution is generated with a time step of \(dt=1ms\) and undersampled in time to \(dt_{meas}=15ms\), leading to a total of 56 measurements. From the solution of the forward problem, we simulate a PC-MRI acquisition by subsampling into a rectangular measurement mesh with a resolution of \([2mm,2mm,2mm]\) and then applying the process described in Section 2.1 with a \(venc\) of double the maximal velocity. The magnitude is modelled as + +\[M(\boldsymbol{x})= \begin{cases} 1.0\text{if}\boldsymbol{x}\text{is in the lumen of the vessel} \\ 0.5\text{otherwise}. \end{cases}\] (16) + +Finally a complex Gaussian noise \(\varepsilon\in\mathbb{C}^{N}\) is added with a signal-to-noise ratio (SNR) of 15. Fifty independent realizations of the noise were generated. For comparison, we reconstructed velocity measurements from these synthetic measurements using the Berkeley Advanced Reconstruction Toolbox (BART)[11]. BART is a command-line-based software that provides a flexible framework of compressed sensing methods, as well as tools for simulation, pre-processing, and image reconstruction, providing a multitude of different regularization options. In this work, we have used this toolbox for compressed sensing reconstructions of the velocities, using total variation in time as for the regularization. Next, the sampling mask is applied to these simulated frequency space measurements. We take a 2D subsampled mask in the \(x-y-\)plane and sample fully in the \(z-\)direction as in [4]. We consider different subsampling rates \(R= \frac{N_{sampled}}{N_{total}} =8,16,32\), with two different masks: the pseudo-spiral mask and the pseudo-random Gaussian mask, which is sampled according to a Gaussian probability distribution, as shown in Figure 2. For the pseudo-spiral mask, the points are placed evenly on a cartesian grid along a spiral with six turns and a final radius reaching the edge of the mask. diff --git a/outputs/arxiv_math/2503.04108_pg45_pg1_repeat1.md b/outputs/arxiv_math/2503.04108_pg45_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..669e10ab9ed6943c2c60856516773c5e5680c95d --- /dev/null +++ b/outputs/arxiv_math/2503.04108_pg45_pg1_repeat1.md @@ -0,0 +1,5 @@ +Thanks to this technique, we have been able to reveal that the hidden symmetry algebra emerging from the embedding chain \(su(4)\supset su(2)\times su(2)\) gives rise to a quartic polynomial Poisson algebra. As a byproduct, this paper has also demonstrated that the grading method not only simplifies complex expansions in the Poisson bracket relations of a given degree but is, in fact, indispensable for deriving these relations when the degree becomes so high that standard “brute-force” methods are no longer applicable. This ansatz can be generalized and applied to any other reduction chain \(g\supset g^{\prime}\) involving reductive Lie algebras where the subalgebra embedding is not regular, and for which the usual argumentation using the root system is no longer applicable [10]. However, as shown in [39], there are several alternatives to define non-canonical gradings in Lie algebras, that can be conveniently adapted to various non-regular reduction chains (see [40,41] and references therein). Following this argumentation, the reduction chains of interest in physical applications (see e.g. [31, Chapter 12], [42]) examined in the context of the missing label problem can be reevaluated from the perspective of polynomial algebras, allowing us to describe the complete algebraic structure, and potentially allowing us to make more effective of labeling operators to separate degenerate states. For applications in superintegrable systems [43, 44], the precise knowledge of the associated polynomial algebra can be useful for the systematic construction of Hamiltonians admitting constants of the motion of degrees higher than two, providing new hierarchies of systems for which the classical criteria for the separation of variables no longer necessarily apply [45, 46]. It is worthy to be mentioned in this context that the observed relations between polynomial algebras and quasi-exactly solvable problems (see e.g. [47]) could be systematized by means of a detailed analysis of the (hidden) symmetry algebras and their associated polynomial structures. Work in these directions is currently in progress. + +# Acknowledgements + +This work was partially supported by the Future Fellowship FT180100099 and the Discovery Project DP190101529 from the Australian Research Council. RCS acknowledges financial support by the Agencia Estatal de Investigaci´on (Spain) under the grant PID2023-148373NB-I00 funded by MCIN/AEI/10.13039/501100011033/FEDER, UE. The research of DL is partially funded by MUR - Dipartimento di Eccellenza 2023-2027, codice CUP G43C22004580005 - codice progetto DECC23 012 DIP and partially supported by INFN-CSN4 (Commissione Scientifica Nazionale 4 - Fisica Teorica), MMNLP project. DL is a member of GNFM, INdAM. diff --git a/outputs/arxiv_math/2503.04116_pg9_pg1_repeat1.md b/outputs/arxiv_math/2503.04116_pg9_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..1ceac8de4bcf1862f85579a70c9eb9b59d2b13af --- /dev/null +++ b/outputs/arxiv_math/2503.04116_pg9_pg1_repeat1.md @@ -0,0 +1,19 @@ +![](figure) + +Figure \(9.\Gamma_{5}\) + +![](figure) + +Figure \(10.\Gamma_{6}\) + +\[\begin{aligned} (3)\sigma_{1}(\vec{e}_{i})&=e_{i}\text{and}\sigma_{1}(^{\rightarrow-}f_{j})=^{\longleftarrow }f_{j};1\leq i\leq3\text{and}1\leq j\leq7. \\ (4)\sigma_{0}&=C_{1}\ldots C_{5,}\text{where} \\ C_{1}&=(\vec{e}_{1},f_{1}\vec{,}e_{3}\vec{,}\vec{f}_{1}), \\ C_{2}&=(\vec{e}_{2},f_{3}\vec{,}e_{1}\vec{,}\vec{f}_{2}), \\ C_{3}&=(\vec{e}_{3},f_{6}\vec{,}e_{2}\vec{,}\vec{f}_{5}), \\ C_{4}&=(f_{2}\vec{,}\vec{f}_{4},\vec{f}_{3},f_{4}\vec{)}, \\ C_{5}&=(f_{5}\vec{,}\vec{f}_{7},\vec{f}_{6},f_{7}\vec{)}. \end{aligned}\] + +Now, we count the boundary components of \(\Gamma_{5}\) using the permutation \(\sigma_{\infty}\). The boundary components of the graph \(\Gamma_{5}\) given by: \(\partial_{1}=\vec{e}_{1}\vec{f}_{2}\vec{f}_{4}f⃗\) So, there is only one boundary component. + +\[\vec{e}_{2}⃗_{2}\vec{f}_{5}\vec{f}_{7}f_{5}\vec{e}⃗_{3}\vec{f}_{1}e_{3}\vec{f}_{6}\vec{f}_{7}⃗\vec{f}_{6}e_{2}\vec{f}_{3}\vec{f}_{4}⃗\vec{f}_{3}e_{1}\vec{f}_{1}⃗\] + +1. + +## 4. Upper Bound On Mapping Class Orbits of Maximum Size Filling + +Let I(S g) represent the set of all isotopy classes of simple closed curves on \(S_{g}.I(S_{g})^{n}\) denotes the Cartesian product of \(n\in\mathbb{N}\) copies of I(S g). Define an equivalence relation \(\sim\) on I(S g)n by diff --git a/outputs/arxiv_math/2503.04124_pg14_pg1_repeat1.md b/outputs/arxiv_math/2503.04124_pg14_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..c0ced3333bf0e279b1295f91a86e53de3fa8b0a2 --- /dev/null +++ b/outputs/arxiv_math/2503.04124_pg14_pg1_repeat1.md @@ -0,0 +1,17 @@ +whose center vertex is \(x\), by adding an edge \(vx\) and a pendent 4-cycle at vertex \(v\). In that case, by Proposition 2.6, there is a hop dominating set \(S\) of \(H,\text{such that}x\in S\text{and}|S|\leq \frac{2(n-4)}{5}\) , which implies that \(S\cup\{v\}\) is a hop dominating set \(S\) of \(G\), a contradiction. Thus such a path \(P\) exists. Note that \(k\geq2\). + +## Claim 3.10. \(k\in\{2,3\}\) + +Proof. Suppose that \(k\geq4\). Let \(G^{\prime}=(G-xx_{1})+x_{1}x_{4}\). Note that \(x_{1}x_{2}x_{3}x_{4}v_{1}\) is a pendent 4-cycle at \(x_{4}\text{in}G^{\prime}\). Note that \(|V(G^{\prime})|=n,|E(G^{\prime})|=|E(G)|,\text{and}G^{\prime}\) has more pendent 4-cycles than \(G\). Suppose that \(\deg_{G}(x)\geq3.\text{Then}\delta(G^{\prime})\geq2\text{and}G^{\prime}\) has no connected component in \(\mathcal{B}\). There is a minimum hop dominating set \(S^{\prime}\) of \(G^{\prime}\text{such that}x_{4},v,x\in S^{\prime}\text{and}N_{G}(x_{4})\cap S^{\prime}\neq \emptyset\) by Proposition 2.7. By the choice of \(G,|S^{\prime}|\leq \frac{2n}{5}\) . Thus \(S^{\prime}\) is a hop dominating set of \(G\), which is a contradiction. Suppose that \(\deg_{G}(x)=2\). Then let \(H\) be the connected component of \(G^{\prime}\) containing \(x_{4}\). Note that \(|V(H)|\leq n-5\). There is a minimum hop dominating set \(S^{\prime}\) of \(G^{\prime}\text{such that}x_{4}\in S^{\prime}\), \(N_{G^{\prime}}(x_{4})\cap S^{\prime}\neq \emptyset\) by Proposition 2.7. If \(S^{\prime}\) contains \(x_{1}\), then we replace \(x_{1}\text{with}x_{3}\) so that \(S^{\prime}\) does not have \(x_{1}.\text{Then}S^{\prime}\cup\{v,x\}\) is a minimum hop dominating set of \(G\), a contradiction. Let \(G^{\prime}=G-\{x_{1},\ldots,x_{k-1}\}\). Suppose that \(\deg_{G}(x)\geq3.\text{Then}\delta(G^{\prime})\geq2.\text{If}G^{\prime}\) has no connected component in \(\mathcal{B}\), then take a minimum hop dominating set \(S^{\prime}\) of \(G^{\prime}\text{such that}v,x\in S^{\prime}\) by Proposition 2.7, which implies that \(S^{\prime}\) is a minimum hop dominating set of \(G\), a contradiction. Thus, \(G^{\prime}\) has two connected components \(D_{1}\text{and}D_{2}\). We may assume that \(D_{1}\not\in \mathcal{B}\text{and}D_{2}\in\mathcal{B}\). There is a hop dominating set \(S_{1}\) of \(D_{1}\text{such that}v,x\in S_{1}\text{and}|S_{1}|\leq \frac{2|V(D_{1})|}{5}\) by Proposition 2.7. If \(k=3\text{or}D_{2}\neq C_{8}\), then by (3.1), there is a minimum hop dominating set \(S_{2}\) of \(D_{2}\) such that \(|S_{2}|\leq^{\underline{2|V(D_{2})|+2(k-1)}}\) a path \(D_{2}-x_{2}\text{such that}|S_{2}|=3\leq \frac{2|V(D_{2})|+2(k-1)}{5}\) . Note that \(S_{1}\cup S_{2}\) is a hop dominating set of \(G\). Then + +. If \(k=2\text{and}D_{2}=C_{8}\), then we take a minimum hop dominating set \(S_{2}\) of + +5 + +\[|S_{1}\cup S_{2}|\leq \frac{2|V(D_{1})|+2|V(D_{2})|+2(k-1)}{5} = \frac{2n}{5} ,\] + +which is a contradiction. Suppose that \(\deg_{G}(x)=2\). Let \(H\) be the connected component containing \(x_{k}.\text{Then}|V(H)|\leq n-6.\text{If}H\neq C_{8}\), then for a minimum hop dominating set \(S\) of \(H\), we have \(|S|\leq \frac{2(n-6)+2}{5}\) by (3.1), and so \(S\cup\{v,x\}\) is a hop dominating set of \(G\) whose size is at most \(\frac{2n}{5}\) , a contradiction. Thus \(H=C_{8}.\text{Then}G\) is one of the last two graphs in Figure 6, which shows that \(\gamma_{h}(G)\leq \frac{2n}{5}\) . This completes the proof of our main theorem. + +## 4 An open problem + +It would be a natural extension to consider giving a sharp upper bound on the hop domination number for a graph with a large girth. So we propose the following open problem. diff --git a/outputs/arxiv_math/2503.04147_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.04147_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..254b7f7549ff32530cd16fd94cfba8f1d9a4dccf --- /dev/null +++ b/outputs/arxiv_math/2503.04147_pg3_pg1_repeat1.md @@ -0,0 +1,53 @@ +Transmitter with 𝑁௧ fluid antennas + +## 1 2 + +![](figure) + +Fig. 1. System model. + +● WET + +↓ Data Receiver + +↓ Energy Receiver + +finally, our paper is concluded in Section VI. Notations: In this paper, boldface upper letters, boldface lower letters, and lower letters denote matrices, vectors, and scalars, respectively. \(\mathbb{C}^{x\times y}\text{and}\mathbb{R}^{x\times y}\) represent the set of complex matrices and real matrices with the dimension of \(x\times y\), respectively. \(|x|\) represents the amplitude of a complex number + +\(x.||\boldsymbol{x}||\text{and}||\boldsymbol{x}||_{1}\) denote the 2-norm and 1-norm of a vector \(\boldsymbol{x}\), respectively. The notation \((\cdot)^{T}\text{and}(\cdot)^{H}\) refer to the transpose and the conjugate transpose of a vector or matrix, while E represents the expectation operator. + +## II. SYSTEM MODEL + +As shown in Fig. 1, we consider a multiple FAs assisted IDET system, which is comprised of a transmitter, a DR and an ER. The transmitter is equipped with \(N_{t}\) FAs, each of which has the same size and parameters without loss of generality. The DR and ER are both equipped with a single traditional antenna due to the limited hardware space. We assume that the transmit antennas are far apart in space and the channels of different antennas are fully independent, while the channels among different ports in the same antenna are strongly spatially correlated. The set of FAs and FA’s port are denoted by \(\mathcal{N}_{t}= \{1,\cdots ,N_{t}\}\text{and}\mathcal{N}=\{1,\cdots ,N\}\), respectively. + +## A. Fluid antenna model + +In this paper, we consider the practical architecture of the liquid metal-based fluid antenna. The length of fluid antenna is \(W\lambda,\text{where}\lambda\) is the wavelength of RF signal and \(W\) is a scaling constant. The MEMS is adopted in each fluid antenna to drive the metal droplet to move. Specifically, due to the electrocapillary effect, the motion of metal droplet is achieved by applying a voltage gradient along the FA, resulting in a phenomenon known as continuous electrowetting [24]. According to [5], the average velocity of the metal droplet motion is given by + +\[u= \frac{q}{6\mu} \frac{D}{L} \Delta\phi,\] (1) + +where \(q\) is the initial charge in the electrical double layer for EGaIn, \(\mu\) denotes the viscosity of EGaIn, \(D\text{and}L\) are the thickness and length of metal droplet, respectively. \(\Delta\phi\) represents the voltage differential resulting from the flow of current through the narrow electrolyte layer between EGaIn and the wall, which is much smaller than the voltage \(U\) applied between two ends of the FA [25]. Therefore, the moving delay from \(i-\)th port to \(j-\)th port can be expressed as + +\[\tau= \frac{W\lambda}{u} \frac{|i-j|}{N-1} .\] (2) + +## B. Wireless channel model + +The block fading channel model is assumed between the transmitter and receivers, implying that the channel remains constant during a coherence time \(T\) but varies from one frame to another. Since the ports are located much closer in the same FA, strong correlation exists between the channels of different ports. Following Jake’s two dimensional correlation model [26], the correlation between any two ports can be represented by a matrix \(\boldsymbol{J}\), which is given by + +\[\boldsymbol{J}= \begin{bmatrix} J_{1,1} & J_{1,2} & \cdots & J_{1,N} \\ J_{2,1} & J_{2,2} & \cdots & J_{2,N} \\ \ddots & \ddots & \ddots & \ddots \\ J_{N,1} & J_{N,2} & \cdots & J_{N,N} \end{bmatrix} ,\] (3) + +where \(J_{i,j}=J_{0}( \frac{2\pi(i-j)}{N-1} W)\text{and}J_{0}(\cdot)\) is the zeroth-order Bessel function of the first kind. The channels between \(i-\)th FA and DR and ER in the \(t-\)th block are denoted as \(\boldsymbol{h}_{i}(t)\) and \(\boldsymbol{g}_{i}(t)\), which can be modeled as + +\[\boldsymbol{h}_{\boldsymbol{g}_{i}^{i}}(_{(}t_{t})_{)}=_{=} \sqrt{\frac{\frac{1}{PL_{h,i}}}{\frac{1}{PL_{g,i}}}} \boldsymbol{U}_{\boldsymbol{U}} \sqrt{\frac{\boldsymbol{\Lambda}}{\boldsymbol{\Lambda}}} \boldsymbol{z}_{\boldsymbol{z}_{g}^{h}}(_{(}t_{t})_{)},_{,}\] (4) + +where \(PL_{h,i}\text{and}PL_{g,i}\) are the path-loss from \(i-\)th FA to DR and ER, respectively. \(\boldsymbol{U}\text{and}\boldsymbol{\Lambda}\) are a unitary matrix composed of the eigenvectors of \(\boldsymbol{J}\) and a diagonal matrix consisting of the eigenvalues of \(\boldsymbol{J}\), which satisfy \(\boldsymbol{J}=\boldsymbol{U}\boldsymbol{\Lambda}\boldsymbol{U}^{T}\) . In addition, \(\boldsymbol{z}_{h}(t)\in\mathbb{C}^{N\times1}\text{and}\boldsymbol{z}_{g}(t)\in\mathbb{C}^{N\times1}\) are complex Gaussian random vector with each element having zero mean and variance of 1. On the other hand, the channels of different transmission frames are temporally correlated, which can be expressed as + +\[\boldsymbol{h}_{\boldsymbol{g}^{i}}(_{(_{i}}t)_{t)}=_{=}\rho_{\rho}\boldsymbol{h}_{\boldsymbol{g}_{i}^{i}}(_{(}t_{t}-_{-}1_{1})_{)}+_{+} \sqrt{\frac{\frac{1-\rho^{2}}{PL_{h,i}}}{\frac{1-\rho^{2}}{PL_{g,i}}}} \boldsymbol{\delta} \boldsymbol{\delta},\] (5) + +where \(\rho\) is the temporal correlation factor and \(\boldsymbol{\delta}\sim\mathcal{C}\mathcal{N}(0,\boldsymbol{I}_{N})\) is the random difference with an identity matrix \(\boldsymbol{I}_{N}\). + +## C. Signal model + +Before the transmitter delivering wireless signal to the DR and ER, an appropriate antenna port should be selected on each FA. Assuming that the initial port of \(i-\)th FA is \(k_{i,0}(t)\) in the \(t-\)th block, the port switching delay on the \(i-\)th antenna from the initial port to the selected port \(k_{i}(t)\) is expressed as + +\[\tau_{i}(t)= \frac{W\lambda}{u} \frac{|k_{i}(t)-k_{i,0}(t)|}{N-1} .\] (6) diff --git a/outputs/arxiv_math/2503.04182_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.04182_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..fc3734690a52056188285a27cb8b5ebf0a759c51 --- /dev/null +++ b/outputs/arxiv_math/2503.04182_pg3_pg1_repeat1.md @@ -0,0 +1,27 @@ +Prior to proceed let us consider the the product \(D_{p}x\). Since \(D_{p}\) is an \(n\times n\) matrix with entries in \(\mathbb{Q}_{p}\text{and}x\) is an \(n-\)tuple with entries in \(\mathbb{Q}_{p}\), the result \(D_{p}x\) is an \(n-\)tuple with entries in \(\mathbb{Q}_{p}\). Formally, if \(D_{p}=(d_{ij})\text{and}x=(x_{1},x_{2},\ldots,x_{n})\), then the \(i-\)th component of \(D_{p}x\) is given by: + +\[(D_{p}x)_{i}=\sum_{j=1}^{n}d_{ij}x_{j}\in\mathbb{Q}_{p}\] + +The p-adic norm \(|\cdot|_{p}\) is applied component-wise to the resulting \(n-\)tuple \(D_{p}x\). For each component \((D_{p}x)_{i}\) of \(D_{p}x\), we have: + +\[|(D_{p}x)_{i}|_{p}= \frac{1}{p^{\text{ord}_{p}((D_{p}x)_{i})}}\] + +where \(\text{ord}_{p}((D_{p}x)_{i})\) is the highest power of \(p\) dividing \((D_{p}x)_{i}\). Since the p-adic norm \(|\cdot|_{p}\) maps elements of \(\mathbb{Q}_{p}\) to \(\mathbb{Q}_{p}\), applying it component-wise to an \(n-\)tuple will result in another \(n-\)tuple with entries in \(\mathbb{Q}_{p}\). Therefore, \(|D_{p}x|_{p}\) is an \(n-\)tuple with entries in \(\mathbb{Q}_{p}\). Thus, \(\delta_{p}(x)=|D_{p}x|_{p}\in\mathbb{Q}_{p}^{n}\), hence, the map \(\delta_{p}\) is well-defined. As we have seens previous we can define the \(p-\)adic Ducci sequence the sequence + +## Definition 0.4 \((p-\)adic Ducci sequence). + +\[x,\delta_{p}(x),\delta_{p}^{2}=\delta_{p}(\delta_{p}(x))=\delta_{p}\circ\delta_{p},\ldots,\delta_{p}^{n}(x),\ldots\] + +with initial seed \(x\in\mathbb{Q}_{p}^{n}\) respect to \(D_{p}\in\mathbb{M}_{n\times n}(\mathbb{Q}_{p})\). In this p-adic context, we define a p-adic Ducci map associated with a \(p-\)adic matrix \(D_{p}\) as a function that maps a p-adic vector \(x\) to \(|D_{p}x|_{p},\text{where}|\cdot|_{p}\) is applied elementwise. Without loss of generality we can define the following For convergence to zero, we analyze when \(x_{k}\rightarrow0\text{as}k\rightarrow\infty\). Prior to move forward we recall the following + +Remark 1. In the field of \(p-\)adic numbers \(\mathbb{Q}_{p}\), the \(p-\)adic absolute value \(|x|_{p}\) of a nonzero element \(x\) is defined as: + +\[|x|_{p}=p^{-v_{p}(x)},\] + +where \(v_{p}(x)\) is the \(p-\)adic valuation of \(x\). The valuation \(v_{p}(x)\) is the highest power of \(p\) that divides \(x\text{in}\mathbb{Q}_{p}\). If \(|x|_{p}<1\text{and}x\neq 0\), this means that \(v_{p}(x)>0\), so \(x\) is divisible by \(p\). The possible values of \(|x|_{p}\) in this case are: + +\[|x|_{p}=p^{-k},\text{where}k\in\mathbb{N}\text{and}k\geq1.\] + +Thus, the possible values of \(|x|_{p}\text{when}|x|_{p}<1\) are: + +\[\left\{p^{-1},p^{-2},p^{-3},\ldots\right\}.\] diff --git a/outputs/arxiv_math/2503.04189_pg11_pg1_repeat1.md b/outputs/arxiv_math/2503.04189_pg11_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..f26a3af523e51e704067e6c4e248cec2b6ef0b74 --- /dev/null +++ b/outputs/arxiv_math/2503.04189_pg11_pg1_repeat1.md @@ -0,0 +1,37 @@ +with + +\[\rho(y,x,\eta^{\prime})=\varphi(y,\eta^{\prime})-\varphi(x,\eta^{\prime})-(y-x)\varphi_{x}(x,\eta^{\prime}),\] + +\(\rho\) vanishes at order \(\geq2\text{in}y=x\). For checking (3.10), we see + +\[Q(ae^{i \frac{\varphi}{h}})=\int\int e^{i \frac{1}{h} ((x^{\prime}-y^{\prime})\theta^{\prime}+\varphi(y,\eta^{\prime}))} \frac{q(x,\theta^{\prime})}{(2\pi h)^{n-1}} a(y,\eta^{\prime})dy^{\prime}d\theta^{\prime}.\] (3.11) + +on \(\text{supp}(qa)\), so for + +a having compact support and \(\varphi\) being a symbol, one has \(|\varphi_{y}^{\prime}|\leq C^{\prime} |\theta^{\prime}|\geq C>0\), large, the phase of \((3.11)H=(x^{\prime}-y^{\prime})\theta^{\prime}+\varphi(y,\eta^{\prime})\) satisfies + +\[|H_{y^{\prime}}^{\prime}|\geq c(1+|\theta^{\prime}|),\text{for}|\theta^{\prime}|\geq C>0,\text{large}\] + +and if we split the integrand of (3.11) in \(qa=\chi(\theta^{\prime})qa+(1-\chi(\theta^{\prime}))qa\), we integrate the second term by parts and obtain + +\[Q(ae^{i \frac{\varphi}{h}})=Q_{\varphi}^{\prime}(a)e^{i \frac{\varphi}{h}}+R(a),\] (3.12) + +where \(Q_{\varphi}^{\prime}\) is a \(\mathcal{G}^{s}\) symbol of order \(\tilde{S}_{s}^{m,k+1}\) having the expansion (3.10) by the stationary phase lemma as \(R(a)\) is an \(\mathcal{O}_{s}(h^{\infty})\) remainder. It is easy to see in view of these arguments that \(a_{1}\in\tilde{S}_{s}^{m-1,k}\). Moreover, it is to be observed that the above expansion is only a formal Gevrey \(2s-1\) symbol. The microlocal invertibility of FIO reduces to the PDO case. We refer to [5] for a proof of the Gevrey elliptic result in classes \(S_{s}^{m}\) . For proving Theorem 2, we rewrite (3.8) in the form + +\[(hD_{x_{1}}+Q(x,hD_{x};h))Fu=F(hD_{x_{1}}+Q^{\prime})u,\] + +close to \((x_{0},\xi_{0};x_{0},\xi_{0})\) for some PDO \(Q^{\prime}(x,hD_{x};h)\) of bi-order \((-1,0)\) in using a left microlocal inverse of \(F\) close to \((x_{0},\xi_{0}^{\prime})\). Indeed, we compute \(FF^{*}\text{and}F^{*}F\) , and one has writing \(y=(x_{1},y^{\prime})\). One has, following Eskin [8], + +\[FF^{*}u(x,h)= \frac{1}{(2\pi h)^{n-1}} \int\int e^{\frac{i}{h} (\varphi(x,\xi^{\prime})-\varphi(y,\xi^{\prime}))}a(x,\xi^{\prime}) \overline{a(y,\xi^{\prime})} u(x_{1},y^{\prime})dy^{\prime}d\xi^{\prime},\] + +\(\varphi(x,\xi^{\prime})\) having been obtained in (3.6). We split the integral above into two terms. The first is a \(h-\text{PDO}\), the second is a smoothing operator. First, we note that the map: + +\[(x,y^{\prime},\xi^{\prime})\rightarrow(x,y^{\prime},\Sigma(x,y^{\prime},\xi^{\prime})),\] + +with + +\[\Sigma(x,y^{\prime},\xi^{\prime})=\int_{0}^{1}\varphi_{x^{\prime}}^{\prime}(x_{1},y^{\prime}+t(x^{\prime}-y^{\prime}),\xi^{\prime})dt\] (3.13) + +is a \(\mathcal{G}^{s}-\)diffeo in a neighbourhood of \((x_{0},y_{0}^{\prime},\eta_{0}^{\prime})\text{with}|x_{1}|\leq\delta,|x^{\prime}-y^{\prime}|\leq\delta,0<\delta\) small, close to the identity. Let \((x,y^{\prime},\eta^{\prime})\rightarrow(x,y^{\prime},\Sigma^{-1}(x,y^{\prime},\eta^{\prime}))\) be an inverse map. One has obviously \(\varphi(x,\xi^{\prime})-\varphi(y,\xi^{\prime})=\Sigma(x,y^{\prime},\xi^{\prime})(x^{\prime}-y^{\prime}),\) and + +\[FF^{*}u(x,h)=K_{1}u(x,h)+K_{2}u(x,h),\] (3.14) diff --git a/outputs/arxiv_math/2503.04226_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.04226_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..699e731d1a65ee9f4a100952a048f66731ff5e16 --- /dev/null +++ b/outputs/arxiv_math/2503.04226_pg3_pg1_repeat1.md @@ -0,0 +1,7 @@ +(resp. dual) closedness criterion. To this aim, let us recall the concept of closedness regarding a set [1]. Given two subsets \(U\text{and}V\) of a topological space \(Z,\) one says that \(U\) is closed regarding \(V\text{if} \overline{U} \cap V=U\cap V,\text{where} \overline{U}\) is the closure of \(U\text{in}Z.\) In particular, Given \(z\in Z,U\) is closed regarding \(\{z\}\) if and only if either \(z\in U\text{or}z\in/ \overline{U} ,U\) is closed regarding \(V\) if and only if \(U\) is closed regarding \(\{z\}\text{for all}z\in V,\text{and}U\) is closed if and only if it is closed regarding \(Z.\) The term characterization of Farkas’ lemma was introduced by V. Jeyakumar, S. Kum, and G. M. Lee [13] and that of characterization of stable Farkas’ lemma by V. Jeyakumar and G. M. Lee [14], both papers published in 2008. Their characterization of stable Farkas’ lemma consisted in the closedness of certain cone associated with the given conic system. After the introduction of the concept of closedness regarding a given set by Boţ in 2010, successive characterizations of (stable) Farkas’ lemma in terms of closedness of certain set regarding another set were obtained for different types of systems, e.g., systems involving convex, nonconvex composite functions [8], [9], [10], systems involving vector-valued functions ([4] and [6]), systems involving a family of finite subsets of the given index set [7], etc. The characterization of stable Farkas’ lemma for nonconvex composite semi-infinite programming has been considered in [16] and [17]. Considering feasible sets of the form \(C\cap\mathbb{A}^{-1}(D),\) with the convex set \(D\) not being necessarily a cone, is the main novelty of this paper, which is organized as follows. Section 2 characterizes the equivalence \((\mathcal{A})\Longleftrightarrow (\mathcal{B})\) in terms of the closedness regarding \(\{(0_{X},0_{Y},-1)\}\text{and}\{(0_{Y},-1)\}\) of certain subsets of the primal spaces \(X\times Y\times\mathbb{R}\) and \(Y\times\mathbb{R},\) respectively (see Theorems 1 and 2). Section 3, in turn, characterizes the equivalence \((\mathcal{A})\Longleftrightarrow (\mathcal{B})\) in terms of the closedness regarding \(\{(0_{X^{\prime}},0)\}\) of certain subset of the dual space \(X^{\prime}\times\mathbb{R}\) (Theorem 11) as well as the non-emptyness of the feasible set \(C\cap\mathbb{A}^{-1}(D)\) (Proposition 9) and the stable Farkas’ lemma (Proposition 16). Section 4 is devoted to linear infinite systems. Section 5 characterizes Farkas-type lemmas oriented to identify the minima of convex (resp. concave) functions on \(C\cap\mathbb{A}^{-1}(D),\) reason why they are known in the literature as convex (resp. concave) Farkas’ lemmas. More in detail, the convex Farkas’ lemmas provided in Section 3 determine when a convex feasible set of the form \(C\cap\mathbb{A}^{-1}(D)\) is contained in the reverse-convex set \(\{x\in X:f(x)\geq0\},\text{where}f\) is convex. Similarly, Section 5 characterizes the containment of \(C\cap\mathbb{A}^{-1}(D)\) in the sublevel set of a convex function \(f\) (Theorem 23). Finally, Section 6 shows applications to constrained convex minimization problems (optimization, strong duality and stable strong duality theorems) and to functional approximation by polynomials (Farkas-type lemmas). + +## 2 Primal closedness characterizations + +Recall that \(f:X\longrightarrow \overline{\mathbb{R}} (f\in \overline{\mathbb{R}}^{X}\) in short) is a proper convex function. We consider the vector-valued mapping \(H:X\times Y\times\) dom \(f\longrightarrow X\times Y\times\mathbb{R}\) such that + +\[H(x,y,v)=(x-v,\mathbb{A}x-y,f(v)),\] diff --git a/outputs/arxiv_math/2503.04228_pg5_pg1_repeat1.md b/outputs/arxiv_math/2503.04228_pg5_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..1697cbf07b81da3e5f01b1b8482d7878a89f4926 --- /dev/null +++ b/outputs/arxiv_math/2503.04228_pg5_pg1_repeat1.md @@ -0,0 +1,43 @@ +## 2 Simple Proof + +This section presents a simple proof of the Apex Minor Theorem. The next lemma is the key. A centre of a graph \(G\) is a vertex \(\alpha\in V(G)\text{such that}\max\{\text{dist}(\alpha,v):v\in V(G)\}\) is equal to the radius of \(G\). Lemma 10. Let \(A\) be a planar graph such that \(A-z\) is a minor of the \(k\times k\) grid for some \(z\in V(A)\). Let \(G\) be an \(A-\text{minor}-\)free graph with radius \(r\) and centre \(\alpha.\text{Then}G-\alpha\) has no + +\(k^{r}\times k^{r}\) grid minor. + +Proof. We proceed by induction on \(r\). In the \(r=0\) case, \(V(G-\alpha)=\emptyset\) and the result + +holds. Now consider the radius \(r\) case, and assume the radius \(r-1\) case holds. Let + +\(V_{i}:=\{x\in V(G):\text{dist}_{G}(x,\alpha)=i\}\text{for}i\in\{0,\ldots,r\}\). So \(V_{0}=\{\alpha\}\text{and}V_{0}\cup\cdots \cup V_{r}=V(G)\). + +Suppose for the sake of contradiction that \(G-\alpha\) has a \(k^{r}\times k^{r}\) grid minor. Partition + +this \(k^{r}\times k^{r}\) grid into \(k\times k\) subgrids, so that contracting each subgrid to a vertex gives a + +\[k^{r-1}\times k^{r-1}\text{grid}.\] + +First suppose that some \(k\times k\) subgrid is contained in \(V_{r}\). Thus \(H\) is a minor of \(G[V_{r}]\). By + +\(V_{0}\cup\cdots \cup V_{r-1}\) to a vertex \(w\). Since every vertex in \(V_{r}\) has a neighbour in \(V_{r-1}\), every vertex + +construction, \(G[V_{0}\cup\cdots \cup V_{r-1}]\) is connected. Let \(G^{\prime}\) be obtained from \(G\) by contracting in \(G^{\prime}\) is at distance at most 1 from \(w\). Since \(H\) is a minor of \(G[V_{r}],A\) is a minor of \(G^{\prime}\) and thus of \(G\), which is a contradiction. + +Now assume that every \(k\times k\) subgrid intersects \(V_{1}\cup\cdots \cup V_{r-1}\). Let \(G^{\prime}\) be obtained from \(G\) by contracting each subgrid to a vertex, and deleting any vertices in \(V_{r}\) not in one of the + +\(k^{r-1}\times k^{r-1}\) grid minor. Since \(G^{\prime}\) is a minor of \(G,G^{\prime}\) is \(A-\text{minor}-\text{free}\). Hence \(G^{\prime}\) contradicts + +subgrids. So every vertex in \(G^{\prime}\) is at distance at most \(r-1\) from \(\alpha,\text{and}G^{\prime}-\alpha\) contains a the assumed truth of the \(r-1\) case. + +Therefore, \(G-\alpha\) has no \(k^{r}\times k^{r}\) grid minor. + +Lemmas 2 and 10 imply that for every apex graph \(A\text{with}t\) vertices, every \(A-\text{minor}-\)free + +graph \(G\) with radius \(r\) has no \((2t-2)^{r}\times(2t-2)^{r}\) grid minor. Equation (1) implies that + +\(\text{tw}(G)\in O^{*}((2t)^{9r})\). This completes our simple proof of the Apex Minor Theorem by Eppstein [18] (Theorem 4). + +## 3 Polynomial Upper Bound + +This section proves Theorem 5 from Section 1. We need the following straightforward lemma. + +Lemma 11. If a graph \(G\) is a minor of the \(k\times\ell\) grid, then there is an \(H-\text{model}(B_{u}:u\in V(H))\) in the \(2k\times2\ell\) grid \(J\), such that for each vertex \(u\in V(H)\) there is a vertex \(h_{u}inB_{u}\) incident to no edge of \(J\) representing an edge of \(H\). diff --git a/outputs/arxiv_math/2503.04238_pg26_pg1_repeat1.md b/outputs/arxiv_math/2503.04238_pg26_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..0897785df85826dae17b84a693076af4117e9d3c --- /dev/null +++ b/outputs/arxiv_math/2503.04238_pg26_pg1_repeat1.md @@ -0,0 +1,31 @@ +for the Dirichlet form associated to sticky Brownian motion. Denoting \(a_{0}= \frac{1}{2+\omega L}\) and \(b_{0}=\omega a_{0},\text{for all}s>0\) we have + +\[\begin{aligned} \int f^{2}d\mu&=a_{0}(f(0)^{2}+f(L)^{2})+b_{0}\int_{0_{L}}^{L}f^{2}dx, \\ &\leq a_{0}(f(0)^{2}+f(L)^{2})+b_{0}s\int_{0}(f^{\prime})^{2}dx+b_{0}\beta(s)\left(\int_{0}^{L}|f|dx\right)^{2}, \\ &\leq b_{0}s\int_{0}^{L}(f^{\prime})^{2}dx+b_{0}\max(b_{0}^{-2},a_{0}^{-1})\beta(s)\left(\int|f|d\mu\right)^{2}. \end{aligned}\] + +Therefore the sticky Brownian satisfies a super Poincaré inequality. Then by [Wan00, Th. 5.1], it has an empty essential spectrum. Now, by [BGL14, Th. A.6.4], the resolvent is compact and thus the generator has discrete spectrum. + +Corollary 19. Choosing \(T=m^{-1/2}\), the transition semigroup of the RTP process is exponentially contractive in \(T\) -average with rate + +\[\nu=\Omega\left( \frac{\omega}{1+(\omega L)^{2}} \right).\] + +Note that the relaxation time corresponding to this decay rate is of the same order as the mixing time obtained in [GHM24]. It reveals the existence of two regimes controlled by the parameter \(\omega L\). In the ballistic regime \(\omega L\ll1\), velocity flips are rare, leading to a fast exploration of the position space \(\mathcal{S}\) and a comparatively slow exploration of the velocity space \(\mathcal{V}\). This results in the scaling \(\nu\propto\omega\). On the contrary, in the diffusive regime \(\omega L\gg1\), the high frequency of velocity flips makes the exploration of \(\mathcal{V}\) faster than the exploration of \(\mathcal{S}\). This leads to the scaling \(\nu\propto\omega^{-1}L^{-2}\). + +Proof. We begin by verifying Assumption (A). Recall that \(\text{Dom}(\mathcal{L}_{C^{0}})\) is a core of \(\mathcal{L}\) by Theorem 7. For all \(f\in\text{Dom}(\mathcal{L}_{C^{0}})\) we have \(\hat{\mathcal{L}}_{v}(f\circ\pi)=0\) hence \(\hat{\mathcal{L}}_{\text{tr}}\) is a lift of \(\mathcal{L}\) by Remark 8. Furthermore, for \(f\in\text{Dom}(\mathcal{L}_{C^{0}})\) one has + +\[\hat{\mathcal{L}}_{\text{tr}}^{*}(f\circ\pi)(x,v)=-v1_{\{00,b\geq0,\] + +to study the monotonicity property of the operator V := II + IV. In this way, we conveniently use the relation \(\partial_{x}G(t,s)=-bG(t,s),\text{for}0\leq s0,b_{i}\geq0,i\in\{1,\cdots ,n\},n\in\mathbb{N}.\] + +The answer turns out to be negative. Indeed, assume without loss of generality that the mean– reversion speeds \((b_{i})_{i}\) differ from one another. Then, in this case, \(k\equiv\sum_{i=1}^{N}a_{i}=:A\) and + +\[\partial_{x}G(t,s)=-\sum_{i=1}^{n}a_{i}b_{i}\exp\{-b_{i}(t-s)\},0\leq s0\), \(mm\). Let \(\delta>0\) be such that \(\ell(\gamma|_{[\delta,1]})>m\). But \(\gamma(\delta)\gg q\) hence \(\gamma(\delta)\gg p\), which implies \(d_{p}(r)\geq\ell(\gamma|_{[\delta,1]})>m\). Since + +\[m0\) so \(d_{p}(r)>0\) which implies \(r\in I^{+}(p)\). By the arbitrariness of \(r,I^{+}(q)\subset I^{+}(p)\). We recall [43, Def. 4.46] that a spacetime is future distinguishing if \(I^{+}(p)= I^{+}(q)\Rightarrow p=q\) (and similarly in the past case). A spacetime is weakly distinguishing if \(I^{+}(p)=I^{+}(q)\text{and}I^{-}(p)=I^{-}(q)\Rightarrow p=q\), cf. [43, Def. 4.47]. A spacetime is distinguishing if \(I^{+}(p)=I^{+}(q)\text{or}I^{-}(p)=I^{-}(q)\Rightarrow p=q\), cf. [43, Def. 4.59]. It is convenient to introduce the following notions + +Definition 2.2. We say that the the Lorentzian distance future distinguishes events (the spacetime is future \(d-\text{distinguishing})\text{if}d_{p}=d_{q}\Rightarrow p=q\), and that is past distinguishes events if \(d^{p}=d^{q}\Rightarrow p=q\). The spacetime is \(d-\) distinguishing if it is both future and past \(d-\text{distinguishing}\), namely: \(d^{p}=d^{q}\) or \(d_{p}=d_{q}\Rightarrow p=q\). We say that the spacetime is weakly \(d-\)distinguishing if \(d^{p}=d^{q}\text{and}d_{p}=d_{q}\Rightarrow p=q\). Note that \(d-\)distinction implies ‘future or past \(d-\text{distinction}’\) which implies weak \(d-\text{distinction}\). Remark 2.3. Weak \(d-\)distinction is called just \(d-\)distinction (or it is said that the Lorentzian distance distinguishes events) in [45, 13] as this is the only version that plays a role in those works. An immediate consequence of Prop. 2.1 is + +Corollary 2.4. On a spacetime \((M,g)\) (future/past/weak) \(d-\)distinction is equivalent to (resp. future/past/weak) distinction. diff --git a/outputs/arxiv_math/2503.04397_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.04397_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..f311ea6b29f7fcb3b63cb045158e8ce2c788a22f --- /dev/null +++ b/outputs/arxiv_math/2503.04397_pg2_pg1_repeat1.md @@ -0,0 +1,17 @@ +grate the rotatable STAR-RIS into MEC systems [14]. For example, although RIS configurations have been intensively studied, existing algorithms are not suitable of the optimization of STAR-RIS configurations since they must be designed based on different operating protocols, namely energy splitting (ES), mode switching (MS), and time switching (TS) [9]. Meanwhile, STAR-RIS introduces more adjustable parameters beyond phase shift, i.e., the amplitude coefficients for ES/MS protocol, and time allocation for TS protocol, all of which are highly coupled with STAR-RIS orientation and computation resource allocation for MEC [15]. In addition, although existing studies have explored the orientation optimization of RIS/STAR-RIS, most of them focus on static- or single-UD environments [16], [17], which may not fully capture the complexities arising from multiple moving UDs in MEC systems. Therefore, this paper proposes a rotatable STAR-RIS scheme to account for dynamic environments and enhance MEC performance. The primary contributions of this paper are outlined as follows. + +- We propose a novel rotatable STAR-RIS-assisted MEC system with multiple moving UDs, where the STAR-RIS can dynamically adjust its orientation to accommodate UDs mobility. Considering the characteristics of STAR-RIS, the energy consumption minimization problem is formulated for three operating protocols, jointly optimizing STAR-RIS configurations, orientation, computation resource allocation, transmission power, and task offload-ing strategies. + +- To account for practical scenarios, we incorporate discrete phase shift, STAR-RIS elements grouping, and UDs mobility into our model. Thus, the energy consumption minimization problem is formulated as a time-vary problem. Considering this problem is difficult to handle using traditional algorithms, we propose a sample-efficient deep reinforcement learning (DRL) algorithm based on soft actor-critic (SAC) to adapt to the dynamic environment and achieve near-optimal solutions. + +- Simulation results validate that the proposed approach outperforms representative DRL algorithms in both convergence and performance. Additionally, the proposed rotatable STAR-RIS scheme significantly reduces energy consumption compared to the fixed scheme. + +The rest of the paper is structured as follows. Related works are discussed in Section II. In Section III, the system model and formulated energy consumption minimization problems are presented. In Section IV, we detail the proposed solution algorithms. In Section V, performance evaluation is conducted to demonstrate the validation of our proposed system. Finally, Section VI concludes this paper. Notation: Throughout this paper, non-boldface, boldface lowercase, and boldface uppercase letters denote scalar, vector, + +and matrix, respectively; \(|\cdot|,\Vert\cdot\Vert_{2},\text{and}(\cdot)^{\mathrm{H}}\) denote absolute sent expectation, a diagonal matrix with vector a on the main diagonal, and complex Gaussian distribution with zero mean value and variance \(\sigma^{2}\), respectively. + +value, \(\ell_{2}-\text{norm}\), respectively; \(\mathbb{E}(\cdot),\text{diag}(\mathbf{a}),\mathcal{C}\mathcal{N}(0,\sigma^{2})\) repre- + +## II. RELATED WORKS + +1. MEC: In recent years, MEC technologies have garnered significant attention from academic researchers and been applied in various domains [18]. By provisioning extensive computation resource at edge, MEC facilitates low-latency wireless access, thereby enabling lightweight terminals with limited processing capability to efficiently handle computing-intensive tasks. Furthermore, researchers have conducted in-depth optimizations of MEC systems with diverse objectives. For example, [19] proposed a wireless powered MEC system featuring binary computation offloading, utilizing block coordinate descent (BCD) and alternating direction method of multipliers to maximize computation rate by optimizing the selection of computing mode and allocation of transmission time. In scenarios where UDs are highly sensitive to latency, [20] focused on weighted total latency minimization and proposed an innovative approach jointly designing communication, computation, and caching configurations in MEC systems. To improve task execution efficiency, [21] investigated the energy efficiency maximization problem for task offloading in a two-tier MEC network. Nevertheless, the aforementioned studies mainly concentrate on task offloading for fixed UDs, which may limit their applicability in practical scenarios. To address this, [22] introduced a mobility-aware MEC network and employed reinforcement learning (RL) to jointly optimize task offloading and migration schemes aiming to maximize system revenue. Additionally, to further extend MEC network coverage and improve deployment flexibility, [23] incorporated unmanned aerial vehicles (UAVs) into MEC systems to maximize computation rate. 2) RIS and STAR-RIS: Thanks to its advantageous features, RIS has garnered substantial attention from both industry and academia. For example, [24] jointly designed phase shifts, offloading strategies, and transmission power based on BCD and Lagrange multiplier method to minimize weighted total latency. To further ensure task offloading security, [25] proposed a RIS-assisted MEC system framework, and investigated the problem of maximizing the minimum computation efficiency, optimizing local computation frequencies, IoT devices’ transmission power and RIS phase shifts to meet secure computation rate demands. For moving UDs, [26] aimed to minimize energy consumption through jointly designing RIS phase shifts and UDs’ transmission power. Despite these appealing features of RIS, the spatial restriction of transmitters and receivers imposes challenges in practical implementation, prompting the development of STAR-RIS [9], [10]. Currently, the study of STAR-RIS is still in its early stages. In [27], Xu et al. proposed the STAR-RIS concept, introducing standard hardware and channel models covering both near-field and far-field scenarios. Then, the operating protocols of STAR-RIS were further investigated in [10], analyzing the constraints, merit, and demerit of each protocol. Furthermore, in [28], STAR-RIS was introduced into MEC systems, where the STAR-RIS was operated under the TS protocol, with the objective of minimizing the total energy consumption of UDs, involving the joint optimization of STAR-RIS transmission/reflection time and coefficients. diff --git a/outputs/arxiv_math/2503.04407_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.04407_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..e12f2b7d16dbe85e16fd310d9c8ccfb952cfa799 --- /dev/null +++ b/outputs/arxiv_math/2503.04407_pg3_pg1_repeat1.md @@ -0,0 +1,38 @@ +| ● ← | ● | ● | ● | ● ● | +| --- | --- | --- | --- | --- | +| M | M | Movable Antennas | M | Fixed Antennas | +| RF Chain | RF Chain | RF Chain | RF Chain | RF Chain RF Chain | + +Radar Signal Processing + +Transmitter + +Fig. 1. Proposed MA-enabled FH-MIMO radar system. + +achieve improved performance in various tasks, e.g., if better resolution of target angles (velocity or distance) is required, the antenna positions can be adjusted to improve the performance of the ambiguity function in the angular (Doppler or delay) domain. The remainder of this paper is organized as follows. Section II describes the considered MA-enabled FH-MIMO radar system model and the corresponding ambiguity function. In Section III, we analyze the relationship between the ambiguity function and the antenna positions. In Section IV, we propose a low complexity algorithm to solve the antenna position optimization problem. Section V provides the numerical results and discussions. Finally, we conclude this paper in Section VI. Notations: Scalars, vectors and matrices are respectively denoted by lower/upper case, boldface lower case and boldface upper case letters. For an arbitrary matrix \(\mathbf{A},\mathbf{A}^{T},\mathbf{A}^{*}\) and \(\mathbf{A}^{H}\) denote its transpose, conjugate and conjugate transpose respectively. \(\Vert\cdot\Vert\) denotes the Euclidean norm of a complex vector, and \(|\cdot|\) denotes the absolute value of a complex scalar. \(\lceil\cdot\rceil\) represents the round-up operator. For a complex number \(x,\Re\{x\}\) denotes its real part and \(\angle x\) denotes its angle. I and 0 denote an identity matrix and an all-zero vector with appropriate dimensions, respectively. \(\mathbb{C}^{n\times m}\) denotes the space of \(n\times m\) complex matrices. + +## II. SIGNAL MODEL + +## A. MA-Enabled FH-MIMO Radar System + +As shown in Fig. 1, we consider an MA-enabled FH-MIMO radar system equipped with a colocated transmitter and receiver, which are comprised of linear arrays with \(M_{t}\) and \(M_{r}\) antennas, respectively. Since the transmit antenna array is the primary factor that affects the ambiguity function, we assume that the transmit antennas are movable while the receive antenna positions are fixed with half wavelength spacing for ease of analysis [8]. Each MA is attached to an electrical machinery, such that the interval between two adjacent antennas can be dynamically adjusted [26]. Let \(d_{t,i} (d_{r,i}),1\leq i\leq M_{t}-1(1\leq i\leq M_{r}-1)\) denote the interval between the \((i-\) 1)-th and \(i-\)th transmit (receive) antenna and define \(d_{t,0}=0,d_{r,0}=0,d_{r,i}= \frac{\lambda}{2} ,i\in\mathcal{M}_{r}≜ [1,2,\cdots ,M_{r}-1]\). Then, the transmit antenna position vector can be denoted by \(\mathbf{x}_{t}=[x_{t,0},x_{t,1},\cdots ,x_{t,M_{t}-1}]^{T}\) , where + +Radar Signal Processing + +Receiver + +\[\tfrac{\pi}{\lambda} x_{t,M_{t}-1}\sin\alpha_{]}T\] + +\(x_{t,0}=0\text{and}x_{t,m}=\sum_{i=0}^{m}d_{t,i},m\in\mathcal{M}_{t}≜[1,2,\cdots ,M_{t}-1]\). Similarly, the receive antenna position vector can be expressed as \(\mathbf{x}_{r}=[x_{r,0},x_{r,1},\cdots ,x_{r,M_{r}-1}]^{T}\) , where \(x_{r,0}=0\) and \(x_{r,m}= \frac{\lambda}{2} m,m\in\mathcal{M}_{r}\). Accordingly, the steering vectors of the transmit and receive antenna arrays are respectively given by \(\mathbf{a}(\mathbf{x}_{t},\alpha)=[1,e^{j \frac{2\pi}{\lambda} x_{t,1}\sin\alpha},\cdots ,e^{j^{\underline{2\pi}}}\) and \(\mathbf{b}(\mathbf{x}_{r},\alpha)=[1,e^{j\pi\sin\alpha},\cdots ,e^{j\pi(M_{r}-1)\sin\alpha}]^{T}\) , where \(\lambda\) denotes the signal wavelength and \(\alpha\) is the steering angle of the array. Consider a target at \((\hat{\tau},\hat{v},\theta),\text{where}\hat{\tau}\) denotes the delay corresponding to the target range, \(\hat{v}\) is the Doppler frequency of the target and \(\theta\in[- \frac{\pi}{2} , \frac{\pi}{2} ]\) represents the direction angle of the target. Then, the received signal can be represented by + +\[\mathbf{y}_{\hat{\tau},\hat{v},\theta}(t)=\mathbf{a}(\mathbf{x}_{t},\theta)^{T}\varphi(t-\hat{\tau})\mathbf{b}(\mathbf{x}_{r},\theta)e^{j2\pi\hat{v}t}+\mathbf{n}(t),(1)\] + +where \(\varphi(t)=[\phi_{0}(t),\phi_{1}(t),\cdots ,\phi_{M_{t}-1}(t)]^{T},\boldsymbol{n}(t)= [n_{0}(t),n_{1}(t),\cdots ,n_{M_{r}-1}(t)]^{T},\phi_{m}(t)\) represents the FH waveform transmitted from the \(m-\)th transmit antenna and \(n_{m}(t)\) denotes the Gaussian noise received by the \(m-\)th receive antenna. As the FH waveform, the pulse width \(T_{w}\) is divided into \(Q\) sub-pulses of width \(\Delta_{t}=T_{w}/Q\) each [35]. Therefore, the \(m-\)th FH waveform during each pulse can be further expressed as [8] + +\[\phi_{m}(t)=\sum_{q=0}^{Q-1}e^{j2\pi c_{m,q}\Delta_{f}t}s(t-q\Delta_{t}),\] (2) + +where \(c_{m,q}\in\mathbb{K}\) is the FH code with \(\mathbb{K}≜\{1,2,\cdots ,K\}\) being the set of available hops, \(\Delta_{f}\) represents the frequency hopping interval and \(s(t)\) represents the pulse function which is defined as + +\[s(t)= \begin{cases} 1, & 00\) as in Theorem 1.2. For \(\epsilon\in\mathbb{R},|\epsilon|\leq\epsilon^{*}\), let \(v_{\epsilon}^{T}\) + +be defined as in Eq. (1.21) and let \((s,k)\in\mathbb{R}\times\mathbb{N}\) satisfy + +\[\frac{n}{2} -1n.\] (1.22) + +Then there exist \(\omega>0\text{and}0< \overline{\epsilon} \leq\epsilon^{*}\) such that for every \(\epsilon\in\mathbb{R},|\epsilon|\leq \overline{\epsilon}\) there are \(\delta>0\text{and}M>1\) such that the following holds: For any pair of radial, real-valued functions + +\(\varphi_{0},\varphi_{1}\in\mathcal{S}(\mathbb{R}^{n})\) satisfying + +\[\Vert(\varphi_{0},\varphi_{1})\Vert_{\dot{H}^{s}\cap\dot{H}^{k}(\mathbb{R}^{n})\times\dot{H}^{s-1}\cap\dot{H}^{k-1}(\mathbb{R}^{n})}< \frac{\delta}{M} ,\] (1.23) + +there exists \(T=T_{\epsilon}\in[1-\delta,1+\delta]\) and a unique radial solution \(v\in C^{\infty}([0,T)\times\mathbb{R}^{n})\) to Eq. (1.14) with + +\[v(0,\cdot)=v_{\epsilon}^{1}(0,\cdot)+\varphi_{0},\partial_{t}v(0,\cdot)=\partial_{t}v_{\epsilon}^{1}(0,\cdot)+\varphi_{1}.\] + +Moreover, \(v\) blows up at \((T,0)\) and can be decomposed as + +\[v(t,x)=v_{\epsilon}^{T}(t,x)+ \frac{1}{T-t} \varphi\left(\log\left( \frac{T}{T-t} \right), \frac{x}{T-t} \right)\] + +for all \((t,x)\in[0,T)\times\mathbb{R}^{n},\text{where}\varphi\in C^{\infty}([0,T)\times\mathbb{R}^{n})\) is radially symmetric and satisfies + +\[\Vert\varphi(-\log(T-t)+\log T,\cdot)\Vert_{\dot{H}^{r}(\mathbb{R}^{n})}≲\delta(T-t)^{\omega}\] (1.24) + +for all \(r\in[s,k]\). Furthermore, + +\[\Vert(\partial_{0}+\Lambda+1)\varphi(-\log(T-t)+\log T,\cdot)\Vert_{\dot{H}^{r-1}(\mathbb{R}^{n})}≲\delta(T-t)^{\omega}.\] (1.25) + +Finally, we rephrase the results of Theorem 1.6 in terms of normal coordinates using the equivalence of norms of corotational maps and their radial profiles, see [23], to obtain Theorem 1.3. diff --git a/outputs/arxiv_math/2503.04430_pg9_pg1_repeat1.md b/outputs/arxiv_math/2503.04430_pg9_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..7bd01f1550e90adbe752010c420539399e1cd6c8 --- /dev/null +++ b/outputs/arxiv_math/2503.04430_pg9_pg1_repeat1.md @@ -0,0 +1,49 @@ +Repeatedly applying the same trick from right to left we obtain that + +\[Y_{1}\vee(\cdots (Y_{n-1}\vee Y_{n}))=q_{2}^{n}(x,y_{1}[1],\ldots,y_{n}[1]).\] + +Proceeding similarly for every component of \((\star)\), we get that + +\[\begin{aligned} q_{m}^{n}(\varphi(x),\varphi(y_{1}),\ldots,\varphi(y_{n}))&=(q_{2}^{n}(x,y_{1}[1],\ldots,y_{n}[1]),\ldots,q_{2}^{n}(x,y_{1}[m],\ldots,y_{n}[m])) \\ &=(q_{m}^{n}(x,y_{1},\ldots,y_{n})[1],\ldots,q_{m}^{n}(x,y_{1},\ldots,y_{n})[m]) \\ &=\varphi(q_{m}^{n}(x,y_{1},\ldots,y_{n})) \end{aligned}\] + +as desired. + +- + +As a consequence the following representation theorem holds. + +Corollary 3.12. Every partition clone T is isomorphic to the partition-of-unity clone of the Boolean algebra \(T(2)\). + +## 4. Hypervarieties + +We recall the following definitions for an algebra A: + +\(\bullet\mathbf{A}\) is directly indecomposable if it cannot be expressed as an isomorphic product of two nontrivial algebras; + +\(\bullet\mathbf{A}\) is subdirectly ireeducible if its lattice of congruences possesses a unique nontrivial minimal element; + +\(\bullet\mathbf{A}\) is simple if its lattice of congruences is isomorphic of the two-element chain \(\{\Delta<\nabla\}\). + +Stone famously proved that \(2=\{0,1\}\) is the unique nontrivial directly indecomposable Boolean algebra. Consequently, 2 is also the only subdirectly irreducible Boolean algebra (and, incidentally, the only simple Boolean algebra). By Theorem 2.1, this implies that every Boolean algebra is isomorphic to a subdirect power of 2. As a consequence, the variety of Boolean algebras is generated by 2. + +The first purpose of this section is to lift these considerations to the theory of partition clones. Here the role of the Boolean algebra 2 is played by the clone of projections, which corresponds to 2 in the equivalence of Theorem 3.11. + +## Lemma 4.1. Let T be a partition clone. The following are equivalent: + +1. T is simple; + +2. T is subdirectly irreducible; + +3. T is directly indecomposable; + +\((4)\mathbf{T}\simeq\mathbf{N}\). + +\(/0,1\}\). Letting \(c:=b_{i}\), and consequently having \(\neg c=\bigvee\{a_{k}:k\neq i\}\), it is a standard + +Proof. The implications \((1)\Longrightarrow (2)\text{and}(2)\Longrightarrow (3)\) are standard. (It is easy to see that the argument for single-sorted algebras lifts to the many-sorted case; at any rate, cf. [25, Proposition 4.8]). We now prove that \(\neg(4)\Longrightarrow \neg(3)\) Up to passing to coordinates, we can assume that the clone T is the partition-of-unity clone of a Boolean algebra \(B\). Let \(a:=(b_{1},\ldots,b_{n})\in T(n),\text{with}a\) different from \(e_{1}^{n},\ldots,e_{n}^{n}\). This implies that there exists \(i\text{such that}b_{i}\in\{\) fact regarding Boolean algebras that \(B\simeq B/\theta(c,1)\times B/\theta(c,0),\text{where}\theta(c,1)\) (resp. \(\theta(c,0))\) is the smallest congruence identifying \(c\) and 1 (resp. \(c\) and 0). To conclude the argument, observe that if \((b_{1},\ldots,b_{n})\in T(n)\), then, \((b_{1},\ldots,b_{n},0,\ldots,0)\in T(m)\text{for all}m>n\). Therefore, for each \(n\) we have a decomposition of \(T(n)\) in (at least) two factors and T is not directly indecomposable. To prove that \((4)\Longrightarrow (1)\), assume that \(\mathbf{T}\simeq\mathbf{N}\), and let \(\theta\) be a congruence of T different than \(\Delta\). Then there is \(n\geq2\text{and}1\leq i\neq j\leq n\text{such that}(e_{i}^{n},e_{j}^{n})\in\theta\). But then for all \(m\in\omega\text{and all}x,y\in T(m)\) + +\[(q_{m}^{n}(e_{i}^{n},z_{1},\ldots,z_{n}),q_{m}^{n}(e_{j}^{n},z_{1},\ldots,z_{n}))\in\theta\] + +where \(z_{1},\ldots,z_{n}\) are any elements of \(T(m)\text{such that}z_{i}=x\text{and}z_{j}=y\), so that, in light of (C2), \((x,y)\in\theta\), i.e. \(\theta=\nabla\). + +- diff --git a/outputs/arxiv_math/2503.04433_pg10_pg1_repeat1.md b/outputs/arxiv_math/2503.04433_pg10_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..ad2dfb485913b9f8fd9fe90826b02a3160ac319a --- /dev/null +++ b/outputs/arxiv_math/2503.04433_pg10_pg1_repeat1.md @@ -0,0 +1,31 @@ +Equation (36) is a special form, for the zero structural damping case d = 0, of the following: + +\[\mathbf{a}_{\mathbf{m}}\ddot{\mathbf{q}}+(\rho V\mathbf{b}+\mathbf{d})\dot{\mathbf{q}}+\left(\rho V^{2}\mathbf{c}+\mathbf{e}\right)\mathbf{q}=\mathbf{0}\] (37) + +where am is the mass, or inertia, matrix, b is the aerodynamic damping matrix, c is the aerodynamic stiffness matrix, d is the structural damping and e is the stiffness matrix. Now, considering the availability of modal data and assuming that the measured damping is dependent solely on the structure itself, d can be built from the uncoupled modal damping assumption in [52]: + +\[\mathbf{d}=\phi^{-T}\mathbf{d}_{n}\phi^{-1}\text{for}\mathbf{d}_{n}=2\zeta_{n}\omega_{n}a_{n}\] (38) + +where the subscript n identifies the uncoupled matrix, \(\omega_{n}\) the natural frequency, \(\zeta_{n}\) the damping ratio and \(\phi\) the mode shape. Hence, Equations (36) and (38) can be combined to assemble Equation (37). Given full knowledge of the wing geometric characteristics, three properties remain to be defined: \(M_{\dot{\theta}}\), EI, and GJ. \(M_{\dot{\theta}}\) is the unsteady aerodynamics term and it is defined from oscillatory aerodynamics [49]: + +\[M_{\dot{\theta}}=2\pi\left[- \frac{k}{2} \left( \frac{1}{2} -a\right)+kF\left(a+ \frac{1}{2} \right) \frac{1}{2} -a\Bigg)++ \frac{G}{k} \left( \frac{1}{2} +a\right)\right]\] (39) + +where fk is the reduced frequency, a is the ratio between c and the flexural axis position, and F and G are, respectively, the real and imaginary part of Theodorsen’s function, C( fk), such that: + +\[C(f_{k})=F(f_{k})+jG(f_{k})= \frac{H_{1}^{(2)}(f_{k})}{H_{1}^{(2)}(f_{k})+jH_{1}^{(2)}(f_{k})}\] (40) + +where H(2)n ( fk) are Hankel functions of the second kind and j is the imaginary number. Concerning the bending and torsional stiffness, EI and GJ, let us consider the still air case where b and c are zero. Equation (37) then becomes a simple mass-spring-damper system: + +\[\mathbf{a}_{\mathbf{m}}\ddot{\mathbf{q}}+\mathbf{d}\dot{\mathbf{q}}+\mathbf{e}\dot{\mathbf{q}}=\mathbf{0}\] (41) + +The natural frequencies can then be easily extracted through eigenanalysis. Hence, by having a set of experimental \(\omega_{n}\) it is possible to define the EI and GJ of the equivalent system by minimising its squared difference to the experimental \(\omega_{n}\). Thus, by using this and the identified \(\zeta_{n}\) for Equation (38), an aeroelastic model can be defined starting from experimental data. To study the system stability, the eigenanalysis of Equation (41) can be solved iteratively with the well-known p-k method [53] to find the divergence and flutter onset speeds. The p-k method is based on the hypothesis that pure harmonic aerodynamics can be used as a good approximation for lightly damped harmonic motions. This allows the computation of the aerodynamic transfer matrix at a complex frequency \(p=\delta\pm\) jk, such that \(p\approx\) jk. In simple terms, the real part, i.e. the damping, is neglected. A widely accepted workflow of the p-k method [49] can be summarised as follow: + +1. Initiate an estimation, usually the still air value, of p, said \(p_{0}=\delta\pm jk_{0}\) + +2. Evaluate the aerodynamics, in our case \(M_{\dot{\theta}}\) + +3. Solve the eigenvalue \((\lambda)\) problem for Equation (37) and obtain a new set of \(\lambda,p_{1}=\delta\pm jk_{1}\) + +4. Iterate between 2 and 3 until \(k_{n}\approx k_{n-1}\) + +From the \(\lambda\) obtained after convergence, it is possible to build \(\omega_{n},\zeta_{n},\text{real}(\lambda)\text{and imag}(\lambda)\) vs air-speed \((U_{\infty})\) plots, which can be used to graphically portray divergence or flutter speed, whichever is detected first. Particularly, critical speeds are identified for \(\zeta_{n}\) approaching zero or for \(\text{real}(\lambda)\) zero crossings, since both cases are interpreted as instability in the system. Particularly, for flutter, only the \(\text{real}(\lambda)\) zero crossing condition needs to be satisfied, while for divergence, \(\text{imag}(\lambda)\) must be zero. diff --git a/outputs/arxiv_math/2503.04438_pg16_pg1_repeat1.md b/outputs/arxiv_math/2503.04438_pg16_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..f29dae15f10e3191da6ca890211b254da416fa7f --- /dev/null +++ b/outputs/arxiv_math/2503.04438_pg16_pg1_repeat1.md @@ -0,0 +1,29 @@ +Using the universal property of \(\text{THH}(k)\) as a commutative algebra, we find that \(k\otimes_{k\otimes\text{THH}(k)} k\simeq\text{THH}(k)\otimes(k\otimes_{\text{THH}(k)}k)\), and using now Bökstedt’s equivalence \(\text{THH}(k)\simeq k[\Omega S^{3}]\), we find \(k\otimes_{\text{THH}(k)}k\simeq k[S^{3}]\), so that, in total, + +\[\text{map}_{k\otimes\text{THH}(k)}(k,k)\simeq\prod_{n}(\Omega^{2n}k\oplus\Omega^{2n+3}k)\] + +To get \(k-\)linear endomorphisms of \(\text{HH}_{k}\), one simply adjoins \([S^{1}]\) to this. + +◁ + +Remark 3.4. In the previous examples, it is not clear to the author what the extra operations “do”. Similarly to the situation in Remark 2.15, we do not know how to “name” them, and it would probably be worthwhile to spend time figuring out what these operations do, or are. + +◁ + +## 4 Operations on THH of \(O-\)algebras + +The goal of this section is to initiate the study of endomorphisms of THH viewed as a functor Alg\((_{O}\text{Cat}^{\text{perf}})\rightarrow\) Sp, that is, to study the extra operations that arise on \(\text{THH}(C)\text{when}C\) has a particular kind of multiplicative structure encoded by a one-colored \(\infty-\text{operad}O\). Our main result in this section is Theorem B. As is clear from the statement of Theorem B, the case of endomorphisms of THH viewed as a functor on Alg\((_{O}\text{Cat}^{\text{perf}})\) is more subtle. We first describe the general approach to this question, and then specialize to get the precise results that we claimed. We let \(U:\text{Alg}_{O}(\text{Cat}^{\text{perf}})\rightarrow\) Catperf denote the forgetful functor, with a left adjoint \(F\) such that \(UF\simeq\bigoplus_{n}(O(n)\otimes(-)^{\otimes n})_{h\Sigma_{n}}\) [Lur12, Proposition 3.1.3.13]. The idea now is that by general adjunction nonsense, + +\[\text{Map}_{\text{Fun}(\text{Alg}_{O}(\text{Cat}^{\text{perf}}),\text{Sp})}(\text{THH}\circ U,\text{THH}\circ U)\simeq\text{Map}_{\text{Fun}(\text{Cat}^{\text{perf}},\text{Sp})}(\text{THH},\text{THH}\circ UF)\] + +and so we are “left with” understanding THH \(\circ UF\) . For this, we compute each of the individual terms of + +\[\text{THH}\circ UF=\bigoplus_{n}\text{THH}((O(n)\otimes(-)^{\otimes n})_{h\Sigma_{n}})\] + +Proposition 4.1. Let \(O\) be a space with \(a\Sigma_{n}-\text{action}\). There is an equivalence, natural in \(C\in\) Catperf: + +\[\text{THH}((O\otimes C^{\otimes n})_{h\Sigma_{n}})\simeq\bigoplus_{\sigma\in\Sigma_{n}/\text{conj}}(O^{\sigma}\otimes\text{THH}(C)^{\otimes n(\sigma)})_{hC(\sigma)}\] + +where for \(\sigma\in\Sigma_{n},n(\sigma)\) is the number of cycles appearing in \(\sigma,C(\sigma)\) is the centralizer of \(\sigma\) in \(\Sigma_{n}\text{and}O^{\sigma}=L(O_{h\Sigma_{n}})\times_{LB\Sigma_{n}}\{\sigma\}\) with its residual \(C(\sigma)-\text{action}^{13}\). + +13Here, \(L\) denotes the free loop space, i.e. \(\text{Map}(S^{1},-)\). diff --git a/outputs/arxiv_math/2503.04448_pg10_pg1_repeat1.md b/outputs/arxiv_math/2503.04448_pg10_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..3aa189f59d0f596895eaf9f8b2f4e71fcc7bb9df --- /dev/null +++ b/outputs/arxiv_math/2503.04448_pg10_pg1_repeat1.md @@ -0,0 +1,21 @@ +Lemma 3. The average number of waiting customers in the polling model under the exhaustive service discipline is given by: + +\[\mathbb{E}[L]= \frac{\lambda\mathbb{E}[K]}{2(1-\rho)} \left(\alpha+\rho \frac{\mathbb{E}[B^{2}]}{\mathbb{E}[B]} + \frac{\mathbb{E}[B]\mathbb{E}[K(K-1)]}{\mathbb{E}[K]} \right).\] (16) + +Proof. We consider the related FCFS system, and denote the variables related to this system with a superscript FCFS. Remark that the waiting time of a tagged customer under the FCFS model is given by the sum of: (i) the travel time, \(T\) , to the tagged customer; (ii) the residual service time of the customer in service, if any, at the arrival instant of the tagged customer; (iii) the service time of all waiting customers present at the time of arrival and + +(iv) the service time of all customers arriving in the same batch, who are served before the tagged customer. Hence: + +\[\mathbb{E}\left[W^{\text{FCFS}}\right]=\mathbb{E}[T]+\rho \frac{\mathbb{E}[B^{2}]}{2\mathbb{E}[B]} +\mathbb{E}\left[L^{\text{FCFS}}\right]\mathbb{E}[B]+ \frac{\mathbb{E}[B]\mathbb{E}[K(K-1)]}{2\mathbb{E}[K]} .\] + +One can now apply Little’s law, \(\mathbb{E}[L^{\text{FCFS}}]=\lambda\mathbb{E}[K]\mathbb{E}[W^{\text{FCFS}}]\) to find a first expression for the mean number of waiting customers in the system. It remains to find the average travel time \(\mathbb{E}[T]\) to a waiting customer. The travel time under the FCFS service policy is hard to analyse, as this depends on the state of the system at the arrival instant, as well as future arrivals. Instead, we remark that the average travel time to a tagged customer is the same for the FCFS model and polling model. This is due to the fact that the sum of the travel times to all customers is the same. We can therefore focus on the original system to derive \(\mathbb{E}[T]\). Recall (1), stating that the server is within a small interval \([x,x+dx]\) with probability \([\rho\pi(x)+1-\rho]dx\) . Therefore, we have: + +\[\mathbb{E}[T]=\alpha\int_{x=0}^{1}\big[\rho\pi(x)+1-\rho]\mathbb{E}[d(x,X_{1})]dx=\alpha\rho\mathbb{E}[d(X_{2},X_{1})]+\alpha(1-\rho)\mathbb{E}[d(U,X_{1})],\] + +where \(X_{1},X_{2}\) denote the locations of two arbitrary customers and \(U\) denotes a \(\text{Uniform}[0,1]\) random variable. As the average distance from a uniform point on the circle to any point equals \(1/2\), also \(\mathbb{E}[d(U,X_{1})]=1/2\) by the independence of \(X_{1}\text{and}U\). Due to the symmetry, we further have \(\mathbb{E}[d(X_{1},X_{2})]=\mathbb{E}[d(X_{2},X_{1})]\). Combined with the fact that \(d(X_{1},X_{2})+ d(X_{2},X_{1})=1\), we find: \(\mathbb{E}[d(X_{2},X_{1})]=1/2\). The proof is now finished by substituting \(\mathbb{E}[T]=\alpha/2\) in the expression for the waiting time. + +Remark 4. The mean number of waiting customers in the system is not affected by the arrival location distribution. Using Eliazar’s limit argument (Eliazar, 2005), this is a consequence of the pseudo-conservation law for discrete polling models, cf. Equation (3.21) of Boxma (1989), stating that the weighted (by load of queue) sum of expected waiting times is indifferent to the arrival location distribution, \(\lambda_{i}/(\sum_{j}\lambda_{j})\). + +### 4.2 Integral equation for spread of customers + +We now turn to \(f(x,y)\), see (14), describing the average spread of customers on the circle. This characteristic is essential to the mean-value analysis of polling models on a circle and diff --git a/outputs/arxiv_math/2503.04465_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.04465_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..43c824d727fcaaf02cfe63adf2255de8ba0bdfbb --- /dev/null +++ b/outputs/arxiv_math/2503.04465_pg3_pg1_repeat1.md @@ -0,0 +1,7 @@ +Many examples satisfying (H) are given in Section 2.2, including normal and Cauchy random variables. Moreover, a second contribution is that, as we show in Theorem 4.2, the averaged exact controllability does not hold in the \(L^{2}\) set up, which is a surprising result because one would expect that such property would be inherited from the exact controllability properties of the Schrödinger equation. Finally, a third contribution, as we show in Theorem 4.4, is that the preimage of 0 under \(\xi\mapsto\) + +\(y(T,\cdot;\xi;y_{0};u)\) is finite for all \(y_{0}\in L^{2}(G)\setminus\{0\}\text{and}u\in L^{2}((0,T)\times G_{0})\). In particular, the event: “the system (1.1) is simultaneous null controllable” is negligible for absolutely continuous random variables, that is, + +\[\mathbb{P}[\omega:y(T,\cdot;\alpha(\omega);y_{0};u)=0]=0.\] + +A brief overview of the related literature. Regarding the control of averaged properties, the problem and its abstract formulation were first presented in the work [30], where finite-dimensional systems were considered. The pioneering study on averaged controllability in the context of partial differential equations (PDEs) was introduced in [23]. In this study, the authors addressed the problem of controlling the average of PDEs and examined the properties of the transport, heat, and Schrödinger equations in specific scenarios. For the Schrödinger equation, they analyzed the equation when the diffusivity follows uniform, exponential, normal, Laplace, chi-square, and Cauchy probability laws. In [31] the continuous average of the heat equation was considered. Further contributions, such as [17] and [13] investigated perturbations of probability density functions modeled by Dirac masses. In [8] and [4] the controllability of the heat equation with random diffusivity was studied. In the second paper, it was discovered that some probability density functions caused fractional dynamics for the heat equation. Moreover, we would like to point out that there are many known results for lower-order random terms, as shown in the survey [22] and the book [21], and some of them are recent results for nonlinear parabolic stochastic equations such as in [10] and [11]. Regarding the controllability of the Schrödinger equation, there are many classical papers such as [14], [24], [27], [7], and [2], involving geometric control conditions. As for simultaneous control, we would like to highlight [26], where there is a bilinear control, and [18], where cascade-like systems are studied. Also, for stochastic equations with random lower-order terms, we may highlight, for example, [20]. Finally, as mentioned above, for random diffusivity, it was studied in [23] for specific probability laws. Our contribution to the literature is to obtain the averaged controllability of the Schrödinger equation with random diffusivity in a more general setting and to determine how much we can generalize the known results. Structure of this paper. In Section 2, we introduce several key concepts along with their characterization, and analyze the main assumption of the paper. Section 3 is devoted to proving the main result on the averaged null controllability of the system (1.1). In Section 4, we prove the lack of exact average for general absolutely continuous random variables and of simultaneous controllability. In Section 5, we present a numerical method to validate our theoretical result in the case of Cauchy and normal distributions. Finally, Section 6 is dedicated to the conclusion and outlines several potential directions for future research. Notation: Throughout this paper, we adopt some standard notations and usual abbreviations in diff --git a/outputs/arxiv_math/2503.04466_pg9_pg1_repeat1.md b/outputs/arxiv_math/2503.04466_pg9_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..399ff09d05fef3ea0f01e3a840ab722563f53fac --- /dev/null +++ b/outputs/arxiv_math/2503.04466_pg9_pg1_repeat1.md @@ -0,0 +1,31 @@ +## 2 Optimal control of second-order systems + +The optimal control problem for second-order systems, under sufficient differentiability conditions can be treated exactly as in Eq. (3). There, the augmented running cost featured inside the integral was a function on \(T^{*}\mathcal{M}\oplus_{\mathcal{M}}\mathcal{E}\). In the second-order case, this naturally leads to an augmented running cost on \(T^{*}T\mathcal{Q}\oplus_{T\mathcal{Q}}\mathcal{E}\). Assuming adapted local coordinates \((q,v,\lambda_{q},\lambda_{v})\text{on}T^{*}T\mathcal{Q}\), and a generic controlled SODE \(X=v\partial_{q}+f(q,v,u)\partial_{v}\), Eq. (3) transforms into + +\[\begin{aligned} \tilde{J}_{1}(x,\lambda,u)&=\phi(q(T),v(T)) \\ +\int_{0}^{T}\left[C(q(t),v(t),u(t))+\langle(\lambda_{q}(t),\lambda_{v}(t)),(\dot{q}(t)-v(t),\dot{v}(t)-f(q(t),v(t),u(t)))\rangle_{T\mathcal{Q}}\right]dt. \end{aligned}\] (7) + +We refer to this as the first-order version of the optimal control problem since the controlled SODE appears as a first order system. The second-order constraint, i.e. \(\dot{q}=v\), may be added implicitly, leading to a new augmented cost function + +\[\tilde{J}_{2}(y,u)=\phi(q(T),\dot{q}(T))+\int_{0}^{T}\left[C(q(t),\dot{q}(t),u(t))+\kappa(t)^{\top}(\ddot{q}(t)-f(q(t),\dot{q}(t),u(t)))\right]dt.(8)\] + +Here, the curve \(y=(q,\kappa)\) is a curve on \(T^{*}\mathcal{Q}\). One must also understand \(u\) to denote the curve defined on the fibers along the tangent lift of \(q\), i.e. \((q(t),\dot{q}(t),u(t))\in\mathcal{E}\). The remaining constraint is now to be interpreted as a function on \(T^{(2)}\mathcal{Q}\oplus_{T\mathcal{Q}}\mathcal{E}\). Taking variations we find that the necessary conditions for optimality provided by \(\tilde{J}_{1}\) are + +- (state dynamics) \(\dot{q}(t)=v(t)\), \(\dot{v}(t)=f(x(t),v(t),u(t))\), + +\[\bullet(\text{adjoint dynamics})\dot{\lambda}^{\dot{\lambda}_{vq}(}(^{t}t^{)})^{=_{\top}^{\top}}=^{D}D_{2}^{C_{1}}C^{(}(^{q}q^{(}(^{t}t^{)})^{,},^{v}v^{(}(^{t}t^{)})^{,},^{u}u^{(}(^{t}t^{)})^{)})^{-}-^{\lambda}\lambda_{v}^{(_{v}}(^{t}t^{)})^{D_{\top}^{\top}}D_{2}^{f_{1}}f^{(}(^{q}q^{(}(^{t}t^{)})^{,},^{v}v^{(}(^{t}t^{)})^{,},^{u}u^{(}(^{t}t^{)})^{)})^{,}-\lambda_{q}(t)^{\top},\] + +- (maximization) \(0=D_{3}C(q(t),v(t),u(t))-\lambda_{v}(t)^{\top}D_{3}f(q(t),v(t),u(t))\), + +\[\begin{aligned} \bullet(\text{transversality})\lambda_{q}(T)^{\top}&=-D_{1}\phi(q(T),v(T)), \\ \lambda_{v}(T)^{\top}&=-D_{2}\phi(q(T),v(T)); \end{aligned}\] + +while those provided by \(\tilde{J}_{2}\) are + +- (state dynamics) \(\ddot{q}(t)=f(q(t),\dot{q}(t),u(t))\), + +\[\bullet(\text{adjoint dynamics})\ddot{\kappa}(t)^{\top}=_{-} \frac{d}{D^{dt}}_{1} \left[_{C}D_{(^{2}q}C_{(t}(_{)}q_{,}(_{\dot{q}}t_{(})_{t},_{)}q_{\dot{,}}(_{u}t)_{(},_{t)}u_{)}(t_{+}))_{\kappa}-_{(t)}\kappa_{\top}(t_{D})_{1}^{\top}D_{f(^{2}q}f_{(}(_{t}q_{),}(_{\dot{q}}t)_{(},_{t}q_{\dot{)},}(_{u}t)_{(},_{t}u_{))}(_{,}t))\right]\] + +- (maximization) \(0=D_{3}C(q(t),\dot{q}(t),u(t))-\kappa(t)^{\top}D_{3}f(q(t),\dot{q}(t),u(t))\), + +\[\begin{aligned} \bullet(\text{transversality})\kappa(T)^{\top}&=-D_{2}\phi(q(T),\dot{q}(T)), \\ \dot{\kappa}(T)^{\top}&=D_{1}\phi(q(T),\dot{q}(T)) \\ +D_{2}C(q(T),\dot{q}(T),u(T))+D_{2}\phi(q(T),\dot{q}(T))D_{2}f(q(T),\dot{q}(T),u(T)). \end{aligned}\] + +These, lead to the following easy to check Theorem 2.1. The necessary optimality conditions provided by \(\tilde{J}_{1}\text{and}\tilde{J}_{2}\) are equivalent under the identification \(\lambda_{v}=\kappa\). diff --git a/outputs/arxiv_math/2503.04467_pg7_pg1_repeat1.md b/outputs/arxiv_math/2503.04467_pg7_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..49caf793a40b8a42434484ad8acc5b9371315ed7 --- /dev/null +++ b/outputs/arxiv_math/2503.04467_pg7_pg1_repeat1.md @@ -0,0 +1,35 @@ +Assuming \(u_{1},u_{2}\) are solutions of the equation, let \(u=u_{1}-u_{2}\): + +\[\begin{aligned} \begin{cases} u+vAu=B(u_{2})-B(u_{1}) & \infty & 2 \\ \overline{dt} & , & u\in L & (0,T;V)\cap L(0,T,D(A)) \end{cases} \{_{d} \\ \{u(0)&=0 \end{aligned}\] + +By lemma 2.8: + +Then we observe: + +and + +\[\frac{1}{2} \frac{d}{dt} \|u_{m}\|_{L^{2}}^{2}+va(u,u)=(B(u_{2})-B(u_{1}),u_{2}-u_{1})\] + +\[B(u_{1})-B(u_{2})=B(u_{1},u)+B(u,u_{2})\] + +\[\begin{aligned} \{(B(u_{1},u),u)&=0 \\ \{\{|(B(u,u_{2}),u)|&=|\int_{\Omega}u\cdot\nabla u_{2}\cdot udxdy|\leq\|u\|_{L^{\infty}}\|\nabla u_{2}\|_{V}\|u\|_{L^{2}} \end{aligned}\] + +2 + +Due to \(H^{1}(\Omega)\hookrightarrow\rightarrow L^{\infty}(\Omega)\), with Agmon’s inequality, \(|(B(u,u_{2}),u)|\leq c\|u\|_{L^{2}}^{3/2}\|u_{2}\|_{V}\|u\| \frac{1}{V}\) . By Young inequality, \(|(B(u,u_{2}),u)|\leq \frac{v}{2} \|u\|_{V}^{2}+ \frac{c^{2}}{2v} \|u\|_{H}^{2}\|u_{2}\|_{V}^{2}\) . Hence, + +\[\frac{d}{dt} \|u\|_{H}^{2}\leq \frac{c^{2}}{v} \|u\|_{H}^{2}\|u_{2}\|_{V}^{2}\] + +By Gronwall inequality, we obtain: + +\[\|u(t)\|_{H}^{2}\leq\|u(0)\|_{H}^{2}e^{\frac{c2}{v} ´_{0t}\|u_{2}(s)\|_{V}^{2}ds}=0\] + +Hence, the solution is unique. And the continuous dependence of the solution on \(f\) and the initial value \(u_{0}\) can be deduced by the prior estimation. + +## 3 Application + +The Burgers equation, originally derived as a simplified model of fluid dynamics, has found widespread applications in various fields, including statistical mechanics, turbulence modeling, and transportation systems. Its nonlinearity and ability to describe shock formation and energy dissipation make it a powerful tool for analyzing complex systems. In transportation, the Burgers equation is particularly useful for modeling traffic flow dynamics, where it helps capture phenomena such as congestion, wave propagation, and the formation of traffic jams. + +### 3.1 Introduction of Detailed Application in Transportation Systems + +The application of the Burgers equation in transportation systems has been extensively studied. For instance, Payne (1971, 1979) utilized the Burgers equation to develop mathematical models for traffic flow, focusing on the propagation of shock waves and the behavior of traffic under varying conditions. These models provided insights into the dynamics of traffic congestion and helped improve traffic management strategies. Kühne (1984) further expanded on this work by applying the Burgers equation to simulate traffic flow in urban networks. His studies demonstrated how the equation could be used to predict the formation and dissipation of traffic jams, offering valuable tools for urban planning and traffic control. The Burgers equation’s ability to describe the nonlinear interactions between vehicles and the environment makes it a cornerstone in traffic flow theory. By incorporating viscosity and external forcing terms, researchers can model the effects of road conditions, driver behavior, and traffic regulations on flow dynamics. diff --git a/outputs/arxiv_math/2503.04471_pg7_pg1_repeat1.md b/outputs/arxiv_math/2503.04471_pg7_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..8581cea341d714a92d5a78cf22541593306973db --- /dev/null +++ b/outputs/arxiv_math/2503.04471_pg7_pg1_repeat1.md @@ -0,0 +1,15 @@ +nowhere dense sets), which are classically equivalent to each other. However, the combinatorial definition of \(\kappa-\)meagre sets coincides with the topological notion if and only if \(\kappa\) is countable or inaccessible, as was shown in [BHZ, §4]. Even though the ideals are distinct if \(\kappa\) is uncountable but not inaccessible, it is an open question whether the cardinal invariants given by the combinatorial \(\kappa-\)meagre ideal are consistently distinct from those given by the topological \(\kappa-\)meagre ideal: + +Question 2.3. Let the combinatorially \(\kappa-\)meagre ideal \(\mathcal{M}_{\kappa}^{\mathrm{c}\text{omb}}\) be defined as in [BHZ, Def. 4.4]. Is \(\text{non}(\mathcal{M}_{\kappa}^{\mathrm{c}\text{omb}})<\text{non}(\mathcal{M}_{\kappa})or\text{cov}(\mathcal{M}_{\kappa})<\text{cov}(\mathcal{M}_{\kappa}^{\mathrm{c}\text{omb}})\) consistent for \(\kappa\) that are not inaccessible? + +The characterisation of \(\text{cof}(\mathcal{N})\text{and add}(\mathcal{N})\) from Theorem 2.1 cannot be generalised to the higher context, not just because we would need a higher analogue of \(\mathcal{N}\) , but also because the cardinality of \(d_{\kappa}^{h}(\in^{*})\) is highly dependent on the choice of \(h\in^{\kappa}\kappa\text{when}\kappa\) is inaccessible, as we will discuss in Section 3.2. For successor \(\kappa\), Theorem 2.1 looks very different: Theorem 2.4 ([Hyt06], [MS, Thm. 4.6]). If \(\kappa\) is a regular successor, then \(b_{\kappa}(✟=^{✟^{\infty}})=b_{\kappa}^{h}(\in^{*})=b_{\kappa}\) and \(d_{\kappa}=d_{\kappa}^{h}(\in^{*})\geq d_{\kappa}(✟=^{✟^{\infty}})\). If we also assume \(\kappa^{<\kappa}=\kappa\), then the last inequality becomes an equality. Question 2.5 ([MS, §4]). Is \(d_{\kappa}(✟=^{✟^{\infty}})\kappa\)? + +It was observed by Truss and Miller that the cardinal characteristics \(\text{add}(\mathcal{M})\text{and cof}(\mathcal{M})\) can be expressed in terms of the other cardinals of the Cichoń diagram. + +Theorem 2.6 ([Tru77], [Mil81]). \(\text{add}(\mathcal{M})=\min\{b,\text{cov}(\mathcal{M})\}\text{and}\text{cof}(\mathcal{M})=\max\{d,\text{non}(\mathcal{M})\}\). + +The same characterisation generalises to the higher Cichoń diagram if we make strong enough assumptions on \(\kappa\). We note that this is not provable for all uncountable \(\kappa\). For example, Brendle [Bre22] showed that \(\kappa^{<\kappa}<\text{cof}(\mathcal{M}_{\kappa})\), and thus \(\max\{d_{\kappa},\text{non}(\mathcal{M}_{\kappa})\}<\text{cof}(\mathcal{M}_{\kappa})\) is consistent, and holds whenever \(\max\{d_{\kappa},\text{non}(\mathcal{M}_{\kappa})\}\leq\kappa^{<\kappa}\). + +Theorem 2.7 ([Bre22]). \(\text{add}(\mathcal{M}_{\kappa})=\min\{b_{\kappa},\text{cov}(\mathcal{M}_{\kappa})\}\text{and}\text{cof}(\mathcal{M}_{\kappa})\geq\max\{d_{\kappa},\text{non}(\mathcal{M}_{\kappa})\}\). Under assumption of \(\kappa^{<\kappa}=\kappa\) the inequality becomes an equality. + +As for the other eight cardinal characteristics of the classical Cichoń diagram, they appear to be as independent as is possible in the following sense: any assignment of the cardinalities \(\aleph_{1}\) and \(\aleph_{2}\) to the cardinals of the classical Cichoń diagram compatible with both Theorem 2.6 and the arrows in the diagram, is consistent (see [BJ95, Sections 7.5 and 7.6] for each of these cases). This implies that the classical Cichoń diagram is complete, as no arrows are missing from the diagram. In fact, Goldstern, Kellner, Mejıá, and Shelah [GKMS21; GKS19] showed that all eight cardinal characteristics can have mutually different cardinalities within the same model. diff --git a/outputs/arxiv_math/2503.04486_pg21_pg1_repeat1.md b/outputs/arxiv_math/2503.04486_pg21_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..623a87a7bea75e4307456a65054ea997f8426235 --- /dev/null +++ b/outputs/arxiv_math/2503.04486_pg21_pg1_repeat1.md @@ -0,0 +1,63 @@ +# E ON THE BEST CURVATURE SPLITTING PROBLEM (see Section 4) + +In Section 4 we examined how to achieve the optimal splitting for a given objective \(F=f_{1}-f_{2}\) by subtracting the same curvature term \(\lambda \frac{\Vert\cdot\Vert^{2}}{2}\) in both functions. We concluded that if \(\mu_{1}\leq\mu_{2}\), the best splitting is achieved by shifting the lower curvature of \(f_{1}\) to 0, i.e., make it convex. Conversely, when \(\mu_{1}>\mu_{2}\), the best splitting is achieved for some \(f_{2}\) weakly convex, with lower curvature \(\mu_{2}-\lambda<0\). Within this section, we provide additional numerical experiments. In Figure 10 we provide an example with fixed curvature bounds of a nonconvex-nonconcave objective function \(F\) , namely \(\mu_{F}=-0.5\text{and}L_{F}=1.5\), and illustrate all possible regimes after one iteration. Note that \(\mu_{F}=\mu_{1}-L_{2}\text{and}L_{F}=L_{1}-\mu_{2}\); further on, we examine the regimes based on the ranges of \(L_{2}\text{and}\mu_{2}\). The condition \(\mu_{1}\geq0\) implies that \(L_{2}=\mu_{1}-\mu_{F}\geq-\mu_{F}\) , while the condition \(\mu_{1}+\mu_{2}>0\text{that}\mu_{2}>-\mu_{1}=L_{2}+\mu_{F}\) . The contour lines represent the values of the denominators \(p_{i},\text{with}i=1,2,3,4\). The red points mark the initial curvature values of \(L_{2}\text{and}\mu_{2}\), whereas the green dots indicate the points with the largest possible \(p_{i}\) obtained through the optimal choice of \(\lambda\). Since these shifts are linear in \(\lambda\), the dashed lines connecting the dots have a slope of one. + +For example, in the case where \(L_{2}=1\text{and}\mu_{2}=0.75\), we have \(\mu_{1}<\mu_{2}\) and the best splitting is obtained by shifting to the lowest possible value of \(L_{2}\), corresponding to \(\mu_{1}=0\). In all other examples, \(\mu_{1}>\mu_{2}\) and the optimal splittings are found within regime \(p_{3},\text{where}\mu_{2}<0\). + +p4 + +3 + +2.5 + +2 + +p3 + +1.5 + +1 + +p2 + +0.5 + +0 + +p1 + +-0.5 + +-1 + +-- + +-1.5 0.5 + +| B = 0 L2 = 72 | | | | 0.5 0 0.25 0.5 0 | +| --- | --- | --- | --- | --- | +| | | | 0.75 0.25 | | +| | | | 0.5 0 | | +| | | | 0.75 0.25 | | +| | | | 0.5 0 | | +| | | 0.5 0 1 0.75 0.25 0.75 0.25 | | 0.75 | +| | 0.5 0 1 0.5 0 1.25 1 0.75 0.25 1.25 1.5 | | | 0.75 | +| 1.5 1.75 0.5 0 1.25 1 0.75 0.25 0.5 0 1.25 1 2 0.75 0.25 | [icon] | 1 | [icon] | | +| 2 2.5 1.75 | 1.5 | 1.25 | 1 1 | 1 | +| 1.5 1.75 0.25 1.25 0.751 0.5 2 0 1.5 0.25 1.75 1.25 0.75 0.5 1 ● | 1.5 1.25 | 1.25 1.25 ● | | 1 ✓ | +| | 0 0.75 1 0.25 0.5 0 | 0.75 1 1 0.5 0.25 0.75 0.5 | 1 | 1 0.75 | +| | | 0 0.25 0 ● | 0.75 0.5 0.25 | 0.75 0.5 0.5 0.25 | + +1 + +1.5 + +2 + +2.5 + +3 + +\(L_{2}\) + +Figure 10: All regimes for a fixed objective function \(F\text{with}\mu_{F}=-0.5\text{and}L_{F}=1.5\), along with several mappings of the optimal splittings, shown as transitions from red to green dots along dashed lines with a slope of 1. diff --git a/outputs/arxiv_math/2503.04488_pg9_pg1_repeat1.md b/outputs/arxiv_math/2503.04488_pg9_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..ae8bacfccd509b1bf7f0ab42ea74935f92c5db8d --- /dev/null +++ b/outputs/arxiv_math/2503.04488_pg9_pg1_repeat1.md @@ -0,0 +1,45 @@ +by \(U(X)\) in Alt. In fact, one may verify that the semi-direct product \(B\ltimes_{\varphi}U(X)\) with multiplication + +\[(b,x)\cdot(b^{\prime},x^{\prime})=(bb^{\prime},xx^{\prime}+\varphi(b)x^{\prime}+x\varphi(b^{\prime}))\] + +is an alternative algebra. We thus have a natural isomorphism + +\[\text{SplExt}(-,U(X))\cong\text{Hom}_{\mathbf{Alt}}(-,U(X))\] + +and we can state the following. Theorem 3.2. The category Alt1 is action representable with the actor of a unitary alternative algebra \(X\) being isomorphic to \(X\) itself. + +- + +## 4. Poisson algebras + +The aim of this section is to prove that the categories Pois1 of unitary Poisson algebras and CPois1 of unitary commutative Poisson algebras over a field \(\mathbb{F}\) with char\((\mathbb{F})\neq2\) are action representable. We recall that a Poisson algebra is a vector space \(X\) over \(\mathbb{F}\) equipped with two bilinear multiplications + +\[\cdot:X\times X\rightarrow X\text{and}[-,-]:X\times X\rightarrow X\] + +such that \((X,\cdot)\) is an associative algebra, \((X,[-,-])\) is a Lie algebra and the Poisson identity holds: + +\[[x,yz]=[x,y]z+y[x,z],\forall x,y,z\in X.\] + +A Poisson algebra is said to be commutative (resp. unitary) if the underlying associative algebra is commutative (resp. unitary). It was proved in [6] that for any Poisson algebra \(X\) there exists a natural monomorphism of functors + +\[\tau:\text{SplExt}(-,X)\text{Hom}_{\mathbf{Alg}^{2}}(\tilde{U}(-),[X]),\] + +where Alg2 denotes the category of algebras with two non-necessairly associative bilinear operations, \(\tilde{U}\) : Pois \(\rightarrow\mathbf{Alg}^{2}\) is the forgetful functor and + +\[\begin{aligned} [X]&=\{f=(f*-,-*f,[f,-])\in\text{Bim}(X)\times\text{Der}(X)|\cdots \\ \cdots |f*[x,y]&=[f*x,y]-[f,y]x,[x,y]*f=[x*f,y]-x[f,y],[f,xy]=[f,x]y+x[f,y]\} \end{aligned}\] + +is the universal strict general actor of \(X\), which is endowed with the bilinear multiplications + +\[f\cdot g=(f*(f^{\prime}*-),(-*f)*f^{\prime}),f*[f^{\prime},-]+[f,-]*f^{\prime})\] + +and + +\[[f,g]=(f*[f^{\prime},-]-[f^{\prime},f*-],[f^{\prime},-]*f-[f^{\prime},-*f],[f,[f^{\prime},-]]-[f^{\prime},[f,-]]).\] + +Furthermore, a morphism \(\varphi=(\varphi_{1},\varphi_{2},\varphi_{3}):\tilde{U}(B)\rightarrow[X]\) belongs to \(\text{Im}(\tau_{B})\) if and only if \((\varphi_{1},\varphi_{2}):(B,\cdot)\rightarrow\text{Bim}(X)\) is an acting morphism in Assoc. If \(X\) has trivial annihilator or \((X^{2},\cdot)=(X,\cdot)\), then eq. (3.1) holds in Bim\((X)\) and we have a natural isomorphism + +\[\text{SplExt}(-,X)\cong\text{Hom}_{\mathbf{Alg}^{2}}(\tilde{U}(-),[X]).\] + +This happens, for instance, when \(X\) is a unitary Poisson algebra. In this case, it is possible to prove that \([X]\) is a Poisson algebra. Indeed, by the results in Section 3, the universal strict general actor \([X]\) may be described as the subalgebra of all pairs \((\alpha,[f,-])\in X\times\text{Der}(X)\) such that + +\[\begin{aligned} \alpha[x,y]&=[\alpha x,y]-[f,y]x, \\ [x,y]\alpha&=[x\alpha,y]-x[f,y] \end{aligned}\] diff --git a/outputs/arxiv_math/2503.04493_pg31_pg1_repeat1.md b/outputs/arxiv_math/2503.04493_pg31_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..9eb105f33ef61e464baf8a576e9c8676a2edb472 --- /dev/null +++ b/outputs/arxiv_math/2503.04493_pg31_pg1_repeat1.md @@ -0,0 +1,25 @@ +\[\phi_{2}=a_{x}^{2}+a_{y}^{2}- \frac{(a_{x}b_{x}+a_{y}b_{y})^{2}}{b_{x}^{2}+b_{y}^{2}} .\] + +To calculate the curve-averaged value of the basis function of the \(p^{2}\) polynomial in Eq. 19, the following integral needs to be calculated + +\[\int_{0}^{1}(\xi+\phi_{1})^{p} \sqrt{\phi_{0}(\xi+\phi_{1})^{2}+\phi_{2}} d\xi=( \frac{\phi_{2}}{\phi_{0}} )^{\frac{\mathrm{p}+1}{2}} \sqrt{\phi_{2}} \int_{\theta_{0}}^{\theta_{1}}\tan^{\mathrm{p}}\theta\sec^{3}\theta d\theta,\] (37) + +where \(\theta_{0}=\arctan( \sqrt{\frac{\phi_{0}}{\phi_{2}}} \phi_{1}),\theta_{1}=\arctan( \sqrt{\frac{\phi_{0}}{\phi_{2}}} (1+\phi_{1})).\text{If}p\) is an odd integer \(p=2k+1\), + +\[\int\tan^{2k+1}\theta\sec^{3}\theta d\theta=\int(\sec^{2}\theta-1)^{\mathrm{k}}\sec^{2}\theta\text{dsec}\theta.\] (38) + +If \(p\) is an even integer \(p=2k\), + +\[\int\tan^{2k}\theta\sec^{3}\theta d\theta=\int(\sec^{2}\theta-1)^{\mathrm{k}}\sec^{2}\theta d\theta,\] (39) + +and + +\[I_{n}=\int\sec^{n}\theta d\theta= \frac{1}{\mathrm{n}-1} (\tan\theta\sec^{\mathrm{n}-2}\theta+(\mathrm{n}-2)\mathrm{I}_{\mathrm{n}-2}).\] + +# B. Quadratic integral on curved triangle + +The isoparametric transformations of the quadratic triangular cell Eq. 30 is + +\[\begin{aligned} \frac{\partial x}{\partial\xi} &=a_{0}\xi+a_{1}\eta+a_{2}, \\ \frac{\partial x}{\partial \frac{\partial^{\eta}y}{\partial\xi}} &=_{=}a_{b_{0}^{3}}\xi_{\xi}+_{+}a_{b_{1}^{4}}\eta_{\eta}+_{+}a_{b_{2}^{5},}, \\ \frac{\partial y}{\partial\eta} &=b_{3}\xi+b_{4}\eta+b_{5}, \end{aligned}\] (40) + +\[\begin{aligned} \text{where}a_{0}&=4(x_{0}+x_{1}-2x_{3}),a_{1}=4(x_{0}-x_{3}+x_{4}-x_{5}),a_{2}=-3x_{0}-x_{1}+4x_{3},a_{3}= \\ 4(x_{0}-x_{3}+x_{4}-x_{5}),a_{4}&=4(x_{0}+x_{2}-2x_{5}),a_{5}=-3x_{0}-x_{2}+4x_{5},b_{0}=4(y_{0}+y_{1}-2y_{3}),b_{1}= \\ 4(y_{0}-y_{3}+y_{4}-y_{5}),b_{2}&=-3y_{0}-y_{1}+4y_{3},b_{3}=4(y_{0}-y_{3}+y_{4}-y_{5}),b_{4}=4(y_{0}+y_{2}-2y_{5}),b_{5}= \\ -3y_{0}-y_{2}+4y_{5}. \end{aligned}\] diff --git a/outputs/arxiv_math/2503.04494_pg21_pg1_repeat1.md b/outputs/arxiv_math/2503.04494_pg21_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..38a64aab5385c22ded939ab46ba8bf4db2f3c310 --- /dev/null +++ b/outputs/arxiv_math/2503.04494_pg21_pg1_repeat1.md @@ -0,0 +1,19 @@ +this is true for the classical Tate curve. To glue each T \(\hat{}\) with the rational Tate curve TQ, we will use an arithmetic fracture square, so the Cartesian diagram of \(\mathbf{E}_{\infty}-\)rings + +\[\mathrm{B}\prod_{p}\widehat{\mathrm{B}}\] + +\[\mathrm{B}_{\mathbf{Q}}\left(\prod_{p}\widehat{\mathrm{B}}\right)_{\mathbf{Q}}.\] + +As the functor Ell\((^{\text{or}}-)^{\simeq}\) : CAlg \(\rightarrow\) S is corepresented by TMF when restricted to complex periodic \(\mathbf{E}_{\infty}-\)rings by Th.5.4, we have equivalences of spaces + +\[\begin{aligned} \text{Ell}^{\text{or}}(\mathrm{B})^{\simeq}&\simeq\text{Map}_{\text{CAlg}}(\text{TMF},\mathrm{B})\simeq\text{Map}_{\text{CAlg}}(\text{TMF},\mathrm{B}_{\mathbf{Q}})\times\hat{}\text{Map}_{\text{CAlg}}(\text{TMF},\prod_{p}\widehat{\mathrm{B}}) \\ \text{Map}_{\text{CAlg}}(\text{TMF},(\prod_{p}\mathrm{B})_{\mathbf{Q}}) \\ &\simeq\text{Ell}^{\text{or}}(\mathrm{B}_{\mathbf{Q}})_{\text{Ell}^{\text{or}}((}^{\simeq}\prod_{p}\times_{\widehat{\mathrm{B}})_{\mathbf{Q}})^{\simeq}}\text{Ell}^{\text{or}}(\prod_{p}\widehat{\mathrm{B}})^{\simeq}. \end{aligned}\] + +We already have the desired oriented elliptic curves T \(\hat{}\) over each \(\widehat{\mathrm{B}}\) and TQ over BQ, so to obtain T over B, it suffices to produce an isomorphism between oriented elliptic curves over \((\prod_{p}\widehat{\mathrm{B}})_{\mathbf{Q}}\). To do this, consider the equivalences of spaces + +\[\begin{aligned} \text{Ell}^{\text{or}}\left((\prod_{p}\widehat{\mathrm{B}})_{\mathbf{Q}}\right)^{\simeq}&\simeq\text{Map}_{\text{CAlg}}\left(\text{TMF},(\prod_{p}\widehat{\mathrm{B}})_{\mathbf{Q}}\right)\simeq\text{Map}_{\text{CAlg}_{\mathbf{Q}}}\left(\text{TMF}_{\mathbf{Q}},(\prod_{p}\widehat{\mathrm{B}})_{\mathbf{Q}}\right) \\ &\subseteq\text{Map}_{\text{CAlg}_{\mathbf{Q}}}\left(\text{tmf}_{\mathbf{Q}},(\prod_{p}\widehat{\mathrm{B}})_{\mathbf{Q}})\right)\simeq\Omega^{\infty+8}\left((\prod_{p}\widehat{\mathrm{B}})_{\mathbf{Q}}\right)\times\Omega^{\infty+12}\left((\prod_{p}\widehat{\mathrm{B}})_{\mathbf{Q}}\right), \end{aligned}\] (5.5) + +where we use that tmfQ is the free \(\mathbf{E}_{\infty}-\mathbf{Q}-\)algebra on two generators \(c_{4}\text{and}c_{6}\) in degrees 8 and + +12, respectively, that \(\text{TMF}_{\mathbf{Q}}\simeq\text{tmf}_{\mathbf{Q}}[\Delta^{-1}]\text{for}\Delta=(c_{4}^{3}-c_{6}^{2})/1728\), and that the containment above is a monomorphism onto those path components where the image of \(\Delta\) is invertible. Both the base-change of T \(\hat{}\) and T Q to \((\prod_{p}\widehat{\mathrm{B}})_{\mathbf{Q}}\) are chosen such that the associated map from tmfQ sends \(c_{4}\text{and}c_{6}\) to their \(q-\)expansion maps. From the equivalences above, this shows that these oriented elliptic curves are equivalent over \((\prod_{p}\widehat{\mathrm{B}})_{\mathbf{Q}}\). The fact that \((\prod_{p}\widehat{\mathrm{B}})_{\mathbf{Q}}\) has homotopy groups concentrated in even degrees also shows that the choice of equivalence is unique up to homotopy; even up to 3-homotopy. This yields the oriented elliptic curve T over B. + +To compute the formal group associated with T, notice that B is an \(\mathbf{E}_{\infty}-\text{KU}-\text{algebra}\), as it then follows that the formal group associated with T is \(\widehat{\mathbf{G}}_{m}\). Indeed, by [Lur18b, Pr.4.3.23], an orientation on T is an equivalence between its associated formal group and the Quillen formal group over B. As the Quillen formal group of a complex-periodic \(\mathbf{E}_{\infty}-\)ring satisfies base-change, we obtain the desired result of the Quillen formal group \(\widehat{\mathbf{G}}_{m}\) over KU. Alternatively, one can use the fact that the space of oriented formal groups over a complex-periodic diff --git a/outputs/arxiv_math/2503.04498_pg27_pg1_repeat1.md b/outputs/arxiv_math/2503.04498_pg27_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..167683f2878a2d70678137bd54ce563158f6c2d3 --- /dev/null +++ b/outputs/arxiv_math/2503.04498_pg27_pg1_repeat1.md @@ -0,0 +1,25 @@ +[5] Bocong Chen et al. "Constacyclic codes over finite fields". In: Finite Fields and Their Applications 18.6 (2012), pp. 1217-1231. + +[6] Abdelghaffar Chibloun, Hassan Ou-azzou, and Mustapha Najmeddine. "On isometry and equivalence of constacyclic codes over finite chain rings". In: Cryptography and Communications 17.1 (2024), pp. 1-25. + +[7] Alexandre Fotue-Tabue, Edgar Martínez-Moro, and J Thomas Blackford. "On polycyclic codes over a finite chain ring". In: Advances in Mathematics of Communications 14.3 (2020), pp. 455-466. + +[8] Markus Grassl. Bounds on the minimum distance of linear codes and quantum codes. Online available at http://www. codetables.de. Accessed on 2025-02-20. 2007. + +[9] R Lidl. Introduction to Finite Fields and their Applications. Cambridge University Press, 1994. + +[10] Sergio R López-Permouth, Benigno R Parra-Avila, and Steve Szabo. "Dual generalizations of the concept of cyclicity of codes." In: Advances in Mathematics of Communications 3.3 (2009), pp. 227-234. + +[11] Hassan Ou-Azzou and Mustapha Najmeddine. "On the algebraic structure of polycyclic codes". In: Filomat 35.10 (2021), pp. 3407-3421. + +[12] Hassan Ou-azzou, Mustapha Najmeddine, and Nuh Aydin. "On isometry and equivalence of skew constacyclic codes". In: Discrete Mathematics 348.1 (2025), p. 114279. + +[13] W.W. Peterson and E.J. Weldon. Error-correcting Codes. Cambridge, MA, 1972. ISBN: 9780262160391. + +[14] E Prange. Cyclic Error-Correcting Codes in Two Symbols. Tech. rep. TN-57-103, 1957. + +[15] L Rédie. "A short proof of a theorem of Št. Schwarz concerning finite fields". In: Časopis Pro Pěstování Matematiky a Fysiky 75.4 (1950), pp. 211-212. + +[16] Steven Roman. Field Theory. Vol. 158. Springer Science & Business Media, 2005. + +[17] Štefan Schwarz. "On the reducibility of binomial congruences and on the bound of the least integer belonging to a given exponent \(\mod p^{\prime\prime}\). In: Časopis Pro Pěstování Matematiky a Fysiky 74.1 (1949), pp. 1-16. diff --git a/outputs/arxiv_math/2503.04523_pg4_pg1_repeat1.md b/outputs/arxiv_math/2503.04523_pg4_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..bd8d05ec20c5ba1fd400fb628e9a4ea1cd28ca35 --- /dev/null +++ b/outputs/arxiv_math/2503.04523_pg4_pg1_repeat1.md @@ -0,0 +1,26 @@ +GRAP-R method to consider the nearby rank-deficient points where apocalypse often occurs. Line search along approximate projections contributes to an Armijo condition. Both steps protect the sequences generated by GRAP-R from converging to non-stationary points, which renders GRAP-R apocalypse-free. + +\[\text{Goal}:f(\mathcal{Y})-f(\mathcal{X})\leq-\delta( \underline{\mathcal{X}} )<0,\text{for all}\Vert\mathcal{X}- \underline{\mathcal{X}} \Vert_{\mathrm{F}}\leq\varepsilon( \underline{\mathcal{X}} )\] + +| X ● | (rf)GRAP-R | | | +| --- | --- | --- | --- | +| | [icon] | Y Goal ← | | +| | [icon] | f(Y) = arg minˆr f(Yˆr) ↑ | f(X ) | +| | X ε(X ) | Minimum | Continuity | +| | | f(Yˆr) Armijo conditions ← | f(PHOˆr(X )) | + +Fig. 2. Definition and validation of the sufficient decrease condition in convergence analysis. + +In a neighborhood of a non-stationary point \(\underline{\mathcal{X}}\) , the function value decreases by more than \(\delta( \underline{\mathcal{X}} )\) from \(\mathcal{X}\) to \(\mathcal{Y}\). The roadmap illustrates how the (rf)GRAP-R method achieves the sufficient decrease condition. + +Alternatively, we consider retraction-free search directions that line searches never leave \(\mathcal{M}_{\leq\mathbf{r}}\). In a similar manner, we adopt the framework in Figure 1 and propose a new retraction-free GRAP-R method (rfGRAP-R). In contrast with the GRAP-R method where all singular values of unfolding matrices are required, the rfGRAP-R method only detects the smallest singular value of unfolding matrices, which is in favor of saving computational costs. We prove that the rfGRAP-R method also converges to stationary points by verifying a sufficient decrease condition in Figure 2. In addition, numerical results validate that the proposed methods indeed converge to stationary points, and the rfGRAP-R method performs better than the GRAP-R method due to the simpler rank-decreasing mechanism. Organization. We introduce the geometry of Tucker tensor varieties and notation for tensor operations in section 2. In section 3, we provide an explicit representation of the normal cone of Tucker tensor varieties, and propose approximate projections. Based on two different projections, we propose two first-order methods and prove that all accumulation points of sequences generated by these methods are stationary in sections 4 and 5. Section 6 reports the numerical performance of proposed methods in tensor completion. Finally, we draw the conclusion in section 7. + +2. Geometry of Tucker tensor varieties. In this section, we introduce the + +preliminaries of variational analysis for low-rank matrices. Then, we describe the notation in tensor operations and Tucker tensor varieties. + +Given a nonempty subset \(C\) of \(\mathbb{R}^{n_{1}\times n_{2}\times\cdots \times n_{d}}\), the (Bouligand) tangent cone + +\[\begin{aligned} \text{of}C\text{at}\mathcal{X}&\in C\text{is defined by T}_{\mathcal{X}}C:=\{\mathcal{V}\in\mathbb{R}^{n_{1}\times n_{2}\times\cdots \times n_{d}}:\exists t^{(i)}\rightarrow0,\mathcal{X}^{(i)}\rightarrow \\ \mathcal{X}\text{in}C,\mathrm{s}.\mathrm{t}.(\mathcal{X}^{(i)}-\mathcal{X})/t^{(i)}&\rightarrow\mathcal{V}\}.\text{The set N}C=(\mathrm{T}C)^{\circ}:=\{\mathcal{W}\in\mathbb{R}^{n_{1}\times n_{2}\times\cdots \times n_{d}}: \end{aligned}\] + +\(\mathcal{X}\text{in}C,\mathrm{s}.\mathrm{t}.(\mathcal{X}^{(i)}-\mathcal{X})/t^{(i)}\rightarrow\mathcal{V}\}\). The set \(\mathrm{N}_{\mathcal{X}}C=(\mathrm{T}_{\mathcal{X}}C)^{\circ}:=\{\mathcal{W}\in\mathbb{R}^{n_{1}\times n_{2}\times\cdots \times n_{d}}\) : \(\langle\mathcal{W},\mathcal{V}\rangle\leq0\text{for all}\mathcal{V}\in\mathrm{T}_{\mathcal{X}}C\}\) is called the normal cone of \(C\) at \(\mathcal{X}\) . Note that if \(C\) is a manifold, the tangent cone \(\mathrm{T}_{\mathcal{X}}C\) (normal cone \(\mathrm{N}_{\mathcal{X}}C)\) is a linear space and referred to as the tangent space (normal space). Given a continuously differentiable function \(f\) on \(C\), a point \(x\in C\) is stationary of \(f\text{if}\Vert\mathrm{P}_{\mathrm{T}_{x}C}(-\nabla f(x))\Vert_{\mathrm{F}}=0,\text{or}-\nabla f(x)\in\mathrm{N}_{x}C\) equivalently. diff --git a/outputs/arxiv_math/2503.04527_pg19_pg1_repeat1.md b/outputs/arxiv_math/2503.04527_pg19_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..982aa2b129b439e66ead5778b4dd986e3a139993 --- /dev/null +++ b/outputs/arxiv_math/2503.04527_pg19_pg1_repeat1.md @@ -0,0 +1,43 @@ +# Materials and Methods + +## Coupling disease surveillance and epidemic dynamics + +Our coupled behavioral-epidemiological model depicted in Figure 1 assumes the epidemic initially + +spreads among a naive population with a Susceptible-Infected-recovered scheme formalized by + +equation (S1) + +\[\begin{aligned} S_{I}¤_{¤}&=_{=\beta}-\beta_{S} \frac{S_{I}}{N} \frac{I}{N_{-}} ,\gamma I, \\ R¤&=\gamma I, \end{aligned}\] (S1) + +where \(\beta\) corresponds to the transmission rate, and \(\gamma\) is the recovery rate. We couple the disease + +progression model with a surveillance model formulated using a probabilistic approach to compute + +the expected time required to identify at least a single infected individual using power calculations. + +Assuming a homogeneous mixing population and a random sampling strategy, we compute the + +cumulative probability of detecting at least one infectious individual. The proposed disease detection + +model incorporates the dynamic progression of the epidemic into the daily probability of detecting + +infectious individuals. The probability of detecting at least a single positive case by time \(t\), for a + +disease prevalence level of \(\phi_{s}\) on day \(s\), for surveillance effort of \(N_{s}\) daily tested individuals, is + +given by + +\[\mathbb{P}_{det}=1-\prod_{s=1}^{t}(1-\phi_{s})^{N_{s}}.\] (S2) + +Finally, we let the disease detection time \((\tau_{det})\) to be reached at the time at least a single infectious + +individual is detected with at least 95% confidence, \(\mathbb{P}_{det}=0.95\). The detection time determines the disease prevalence threshold \((P^{*})\) at which health emergency declarations are issued. Note that \(P^{*}\) + +is not arbitrarily selected; instead it depends on the preselected surveillance effort. + +## Disease surveillance triggers interventions, risk information spread and adaptive human behavior + +Upon health emergency declaration, we assume the onset of centralized intervention quarantining + +a fraction \(\phi\) newly infected individuals \((Q)\), and that the susceptible population is divided into the diff --git a/outputs/arxiv_math/2503.04535_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.04535_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..2f83af3fe3e6c4f111b8ae8ef0d957c6b342a28d --- /dev/null +++ b/outputs/arxiv_math/2503.04535_pg2_pg1_repeat1.md @@ -0,0 +1,11 @@ +for m sufficiently large and divisible. Asymptotically, \(W_{\bullet}^{t}\) records the hypersurfaces that appear as tangent cones of sections of L with multiplicity t at x. The challenge is to provide an intrinsic geometric interpretation of the family of graded linear series \(W_{\bullet}^{t}.\text{When}X=C_{1}\times\ldots\times C_{n}\), intersection theory on BlxX implies that \(W_{\bullet}^{t}\) is contained in a family of complete linear series associated to divisors on \(X_{\Sigma}\), the blow-up of \(\mathbb{P}^{n-1}\) in n points in general linear position. We show that this inclusion is asymptotically an equality. For this we use that \(X_{\Sigma}\) is a toric variety and the asymptotic behaviour of complete linear series on them can be managed through convex geometry. As a consequence, the vertical slices of \(\Delta_{x}(L)\) are NObodies associated to a ray of divisors on \(X_{\Sigma}\) with respect to a generic (strict transform) linear flag. Lastly, the global (gluing all \(t\geqslant0)\) NObody on this special ray of divisors on \(X_{\Sigma}\) is exactly the convex set described in the statement, as we are able to prove the existence of sections whose valuation are the vertices of the simplex. For box-products of arbitrary degrees the vertical slices of \(\Delta_{x}(L)\) are still a family of NObodies on \(X_{\Sigma}\) with respect to general linear flags. But it is harder to guess the vertices. It seems an interesting problem to find an algorithm that starts with a polytope in \(\mathbb{R}^{n-1}\) that is a NObody on a toric variety with respect to a T-invariant flag, and construct another polytope, that will be a NObody on the same toric space but with respect to a generic linear flag. In dimension \(n\leqslant3\), we can find non-trivial sections on \(X_{\Sigma}\), whose valuations provide the vertices of a slice of the global NObody on \(X_{\Sigma}\) with respect to general linear flags. More precisely: + +Theorem 1.2. Let \(X=C_{1}\times\ldots\times C_{n}\) be a product of smooth complex projective curves, polarized by a box product \(L=L_{1}⊠\ldots⊠L_{n}\) with deg Li = di. Assume \(d_{1}\geqslant d_{2}\geqslant\ldots\geqslant d_{n}>0\). Let \(x\in X\) be any point. + +- If n = 2, then \(\Delta_{x}(L)\) is the isosceles trapezoid with vertices \((0,0),(d_{2},d_{2}),(d_{1}+d_{2},0),(d_{1},d_{2})\). + +- If n = 3, then \(\Delta_{x}(L)\) is the convex hull of the 9 points \((0,0,0),(d_{3},d_{3},0),(d_{2},d_{3},d_{2}-d_{3}),(d_{1},d_{3},d_{2}-d_{3})\), \((d_{1}+d_{2}-d_{3},d_{3},0),(d_{2}+d_{3},0,d_{2}+d_{3}),(d_{1}+d_{3},0,d_{2}+d_{3}),(d_{1}+d_{2},0,2d_{3}),\text{and}(d_{1}+d_{2}+d_{3},0,0)\), + +In both cases, the shapes degenerate to the simplex in Theorem 1.1 when di = 1 for all i, while when the di are not all equal the shapes are not simplicial. + +It turns out that the verticies of our iNObodies from Theorem 1.1 and Theorem 1.2 happen exactly when there is a change in the associated base loci of the graded linear series \(W_{\bullet}^{t}\). These ideas are generalized by the authors in [FL25b] on any projetcive variety X by connecting three aspects of local positivity of a line bundle L on X at a point \(x\in X\). On one hand we have infinitesimal successive minima, then jets and finally polytopal upper and lower bounds on the generic iNObodies. Moreover, in [FL25c] the authors study the shape of generic iNObodies on Jacobians in small dimensions. The main issue here is realizing at least asymptotically the family \(W_{\bullet}^{t}\) as complete graded linear series on a modification of \(\mathbb{P}^{n-1}\). For example, for non-hyperelliptic Jacobian threefolds, \(W_{\bullet}^{t}\) becomes complete on a birational model of \(\mathbb{P}^{2}\) that is far from being toric, while for the hyperelliptic case we may need to go to the Zariski–Riemann space. The paper is organized as follows. In Section 2 we review the Newton–Okounkov body construction and their basic properties. In Section 3 we use intersection theory, positivity, toric geometry and Euclidean volume computations to understand the vertical slices as Newton–Okounkov bodies on \(X_{\Sigma}\). In Section 4 we treat the case of dimension \(n\leqslant3\). In Section 5 we use particular incomplete linear systems on \(X_{\Sigma}\) to prove that the vertices of the simplex from Theorem 1.1 do belong to the iNObody, concluding its proof. In Section 6 we state and motivate some open questions. diff --git a/outputs/arxiv_math/2503.04536_pg9_pg1_repeat1.md b/outputs/arxiv_math/2503.04536_pg9_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..d3612cc830c38210e219182a794defbaf7ab7890 --- /dev/null +++ b/outputs/arxiv_math/2503.04536_pg9_pg1_repeat1.md @@ -0,0 +1,23 @@ +\(\sigma(y_{i},t+s)\) as functions of s. These solve \(\partial_{s}\sigma(y_{i},t+s)=F\left(\sigma(y_{i},t+s),t+s\right)\) and \(\sigma(y_{1},\) t+0) \(=\sigma(y_{2},\) t+0). Again by uniqueness, \(\sigma(y_{1},\) t+s) \(=\sigma(y_{2},\) t+s) for all \(|s|<\delta\), for some \(\delta>0\), that is, \((t-\delta,t+\delta)\subset J\) and so J is open. + +## 2.3. Minimization flows and OT. Here we use the approach from [AHT03]. Let T : \((\Omega_{0},\rho_{0})\rightarrow(\Omega_{1},\rho_{1})\) be a measure preserving map, i.e., + +\[\int_{T^{-1}(E)}\rho_{0}(x)dx=\int_{E}\rho_{1}(x)dx\] + +for each Borel set \(E\subset\Omega_{1}\). Set \(\sigma_{t}(\cdot)=\sigma(\cdot,t)\text{where}\sigma\) is as before, the flow corresponding to a vector field F satisfying (2.2). The family of maps \(T\circ(\sigma_{t})^{-1}\) : \((\Omega_{0},\rho_{0})\rightarrow(\Omega_{1},\rho_{1})\) are measure preserving since + +\[\left(T\circ(\sigma_{t})^{-1}\right)^{-1}(E)=\sigma_{t}\circ T^{-1}(E),E\subset\Omega_{1}\] + +and + +\[\begin{aligned} \int_{(T\circ(\sigma_{t})^{-1})^{-1}(E)}\rho_{0}(x)dx&=_{(^{-1}E)}\rho_{0}(x)dx \\ &=\int_{ET^{-1}(E)}^{\sigma_{t}\circ T}\rho_{0}(x)dx\text{since}\sigma_{t}\text{preservesthemeasure} \\ &=\rho_{1}(x)dx. \end{aligned}\] + +Consider the function of t + +\[G(t)=\int_{\Omega_{0}}c\left(x,T\circ(\sigma_{t})^{-1}(x)\right)\rho_{0}(x)dx;\] + +here \(c\left(x,y\right)\) is a general cost. Making the change of variables \(x=\sigma(z,t)\) yields + +\[\begin{aligned} G(t)&=c(\sigma(z,t),T(z))\rho_{0}(\sigma(z,t))J_{\sigma}(z)dz \\ \int_{\Omega_{0}}^{\Omega_{0}} \\ &=c(\sigma(z,t),T(z))\rho_{0}(z)dz \end{aligned}\] + +since \(\rho_{0}(\sigma(z,t))J_{\sigma}(z)=\rho_{0}(z)\) because \(\int_{\sigma(E,t)}\rho_{0}(x)dx=\int_{E}\rho_{0}(x)dx\) for all t and all \(E\subset\Omega_{0}\). If T is an optimal map with respect to the cost c, then \(G^{\prime}(t)=0\) when diff --git a/outputs/arxiv_math/2503.04555_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.04555_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..555a3480efa5f702e4d4d0de17e443aa52581ec3 --- /dev/null +++ b/outputs/arxiv_math/2503.04555_pg2_pg1_repeat1.md @@ -0,0 +1,17 @@ +actions, extending the original framework. Other examples include the work of Gnilke and Zumbrägel [15], who linked Maze et al.’s ideas to recent advances in isogeny-based cryptography. Additionally, [16] introduced a key-exchange protocol based on twisted group rings, incorporating a group key-exchange agreement. In this research, Grigoriev and Shpilrain proposed the use of tropical semiring for public key exchange [2], [3] and for digital signatures [4]. Nevertheless, the first attempt was analyced by [5], where it was introduced the so called Kotov-Ushakov attack, an heuristic attack that knowadays has become the standar attack against tropical cryptography. In [6], the authors propose a new deterministic attack against a public key exchange protocol based on tropical semiring, that improve the Kotov-Ushakov attack in the sense that it can be used in the sames scenarios, but with a deterministic output. In addition, the other two tropical cryptographyc protocols proposed by Grigoriev and Shpilrain has been proven not secure in [9], [7] and [8]. Far from being abandoned, this ideas has been further explored. In [17], the authors recently proposed the use of triad matrix semiring. This semiring is a generalization of tropical semiring where elements are a vector of 3 entries, and with a modified addition and multiplication that endow them with the structure of semiring. Based on them, the authors stablish a public key exchange protocol that use matrix with entries over triad semiring. + +## 2 Results + +In this paper, we introduce an isomorphism between triad semiring and circuland matrix over tropical semiring. As a result, it is possible to satblish an isomorhism between triad matrix semiring and tropical matrix semiring, and therefore reinterpretate the public key exchange in terms of tropical matrix. This prove that the previous protocol can be seen as the Stickel protocol where instead of taking a polynomial, only a monomial is used. Finally, this protol is suscentible to the attack introduced in [18] . + +## 3 Preliminaries + +In this secition we will introduce some basic background on tropical semiring as well as triad semiring. + +Definition 1 A semiring \(R\) is a non-empty set together with two operations + and · such that \((S,+)\) is a commutative monoid, \((S,\cdot)\) is a monoid and the following distributive laws hold: + +\[\begin{aligned} a(b+c)&=ab+ac \\ (a+b)c&=ac+bc \end{aligned}\] + +We say that \((R,+,\cdot)\) is additively idempotent if \(a+a=a\text{for all}a\in R\). + +Definition 2 Let \(R\) be a semiring and \((M,+)\) be a commutative semigroup with identity \(0_{M}.M\) is a right semimodule over \(R\) if there is an external operation \(\cdot:M\times R\rightarrow M\) such that diff --git a/outputs/arxiv_math/2503.04567_pg38_pg1_repeat1.md b/outputs/arxiv_math/2503.04567_pg38_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..a6b4f1a1d0601fd61686184a357744c227b80263 --- /dev/null +++ b/outputs/arxiv_math/2503.04567_pg38_pg1_repeat1.md @@ -0,0 +1,23 @@ +If there are no external electromagnetic forces (or if the external electromagnetic forces are time independent), the classical model conserves energy, so that it does not have any hyperbolic orbits. On the other hand, in time dependent (e.g. periodic) external fields, many interesting models (e.g. ion traps, mirror magnet machines) have many hyperbolic orbits [Gho96, Kaj22]. In this paper, we show that near hyperbolic orbits of the unperturbed model (59), i.e. \(\varepsilon=0\) in (60) 2, and if the perturbed model avoids the singularities and it has bounded delays, Theorem 4.8 applies. Moreover, constructed solutions are similar to those hyperbolic orbits for the unperturbed model (59), as long as \(0<\varepsilon\ll1\) in (60). See Theorem 7.3 for a precise formulation. Before proceeding to the detailed analysis, let us make some comments on the equations and their physical properties. + +1. The expressions defining the forces are algebraic expressions (arithmetic operations and square roots). They have singularities when there are collisions \((q_{i}(t)=q_{j}(t)\text{for some}i\neq j)\) or when a particle reaches the speed of light \((|\dot{q}_{i}(t)|=c\text{for some}i)\). + +We will assume that the unperturbed solutions we consider are a finite distance away from these singularities so that the expressions given the second derivatives and given the positions and velocities are smooth functions in a neighborhood of the unperturbed solution. See Definition 7.2. + +2. The delays \(\tau_{ij}\) or advances \(\sigma_{ij}\) as in (60) involve solving implicit equations that involve the pairwise trajectories \(q_{i}(t)\text{and}q_{j}(t)\). + +The physical meaning of the delays/advances are the time it takes a light signal to travel from the particle \(i\) to the particle \(j\). Since the speed of the particles is bounded away from the speed of light, this time exists and is unique. Nevertheless, finding the delay requires to solve an equation that depends on the pairwise trajectories. See (60). + +Fortunately, for the method used in this paper, the main property needed is that \(\tau_{ij}\text{and}\sigma_{ij}\) are uniformly smooth assuming that the trajectories are smooth (and that they are away from the singularities). Note that in general \(\tau_{ij}\neq\tau_{ji}\) even if \(\tau_{ij}-\tau_{ji}=O(\varepsilon^{2})\). Indeed, after expanding the solution of (60) up to first order, i.e. + +\[\tau_{ij}(t)=\varepsilon|q_{i}(t)-q_{j}(t)|+\varepsilon^{(_{2}}q_{i}(t)-q_{j}(t)^{)}\cdot\dot{q}_{j}(t)+O(\varepsilon^{3}).\] (62) + +(hint: It is easy to consider the expansions for \(\tau_{ij}^{2}\) obtained by squaring both sides of (60) and express the square of length as an inner product). The above derivation (62) of an approximate form of the delay is purely formal. Note that it is only valid for differentiable \(q\) and that the error incurred in the approximation depends on higher derivatives of \(q\). Of course in a delay equation, modifying the form of the delay is a very singular perturbation so, substituting the approximation above may lead to equations with different solutions. + +In Theorem 4.8 the main objects are sets of uniformly differentiable \(q\). In these sets of uniformly differentiable functions, the approximations in (62) are uniform (in a norm involving one less derivative than the uniform). + +3. Theorem 4.8 also applies to many modifications of the model and produces solutions. For example, ignoring The Figtgerald contractions [Ver16] or changing the delay by its state dependent approximation (62) [Dri84] (they are formally second order in \(\varepsilon)\). The exact solutions of these models obtained applying Theorem 4.8 will be approximate solutions of (59). + +In theoretical Physics, there are also other methods to produce approximate solutions using formal power series [CCdlL20, GdlLY25, MS78] or using numerical approximations. + +2For \(\varepsilon=0\), the delays and advances vanish so that (59) is an ODE. diff --git a/outputs/arxiv_math/2503.04577_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.04577_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..0fe18269dc79b843718e6de79ee4fff1257fe341 --- /dev/null +++ b/outputs/arxiv_math/2503.04577_pg2_pg1_repeat1.md @@ -0,0 +1,27 @@ +A closely related concept is the proximal-point operator [31], which plays a fundamental role in the definition of the Moreau envelope and in developing proximal-based iterative schemes for minimizing the cost function \(\varphi\), which is defined as + +\[\mathbf{prox}_{\gamma\varphi}(x):=\operatorname*{arg\,min}_{y\in\mathbb{R}^{n}}\left(\varphi(y)+ \frac{1}{2\gamma} \Vert x-y\Vert^{2}\right).\] (1.2) + +The proximal operator enjoys several favorable properties under convexity, contributing to its popularity for + +developing numerical methods: (i) prox\(\gamma\varphi\) is single-valued at each \(x\in\mathbb{R}^{n}\); (ii) it is Lipschitz continuous with the constant 1; (iii) it satisfies the identity + +\[\operatorname*{arg\,min}_{x\in\mathbb{R}^{n}}\varphi(x)=\mathbf{Fix}(\mathbf{prox}_{\gamma\varphi}),\] + +where the latter set denotes the set of fixed points of prox\(\gamma\varphi\); see, e.g., [8, Chapter 12]. The generic proximalpoint method is given by \(x^{k+1}=\mathbf{prox}_{\gamma\varphi}(x^{k})\), satisfying \(x^{k}\rightarrow x^{*}\in\operatorname*{arg\,min}_{x\in\mathbb{R}^{n}}\varphi(x)\) under the convexity. For convex \(\varphi\), it holds that + +\[\mathbf{prox}_{\gamma\varphi}(x^{k})=x^{k}-\gamma\nabla\varphi_{\gamma}(x^{k}).\] (1.3) + +implying that the proximal method can be interpreted as a gradient descent with constant step-size \(\gamma\); cf. [36]. Thanks to the simple structure and low memory requirements, proximal-point methods have received increasing attention, following the seminal works by Martinet [26, 27]; see, e.g., [3, 18, 24, 35, 36, 42] and references therein. In the nonconvex setting, the fundamental properties of the proximal operator and Moreau envelope require involved assumptions such as prox-regularity and calmness of the cost in the local or global setting for which we invite the readers to study [37, 39]. Recent studies of proximal-point methods and the Moreau envelope have uncovered the further potential of these approaches in the presence of high-order regularization terms; see, e.g., [2, 3, 20–22, 34, 35, 47]. In these developments, as opposed to the classical case, the quadratic regularization in (1.1) and (1.2) is replaced by the + +term \(\frac{1}{p} \Vert x-y\Vert^{p}\text{for}p>1\), leading to a much more flexible setting. In particular, in [20] an inexact two-level smoothing optimization framework (ItsOPT) is introduced for general nonsmooth and nonconvex optimization problems. This framework consists of two levels: (i) at the lower level, the high-order proximal auxiliary problems are solved inexactly to produce an inexact oracle for HOME; (ii) at the upper level, an inexact zero-, first- or second-order method is developed to minimize HOME. Additionally, the framework has been adapted for solving nonsmooth weakly convex optimization problems in [21]. The crucial role of the basic properties of HOME, its differentiability, and its weak smoothness in these methodologies motivates the quest to study fundamental and differential properties of HOME in the nonconvex setting. + +### 1.1 Contribution + +Our contributions are threefold: + +(i) Fundamental properties of HOPE and HOME. We derive several fundamental properties of the high-order Moreau envelope (HOME) and the corresponding proximal operator (HOPE), including coercivity and sublevel set relationships between \(\varphi\text{and}\varphi_{\gamma}\) (cf., Propositions 9 and 10, Corollary 11), facilitating to design algorithms on \(\varphi_{\gamma}\). Further, we introduce the notion of \(p-calmness\), a key condition for deriving the differentiability of HOME, and characterize its relationships with classical reference points (cf., Theorem 15). + +(ii) Differentiability and weak smoothness of HOME. We comprehensively analyze the differentiability and weak smoothness of HOME for nonsmooth and nonconvex functions under \(p-\text{calmness and}q-\text{prox}-\)regularity assumptions (see Definitions 13 and 18), with \(p>1\). It is shown that HOME is continuously differentiable when \(q\geq2\text{and}p\in(1,2]\text{or}2\leq p\leq q\) (cf. Theorems 25 and 26), and weakly smooth with H¨older-continuous gradients under broader conditions (cf. Theorems 29 and 30). However, for \(p\leq q\) the differential properties of HOME remain open to us, which is summarized in Subfigure (a) of Figure 1. Furthermore, the relationship among reference points of \(\varphi\text{and}\varphi_{\gamma}\) are clarified (cf. Corollary 27 and Remark 28.), as emphasized in Subfigure (b) of Figure 1. + +(iii) The high-order proximal-point algorithm (HiPPA). The HiPPA algorithm is introduced and a very simple convergence analysis to proximal fixed points using properties of HOME is studied (cf. Theorem 31). Preliminary numerical tests of HiPPA on Nesterov-Chebyshev-Rosenbrock functions demonstrate its promising potential for solving nonsmooth and nonconvex optimization. diff --git a/outputs/arxiv_math/2503.04578_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.04578_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..d910df325303a7b7c5a5a4c88fef9365076f0bc0 --- /dev/null +++ b/outputs/arxiv_math/2503.04578_pg2_pg1_repeat1.md @@ -0,0 +1,9 @@ +for all \(\xi\in L^{2}(M,\mu)\text{with}\int_{M}\xi(x)d\mu(x)=0\text{and}s\in S\setminus\{e\},\text{where}\pi\) is the unitary representation \(\pi:G\rightarrow U(L^{2}(M,\mu))\) induced by the measure preserving action. Instead of considering the entire warped cone, he considered the coarse disjoint union of sections of \(\mathcal{O}_{G}M\) at \(t(n)’\mathrm{s},\text{where}\{t(n)\}_{n\in\mathbb{N}}\) is an increasing sequence of positive reals converging to infinity and we call it a warped system. Then he showed that an action has spectral gap if and only if for any (equivalently, some) sequence \(\{t(n)\}_{n\in\mathbb{N}}\), the corresponding warped system is quasi-isometric to an expander. In [WY12, WY14], Willett and Yu introduced a coarse invariance for metrics spaces, so called geometric property (T), which is an obstruction to the surjectivity of the maximal coarse Baum-Connes assembly map. Roughly speaking, it is the existence of spectral gap for the Laplacian in the maximal Roe algebra associated with the space. So, for a coarse disjoint union of finite graphs, it is stronger than being an expander in the sense that the Laplacian has spectral gap in the Roe algebra. They showed that for a box space \(X\) of a group \(G,X\) has geometric property (T) if and only if \(G\) has property (T). In [WY14], geometric property (T) was defined for discrete metric spaces, but to study it for warped systems, we adopt the non-discrete analogue formulated in [Win21]. If the base space is a compact Riemannian manifold \(M\) and the action is Lipschitz, the associated warped cone can be approximated by a graph with uniformly bounded degree as Lemma 2.7 and Lemma 2.8. Geometric property (T) in the sense of [Win21] is equivalent to the geometric property (T) of its discretization in the sense of [WY14]. In [Win21, Question 11.2], Winkel asked the following question: + +Question 1.1. If \(G\) has property \((T),M\) is a compact Riemannian manifold and \(G↷M\) is an ergodic action by diffeomorphisms, then does the warped system has geometric property (T)? + +In this paper, we provide a negative answer to the question. The following is the main result. + +Theorem 1.2. Let \(G\) be a finitely generated group, and \(M\) a compact Riemannian manifold with the Riemannian measure \(\mu\) and a \(G-\)action by homeomorphisms. Assume that \(G↷M\) is an ergodic, isometric and free action and \(\mu\) is \(G-\text{invariant}\). Then, for any sequence \(\{t(n)\}_{n\in\mathbb{N}}\) converging to infinity, the associated warped system \(\bigsqcup M\times\{t(n)\}\) does not have geometric property (T). + +While numerous studies, such as those in [Roe05, DN19, Saw19], have explored the similarities between the large-scale geometric properties of box spaces and warped cones, the above result shows that their behavior can differ significantly under maximal representations diff --git a/outputs/arxiv_math/2503.04583_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.04583_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..3ceb46598acdb2870129d661c4dae7fc0349285a --- /dev/null +++ b/outputs/arxiv_math/2503.04583_pg3_pg1_repeat1.md @@ -0,0 +1,17 @@ +Theorem 1. Deadlock-free routing can be constructed for a network \(G=(V,E)\) if and only if \(G\) contains two edge-disjoint directed trees rooted at the same vertex \(v\), one tree directed into \(v\) and the other directed away from \(v\). + +Sufficiency of the two-tree condition of Theorem 1 for the existence of deadlock-free routing follows from the fact that given such trees, any message can be routed by first sending it to the common root \(v\) along the first tree, and then sending it to its target along the second tree. The two-tree condition was known to be sufficient for deadlock-free routing. For example, [BS92; JMS94] established this fact, while [Aba97, Section 4.4] uses such two trees to construct a deadlock-free routing for a 4x4 mesh. However, to the best of our knowledge, the necessity of this condition has neither been recognized nor proven. To prove necessity we explicitly construct the two required trees from a network with a given deadlock-free routing. Using the result of [DS87], the dependency graph induced by the deadlock-free routing is acyclic, so there is a dependency-respecting total ordering of the graph edges \(E\) such that every vertex is reachable from every other vertex through a sequence of edges that are ascending according to the total order. Given a total ordering of the edges, we introduce three definitions: a global attractor, the attraction number and the attraction subgraph. + +Definition. A vertex in a directed graph is called a global attractor if it is reachable from all other vertices. + +Definition. Given a strongly connected directed graph \(G=(V,E)\) with a total ordering of its edges \(E=(e_{1},...,e_{m})\), we define its attraction number \(s\) to be the smallest integer such that there exists a vertex \(v\in V\) reachable from all other vertices using only the edges in the prefix \(\{e_{1},...,e_{s}\}\). + +The existence of such an \(s\) follows from the fact that \(G\) is strongly connected. We note that the attraction number is the minimal size of the prefix of the edges needed for the graph to have a global attractor. For the purpose of this definition, reachability using edges \(\{e_{1},...,e_{s}\}\) is not restricted to paths that respect the total ordering of the edges. + +Definition. Given a strongly connected directed graph \(G=(V,E)\) with a total ordering of its edges \(E=(e_{1},...,e_{m})\), we define its attraction subgraph to be the subgraph \(G_{s}=(V,E_{s})\) with the same vertices but only the first \(s\) edges in \(E:E_{s}=(e_{1},...,e_{s})\), where \(s\) is the attraction number of the graph. + +By definition, the attraction subgraph has a global attractor. + +Lemma 1.1. Let \(G=(V,E)\) be a strongly connected directed graph with a total ordering of its edges \(E=(e_{1},...,e_{m})\) that is consistent with the dependencies induced by a deadlock-free routing. If there is only a single global attractor \(v\) in the attraction subgraph \(G_{s}\), then the two-trees required by Theorem 1 exist in \(G\). + +Proof. If \(v\) is the only vertex reachable from all vertices in \(G_{s}\), no other vertex in \(G_{s}\) can be reachable from it, so there are no outbound edges of \(v\text{in}G_{s}\). Therefore, routing diff --git a/outputs/arxiv_math/2503.04590_pg7_pg1_repeat1.md b/outputs/arxiv_math/2503.04590_pg7_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..fcb9030f075f403e79c51a63b593dcb2e802205f --- /dev/null +++ b/outputs/arxiv_math/2503.04590_pg7_pg1_repeat1.md @@ -0,0 +1,25 @@ +Then the operator \(T(u)\equiv f(u)-P_{\Phi(u)}(f(u)-\alpha u)\) is Lipschit continuous. Proof. By the non-expansion property of the projection operator and (2.6) we have that + +\[\begin{aligned} \Vert T(u)-T(v)\Vert&=\Vert f(u)-P_{\Phi(u)}(f(u)-\alpha u)-(f(v)-P_{\Phi(v)}(f(v)-\alpha v))\Vert \\ &\leq\Vert f(u)-f(v)\Vert \\ +\Vert P_{\Phi(u)}(f(u)-\alpha u)-P_{\Phi(u)}(f(v)-\alpha v)+P_{\Phi(u)}(f(v)-\alpha v)-P_{\Phi(v)}(f(v)-\alpha v)) \\ &\leq L\Vert u-v\Vert+\Vert f(u)-f(v)\Vert+\alpha\Vert u-v\Vert+\mu\Vert u-v\Vert \\ &\leq(2L+\alpha+\mu)\Vert u-v\Vert. \end{aligned}\] + +□ + +This implies that T is Lipschit continuous. + +## 3. FINITE-TIME STABILITY ANALYSIS + +In this section we will analyze the finite-time stability for IQVIPs (1.1). By doing so we propose a new first order dynamical system associated with the problem (1.1) such that a solution of (1.1) becomes an equilibrium point of the dynamical system. Under mild conditions for the parameters, we demonstrate that the proposed dynamical system is finite-time stable. We first recall a characterization for the finite-time stability of solutions of a dynamical system, given by Bhat and Bernstein in [5]. This result will play an important role in studying finite-time convergence. Theorem 3.1. [5] (Lyapunov condition for finite-time stability). Assume that there exists a continuously differentiable function V : \(D\rightarrow\mathbb{R},whereD\subseteq\mathbb{R}^{n}\) is a neighborhood of the equilibrium point \(u^{*}\) for (2.1) and an open neighborhood \(U\subseteq D\) of \(u^{*}\) such that + +\[\dot{V}(u)\leq-K.\left(V(u)\right)^{p}\forall u\in U\setminus\{u^{*}\},\] + +where \(K>0andp\in(0,1).\) Then, the equilibrium point \(u^{*}\) of (2.1) is finite-time stable equilibrium point of (2.1). Moreover, the settling time T satisfies + +\[T(u(0))\leq \frac{V(u(0))^{1-p}}{K(1-p)}\] + +for any \(u(0)\in U\). In addition, if \(D=\mathbb{R}^{n}\), then the equilibrium point \(u^{*}\) of (2.1) is globally finite-time stable. In order to obtain a finite-time stability of solutions of inverse quasi-variational inequality problem (1.1) we now present a novel first-order dynamical system associated with the problem (1.1). Our proposed dynamical system is as follows: + +\[\dot{u}(t)=-\sigma \frac{f(u)-P_{\Phi(u)}(f(u)-\alpha u)}{\Vert f(u)-P_{\Phi(u)}(f(u)-\alpha u)\Vert^{\frac{\gamma-2}{\gamma-1}}} ,\] (3.1) + +where \(\sigma>0\) is a scalar tuning gain, and \(\gamma>2,\alpha>0\) are two design parameters. It is worth noting that for all \(u\in\mathbb{R}^{n}\text{suchthat}f(u)-P_{\Phi(u)}(f(u)-\alpha u)=0\), the expression on the right-hand side of (3.1) is still well defined for all \(\gamma>2\). In fact, it is equal to zero. Indeed, we have + +\[\Bigg\| \frac{f(u)-P_{\Phi(u)}(f(u)-\alpha u)}{\Vert f(u)-P_{\Phi(u)}(f(u)-\alpha u)\Vert^{\frac{\gamma-2}{\gamma-1}}} \Bigg\|=\Vert f(u)-P_{\Phi(u)}(f(u)-\alpha u)\Vert^{\frac{1}{\gamma-1}}=0.\] diff --git a/outputs/arxiv_math/2503.04604_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.04604_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..28d4c1a20a4712cb97b7cd5f7d6b6ec16781be38 --- /dev/null +++ b/outputs/arxiv_math/2503.04604_pg3_pg1_repeat1.md @@ -0,0 +1,13 @@ +Hence, the class of inhomogeneous Calderón–Zygmund singular integral operators is smaller than the class of Calderón–Zygmund singular integral operators. This kind of operator includes some of the pseudo-differential operators, see [3, 7]. Then, if \(T\) is an inhomogeneous Calderón–Zygmund singular integral operator, the authors in [3,4] found necessary and sufficient conditions fot the boundedness of such operators on \(h^{p}(\mathbb{R}^{n})\) for all \(00\), and a constant \(C>0\) such that for all balls \(B:=B(c,r)\) we have + +\[(1.5)_{\|}^{\|_{\|_{\|}}(} \frac{|x-c|}{r} )_{n(-)+}^{\frac{1}{p} \frac{1}{s} \frac{\varepsilon}{s}}\|K(x,y)-P_{K,B}(x,y)\|_{L^{q}(B,dy/|B|)_{\|}L^{s}([B(c,2r)]^{c},dx)}^{\|_{\|_{\|}}}\leq Cr^{n(s^{-1}-1)},\] + +\[\text{where}P_{K,B}(x,y)\equiv0ifr\geq1;\] diff --git a/outputs/arxiv_math/2503.04607_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.04607_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..a33ec639bacdb06c83b99ff0ecfe80e6aaf0022e --- /dev/null +++ b/outputs/arxiv_math/2503.04607_pg2_pg1_repeat1.md @@ -0,0 +1,9 @@ +Theorem 1.3. Suppose \(X\) admits a weakly reducible genus-three trisection \(\mathcal{T}\) . Then either \(\mathcal{T}\) is reducible, or \(\mathcal{T}\) contains a five-chain. In particular, \(X\) is diffeomorphic to a spun lens space \(S_{p}\) or its sibling \(S_{p}^{\prime},S^{4}\), or a connected sum of copies of \(\pm\mathbb{C}\mathbb{P}^{2},S^{1}\times S^{3},\text{and}S^{2}\times S^{2}\). A trisection \(\mathcal{T}\) is determined up to diffeomorphism by its spine, the union of three 3-dimensional handlebodies \(H_{\alpha}\cup H_{\beta}\cup H_{\gamma}\). A trisection \(\mathcal{T}\) is said to be reducible if there is a curve \(c\) bounding disks in all three handlebodies; in this case, we can split \(\mathcal{T}\) into a connected sum of smaller-genus trisections. In a foundational result from Heegaard theory, Casson and Gordon proved that if a 3-manifold \(Y\) admits a weakly reducible Heegaard splitting, then either the splitting is reducible or \(Y\) contains an essential surface [CG87]. We adapt weak reducibility to the setting of trisections, providing further evidence that key ideas from Heegaard theory can inform new insights in dimension four via trisections. We say that \(\mathcal{T}\) is weakly reducible if there are disjoint non-separating curves \(c\text{and}c^{\prime}\) such that \(c\) bounds a disk in one of the three handlebodies, and \(c^{\prime}\) bounds a disk in the other two. See [ST94] for additional discussion of weak reduction in the context of Heegaard splittings. A five-chain is a specific sequence of five curves bounding disks in each of the three handlebodies such that consecutive curves intersect once and other pairs of curves are disjoint. A more rigorous definition can be found in Section 5, in which we show that if \(\mathcal{T}\) admits a five-chain, then \(X\) can be obtained by surgery on a loop \(\ell\) in a manifold \(X^{\prime}\) admitting a lower-genus trisection \(\mathcal{T}^{\prime}\). The proof of Theorem 1.3 relies heavily on ideas from the theory of Heegaard splittings and uses a variety of tools and techniques, including connections between 4-manifolds and Dehn surgery on knots in 3-manifolds, weak reduction and thin position of Heegaard splittings, and many ideas arising in the proof of Theorem 1.1. In the spirit of classifying as many genus-three trisections as possible, we prove a stronger version of Theorem 1.3 using the same techniques. Given a trisection \(\mathcal{T}\text{for}X\) with spine \(H_{\alpha}\cup H_{\beta}\cup H_{\gamma}\), a dependent triple for \(\mathcal{T}\) consists of three pairwise disjoint non-separating curves \(\alpha_{1},\beta_{1},\text{and}\gamma_{1}\) bounding disks in \(H_{\alpha},H_{\beta},\text{and}H_{\gamma}\), respectively, such that the homology classes \([\alpha_{1}],[\beta_{1}],\text{and}[\gamma_{1}]\) are linearly dependent in \(H_{1}(\Sigma)\). + +Theorem 1.4. Suppose a genus-three trisection \(\mathcal{T}\) of \(X\) admits a dependent triple. Then either \(\mathcal{T}\) is reducible, or \(\mathcal{T}\) contains a five-chain. In particular, \(X\) is diffeomorphic to a spun lens space \(S_{p}\) or its sibling \(S_{p}^{\prime},S^{4}\), or a connected sum of copies of \(\pm\mathbb{C}\mathbb{P}^{2},S^{1}\times S^{3}\), and \(S^{2}\times S^{2}\). + +Corollary 1.5. If \(X\) is simply-connected and admits a genus-three trisection \(\mathcal{T}\) with a dependent triple, then \(X\) is diffeomorphic to \(S^{4}\) or a connected sum of copies of \(\pm\mathbb{C}\mathbb{P}^{2}\), and \(S^{2}\times S^{2}\). + +Corollary 1.5 is notable in light of the trisection genus additivity problem, which asks whether \(g(X_{1}\#X_{2})=g(X_{1})+g(X_{2})\). In particular, there are examples of exotic \(\mathbb{C}\mathbb{P}^{2}\#^{2} \overline{\mathbb{C}\mathbb{P}}^{2}\) (manifolds \(X\) homeomorphic but not diffeomorphic to \(\mathbb{C}\mathbb{P}^{2}\#^{2} \overline{\mathbb{C}\mathbb{P}}^{2} )\) [AP10, FS11]. If trisection genus is additive, then the fact that there is some \(k\text{such that}X\#^{k}(S^{2}\times S^{2})\) is diffeomorphic to \(\mathbb{C}\mathbb{P}^{2}\#^{2} \overline{\mathbb{C}\mathbb{P}}^{2} \#^{k}(S^{2}\times S^{2})\) [Wal64] implies that \(g(X)=g(\mathbb{C}\mathbb{P}^{2}\#^{2} \overline{\mathbb{C}\mathbb{P}}^{2} )=3\). Corollary 1.5 can be interpreted as evidence that no exotic \(\mathbb{C}\mathbb{P}^{2}\#^{2} \overline{\mathbb{C}\mathbb{P}}^{2}\) admits a genus-three trisection, which would imply that trisection genus is not additive. For more information on the additivity of trisection genus and exotic pairs, see Section 1.3 of [LCM22]. + +Remark 1.6. The only progress on the genus-three classification problem appears in work of the first author and Moeller in [AM22]. In Section 7, they prove that a special class of genus-three trisections, called Farey trisections, correspond to standard 4-manifolds (and diff --git a/outputs/arxiv_math/2503.04612_pg8_pg1_repeat1.md b/outputs/arxiv_math/2503.04612_pg8_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..b949e32f422bb3956fea739dc298586b90d3752b --- /dev/null +++ b/outputs/arxiv_math/2503.04612_pg8_pg1_repeat1.md @@ -0,0 +1,15 @@ +is continuous. Since the projective space \(\mathbb{R}\mathrm{P}^{1}\) is compact, the function is bounded. Given \(\omega=(\omega)_{n\in\mathbb{Z}}\in\Omega\), write \(\omega=(\omega_{-},\omega_{+}),\text{where}\omega_{-}=(\omega)_{n<0}\) and \(\omega_{+}=(\omega)_{n\geq0}\). For one-step cocycles, the first Oseledets direction \(\mathbf{E}_{1}(\omega)\) depends only on \(\omega_{-}\), while the second direction \(\mathbf{E}_{2}(\omega)\) depends only on \(\omega_{+}\). So we can denote them as \(\mathbf{E}_{1}(\omega_{-})\text{and}\mathbf{E}_{2}(\omega_{+})\). Let \(\mu_{\pm}\) denote the push-forward of \(\mu\) under the projection \(\omega\mapsto\omega_{\pm}.\text{Then}\mu\) coincides with the product measure \(\mu_{-}\otimes\mu_{+}\). Therefore, by Fubini–Tonelli theorem, \(\int|\log\sin\angle(\mathbf{E}_{1},\mathbf{E}_{2})|d\mu\) equals the double integral + +\[\int\int|\log\sin\angle(\mathbf{E}_{1}(\omega_{-}),\mathbf{E}_{2}(\omega_{+}))|d\mu_{-}(\omega_{-})d\mu_{+}(\omega_{+})\] (32) + +which is finite since the inner integral is bounded. This concludes the proof of Theorem 2 in the strongly irreducible case. Next, we must deal with the cases where \(\Gamma_{\nu}\) permutes a finite set \(S\) of one-dimensional subspaces of \(\mathbb{R}^{2}\). If \(\#S\geq3\), then a cross-ratio argument shows that the projection of \(\Gamma_{\nu}\) to \(\text{PGL}_{2}(\mathbb{R})\) is finite. This implies that \(\Gamma_{\nu}\) consists of linear conformal mappings with respect to some inner product on \(\mathbb{R}^{2}\), which is incompatible with the hypothesis \(\lambda_{1}(\nu)>\lambda_{2}(\nu)\). If \(\#S=2\), then the hypothesis \(\lambda_{1}(\nu)>\lambda_{2}(\nu)\) implies that \(\{\mathbf{E}_{1}(\omega),\mathbf{E}_{2}(\omega)\}= S\mu-\text{a.e}\)., and in particular the angle between Oseledets subspaces is logintegrable. It remains to consider the case \(\#S=1\). Taking inverses we may assume that the stable subspace \(\mathbf{E}_{2}(\omega)\) belongs to \(S\text{for}\mu-\text{a.e.}\omega\in\Omega\). With a change of basis we further assume that all \(\nu-\)almost every matrix is upper triangular. Multiplying the cocycle matrices by a log-integrable factor, we can assume that the lower right entry is always 1. That is, the function \(F\) is of the form: + +\[F(\omega)= \begin{pmatrix} a(\omega_{0}) & b(\omega_{0}) \\ 0 & 1 \end{pmatrix} ,\] (33) + +where \(\log|a|\text{and}\log|b|\) are square-integrable and \(\int\log|a|d\pi<0\). By Lemma 5, the cocycle has Lyapunov exponents \(\lambda_{1}=0\text{and}\lambda_{2}=\int\log|a|d\pi\), and the corresponding Oseledets directions are spanned by the vectors \((X(\omega),1)^{\top}\) and \((1,0)^{\top}\), where + +\[X(\omega)=\sum_{n=0}^{\infty}a(\omega_{-1})a(\omega_{-2})\cdots a(\omega_{-n})b(\omega_{-n-1}).\] (34) + +Writing \(\varphi(\omega_{0}):=-\log a(\omega_{0}),\psi(\omega_{0}):=\log b(\omega_{0})\), we have: + +\[(35)X(\omega)=\sum_{n=0}^{\infty}\pm e^{S_{n}(\omega)}\text{where}S_{n}(\omega):=\psi(\omega_{-n-1})-\sum_{i=0}^{n-1}\varphi(\omega_{-i}).\] diff --git a/outputs/arxiv_math/2503.04620_pg35_pg1_repeat1.md b/outputs/arxiv_math/2503.04620_pg35_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..bc8d274e5bdaadd338330b59652b59544132e118 --- /dev/null +++ b/outputs/arxiv_math/2503.04620_pg35_pg1_repeat1.md @@ -0,0 +1,44 @@ +where we define + +\[\tilde{a}_{k}= \begin{cases} 0, & \text{for}k=0 & \text{and} & \tilde{b}_{k}=-a_{k}k,\text{for}k=1,2,\ldots,r. \\ b_{k}k, & \text{for}k=1,2,\ldots,r \end{cases}\] (181) + +It is assumed that both \(a_{r}\text{and}b_{r}\) are nonzero; otherwise, they should be removed, and \(r\) should be reduced. + +## Step 2: Define the coefficients \(h_{j}\) as + +\[h_{j}= \begin{cases} \tilde{a}_{r-j}+i\tilde{b}_{r-j}, & j=0,1,\ldots,r-1, \\ 2\tilde{a}_{0}, & j=r, \\ \tilde{a}_{j-r}-i\tilde{b}_{j-r}, & j=r+1,r+2,\ldots,2r. \end{cases}\] (182) + +Since \(\tilde{a}_{0}=0\), it follows that \(h_{r}=0\). + +Step 3: Next, define a \(2r\times2r\) matrix B with entries \(B_{kj}\) as + +\[B_{kj}= \begin{cases} \delta_{k,j-1}, & \text{for}k=1,2,\ldots,2r-1, \\ \underline{h_{j-1}} \\ - & , & \text{for}k=2r, \\ \overline{\tilde{a}_{r}-i\tilde{b}_{r}} \end{cases}\] (183) + +where \(\delta_{k,j-1}\) is the Kronecker delta function. For example, when \(r=2\), the matrix B is + +\[\mathbf{B}=\left[_{-} \frac{0_{0_{0_{\tilde{a}_{2}+i\tilde{b}_{2}}}}}{\tilde{a}_{2}-i\tilde{b}_{2}} - \frac{1_{0_{0_{\tilde{a}_{1}+i\tilde{b}_{1}}}}}{\tilde{a}_{2}-i\tilde{b}_{2}} 0^{0_{1_{0}}}- \frac{0_{0_{1_{\tilde{a}_{1}-i\tilde{b}_{1}}}}}{\tilde{a}_{2}-i\tilde{b}_{2}} \right].\] (184) + +| explicitly | | | | | | +| --- | --- | --- | --- | --- | --- | +| 0 | 1 | 0 | 0 | | | +| 0 | 0 | 1 | 0 | | | +| B = 0 | 0 | 0 | | . | (184) | +| − a˜2+i˜b2 | a˜1+i˜b1 | | a˜1−i˜b1 1 | | | +| | − | 0 − | | | | +| a˜2−i˜b2 | a˜2−i˜b2 | | a˜2−i˜b2 | | | + +Note that B has a significant sparse structure, with at most \(4r-2\) non-zero elements. + +Step 4: Let the eigenvalues of B be denoted by \(z_{t}\in\mathbb{C}.[24\), Theorem 2] shows that the roots (which may be complex) of \(f^{\prime}(x)=0\) are given by \(x_{t}=-i\log(z_{t})\) where the complex logarithm is defined as \(\log(z)=\log|z|+i(\arg(z)+2\pi m),\forall m\in\mathbb{Z}\). Therefore, the final roots are + +\[x_{t}=(\arg(z_{t})+2\pi m)-i\log|z_{t}|,t=1,2,\ldots,2r,\forall m\in\mathbb{Z}.\] (185) + +Since we are only interested in the real roots of \(f^{\prime}(x)\), these real roots correspond to the eigenvalues \(z_{t}\) lying on the unit circle. This simplifies to + +\[x_{t}=\arg(z_{t})+2\pi m,\text{when}|z_{t}|=1.\] (186) + +By taking \(x_{k}\) modulo \(2\pi\), the final real roots can be obtained. + +Step 5: The global minimizer is the value of \(x_{t}\) that yields the smallest \(f(x)\) among these points. + +This method uses the inherent properties of trigonometric polynomials to transform the problem of finding the global minimizer of \(f(x)\) into an equivalent problem of determining all eigenvalues with modulus equal to 1 of a sparse non-Hermitian matrix B. Compared to directly using global optimization solvers (e.g., differential evolution), which are typically heuristic algorithms, this eigenvalue approach guarantees the identification of the global minimum, thereby avoiding the risk of getting trapped in local minima. Although eigenvalue problems may appear complex, in practical applications, the integer \(r\) is usually small, making it feasible to solve the eigenvalues of small matrices both efficiently and accurately. diff --git a/outputs/arxiv_math/2503.04623_pg54_pg1_repeat1.md b/outputs/arxiv_math/2503.04623_pg54_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..b9d70c81b233a249ccab00c61d7de949fc3a72ae --- /dev/null +++ b/outputs/arxiv_math/2503.04623_pg54_pg1_repeat1.md @@ -0,0 +1,43 @@ +Lemma 8.1.6. Suppose \(\ell/=2\text{and}(\ell,n!)=1\) if we are in case O1. If \(\phi\in\Phi^{\text{ss}}(G^{*}, \overline{\mathbb{F}_{\ell}} )\) is a generic + +Similarly, if \(\phi^{♯}\in\Phi^{\text{ss}}(G^{♯}, \overline{\mathbb{F}_{\ell}} )\) is a generic semi-simple toral \(L-\text{parameter},\text{and}\mu^{♯}\in X_{\bullet}(G^{♯})\) is a + +The standard representation Std\(\hat{}\) of \(\text{Sp}(2n(G^{*}))\text{or SO}(2n(G^{*}))\) has weights given by Weyl orbits of + +semi-simple toral \(L-\text{parameter} \underline{,a} nd\mu\in X_{\bullet}(G^{*})\) is a dominant cocharacter, then \(\phi\) is \(\mu-\text{regular}\). dominant cocharacter such that \((\ell,n!)=1\) if we are in case O1, then \(\phi^{♯}\) is \(\mu^{♯}-\text{regular}\). In particular, \(\mu^{♯}\) can be chosen to be not fixed by any non-trivial element of \(W_{G^{♯}}\). Proof. By base change [HL24, Lemma 4.22] and the isomorphism Equation (8.1), it suffices to prove for the general linear group \(\text{GL}(n)\), the split special orthogonal group \(\text{SO}(d(G))\) and the split general spin group \(\text{GSpin}(d(G))\). For general linear groups (i.e., in case U) this is proved in [HL24, Lemma 4.22], and the argument also works for G \(\in\{\text{SO}(d(G)),\text{GSpin}(d(G))\}\): \(\omega_{1}\). The standard representation extends to a standard representation of \(\text{GSp}(2n(G^{*}))\text{or GSO}(2n(G^{*}))\), and we write the highest weight of the standard representation of \(\text{GSp}(2n(G^{*}))\text{or GSO}(2n(G^{*}))\) as \(\omega_{1}^{♯}\). Then it is clear that difference of the weights appearing in Std\(\hat{}\) are coroots of G, thus \(\phi\) is strongly \(\mu-\text{regular and}\phi^{♯}\) is strongly \(\mu^{♯}-\text{regular}\), by definition of genericity. Thus, they are also \(\mu-\)regular (resp. \(\mu^{♯}-\text{regular})\) by the proof of [Ham24, Theorem 10.10]. For other cocharacters, we first recall that with \(\overline{\mathbb{Q}_{\ell}} -\text{coefcfi ients}\), the highest weight tilting module \(T_{\omega_{i}}\) in [GW09, §5.5.2], and it extends to a standard representation of \(\text{GSp}(2n(G^{*}))\) with highest weight denoted by \(\omega_{i}^{♯}\), and the same is true with \(\overline{\mathbb{F}_{\ell}} -\text{coefcfi ients}\), where each highest weight tilting module + +associated to \(\omega_{i}\in X_{\bullet}(\text{Sp}(2n(G^{*})))\) is realized on the space of harmonic elements in \(\wedge^{i}(\text{Std})\) as defined + +associated to \(\omega_{i}^{♯}\) of \(\text{GSp}(2n(G^{*}))\) is a direct sum of \(\wedge^{i}(\text{Std})\) , by our assumption on \(\ell\), cf. [Jan03, Page + +286-287] and [Ham24, §10.1, Appendix B.2]. Similarly, with \(\overline{\mathbb{Q}_{\ell}} -\text{coefcfi ients},\wedge^{i}(\text{Std})\) is isomorphic to + +the highest weight tilting module \(T_{\omega_{i}}\) associated to \(\omega_{i}\in X_{\bullet}(\text{SO}(2n(G^{*})))\text{for}1\leq i\leq n(G^{*})-2\), + +\(\omega_{n(G^{*})-1}+\omega_{n(G^{*})}\in X_{\bullet}(\text{SO}(2n(G^{*})))\). On the other hand, \(\wedge^{n(G^{*})-1}(\text{Std})\) is isomorphic to the direct + +and \(\wedge^{n(G^{*})-1}(\text{Std})\) is isomorphic to the highest weight tilting module \(T_{\omega_{n(G}*_{)-1}+\omega_{n(G}*_{)}}\) associated to sum of highest weight tilting modules \(T_{2\omega_{n(G}*_{)-1}}\text{and}T_{2\omega_{n(G}*_{)}}\), cf. [GW09, Theorem 5.5.13], and these extends to a standard representation of \(\text{GSp}(2n(G^{*}))\) with highest weight denoted by + +\[\omega_{1}^{♯},\ldots,\omega_{n-2}^{♯},\omega_{n-1}^{♯}+\omega_{n-2}^{♯},2\omega_{n-1}^{♯},2\omega_{n-2}^{♯},\] + +respectively. The same is true with \(\overline{\mathbb{F}_{\ell}} -\text{coefcfi ients}\), where each highest weight tilting module associated + +to the above cocharacters of \(\text{GSp}(2n(G^{*}))\) is a direct sum of \(\wedge^{i}(\text{Std})\) , by our assumption on \(\ell\), cf. [Jan03, Page 286-287] and [Ham24, §10.1, Appendix B.2]. Now all these highest weight tilting modules of \(G^{♯}\) with fundamental weights \(\mu^{♯}\) appear as direct summand of tensor products of Std\(\hat{}\), so \(\phi^{♯}\) is \(\mu^{♯}-\)regular by [Ham24, Proposition 10.12]. Finally, we can choose \(\mu^{♯}\) appropriately such that under the isomorphism Equation (8.1) + +\[G \tfrac{♯}{\mathbb{Q}_{p}} \cong \begin{cases} \prod & * \\ \text{GL}(1)_{\overline{\mathbb{Q}}}\times & \text{GSpin}_{v\in\text{Hom}(K, \overline{\mathbb{Q}} )}(V & \otimes_{K,v} \overline{\mathbb{Q}_{p}} ) & \text{in case O} \\ p & \prod & p & , \\ \text{GL}(1)_{\overline{\mathbb{Q}_{p}}}\times & \text{GU}_{v\in\text{Hom}(K, \overline{\mathbb{Q}_{p}} )}(V^{*}\otimes_{K_{1},v} \overline{\mathbb{Q}_{p}} ) & \text{in case U} \end{cases}\] + +it is of the form \((0,\mu^{♯\prime},\ldots,\mu^{♯\prime})\) (i.e., trivial on the GL(1) factor and identical on the other factors), where \(\mu^{♯\prime}\) is not fixed by any non-trivial Weyl group element. Then \(\mu^{♯}\) is not fixed by any non-trivial Weyl group element. + +- + +8.2. Perverse \(t-\)exactness and vanishing results. In this subsection, we prove a perverse \(t-\)exactness result for Hecke operators, and deduce a vanishing result for cohomology of Shimura varieties with torsion coefficients. We use notations related to BunG defined in Section 3.1. For any reductive group G over a non-Archimedean local field \(K\) of characteristic 0 and any open substack \(U\subset\) BunG, there exists a perverse \(t-\text{structure on D}_{\text{lis}}(\text{Bun}_{\mathrm{G}}, \overline{\mathbb{F}_{\ell}} )\) defined as follows [HL24, Definition 4.11]: For each b \(\in B(\mathrm{G})\), denote \(d_{\mathrm{b}}:=\langle2\rho_{\mathrm{G}},\nu_{\mathrm{b}}\rangle\text{where}\nu_{\mathrm{b}}\) is the slope homomorphism of b, + +then an object \(A\) is contained in \(\mathrm{D}^{p\leq0}(U, \overline{\mathbb{F}_{\ell}} )\text{if}i_{\mathrm{b}}^{*}A\in\mathrm{D}^{\leq d_{\mathrm{b}}}(\mathrm{G}_{\mathrm{b}},\Lambda),\text{and}A\) is contained in \(\mathrm{D}^{p\geq0}(U, \overline{\mathbb{F}_{\ell}} )\) if We also need the notion of universally locally acyclic (ULA) objects [FS24, Definition IV.2.31]: The + +\(i_{\mathrm{b}}^{!}A\in\mathrm{D}^{\geq d_{\mathrm{b}}}(\mathrm{G}_{\mathrm{b}},\Lambda)\). Here we recall that \(i_{\mathrm{b}}\) is the inclusion BunbG \(\subset\) BunG. + +full subcategory \(\mathrm{D}^{\text{ULA}}(\text{Bun}_{\mathrm{G}}, \overline{\mathbb{F}_{\ell}} )\subset\mathrm{D}_{\text{lis}}(\text{Bun}_{\mathrm{G}}, \overline{\mathbb{F}_{\ell}} )\) consists of objects \(A\text{such that}i_{\mathrm{b}}^{*}A\in\mathrm{D}^{\text{adm}}(\mathrm{G}_{\mathrm{b}}, \overline{\mathbb{F}_{\ell}} )\) for each b \(\in B(\mathrm{G})\). We import Setup 7.2.2. In particular, \(F\) is a totally real number field unramified at a prime \(p,\mathbf{G}\) is + +a special orthogonal or unitary group over \(F\text{with}G^{*}=\mathbf{G}\otimes_{F}(F\otimes\mathbb{Q}_{p}),\text{and}G^{♯}\) is a central extension + +of G with \(G^{♯}\). Then we have the following local result on the perverse \(t-\)exactness of Hecke operators, which generalizes [HL24, Corollary 4.24] to special orthogonal groups and unitary groups: diff --git a/outputs/arxiv_math/2503.04646_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.04646_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..6d4d2bb37b440bf2b19f23accf8e9c8d9c9afe5c --- /dev/null +++ b/outputs/arxiv_math/2503.04646_pg2_pg1_repeat1.md @@ -0,0 +1,15 @@ +regularization) that make the computation of the Wasserstein distance more efficient and scalable. + +DRO frameworks are explicitly designed for optimizing in the presence of uncertainties. For this reason, it is natural to draw a connection to inverse problems, where the measurement process \(H:X\rightarrow Y\) is often imperfect and the observation \(y=Hx\) is corrupted by different types of noise. Despite this analogy, the relation between DRO approaches, inverse problems reconstruction, and regularization is currently substantially unexplored. In this paper, we aim to make a first step to bridge this gap by designing a general DRO framework that is flexible to be used in inverse problems. In particular, given an in-sample estimation of the data acquisition process \(\mu^{*}\in P(X\times Y)\) (often given as an empirical estimation) we define a perturbation-aware DRO framework, where the ambiguity set is constrained to all possible perturbations of \(\mu^{*}\) that belong to a given class \(K\subset P(X\times Y)\). This framework builds on Bayesian variants of DRO [15, 24]. It allows - by making suitable choices of \(K-\) to enforce the desired robustness in the reconstruction and is ultimately linked to specific choices for Tikhonov-type regularization. For example, by choosing \(K\subset P(X)\), one could enforce robustness to perturbations of the input, while by choosing \(K\subset\{\mu\in P(X\times Y):\mu_{X}^{*}\in P(X)\) is the \(X-\)marginal of \(\mu\}\), one instead enforces robustness with respect to perturbations in the conditional distribution of \(Y|X\), thus mimicking the uncertainty of the data acquisition process. Moreover, by further specifying \(K\) one can recover robustness to prescribed distributions (e.g. Gaussian, Poisson) and impose desired anisotropies. + +In practice, we define the perturbation-aware DRO as a min-max problem where the ambiguity set includes all perturbations of \(\mu^{*}\text{in}K\) that additionally belong to the entropy-regularized Wasserstein ball. To implement an efficient numerical method to compute reconstructions of inverse problems we derive a (weak) dual formulation for the perturbation-aware DRO. Although this provides only an upper bound for the problem, we believe that under suitable assumptions a strong duality result should hold. However, we leave this proof up to future research. This allows us to rely on a biased stochastic mirror descent approach [17], where we follow [22] which was designed for entropy-regularized Wasserstein DRO with no perturbation constraints. To showcase the effectiveness of our framework for inverse problems, we present several examples. First, we set \(K\) as the family of multivariate Gaussian perturbations in \(Y|X\) with varying (scalar) variance. Subsequently, we consider the case where the Gaussian perturbations can have an arbitrary covariance matrix, allowing for anisotropic perturbations and robustness. Finally, we apply our method to solve image deconvolution problems with Gaussian and Poisson noise. + +## Main contributions. The main contributions of this paper can be summarized as follows: + +1. We introduce a perturbation-aware DRO framework, where perturbations of an in-sample estimated distribution are constrained to be in a set \(K\subset P(X\times Y)\). + +2. We show how this framework naturally applies to inverse problems and is amenable to different kinds of robustness, resulting in a flexible inverse that can be applied to measurements corrupted by any type of noise. + +3. We derive a weak duality formula for the problem, enabling the design of efficient numerical schemes. Under suitable assumptions, we believe strong duality will hold. + +4. We demonstrate the effectiveness of our method for matrix inversion tasks and deconvolution problems, by enforcing robustness in the reconstruction through different diff --git a/outputs/arxiv_math/2503.04649_pg14_pg1_repeat1.md b/outputs/arxiv_math/2503.04649_pg14_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..5feb28e1ed41595fc135a5ae98fa5b10f72f7b75 --- /dev/null +++ b/outputs/arxiv_math/2503.04649_pg14_pg1_repeat1.md @@ -0,0 +1,25 @@ +The \(r\) is a positive scalar function and the \(\boldsymbol{r}\) is the unit vector from the origin to the point on the sphere with spherical coordinates \((\theta,\phi)\). In practice, when performing coordinate-based calculations at least two charts are required for radial manifolds in order to overcome singularities that occur from the topology in the surface coordinate frames, such as poles when using spherical angles. To this end, when using our numerical methods we consider two coordinate charts as in [25, 26]. The first chart we call Chart A and has coordinate singularities at the north and south pole, while the latter chart we call Chart B which has coordinate singularities at the east and west poles, see [25, 26]. In practice, for a given \((\theta,\phi)\in[0,2\pi)\times[0,\pi]\), we typically restrict usage of a chart for \(\phi\in[ \frac{\pi}{5} , \frac{4\pi}{5} ]\). For chart A, we parameterize the manifold in the embedding space \(\mathbb{R}^{3}\) as + +\[\boldsymbol{x}(\hat{\theta},\hat{\phi})=r(\hat{\theta},\hat{\phi})\boldsymbol{r}(\hat{\theta},\hat{\phi}),\boldsymbol{r}(\hat{\theta},\hat{\phi})=\left[\sin\left(\hat{\phi}\right)\cos\left(\hat{\theta}\right),\sin\left(\hat{\phi}\right)\sin\left(\hat{\theta}\right),\cos\left(\hat{\phi}\right)\right].\] (24) + +For Chart B, we use + +\[\boldsymbol{x}(\bar{\theta},\bar{\phi})=r(\bar{\theta},\bar{\phi})\boldsymbol{r}(\bar{\theta},\bar{\phi}),\boldsymbol{r}(\bar{\theta},\bar{\phi})=\left[\cos\left(\bar{\phi}\right),\sin\left(\bar{\phi}\right)\sin\left(\bar{\theta}\right),\sin\left(\bar{\phi}\right)\cos\left(\bar{\theta}\right)\right].\] (25) + +Using these parametrizations, we can compute the basis \(\partial_{\phi},\partial_{\theta}\) for the tangent space as + +\[\boldsymbol{\sigma}_{\phi}(\theta,\phi)=r_{\phi}(\theta,\phi)\boldsymbol{r}(\theta,\phi)+r(\theta,\phi)\boldsymbol{r}_{\phi}(\theta,\phi),\] (26) + +\[\boldsymbol{\sigma}_{\theta}(\theta,\phi)=r_{\theta}(\theta,\phi)\boldsymbol{r}(\theta,\phi)+r(\theta,\phi)\boldsymbol{r}_{\theta}(\theta,\phi).\] (27) + +Expressions for \(\boldsymbol{r}_{\phi},\boldsymbol{r}_{\theta}\) can be found using equations 24 and 25 depending on which chart is being used. These can also be used to compute all the relevant quantities described in Appendix A. We use radial manifolds in training and validation of the GNPs. We use spherical harmonics to generate radial functions \(r(\theta,\phi)\) for the radial manifolds. In order to have a rich class of shapes for training, we consider a range of complexities for the radial functions. We sample complex coefficients \(a_{l}^{m}\text{for}m=0,\ldots,l\) from a normal distribution with mean 0 and standard deviation \(\frac{1}{l}\) , to obtain the radial function + +\[r(\theta,\phi)=\sum_{l=0m}^{L}\sum_{=-l}^{l}a_{l}^{m}Y_{l}^{m}(\theta,\phi),\] (28) + +where \(Y_{l}^{m}\) are the spherical harmonics. We choose \(a_{l}^{m}\text{for}m<0\) to ensure that \(r(\theta,\phi)\) is real-valued. In order to have a range of complexities in the geometry, we truncate the series at \(L=3,6,8,10,12,15,18,22\). Further, we translate and scale \(r(\theta,\phi)\) so that it has mean 1 and satisfies \(0.7\leq r(\theta,\phi)\leq1.3\text{for all}(\theta,\phi)\). For validation of the GNPs, we also consider toroidal manifolds. These are surfaces that are diffeomorphic to a torus. We consider the general parametrizations of + +\[\boldsymbol{\sigma}(u,v)=((a(u,v)\cos(v)+b(u,v))\cos(u),(a(u,v)\cos(v)+b(u,v))\sin(u),a(u,v)\sin(v)),(29)\] + +for \(u,v\in[0,2\pi)\). One can choose \(a(u,v)=a,b(u,v)=b\) to obtain a standard torus. We consider functions \(a(u,v),b(u,v)\) of the form + +\[a_{b(}(_{u}u_{,},_{v}v_{)})_{=}=a_{b_{0}^{0}+}+_{r}r_{\sin_{1}^{0}}\sin_{(}(_{A}A_{u_{1}^{0}}u_{)})_{\cos}\cos_{(}(_{B}B_{v_{1}^{0}}v_{)})_{.},\] (30) diff --git a/outputs/arxiv_math/2503.04667_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.04667_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..aa72ef4efb0f839abf85a9bc508a540c19adbef3 --- /dev/null +++ b/outputs/arxiv_math/2503.04667_pg3_pg1_repeat1.md @@ -0,0 +1,27 @@ +head \(p_{\phi_{t}}(z_{t}|z)\), and the corresponding prediction variable \(\hat{Y}_{t}\) is non-parametric mapping of \(Z_{t}\). Then, define information flow (Shwartz-Ziv and Tishby 2017; Goldfeld et al. 2019) in neural networks as a Markov chain shared by all tasks, i.e., \(Y_{t}\rightarrow X\rightarrow Z\rightarrow Z_{t}\rightarrow\hat{Y}_{t}\) for task \(t\). + +# Methodology + +To ensure sufficiency for target tasks and mitigate the negative effects of redundant features, we propose a principled multi-task representation learning framework (InfoMTL) that extracts noise-invariant sufficient representations for all tasks. It contains two learning principles: shared information maximization (SIMax) and task-specific information minimization (TIMin), which constrain the amount of shared and task-specific information in the multi-task learning process. + +## Shared Information Maximization Principle + +In the MTL paradigm, the mutual information between the input \(X\) and the shared representations \(Z\) is typically reduced implicitly (Figure 1(a)) or explicitly (Figure 1(b)) under the information bottleneck (IB) theory (Shwartz-Ziv and Tishby 2017; Kawaguchi et al. 2023). However, directly reducing redundancy for one task is prone to losing necessary information for others. As a result, the learned shared representations \(Z\) usually suffer from task-specific insufficiency. To alleviate the insufficiency issue, a shared information maximization (SIMax) principle is proposed to learn more sufficient shared representations for all target tasks. The SIMax principle simultaneously maximizes the mutual information between the input \(X\) and the shared representations \(Z\) for all tasks, as well as the mutual information between the shared representations \(Z\) and the target task \(Y_{t}\) for each task \(t\). It can be formulated as the maximization of the following Lagrangian, + +\[\max\sum_{t=1}^{|T|}[I(Y_{t};Z)]+\alpha I(X;Z),\] (1) + +subject to the Markov constraint, i.e., \(Y_{t}\rightarrow X\rightarrow Z\rightarrow Z_{t}\rightarrow\hat{Y}_{t}.\alpha\) is a parameter that balances the trade-off between the informativeness of \(Z\text{for}X\text{and}Y_{t}\). In Equation (1), the second term promotes the task-agnostic shared representations \(Z\) preserves as much information as possible about the input \(X\). It ensures the sufficiency of the shared representations \(Z\) for all potential targets \(Y\) . The first term encourages the shared representations \(Z\) to capture the necessary information relevant to the target \(Y_{t}\) for each task \(t\). As shown in Figure 1(b) and (c), unlike IB-based MTL methods (Qian, Chen, and Gechter 2020; de Freitas et al. 2022) that explicitly minimize input-representation information, our InfoMTL with SIMax handles the information in an opposite manner to avoid the insufficiency issue arising from representation compression in the multi-task paradigm. Implementation of SIMax The implementation of SIMax contains two terms, i.e., maximizing \(\sum_{t=1}^{|T|}[I(Y_{t};Z)]\) and \(I(X;Z)\). Firstly, we maximize the lower bound of \(I(Y_{t};Z)\) + +by estimating the conditional entropy of the target \(Y_{t}\) given the shared representations \(Z\). Following Kolchinsky, Tracey, and Wolpert (2019) and Hu et al. (2024a), we use cross-entropy (CE) as the estimator for each classification task \(t\). Secondly, to maximize \(I(X;Z)\), we use the InfoNCE estimator (Gutmann and Hyvärinen 2010; van den Oord, Li, and Vinyals 2018) to optimize the lower bound of \(I(X;Z)\) during the multi-task encoding process. According to the information flow of \(Z\leftarrow X\rightarrow Z^{\prime}\), the Markov chain rule states that \(I(X;Z)\geq I(Z;Z^{\prime})\). For the InfoNCE estimation of maximizing \(I(Z;Z^{\prime})\), the selection of positive and negative samples as well as the implementation of noise-contrastive loss are consistent with Hu et al. (2024b). The optimization objective of SIMax for MTL can be: + +\[\begin{aligned} \mathcal{L}_{\text{SIMax}}&=\mathbb{E}_{z\sim p_{\theta}(z|x)}\{\mathbb{E}_{t\sim T}[-\log q_{\phi_{t}}(y_{t}|z)] \\ -\alpha\log \frac{\exp(\text{sim}(z,z^{+})/\tau)}{\sum_{z^{\prime}\in\mathcal{B}^{+}}\exp(\text{sim}(z,z^{\prime})/\tau)_{(}} 2\}_{)}, \end{aligned}\] + +where \(p_{\theta}(z|x)\) is a shared encoder with parameters \(\theta\). \(q_{\phi_{t}}(y_{t}|z)\) is a task-specific decoder with parameters \(\phi_{t}\) for task \(t\), and its output distribution is adapted for task prediction by a non-parametric function (e.g., Softmax operation for classification). \(z^{+}\) refers to the positive key of \(z\), generated by dropout in \(p_{\theta}(z|x).\mathcal{B}^{+}\) represents the set of positive keys in the current batch \(\mathcal{B}.\text{sim}(\cdot)\) is a pairwise similarity function, i.e., cosine similarity. \(\tau>0\) is a scalar temperature parameter that controls the sharpness of the probability distribution, which is applied during the Softmax operation. + +## Task-specific Information Minimization Principle + +In multi-task scenarios, redundant information often differs across tasks, such that information beneficial for one task may become redundant for another. Directly applying the IB principle (Tishby, Pereira, and Bialek 1999; Tishby and Zaslavsky 2015) to compress the task-specific redundancy for one task can result in the loss of necessary information for other tasks. Consequently, both the shared representations \(Z\) and the task-specific output representations \(Z_{t}\) for a given task \(t\) often contain redundant features that are irrelevant to the specific task. To better alleviate the redundancy issue, a task-specific information minimization (TIMin) principle is designed to mitigate the negative effect of potential redundant features in the inputs for the target task. It can compress task-irrelevant redundant information and preserve necessary information relevant to the target for multi-task prediction. The TIMin principle minimizes the mutual information between the input \(X\) and the task-specific output representations \(Z_{t}\), while maximizing the mutual information between the output representations \(Z_{t}\) and the corresponding target task \(Y_{t}\) for each task \(t\). The principle can be formulated as the maximization of the following Lagrangian, + +\[\max\sum_{t=1}^{|T|}[I(Y_{t};Z_{t})-\beta I(X;Z_{t})],\] (3) + +subject to \(Y_{t}\rightarrow X\rightarrow Z\rightarrow Z_{t}\rightarrow\hat{Y}_{t}\) for each task \(t.\beta\) is a trade-off parameter of the compression of \(Z_{t}\) from the input \(X\) and the informativeness of \(Z_{t}\text{for}Y_{t}\). diff --git a/outputs/arxiv_math/2503.04674_pg19_pg1_repeat1.md b/outputs/arxiv_math/2503.04674_pg19_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..e541692865c5bd1e11239803dd92556b8698dc49 --- /dev/null +++ b/outputs/arxiv_math/2503.04674_pg19_pg1_repeat1.md @@ -0,0 +1,21 @@ +Example 6.2. Consider the two-dimensional semilinear parabolic problem + +\[\frac{\partial u}{\partial t} (t,x,y)- \frac{\partial^{2}u}{\partial x^{2}} (t,x,y)- \frac{\partial^{2}u}{\partial y^{2}} (t,x,y)= \frac{1}{1+u(t,x,y)^{2}} + \frac{1}{1+u( \tfrac{t}{2} - \tfrac{1}{2} ,x,y)^{2}} ,\] (6.63) + +for \((x,y)\in[0,1]^{2}\text{and}t\in[0,3]\), subject to homogeneous Dirichlet boundary conditions. The initial condition is given by \(\phi(t,x,y)=\mathrm{e}^{-t}x(1-x)y(1-y)\text{for}t\in[- \frac{1}{2} ,0]\). We apply standard finite differences with \(n=200\) grid points to discretize the problem in each spatial direction. In this example the exact solution is unknown. The reference solution is computed by the ERKC-C method with Gauss collocation points \((s=3)\) using the constant step size \(h=2^{-11}\). The errors of the ERKC-I and ERKC-C methods in the \(L^{\infty}(\Omega)\) and the \(L^{2}(\Omega)\) norm at final time \(T=3\) are presented in Figures 3 and 4. The figures confirm the theoretical analysis. Moreover, a comparison of computational cost of the ERKC methods with Gauss collocation points \((s=3)\) is presented in Table 1. For step sizes \(h=2^{-k}\text{with}k=3,4,\ldots,9\), the ERKC-I method consistently reduces the CPU time by approximately 12% compared to the ERKC-C method, demonstrating its computational efficiency across a range of step sizes. + +| 10-6 | ● | | 10-6 | ● | | +| --- | --- | --- | --- | --- | --- | +| | Gauss (s=1) | ● | | Gauss (s=1) | ● | +| | Radau IIA (s=2) | ● | | Radau IIA (s=2) | ● | +| | Gauss (s=2) | ● | | Gauss (s=2) | | +| 10-8 | Radau IIA (s=3) | ● | 10-8 | Radau IIA (s=3) | ● | +| | Gauss (s=3) | ● [icon] | | Gauss (s=3) | ● | +| 10-10 | ● ● | [icon] | 10-10 | ● ● | [icon] ✗ | +| 10-12 | | [icon] | 10-12 | [icon] | [icon] ✗ ✓ | +| 10-14 | [icon] | [icon] | 10-14 | ✗ | [icon] | +| [icon] | [icon] | slope = 2 slope = 3 slope = 4 | [icon] | [icon] | slope = 2 slope = 3 slope = 4 | +| 10-16 | | | 10-16 | | | +| | 10-2 | 10-1 h | | 10-2 | 10-1 h | + +Figure 3: The convergence rates of ERKC-I methods (in the left panel) and ERKC-C methods (in the right panel) for (6.63). The errors are measured at \(T=3\) in the \(L^{\infty}(\Omega)\) norm. diff --git a/outputs/arxiv_math/2503.04678_pg20_pg1_repeat1.md b/outputs/arxiv_math/2503.04678_pg20_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..7eb8d49185b416b11329d850531678fef34b78f3 --- /dev/null +++ b/outputs/arxiv_math/2503.04678_pg20_pg1_repeat1.md @@ -0,0 +1,27 @@ +which has 2n distinct blocks of width \(\geq2^{k-1}\). The blocks are distinct since \(3d-\mu_{n}>\ldots>3d-\mu_{1}>d>\mu_{1}>\ldots>\mu_{n}\). By Lemma 4.6, the degree \(d^{\prime}\) obtained after this sequence of \(\ell-1=n2^{k-2}*-\)seeds is at most + +\[\begin{aligned} d^{\prime}&\leq\left( \frac{n2^{k-1}+3}{3} \right)(5d)\leq5n2^{k}d. \\ □ \end{aligned}\] + +## 5.2. Lower bound on length increase. Here we prove the lower bound in Theorem 2.5, which we recall in the next proposition. + +Proposition 5.2. There exists a sequence of proper homaloidal types \((\mathbf{y}_{N})_{N\geq1}\) of degrees \(d(\mathbf{y}_{N})\leq\) 2N210N, such that \(s(\mathbf{y}_{N})\geq2^{N-1}\). In particular, for any \(c< \sqrt{\ln(2)}\) and N large enough, + +\[s(\mathbf{y}_{N})\geq\exp(c \sqrt{\ln(d(\mathbf{y}_{N}))} )\] + +Proof. Let \(N\geq1\) be an integer parameter. We begin by constructing recursively an auxiliary finite sequence of homaloidal types \((\mathbf{x}_{0,N},\ldots,\mathbf{x}_{N-1,N})\) whose last element will be yN. If we start with the de Jonquières homaloidal type + +\[\mathbf{x}_{0,N}:=(2^{N-1}+1;2^{N-1},(1)^{2^{N}})\] + +which has 1 block of width 2N, we can apply Lemma 5.1 successively \(N-1\) times with n = 2i and \(k=N-i,\text{where}i=0,\ldots,N-2\). Doing so, we obtain a sequence of homaloidal types \((\mathbf{x}_{i,N})=(\mathbf{x}_{0,N},\ldots,\mathbf{x}_{N-1,N})\) of degrees \(d_{0,N}= 2^{N-1}+1,d_{1,N},\ldots,d_{N-1,N}\) such that each \(\mathbf{x}_{i,N}\) has 2i distinct blocks of width \(\geq2^{N-i}\). The successive degrees satisfy the inequality + +At the end, + +\[d_{i+1,N}\leq5\cdot2^{N-i}\cdot2^{i}\cdot d_{i,N}=5\cdot2^{N}\cdot d_{i,N}.\] + +\[d_{N-1,N}\leq5^{N}2^{N^{2}}(2^{N-1}+1)\] (5.1) + +\[\leq2^{N^{2}}10^{N}.\] (5.2) + +We now define the homaloidal type \(\mathbf{y}_{N}:=\mathbf{x}_{N-1,N}\), so \(d(\mathbf{y}_{N})\leq\) 2N210N. Its seedbed s(yN) is larger than the number \(2^{N-1}\) of blocks of width 2 that have been created. The last statement follows from the fact that when N is large, we have + +\[N\geq \sqrt{\frac{\ln(d(\mathbf{y}_{N}))}{\ln(2)}} (1+o(1)),\] (5.3) diff --git a/outputs/arxiv_math/2503.04690_pg4_pg1_repeat1.md b/outputs/arxiv_math/2503.04690_pg4_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..d7062883fffd4457ba05ddc2f5cf7ef2996d2c28 --- /dev/null +++ b/outputs/arxiv_math/2503.04690_pg4_pg1_repeat1.md @@ -0,0 +1,47 @@ +## 2 Methods + +### 2.1 Video Pre-processing + +An unprocessed video of a plume that is recorded with ideal background conditions.1 All frames are converted to gray scale, that is \(n\times d\) arrays taking on values between 0 and 1 (or 0 to 255). Two main steps are used in pre-processing prior to applying our model: background subtraction and Gaussian blurring. + +Background subtraction. A fixed background subtraction method is applied to isolate the plume, where the first \(k_{\text{fixed}}\) subtraction frames of video (where no plumes are have formed) are used to create an average background image. The average image is subtracted against the remaining frames to create the isolated plume frames. Once the plume dynamics have been isolated from background across selected frames, Gaussian blurring is applied. + +Gaussian Blurring. Two separately tuned Gaussian filters are applied—a temporal and spatial filter, respecitvely. First, a Gaussian filter is applied across the time series of frames to add a time blur, reducing the high resolution of the plume dynamics. Second, a Gaussian filter is applied to each frame independently to coarse grain the image, adding spatial blur. + +### 2.2 Coarse-graining for Centerline and Edge modeling + +We denote the output of data pre-processing as \(\boldsymbol{Z}=[\boldsymbol{z}(t_{0}),\ldots,\boldsymbol{z}(t_{K})]\), where background subtraction and Gaussian filters have been applied. We extract a time series of second-order polynomial coefficients that model the center path of the plume for each frame. Additionally, we learn the parameters of a growing sinusoidal function that best characterizes the spread, or edge-model, of the plume. We theorize there exists a connection between the Kevin Helmholtz shear velocity and the sinusoidal frequency. + +Each frame \(\boldsymbol{z}(t_{i})\) is converted to a reduced order model describing the center and edge paths in a three step procedure: (i) contour detection, (ii) a concentric circle search, and (iii) regression. For each array, \(\boldsymbol{z}(t_{i})\), image recognition techniques are used to identify the plume contour and subsequently search along concentric circles, centered at the leak source, to identify the path of highest density, where we denote raw pixel value as a proxy for plume density, and the edge paths, as seen in Fig. 2. + +![](figure) + +![](figure) + +![](figure) + +![](figure) + +![](figure) + +Data reduction process for plume film + +(a) (b) (c) (d) (e)[^inferred] + +![](figure) + +Data reduction process for plume film + +(a) (b) (c) (d) (e)[^inferred] + +Figure 2: The plume point reduction step converts a frame of video into a scatter of edge and center points: + +(a) the raw frame image, (b) background subtraction, (c) contour selection, and finally (d) Along various ranges from origin, identify the max intensity (center) and intersection (edge) points with contours. (e) displays the final points on the original frame. + +(i) Contour Detection. We apply a binary threshold to identify the contours outlining the plume for each array \(\boldsymbol{z}(t_{i})\). Hyperpameter selection for the thresholding is done by using opencv’s Otsu’s binarization. Optimal global threshold selection is performed by inspecting the image histogram for each array \(\boldsymbol{z}(t_{i})\). The \(n\) largest contours, by area, are then selected for remainder of the pipeline. We denote the identified plume at time \(t_{k}\text{as}\boldsymbol{\psi}_{k}\). + +(ii) Concentric Circle Search. A search is performed along a set of \(\ell\) concentric circles, centered at the plume leak source, with incrementally growing radii of \(r_{i}=r\cdot i\text{for}i=1,\ldots,\ell\text{where}r\) is some fixed + +1A solid black background is used when filming plume dynamics. + +[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter. diff --git a/outputs/arxiv_math/2503.04701_pg26_pg1_repeat1.md b/outputs/arxiv_math/2503.04701_pg26_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..1715220e4e04d103a2c5a6a0eef7202fa0ac31aa --- /dev/null +++ b/outputs/arxiv_math/2503.04701_pg26_pg1_repeat1.md @@ -0,0 +1,27 @@ +For the Fourier expansion, we use the same truncation mode for both the bundle and the manifold, as we can always take the maximum between the two when needed. The coefficients used in these computations are available in the GitHub repository associated with this paper [38]. The zero-finding problems (27), (44), and (61) define well-conditioned maps that can be used with root-finding algorithms to refine numerical approximations. In practice, after obtaining an approximate solution, we apply Newton’s method to these maps to improve the accuracy of our numerical approximations. The required level of accuracy varies depending on the proof. In some cases, smaller residuals are necessary, while in others, lower accuracy suffices. Our proof consists of three stages: the proof of the tangent bundle, the proof of the stable manifold attached to the periodic orbit and the validation of the solution of the soliton boundary value problem. We begin by obtaining a solution to the bundle problem. Our numerical approximation \(\bar{x}_{\mathrm{F}}\in X_{\mathrm{F}}\) of the bundle problem satisfies that \(\bar{\lambda}<0\) and the coefficients of \(\bar{v}\) satisfy the symmetry condition + +\[\bar{v}_{k}^{(i)}=[\bar{v}_{-k}^{(i)}]^{*},\text{for}i=1,2,k\in\mathbb{Z}.\] (72) + +Observe that this condition can be explicitly enforced in our computational implementation. We fix a norm weight \(\nu=1.05\) for the norm in (15), and a scaling factor \(l=0.5\) for the phase condition (26). The computable bounds \(Y(\bar{x}_{\mathrm{F}})=2.6879100002352747\times10^{-13},Z_{1}(\bar{x}_{\mathrm{F}})=0.3465291783592818\), and \(Z_{2}(\bar{x}_{\mathrm{F}})=14.980732463866438\), defined in (32), (33), and (41), satisfy the following inequalities: + +\[Z_{1}(\bar{x}_{\mathrm{F}})<1\text{and}Z_{2}(\bar{x}_{\mathrm{F}})< \frac{(1-Z_{1}(\bar{x}_{\mathrm{F}}))^{2}}{2Y(\bar{x}_{\mathrm{F}})} .\] + +Applying Theorem 7, we obtain a unique zero of the validation map (27), + +\[x_{\mathrm{F}}^{d}=^{\text{ef}}(\lambda,v^{(1)},v^{(2)}),\] + +within the ball \(B_{X_{\mathrm{F}}}(\bar{x}_{\mathrm{F}},r_{\mathrm{F}})\) with radius \(r_{\mathrm{F}}=4.122891017172993\times10^{-13}\). This zero corresponds to a solution + +\[(\lambda,v^{(1)},v^{(2)},0,0)\in\mathbb{R}\times S_{\mathrm{F}}^{4},\] + +which satisfies the sequence equation (23) and represents a real solution of (11). With this result, we can now evaluate the validation map and its derivative for the parameterization of the stable manifold attached to the periodic orbit described by (44). The second stage of our proof corresponds to the construction of this parameterization. We have that in this case, for \(r_{\mathrm{T}}^{*_{\mathrm{F}}}=10^{-3}\) , the bounds \(Y(\bar{w},r_{\mathrm{F}})=6.327932449800631\times10^{-9},Z_{1}(\bar{w},\bar{x}_{\mathrm{F}})= 0.9583731072113382,\text{and}Z_{2}(\bar{w},r_{\mathrm{T}}^{*_{\mathrm{F}}})=104.77593347038471\) defined in (48), (50), and (53) satisfy + +\[Z_{1}(\bar{w},\bar{x}_{\mathrm{F}})<1\text{and}Z_{2}(\bar{w},r_{\mathrm{T}}^{*_{\mathrm{F}}})n\}|=k\). We denote the set of \((k,n)-\)bounded affine permutations by \(\text{BA}(k,n)\). We say that \(f\) is a bounded \(n-\)affine permutation if there exists \(k\text{such that}f\) is a \((k,n)-\)bounded affine permutation. + +Lemma 3.1.5. The sets \(\text{GN}(k,n)\text{and}\text{BA}(k,n)\) are in natural bijection. + +In the proof, we will need the following definition. + +Definition 3.1.6. We define the cyclic order \(\leq_{i}\) on the set \([n]:i<_{i}i+1<_{i}i+2<_{i}\cdots <_{i} n<_{i}1<_{i}\cdots <_{i}i-1\). + +Proof of Lemma 3.1.5. This is essentially a combination of [19, Corollary 3.13] and [23, Remark 2.4]. For the reader’s convenience and to fix notation, we provide the bijection \(\varphi\) : \(\text{GN}(k,n)\rightarrow\text{BA}(k,n)\). Let us first describe \(\bar{f}_{\mathcal{I}}:=\varphi(\mathcal{I})\pmod{n}\) through its inverse \(\bar{f}_{\mathcal{I}}^{-1}\) : we have \(\bar{f}_{\mathcal{I}}^{-1}(i)=i\text{if}I_{i-1}=I_{i},\text{and}\bar{f}_{\mathcal{I}}^{-1}(i)=j\text{if}I_{i-1}\setminus I_{i}=\{j\}\). To lift this to a bounded affine permutation, we set \(f_{\mathcal{I}}(i)=i+n\text{if}I_{i-1}=I_{i}\text{and}i\in I_{i},\text{and}f_{\mathcal{I}}(i)=i\text{if}i\not\in I_{i}\). On the other hand, if \(f:\mathbb{Z}\rightarrow\mathbb{Z}\) is a \(k-\)bounded affine permutation, let \(I=\{i\in[n]|f(i)=i+n\}\), and \(\bar{f}=f\pmod{n}.\text{Then}I_{i}\) consists of \(I\) together with the elements \(\{a\in[n]|a<_{i}\bar{f}(a)\}\). The collection \(\mathcal{I}_{f}=(I_{1},\ldots,I_{n})\) is a Grassmann necklace, and \(f\mapsto\mathcal{I}_{f},\mathcal{I}\mapsto f_{\mathcal{I}}\) are inverse bijections. Now, we associate a Grassmann necklace and a bounded affine permutation to an element \(V\) in the Grassmannian \(\text{Gr}(k,n)\) following [19, 24]. We represent \(V\) by a \(k\times n\) matrix of rank \(k\), up to row operations, and denote by \(v_{1},\ldots,v_{n}\in\mathbb{C}^{k}\) the columns of \(V\) . Furthermore, we define \(v_{i}\text{for all}i\in\mathbb{Z}\) by setting \(v_{i+n}:=(-1)^{k-1}v_{i}\). Given an (ordered) \(k-\)element subset + +- diff --git a/outputs/arxiv_math/2503.04932_pg20_pg1_repeat1.md b/outputs/arxiv_math/2503.04932_pg20_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..09478e71410ddc0332768c70fe51c4741197f99f --- /dev/null +++ b/outputs/arxiv_math/2503.04932_pg20_pg1_repeat1.md @@ -0,0 +1,41 @@ +![Line plot showing L^1 error versus \( \lambda \) for different methods: bEuler, DIRK2, DIRK3, Order 1, Order 2, Order 3. The y-axis is labeled 'L^1 error' (on a log scale), and the x-axis is labeled \( \lambda \). The legend identifies each line by method name.](figure)[^inferred] + +![Line plot showing L^1 error versus \( \lambda \) for different methods: bEuler, DIRK2, DIRK3, Order 1, Order 2, Order 3. The y-axis is labeled 'L^1 error' (on a log scale), and the x-axis is labeled \( \lambda \). The legend identifies each line by method name.](figure)[^inferred] + +Fig. 2: Error plot for (66) with initial condition + +(67) using backward Euler, DIRK2 and DIRK3. + +![Line plot showing rank versus time for three curves labeled r_1, r_2, r_3, and their average (r_1 + r_2 + r_3)/3. The y-axis is labeled 'rank' and ranges from 0 to 11. The x-axis is labeled 't' and ranges from 0 to 15. A legend indicates the color and style for each curve.](figure)[^inferred] + +![Line plot showing L^1 error versus \( \lambda \) for different schemes and orders. The y-axis is labeled 'L^1 error' (log scale), and the x-axis is labeled \( \lambda \) (log scale). Three solid lines represent IMEX111, IMEX222, and IMEX443. Three dashed lines represent Order 1, Order 2, and Order 3.](figure)[^inferred] + +![Line plot showing L^1 error versus \( \lambda \) for different schemes and orders. The y-axis is labeled 'L^1 error' (log scale), and the x-axis is labeled \( \lambda \) (log scale). Three solid lines represent IMEX111, IMEX222, and IMEX443. Three dashed lines represent Order 1, Order 2, and Order 3.](figure)[^inferred] + +Fig. 3: Error plot for (69) with initial condition \(u_{0}(x,y,z,)=\sum_{k=1}^{2}\sin(kx)\sin(ky)\sin(kz)\) using IMEX111, IMEX222 and IMEX443. + +![Line plot showing rank vs t for r_1, r_2, r_3, and (r_1 + r_2 + r_3)/3. The y-axis is labeled 'rank' and the x-axis is labeled 't'. The legend indicates four lines: r_1 (blue), r_2 (green), r_3 (magenta), and (r_1 + r_2 + r_3)/3 (dashed magenta).](figure)[^inferred] + +![Line plot showing rank vs t for r_1, r_2, r_3, and (r_1 + r_2 + r_3)/3. The y-axis is labeled 'rank' and ranges from 0 to 11. The x-axis is labeled 't' and ranges from 0 to 15. A legend indicates blue for r_1, green for r_2, magenta for r_3, and dashed magenta for (r_1 + r_2 + r_3)/3.](figure)[^inferred] + +![](figure) + +Fig. 4: The multilinear \(\text{rank}(r_{1},r_{2},r_{3})\) and average of the multilinear \(\text{rank}(r_{1}+r_{2}+r_{3})/3\) of the solution to (66) with initial condition (67) using backward Euler (left), DIRK2 (middle) and DIRK3 (right). + +As seen in Figure 3, the expected accuracies are observed for the RAIL scheme when using IMEX111, IMEX222 and IMEX443. We used a mesh size \(N=80\), tolerance \(\varepsilon=10^{-6}\), final time \(T_{f}=0.3,\text{and}\lambda\) ranging from 0.1 to 1. Despite observing the expected accuracy, the \(L^{1}\) error for the first-order scheme (and even the second-order scheme) is quite large. However, recall that we do not scale the \(L^{1}\) error by the measure of the domain, which in this case would be \(|\boldsymbol{\Omega}|=(2\pi)^{3}\); scaling by the measure of the domain would provide a better comparison against the \(L^{\infty}\) error which is not as large. + +# Example 3 (Rigid body rotation with diffusion, about \(\hat{z})\) + +\[u_{t}-yu_{x}+xu_{y}=d(u_{xx}+u_{yy}+u_{zz})+c(x,y,z,t),x,y,z\in(-2\pi,2\pi)\] (71) + +where the flow field describes rotation about the vector \(\hat{z}\). To test the accuracy of the scheme, we use the manufactured solution \(u(x,y,z,t)=\exp(-(x^{2}+2y^{2}+3z^{2}+3dt))\text{with}d=1/3\), for which the source term \(c(x,y,z,t)\) offsets the rotation and is + +\[c(x,y,z,t)=e^{-(x^{2}+2y^{2}+3z^{2}+3dt)}\left(-2xy-d(-9+4x^{2}+16y^{2}+36z^{2})\right).\] (72) + +As seen in Figure 5, the expected accuracies are observed for the RAIL scheme when using IMEX111, IMEX222 and IMEX443. We used a mesh size \(N=80\), tolerance \(\varepsilon=10^{-8}\), final time \(T_{f}=0.3,\text{and}\lambda\) ranging from 0.5 to 2. When a low-rank source term is involved, we must express it in a Tucker tensor format. By inspection, it is straightforward for one to write down a Tucker decomposition of (72). To test the rank of the solution, we set \(d=1/12,c(x,y,z,t)=0\), double the speed of the rotation, + +\[u_{t}-2yu_{x}+2xu_{y}= \frac{1}{12} (u_{xx}+u_{yy}+u_{zz}),x,y,z\in(-2\pi,2\pi)\] (73) + +and set the initial condition to \(u_{0}(x,y,z)=\exp(-(x^{2}+9y^{2}+z^{2}))\). The solution rotates counterclockwise about the positive \(z-\)axis while slowly diffusing. Theoretically, the exact multilinear rank should be + +[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter. diff --git a/outputs/arxiv_math/2503.04950_pg23_pg1_repeat1.md b/outputs/arxiv_math/2503.04950_pg23_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..7d58fda0ea2977c0f99489ecdf40672855b3b243 --- /dev/null +++ b/outputs/arxiv_math/2503.04950_pg23_pg1_repeat1.md @@ -0,0 +1,63 @@ +| 1 | 4 | 3 | | +| --- | --- | --- | --- | +| 0 | 2 | 1 | | +| 0 | 3 | 0 | 1 | + +Figure 6: An \(\mathbb{N}-\)labeling of \(\nu=(4,3,3)\). The descents occur at cells \((2,3),(3,1),(3,2),(3,3)\) yielding \(\text{maj}(\sigma)=2+1+1+1=5\). The inversion triples are formed by \(((3,2),(3,1),(3,3)),((1,2),(1,3)),\text{and}((1,2),(1,4))\) yielding \(\text{inv}(\sigma)=3\). + +The reading order of a diagram \(\nu\) is the order on cells that goes left-to-right along rows starting from the top row and working down. The standardization of a labeling \(\sigma:\nu\rightarrow\mathbb{N}\) with content \(\alpha=(\alpha_{0},\alpha_{1},\cdots )\), denoted by \(\text{st}(\sigma)\), is an injective labeling \(\sigma:\nu\rightarrow\{1,\cdots ,|\nu|\}\) that labels the 0’s appearing in \(\nu1,2,\cdots ,\alpha_{0}\) in the order that they appear in the reading order. The 1’s appearing in \(\nu\) are labeled \(\alpha_{0}+1,\cdots ,\alpha_{0}+\alpha_{1}\) in reading order and so on. A triple consists of cells with row-column coordinates of the form \(u=(r,c)\), \(v=(r-1,c)\text{and}w=(r,c+k)\text{for}r,c,k\geq1,\text{where}u,w\) must be cells of \(\nu\), but \(v\) is allowed to be a cell immediately below \(\nu\). Extend \(\text{st}(\sigma)\) to cells immediately below \(\nu\) by labeling them \(-\infty\). A triple is an inversion triple of \(\sigma\) if the labels \(\text{st}(\sigma)(u),\text{st}(\sigma)(w),\text{st}(\sigma)(v)\) are decreasing clockwise. The statistic \(\text{inv}(\sigma)\) counts the number of inversion triples of \(\sigma\). + +A descent of \(\sigma\) is a cell \(u=(r+1,c)\text{in}\nu,\text{such that}v=(r,c)\) is also in \(\nu\text{and}\sigma(u)>\sigma(v)\). Letting \(\text{Des}(\sigma)\) denote that set of descents we define the major index statistic + +\[\text{maj}(\sigma):=\sum_{u\in\text{Des}(\sigma)}(\text{leg}(u)+1),\] + +where \(\text{leg}(u)\) is the number of cells of \(\nu\) in the same column as \(u\) and above \(u\). See Fig 6 for an example. Lemma 5.2.1. The monomial coefficient \(\langle[q^{i}t^{j}]\tilde{H}_{\mu[n]},h_{\eta[n]}\rangle\) stabilizes once \(n\geq\max(|\mu|+\mu_{1}+|\eta|+i,|\eta|+\eta_{1})\). Proof. Fix partitions \(\mu,\nu\text{and}i,j\) and let \(M^{(n)}\) denote the set of being enumerated in (26). Then + +and we wish to show \(|M^{(n)}|\) stabilizes. + +\[\langle[q^{i}t^{j}]\tilde{H}_{\mu[n]},h_{\eta[n]}\rangle=|M^{(n)}|\] + +Suppose \(n\geq|\mu|+\mu_{1}+|\eta|+i\text{and}n\geq|\eta|+\eta_{1}\) so that \(\mu[n]\text{and}\eta[n]\) are well-defined. Define the map \(\gamma_{n}:M^{(n)}\rightarrow M^{(n+1)}\) as follows. For \(\sigma\in M^{(n)}\), shift the bottom row of the corresponding diagram to the right by one cell and insert a new cell labelled 0 in the bottom left corner. So for instance: + + + + + + + + + + + + + + + + +
102
004310
+ + + + + + + + + + + + + + + + + +
102
0004310
+ +\(\mapsto\) + +\[\gamma_{n}:\] + +We have to check that the map is well-defined and surjective. We claim that the first \(\mu_{1}+1\) entries of the bottom row of any \(\sigma^{\prime}\in M^{(n+1)}\) are 0. Suppose for contradiction that there was a non-zero value \(a\) among these first \(\mu_{1}+1\) cells. Then there are at least \(n+1-|\mu|-(\mu_{1}+1)=n-|\mu|-\mu_{1}\) entries right of \(a\). Of these, at most \(|\eta|-1\) are allowed to be non-zero (since \(a\) is one of the \(|\eta|\) non-zero values in the labeling). Hence \(a\) would contribute at least \(n-|\mu|-\mu_{1}-(|\eta|-1)>i\) inversion triples formed by taking \(a\), the phantom cell below it, and a 0 entry to the right of it. This contradicts that \(inv(\sigma^{\prime})=i\text{for}\sigma^{\prime}\in M^{(n+1)}\). + +This implies that if we let \(\sigma\) be obtained from a \(\sigma^{\prime}\in M^{(n+1)}\) by deleting the bottom left cell and shifting the bottom row to the left by one, then \(\sigma\in M^{(n)}\text{and}\gamma_{n}(\sigma)=\sigma^{\prime}\), which implies that \(\gamma_{n}\) is well-defined and bijective. Indeed, the first \(\mu_{1}\) cells of the bottom row of \(\sigma\) will be labeled 0 and so the status of descents and inversion triples doesn’t change when going from \(\sigma^{\prime}\) to \(\sigma\). Thus, \(\sigma^{\prime}\text{and}\sigma\) have the same inv and maj values. diff --git a/outputs/arxiv_math/2503.04958_pg4_pg1_repeat1.md b/outputs/arxiv_math/2503.04958_pg4_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..bfa5aee552285ff3c3c0c78300b0f6e2c4d0363d --- /dev/null +++ b/outputs/arxiv_math/2503.04958_pg4_pg1_repeat1.md @@ -0,0 +1,21 @@ +Proposition 2.2. Let \(V\neq \{0\}\) be an ordered normed space whose cone has an interior point \(v.\text{Then}v\) is an order unit and the order unit norm \(\Vert\cdot\Vert_{v}\) is weaker than the original norm \(\Vert\cdot\Vert onV\) . Moreover, every positive functional on \(V\) is continuous with respect to both norms. Note that the two norms in Proposition 2.2 need not be equivalent, see, e.g., [6, Example 1.5.12]. For a detailed discussion in ordered Banach spaces, see [4, Proposition 2.11]. For an ordered vector space \((V,V_{+})\), a set \(S^{\prime}\) of positive linear functionals from \(V\) to \(\mathbb{R}\) is said to determine positivity if, for every \(v\in V\) , one has \(v\geq0\text{if}\varphi(v)\geq0\) for all \(\varphi\in S^{\prime}\). Note that, if \(V\) is an order unit space and \(\Sigma\subseteq V_{+}^{\prime}\) is given as in (2.1), then ext \(\Sigma\) determines positivity. We will make use of the following two simple propositions. Proposition 2.3. Let \(X\) be an ordered normed vector space and let \(S^{\prime}\subseteq X_{+}^{\prime}\). The following are equivalent: + +(i) The set \(S^{\prime}\) determines positivity, i.e., for every \(x\in X\), the inequality \(\langle x^{\prime},x\rangle\geq0\) for all \(x^{\prime}\in S^{\prime}\) implies that \(x\geq0\). + +(ii) The positive-linear hull of \(S^{\prime}\) is weak-∗ dense in the dual wedge \(X_{+}^{\prime}\). + +Proof. “(i) \(\Rightarrow\) (ii)” We show the contraposition. Let \(C^{\prime}\subseteq X_{+}^{\prime}\) denote the weak-∗ closure of the positive-linear hull of \(S^{\prime}\) and assume that there exists a point \(x_{0}^{\prime}\in X_{+}^{\prime}\) that is not in \(C^{\prime}\). By the Hahn–Banach separation theorem in the locally convex space \(X^{\prime}\) with the weak-∗ topology, there exists a weak-∗ continuous functional \(x^{\prime\prime}:X^{\prime}\rightarrow\mathbb{R}\) and a number \(\gamma\in\mathbb{R}\text{such that}\langle x^{\prime\prime},x^{\prime}\rangle\geq\gamma>\langle x^{\prime\prime},x_{0}^{\prime}\rangle\text{for all}x^{\prime}\in C^{\prime}\). Due to the weak-∗ continuity of \(x^{\prime\prime}\), there exists an element \(x\in X\text{such that}x^{\prime\prime}\) is the point evaluation at \(x\), see [13, § IV.1.2 on pp. 124–125]. It follows that \(\langle x,x^{\prime}\rangle\geq\gamma>\langle x,x_{0}^{\prime}\rangle\) for all \(x^{\prime}\in C^{\prime}\). Since \(0\in C^{\prime}\), we obtain \(\gamma\leq0\). Therefore, \(x\) is not in \(X_{+}.\text{As}C^{\prime}\) is a wedge, for every \(x^{\prime}\in C^{\prime}\) we have \(\langle x,x^{\prime}\rangle\geq0\), hence \(C^{\prime}\) does not determine positivity. + +“(ii) \(\Rightarrow\) (i)” If \(x\in X\) satisfies \(\langle x^{\prime},x\rangle\geq0\text{for all}x^{\prime}\in S^{\prime}\), then it follows from (ii) that the same inequality holds for all \(x^{\prime}\in X_{+}^{\prime}.\text{As}X_{+}\) is closed, this implies that \(x\in X_{+}\). + +The following is a primal version of Proposition 2.3. As its proof is similar, we omit it. + +Proposition 2.4. Let \(X\) be an ordered normed vector space and let \(S\subseteq X_{+}\). The following are equivalent: + +(i) The set \(S\) determines positivity of functionals in the sense that, for each \(x^{\prime}\in X^{\prime}\), the inequality \(\langle x^{\prime},x\rangle\geq0\text{for all}x\in S\) implies that \(x^{\prime}\geq0\). + +(ii) The positive-linear hull of \(S\) is weakly dense (equivalently, norm dense) in \(X_{+}\). + +If the cone \(V_{+}\) in an ordered vector space \(X\) has a base \(D,\text{then}x\in D\) is extremal if and only if \(x\) is an extreme point of \(D\), see [6, Lemma 1.5.19]. The following result by Krein and Milman is given, e.g., in [13, §II.10.5 on p. 68]. + +Proposition 2.5. Let \(V\) be a locally convex Hausdorff space. Let \(\Sigma\subseteq V\) be compact and convex and let \(S^{\prime}\subseteq\Sigma.If\Sigma= \overline{\text{conv}(S^{\prime})}\) then ext \(\Sigma\subseteq \overline{S^{\prime}}\). diff --git a/outputs/arxiv_math/2503.04993_pg18_pg1_repeat1.md b/outputs/arxiv_math/2503.04993_pg18_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..a7290618230453f2dc223196b092d5bc186ccad0 --- /dev/null +++ b/outputs/arxiv_math/2503.04993_pg18_pg1_repeat1.md @@ -0,0 +1,23 @@ +Thus, the necessary optimality condition for a two-player game provides the conditions under which both players’ controls are optimal in response to each other’s strategies. Each player’s control must satisfy the condition that the derivative of the Hamiltonian with respect to their control vanishes almost everywhere, ensuring that the strategy of each player is optimal given the strategy of the other player. + +Proof. + +Consider the variation of the functional \(J_{1}\) with respect to the control \(u_{1}\). We compute the derivative: + +\[\frac{d}{d\epsilon} J_{1}(u_{1}+\epsilon v_{1},u_{2})\Big|_{\epsilon=0}.\] + +Expanding the objective functional: + +\[\begin{aligned} \frac{d}{d\epsilon} J_{1}(u_{1}+\epsilon v_{1},u_{2})\Big|_{\epsilon=0}&=\lim_{\epsilon\rightarrow0} \frac{1}{\epsilon} \mathbb{E}\Bigg[\int_{R_{Z}}\Big\{f_{1}(\zeta,Y^{(u_{1}+\epsilon v_{1},u_{2})}(\zeta),u_{1}(\zeta)+\epsilon v_{1}(\zeta),u_{2}(\zeta)) \\ -f_{1}(\zeta,Y^{(u_{1},u_{2})}(\zeta),u_{1}(\zeta),u_{2}(\zeta))\Big\}d\zeta+g_{1}(Y^{(u_{1}+\epsilon v_{1},u_{2})}(Z))-g_{1}(Y^{(u_{1},u_{2})}(Z))\Bigg]. \end{aligned}\] + +Using the first-order expansion: + +\[\begin{aligned} \frac{d}{d\epsilon} J_{1}(u_{1}+\epsilon v_{1},u_{2})\Big|_{\epsilon=0}&=\mathbb{E}\Bigg[\int_{R_{Z}} \begin{cases} \underline{\partial f_{1}} & (u_{1},u_{2}) \\ (\zeta,Y & (\zeta),u_{1}(\zeta),u_{2}(\zeta))G_{1}(\zeta) \\ \overline{\partial y} \end{cases} \\ + \frac{\partial f_{1}}{\partial u_{1}} (\zeta,Y^{(u_{1},u_{2})}(\zeta),u_{1}(\zeta),u_{2}(\zeta))v_{1}(\zeta)\Big\}d\zeta+ \frac{\partial g_{1}}{\partial y} (Y^{(u_{1},u_{2})}(Z))G_{1}(Z)\Bigg], \end{aligned}\] + +where \(G_{1}\) is the variation in the system’s state due to the change in control. We now decompose it into two terms: + +\[\frac{d}{d\epsilon} J_{1}(u_{1}+\epsilon v_{1},u_{2})\Big|_{\epsilon=0}=I_{1}+I_{2},\] + +where + +\[\begin{aligned} I_{1}&=\mathbb{E}\Bigg[\int_{R_{Z}}\left( \frac{\partial H_{1}}{\partial y} (\zeta)- \frac{\partial\alpha}{\partial y} (\zeta)p_{1}(\zeta)- \frac{\partial\beta}{\partial y} (\zeta)q_{1}(\zeta)-(L_{1}\star \frac{\partial\alpha}{\partial y} )(\zeta)\right)G_{1}(\zeta)d\zeta \\ +\int_{R_{Z}}\left( \frac{\partial H_{1}}{\partial u_{1}} (\zeta)- \frac{\partial\alpha}{\partial u_{1}} (\zeta)p_{1}(\zeta)- \frac{\partial\beta}{\partial u_{1}} (\zeta)q_{1}(\zeta)-(L_{1}\star \frac{\partial\alpha}{\partial u_{1}} )(\zeta)\right)v_{1}(\zeta)d\zeta\Bigg], \end{aligned}\] diff --git a/outputs/arxiv_math/2503.05000_pg9_pg1_repeat1.md b/outputs/arxiv_math/2503.05000_pg9_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..3e2f2ddc0dad0146ab265274bf2a5929f8737039 --- /dev/null +++ b/outputs/arxiv_math/2503.05000_pg9_pg1_repeat1.md @@ -0,0 +1,33 @@ +time-dependent Hamiltonian vector field of \(\tau^{*} \frac{\partial S_{t}}{\partial t}\) starting at \(d_{x}S_{0}.\) Equation (2.24) is then a plain consequence of the chain rule. Since + +\[\forall f,g\in C^{\infty}(T^{*}M),d\{f,g\}_{\omega}=[df,dg],\] (2.25) + +equation (2.23) is obtained by usual extension from smooth functions to exact 1-forms, and then from closed forms to generic 1-forms using the Leibniz rule. + +Let us illustrate Lemma 2.17. The algebroid bracket \([.,.]\) allows us to obtain iterated derivations of the Hamilton-Jacobi equation (2.21). We spell out the order 2 : + +\[\frac{\partial^{2}\zeta_{t}}{\partial t^{2}} =(\zeta_{t})^{*}[\alpha^{*}\theta,\tau^{*}\zeta_{t}\alpha^{*}\theta],\] (2.26) + +After applying Lemma 2.17 a second time, we obtain an order 3 derivation: + +\[\frac{\partial^{3}\zeta_{t}}{\partial t^{3}} =(\zeta_{t})^{*}([\alpha^{*}\theta,\tau^{*}(\zeta_{t})^{*}[\alpha^{*}\theta,\tau^{*}(\zeta_{t})^{*}\alpha^{*}\theta]]+[[\alpha^{*}\theta,\tau^{*}(\zeta_{t})^{*}\alpha^{*}\theta],\tau^{*}(\zeta_{t})^{*}\alpha^{*}\theta]).(2.27)\] + +## 3 A Pre-Lie approach to Hamiltonian Poisson integrators + +Since \(G\subset T^{*}M,\) the Lagrangian bisections of \(G\) that are close to the identity section are described by graphs of time-dependent closed 1-forms. We can approximate them by using the notion of jets we developed in the previous section. In order to approximate a given Lagrangian bisection by 1-forms, we now introduce an appropriate space \(J_{\xi}^{\infty}\) and a pre-Lie algebra structure on it. We then explain how this pre-Lie algebra encodes expansions of solutions of the Hamilton-Jacobi equation (2.21) through the introduction of Butcher series. In Section 3.3, we exhibit some algebraic simplifications arising in this expansion if the initial condition is chosen to be zero. + +### 3.1 Pre-Lie formalism for Hamilton-Jacobi flows + +In full generality, for \(\xi\in\Omega^{1}(G)\), we are interested in the expansion of the general Hamilton-Jacobi flow + +\[\frac{\partial\zeta}{\partial t} =\zeta_{t}^{*}\xi,\zeta_{0}\in\Omega_{0}^{1}(M).\] (3.1) + +We consider the infinite dimensional real vector subspace \(J_{\xi}^{\infty}\subset\Omega^{1}(G)\) spanned by \(\xi\) and stable w.r.t. the following maps: for any \(f,g\in J_{\xi}^{\infty},\) + +\[\eta(f,g):^{\Omega_{0}^{1}(}\zeta^{M)\rightarrow}\mapsto\zeta^{*}[^{\Omega}f^{1_{0}},^{(}\tau^{M_{*}}\zeta^{)_{*}}g]^{.}\] (3.2) + +The space \(J_{\xi}^{\infty}\) is naturally equipped with the product \(\triangleright\) + +\[h\triangleright\xi(\zeta)=\zeta^{*}[\xi,\tau^{*}\zeta^{*}g],h\triangleright\eta(f,g)=\eta(h\triangleright f,g)+\eta(f,h\triangleright g).\] (3.3) + +This yields a non-associative non-commutative magmatic structure on \((J_{\xi}^{\infty},\triangleright)\). Note that \((J_{\xi}^{\infty},\triangleright)\) is not a Lie algebra as \(\triangleright\) is not antisymmetric. Remark 3.1 \((J_{\xi}^{\infty}\) and jet spaces). The space \(J_{\xi}^{\infty}\) is analogous to the infinite jet space [1, 41, 37] on \(X(M)\) used for the analysis of Runge-Kutta and Lie-group methods. In [31, 27], the accuracy of numerical integrators is studied via the use of Taylor expansions of ODE flows. Given a vector diff --git a/outputs/arxiv_math/2503.05034_pg12_pg1_repeat1.md b/outputs/arxiv_math/2503.05034_pg12_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..3459ee810d181768a99071173f1d688bbb1c06ef --- /dev/null +++ b/outputs/arxiv_math/2503.05034_pg12_pg1_repeat1.md @@ -0,0 +1,41 @@ +Let \((d_{n,k})_{n,k\in\mathbb{N}_{0}}\) be an infinite lower-triangular matrix over \(\mathbb{R}\). If there exists a pair \((d(t),h(t))\) such that the \((n,k)-\)th element of the matrix is defined as: + +\[d_{n,k}=[t^{n}]d(t)h(t)^{k},\] + +where \([t^{n}]\) is an operator that takes the coefficient of the term \(t^{n}\), then, such a matrix is called a Riordan array and we denote it as \(\mathcal{R}(d(t),h(t)).\text{If}\mathcal{R}(d(t),h(t))\text{and}\mathcal{R}(g(t),f(t))\) are Riordan arrays, the multiplication ∗ is defined as: + +\[\mathcal{R}(d(t),h(t))*\mathcal{R}(g(t),f(t)):=\mathcal{R}(d(t)g(h(t)),f(h(t))\] + +which is another Riordan array. Hence, the identity matrix in Riordan array form is \(\mathcal{R}(1,t)\) while the inverse matrix of \(\mathcal{R}(d(t),h(t))\) is given as: + +\[\mathcal{R}(d(t),h(t))^{-1}:=\mathcal{R}\left( \frac{1}{d(\bar{h}(t))} ,\bar{h}(t)\right)\] + +where \(\bar{h}(t)\) is the compositional inverse of \(h(t)\), that is \(h(\bar{h}(t))=\bar{h}(h(t))=t\). Therefore, the set of Riordan arrays forms a group called the Riordan group under multiplication. While Riordan arrays have various useful properties, we primarily utilize the following result called the fundamental theorem of Riordan arrays (FTRA): + +Theorem 1 (Shapiro et al., 1991): Every combinatorial sum representable as a linear combination of the elements in the \(n-\text{throw}d_{n,k},k=0,\ldots\) of Riordan array \(\mathcal{R}(d(t),h(t))\) can be written down as follows: + +\[\text{where}c(t)=\sum_{i=0}^{\infty}c_{i}t^{i}.\] + +\[\sum_{k=0}^{\infty}d_{n,k}c_{k}=\sum_{k=0}^{n}d_{n,k}c_{k}=[t^{n}]d(t)c(h(t))\] + +The type of Riordan array used primarily in this paper is called the Exponential Riordan array, which is a generalization of the Riordan array. It is again defined using a pair of power series \((d(t),h(t))\), but this time exponential power series: + +\[d(t)=\sum_{i=0}^{\infty}d_{i} \frac{t^{i}}{i!} ,d_{0}\neq 0,d_{i}\in\mathbb{R},\] + +\[h(t)=\sum_{i=1}^{\infty}h_{i} \frac{t^{i}}{i!} ,h_{1}\neq 0,h_{i}\in\mathbb{R}.\] + +From here on, we denote the exponential Riordan array defined with respect to exponential power series \(d(t)\text{and}h(t)\) as \(\mathcal{R}_{e}[d(t),h(t)]\). The \((n,k)-\)th element of \(\mathcal{R}_{e}[d(t),h(t)]\) is defined as + +\[d_{n,k}=\left[ \frac{t^{n}}{n!} \right]d(t) \frac{h(t)^{k}}{k!} .\] + +Note that the extraction of the coefficient for the term \(\frac{t^{n}}{n!}\) on an exponential power series has a nice interpretation as an operation of taking the \(n-\)th derivative of the series with respect to \(t\) followed by setting \(t=0\). For example, + +\[d_{2}=\left[ \tfrac{t^{2}}{2!} \right]d(t)= \tfrac{d}{dt}^{2} d(t)\big|_{t=0}.\] + +The FTRA with respect to \(\mathcal{R}_{e}[d(t),h(t)]\) and an exponential series \(c(t)=\sum_{i=0}^{\infty}c_{i} \frac{t^{i}}{i!}\) is defined as + +\[\sum_{k=0}^{\infty}d_{n,k}c_{k}=\sum_{k=0}^{n}d_{n,k}c_{k}=\left[ \frac{t^{n}}{n!} \right]d(t)c(h(t))\] (3.4) + +### 3.3 The Riordan group and the Sheffer group + +The expansion of the polynomial \(f_{\lambda}(\theta;\alpha)\) via the Sheffer A-type zero sequence (Sheffer, 1939) introduced in Section 3.1 is encompassed in the modern theory of umbral interpolation (Costabile et al., 2025) initiated by the work Roman and Rota (1978) and the book of Roman (1984) as the main reference for further developments. Consider a sequence of polynomials \(\{s_{i}(\theta)\}_{i=0}^{\infty}\) (where \(s_{i}(\theta)\) is of degree \(i)\) for a pair of characteristic exponential power series \((d(t),h(t))\) (overlap in notation with the previous section is intentional) with real-valued coefficients provided that \(d_{0}\neq 0,h_{0}=0\) diff --git a/outputs/arxiv_math/2503.05062_pg35_pg1_repeat1.md b/outputs/arxiv_math/2503.05062_pg35_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..7a022fbbd6aec467b2e1e91e9c0f58bc107748a7 --- /dev/null +++ b/outputs/arxiv_math/2503.05062_pg35_pg1_repeat1.md @@ -0,0 +1,51 @@ +[SSB09] + +Georg Schmidt, Vladimir R Sidorenko, and Martin Bossert. Collaborative decoding of interleaved reed–solomon codes and concatenated code designs. IEEE Transactions on Information Theory, 55(7):2991–3012, 2009. + +[SSB10] + +Georg Schmidt, Vladimir R Sidorenko, and Martin Bossert. Syndrome decoding of reed–solomon codes beyond half the minimum distance based on shift-register synthesis. IEEE Transactions on Information Theory, 56(10):5245–5252, 2010. + +[Sti09] + +Henning Stichtenoth. Algebraic function fields and codes, volume 254. Springer Science & Business Media, 2009. + +[Sud97] + +Madhu Sudan. Decoding of Reed Solomon codes beyond the error-correction bound. Journal of complexity, 13(1):180–193, 1997. + +[SW99] + +Mohammad Amin Shokrollahi and Hal Wasserman. List decoding of algebraic-geometric codes. IEEE Transactions on Information Theory, 45(2):432–437, 1999. + +[Tam24] + +Itzhak Tamo. Tighter list-size bounds for list-decoding and recovery of folded Reed-Solomon and multiplicity codes. IEEE Transactions on Information Theory, 70(12):8659–8668, 2024. + +[VH97] + +Conny Voss and T Hoholdt. An explicit construction of a sequence of codes attaining the tsfasman-vladut-zink bound. the first steps. IEEE Transactions on Information Theory, 43(1):128–135, 1997. + +[VZGG13] Joachim Von Zur Gathen and Jürgen Gerhard. Modern computer algebra. Cambridge University Press, 2013. + +Bin Wang, Haibin Kan, and Kenneth W. Shum. Hermitian codes in distributed storage systems + +[WKS15] with optimal error-correcting capacity. In 2015 IEEE International Symposium on Information Theory (ISIT), pages 601–605, 2015. + +Yingquan Wu. Novel burst error correction algorithms for Reed-Solomon codes. IEEE Trans- + +[Wu12] actions on information theory, 58(2):519–529, 2012. + +[YLLW01] Liuguo Yin, Jianhua Lu, K Ben Letaief, and Youshou Wu. Burst-error-correcting algorithm for Reed-Solomon codes. Electronics Letters, 37(11):1, 2001. + +# Appendix A. The proof of Lemma 2.6 + +In this section, we will give a generalized version of Wu’s algorithm that can be applied to RS codes defined on a non-cyclic multipoint set. Let us first recap some notations in syndrome decoding. + +Assume \(\mathcal{P}=\{\xi,\xi\alpha,\ldots,\xi\alpha^{n-1}\}\in\mathbb{F}_{q}^{*}/\langle\alpha\rangle\) is a coset of cyclic group, where \(\alpha\in\mathbb{F}_{q}^{*}\) has order \(n\). The \(\text{RS}[n,k]\) that we concerned is defined as + +\[\text{RS}[n,k]=\{\left(f(\xi),f(\xi\alpha),\ldots,f(\xi\alpha^{n-1})\right)|f(x)\in\mathbb{F}_{q}[x]_{n. \end{cases}\] diff --git a/outputs/arxiv_math/2503.05068_pg47_pg1_repeat1.md b/outputs/arxiv_math/2503.05068_pg47_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..5326c528e03c62696982b579f2b0602dd810fbb0 --- /dev/null +++ b/outputs/arxiv_math/2503.05068_pg47_pg1_repeat1.md @@ -0,0 +1,19 @@ +From [22, Section 3], for every \(\theta\), we have + +\[\mathbb{E}[\Vert U(\boldsymbol{x},k\Delta t;\theta)-U_{n}(\boldsymbol{x},k\Delta t;\theta)\Vert^{2}]\leq K_{k\Delta t,U(\cdot,0),\theta}\lambda_{N+1}^{-1}(\theta),\] (D.15) + +where \(K_{k\Delta t,U(\cdot,0),\theta}\) is a constant depending on time \(k\Delta t\) and the initial condition \(U(\cdot,0),\text{and}\theta\). Without loss of generality, we assume that \(K_{k\Delta t,U(\cdot,0),\theta}\) is non-decreasing in \(k\) (otherwise we can replace \(K_{k\Delta t,U(\cdot,0),\theta}\text{with}\tilde{K}_{k\Delta t,U(\cdot,0),\theta}:= \max_{1\leq i\leq k}K_{i\Delta t,U(\cdot,0),\theta}\). Given the initial condition \(U(\boldsymbol{x},0)\text{and}P_{n}U(\boldsymbol{x},0)\), we denote the probability measures of \(U(\boldsymbol{x},k\Delta t;\theta)\text{and}U_{n}(\boldsymbol{x},T;\theta)\) by \(\nu_{U(\cdot,0)}(k\Delta t)\) and \(\nu_{n,U_{n}(\cdot,0)}(k\Delta t)\), respectively. Moreover, the joint probability measure of \(\left(U(\boldsymbol{x},k\Delta t;\theta),U_{n}(\boldsymbol{x},k\Delta t;\theta)\right)\) has marginal distributions \(\nu_{U(\cdot,0)}(k\Delta t)\) and \(\nu_{n,U_{n}(\cdot,0)}(k\Delta t)\), respectively. From Eq. (D.15), we can deduce that: + +\[\begin{aligned} W_{2}^{2}(\nu_{U(\cdot,0)}(k\Delta t),\nu_{n,U_{n}(\cdot,0)}(k\Delta t))&\leq\mathbb{E}[\Vert U(\boldsymbol{x},k\Delta t;\theta)-U_{n}(\boldsymbol{x},k\Delta t;\theta)\Vert^{2}] \\ &\leq\sup_{\theta,U(\boldsymbol{x},0)}K_{T,U(\cdot,0),\theta}\lambda_{N+1}^{-1}(\theta). \end{aligned}\] + +Furthermore, using the definition of the local squared \(W_{2}\) distance in Eq. (2.4), we have: + +\[\begin{aligned} W_{2,\delta}^{2,\mathrm{e}}\left(U(\cdot,k\Delta t;\theta),U_{n}(\cdot,k\Delta t;\theta)\right)&\leq\sup_{\theta)}\mathbb{E}[\Vert U(\boldsymbol{x},k\Delta t;\theta)-U_{n}(\boldsymbol{x},k\Delta t;\theta)\Vert^{2}] \\ &\leq\sup_{\theta,U(\boldsymbol{x},0)}^{,U(\boldsymbol{x},0}K_{T,U(\cdot,0),\theta}\lambda_{N+1}^{-1}(\theta). \end{aligned}\] + +Similarly, we can conclude that: + +\[W_{2,\delta}^{2,\mathrm{e}}\left(\hat{U}(\cdot,k\Delta t;\hat{\theta}),\hat{U}_{n}(\cdot,k\Delta t;\hat{\theta})\right)\leq\sup_{\hat{\theta},U(\boldsymbol{x},0)}K_{T,U(\cdot,0),\theta}\lambda_{N+1}^{-1}(\hat{\theta}).\] (D.18) + +Given the same initial condition \(U(\boldsymbol{x},0)=\hat{U}(\boldsymbol{x},0)\), using the triangle inequality of the Wasserstein distance [14], we have + +\[\begin{aligned} W_{2,\delta}^{2,\mathrm{e}}(U(\boldsymbol{x},t;\theta),\hat{U}(\boldsymbol{x},t;\hat{\theta}))&\leq3W_{2,\delta}^{2,\mathrm{e}}\left(U(\cdot,k\Delta t;\theta),U_{n}(\cdot,k\Delta t;\theta)\right) \\ +3W_{2,\delta}^{2,\mathrm{e}}\left(\hat{U}(\cdot,k\Delta t;\hat{\theta}),\hat{U}_{n}(\cdot,k\Delta t;\hat{\theta})\right)+3W_{2}^{2,\mathrm{e}}(U_{n}(\boldsymbol{x},t;\theta),\hat{U}_{n}(\boldsymbol{x},t;\hat{\theta}))\big) \\ &\leq3W_{2,\delta}^{2,\mathrm{e}}(U_{n}(\boldsymbol{x},t;\theta),\hat{U}_{n}(\boldsymbol{x},t;\hat{\theta}))+3\sup_{\theta,U(\boldsymbol{x},0)}^{,\delta}K_{T,U(\cdot,0),\theta}\lambda_{N+1}^{-1}(\theta) \\ +3\sup_{\hat{\theta},U(\boldsymbol{x},0)}K_{T,U(\cdot,0),\hat{\theta}}\lambda_{N+1}^{-1}(\hat{\theta}). \end{aligned}\] diff --git a/outputs/arxiv_math/2503.05089_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.05089_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..78e33fd6b972be9d07e56ec025788b32b9fcaad4 --- /dev/null +++ b/outputs/arxiv_math/2503.05089_pg3_pg1_repeat1.md @@ -0,0 +1,21 @@ +be robust enough to generalise to hypergraphs that miss a significant proportion of the edges. Moreover, the proof of the graph case \((r=2)\) from [18] relies on a good understanding of matchings in graphs in form of the Tutte–Berge formula, no analogue of which is available for hypergraphs. Finally, one could try to prove the existence of certain small coloured configurations (“gadgets”) that can be repeatedly removed until the remaining hypergraph has a very specific structure. For instance, in the case of two colours, in [17] it is implicitly proved that either there exist two edges of different colour that share \(r-1\) vertices, or the colouring is almost monochromatic. For general \(q\), one possible gadget would be a set of \(r+q-1\) vertices that contains an edge of each colour. However, there are constructions where all colour classes are large yet there is no such gadget, so this approach seems infeasible as well. We instead follow a new approach and make use of tools from extremal set theory developed in the study of the Erdős matching conjecture (see the overview in Section 2 for more details). Theorem 1.2 also has implications to the discrepancy of perfect matchings in random hypergraphs. The general question in discrepancy theory is whether, given a ground set \(\Omega\), a family + +\(\mathcal{P}\subseteq2^{\Omega}\) and a positive integer \(q\geq2\), there exists a \(q-\)colouring of \(\Omega\) such that each set in \(\mathcal{P}\) contains roughly the same number of elements of each colour. In the setting of (hyper)graphs, the main interest is finding conditions on a hypergraph \(G\) under which any \(q-\text{edge}-\)colouring of \(G\) contains a particular substructure with high discrepancy, meaning that significantly more than a \(1/q-\)proportion of the substructure’s edges are in the same colour. The question goes back to works of Erdős and Spencer [12], and of Erdős, Füredi, Loebl and Sós [11]. Recently, this has been extensively studied for minimum degree conditions forcing perfect matchings with high discrepancy in hypergraphs (see [3, 17, 20, 28]), with the graph case having been considered earlier (see [2, 15, 19]). H`an, Lang, Marciano, Pavez-Signé, Sanhueza-Matamala, Treglown and Zárate- + +Guerén [20, Section 5.3] showed that there exists a constant \(C>0\text{such that if}p\geq C \sqrt{n^{2-r}}\), + +then w.h.p. in every \(q-\)colouring of the edges of \(\mathbb{G}^{(r)}(n,p)\) there is a perfect matching with high discrepancy. However, due to the celebrated result of Johansson, Kahn, and Vu [22], we know + +that a perfect matching already typically exists for \(p\gg n^{-r+1}\log n\) (cf. Theorem 3.5). A natural question, which was asked in [20], is to determine the correct threshold (depending on \(r,q)\) for the property of having a perfect matching with high discrepancy in every \(q-\text{colouring}\). Motivated by this, we prove the following result, which starts working at the threshold for the existence of a perfect matching, and even yields an asymptotically optimal bound on the discrepancy. However, note that we assume the colouring is known a priori, that is, the statement we prove to hold w.h.p. is for a fixed colouring. Theorem 1.4. For all \(r,q\in\mathbb{N}\text{with}r,q\geq2,\text{and all}\mu>0\), there exists \(C>0\) such that, + +provided \(p\geq Cn^{-r+1}\log n\), the following holds. For any \(q-\)colouring of the \(k-\)subsets of \([n]\), the + +\[\frac{n}{r+q-1}\] + +random hypergraph \(G\sim\mathbb{G}^{(r)}(n,p)\) contains w.h.p. a perfect matching with at least \((1-\mu)\) edges of the same colour. + +Organisation. The rest of the paper is organised as follows. The proof of the defect theorem (Theorem 1.3) is discussed in Section 2 and the proof of the transference theorem (Theorem 1.2) in Section 3. The proof of Theorem 1.4 is also discussed there. Finally, we give concluding remarks in Section 4. The proof of the multicolour weak sparse hypergraph regularity lemma is added for completeness in Appendix A. + +Notation. We use standard graph theory notation. We let \(K_{n}^{(r)}\) denote the complete \(r-\)graph on \(n\) vertices. + +We let \([n]\) denote the set \(\{1,\ldots,n\}\) and, given a set \(X\) and an integer \(i\geq0\), we write \(\left(_{i}^{X}\right)\) for the collection of all subsets of \(X\) of size \(i\). For \(a,b,c\in(0,1]\), we write \(a\ll b\ll c\) in our statements to mean that there are increasing functions \(f,g:(0,1]\rightarrow(0,1]\text{such that whenever}a\leq f(b)\text{and}b\leq g(c)\), then the subsequent result holds. diff --git a/outputs/arxiv_math/2503.05104_pg18_pg1_repeat1.md b/outputs/arxiv_math/2503.05104_pg18_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..8cf1811954b45732a487e109d4425ca31e422261 --- /dev/null +++ b/outputs/arxiv_math/2503.05104_pg18_pg1_repeat1.md @@ -0,0 +1,91 @@ +10150 + +U0 U1 + +10100 + +1050 + +| 100 0 0.2 | 0.4 0.6 | 0.8 1 | 1.2 1.4 | 0 | 0.1 0.2 | 0.3 0.4 0.5 0.6 0.7 0.8 | 0.9 1 | 10-3 0 | 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 | 1 | +| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | +| | Time | | 10-4 | | | Time | 10-3 | | Time | 10-3 | +| 0.09 | | | | 9 | 10-3 | | | 5 10-3 | | | +| 0.08 | | | | 8 | | | | | | U0 | +| 0.07 | | | | 7 | | | | | | U1 | +| 0.06 0.05 | | | U0 | 6 | | | U0 | | | | +| | | | U1 | 5 | | | U1 | 4.5 | | | +| 0.04 | | | | 4 | | | | | | | +| 0.03 | | | | 3 | | | | | | | +| | | | | | | | | 4 | | | +| 0.02 | | | | 2 | | | | | | | + +0.01 + +0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 + +0 + +1 10-3 + +Time + +U0 U1 + +10-1 + +10-2 + +0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 + +0 + +0.9 + +1 10-3 + +Time + +U0 U1 + +10-2 + +3.5 + +0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 + +0 + +1 10-3 + +Time + +Fig. 8: The first row: relative errors by explicit L1 scheme; method. For each column, we take \(\alpha=0.3,0.6,0.9\) form left + +### 5.3 A more complicated media + +The second row: relative errors by EI to right. + +also represent the reference solution in Figure 11 + +In our final example, we take nonperiodic media \(\kappa\) as shown in the left of Figure 9. The other settings are the same as the previous semilinear case. We snapshots in Figure 10 with \(\alpha=0.6\). By the relative errors findings has been observed as in the previous cases. The errors of the numerical solutions is slightly larger compared to the previous examples, but they still converge to the reference solution. + +![](figure) + +![](figure) + +| 1 | 1 | | +| --- | --- | --- | +| 0.9 | 0.9 | 0.9 | +| 0.8 | 0.8 | 0.8 | +| 0.7 | 0.7 | 0.7 | +| 0.6 | 0.6 | 0.6 | +| 0.5 | 0.5 | 0.5 | +| 0.4 | 0.4 | 0.4 | +| 0.3 | 0.3 | 0.3 | +| 0.2 | 0.2 | 0.2 | +| 0.1 | 0.1 | 0.1 | +| 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 | 1 0 0 0.2 0.4 | 0.6 0.8 1 | + +Fig. 9: Left: \(\kappa\); Right: \(f_{0}\). + +and Table 3, the similar diff --git a/outputs/arxiv_math/2503.05135_pg5_pg1_repeat1.md b/outputs/arxiv_math/2503.05135_pg5_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..29bd7f6c1187f1ba9c0bdb5d27bcfb2ad8473844 --- /dev/null +++ b/outputs/arxiv_math/2503.05135_pg5_pg1_repeat1.md @@ -0,0 +1,25 @@ +We have the following subsequent corollary. Corollary 1. Let \(C^{\sigma}\) be an unbalanced signed cycle, then \(p^{+}(C^{\sigma})\geq2\). Yu et al. [18] calculated the positive inertia of signed path \(P_{n}^{\sigma}\) . Lemma 6. Let \(P_{n}^{\sigma}\) be a signed path of order \(n,\text{then}p^{+}(P_{n}^{\sigma})=\lfloor \frac{n}{2} \rfloor\). Yu et al. [4, Theorem 3.1] characterized all connected signed graph \(G^{\sigma}\text{with}p^{+}(G^{\sigma})=1\). Lemma 7. Let \(G^{\sigma}\) be a connected signed graph. Then \(p^{+}(G^{\sigma})=1\) if and only if \(G^{\sigma}\) is a balanced complete multipartite signed graph. The length of a signed path (or simply a path) \(P^{\sigma}\) refers to the total number of edges in \(P^{\sigma}\). For any two vertices \(y\text{and}z\), the distance between them, denoted by \(d(y,z)\), is defined as the length of the shortest path between \(y\text{and}z\). Lemma 8. Let \(G^{\sigma}\) be a connected signed graph with girth \(g_{r}\) and let \(C^{\sigma}\) be a shortest cycle in \(G^{\sigma}\). If \(y,y^{\prime}\in V(C^{\sigma})\) and there exists a path \(P^{\sigma}\) of length \(k\) from \(y\) to \(y^{\prime}\) satisfying \((V(P^{\sigma})\setminus\{y,y^{\prime}\})\cap V(C^{\sigma})=\emptyset\), then + +\[\left\lceil \frac{g_{r}}{2} \right\rceil\leq k.\] + +Proof. Remember that \(C^{\sigma}\) have two different paths from \(y\) to \(y^{\prime}\). The total length of both the paths equals the length of \(C^{\sigma}\), i.e., \(g_{r}\). Thus, the shorter length of these paths is at most \(\left\lfloor \frac{g}{2} \right\rfloor\). The shorter of these two paths from \(y\) to \(y^{\prime}\), followed by the path \(P^{\sigma}\), forms a cycle of length at most \(\left\lfloor \frac{g_{r}}{2} \right\rfloor+k\). Therefore, we obtain: + +and consequently + +This concludes the proof. + +\[\begin{aligned} \left\lfloor \frac{g_{r}}{2} \right\rfloor+k&\geq g_{r}, \\ \left\lceil \frac{g_{r}}{2} \right\rceil&\leq k. \end{aligned}\] + +Let \(G^{\sigma}[H^{\sigma}]\) be a subgraph of \(G^{\sigma}\) induced by \(H^{\sigma}\), and let \(x\) be a vertex outside \(H^{\sigma}\). We denote the distance between \(x\text{and}G^{\sigma}[H^{\sigma}]\text{as}:d(x,G^{\sigma}[H^{\sigma}])=\min\{d(x,y)|y\in H^{\sigma}\}\). Next, we define \(N_{j}(G^{\sigma}[H^{\sigma}])\) as: + +\[N_{j}(G^{\sigma}[H^{\sigma}])=\{x\in V(G^{\sigma})\setminus H^{\sigma}|d(x,G^{\sigma}[H^{\sigma}])=j,j=1,2,\ldots,n\}.\] + +The number of vertices in \(N_{j}(G^{\sigma}[H^{\sigma}])\) is denoted by \(|N_{j}(G^{\sigma}[H^{\sigma}])|\). Lemma 9. Let \(G^{\sigma}\) be a connected signed graph with girth \(g_{r}\), and let \(C^{\sigma}\) be a shortest cycle in \(G^{\sigma}\). If \(p^{+}(C^{\sigma})=p^{+}(G^{\sigma}),\text{then}N_{j}(C^{\sigma})=\emptyset\text{for}j\geq2\). Proof. Indeed, we only need to establish that \(N_{2}(C^{\sigma})=\emptyset\). Suppose, for the sake of contradiction, that \(N_{2}(C^{\sigma})\neq \emptyset\). Let \(y^{\prime}\in N_{2}(C^{\sigma})\text{and}y^{\prime}\sim y\in N_{1}(C^{\sigma}).\text{Then},y^{\prime}\) is a pendant vertex of \(G^{\sigma}[V(C^{\sigma})\cup\{y,y^{\prime}\}]\). Consequently, + +\[p^{+}(G^{\sigma})\geq p^{+}(G^{\sigma}[V(C^{\sigma})\cup\{y,y^{\prime}\}])=p^{+}(C^{\sigma})+1>p^{+}(C^{\sigma}),\] + +by Lemmas 1 and 2, which contradicts the assumption that \(p^{+}(C^{\sigma})=p^{+}(G^{\sigma})\). Hence, \(N_{2}(C^{\sigma})=\emptyset\), and therefore, \(N_{j}(C^{\sigma})=\emptyset\text{for}j\geq3\). This completes the proof. + +Smith [6] showed the following characterization. + +Lemma 10. A graph has precisely one positive eigenvalue if and only if the set of its non-isolated vertices constitute a complete multipartite graph. diff --git a/outputs/arxiv_math/2503.05138_pg21_pg1_repeat1.md b/outputs/arxiv_math/2503.05138_pg21_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..60ae38686413eff5d8d0fcafa522b05b26a982c8 --- /dev/null +++ b/outputs/arxiv_math/2503.05138_pg21_pg1_repeat1.md @@ -0,0 +1,45 @@ +which is rewritten as + +\[\begin{aligned} m_{A}\Vert u-u^{h}\Vert_{V}^{2}&\leq\langle Au-Au^{h},u-v^{h}\rangle+\langle Au,v^{h}-u\rangle+\langle Au,v-u^{h}\rangle \\ +\langle Au,u-v\rangle+\langle Au^{h},u^{h}-v^{h}\rangle. \end{aligned}\] (4.40) + +Applying (4.2), + +\[\langle Au,u-v\rangle\leq\Phi(v)-\Phi(u)+I_{\Delta}(\psi^{0}(\gamma_{\psi}u;\gamma_{\psi}v-\gamma_{\psi}u))-\langle f,v-u\rangle.\] + +Applying (4.16), + +\[\langle Au^{h},u^{h}-v^{h}\rangle\leq\Phi(v^{h})-\Phi(u^{h})+I_{\Delta}(\psi^{0}(\gamma_{\psi}u^{h};\gamma_{\psi}v^{h}-\gamma_{\psi}u^{h}))-\langle f,v^{h}-u^{h}\rangle.\] + +Using these inequalities in (4.40), after some rearrangement of the terms, we have + +\[m_{A}\Vert u-u^{h}\Vert_{V}^{2}\leq\langle Au-Au^{h},u-v^{h}\rangle+R_{u}(v^{h},u)+R_{u}(v,u^{h})+I_{\psi}(v,v^{h}),\] (4.41) + +where + +\[R_{u}(v,w):=\langle Au,v-w\rangle+\Phi(v)-\Phi(w)+I_{\Delta}(\psi^{0}(\gamma_{\psi}u;\gamma_{\psi}v-\gamma_{\psi}w))-\langle f,v-w\rangle,\] (4.42) + +\[\begin{aligned} I_{\psi}(v,v^{h}):&=I_{\Delta}(\psi^{0}(\gamma_{\psi}u;\gamma_{\psi}v-\gamma_{\psi}u)+\psi^{0}(\gamma_{\psi}u^{h};\gamma_{\psi}v^{h}-\gamma_{\psi}u^{h})) \\ -I_{\Delta}(\psi^{0}(\gamma_{\psi}u;\gamma_{\psi}v^{h}-\gamma_{\psi}u)+\psi^{0}(\gamma_{\psi}u;\gamma_{\psi}v-\gamma_{\psi}u^{h})). \end{aligned}\] (4.43) + +Let us bound the first and the last two terms on the right hand side of (4.41). First, + +\[\langle Au-Au^{h},u-v^{h}\rangle\leq L_{A}\Vert u-u^{h}\Vert_{V}\Vert u-v^{h}\Vert_{V}.\] + +By the modified Cauchy-Schwarz inequality (2.12), for any \(\epsilon>0\) arbitrarily small, + +\[\langle Au-Au^{h},u-v^{h}\rangle\leq\epsilon\Vert u-u^{h}\Vert_{V}^{2}+c\Vert u-v^{h}\Vert_{V}^{2}\] (4.44) + +for some constant \(c\) depending on \(\epsilon\). Applying the subadditivity of the generalized directional derivative, + +\[\psi^{0}(z;z_{1}+z_{2})\leq\psi^{0}(z;z_{1})+\psi^{0}(z;z_{2})\forall z,z_{1},z_{2}\in\mathbb{R}^{m},\] + +we have + +\[\begin{aligned} \psi^{0}(\gamma_{\psi}u;\gamma_{\psi}v-\gamma_{\psi}u)&\leq\psi^{0}(\gamma_{\psi}u;\gamma_{\psi}v-\gamma_{\psi}u^{h})+\psi^{0}(\gamma_{\psi}u;\gamma_{\psi}u^{h}-\gamma_{\psi}u), \\ \psi^{0}(\gamma_{\psi}u^{h};\gamma_{\psi}v^{h}-\gamma_{\psi}u^{h})&\leq\psi^{0}(\gamma_{\psi}u^{h};\gamma_{\psi}v^{h}-\gamma_{\psi}u)+\psi^{0}(\gamma_{\psi}u^{h};\gamma_{\psi}u-\gamma_{\psi}u^{h}). \end{aligned}\] + +Thus, + +\[\begin{aligned} I_{\psi}(v,v^{h})&\leq I_{\Delta}(\psi^{0}(\gamma_{\psi}u^{h};\gamma_{\psi}v^{h}-\gamma_{\psi}u)-\psi^{0}(\gamma_{\psi}u;\gamma_{\psi}v^{h}-\gamma_{\psi}u)) \\ +I_{\Delta}(\psi^{0}(\gamma_{\psi}u;\gamma_{\psi}u^{h}-\gamma_{\psi}u)+\psi^{0}(\gamma_{\psi}u^{h};\gamma_{\psi}u-\gamma_{\psi}u^{h})). \end{aligned}\] + +By (4.8) and (4.12), + +\[\begin{aligned} I_{\Delta}(\psi^{0}(\gamma_{\psi}u;\gamma_{\psi}u^{h}-\gamma_{\psi}u)+\psi^{0}(\gamma_{\psi}u^{h};\gamma_{\psi}u-\gamma_{\psi}u^{h}))&\leq\alpha_{\psi}I_{\Delta}(|\gamma_{\psi}u-\gamma_{\psi}u^{h}|^{2}) \\ &\leq\alpha_{\psi}c_{\Delta}^{2}\Vert u-u^{h}\Vert_{V}^{2}. \end{aligned}\] diff --git a/outputs/arxiv_math/2503.05140_pg22_pg1_repeat1.md b/outputs/arxiv_math/2503.05140_pg22_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..88223a73c4d103f7009f808230493d351f5f913c --- /dev/null +++ b/outputs/arxiv_math/2503.05140_pg22_pg1_repeat1.md @@ -0,0 +1,19 @@ +## Remark 4.2. For any \(\epsilon>0\), a computation yields + +\[\tilde{s}_{C,\epsilon}(p,q,r)= \begin{cases} - \tfrac{2}{q} +\epsilon, & \text{for}q\geq3p^{\prime}\text{and} \tfrac{1}{p} \leq \tfrac{1}{4} ; \\ \tfrac{2}{p} - \tfrac{2}{q} - \tfrac{1}{2} +\epsilon, & \text{for}q\geq3p^{\prime}\text{and} \tfrac{1}{4} < \tfrac{1}{p} < \tfrac{1}{2} - \tfrac{1}{q} ; \\ \tfrac{1}{p} - \tfrac{3}{q} +\epsilon, & \text{for}q\geq3p^{\prime}\text{and} \tfrac{1}{p} \geq \tfrac{1}{2} - \tfrac{1}{q} ; \\ \tfrac{3}{2p} - \tfrac{3}{2q} - \tfrac{1}{2} +\epsilon, & \text{for}p^{\prime} \tfrac{2}{3p} - \tfrac{1}{6} , & \text{for}q\geq3p^{\prime}; \\ \tfrac{1}{q} > \tfrac{1}{p} - \tfrac{1}{3} , & \text{for}p^{\prime} \tfrac{2}{p} -1, & \text{for}q\leq p^{\prime}. \end{cases}\] + +\[\Big(_{\frac{1}{q}}-\] + +Combining \(1\leq r0\), the region of \(( \frac{1}{p} , \frac{1}{q} )\) in Theorem 1.4 for this case is illustrated in Figure 5. Here, we assume \(1\leq r\leq\) 2,3 and the figures for other ranges of r can be constructed similarly. + +![A diagram showing points in the plane labeled by their coordinates (1/q, 1/p), including points O(0,0), (1/4,0), (1/2,1/6), (2/3,1/3), (1/r, 2-r/r), (1/r, 1/r), and (1,1). The axes are labeled 1/q and 1/p, and there are lines and dashed lines indicating regions labeled ω and ω=2.](figure)[^inferred] + +![A diagram showing points in the plane labeled by their coordinates (1/q, 1/p), including points O(0,0), (1/4,0), (1/2,1/6), (2/3,1/3), (1/r, 2-r/r), (1/r, 1/r), and (1,1). The axes are labeled 1/q and 1/p, and there are lines and dashed lines indicating regions labeled ω and ω=2.](figure)[^inferred] + +Figure 5: The region of boundedness in (1.4) for the case \(1\leq r0\). From \(\mathbb{E}X \frac{1}{k}\). Introduce the random variable \(N\) by + +\[N=\min\left\{n:\forall m\geq n, \frac{A(m)}{m} > \frac{1}{k} \right\}.\] + +Since \(\sum_{n=1}^{\infty}P(N=n)=1\), it follows that \(\exists n_{0}\text{such that}P(N=n_{0})>0\). We now distinguish two cases. Case 1: \(1\in SuppX\). We have + +\[P(N=n_{0}\text{for the sequence generated by}\{X_{i}\}_{i>n_{0}})>0,\] + +and since \(1\in SuppX\), we have + +\[P(X_{1}=X_{2}=\cdots =X_{n_{0}}=1)>0.\] + +Since these two events deal with distinct parts of the i.i.d. sequence \(\{X_{i}\}_{i>n_{0}}\), with positive probability both events occur simultaneously. Now, let us evaluate the Schnirelmann density \(\sigma(\{W_{i}\}_{i\geq1})\) for such sequences of gaps. We have + +\[\frac{A(n)}{n} \geq \begin{cases} 1 & \text{if}n\leq n_{0} \\ \frac{1}{2} & \text{if}n_{0}2n_{0} \end{cases}\] + +| | ≥ 1 2 | if n0 < n ≤ 2n0 , | +| --- | --- | --- | +| n | 1 | | +| | k | if n > 2n0 | + +where the first two inequalities follow from the choice of the first \(n_{0}\) gaps and the last one from the condition \(N=n_{0}\). Thus we get \(P\left(\sigma(W)\geq \frac{1}{k} \right)>0\) and Mann’s theorem implies asymptotic \(\leq k-\text{comp}-\) leteness with the same positive probability. + +\[\text{Case}2:1\not\in SuppX.\] diff --git a/outputs/arxiv_math/2503.05181_pg1_pg1_repeat1.md b/outputs/arxiv_math/2503.05181_pg1_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..c259ccbf96e42a98feb39d1b3ac0788fe24c7125 --- /dev/null +++ b/outputs/arxiv_math/2503.05181_pg1_pg1_repeat1.md @@ -0,0 +1,31 @@ +# A Gap Penalty Reformulation for Mathematical Programming with Complementarity Constraints: Convergence Analysis + +Kangyu Lin and Toshiyuki Ohtsuka + +Abstract—Our recent study [1] proposed a new penalty method to solve the mathematical programming with complementarity constraints (MPCC). This method reformulates the MPCC as a parameterized nonlinear programming (NLP) called gap penalty reformulation and solves a sequence of gap penalty reformulations with an increasing penalty parameter. This letter studies the convergence behavior of the new penalty method. We prove that it converges to a strongly stationary point of MPCC, provided that: (1) The MPCC linear independence constraint qualification holds; (2) The upper-level strict complementarity condition holds; + +(3) The gap penalty reformulation satisfies the second-order necessary conditions in terms of the second-order directional derivative. Since the strong stationarity is used to identify the local minimum of MPCC, our convergence analysis indicates that the new penalty method is capable of finding an MPCC solution. + +## A. Background + +## I. INTRODUCTION + +This study considers a special yet common class of nonlinear programming (NLP), known as mathematical programming with complementarity constraints (MPCC). Discrete variables defining logical decisions can be represented by continuous variables with complementarity constraints. Thus, MPCC can model various optimization problems with combinatorial characteristics originating from engineering and economics, such as process optimization with discrete events, trajectory optimization with contacts, and bilevel optimization [2]. However, complementarity constraints also pose significant challenges for solving the MPCC, where almost all constraint qualifications (CQs) are violated at any feasible point of MPCC. The lack of constraint regularity leads to two difficulties: First, MPCC solutions can not be characterized by the optimality conditions for a standard NLP problem; Second, MPCC cannot be solved using standard NLP solution methods. To characterize the MPCC solution, various MPCC-tailored concepts have been proposed [3], where some of which are reviewed in Section II-A. To solve the MPCC efficiently, many MPCC-tailored solution methods, such as relaxation methods and penalty methods, have been proposed (Chapter 11 in [2]). These methods do not solve the MPCC directly. Instead, they reformulate the complementarity constraints as parameterized costs or constraints, solve a sequence of well-defined parameterized NLP, and then present a rigorous convergence analysis showing that the sequence of solutions to the parameterized + +This work was supported in part by the JSPS KAKENHI Grant Number JP22H01510 and JP23K22780. Kangyu Lin was supported by the CSC scholarship (No. 201906150138). The authors are with the Systems Science Course, Graduate School of Informatics, Kyoto University, Kyoto, Japan. Corresponding author: Kangyu Lin. Email: k-lin@sys.i.kyoto-u.ac.jp, ohtsuka@i.kyoto-u.ac.jp. + +NLP can finally converge to an MPCC solution. These MPCCtailored solution methods are practical because they are easy to implement using state-of-the-art NLP software. Our recent study [1] proposed a new penalty method called gap penalty method, which reformulates the MPCC as a parameterized NLP called gap penalty reformulation and solves a sequence of gap penalty reformulations with an increasing penalty parameter. Compared to other penalty methods, gap penalty reformulation exhibits certain convexity structures that can be exploited by a dedicated Hessian regularization method. Gap penalty method has been shown to be practically effective in [1], but its theoretical convergence has not been analyzed. + +## B. Contribution + +This study builds upon the work in [1] by providing a rigorous convergence analysis for the gap penalty method. The contributions of this study are mainly in two aspects. First, since gap penalty reformulation involves a Lipschitz continuously differentiable function, we characterize its solutions using second-order directional derivatives rather than the Hessian matrix as in standard NLP theory. These results are critical for the subsequent convergence analysis. Second, we prove that the gap penalty method converges to a strongly stationary point of MPCC under certain standard MPCC-tailored assumptions. Our convergence analysis theoretically confirms that the gap penalty method is capable of finding an MPCC solution. + +## C. Outline and notation + +The remainder of this study is organized as follows: Section II reviews some MPCC-tailored concepts and the gap penalty reformulation; Section III presents the optimality conditions of the gap penalty reformulation; Section IV presents a detailed convergence analysis; and Section V concludes this study. Given two variables \(x,y\in\mathbb{R}^{n}\), we denote the element-wise complementarity conditions between \(x\text{and}y\) by \(0\leq x\perp y\geq0\), i.e., \(x,y\geq0\text{and}x\odot y=0,\text{with}\odot\) the Hadamard product. The complementarity problem is to find a pair \((x,y)\) that satisfies \(0\leq x\perp y\geq0\). Given a differentiable function \(f(x)\), we denote its Jacobian by \(\nabla_{x}f\in\mathbb{R}^{m\times n}\text{with}f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}\), and its Hessian by \(\nabla_{xx}f\in\mathbb{R}^{n\times n}\text{with}f:\mathbb{R}^{n}\rightarrow\mathbb{R}\). We say that function \(f(x)\) is \(k-\)th Lipschitz continuously differentiable \((LC^{k}\) in short) if its \(k-\)th derivative is Lipschitz continuous. + +## II. MATHEMATICAL PROGRAMMING WITH COMPLEMENTARITY CONSTRAINTS + +## A. MPCC-tailored concepts + +We review some MPCC-tailored concepts that are relevant to our subsequent convergence analysis. Consider the MPCC diff --git a/outputs/arxiv_math/2503.05183_pg15_pg1_repeat1.md b/outputs/arxiv_math/2503.05183_pg15_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..1422167fda63c93d70a6c715edc35b590304865e --- /dev/null +++ b/outputs/arxiv_math/2503.05183_pg15_pg1_repeat1.md @@ -0,0 +1,39 @@ +It is evident that G1 and G2 are Lipschitz continuous on any bounded set. From Lemma 2(3), we know that \(\mathcal{W}^{t}\) is bounded. Hence, we obtain + +\[\big\|V^{t+1}\big\|\leq\varpi\big\|\mathcal{W}^{t+1}-\mathcal{W}^{t}\big\|,\] (3.33) + +- where \(\varpi=\max_{i\in[6]}\{\rho_{i}+l_{g}\}\) and lg is the maximum Lipschitz constant of G1 and G2. + +Lemma 4 The function \(F(\mathcal{W})\) is a Kurdyka-Łojasiewicz (KL) function. + +Proof According to [5], the Frobenius norm functions \(\frac{\lambda_{1}}{2} \Vert B\Vert^{2}, \frac{\lambda_{3}}{2} \Vert\mathcal{H}-\mathcal{C}\times_{3}B-\mathcal{E}_{1}\Vert^{2},\text{and} \frac{\lambda_{6}}{2} \Vert\mathcal{C}-\mathcal{D}* \mathcal{Z}^{T}-\mathcal{E}_{2}\Vert^{2}\) are semi-algebraic functions. The terms \(\lambda_{2}\Vert\mathcal{E}_{1}\Vert_{11\phi}\text{and}\lambda_{5}\Vert\mathcal{E}_{2}\Vert_{11\phi}\) are also semi-algebraic functions because they are composed of semi-algebraic operations, including the Frobenius norm, minimum, and finite summation. Similarly, \(\lambda_{4}\Vert\mathcal{Z}\Vert_{F,1}^{\psi}\), which involves only the Frobenius norm and finite summation, is semi-algebraic. The functions \(\delta_{\mathbb{B}}(B),\delta_{\mathbb{C}}(\mathcal{C}),\text{and}\delta_{\mathbb{D}}(\mathcal{D})\) are semi-algebraic because they are indicator functions with semi-algebraic sets [5]. Hence, \(F(\mathcal{W})\) is semi-algebraic because it is a finite sum of semi-algebraic functions. Additionally, since \(F(\mathcal{W})\) is a proper continuous function, it follows from [5, Theorem 3] that \(F(\mathcal{W})\) is a KL function. This completes the proof. + +- + +Finally, we provide a theoretical guarantee for the convergence of Algorithm 1. + +Theorem 4 Consider the sequence \(\{\mathcal{W}^{t}\}\) obtained by Algorithm 1. Assuming that either \(\mathcal{Z}^{t}or\mathcal{E}_{2}^{t}\) is bounded, the sequence \(\{\mathcal{W}^{t}\}\) converges to a critical point of \(F(\mathcal{W})\). + +Proof We begin by noting that Lemma 2(3) establishes the boundedness of the sequence \(\{\mathcal{W}^{t}\}\). Given this boundedness, the Bolzano-Weierstrass theorem ensures the existence of a convergent subsequence. Furthermore, by exploiting the continuity of \(F(\mathcal{W})\), along with the results derived from Lemmas 2, 3, □ + +4, we can rigorously establish the desired conclusion as articulated in [2, Theorem 2.9]. + +## 3.7. Rank reduction strategy with validation mechanism + +In this subsection, we propose a rank reduction strategy to decrease r, which is related to the dimensions of \(\mathcal{D}\in\mathbb{R}^{n_{1}\times r\times b}\text{and}\mathcal{Z}\in\mathbb{R}^{n_{2}\times r\times b}\), thereby reducing the complexity of Algorithm 1. Unlike most decomposition methods that set a small initial r0, we set \(r^{0}=\min\{n_{1},n_{2}\}\) and leverage the algorithm’s inherent capability to adaptively reduce rt. This approach is initially supported by Theorem + +2. Furthermore, we present the following theorem to substantiate our proposed rank reduction strategy. + +# Theorem 5 + +There exists t# \(\in\mathbb{N}\text{such that}\Gamma(\mathcal{Z}^{t})=\Gamma(\mathcal{Z}^{t+1})\) for each \(t\geq\) t#, where \(\Gamma(\mathcal{Z})= \{j|\Vert\mathcal{Z}(:,j,:)\Vert\neq 0,j=1,\ldots,r\}\). + +Proof By Lemma 2 that \(\lim_{t\rightarrow\infty}\Vert\mathcal{Z}^{t+1}-\mathcal{Z}^{t}\Vert=0\), there exists t# \(\in\mathbb{N}\text{such that}\Vert\mathcal{Z}^{t+1}-\mathcal{Z}^{t}\Vert< \min\{(2\hat{\lambda}_{4}(1-p))^{1/(2-p)},\nu\}\) for each \(t\geq\) t#. Proving by contradiction, we assume that there exists \(t\geq\) t# such that \(\Gamma(\mathcal{Z}^{t})\neq \Gamma(\mathcal{Z}^{t+1})\). Then there exists \(j\in r\) such that (a) \(\mathcal{Z}^{t}(:,j,:)\neq 0\text{and}\mathcal{Z}^{t+1}(:,j,:)=0\), or + +(b) \(\mathcal{Z}^{t}(:,j,:)=0\text{and}\mathcal{Z}^{t+1}(:,j,:)\neq 0\). Hence by (3.19), we have for both cases that + +\[\big\|\mathcal{Z}^{t+1}-\mathcal{Z}^{t}\big\|\geq\big\|\mathcal{Z}^{t+1}(:,j,:)-\mathcal{Z}^{t}(:,j,:)\big\|\geq\min\{(2\hat{\lambda}_{4}(1-p))^{1/(2-p)},\nu\},\] + +□ + +which yields a contradiction. The proof is complete. diff --git a/outputs/arxiv_math/2503.05218_pg5_pg1_repeat1.md b/outputs/arxiv_math/2503.05218_pg5_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..58c6eb90dd80d7dc9e1e5839ed61d2aa41d65462 --- /dev/null +++ b/outputs/arxiv_math/2503.05218_pg5_pg1_repeat1.md @@ -0,0 +1,25 @@ +Remark: The condition in (5) is best possible. For instance, there is a tripartite graph \(G\) with densities \(\alpha=1-1/n+\alpha^{\prime}n^{(\delta_{a}-2)},\beta=1-1/n+\beta^{\prime}n^{(\delta_{b}-2)},\gamma=1-1/n+\gamma^{\prime}n^{(\delta_{c}-2)}\) satisfying \(\alpha^{\prime}\beta^{\prime}n^{(\delta_{a}+\delta_{b}-1)}+\gamma^{\prime}n^{(\delta_{c}-1)}=1\) such that \(G\) does not have a triangle-factor. To see this, consider a tripartite graph \(G\) with parts \(A,B\) and \(C\). Denote the edge densities of \(G[A,B],G[A,C]\text{and}G[B,C]\) by \(\alpha,\beta\text{and}\gamma\), respectively. Let \(x\in A\), \(B_{1}\) be a subset of \(B\text{such that}|B_{1}|=\alpha n^{2}-n(n-1)\text{and}C_{1}\) be a subset of \(C\text{such that}|C_{1}|=\beta^{2}-n(n-1)\). Define the edge set of \(G\) to be the set of all edges between \(A\setminus\{x\}\text{and}B,A\setminus\{x\}\text{and}C,\{x\}\text{and}B_{1}\), \(\{x\}\text{and}C_{1}\). + +The rest of the paper is organized as follows. In Section 2, we will introduce notions and calculations needed for our proofs. Section 3 is devoted to prove our main result. In section 4, we will prove the triangle-factor case. + +## 2 Preliminaries + +Let + +\[R=\{(\alpha,\beta,\gamma)\in[0,1]^{3}:\alpha\beta+\gamma>1,\alpha\gamma+\beta>1,\beta\gamma+\alpha>1\},\] + +and + +\[\Delta(\alpha,\beta,\gamma)=\alpha^{2}+\beta^{2}+\gamma^{2}-2\alpha\beta-2\alpha\gamma-2\beta\gamma+4\alpha\beta\gamma.\] + +\(R_{1}\text{and}R_{2}\) is a partition of \(R\) such that + +\[R_{1}=\{(\alpha,\beta,\gamma\in R:\Delta(\alpha,\beta,\gamma)\geq0)\}\] + +and \(R_{2}=R\setminus R_{1}\). Let \(T_{min}(\alpha,\beta,\gamma)\) denote the minimum number of triangles in tripartite graph with the stated densities, divided by \(n^{3}\). From [BJT10], we have the following: + +Theorem 2.1 ( Baber, Johnson and Talbot [BJT10]). If \((\alpha,\beta,\gamma)\in R_{1},\text{then}T_{min}(\alpha,\beta,\gamma)=\alpha+\beta+\gamma-2\). If \((\alpha,\beta,\gamma)\in R_{2},\text{then}T_{min}(\alpha,\beta,\gamma)\leq2 \sqrt{\alpha\beta(1-\gamma)} +2\gamma-2\). Conjecture 2.1 ( Baber, Johnson and Talbot [BJT10]). If \((\alpha,\beta,\gamma)\in R_{2},\text{then}T_{min}(\alpha,\beta,\gamma)=2 \sqrt{\alpha\beta(1-\gamma)} + 2\gamma-2\). For integers \(n>r\geq1\), let \([n]:=\{1,2,\cdots ,n\}\) be the standard \(n\) element set and \(\left(_{r}^{[n]}\right)=\{T\subseteq[n]\) : \(|T|=r\}\) be the collection of all its \(r-\text{subsets}\). An \(n-\text{vertex}r-\)uniform hypergraph \(H\) is a pair \(H=(V,E)\), where \(V:=[n]\text{and}E(H)\subset\left(_{r}^{[n]}\right)\). Let \(\nu(H)\) be the matching number of \(H\), that is, the maximum number of pairwise vertex-disjoint members of \(E(H)\). An \(r-\)uniform hypergraph \(H\) is called \(n-\text{balanced}r-\)partite if \(V(H)\) is partitioned into sets \(V_{1},V_{2},\ldots,V_{r}\text{such that}|V_{1}|=|V_{2}|=\cdots =|V_{r}|=n\) and each hyperedge meets every \(V_{i}\) in precisely one vertex. Aharoni and Howard [AH17] showed the tight hyperedge density needed to guarantee the existence of a matching of size \(k\) in an \(n-\text{balanced}r-\text{partite}r-\)uniform hypergraphs. (see Observation 1.9 in [AH17]) + +Lemma 2.2 (Aharoni and Howard [AH17]). If \(H\) is an \(n-\text{balanced}r-partiter-\)uniform hypergraph and \(e(H)>(k-1)n^{r-1}\text{then}\nu(H)\geq k\) and the bound is tight. + +For a graph \(G\text{and}U\subseteq V(G)\), we use \(G-U\) to denote the subgraph of \(G\) induced by \(V(G)\setminus U\), and we use \(G[U]\) to denote the subgraph of \(G\) induced by \(U\). For a graph \(G\text{and}E\subseteq E(G)\), we use \(H-E\) to denote the graph obtained from \(G\) by deleting \(E\). diff --git a/outputs/arxiv_math/2503.05232_pg22_pg1_repeat1.md b/outputs/arxiv_math/2503.05232_pg22_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..d784b8aae590dc634383923a31ff0bd362ad16c6 --- /dev/null +++ b/outputs/arxiv_math/2503.05232_pg22_pg1_repeat1.md @@ -0,0 +1,31 @@ +where the first inequalities are ensured by Theorem 2. However, the irreducibility condition \((\mathrm{H}\kappa_{irr})\) is not satisfied by \(\kappa^{red}\), so uniqueness is not ensured and Proposition 1 does not apply to guarantee that \(N_{1}^{red}\text{and}N_{2}^{red}\) are non-zero. Adapting the proof of Proposition 1 to each of the strongly connected components of the graph of \(\kappa\), simply reduced to \(\{v_{1}\}\) + +and \(\{v_{2}\}\), shows that for each \(i\in\{1,2\},N_{i}^{red}\) is either zero everywhere or positive on + +\(( \frac{b}{2} ,+\infty)\). The normalization condition + +\[\sum_{i\in\{1,2\}}\int_{0}^{\infty}N_{i}^{red}(x)dx=1\] + +implies that at least for one \(i\in\{1,2\},N_{i}^{red}\) is non-zero. Both \(N_{1}^{red}\text{and}N_{2}^{red}\) cannot be non-zero, otherwise Lemma 2 would work (in equality (14), we need the existence of + +\(i\neq j\text{such that}N_{i}^{red}\neq 0\text{and}N_{j}^{red}\neq 0)\) and we could use it to prove the long-time convergence of system (20) towards a stationary sate (fourth step of Theorem 4). That + +would contradict [6]. The only possibility is that \(N_{1}^{red}\equiv0\text{and}N_{2}^{red}(x)>0\text{for}x\geq \frac{b}{2}\) or the opposite and then \(\lambda=\lambda_{2}\text{or}\lambda=\lambda_{1}\) respectively. In the mixing case, the equations of system \((\text{GF}_{t,v})\) are coupled through their source term: + +\[\begin{cases} \big( & \big) \\ \tfrac{\partial}{\partial t} n_{1}(t,x)+v_{1} \tfrac{\partial}{\partial x} xn_{1}(t,x)+\gamma_{1}(x)n_{1}(t,x)=4\gamma_{2}(2x)n_{2}(t,2x), \\ \big( & \big) \\ \tfrac{\partial}{\partial t} n_{2}(t,x)+v_{2} \tfrac{\partial}{\partial x} xn_{2}(t,x)+\gamma_{2}(x)n_{2}(t,x)=4\gamma_{1}(2x)n_{1}(t,2x), \\ n_{1}(0,x)=n_{1}^{in}(x), & n_{2}(0,x)=n_{2}^{in}(x), \end{cases}\] + +and the irreducibility condition \((\mathrm{H}\kappa_{irr})\), missing in the non-mixing case, is now satisfied. We can thus apply successively Theorem 3, Proposition 1 and Theorem 4 to get eigenelements \((\lambda^{irr},N^{irr},\phi^{irr})\) such that + +\[\sum_{i=1}^{2}\int_{\mathbb{R}^{+}}\big|n_{i}(t,x)e^{-\lambda^{irr}t}-N_{i}^{irr}(x)\big|\phi_{i}^{irr}(x)dx_{t}-_{\rightarrow}\rightarrow_{\infty}0\] + +with \(\lambda^{irr}>0,\text{and for}i\in\{1,2\},N_{i}^{irr}\) positive on \(( \frac{b}{2} ,+\infty)\text{and}\phi_{i}^{irr}>0\text{on}(0,+\infty)\). These two simple cases illustrate that the existence result (Theorem 3) holds for every probability matrix \(\kappa\), in particular reducible ones. The irreducibility condition on \(\kappa\) comes into play to characterize the functions canceling the dissipation of entropy (Lemma 2), which then proves crucial to establish uniqueness of the steady state and convergence towards it. + +### 3.2 Numerical illustration + +Similarly to the previous subsection, we focus here on the special case of linear growth rates to illustrate the convergence result of Theorem 4. We numerically approximate and compare the long-time asymptotics in the presence and absence of mixing in feature. We choose \(M=3\) different features, namely \(\mathcal{V}=\{1,2,3\}\), and approximate on the grid + +\[\mathcal{S}_{N}:=\mathcal{V}\times\{x_{0},\ldots x_{2N}\},x_{m}:=2^{\frac{-m-N}{k}},m\in\{0,\ldots,2N\},k=200,N=2501,\] + +the time-evolution of the following initial data (taken identical for all features) + +\[n^{in}:(v,x)\mapsto Cx^{a}\mathrm{e}^{-bx^{2}},a=30,b=60,C\text{s.t.}\big\|n^{in}\big\|_{L^{1}(\mathcal{V}\times(0,x_{2N}))}=1,\] diff --git a/outputs/arxiv_math/2503.05267_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.05267_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..32db33ba8da787aa6e02614214765f6f953e40c8 --- /dev/null +++ b/outputs/arxiv_math/2503.05267_pg3_pg1_repeat1.md @@ -0,0 +1,17 @@ +problem becomes + +\[\begin{aligned} \begin{cases} \big( & \big) & \big( & \big) \\ u¤_{i}(t)-\nabla\cdot\alpha(t)\nabla u_{i}(t)+\nabla\cdot w(t)+\beta(t)u_{i}(t)=f_{i}(t) & \text{in}\Omega_{i}(t), \\ u_{i}(t)=0 & \text{on}\partial\Omega_{i}(t)\setminus\Gamma(t) \end{cases} \\ \{\text{for}i&=1,2, \\ \begin{cases} u_{1}(t)=u_{2}(t) & \text{on}\Gamma(t), \\ \alpha(t)\nabla u_{1}(t)\cdot\nu_{1}(t)+\alpha(t)\nabla u_{2}(t)\cdot\nu_{2}(t)=0 & \text{on}\Gamma(t), \end{cases} \end{aligned}\] (2) + +where \(\nu_{i}(t)\) is the unit outward normal vector of \(\partial\Omega_{i}(t),f_{i}(t)=f(t)|_{\Omega_{i}(t)},\text{and}u_{i}(t)= u(t)|_{\Omega_{i}(t)}\). The non-overlapping domain decompositions can then be derived by approximating the transmission problem. For example, consider the classic Robin–Robin method, first introduced in [27]. By taking linear combinations of the last two equations in (2), one has the equivalent Robin conditions + +\[\alpha(t)\nabla u_{1}(t)\cdot\nu_{i}(t)+s_{0}u_{1}(t)=\alpha(t)\nabla u_{2}(t)\cdot\nu_{i}(t)+s_{0}u_{2}(t)\text{on}\Gamma(t)\text{for}i=1,2,\] + +and a method parameter \(s_{0}>0\). Alternating between the subdomains then gives the Robin– Robin method as computing \((u_{1}^{n},u_{2}^{n})\text{for}n=1,2,\ldots\) with + +\[\begin{cases} u¤_{1}^{n}(t)-\nabla\cdot\alpha(t)\nabla u_{1}^{n}(t)+\nabla\cdot w(t)+\beta(t)u_{1}^{n}(t)=f_{1}(t) & \text{in}\Omega_{1}(t), \\ u_{1}^{n}(t)=0 & \text{on}\partial\Omega_{1}(t)\setminus\Gamma(t), \\ \alpha(t)\nabla u_{1}^{n}(t)\cdot\nu_{1}(t)+s_{0}u_{1}^{n}(t)= \\ \alpha(t)\nabla u_{2}^{n-1}(t)\cdot\nu_{1}(t)+s_{0}u_{2}^{n-1}(t) & \text{on}\Gamma(t), \\ \big( & \big) & \big( & \big) \end{cases} \begin{cases} u¤_{2}^{n}(t)-\nabla\cdot\alpha(t)\nabla u_{2}^{n}(t)+\nabla\cdot w(t)+\beta(t)u_{2}^{n}(t)=f_{2}(t) & \text{in}\Omega_{2}(t), \\ u_{2}^{n}(t)=0 & \text{on}\partial\Omega_{2}(t)\setminus\Gamma(t), \\ \alpha(t)\nabla u_{2}^{n}(t)\cdot\nu_{2}(t)+s_{0}u_{2}^{n}(t)= \\ \alpha(t)\nabla u_{1}^{n}(t)\cdot\nu_{2}(t)+s_{0}u_{1}^{n}(t) & \text{on}\Gamma(t). \end{cases} \{(3)\] + +Here, \(u_{2}^{0}\) is an initial guess and \(u_{i}^{n}(t)\) approximates \(u_{i}(t)=u(t)|_{\Omega_{i}(t)}\). Note that the Robin– Robin method is sequential, but the computation of each \(u_{i}^{n}\) can be implemented in parallel when \(\Omega_{i}(t)\) is a union of nonadjacent subdomains, as is the case in Figure 1. The well posedness of parabolic equations on evolving domains can be derived via the framework [3], which relies on a variational formulation where the standard Sobolev– Bochner solution space + +\[H^{1}\left((0,T);H^{-1}(\Omega)\right)\cap L^{2}\left((0,T);H_{0}^{1}(\Omega)\right)\] (4) + +is generalized to evolving domains \(\Omega(t)\). The framework has also been extended to a Banach space setting [2]. This variational setting constitutes the starting point of the design and analysis of a wide range of finite element methods for equations on evolving domains. The diff --git a/outputs/arxiv_math/2503.05270_pg21_pg1_repeat1.md b/outputs/arxiv_math/2503.05270_pg21_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..80f34776933ef517fd5c8d7785373c76a79d3499 --- /dev/null +++ b/outputs/arxiv_math/2503.05270_pg21_pg1_repeat1.md @@ -0,0 +1,23 @@ +\[\frac{n}{\log n} \exp\left\{- \frac{\alpha_{+}^{2}b}{2} \log n\right\}= \frac{n}{\log n} \exp\left\{- \frac{1}{2} \log n\right\}= \frac{\overline{n}}{\log n} \rightarrow\infty,\text{for}n\rightarrow\infty.\] + +Thus, \(\frac{n}{\log n} \exp\{-g_{n}\}\rightarrow\infty\text{for}n\rightarrow\infty\), which is a contradiction to (20), so the statement is true. + +For a kink Now we set \(\alpha_{+}=0,|\beta_{+}|\geq\delta_{0}\text{and}\psi_{n}=n^{1/3}\Big/\log^{1/3}n\). Then, it holds + +\[\begin{aligned} g_{n}&= \frac{1}{2} \sum_{i=1}^{bn^{2/3}\text{log}^{1/3}n}\left(\beta_{+} \frac{i}{n} \right)^{2}= \frac{\beta_{+}^{2}}{12n^{2}} (2b^{3}n^{2}\log n+3b^{2}n^{4/3}\log^{2/3}n+bn^{2/3}\log^{1/3}n) \\ &= \frac{\beta_{+}^{2}b^{3}\log n}{6} + \frac{\beta_{+}^{2}b^{2}\log^{2/3}n}{4n^{2/3}} + \frac{\beta_{+}^{2}b\log^{1/3}n}{12n^{4/3}} . \end{aligned}\] + +For \(b=\beta_{+}^{-2/3}\), we get + +and + +\[\exp\left\{- \frac{\beta_{+}^{2}b^{2}\log^{2/3}n}{4n^{2/3}} - \frac{\beta_{+}^{2}b\log^{1/3}n}{12n^{4/3}} \right\}\rightarrow1,\text{for}n\rightarrow\infty,\] + +\[\frac{n^{1/3}}{\log^{1/3}n} \exp\left\{- \frac{\beta_{+}^{2}b^{3}\log n}{6} \right\}= \frac{n^{1/3}}{\log^{1/3}n} \exp\left\{- \frac{1}{6} \log n\right\}= \frac{n^{1/6}}{\log^{1/3}n} \rightarrow\infty,\text{for}n\rightarrow\infty.\] + +Thus, \(\left(n^{1/3}\Big/\log^{1/3}\right)\exp\{-g_{n}\}\rightarrow\infty,\text{for}n\rightarrow\infty\), which is a contradiction to (20). + +## 6 Discussion + +We have studied the online detection of changes in segmented linear models with additive i.i.d. Gaussian noise. Our focus is on the minimax rate optimality in estimating the change point as well as computational and memory efficiency. We introduce the online detector FLOC, which offers several practical advantages, including ease of implementation as well as constant computational and memory complexity for every newly incoming data point — crucial attributes for effective online algorithms. From a statistical perspective, FLOC achieves minimax optimal rates for detecting changes in both function values (i.e. jump) and slopes (i.e. kink). We believe that this is of particular practical benefit, as in many applications the type of change is not always clear beforehand. Notably, our results reveal a phase transition between the jump and kink scenarios, which echo the understanding in the offline setup (Goldenshluger et al., 2006, Frick et al., 2014b, Chen, 2021; see also Table 1). The FLOC detector is specifically designed to achieve asymptotically minimax optimal rates. While the constants involved have not been fully optimized and could likely be improved, we preliminary guidance for tuning FLOC to improve its empirical performance in finite-sample settings. Alternative approaches for parameter tuning could further enhance the performance of FLOC. For instance, theoretical insights, such as the limiting distribution of detection delay provided by Aue et al. (2009), could guide the selection of thresholds to satisfy specified bounds on type II error, and bootstrapping methods, as introduced by Huškov´a and Kirch (2012), could be adapted to improve performance, particularly in small-sample scenarios. The current implementation of FLOC relies on sufficient historical data to accurately estimate the pre-change signal. As a practical extension, an adaptive approach could be developed to incrementally update the signal estimate as new observations become available. + +Monitoring simultaneously jumps and kinks can enhance detection power compared to conventional approaches that focus solely on mean changes, as demonstrated in our analysis of excess mortality data. However, practitioners should be aware that the Gaussian noise and linear signal assumptions may be strongly violated in certain real-world applications. Enhancing the robustness of FLOC to accommodate broader noise distributions and signal structures, represents a promising direction for future research. For example, in the application diff --git a/outputs/arxiv_math/2503.05275_pg9_pg1_repeat1.md b/outputs/arxiv_math/2503.05275_pg9_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..d701973c5259d4eb267013b53355203faafbaa68 --- /dev/null +++ b/outputs/arxiv_math/2503.05275_pg9_pg1_repeat1.md @@ -0,0 +1,47 @@ +Let \(p:=\lfloor\theta n\rfloor\). Take \(4p\) mutually disjoint vertex sets \(S_{1},S_{2},\ldots,S_{2p},T_{1},T_{2},\ldots,T_{2p}\) from \(V(H)\setminus\) + +\(V(P^{\prime})\text{with}i_{\mathcal{P}}(S_{i})=(m,k-\ell-m)\text{and}i_{\mathcal{P}}(T_{i})=(m-1,k-\ell-m+1)\text{for}i\in[2p]\). We denote + +the union of these \(S_{i}\text{and}T_{i},i\in[2p]\) by \(R_{1}\). So \(R_{1}\subseteq V(H)\setminus V(P^{\prime}),|R_{1}|=4(k-\ell)p\) and + +\(i_{\mathcal{P}}(R_{1})=2p(m,k-\ell-m)+2p(m-1,k-\ell-m+1)\). Thus \(P^{\prime}\) can absorb \(R_{1}\), that is, there + +exists an \(\ell-\text{path}P\) with the same ordered ends as \(P^{\prime},\text{where}V(P)=V(P^{\prime})\cup R_{1}\). Now we show that \(P\) is the desired absorbing \(\ell-\text{path}\). Note that \(|V(P)|\leq\gamma n\). Fix any set \(X\subseteq V(H)\setminus V(P)\) with \(|X|\leq p\text{and}(k-\ell)||X|\) as required by the lemma. Suppose further that \(i_{\mathcal{P}}(X)=(t,s)= x(m,k-\ell-m)+y(m-1,k-\ell-m+1)\). Then we have + +\[\begin{cases} x= \underline{t-} \tfrac{(m-1)(t+s)}{k-\ell} \\ y= \tfrac{(t+s}{k-} \tfrac{)m}{\ell} -t. \end{cases}\] + +Since \(t+s=|X|,(k-\ell)||X|\text{and}m/(k-\ell)\leq1\), we get that \(x,y\) are integers and \(|x|,|y|\leq|X|<2p\). Thus \(i_{\mathcal{P}}(X\cup R_{1})=(x+2p)(m,k-\ell-m)+(y+2p)(m-1,k-\ell-m+1),\text{where}x+2p>0,y+2p>0\) + +and \(|X\cup R_{1}|\leq4(k-\ell)p+p\leq(k-\ell)\beta^{2k-2\ell+4}n\). So \(P^{\prime}\) can absorb \(X\cup R_{1}\), that is, \(P\) can absorb □ + +\(X\). + +To complete the proof of Lemma 3.5, it remains to prove Claim 3.9. + +Proof of Claim 3.9. Let \(S:=\{v_{1},v_{2},\ldots,v_{k-\ell}\}\) be a \((k-\ell)-\text{set}\). Fix a \(k-\text{partite}k-\text{graph}A(k,\ell)\) + +on \([V_{1}^{A},V_{2}^{A},\ldots,V_{k}^{A}]\) satisfying Proposition 3.7 with \(|A(k,\ell)|=r\leq k^{4}\text{and}V(A(k,\ell))=S^{\prime}\cup X\), + +where \(|S^{\prime}|=k-\ell\). Without loss of generality, suppose \(|S^{\prime}\cap V_{i}^{A}|=1\text{for}i\in[k-\ell]\). Let + +\(b:=(4k-2\ell-1)(k-\ell)+r\). Since \((a,k-a)\in I_{\mathcal{P}}^{\mu}(H^{\prime})\), the number of edges whose index vectors + +are \((a,k-a)\) is at least \(\mu n^{k}\). Note that \(t_{1}:=|S\cap V_{1}|\leq a\text{and}|S\cap V_{2}|\leq k-a\) by \(S\in\mathcal{S}\). Since \(A(k,\ell)\) is \(k-\text{partite}\), by the supersaturation result (see [6]) on the subgraph of \(H\) that consists of all + +\(V_{t_{1}+1}^{A},\ldots,V_{k-\ell}^{A}\subseteq V_{2}\). For such a copy \(A\) of \(A(k,\ell)\), we denote by \(S_{A}^{\prime}\) as the set of \(k-\ell\) vertices + +edges of index vector \((a,k-a),H\) contains \(\beta n^{r}\) copies of \(A(k,\ell)\) each with \(V_{1}^{A},\ldots,V_{t_{1}}^{A}\subseteq V_{1}\) and given in Proposition 3.7 (2). So \(i_{\mathcal{P}}(S_{A}^{\prime})=i_{\mathcal{P}}(S)\) for each such \(A\). Consider a copy of \(A(k,\ell)\text{in}H\) which we denote by \(A\). Note that each of \(V_{1},V_{2}\) is \((\beta,\) 2)-closed + +in \(H\). Without loss of generality, suppose \(S_{A}^{\prime}=\{w_{1},w_{2},\ldots,w_{k-\ell}\}\text{such that}v_{i},w_{i}\) are \((\beta,2)-\) + +reachable for \(i\in[k-\ell]\) by \(i_{\mathcal{P}}(S_{A}^{\prime})=i_{\mathcal{P}}(S)\). By the definition of reachability, for each \(i\in[k-\ell]\), + +there are at least \(\beta n^{4k-2\ell-1}(4k-2\ell-\) 1)-sets \(T_{i}\) such that there exist \(\ell-\text{paths}P_{i}^{1},P_{i}^{2},P_{i}^{3},P_{i}^{4}\) with + +\(V(P_{i}^{1}\cup P_{i}^{2})=T_{i}\cup\{v_{i}\}\text{and}V(P_{i}^{3}\cup P_{i}^{4})=T_{i}\cup\{w_{i}\},\text{where}P_{i}^{1}\) has the same ends as \(P_{i}^{2}\) , and + +as an ordered set. Among them, at most \((k-\ell)n^{b-1}\) of them intersect \(S\) and at most \(b^{2}n^{b-1}\) of + +\(P_{i}^{3}\) has the same ends as \(P_{i}^{4}\) . So there are at least \(\beta^{k-\ell+1}n^{b}\) choices for \(A\cup T_{1}\cup T_{2}\cup\cdots \cup T_{k-\ell}\) them contain repeated vertices. Thus there are at least \(\beta^{k-\ell+1}n^{b}/2b-\)tuples avoiding \(S\) such that \(A,T_{1},T_{2},\ldots,T_{k-\ell}\) are pairwise vertex-disjoint. Now it remains to show that the \(b-\)tuple corresponding to \(A\cup T_{1}\cup T_{2}\cup\cdots \cup T_{k-\ell}\) is an \(S-\text{absorber}\). Firstly, \(T_{i}\cup\{w_{i}\},i\in[k-\ell]\) spans two vertex-disjoint \(\ell-\)paths of length two, which together with the spanning \(\ell-\text{path in}A\setminus\{w_{1},w_{2},\ldots,w_{k-\ell}\}\) form a family of \(2k-2\ell+1\ell-\)paths which span \(V(A)\cup T_{1}\cup T_{2}\cup\cdots \cup T_{k-\ell}\). Secondly, \(H[T_{i}\cup\{v_{i}\}]\) forms two vertex-disjoint \(\ell-\)paths of length two for \(i\in[k-\ell]\), which together with the spanning \(\ell-\text{path in}A\) gives a family of \(2k-2\ell+1\ell-\)paths which \(\text{span}S\cup V(A)\cup T_{1}\cup T_{2}\cup\cdots \cup T_{k-\ell}\) and have the same ends as the family of \(\ell-\)paths above. So the \(b-\)tuple corresponding to \(A\cup T_{1}\cup T_{2}\cup\cdots \cup T_{k-\ell}\) is an \(S-\)absorber (cf. Figure 1). 3.3. Proofs of Theorem 1.4 and Theorem 1.5. We prove Theorem 1.4 by following the common approach of absorption (cf. [18, 27]). That is, we decompose the proof in the usual way into the absorbing path lemma, the reservoir lemma, the connecting lemma and the path cover lemma. We will use these lemmas to find an absorbing path and a reservoir firstly, and then we cover the majority of vertices by vertex-disjoint \(\ell-\text{paths}\). We connect up all these \(\ell-\)paths to form an \(\ell-\text{cycle}\). Finally, we absorb the leftover vertices into the absorbing path, thereby completing a Hamilton \(\ell-\text{cycle}\). + +- diff --git a/outputs/arxiv_math/2503.05276_pg14_pg1_repeat1.md b/outputs/arxiv_math/2503.05276_pg14_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..9c1b5f53115f29097d97f1fe8f9c0046fc924237 --- /dev/null +++ b/outputs/arxiv_math/2503.05276_pg14_pg1_repeat1.md @@ -0,0 +1,13 @@ +To address these issues, we introduce Constrained Reinforcement Learning (CRL), a reinforcement learning algorithm specifically designed for constrained MDPs. This algorithm, operating online, learns the values of post-decision states, \(V_{t}(s)\), while also enabling constrained action selection within the feasible mixed-integer space using solvers like Gurobi. Similar to Eq. (10), which defines the Bellman equation for the pre-decision state \(x\), we now formulate the Bellman function for the post-decision state \(s\) as follows: + +\[V_{t}(s)=\sum_{\phi t\in\Phi t}\mathrm{P}\{O_{t}=\phi_{t}\}\left(c_{s}(\phi_{t})+\min_{at\in At(x)}\{c_{x}(a_{t})+V_{t+1}(s^{\prime})\}\right).\] (11) + +As values of pre- and post-decision states, \(V(x)\text{and}V(s)\), there also exist traditionally adapted Q-values, \(Q(x,a)\), which represent the expected total cost when taking action \(a\) in state \(x\) and following the optimal policy thereafter. In our context, this is given by: + +\[Q(x,a)=c_{x}(a)+\sum_{\phi\in\Phi}\mathrm{P}\{O=\phi\}(c_{s}(\phi)+V(x^{\prime}))\] (12) + +Note that \(t\) is omitted in Eq. (12) since, in the infinite-horizon setting, \(Q_{t}(x,a_{t}),V_{t}(x),\text{and}V_{t}(s)\) converge to \(Q(x,a),V(x),\text{and}V(s)\). Q-learning is effective in traditional RL problems because immediate action costs are often uncertain or challenging to estimate. However, as in most operations research problems, our costs are explicitly defined, such as \(c_{x}(a)\), including the selling profit and transportation costs. Instead of following the conventional approach of policy improvement through Q-learning, we leverage this structure by separating the Q-value into its easily calculable immediate costs of \(c_{x}(a)\), and predicting only \(V(s)\), given as: + +\[Q(x,a)=c_{x}(a)+V(s).\] (13) + +This separated approach reduces unnecessary computational overhead in estimating current action costs, and helps the model to solely focus learning \(V(s)\). Moreover, as demonstrated in previous research (Sun et al. 2022), learning post-decision state values is generally a more effective approach compared to learning of the action values. Consequently, our CRL holds significant potential for broader application in other constrained MDPs commonly encountered in operations research. The proposed algorithm is detailed in Algorithm 1. It focuses on learning an approximation to \(V(s)\approx\hat{v}_{w}(s):=w^{⊺}\psi(s)\), using a differentiable value function parameterization with features, \(\psi_{w}(s)\). The estimated value for the post-decision state is calculated as a weighted linear combination of these features. The algorithm iteratively refines the feature weights, \(w\), to improve the action selection process. diff --git a/outputs/arxiv_math/2503.05311_pg1_pg1_repeat1.md b/outputs/arxiv_math/2503.05311_pg1_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..42989e349fc2102e10f4263d6a5bf81e5b1ca735 --- /dev/null +++ b/outputs/arxiv_math/2503.05311_pg1_pg1_repeat1.md @@ -0,0 +1,19 @@ +# UPPER TAIL BOUNDS FOR IRREGULAR GRAPHS + +ANIRBAN BASAK AND SHAIBAL KARMAKAR + +Abstract. In this note we consider the upper tail large deviations of subgraph counts for irregular graphs H in \(\mathbb{G}(n,p)\), the sparse Erd˝os-Rényi graph on \(n\) vertices with edge connectivity probability \(p\in(0,1)\). For \(n^{-1/\Delta}\ll p\ll1,\text{where}\Delta\) is the maximum degree of H, we derive the upper tail large deviations for any irregular graph H. On the other hand, we show that for \(p\text{such that}1\ll n^{v_{\mathrm{H}}}p^{e_{\mathrm{H}}}\ll(\log n)^{\alpha_{\mathrm{H}}^{*}/(\alpha_{\mathrm{H}}^{*}-1)}\), where \(v_{\mathrm{H}}\text{and}e_{\mathrm{H}}\) denote the number of vertices and edges of H, and \(\alpha_{\mathrm{H}}^{*}\) denotes the fractional independence number, the upper tail large deviations of the number of unlabelled copies of H in \(\mathbb{G}(n,p)\) is given by that of a sequence of Poisson random variables with diverging mean. Restricting to the \(r-\)armed star graph we further prove a localized behavior in the intermediate range of \(p\) (left open by the above two results) and show that the mean-field approximation is asymptotically tight for the logarithm of the upper tail probability. + +## 1. Introduction and main results + +The classical large deviation theory traditionally deals with large deviations of linear functions of independent and identically distributed (i.i.d.) random variables. Probably the simplest non-trivial problem that falls outside realm of the classical theory is the large deviations of triangle counts in an Erdős-Rényi random graph, to be denoted by \(\mathbb{G}(n,p)\), with vertex set \([[n]]:=\{1,2,\ldots,n\}\) and edge connectivity probability \(p=p_{n}\in(0,1)\), where each pair of vertices is connected with a probability \(p\) independently of every other pair. In the last fifteen years there have been immense interest in studying large deviations subgraph (in particular triangle) counts in Erdős-Rényi random graphs, both in the dense regime, i.e. \(p≍1\) (see the monograph [9] and the references therein), and in the sparse regime, i.e. \(p\ll1\), e.g. see [2, 7, 10, 12, 13, 14] (we refer the reader to Section 1.5 for all the notation). Recently Harel, Mousset, and Samotij [15] showed that the speed and the rate function of upper tail of cliques (i.e. complete subgraphs) on \(r\) vertices in \(\mathbb{G}(n,p)\) undergoes a transition at the threshold + +\(p^{(r-1)/2}≍n^{-1}(\log n)^{1/(r-2)}.\text{For}p^{(r-1)/2}\ll n^{-1}(\log n)^{1/(r-2)}\) (and \(p^{(r-1)/2}\gg n^{-1}\) so that the expected number of cliques on \(r\) vertices diverges) the speed and the rate function are given by those for a sequence of Poisson random variables with diverging means, and thus that regime of \(p\) is rightly termed as the Poisson + +regime. In contrast, for \(p^{(r-1)/2}n^{-1}(\log n)^{1/(r-2)}\ll p^{(r-1)/2}\ll1\) the large deviation event is primarily due to the presence of some localized structures (of microscopic size) in the graph, and therefore that regime is termed as the localized regime. It was conjectured in (an earlier version of) the work [15] that the transition between the localized and the Poisson regimes, for the upper tail of all connected regular subgraph counts, should occur at the + +threshold \(p^{\Delta/2}≍n^{-1}(\log n)^{1/(v_{\mathrm{H}}-2)},\text{where}\Delta\) is the common degree of the regular graph H in context and \(v_{\mathrm{H}}\) denotes the number of vertices in H. In [15] the Poisson behavior was proved in the predicted Poisson + +regime, and the localized behavior for \(p\text{such that}n^{-1/2-o(1)}\leq p^{\Delta/2}\ll1\) (and under the additional + +assumption that H is non bipartite in the extended \(\text{range}n^{-1}(\log n)^{\Delta v_{\mathrm{H}}^{2}}\ll p\ll1)\). The more recent work [5] extends the localized behavior for all regular graph in the sub regime left open by [15]. Therefore, [15] and [5] together settle the problem of upper tail large deviations of regular subgraph counts in sparse Erdős-Rényi graphs (albeit a couple of boundary cases). Let us also mention in passing that the speed and the rate function in the localized regime turns out to be the solution of an appropriate mean field variational problem. See Section 1.4 below for a further discussion on this. In this short article we investigate whether a transition threshold between the localized and the Poisson regimes exists, and whether in the localized regime the large deviation probability can be expressed as the solution of some mean field variational problem for the upper tail of irregular subgraph (i.e. there exists at least two vertices with unequal degrees) counts in sparse Erdős-Rényi graphs. diff --git a/outputs/arxiv_math/2503.05323_pg16_pg1_repeat1.md b/outputs/arxiv_math/2503.05323_pg16_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..98062c7a1b6875e28fc6ec22d0ad0c22644ce249 --- /dev/null +++ b/outputs/arxiv_math/2503.05323_pg16_pg1_repeat1.md @@ -0,0 +1,29 @@ +## Appendix A. Proof of Claim 6 + +Fix \(\tilde{\epsilon}>0\) and note that for any \(i\neq j\in[n]\), we have + +\[\begin{aligned} \mathbb{E}\left[ \frac{1}{(|\lambda_{j}-\lambda_{i}|+n^{-1.5-\tilde{\epsilon}})^{2}} \right] \\ &=\mathbb{P}\left( \frac{1}{(|\lambda_{j}-\lambda_{i}|+n^{-1.5-\tilde{\epsilon}})^{2}} >x\right)dx \\ &=\int_{0^{0_{n}}n_{3}^{3}}^{0^{0_{\infty}}\infty}\mathbb{P}\left(|\lambda_{j}-\lambda_{i}|< \frac{1}{\sqrt{x}} -n^{-1.5-\tilde{\epsilon}}\right)dx \\ &\leq=_{\mathbb{P}^{+2\tilde{\epsilon}}}^{+2\tilde{\epsilon}}\mathbb{P}\left(_{|}|_{\lambda}\lambda_{-_{j}^{j}}-_{\lambda}\lambda_{|_{i}^{i}}|_{<}< \frac{1}{\frac{1 \sqrt{x}}{\sqrt{x}}} \right)-_{d}n_{x^{-}\leq^{1.5}2^{-}c_{0}^{\tilde{\epsilon}}}\bigg)_{n}d_{2}x+\int_{.5+\tilde{\epsilon}_{,^{n}\infty_{3+2\tilde{\epsilon}}}}\mathbb{P}\left(|\lambda_{j}-\lambda_{i}|< \frac{1}{\sqrt{x}} -n^{-1.5-\tilde{\epsilon}}\right)dx \end{aligned}\] + +where the last inequality holds by (Nguyen et al., 2017, Corollary 2.2) for some constant \(c_{0}>0\). Using the above inequality, we get + +\[\mathbb{E}\left[_{i\neq j\in}\sum_{[n]:|i-j|\leq2} \frac{1}{(|\lambda_{j}-\lambda_{i}|+n^{-1.5-\tilde{\epsilon}})^{2}} \right]\leq8c_{0}n^{3.5+\tilde{\epsilon}}.\] + +Now, by the Markov’s inequality, with probability \(1-n^{-\tilde{\epsilon}}\), we get + +\[\sum_{i\neq j\in[n]:|i-j|\leq2} \frac{1}{(|\lambda_{j}-\lambda_{i}|+n^{-1.5-\tilde{\epsilon}})^{2}} \leq8c_{0}n^{3.5+2\tilde{\epsilon}}.\] + +As \(\min_{i\in[n-1]}|\lambda_{i+1}-\lambda_{i}|\geq n^{-1.5-\tilde{\epsilon}}\) with probability \(1-o(1)\) by (Feng et al., 2019, Corollary 1), we get + +\[\sum_{i\neq j\in[n]:|i-j|\leq2} \frac{1}{(\lambda_{j}-\lambda_{i})^{2}} \leq\sum_{i\neq j\in[n]:|i-j|\leq2} \frac{4}{(|\lambda_{j}-\lambda_{i}|+n^{-1.5-\tilde{\epsilon}})^{2}} \leq32c_{0}n^{3.5+2\tilde{\epsilon}}.(9)\] + +Now, we focus on bounding the terms for which \(|i-j|\geq3\). By (Nguyen et al., 2017, Corollary + +2.5) and union bound, we have \(\min_{i\in[n-3]}|\lambda_{i+3}-\lambda_{i}|\leq n^{-6/5-\tilde{\epsilon}/5}\) with probability \(1-c_{0}n^{-\tilde{\epsilon}}\) for some \(c_{0}>0\). Using this bound, we get + +\[\begin{aligned} \sum_{i,j\in[n]:|i-j|\geq3} \frac{1}{(\lambda_{i}-\lambda_{j})^{2}} &\leq \frac{1}{\min_{i\in[n-3]}(\lambda_{i+3}-\lambda_{i})^{2}} \sum_{i,j\in[n]:|i-j|\geq3}\left\lfloor \frac{|j-i|}{3} \right\rfloor^{-2} \\ &\leq6n^{17/5+2\tilde{\epsilon}/5}\sum_{i=1}^{n} \frac{1}{i^{2}} \leq\pi^{2}n^{17/5+2\tilde{\epsilon}/5}. \end{aligned}\] + +Now, combining the above inequality with (9), we get with probability \(1-o(1)\) + +\[\sum_{i,j\in[n]:i\neq j} \frac{1}{(\lambda_{j}-\lambda_{i})^{2}} \leq32c_{0}n^{3.5+2\tilde{\epsilon}}+\pi^{2}n^{17/5+2\tilde{\epsilon}/5}\leq n^{3.5+3\tilde{\epsilon}},\] + +where the last inequality holds for \(n\) large enough. Picking \(\tilde{\epsilon}=\epsilon/6\) completes the proof. diff --git a/outputs/arxiv_math/2503.05329_pg4_pg1_repeat1.md b/outputs/arxiv_math/2503.05329_pg4_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..bf0c76f98bd317e96b9ef1e1039bfd763ea71776 --- /dev/null +++ b/outputs/arxiv_math/2503.05329_pg4_pg1_repeat1.md @@ -0,0 +1,37 @@ +(1.2) + +\[C(X_{0})\] + +\[C(X_{1})\otimes M_{r(1)}\] + +\[C(X_{1})\otimes M_{r(1)}\] + +\[C(X_{2})\otimes M_{r(2)}\] + +\[C(X_{2})\otimes M_{r(2)}\] + +\[C(X_{2})\otimes M_{r(2)}\] + +\[C(X_{2})\otimes M_{r(2)}\] + +\[C(X_{3})\otimes M_{r(3)}\] + +\[C(X_{3})\otimes M_{r(3)}\] + +\[C(X_{3})\otimes M_{r(3)}\] + +\[C(X_{3})\otimes M_{r(3)}\] + +\[C(X_{3})\otimes M_{r(3)}\] + +\[C(X_{3})\otimes M_{r(3)}\] + +\[C(X_{3})\otimes M_{r(3)}\] + +\[C(X_{3})\otimes M_{r(3)}\] + +\[C(Y_{0})C(Y_{1})\otimes M_{r(2)}C(Y_{2})\otimes M_{r(2)}C(Y_{3})\otimes M_{r(3)}\] + +![](figure) + +The radius of comparison of the resulting AH algebra is shown to be the maximum of the two. Up to composing with inner automorphisms, which are needed to ensure consistency, the actions on the top levels are cyclic permutations, whereas the action on the bottom part is trivial. In essence, we are merging an action which is a souped-up version of the odometer action - which has the Rokhlin property - with an inner action on the bottom part.Since no general result applies here, we analyze the crossed product directly: the crossed product is an AH algebra too, and its radius of comparison is shown to be the same as the one coming from the bottom row. The rest of the paper constitutes the proof of Theorem 1.1. In light of Remark 1.2, we only prove explicitly the case in which \(0 + + +this ODE we obtain + + + +xdx + +dx + + +(4.3) +us(x, y) = Z x Z 1 0 c − t dt, c = +Z 1 +. + + + +a(t, y) 0 a(x, y) +0 +a(x, y) + + + +Here we take + +\[a^{s}(x,\boldsymbol{y})=\exp\left(\sum_{j=1}^{s} \frac{1}{j^{2}} \sin(2j\pi x)y_{j}\right),\] + +with \(y_{1},\ldots,y_{s}^{i.}\sim^{i.d.}N(0,1)\). We are interested in computing the expectation \(\mathbb{E}_{\boldsymbol{y}}[F(\boldsymbol{y})]\), where + +\[F(\boldsymbol{y})=G(u^{s}(\cdot,\boldsymbol{y}))=u^{s}(x_{0},\boldsymbol{y}),\] + +and \(x_{0}\in\{ \frac{1}{3} , \frac{2}{3} \}.\) According to \([10],F\) lies in the unanchored weighted Sobolev space with + +\[\phi(x)= \frac{1}{\sqrt{2\pi}} e^{- \frac{x2}{2}},\psi_{j}^{2}(x)=e^{-2\alpha_{j}|x|},\alpha_{j}>0.\] (4.4) + +We take \(s=30\) and compute the MAEs of the estimators obtained by the MC method, the randomly shifted lattice rule with the CBC algorithm, and the median QMC method. To calculate the integrals in (4.3) for any given \(\boldsymbol{y}\in\mathbb{R}^{s}\), we use the 4th-order Gauss-Legendre formula with 200 nodes. The exact value of \(\mathbb{E}_{\boldsymbol{y}}[F(\boldsymbol{y})]\) are estimated by using 221 points from the nested scrambled Sobol’ sequence averaged over 10 independent replications. Similar to Example 2, for the median QMC method, we take the median of \(k=11\) independent QMC estimators, each utilizing \(N\) points, while for the MC method and the randomly shifted lattice rule with the CBC method, we use \(k\times N\) points per method. Furthermore, for the CBC method, we choose the weight parameters and the weight functions as recommended in [10] . We set \(\lambda=0.55\) and \(b_{j}= \frac{1}{j^{2}} \text{for}j=1,\ldots,s\). For the weight functions \(\psi_{j}^{2}\) in (4.4), we take + +\[\alpha_{1}= \frac{1}{2} \left(b_{1}+ \sqrt{b_{1}^{2}+1- \frac{1}{2\lambda}} \right),\] + +and + +\[\alpha_{j}= \frac{1}{2} \left(b_{2}+ \sqrt{b_{2}^{2}+1- \frac{1}{2\lambda}} \right),2\leq j\leq s.\] diff --git a/outputs/arxiv_math/2503.05342_pg5_pg1_repeat1.md b/outputs/arxiv_math/2503.05342_pg5_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..424acbeca934f6f7c49d6653a7c6aaeee8037777 --- /dev/null +++ b/outputs/arxiv_math/2503.05342_pg5_pg1_repeat1.md @@ -0,0 +1,19 @@ +It is known that the fundamental group of the boundary \(\partial V\) of the solid torus \(V\) is the free abelian group on two generators \(\mathbb{Z}\times\mathbb{Z}\). Therefore one can define a longitude for \(V\) as a simple, closed curve on \(\partial V\) that is of the form \((k,1)\) in the fundamental group of the boundary \(\partial V,k\in\mathbb{Z}\). A meridian is a simple, closed curve of the form \((1,0)\). + +Definition 3.1. Let \(D\) be a diagram of an oriented link \(L\) and let \(c_{1},c_{2}\) be two components of \(D\). The linking number between \(c_{1}\text{and}c_{2}\) is the total number of positive crossings minus the total number of negative crossings, divided by 2, over all crossings between \(c_{1}\text{and}c_{2}\). A positive crossing is when one has to rotate the over arc counterclockwise, in order to align the two arcs involved, so that they both point to the same direction. A negative crossing is when the upper strand has to rotate clockwise. The linking number counts the algebraic number of times one component winds around the other one. Further, it is an ambient isotopy invariant as it respects Reidemeister II and III moves and it is not affected by the Reidemeister I move. Thus we talk about the linking number of two link components as the linking number between them in any diagram representative of the link. + +![](figure) + +![](figure) + +Figure 5: The two types of crossing in a link diagram. + +Definition 3.2 (Framing). Let \(K\) be an oriented knot in \(S^{3}\) (or \(\mathbb{R}^{3}).\text{Then}K\) has a regular neighbourhood in \(S^{3}\) that is a solid torus \(S^{1}\times D^{2}=V\) , where \(K\) is the core of \(V\) . Let \(K_{0}\) be a longitude in \(V\) with the same orientation as \(K\). The linking number \(lk(K,K_{0})\) between \(K\text{and}K_{0}\), is called a framing of \(K\). For an example view Fig. 6a. A framed link of \(c-\)components is a link of whom each one of the \(c\) components is a framed knot. To a framed link of \(c\) components one can assign a framing vector \(\vec{\lambda}=(\lambda_{1},\ldots,\lambda_{c}),\text{where}\lambda_{i}\in\mathbb{Z}\) is the framing of the \(i-\)th component. + +![](figure) + +![](figure) + +Figure 6: Different framing representations of a trefoil with framing -4. + +We note that a framing for a knot can be any integer, as it also depends on how many times the longitude follows a meridian of \(V\) , which is the first coordinate in the fundamental group \(\mathbb{Z}\times\mathbb{Z}\) of the torus \(\partial V\) . diff --git a/outputs/arxiv_math/2503.05348_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.05348_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..7b72fabb5641eceefeda678966bd125f1fc17a67 --- /dev/null +++ b/outputs/arxiv_math/2503.05348_pg3_pg1_repeat1.md @@ -0,0 +1,26 @@ +can be seen in [4, 6, 12]. Throughout the paper, ‘increasing’ means ‘non-decreasing’ and ‘decreasing’ means ‘non-increasing’. Whenever we use a derivative, an expectation or a conditional distribution, we are tacitly assuming that they exist. If \(X,Y\in\mathcal{L}_{1}\), then we say that: + +- \(X\) is less than \(Y\) in the stochastic order (or in the first-order stochastic dominance), shortly written as \(X\leq_{ST}Y\) , if \(\bar{F}_{X}\leq\bar{F}_{Y}\) or, equivalently, if \(E(u(X))\leq E(u(Y))\) for all increasing functions \(u\) such that these expectations exist. Then \(X=_{ST}Y\) represents equality in law (distribution). + +- \(X\) is less than \(Y\) in the increasing concave order (or in the second-order stochastic dominance), shortly written as \(X\leq_{ICV}Y\) , if \(E(u(X))\leq E(u(Y))\) for all increasing concave functions \(u\) such that these expectations exist or, equivalently, if + +\[\int_{-\infty}^{t}F_{X}(x)dx\geq\int_{-\infty}^{t}F_{Y}(x)dx\text{for all}t.\] + +- \(X\) is less than \(Y\) in the increasing convex order (or in the stop-loss order), shortly written as \(X\leq_{ICX}Y\) , if \(E(u(X))\leq E(u(Y))\) for all increasing convex functions \(u\) such that these expectations exist or, equivalently, if + +\[\int_{t}^{\infty}\bar{F}_{X}(x)dx\leq\int_{t}^{\infty}\bar{F}_{Y}(x)dx\text{for all}t.\] + +The function \(\pi(t)=\int_{t}^{\infty}\bar{F}(x)dx\) is called stop-loss function (see e.g. [6], p. 19). Hence, \(X\leq_{ICX}Y\) holds if and only if \(\pi_{X}\leq\pi_{Y}\) . It is also equivalent to \(-X\geq_{ICV}-Y\) . + +- \(X\) is less than \(Y\) in the convex order, shortly written as \(X\leq_{CX}Y\) , if \(E(u(X))\leq E(u(Y))\) for all convex functions \(u\) such that these expectations exists or, equivalently, if \(\pi_{X}\leq\pi_{Y}\text{and}\lim_{t\rightarrow-\infty}\pi_{X}(t)-\pi_{Y}(t)=0\). It implies \(E(X)=E(Y)\) and \(Var(X)\leq Var(Y)\). It is also equivalent to \(X\leq_{ICX}Y\text{and}E(X)=E(Y)\) (see [6], p. 19) and to \(X\geq_{ICV}Y\text{and}E(X)=E(Y)\). + +- \(X\) is less than \(Y\) in the convolution order, shortly written as \(X\leq_{CONV}Y\) if \(Y=_{ST}X+Z\text{where}Z\) is a nonnegative random variable independent of \(X\) (see [12], p. 70 and [13]). + +The relationships between these orders are summarized as follows: + +\[\begin{aligned} X&\leq_{CONV}Y\Rightarrow X\leq_{ST}Y\Rightarrow X\leq_{ICX}Y\Leftarrow X\leq_{CX}Y \\ ⇓⇓⇓ \\ X&\geq_{CX}Y\Rightarrow X\leq_{ICV}Y\Rightarrow E(X)\leq E(Y)E(X)=E(Y) \end{aligned}\] + +| X ≤CONV Y | ⇒ X ≤ST | Y | ⇒ | X ≤ICX Y | ⇐ | X ≤CX Y | +| --- | --- | --- | --- | --- | --- | --- | +| | ⇓ | | | ⇓ | | ⇓ | +| X ≥CX Y | ⇒ X ≤ICV | Y | ⇒ | E(X) ≤ E(Y ) | | E(X) = E(Y ) | diff --git a/outputs/arxiv_math/2503.05358_pg8_pg1_repeat1.md b/outputs/arxiv_math/2503.05358_pg8_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..d4b98754374ef06372c46db857aaeaaf1b5c55cb --- /dev/null +++ b/outputs/arxiv_math/2503.05358_pg8_pg1_repeat1.md @@ -0,0 +1,37 @@ +comet \((t_{4})\). This is possible because the optimal flyby parameters (depth and inclination of the planetocentric hyperbola) can be computed directly from \(t_{2}\) and \(t_{4}\), thus reducing the dimensionality of the problem. + +#### 3.1.1. Earth-GA segment + +For a given flyby date, the ecliptic longitude of the GA planet is computed + +from the ephemeris as + +\[l_{2}^{F}=\Omega^{F}+\omega^{F}+n^{F}(t_{2}-t_{p}^{F}),\] (1) + +where \(n^{F}\) is the mean motion of the planet. The minimum eccentricity of the + +conics that connect Earth with the planet is determined by the condition of + +being able to reach the planet: + +\[a_{\text{min}}= \frac{r^{E}+r^{F}}{2} \rightarrow e_{\text{min}}=1- \frac{r^{E}}{a_{\text{min}}} .\] (2) + +The upper bound of eccentricity is set by the maximum characteristic energy + +\((C3)\) that the launcher is capable of: + +\[v_{1,\text{max}}= \sqrt{C3} +v^{E}= \sqrt{C3} + \sqrt{\frac{\mu^{S}}{r^{E}}} ,\] (3) + +\[E_{\text{max}}= \frac{v_{1,\text{max}}^{2}}{2} - \frac{\mu^{S}}{r^{E}} \rightarrow a_{\text{max}}= \frac{-\mu^{S}}{2E_{\text{max}}} \rightarrow e_{max}=1- \frac{r^{E}}{a_{\text{max}}} ,\] (4) + +where \(E\) denotes the mechanical energy and \(\mu\) is the gravitational parameter. + +Due to the condition of departure from the perihelion, the eccentricity determines the transfer angle between Earth and the planet + +\[\cos(\theta_{12})= \frac{1}{e} \left( \frac{p}{r^{F}} -1\right),\] (5) + +with \(p=a(1-e^{2})\) denoting the semilatus rectum. Likewise, the transfer time + +\(t_{12}\) (only the elliptic case is shown for the sake of brevity) can be computed from \(e\) + +\[cos(E)= \frac{e+\cos(\theta_{12})}{1+e\cos(\theta_{12})} \rightarrow M=E-e\sin(E)\rightarrow t_{12}=M \sqrt{\frac{a^{3}}{\mu^{S}}} ,(6)\] diff --git a/outputs/arxiv_math/2503.05360_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.05360_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..d48ad25d9beda27cf97d482f88f2a7e6b6ef3194 --- /dev/null +++ b/outputs/arxiv_math/2503.05360_pg2_pg1_repeat1.md @@ -0,0 +1,21 @@ +The key idea underlying this simulation technique is the systematic translation of formulas into basic sentences. Suppose \(\vdash\varphi\text{and}\varphi\) contains a subformula \(a\wedge b\), where \(a,b\in\mathbb{B}\). Following the natural deduction rules governing conjunction conjunction, Sandqvist makes sure to include the following rules in the “specifically tailored” base \(\mathscr{N}\text{for}\varphi\): + +\[\frac{ab}{r} \frac{r}{a} \frac{r}{b}\] + +where \(r\) is a fresh basic sentence representing \(a\wedge b\). This means that \(r\) behaves in \(\mathscr{N}\text{as}a\wedge b\) behaves in Gentzen’s NJ [1] — that is, as in intuitionistic sentential logic. More generally, each subformula \(\chi\) of \(\varphi\) is assigned a corresponding basic counterpart \(\chi^{♭}\), e.g., \(r=(a\wedge b)^{♭}\). Sandqvist establishes their equivalence within \(\mathscr{N}\) : + +\[\chi\Vdash_{\mathscr{N}}\chi^{♭}\text{and}\chi^{♭}\Vdash_{\mathscr{N}}\chi.\] + +Since we assume \(\Vdash\varphi\), it follows that \(\Vdash\varphi^{♭}\), and given that every rule in \(\mathscr{N}\) corresponds to an intuitionistic natural deduction rule, we conclude \(\vdash\varphi\), as required. The flattening technique employed here is strikingly different from standard completeness proofs. However, as the saying goes, “all theorems were already proved in the Soviet Union”. In particular, this approach bears a strong resemblance to a method developed by Mints [11, 12, 13, 14, 15] from the 1980s for evaluating intuitionistic consequence via resolution systems. Mints’ approach systematically transforms proof systems for various logics, including intuitionistic logic, into resolution-based systems while preserving derivational structure. This extends earlier work by Maslov [10] on resolution calculi for classical predicate logic. Mints defines a class of formulas, clauses, with a particularly simple structure (details below). Given a formula \(\varphi\), he constructs a corresponding set of clauses \(M\) such that for some designated basic sentence \(g\), + +\[\vdash\varphi\text{iff}M\vdash g(\dagger)\] + +Each subformula \(\chi\) of \(\varphi\) is associated with an basic counterpart \(\chi^{♭},\text{and}M\) is constructed by adding clauses ensuring that \(\chi^{♭}\) correctly encodes the logical behaviour of \(\chi\). For instance, if \(\chi=\chi_{1}\wedge\chi_{2}\), the following clauses are introduced: + +\[\chi_{1}^{♭}\wedge\chi_{2}^{♭}\rightarrow\chi^{♭},\chi^{♭}\rightarrow\chi_{1}^{♭},\chi^{♭}\rightarrow\chi_{2}^{♭}.\] + +The key observation is that relative to such clauses, + +\[(\chi_{1}\wedge\chi_{2})^{♭}\leftrightarrow\chi_{1}^{♭}\wedge\chi_{2}^{♭}.\] + +Thus, reasoning about \(\varphi\) reduces to reasoning about formulas with at most three atomic components. In \((\dagger),M\) is the clause set encoding \(\varphi\text{and}g=\varphi^{♭}\). A natural correspondence emerges between bases and clausal systems. Specifically, for any formula \(\varphi\), Sandqvist’s \(\mathscr{N}\) and Mints’ \(M\) coincide when identical atomic names are used for subformulas. More significantly, this bijection establishes a pointwise correspondence between support in a base and proof search in a clause set, aligning validity in B-eS for \(\varphi\) with resolution in \(M\text{for}g\). Consequently, the soundness and completeness of intuitionistic logic with respect to Sandqvist’s B-eS both follow as corollaries of Mints’ theorem. diff --git a/outputs/arxiv_math/2503.05390_pg14_pg1_repeat1.md b/outputs/arxiv_math/2503.05390_pg14_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..ef8b7fe5447242861019dd95efcfb49d49e1d12a --- /dev/null +++ b/outputs/arxiv_math/2503.05390_pg14_pg1_repeat1.md @@ -0,0 +1,21 @@ +\[\begin{aligned} \sum_{j=0}^{\infty}\int_{2^{-j-1}\delta<|x-y|\leq2^{-j}}≲^{\delta} \frac{1}{\delta|x-_{2}y_{-}|_{_{j}^{1}}^{d-}} |W^{\frac{1}{p}}(x)W_{|W^{- \frac{1}{\frac{p_{1}}{p}}}}(_{(}y_{x})_{)}W_{W^{-}}^{\frac{1}{p}}(_{\frac{1}{p}}y_{(})_{y}\tilde{f}_{)}(_{W}y) \frac{|_{1}}{p} d(y^{y})\tilde{f}(y)|dy \\ j\sum^{\infty}&=0|x-y|\leq2^{-j}\delta \\ ≲\delta M_{W}(W^{\frac{1}{p}}\tilde{f})(x), \end{aligned}\] + +where \(M_{W}\) is the matrix-weighted maximal function (4). Due to the boundedness of the matrix-weighted maximal function we get + +\[\|[b,T_{\Omega}]\tilde{f}-[b,T_{\Omega,\delta}]\tilde{f}\|_{L^{p}(W)}≲_{b,\Omega}\delta\|M_{W}(W^{\frac{1}{p}}\tilde{f})\|_{L^{p}(\mathbb{R}^{d})}≲_{W}\delta\|\tilde{f}\|_{L^{p}(W)}\] + +Since \(\delta\) can be arbitrarily small, it suffices to show that \([b,T_{\Omega,\delta}]:L^{p}(W)\rightarrow L^{p}(W)\) is compact. We note that the truncated kernel satisfies + +\[|K_{\Omega,\delta}(x,y)-K_{\Omega,\delta}(x^{\prime},y)|≲ \frac{|x-x^{\prime}|}{|x-y|^{d+1}} ,2|x-x^{\prime}|\leq|x-y|.\] (5) + +By [17, Corollary 6.3] we have that \([b,T_{\Omega}]\) is \(L^{p}(W)\rightarrow L^{p}(W)\) bounded, and hence + +\[\begin{aligned} \|[b,T_{\Omega,\delta}]\tilde{f}\|_{L^{p}(W)}&\leq\|[b,T_{\Omega,\delta}]\tilde{f}-[b,T_{\Omega}]\tilde{f}\|_{L^{p}(W)}+\|[b,T_{\Omega}]\tilde{f}\|_{L^{p}(W)} \\ ≲_{b,W,\Omega}\|\tilde{f}\|_{L^{p}(W)}&<\infty, \end{aligned}\] + +which takes care of the first part of Theorem 5.1. Considering the second part, we suppose that \(Q\) is a cube centered at the origin such that \(\text{supp}b\subset Q.\text{Then for}x\) sufficiently far away from the origin we have + +\[\begin{aligned} |W^{\frac{1}{p}}(x)[b,T_{\Omega,\delta}]\tilde{f}(x)|^{p}&=\bigg|W^{\frac{1}{p}}(x)\int_{\mathbb{R}^{d}}b(y)K_{\Omega,\delta}(x,y)\tilde{f}(y)dy\bigg|^{p} \\ ≲_{b,\Omega}|W^{\frac{1}{p}}(x)|_{op}^{p} \frac{1}{|x|^{dp}} \int_{Q}|\tilde{f}(y)|dy\Bigg)^{p} \\ &\leq|W(x)|_{op} \frac{1}{|x|^{dp}} \Bigg(\int_{Q}\left(|\tilde{f}(y)|dy\right)^{p}, \end{aligned}\] + +where the last inequality is true due to the Cordes inequality. Moreover, an application Hölder’s inequality and another application of the Cordes inequality yields + +\[\int_{Q}|\tilde{f}(y)|dy\leq\int_{Q}|W^{- \frac{1}{p}}(y)|_{op}|W^{\frac{1}{p}}(y)\tilde{f}(y)|dy\leq\left(\int_{Q}|W^{- \frac{p\prime}{p}}(y)|_{op}dy\right)^{\frac{1}{p\prime}}\|\tilde{f}\|_{L^{p}(W)}.\] diff --git a/outputs/arxiv_math/2503.05392_pg19_pg1_repeat1.md b/outputs/arxiv_math/2503.05392_pg19_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..e1ba3cf2f6b4b16cbd8f5a5abd1b910ec278bfec --- /dev/null +++ b/outputs/arxiv_math/2503.05392_pg19_pg1_repeat1.md @@ -0,0 +1,51 @@ +3. 1.2. Results for the \(L_{\psi}\) general affine surface areas. In the case of the \(L_{\psi}\) general affine surface areas, we deduce the following results. Their proofs follow in completely analogous fashion to those for the \(L_{\phi}\) general surface areas in the results above, and hence are omitted. + +Theorem 3.6. Let \(\psi\in\text{Conv}(0,\infty)\text{and}a,b\geq0,\text{with}a,b\) not both zero. Suppose that the and assume that \(K_{1},K_{2}\) have positive Gaussian curvature almost everywhere. Then + +function \(G(t)=\psi(t^{n+1})\) is convex for \(t\in(0,\infty)\). Let \(K_{1},K_{2}\in K_{o}^{n}\) be such that \(K_{1}∣e_{n}^{\perp}=K_{2}∣e_{n}^{\perp}\), + +\[\text{as}_{\psi}(a◇_{e_{n}}K_{1}|_{e_{n}}b◇_{e_{n}}K_{2})\leq \frac{a}{a+b} \cdot\text{as}_{\psi}((a+b)◇_{e_{n}}K_{1})+ \frac{b}{a+b} \cdot\text{as}_{\psi}((a+b)◇_{e_{n}}K_{2}).\] + +In the special case \(a+b=1\), we obtain the following + +Corollary 3.3. Let \(\psi\in\text{Conv}(0,\infty)\text{and}\lambda\in[0,1]\) be given. Suppose that the function \(G(t)= K_{1},K_{2}\) have positive Gaussian curvature almost everywhere. Then we have + +\(\psi(t^{n+1})\) is convex for \(t\in(0,\infty)\). Let \(K_{1},K_{2}\in K_{o}^{n}\) be such that \(K_{1}∣e_{n}^{\perp}=K_{2}∣e_{n}^{\perp}\), and assume that + +\[\text{as}_{\psi}(\lambda◇_{e_{n}}K_{1}|_{e_{n}}(1-\lambda)◇_{e_{n}}K_{2})\leq\lambda\text{as}_{\psi}(K_{1})+(1-\lambda)\text{as}_{\psi}(K_{2}).\] + +Remark 3.4. Let \(K\in K_{o}^{n}\text{and}\psi\in\text{Conv}(0,\infty)\), and suppose that the function \(G(t)=\phi(t^{n+1})\) is almost everywhere, then + +convex for \(t\in(0,\infty)\). Ye [49, Theorem 3.10] proved that if \(K\) has positive Gaussian curvature + +\[\begin{aligned} \text{as}_{\psi}(K)&\geq\text{as}_{\psi}(S_{e_{n}}(K)). \\ &= \end{aligned}\] + +Again recalling that \(\frac{1}{2} ◇_{e_{n}}K|_{e_{n}} \frac{1}{2} ◇_{e_{n}}R_{e_{n}}(K_{1})=S_{e_{n}}(K)\), we see that Corollary 3.3 extends [49, Theorem 3.10] from the case of the Steiner symmetral of a single convex body to the graph sum of two convex bodies. + +Corollary 3.4. Let \(\psi\in\text{Conv}(0,\infty)\) be given. Let \(K_{1},K_{2}\in K_{o}^{n}\) be convex bodies with curvature + +functions and which satisfy \(K_{1}∣e_{n}^{\perp}=K_{2}∣e_{n}^{\perp}\). If the function \(G(t)=\psi(t^{n+1})\) is convex for + +\(t\in(0,\infty)\), then + +\[\lambda\text{as}_{\psi}^{*}(K_{1})+(1-\lambda)\text{as}_{\psi}^{*}(K_{2})\geq\text{as}_{\psi}^{*}([\lambda◇_{e_{n}}K_{1}^{\circ}|_{e_{n}}(1-\lambda)◇_{e_{n}}K_{2}^{\circ}]^{\circ}).\] + +Remark 3.5. Let \(\psi\in\text{Conv}(0,\infty)\) and suppose that \(G(t)=\psi(t^{n+1})\) is convex for \(t\in(0,\infty)\). + +Ye [49, Corollary 3.11] proved that for every \(K\in K_{o}^{n}\) with a curvature function and for which + +\([S_{e_{n}}(K^{\circ})]^{\circ}\) has a curvature function, we have + +\[\text{as}_{\psi}^{*}(K)\geq\text{as}_{\psi}^{*}([S_{e_{n}}(K^{\circ})]^{\circ}).\] + +Thus, in the special case \(K_{2}=R_{e_{n}}(K_{1})\), Corollary 3.4 reduces to [49, Corollary 3.11]. + +3. 2. Minkowski’s first inequality for the \(L_{p}\) affine surface area of a graph sum. For + +\(K\in K_{o}^{n}\text{and}p\neq-n\), the \(L_{p}\) affine surface area as\((_{p} K )\) is defined by + +\[\text{as}_{p}(K)=\int_{\partial K} \frac{\kappa_{K}(x)^{\frac{p}{n+p}}}{\langle x,N_{K}(x)\rangle^{\frac{n(p-1)}{n+p}}} d\mu_{\partial K}(x),\] (3.22) + +provided the integral exists. The \(L_{p}\) affine surface area is a key affine invariant in the \(L_{p}\) + +Brunn–Minkowski theory. It was introduced by Lutwak [30] for \(p>1\), and by Schütt and diff --git a/outputs/arxiv_math/2503.05396_pg5_pg1_repeat1.md b/outputs/arxiv_math/2503.05396_pg5_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..345d160bb36a5a8a2d88088dc45209e683690631 --- /dev/null +++ b/outputs/arxiv_math/2503.05396_pg5_pg1_repeat1.md @@ -0,0 +1,37 @@ +the function \(\mathcal{Q}_{n_{1},n_{2}}(k)\) which has expected density \(\approx1/N\) into roughly \(N\) many harmonics \(\lambda_{\chi\xi^{\ell}}(k)\). + +Invoking Conjecture \(\mathrm{C}_{1}(\varepsilon)\), we obtain for \(\chi\xi^{\ell}\neq 1\) + +\[\sum_{k≍NX}\Upsilon_{n_{1},n_{2}}(k)\lambda_{\chi\xi^{\ell}}(k)\ll X^{1/3+2\varepsilon/3},\] + +\[k≍NX\] + +which is sufficient provided that \(N0\},H_{-}=\{y\in\mathbb{S}^{2}:y_{3}<0\}\text{and}\mathbb{S}^{1}=\{y\in\mathbb{S}^{2}:y_{3}=0\}. \\ \text{Consider now the projections}f:\mathbb{R}^{2}&\rightarrow H\text{given by} \end{aligned}\] + +Consider now the projections \(f_{\pm}:\mathbb{R}^{2}\rightarrow H_{\pm}\) given by + +\[f_{\pm}(x_{1},x_{2})=\pm\Delta(x_{1},x_{2})(x_{1},x_{2},1),\] + +where \(\Delta(x_{1},x_{2})=(x_{1}^{2}+x_{2}^{2}+1)^{- \frac{1}{2}}\) . These two maps define two copies of \(X\), one + +copy \(X^{+}\text{in}H_{+}\) and one copy \(X^{-}\text{in}H_{-}\). Consider the vector field \(X^{\prime}=X^{+}\cup X^{-}\) The Poincaré compactified vector field \(p(X)\) is the analytic extension of \(X^{\prime}\) from + +defined in \(\mathbb{S}^{2}\setminus\mathbb{S}^{1}\). Note that the infinity of \(\mathbb{R}^{2}\) is identified with the equator \(\mathbb{S}^{1}\). + +\(\mathbb{S}^{2}\setminus\mathbb{S}^{1}\) to \(\mathbb{S}^{2}\) given by \(y_{3}^{n-1}X^{\prime}\). The Poincaré disk \(\mathbb{D}\) is the projection of the closed + +northern hemisphere to \(y_{3}=0\) under \((y_{1},y_{2},y_{3})\mapsto(y_{1},y_{2})\) (the vector field given by this projection is also denoted by \(p(X))\). Note that to know the behavior \(p(X)\) near \(\mathbb{S}^{1}\) is the same than to know the behavior of \(X\) near the infinity. We define + +the local charts of \(\mathbb{S}^{2}\) by \(U_{i}=\{y\in\mathbb{S}^{2}:y_{i}>0\}\text{and}V_{i}=\{y\in\mathbb{S}^{2}:y_{i}<0\}\) for + +\(i\in\{1,2,3\}\). In these charts we define \(\phi_{i}:U_{i}\rightarrow\mathbb{R}^{2}\text{and}\psi_{i}:V_{i}\rightarrow\mathbb{R}^{2}\) by + +\[\phi_{i}(y_{1},y_{2},y_{3})=-\psi_{i}(y_{1},y_{2},y_{3})=\left( \frac{y_{m}}{y_{i}} , \frac{y_{n}}{y_{i}} \right),\] + +where \(m\neq i,n\neq i\text{and}m0\) and a very weak solution for \(-\min(1,\xi)<\lambda\leq0\), where + +\[\xi=\min\{\text{Re}\lambda>0:\lambda\text{satisfies}(3.2)\}\] + +in the case of Dirichlet boundary conditions. The weak solution \((u,p)\in H^{1}(\Omega)\times L_{0}^{2}(\Omega)\) is defined by + +\[\begin{aligned} (\nabla u,\nabla v)-(\nabla\cdot v,p)&=0\forall v\in H_{0}^{1}(\Omega), \\ (\nabla\cdot u,q)&=0\forall q\in L_{0}^{2}(\Omega). \end{aligned}\] + +The very weak solution \((y,p)\in L^{2}(\Omega)\times P^{\prime}\text{with}P=\{v\in H^{1}(\Omega)\cap L_{0}^{2}(\Omega):r^{-1}v\in L^{2}(\Omega)\}\) is defined by + +\[(u,-\Delta v+\nabla q)-(\nabla\cdot v,p)=\langle u,qn-\partial_{n}v\rangle_{\Gamma}\forall(v,q)\in\mathcal{V}\] + +\[\text{where}\mathcal{V}:=\{(v,q)\in H_{0}^{1}(\Omega)\times L_{0}^{2}(\Omega):-\Delta v+\nabla q\in L^{2}(\Omega),\nabla\cdot v\in P\},\text{see}[\text{ALP}24].\] diff --git a/outputs/arxiv_math/2503.05442_pg14_pg1_repeat1.md b/outputs/arxiv_math/2503.05442_pg14_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..6d1a3c3167b292fe71806115ec07793ea193ff05 --- /dev/null +++ b/outputs/arxiv_math/2503.05442_pg14_pg1_repeat1.md @@ -0,0 +1 @@ +O2 Exactly one vertex in T has outgoing neighbor in \(V(BS_{n}\setminus(BS_{n-1}^{1}\oplus\) BS \(\oplus_{n-1}^{2} BS_{n-1}^{3}))\). Refer to Figure 4 Subcase 3.2 O2 , w.l.o.g, assume that b has an outgoing neighbor \(b^{\prime}inV(BS_{n}\setminus(BS_{n-1}^{1}\oplus\) BS \(\oplus_{n-1}^{2}\) BS \()_{n-1}^{3}),\{c^{+}\}\subseteq V(BS_{n-1}^{1}),\{a^{-},c^{-}\}\subseteq V(BS_{n-1}^{2})\), \(\{a^{+}\}\subseteq V(BS_{n-1}^{3})\). Similarly to the proof in Subcase 3.1 in Part 1, we can find \(M_{i}\subseteq H_{i}\text{such that}|M_{i}|=2k-1(i=1,3)andM_{2}\subseteq H_{2}\text{such that}|M_{2}|=2k\). Since \(\kappa(BS_{n-1})=4k+1\), we can construct one 4k-fan from a to \(M_{1}\cup M_{2}\cup\{c^{+}\}\), one 4k-fan from b to \(M_{1}^{+}\cup M_{3}\cup\{a^{-},c^{-}\}\), one 4k-fan from c to \(M_{2}^{+}\cup M_{3}^{+}\cup\{a^{+}\}\). Thus, we obtain the desired paths structure. Let \(b_{1}=b^{\prime}\). Additionally, there must exist \(b_{2}\in N_{BS_{n-1}^{2}}(b)\) such that b2 does not appear on any of \((b,M_{1}^{+}\cup M_{3}\cup\{a^{-},c^{-}\})-\text{paths}\), since \(\kappa(BS_{n-1}^{2}-\{b_{2}\})=\) (2n \(-5)-1=4kand|M_{1}^{+}\cup M_{3}\cup\{a^{-},c^{-}\}|=4k\). O3 Exactly two vertices in T have outgoing neighbor in \(V(BS_{n}\setminus(BS_{n-1}^{1}\oplus BS_{n-1}^{2}\oplus BS_{n-1}^{3}))\). Refer to Figure 4 Subcase 3.2 O3 , w.l.o.g, assume that \(\{a^{-}\}\subseteq V(BS_{n-1}^{2}),\{a^{+}\}\subseteq V(BS_{n-1}^{3})andb,c\) have outgoing neighbors \(b^{\prime},c^{\prime}inV(BS_{n}\setminus(BS_{n-1}^{1}\oplus\) BS \(\oplus_{n-1}^{2}\) BS \()_{n-1}^{3})\), respectively. Similarly to the proof of Subcase 3.1 in Part 1, we can find \(M_{i}\subseteq H_{i}suchthat|M_{i}|=2k-1(i=1,3)andM_{2}\subseteq H_{2}suchthat|M_{i}|=2k+1\). In addition, there must exist one \((b^{\prime},c^{\prime})-\)path in BS \(\setminus_{n}(BS_{n-1}^{1}\oplus\) BS \(\oplus_{n-1}^{2}\) BS \(n^{3}-1^{)}\). Since \(\kappa(BS_{n-1})=4k+1\), we can construct one 4k-fan from a to \(M_{1}\cup M_{2}\), one (4k − 1)-fan from b to \(M_{1}^{+}\cup M_{3}\cup\{a^{-}\}\), one (4k + 1)-fan from c to \(M_{2}^{+}\cup M_{3}^{+}\cup\{a^{+}\}\). Thus, we obtain the desired paths structure. Moreover, there must exist \(b_{1},b_{2}\in N_{BS_{n-1}^{2}}(b)suchthatb_{1},b_{2}do\) not appear on any of \((b,M_{1}^{+}\cup M_{3}\cup\{a^{+}\})-\text{paths}\), since \(\kappa(BS_{n-1}^{2}-\{b_{1},b_{2}\})=\) (2n \(-5)-2=4k-1\text{and}|M_{1}^{+}\cup M_{3}\cup\{a^{+}\}|=4k-1\). O4 Each vertex in T has at least one outgoing neighbor in \(V(BS_{n}\setminus(BS_{n-1}^{1}\oplus BS_{n-1}^{2}\oplus BS_{n-1}^{3}))\). Refer to Figure 4 Subcase 3.2 O4 , w.l.o.g, assume that \(a,b\) and c have outgoing neighbors \(a^{\prime},b^{\prime}andc^{\prime}inV(BS_{n}\setminus(BS_{n-1}^{1}\oplus\) BS \(\oplus_{n-1}^{2}\) BS \()_{n-1}^{3})\), respectively. Since \(a^{\prime},b^{\prime}andc^{\prime}\) cannot all be equal, suppose that \(b^{\prime}\Leftrightarrow a^{\prime}andb^{\prime}\Leftrightarrow c^{\prime}\). Similarly to the proof in Subcase 3.1 in Part 1, we can find \(M_{i}\subseteq H_{i}\text{such that}|M_{i}|=2k(i=1,3)\) and \(M_{2}\subseteq H_{2}suchthat|M_{2}|=2k+1\). In addition, there must exist one \((a^{\prime},c^{\prime})-\)path in BS \(\setminus_{n}(BS_{n-1}^{1}\oplus\) BS \(\oplus_{n-1}^{2}\) BS \(n^{3}-1^{)}\). Since \(\kappa(BS_{n-1})=4k+1\), we can construct one (4k + 1)-fan from a to \(M_{1}\cup M_{2}\), one 4k-fan from b to \(M_{1}^{+}\cup M_{3}\), one (4k + 1)-fan from c to \(M_{2}^{+}\cup M_{3}^{+}\) . Thus, we obtain the desired paths structure. Let \(b_{1}=b^{\prime}\). Additionally, there must exist \(b_{2}\in N_{BS_{n-1}^{2}}(b)\) such that b2 does not occur on any of \((b,M_{1}^{+}\cup M_{3})-\text{paths}\), since \(\kappa(BS_{n-1}^{2}-\{b_{2}\})=\) (2n \(-5)-1=4kand|M_{1}^{+}\cup M_{3}|=4k\). Combining the results of Part 1 and Part 2, the proof is completed. □ diff --git a/outputs/arxiv_math/2503.05468_pg21_pg1_repeat1.md b/outputs/arxiv_math/2503.05468_pg21_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..400a2e716adbfbc56858a91ca1ed3cb938f1682e --- /dev/null +++ b/outputs/arxiv_math/2503.05468_pg21_pg1_repeat1.md @@ -0,0 +1,23 @@ +We can then assign a label from the infinite Ulam-Harris tree \(\mathcal{I}:=\bigcup_{n\in\mathbb{N}_{0}}\mathbb{N}^{n}\) to each individual \(u=(u_{1},\ldots,u_{n})=u_{1}\ldots u_{n}\in\mathcal{I}\) according to its ancestral line, namely, + +\[\emptyset\rightarrow u_{1}\rightarrow u_{1}u_{2}\rightarrow...\rightarrow u_{1}...u_{n}=u,\] + +where \(u_{1}\) is the \(u_{1}-\)th child of the ancestor, \(u_{1}u_{2}\) is the \(u_{2}-\)th child of \(u_{1}\), and so on. We write \(|u|=n\text{if}u\in\mathbb{N}^{n}\). Now let \(\boldsymbol{\xi}_{u}=(\xi_{u}^{i,j})_{i,j=1,\ldots,p},u\in\mathcal{I}\setminus\{\emptyset\}\) be a family of i. i. d. copies of \(\boldsymbol{\xi}=:\boldsymbol{\xi}_{\emptyset}\). We suppose that \((\Omega,\mathcal{A},\mathbf{P})\) is a probability space on which all \(\boldsymbol{\xi}_{u},u\in\mathcal{I}\) are defined and i. i. d. and independent of \(\tau(\emptyset)\), the random variable that gives the ancestor’s type. We write \(\mathbf{P}^{i}\) if the ancestor’s type is \(i\in[p]\). We denote the associated expected value operators by E and \(\mathbf{E}^{i}\), respectively. Each individual \(u\in\mathcal{T}\) that is ever born has a clearly defined type \(\tau(u)\in[p]\) and a time of birth \(S(u)\), namely, \(S(\emptyset)=0\) and, recursively, + +\[S(uk)=S(u)+X_{u,k},u\in\mathcal{I},k\in\mathbb{N}.\] + +For completeness, we write \(S(u)=\infty\) if individual \(u\) is never born. Now suppose that for each \(u\in\mathcal{I}\) there exist a random vector \(\boldsymbol{\zeta}_{u}=(\zeta_{u}^{1},\ldots,\zeta_{u}^{p})^{\mathrm{T}}\) taking values in \([0,\infty]^{p}\) and a product-measurable, separable random characteristic \(\varphi_{u}=(\varphi_{u}^{1},\ldots,\varphi_{u}^{p})^{\mathrm{T}}\) : \(\Omega\times\mathbb{R}\rightarrow\mathbb{R}^{p}\) (taking values in the space of \(p-\)dimensional real column vectors) such that the \((\boldsymbol{\xi}_{u},\boldsymbol{\zeta}_{u}),u\in\mathcal{I}\) are i.i.d. and the \(\varphi_{u},u\in\mathcal{I}\) are identically distributed. Further, we assume throughout that for every \(n\in\mathbb{N}\), the \(\varphi_{u},u\in\mathbb{N}^{n}\) are independent and independent of the \((\boldsymbol{\xi}_{u},\boldsymbol{\zeta}_{u},\varphi_{u}),|u|0,\text{if}m\geq m(u)\) for a suitable \(m(u)\in\mathbb{N}\), + +| k | k 1 − γ | k | +| --- | --- | --- | +| k=s | k=s+1 | k=s | + +\[\begin{aligned} \tilde{\pi}\Big(-\sum_{k=s}^{r-1} \frac{1}{k} ,-\sum_{k=s+1}^{r-1} \frac{1}{k} \Big]&= \frac{\tilde{\beta}}{1-\gamma} \exp\left(-(1-\gamma)\sum_{k=s}^{r-1} \frac{1}{k} \right)(e^{\frac{1-\gamma}{s+1}}-1) \\ &\leq\left( \frac{\tilde{\beta}}{s+1} + \frac{C}{(bum)^{2}} \right)\exp\left(-(1-\gamma)(\log\left( \tfrac{r-1}{s-1} \right)- \tfrac{C}{bum} )\right) \\ &\leq\beta s^{-\gamma}r^{\gamma-1}. \end{aligned}\] + +Hence, using that \(1-e^{-x}\leq x\), we get (5). diff --git a/outputs/arxiv_math/2503.05476_pg14_pg1_repeat1.md b/outputs/arxiv_math/2503.05476_pg14_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..1578f0f0325bffa94f405cbe9d325c3477c21a5e --- /dev/null +++ b/outputs/arxiv_math/2503.05476_pg14_pg1_repeat1.md @@ -0,0 +1,29 @@ +\[\sum_{k\geq n}\mu_{+}^{-1}\] + +\((_{+}\delta^{k})/k\text{for}n\in\mathbb{N}\), and the function \(\psi\) by + +\[\psi(t):=\sum_{n\in\mathbb{N}_{0}}\delta_{(a_{n+1},a_{n}]}^{n_{1}}(t),t\geq0.\] + +Then + +\[F(t)\leq\psi(4t)\text{for all}t\leq t_{0}:= \frac{\delta}{4(2+\delta)} \mu_{+}^{-1}(1).\] + +Proof. The assumptions imply that \(\mu_{+}\) is continuous and strictly increasing and hence is the cumulative mass function of a continuous measure on \([0,\infty)\). Moreover, \(\mu_{+}\) is a bijection of \([0,\infty)\), and its generalized inverse \(\mu_{+}^{-1}\) is simply the inverse function. Let \(\psi_{0}=\psi,\text{and for}n\in\mathbb{N}\), define \(\psi_{n}\) recursively by + +\[\psi_{n+1}(t):=1-\exp\left(-\psi_{n}(t)-\int_{0}^{t/4}\psi_{n}(t-4x)\mu_{+}(dx)\right),t\geq0.\] + +Denote by T denoting the smoothing transform associated with the Poisson point process with intensity measure \(\mu=\delta_{0}+\mu_{+}\) as given by (3.2) and its representation (5.1). By induction, we observe that + +\[1-\psi_{n}(4t)=\mathbf{T}^{n}\left[1-\psi(4(\cdot))\right](t),t\geq0.\] + +Thus, by Theorem 3.4(b), we infer that \(\psi_{n}(4t)\rightarrow F(t)\text{as}n\rightarrow\infty\text{for all}t>0\). To complete the proof, it remains to show that \(\psi_{n}(t)\leq\psi(t)\) holds for all \(t\leq4t_{0}\text{and}n\in\mathbb{N}_{0}\). We will prove the slightly stronger statement \(\psi_{n}(t)\leq\psi(t)\text{for all}t\leq a_{k_{0}}\) via induction on \(n\), where \(k_{0}\in\mathbb{N}_{0}\) is chosen such that + +\[a_{k_{0}+1}<4t_{0}\leq a_{k_{0}}.\] (5.3) + +For \(n=0\), the claim is trivially true (base case). For the inductive step, assume \(\psi_{n}(t)\leq\psi(t)\) for all \(t\leq a_{k_{0}}\text{and some}n\in\mathbb{N}_{0}\). We will show that \(\psi_{n+1}(t)\leq\psi(t)\text{for all}t\in(a_{k+1},a_{k}]\text{and}k\geq k_{0}\). If \(k_{0}=0,\text{for}t>a_{1}\) we clearly have \(\psi(t)=1\geq\psi_{n+1}(t)\). Thus we can assume \(k\geq k_{0}\vee1\). Let \(t\in(a_{k+1},a_{k}]\text{for}k\geq k_{0}\vee1\). Then we have + +\[\begin{aligned} \psi_{n+1}(t)&\leq\psi_{n+1}(a_{k})=1-\exp\left(-\psi_{n}(a_{k})-\int_{a_{k}/}^{a_{k}/4}\psi_{n}(a_{k}-4x)\mu_{+}(dx)\right) \\ &\leq1-\exp\left(-\psi(a_{k})-\int_{0}^{0_{4}}\psi(a_{k}-4x)\mu_{+}(dx)\right) \\ &=1-\exp\left(-(\psi(t)+I_{k})\right) \end{aligned}\] + +where \(I_{k}\) represents the integral in the exponent. To estimate \(I_{k}\), we first note that + +\[\begin{aligned} I_{k}&=\int_{0}^{a_{k}/4}\psi(a_{k}-4x)\mu_{+}(dx)=\sum_{j\geq k}\int_{[a_{k}}\psi(a_{k}-4x)\mu_{+}(dx) \\ -a_{j},a_{k}-a_{j+1})/4 \\ &=\sum_{j\geq k}\delta^{j}\left[\mu_{+}\left( \frac{a_{k}-a_{j+1}}{4} \right)-\mu_{+}\left( \frac{a_{k}-a_{j}}{4} \right)\right] \\ &=\delta^{k}\sum_{j=0}^{\infty}\delta^{j}\left[\mu_{+}\left( \frac{a_{k}-a_{k+j+1}}{4} \right)-\mu_{+}\left( \frac{a_{k}-a_{k+j}}{4} \right)\right] \\ &=\psi(t)\sum_{j=0}^{\infty}\delta^{j}(1-\delta)\mu_{+}\left( \frac{a_{k}-a_{k+j+1}}{4} \right), \end{aligned}\] diff --git a/outputs/arxiv_math/2503.05503_pg13_pg1_repeat1.md b/outputs/arxiv_math/2503.05503_pg13_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..9782450b60e530d5d8d7234fd2a7dc839dc1428e --- /dev/null +++ b/outputs/arxiv_math/2503.05503_pg13_pg1_repeat1.md @@ -0,0 +1,43 @@ +Our analysis is structured as follows. We introduce the correct functional analytic setup in Section 3, where we also prove the Fréchet differentiability of \(\mathcal{J}\text{and}\mathcal{C}\) and give formulas for their derivatives. In Section 4 we provide the proof of Theorem 2.1. + +3. Analysis of the functional \(\mathcal{F}\) + +In this section, we study the regularity properties of the functional \(\mathcal{F}\) and derive and analyze its linearization. First, we introduce the underlying function spaces for functions + +\(\eta:\mathbb{S}^{2}\rightarrow\mathbb{R}\) and recall some important facts on Sobolev spaces on the sphere. + +3. 1. Sobolev spaces on the sphere. We denote by \(\mathrm{L}^{2}(\mathbb{S}^{2})\) the space of square-integrable functions on the sphere equipped with the uniform measure \(d\sigma(x)=\sin(\theta)d\varphi d\theta\) (which + +is the same as the Hausdorff measure on \(\mathbb{S}^{2}\) up to a constant factor) and we write \(\langle\cdot,\cdot\rangle\) + +for the induced L2 scalar product. + +Our analysis is based on the fact that spherical harmonics \(\{Y_{l}^{m}:l\in\mathbb{N}_{0},-l\leq m\leq l\}\) form an orthonormal eigenbasis of the Laplace–Beltrami operator on the sphere \(\Delta_{\mathbb{S}^{2}}\) + +with respect to the \(\mathrm{L}^{2}(\mathbb{S}^{2})\) scalar product. The corresponding eigenvalues \(-l(l+1)\) have the multiplicity \(2l+1\). We have the expansion + +\[f=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\langle f,Y_{l}^{m}\rangle Y_{l}^{m},\] (3.1) + +for any \(f\in\mathrm{L}^{2}(\mathbb{S}^{2})\). We recall that spherical harmonics can be expressed as + +\[Y_{l}^{m}(\theta,\varphi)=c_{l,m}P_{l}^{m}(\cos\theta)e^{im\varphi},\] (3.2) + +where \(c_{l,m}= \sqrt{\frac{(2l+1)}{4\pi} \frac{(l-m)!}{(l+m)!}}\) are positive constants and \(P_{l}^{m}\) are the associated Legendre polynomials. We refer the reader e.g. to [44] for background reading. + +For \(\beta>0\), we define the Sobolev space \(\mathrm{H}^{\beta}(\mathbb{S}^{2})\) as the space of all functions \(f\in\mathrm{L}^{2}(\mathbb{S}^{2})\) with + +\[\Vert f\Vert_{\mathrm{H}^{\beta}(\mathbb{S}^{2})}^{2}=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}(1+l)^{2\beta}|\langle f,Y_{l}^{m}\rangle|^{2}<\infty.\] + +These spaces can be equivalently defined via smooth charts [27], and thus, they arise as + +the trace spaces of \(\mathrm{H}^{\beta+ \frac{1}{2}}(B_{1}(0))\), cf. (3.8). For integer exponents \(\beta\in\mathbb{N}\), they coincide with the classical Sobolev spaces defined via differentiation on the manifold. For notational convenience, we introduce a subspace of \(\mathrm{H}^{\beta}(\mathbb{S}^{2})\) that reflects the symmetric setting we restrict to in (1.23). + +Definition 3.1. Let \(\beta\geq0\). We define + +\[\mathrm{H}_{\text{sym}}^{\beta}(\mathbb{S}^{2}):=\left\{f\in\mathrm{H}^{\beta}(\mathbb{S}^{2}):f=f(\theta)\text{with}f\left( \frac{\pi}{2} -\theta\right)=f\left( \frac{\pi}{2} +\theta\right)\right\},\] + +the subspace of all axisymmetric functions in \(\mathrm{H}^{\beta}(\mathbb{S}^{2})\), which are symmetric in \(x_{3}\). The following characterization will be beneficial for our analysis. + +Lemma 3.2. For all \(\beta\geq0\) we have + +\[\mathrm{H}_{\text{sym}}^{\beta}(\mathbb{S}^{2}):=\left\{f\in\mathrm{H}^{\beta}(\mathbb{S}^{2}):\langle f,Y_{l}^{m}\rangle=0\text{if}l\text{is odd or}m\neq 0\right\}.\] (3.3) diff --git a/outputs/arxiv_math/2503.05506_pg11_pg1_repeat1.md b/outputs/arxiv_math/2503.05506_pg11_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..9db875fe3e8d3b97dbc2da78b9b5bd04c373bb31 --- /dev/null +++ b/outputs/arxiv_math/2503.05506_pg11_pg1_repeat1.md @@ -0,0 +1,25 @@ +![](figure) + +![](figure) + +Figure 8: When \(1\leq j\leq25s\), the edge \(v_{i}^{j}v_{i}^{j+1}\) of \(G\) lies in a cycle of each length in \([3,600s]\) + +Consider the edges \(v_{i}^{1}v_{j}^{1},v_{i}^{50s}v_{j}^{1}\in E_{4},\text{where}1\leq i,j\leq s^{\ell}\). For each \(p\in[3,50s]\), + +\[v_{j}^{1}v_{i}^{1}v_{i}^{2}\ldots v_{i}^{p-2}v_{i}^{50s}v_{j}^{1}\] + +is a cycle of length \(p\) containing \(v_{i}^{1}v_{j}^{1}\text{and}v_{i}^{50s}v_{j}^{1}\) (see Figure 7). Consider the edges \(v^{50s}\) + +\(\in E_{4},\text{where}1\leq i,j\leq s^{\ell}\). For each \(4\leq p\leq50s\), + +\[v_{ij}^{50s50s},v_{i}^{50s+1}v_{j}^{50s}\] + +\[v_{j}^{50s}v_{i}^{50s}v_{i}^{50s+2}\ldots v_{i}^{50s+p-2}v_{i}^{50s+1}v_{j}^{50s}\] + +\[v_{ij}^{50s50s}\text{and}v_{i}^{50s+1}v_{j}^{50s}(\text{see Figure}7).\text{And}v_{j}^{50s}v_{i}^{50s}v_{i}^{50s+1}v_{j}^{50s}\] + +is a \(C_{3}\) + +\[+_{i}^{2}s^{\ell-1}-2s\] + +is a cycle of length \(p\) containing \(v^{50s}\) containing both edges (see Figure 7). Then we can use the similar cycle \(C^{\star}\) as in Claim 8 and the similar replacement to prove every edge in \(E_{4}\) is contained in a cycle of length \(k\text{if}j\not\in \{\tau_{i}^{2},\tau_{i}^{2}+s^{\ell-1},\tau_{i}^{2}+s^{\ell-1}-s,\tau_{i}^{2}+s^{\ell-1}-2s,\tau_{i}^{2}+2s^{\ell-1}- 2s,\ldots,\tau_{i}^{1}\},\text{where}3\leq k\leq600s.\text{When}j\in\{\tau_{i}^{2},\tau_{i}^{2}+s^{\ell-1},\tau_{i}^{2}+s^{\ell-1}-s,\tau_{i}^{2}+s^{\ell-1}-2s,\tau_{i}^{2}+2s^{\ell-1}- 2s,\ldots,v_{\tau_{i}^{1}}\}\), we can replace \(\tau_{i}^{2}\text{with}\hat{\tau}_{i}^{2}:=\tau_{i}^{2}+s\) and add a vertex \(v_{\hat{\tau}_{i}^{2}+s^{\ell-1}-3s}\) between \(v_{\hat{\tau}^{2}}\) and \(v_{\hat{\tau}_{i}^{2}+2s^{\ell-1}-2s}\text{in}C^{\star}\). Then we use a similar replacement to prove such edges in \(E_{4}\) are contained in a cycle of length \(k,3\leq k\leq600s\). Claim 10. Every edge in \(E_{3}\) lies in a cycle of length \(k,3s^{\ell}+100s\leq k\leq v(G)\). Proof of Claim 10. Notice that for every \(3\leq p\leq100s-1\), there is a path of length \(p\) contained in the copy of \(H(s)\text{in}G\) connecting \(v_{i}\text{and}v_{i+1},\text{where}i\in[s^{\ell}]\text{and}v_{i},v_{i+1}\in V(G_{1})\). Moreover, every edge in \(E_{3}\) lies in a path of length \(100s-1\) contained in the copy of \(H(s)\text{in}G\) connecting \(v_{j}\text{and}v_{j+1}\) for some \(j\in[s^{\ell}],\text{where}v_{j},v_{j+1}\in V(G_{1})\). Thus using the Hamilton cycle in \(G_{1}\), we can find a cycle of each length \(k,k\geq3s^{\ell}+100s\) containing \(e\text{for every}e\in E_{3}\). diff --git a/outputs/arxiv_math/2503.05513_pg15_pg1_repeat1.md b/outputs/arxiv_math/2503.05513_pg15_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..a1a86258a3d8a610e11750b32556256de5d17f28 --- /dev/null +++ b/outputs/arxiv_math/2503.05513_pg15_pg1_repeat1.md @@ -0,0 +1,7 @@ +## 5. SEMIPOSITIVE R-PL FUNCTIONS + +Let k be a non-trivially valued non-Archimedean field. We consider a good, strictly k-analytic space X . We refer to 2.5 for the definition of PL functions. Recall from the introduction that an R-PL function \(u:X\rightarrow R\) is called semipositive at \(x\in X\) if the associated residue line bundle \(L_{u}(x)\in\text{Pic}(\text{red}(X,x))_{R}\) is nef, where we refer to \([\text{GJR}21,6.16-6.19]\) for residue line bundles. Equivalently, there is a strictly affinoid neighbourhood U of x such that \(u=h_{L}\) for a formal \(k^{\circ}-\text{model}(U,L)\) of \((U,\mathcal{O}_{U})\text{with}L\) nef \([\text{GJR}21\), Proposition \(7.1]\). We will show first that for an R-PL function on a smooth curve, semipositivity is the same as subharmonicity in the sense of Thuillier’s thesis \([\text{Thu}05\), Définition \(3.1.5]\). Recall that Thuillier defines a subharmonic function on a smooth k-analytic curve X as a semicontinuous function \(u:X\rightarrow R\cup\{-\infty\}\) which is not identically \(-\infty\) on any connected component of X and which has the property that for every harmonic function h on a strictly affinoid domain U of X with \(u|_{\partial U}\leq h|_{\partial U}\), we have \(u|_{U}\leq h\). Proposition 5.1. Let X be a smooth, strictly k-analytic curve and let \(u:X\rightarrow R\) be an R-PL function. Then u is subharmonic at \(x\in X\) in the sense of Thuillier if and only if \(L_{u}(x)\) is nef. Proof. First we treat the case when k is algebraically closed. As explained in 2.5, there exists a compact strictly analytic neighborhood U of x in X , a formal \(k^{\circ}-\text{model}U\) of U, and a formal model \(L\) of \(\mathcal{O}_{U}\text{such that}u|_{U}=h_{L}\). Since k is algebraically closed, we may assume that the special fiber \(U_{s}\) is reduced, which implies that \(U\) is covered by formal affines of the form \(\text{Spf}(\mathcal{A}^{\circ})\) for a strictly k-affinoid algebra \(\mathcal{A}[\text{GJR}21,§2.4]\). It follows from \([\text{Ber}90\), Proposition \(2.4.4]\) that for every irreducible component Y of \(U_{s}\), there is a unique point \(\xi_{Y}\in U\) reducing to the generic point of Y . In the formalism of real valued forms and currents on Berkovich spaces \([\text{CD}12]\), it is well known that \(d^{\prime}d^{\prime\prime}[u]|_{U}\) is the discrete measure on U supported on the points \(\xi_{Y}\) of U for Y an irreducible component of \(U_{s}\), with multiplicity at \(\xi_{Y}\) equal to \(\deg_{L}(Y)\). See \([\text{CD}12,§6.9]\) for a more general statement. Wanner showed that subharmonicity of u on U is equivalent to the positivity of the current \(d^{\prime}d^{\prime\prime}[u]\): see \([\text{Wan}19\), Theorems 1 and 2] and use the fact that X is locally algebraic \([\text{vdP}80]\). Thus \(L_{s}\) is nef if and only if u is subharmonic on U. Suppose that u is subharmonic at x. Shrinking U, we may assume that u is subharmonic on U. This implies that \(L_{s}\) is nef, so it follows from \([\text{GJR}21\), Proposition \(7.1]\text{that}L_{u}(x)\) is nef. Conversely, if \(L_{u}(x)\) is nef, then by \([\text{GJR}21\), Proposition \(7.1]\), we may shrink U so that \(L_{s}\) is nef, which as above implies that u is subharmonic on U. It remains to show that subharmonicity of u at x and numerical effectivity of \(L_{u}(x)\) can be checked after extending scalars to the completion of an algebraic closure \(K= ( \overline{k} )^{\wedge}\). By \([\text{GJR}21\), Proposition \(7.6(2)]\), one can check that \(L_{h}(x)\) is nef after base change to K. The same is true for subharmonicity: this follows directly from Thuillier’s definition and the fact that the base change of a harmonic function remains harmonic \([\text{Thu}05\), Proposition \(2.3.18]\). + +- + +Our next goal is a characterisation of semipositive R-PL functions by restriction to subcurves as in the complex case. We need the following characterization of boundary points: diff --git a/outputs/arxiv_math/2503.05521_pg22_pg1_repeat1.md b/outputs/arxiv_math/2503.05521_pg22_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..59f2fb0535dd791da8e8cb03d201b99ef4e57786 --- /dev/null +++ b/outputs/arxiv_math/2503.05521_pg22_pg1_repeat1.md @@ -0,0 +1,39 @@ +speration function + +\[\tau_{B\times_{f}F}((p,x),(q,y)):=\sup L(\gamma)\] + +where the sup is w.r.t. all future directed causal curves \(\gamma\) that connect \((p,x)\) and \((q,y)\). If there is no such curve, one sets \(\tau((p,x),(q,y))=0.\tau_{B\times_{f}F}\) satisfies the reverse △-inequality + +\[\tau_{B\times_{f}F}((p,x),(q,y))\geq\tau_{B\times_{f}F}((p,x),(r,z))+\tau_{B\times_{f}F}((r,z),(q,y)).\] + +For all \((p,x),(q,y),(r,z)\in I\times F\) . In the following we write also \(\tau_{B\times_{f}F}=\tau\) . The time seperation function has the following properties: + +\(\bullet\tau(y,y^{\prime})=0\text{if}y^{\prime}\text{and}y\) are not cuassally related, + +\(\bullet\tau(y,y^{\prime})>0\text{if}y\ll y^{\prime}\). + +The generalized Lorentzian cone w.r.t. \(-I\), the metric space \(F\) and the function \(f:I\rightarrow(0,\infty)\) is + +\[(I\times F,D,\ll,\leq,\tau)=:-I\times_{f}F.\] + +If \(F\) is an intrinsic metric space, then \(-I\times_{f}F\) is a Lorentzian pre-length space in the sense of [KS18]. Here we collect some properties of \(-I\times_{f}F\) . We always assume that \(F\) is intrinsic. + +\(\bullet-I\times_{f}F\) has the push-up property: if \((s,x)\ll(t,y)\) if and only if there exists a future directed causal curve from \((s,x)\) to \((t,y)\) with + +positive length. Moreover \(I^{+/-}((s,x))\) is open for all \((s,t)\in I\times F\) . + +\(\bullet\text{If}F\) is geodesic, then \(J^{\pm}((s,x))\) is closed for all \((s,x)\in I\times F\) . + +The next theorem is the Lorentzian analogue of Theorem 2.5 above. Theorem 3.1 (Kunzinger, Saemann, Graf, Alexander). Let \(X\) be a strictly intrinsic metric space and let \(\gamma=(\alpha,\beta)\) be a future directed, causal curve that is a maximizer w.r.t. \(Lin-I\times_{f}F\) and parametrized proportional to arclength. Then + +(a) \(\beta\) is a minimizer in \(F\) ; + +(b) (Fiber independence) \(\alpha\) is independent of \(F\) , except for the total height, i.e. the length \(L^{F}(\beta)\) of \(\beta\). More precisely, if \(\hat{F}\) is another strictly intrinsic metric space and \(\hat{\beta}\) is a minimizing geodesic in \(\bar{F}\) with the same length and speed as \(\beta,\text{then}(\alpha,\hat{\beta})\) is a minimizer in + +\(-I\times_{f}\hat{F}\). + +(c) If \(\gamma\) is timelike, then \(\beta\) has speed \(\frac{c_{\gamma}}{2f^{2}\circ\alpha}\) for a constant \(c_{\gamma}\). + +(d) If \(\gamma\) is timelike, then \(- \frac{1}{2} (\alpha^{\prime})^{2}+ \frac{1}{2f^{2}\circ\alpha} =E\) for a constant \(E\). + +We stated above that a generalized cone \(-I\times_{f}F\) over an intrinsic metric space \(F\) is a Lorentzian pre-length space. If \(F\) is locally compact, it was shown in [AGKS23] that \(-I\times_{f}F\) is even a Lorentzian length spaces. Theorem 3.2. Let \(F\) be a locally compact length space. Then \(-I\times_{f}F\) is a strongly causal Lorentzian length space. diff --git a/outputs/arxiv_math/2503.05542_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.05542_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..5b1da3f8ec2f57465897dee738f9b51b9e798814 --- /dev/null +++ b/outputs/arxiv_math/2503.05542_pg2_pg1_repeat1.md @@ -0,0 +1,13 @@ +Consider a potentially high-dimensional linear regression setting under random design, where the number of features \(p\) can possibly exceed the sample size \(n\). We are given i.i.d. observations \((x_{i},y_{i})\), \(i=1,\ldots,n\), of \(\mathbb{R}^{p}-\)valued feature vectors \(x_{i}\text{and}\mathbb{R}-\)valued responses \(y_{i}\) satisfying + +\[y_{i}=x_{i}^{\top}\beta_{0}+\varepsilon_{i},i=1,\ldots,n.\] + +Here, \(\beta_{0}\in\mathbb{R}^{p}\) is the unknown true coefficient vector. The error variables \(\varepsilon_{i}\) satisfy \(\mathbb{E}[\varepsilon_{i}|X_{i}]=0\) and \(\text{Var}(\varepsilon_{i}|X_{i})=\sigma^{2}\) for some noise level \(\sigma>0\). Using the notation \(y:=(y_{1},\ldots,y_{n})^{\top},X:=(x_{1},\ldots,x_{n})^{\top}\) and \(\varepsilon:=(\varepsilon_{1},\ldots,\varepsilon_{n})^{\top}\) for the response vector, the feature matrix and the error vector, respectively, we have the linear model + +\[y=X\beta_{0}+\varepsilon.\] (1.1) + +The ridge regression (RR) estimator of \(\beta_{0}\) is obtained as minimizer of the penalised least squares criterion + +\[\hat{\beta}_{\lambda}^{\text{RR}}:=\operatorname*{arg\,min}_{\beta\in\mathbb{R}^{p}}E_{\lambda}(\beta)\text{with}E_{\lambda}(\beta):= \tfrac{1}{2n} \Vert y-X\beta\Vert^{2}+ \tfrac{\lambda}{2} \Vert\beta\Vert^{2},\] (1.2) + +where \(\lambda\geqslant0\) is the penalty parameter. For \(\lambda=0\) we take the minimum-norm solution for \(\hat{\beta}_{\lambda}^{\text{RR}}\) , see Section 2 below. In practice, the RR estimator \(\hat{\beta}_{\lambda}^{\text{RR}}\) is calculated by iterative solvers, for which standard gradient descent (GD) or conjugate gradients applied to the normal equations (CGNE) are natural choices. The default option for ridge regression in the standard Python package scikit-learn [17] uses conjugate gradients, profiting from its fast numerical convergence. Ali, Kolter and Tibshirani [2] have shown for the unpenalized criterion \((\lambda=0\) in (1.2)) that the regularisation path of gradient flow (GF), the continuous-time analogue of gradient descent, can be very tightly bounded by the regularisation path of ridge regression, with a prediction error at each GF iterate bounded by a small factor times the RR prediction error under a naturally reparametrized ridge penalty \(\lambda^{\prime}\). In Proposition 3.2 below, we generalize this result to gradient flow solving the penalized criterion (1.2). Based on a precise non-asymptotic CG error control, our main result then bounds the prediction error for conjugate gradients after \(t\) iterations by the prediction error for gradient flow at iteration \(\tau_{t}\) (Theorem 3.7). The time reparametrisation \(\tau_{t}\) is genuinely data-dependent due to the nonlinear nature of CG, but explicit in terms of the CG residual polynomial. This comparison result is surprisingly strong and implies that CG has the same regularisation effects as gradient flow and a fortiori ridge regression. The constant involved in the bound only depends on the spectrum of the empirical covariance matrix of the feature vectors \(x_{i}\), which is discussed for polynomial eigenvalue decay, Marchenko-Pastur type spectral distributions and spiked covariance models in Example 3.9. This result is thus in line with other recent comparison and implicit regularisation theorems like Ali, Dobriban and Tibshirani [1] for stochastic gradient flow in regression or Wu et al. [22] for standard gradient descent in logistic regression. The comparison result allows in particular to bound the CG oracle error (the minimal prediction error along the CG regularisation path) by the corresponding GF and RR oracle errors. Establishing that the GD prediction risk is decaying monotonously along the iterates for large penalties \(\lambda\) in Proposition 3.11, another application of the comparison result is the corresponding monotone bound on the CG errors. Since the main results are derived for in-sample prediction risk due to the intricate nonlinear dependencies in the CG iterates, we provide a transfer from in-sample to out-of-sample prediction risk in Proposition 3.13. This high-probability result is essentially valid if the effective rank of the feature covariance matrix is small compared to the sample size. The theoretical results are then illustrated in a high-dimensional simulation and a real data example, where the regularisation paths of CG, GD and RR indeed closely resemble each other, see Section 4. diff --git a/outputs/arxiv_math/2503.05558_pg18_pg1_repeat1.md b/outputs/arxiv_math/2503.05558_pg18_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..88bedab976db57ab96556123238a63cb448d8e43 --- /dev/null +++ b/outputs/arxiv_math/2503.05558_pg18_pg1_repeat1.md @@ -0,0 +1,19 @@ +![](figure) + +![](figure) + +## Michael R. Douglas and Cristofero Fraser-Taliente + +## 718 + +![](figure) + +![](figure) + +Figure 1: Average solution length against number of nodes expanded, for 1000 initial states randomly scrambled between 1000-10000 times. We report results with a ball radius \(R=5\), and no \(T\) -calibration. For beam width 27 or greater, we are able to solve all states. + +![](figure) + +![](figure) + +Figure 2: Average solution length vs beam width. We find a best fit given by \(E=41.9log_{2}(B)^{-1}-1.92,\text{where}B\) is the beam width (trivially related to the number of nodes). Excesses are always even due to parity. We report results with a ball radius \(R=5\), and no \(T\) -calibration. diff --git a/outputs/arxiv_math/2503.05562_pg16_pg1_repeat1.md b/outputs/arxiv_math/2503.05562_pg16_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..ec68fe4591151605b685c44187e35b88c21f6359 --- /dev/null +++ b/outputs/arxiv_math/2503.05562_pg16_pg1_repeat1.md @@ -0,0 +1,60 @@ +## 7 Additional families of graphs + +### 7.1 Asteroidal triple-free graphs + +An asteroidal triple in a graph \(G\) is a set \(T\) of 3 vertices such that \(G[T]\) is independent and every pair of vertices in \(T\) can be joined by a path that avoids the neighborhood of the third vertex in \(T\) . A graph \(G\) is asteroidal triple-free (AT-free) if \(G\) does not contain an asteroidal triple. Various classes of graphs are known to be AT-free, for example, interval graphs, cocomporability graphs and permutation graphs (see e.g. [COS97]). We show that all these classes have a bounded ratio by proving the following theorem. + +Theorem 7.1. For every asteroidal triple-free graph \(G\), we have \(\gamma(G)\leqslant3\cdot\rho(G)\). + +Proof. Our proof is based on the following property. Theorem 7.2. [COS97] Every connected asteroidal triple-free graph contains a pair of vertices such that any path between this pair of vertices is dominating. Let \(G\) be an AT-free graph. By Theorem 7.2 there is a pair of vertices \(u,v\in V(G)\) such that any path between \(u\text{and}v\) is dominating. Let \(\Pi=\pi_{1},\pi_{2},\ldots,\pi_{p}\) be the shortest path between \(u\text{and}v\). Thus, we have \(\gamma(G)\leqslant p\). On the other hand as \(\Pi\) is a shortest path in \(G\), by taking the set of vertices \(\{\pi_{3i+1}|0\leqslant i\leqslant\lfloor p/3\rfloor-1\}\) we get a packing in \(G\). Therefore, \(\rho(G)\geqslant\lfloor p/3\rfloor\geqslant\lfloor\gamma(G)/3\rfloor\). + +### 7.2 Convex graphs + +A bipartite graph \(G=(X\cup Y,E)\) is convex if and only if one of the color classes, without loss of generality \(X\), has an ordering such that for all \(y\in Y\) the vertices adjacent to \(y\) are consecutive in the ordering. We refer to this ordering as convex ordering. + +Theorem 7.3. For every convex graph \(G\), we have \(\gamma(G)\leqslant3\cdot\rho(G)\). + +Proof. We can assume that \(G\) does not have an isolated vertex. We will encode convex graphs using points and intervals in \(\mathbb{R}\). In particular, if the convex ordering of \(X\) is \(x_{1},\ldots,x_{n}\), we use the embedding \(\pi:X\rightarrow\mathbb{R}\) defined as \(\pi(x_{i})=i\), and for each \(y\in Y\) , we associate the interval \(\iota(y)=\{\pi(x):x\in N_{G}(y)\}\), see Figure 4 for an illustration. For subsets \(X^{\prime}\subseteq X\text{and}Y^{\prime}\subseteq Y\) , we use the notation \(\pi(X^{\prime})=\left\{\pi(x):x\in X^{\prime}\right\}\text{and}\iota(Y^{\prime})=\left\{\iota(y):y\in Y^{\prime}\right\}.\) In this encoding, the image of a set \(S\subseteq V(G)\) is a union of points and intervals, corresponding to the vertices in \(S\cap X\text{and}S\cap Y\) , respectively. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
● ● ● x1 x2 x3 x4 x5 x6 x7 x8 ● ● ● ● ●ι(y4)
ι(y3)ι(y5)
● y1 y2 y3 y4 y5 ● ● ● ●ι(y1) →ι(y2)
1 23 4567 8
π(x1) π(x2)π(x3) π(x4)π(x5)π(x6)π(x7) π(x8)
+ +Figure 4: The embeddings \(\pi\text{and}\iota\). diff --git a/outputs/arxiv_math/2503.05572_pg22_pg1_repeat1.md b/outputs/arxiv_math/2503.05572_pg22_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..a75d71d0934103b0995ae2568c9beecf5b8e5f1a --- /dev/null +++ b/outputs/arxiv_math/2503.05572_pg22_pg1_repeat1.md @@ -0,0 +1,9 @@ +permutes the \(\mathbb{Z}_{7}-\)component at \(h\) by a particular even permutation if and only + +if \(hs\neq hs^{\prime}\). Once we can check for equality of nodes, the proof is again similar to the proof of Theorem 4.1, but we check for an accepting tree on the tree obtained by contracting paths. We will construct versions of the \(\phi-\)maps that send information over paths where the tree is not branching, and goodness checks should similarly jump over paths. In the remainder of the proof, we explain how these variants can be constructed. Note that the contracted paths are of length \(n^{d}\) at most. We explain first how to deal with a single length \(\ell\). A minor modification of Equation 1 gives an automorphism \(f\) that performs \(\pi\) in the \(\mathbb{Z}_{7}-\)component at \(h\) if and only if \(ha^{\ell}\) contains \(s\). Namely, we simply replace the conjugating partial shifts of the third track by their \(\ell\)th power. However, we need to check that the intermediate nodes on the path, i.e. \(ha^{\ell^{\prime}}\) for \(\ell^{\prime}<\ell\), do not branch, and that \(ha^{\ell}\) does branch. This can be done since as we showed above, we have in our group automorphisms that permute \(\mathbb{Z}_{7}\) in an arbitrary way depending on whether a given relative node is branching. Specifically, one can use commutator formulas and divide-and-conquer to obtain a polynomial-norm automorphism that cancels the effect of \(f\) if the branching is not correct. The resulting automorphisms can be simply composed for all distinct \(\ell\) to obtain the desired analogs of the \(\phi-\text{maps}\), as their supports are distinct (the \(\ell\)th map can only act nontrivially when the non-branching prefix of the tree starting from \(h\) is of length exactly \(\ell)\). To allow goodness checks to jump over paths, the argument is direct from Barrington’s theorem (taking the maps that check for branching nodes to be to be among the generators). + +## 7 PSPACE-hardness for general groups under the Gap Conjecture + +A sequence \(b_{k}\) is at most polynomial if \(b_{k}=O(k^{d})\text{for some}d\). A function \(b_{k}\) is at least stretched exponential if we have \(b_{k}=\Theta(e^{k^{\beta}})\text{for some}\beta>0,\text{and}\beta\) is called the degree. We again recall the version of Gap Conjecture relevant to us, so let \(\beta\in[0,1)\). Conjecture 7.1 (Conjecture \(C^{*}(\beta)\) of Grigorchuk). A group either has at most polynomial growth, or at least stretched exponential growth with degree \(\beta\). + +We next show that in any group with stretched exponential growth in the above sense, for sufficiently large \(k\) we can find a subtree that fits in a ball of radius polynomial in \(k\), has sufficient branching, and furthermore we can encode this tree in a finite-support configuration of a particular SFT \(X\). We are mainly interested in full shifts, so we will work with SFTs that contain points of finite support, and our first result constructs finite-support points which we will use as markers. This does not require any assumptions on the group. Similar marker constructions can be performed under other (including some strictly weaker) assumptions. See e.g. [2, 19] for such marker constructions. diff --git a/outputs/arxiv_math/2503.05588_pg16_pg1_repeat1.md b/outputs/arxiv_math/2503.05588_pg16_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..3e948211bd94bc40d3a4192c35e08fea678e9cd4 --- /dev/null +++ b/outputs/arxiv_math/2503.05588_pg16_pg1_repeat1.md @@ -0,0 +1,27 @@ +that \(I_{t}\) is surjective. Suppose \(Z\in L^{2}(Y,t,d^{\prime})\) so that \(\Vert Z_{n}-Z\Vert_{2}\rightarrow0forsomeZ_{n}\in\mathcal{E}(Y,t,d^{\prime})\). By isometry we conclude that \(H_{n}:=I_{t}^{-1}(Z_{n})\) is a Cauchy sequence converging to some \(H\in L^{2}(X,t,d^{\prime})\). Finally \(\Vert I_{t}H-Z\Vert=\lim_{n\rightarrow\infty}\Vert Z_{n}-Z\Vert=0\) and therefore \(I_{t}(H)=Z\). Equation (3.11) is evident in the case that \(\gamma\) is a simple function. For general \(\gamma\in L_{t}(X,d^{\prime})\), choose a sequence of simple \(\gamma^{(n)}\text{with}\Vert\gamma^{(n)}-\gamma\Vert_{2}\rightarrow0\). By (3.9) and Itō’s isometry we have + +\[\Vert\Vert_{\Vert}\int_{0}^{t}\gamma^{(n)}(s)dX(s)-\int_{0}^{t}\gamma(s)dX(s)^{\Vert}\Vert_{\Vert2}\rightarrow0,\Vert_{\Vert}\int^{\Vert_{0}^{t}}\gamma^{(n)}(s)dY(s)-\int_{0}^{t}\gamma(s)dY(s)^{\Vert}\Vert_{\Vert2}\rightarrow0\] + +as \(n\rightarrow\infty\). The claim then follows because the continuity of the operator \(I_{t}\) yields + +\[I_{t}\int_{0}^{t}\gamma(s)dX(s)=\lim_{n\rightarrow\infty}I_{t}\int_{0}^{t}\gamma^{(n)}(s)dX(s)=\lim_{n\rightarrow\infty}\int_{0}^{t}\gamma^{(n)}(s)dY(s)=\int_{0}^{t}\gamma(s)dY(s).\] + +## 4 Filtering, smoothing, and prediction + +This section is devoted to optimal linear filtering, prediction and smoothing of partially observed polynomial processes. We let either \(I:=\mathbb{N}orI:=\mathbb{R}_{+}\) and fix a probability space \((\Omega,\mathscr{F},(\mathscr{F}_{t})_{t\in I},\mathbb{P})\) as well as an \(\mathbb{R}^{d}-\)valued adapted process \(X=(X(t))_{t\in I}.IfI=\mathbb{R}_{+}\), we assume \((\mathscr{F}_{t})_{t\in I}\) to be right-continuous. Suppose that the components \(X_{m+1}(t),\ldots,X_{d}(t)\) are observable whereas \(X_{1}(t),\ldots,X_{m}(t)\) are not. We let the subscript \(o\) stand for the observable part of a vector \(x\in\mathbb{R}^{d}\) and let \(H:=(\delta_{m+i,j})_{i=1,\ldots,d-m;j=1,\ldots,d}\), i.e. \(x_{o}:=Hx=(x_{m+1},\ldots,x_{d}).\text{For}\Sigma\in\mathbb{R}^{d\times d}\) we set \(\Sigma_{:,o}:=\Sigma H^{\top}=\Sigma_{1:d,m+1:d},\Sigma_{o,:}:=H\Sigma=\Sigma_{m+1:d,1:d}\) as well as \(\Sigma_{o}:=H\Sigma H^{\top}=\Sigma_{m+1:d,m+1:d}.\) The subscript u standing for the unobservable part of a vector is treated in the same manner. We suppose that \(\mathbb{E}(\Vert X(t)\Vert^{2})<\infty\text{for}t\in I\) and consider the following general filtering problem for fixed \(t\in I\). The goal is to minimise the mean square error \(\mathbb{E}(\Vert X(t)-Y\Vert^{2})\) over all random variables \(Y\) that are measurable with respect to the observable information + +{ } + +\[\mathscr{G}_{t}:=\sigma(X_{o}(s):s\in I,s\leq t).\] (4.1) + +We call the minimiser of (4.1) the optimal filter for \(X\). Regardless of any specific model the optimal filter is then given by the conditional mean \(\hat{X}(t,t):=\mathbb{E}(X(t)|\mathscr{G}_{t}).\) + +### 4.1 Discrete-time linear filtering problems + +Let \(I=\mathbb{N}\). For Gaussian state space models, the optimal filter can be computed recursively: + +Proposition 4.1 (Kálmán filter). Suppose that \(X\) is a linear Gaussian state space model as in Definition 3.1 and set \(C(t):=B(t)B(t)^{\top}\). Let \(\hat{X}(0,-1):=\mathbb{E}(X(0)),\Sigma(0,-1):=Cov(X(0))\) and + +̂ + +\[\begin{aligned} \hat{X}(t+1,t):&=a(t+1)+A(t+1)\hat{X}(t,t), \\ \hat{X}(t,t):&=\hat{X}(t,t-1)+\hat{\Sigma}_{:,o}(t,t-1)\hat{\Sigma}_{o}(t,t-1)^{+}(X_{o}(t)-\hat{X}_{o}(t,t-1)), \\ \hat{\Sigma}(t+1,t):&=A(t+1)\hat{\Sigma}(t,t)A(t+1)^{\top}+C(t+1), \\ \hat{\Sigma}(t,t):&=\hat{\Sigma}(t,t-1)-\hat{\Sigma}_{:,o}(t,t-1)\hat{\Sigma}_{o}(t,t-1)^{+}\hat{\Sigma}_{o,:}(t,t-1) \end{aligned}\] diff --git a/outputs/arxiv_math/2503.05594_pg4_pg1_repeat1.md b/outputs/arxiv_math/2503.05594_pg4_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..7a34ef6db59faceb1a1b238b706979ec0a2af4aa --- /dev/null +++ b/outputs/arxiv_math/2503.05594_pg4_pg1_repeat1.md @@ -0,0 +1,17 @@ +agent is not trading. If \(\rho\) is a.s. symmetric and uniformly positive definite in time, this term induces an exponential decay of the price deviation to zero. This reflects that, following a trade, new orders gradually replenish the order book – a phenomenon known as the resilience effect. We emphasize, however, that our results do not require \(\rho\) to be a.s. symmetric and uniformly positive definite in time. Instead, we impose only weaker assumptions (see, e.g., Assumption 4.3 and Remark 4.4). In particular, our framework also accommodates models where price impact exhibits self-exciting behavior. For a discussion of qualitative effects of such negative resilience in the single-asset situation, we refer to [4]. Furthermore, note that \(\rho\) is not necessarily diagonal, making it another potential source of cross-effects in our model. In addition to reaching the required terminal position \(\xi\), the aim of the agent is to incur minimal costs associated with her trading activities. Let us consider the costs of a block trade at the time \(s\in[0,T]\) when trading according to a strategy \(X\in\mathcal{A}^{fv}(x,d)\) with the associated deviation \(D^{X}\). Observe that immediately prior to \(s\) we have the deviation \(D^{X}(s-)\). By (1) the block trade \(\Delta X(s)\) shifts the deviation \(D^{X}(s-)\) to + +\[D^{X}(s)=D^{X}(s-)+\Delta D^{X}(s)=D^{X}(s-)+\gamma(s)\Delta X(s).\] + +We next take the mid-prices and multiply the mid-prices by the amount of the traded shares. The block trade \(\Delta X(s)\) is thus assigned the costs + +\[\left(D^{X}(s-)+ \tfrac{1}{2} \gamma(s)\Delta X(s)\right)^{\top}\Delta X(s)=D^{X}(s-)\Delta X(s-)+ \tfrac{1}{2} (\Delta X(s))^{\top}\gamma(s)\Delta X(s).\] + +This explanation for the costs of a single block trade motivates the definition of the pathwise costs \(C(x,d,X)\) over the whole trading interval \([0,T]\) by + +\[C(x,d,X)=\int_{[0,T]}(D^{X}(s-))^{\top}dX(s)+ \frac{1}{2} \int_{[0,T]}(\Delta X(s))^{\top}\gamma(s)dX(s).\] + +Then \(E[C(x,d,X)]\) describes the expected costs due to illiquidity when trading according to the finite-variation execution strategy \(X\). This is one of the two summands in the definition of the costs + +\[J^{fv}(x,d,X)=E[C(x,d,X)]+E\left[\int_{0}^{T}(X(s)-\zeta(s))^{\top}\Xi(s)(X(s)-\zeta(s))ds\right](2)\] + +that are to be minimized. The other summand in these costs, \(E[\int_{0}^{T}(X(s)-\zeta(s))^{\top}\Xi(s)(X(s)-\zeta(s))ds]\), can be used to incorporate some kind of risk preference into the model via the choice of the matrix-valued process \(\Xi\), which acts as a penalization, and the \(\mathbb{R}^{n}-\)valued process \(\zeta\), which acts as a running target. We refer to Horst & Xia [36, Section 1.1] for an illustrative example of a possible choice for \(\Xi\). The risk term is a further possible source of cross-effects in our model, since \(\Xi\) does not need to be diagonal. To summarize, the stochastic control problem to minimize the costs (2) over all strategies \(X\in\mathcal{A}^{fv}(x,d)\) models the agent’s task to reach the terminal position \(\xi\) from the diff --git a/outputs/arxiv_math/2503.05606_pg16_pg1_repeat1.md b/outputs/arxiv_math/2503.05606_pg16_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..9b2750110de7ac7655fd117d765567826069503a --- /dev/null +++ b/outputs/arxiv_math/2503.05606_pg16_pg1_repeat1.md @@ -0,0 +1,23 @@ +Next, \(F\) is compact. In fact, let \(B\subset C^{0}([t_{0},T];\mathbb{R}^{d})\) be a bounded set, let us show that \(\overline{F(B)}\) is compact. First, \(F(B)\subset C^{0}([t_{0},T];\mathbb{R}^{d})\) is bounded by (3.22). It remains to prove that for all \(t_{*}\in[t_{0},T]\), the following holds + +\[\sup_{z\in B}|F(z)(t)-F(z)(t_{*})|-_{t}-_{\rightarrow}-_{t}\rightarrow_{*}0.\] (3.23) + +One has immediately from (3.21) and (3.22) that for some \(M_{1}>0\), it holds + +\[|F(z)(t)-F(z)(t_{*})|=|x_{u_{z}}(t)-x_{u_{z}}(t_{*})|\leq(\Lambda_{3}\|x_{u_{z}}\|_{\infty}+\|B\|_{\infty}\|u_{z}\|_{\infty}+\|f\|_{\infty})|t-t_{*}|\leq M_{1}|t-t_{*}|\] + +for every \(t\in[t_{0},T]\). This proves (3.23) and, therefore, that \(F\) is compact by the Ascoli-Arzelà theorem. Finally, the set \(\{z\in C^{0}([t_{0},T];\mathbb{R}^{d})|z=\theta F(z)\text{for some}\theta\in[0,1]\}\) is bounded thanks to (3.21) and (3.22). According to Schaefer’s fixed point theorem, \(F\) admits at least one fixed point \(z\in C^{0}([t_{0},T];\mathbb{R}^{d})\), i.e., \(z=F(z)=x_{u_{z}}\). + +- + +Remark 3.7. It is worth noting that one can also define the map \(F\) with the following control function + +\[v_{z}(t)=B_{z}(t)^{\top}D\Phi_{t,T}^{z}(x_{v_{z}}(t))^{\top}(N_{2}^{z}(v_{z}))^{-1}\left(x^{1}-\Phi_{t_{0},T}^{z}(x^{0})-\int_{t_{0}}^{T}D\Phi_{s,T}^{z}(x_{v_{z}}(s))f(s,z(s))ds\right).\] + +## 4. Conclusion + +This work extends the methodology of control synthesis to general control-affine systems of the form (1.1), thereby advancing the theory of nonlinear controllability and providing a constructive framework for control design that is amenable to both numerical verification and practical implementation. The global controllability results established in Section 2 for systems of the form \(\dot{x}(t)= N_{t}(x(t))+B(t)u(t)\) have broad applicability, including to recurrent neural networks of Hopfieldtype [12], as illustrated in Example 4, and more generally to nonlinear systems in Lur’e form (see, for instance, [8] and references therein). In particular, we derived sufficient conditions—natural nonlinear extensions of classical criteria from the linear setting—that guarantee that the reachable set from any given initial state \(x^{0}\in\mathbb{R}^{d}\) covers the entire space \(\mathbb{R}^{d}\). While these results represent a substantial contribution to the study of nonlinear control, it is important to emphasize that the obtained controllability results are pointwise in nature, in the sense that the sufficient conditions explicitly depend on the initial state \(x^{0}\). A natural avenue for future work is to investigate whether these conditions could be shown to be both necessary and uniform across the state space, thereby providing a complete characterization of global controllability for the class of systems considered. Such a result would mirror the celebrated equivalence between controllability and Gramian invertibility in the linear case, thus achieving a deeper understanding of the nonlinear counterpart. + +## Appendix A. Proofs of some of the results stated in the main text + +In this section, we present the proof of some of the results presented in the main text. We start with the following. diff --git a/outputs/arxiv_math/2503.05610_pg5_pg1_repeat1.md b/outputs/arxiv_math/2503.05610_pg5_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..d3ce218350919bcf8ddae76ac90249d1224c6d50 --- /dev/null +++ b/outputs/arxiv_math/2503.05610_pg5_pg1_repeat1.md @@ -0,0 +1,29 @@ +### 3.1 Zero Infimum Spacing + +The next theorem gives a condition under which the infimum spacing is zero. Here our main result is Theorem 2, where we show that the existence of a nonzero repulsive fixed point of the spectral decimation function with multiplier, larger than the zero fixed point indicates zero infimum spacing in the spectrum. + +Theorem 2. Assume \(\Delta\) admits spectral decimation with spectral decimation + +function \(R\) and suppose \(0,\zeta>0\) are fixed point of \(R\text{with}|R^{\prime}(\zeta)|>\big|R^{\prime}(0)\big|>1\). Then \(\inf\{|\lambda-\lambda^{\prime}|:\lambda\neq \lambda^{\prime},\lambda,\lambda^{\prime}\in\sigma(\Delta)\}=0\). + +Proof. Let \(\phi_{\zeta}\) be the inverse branch of \(R\text{with}\zeta\) in it’s range. Note that \(\zeta\) is in the Julia set of \(R\) and is an attracting fixed point of \(\phi_{\zeta}\) and hence we may choose \(n,\text{and}x_{1},x_{2}\in\sigma(\Delta_{n})\) so that \(\phi_{\zeta}^{m}(x_{j})\rightarrow\zeta,j=1,2\) [HSTZ11]. Therefore, we have, + +\[\Big|\phi_{\zeta}^{m}(x_{1})-\phi_{\zeta}^{m}(x_{2})\Big|=_{=} \begin{vmatrix} (\phi_{\zeta}^{m}(\gamma_{m}))^{\prime} \\ \phi_{\zeta}^{\prime}(\phi_{\zeta}^{m-1}(\gamma \end{vmatrix} |_{m}x)^{1})\Big|^{-}\ldots^{x_{2}}\Big|\phi^{|_{\prime_{\zeta}}}(\gamma_{m})\Big||x_{1}-x_{2}|\] + +Note that \(\phi_{\zeta}\) is a continuous injective map and hence monotone. Therefore, it is no loss to assume, \(\phi_{\zeta}^{k}(x_{1})\leq\phi_{\zeta}^{k}(\gamma_{m})\leq\phi_{\zeta}^{k}(x_{2}),\forall k\). + +Thus, \(\Big|\phi_{\zeta}^{k}(\gamma_{m})-\zeta\Big|\leq\max_{j\in\{1,2\}}\Big|\phi_{\zeta}^{k}(x_{j})-\zeta\Big|\) Hence, by continuity of \(R^{\prime}\), given + +\(\delta>0\) there is \(N\) so that \(m\geq k\geq N\Longrightarrow \big|R^{\prime}(\zeta)\big|-\delta\leq\Big|R^{\prime}(\phi_{\zeta}^{k}(\gamma_{m})\Big|\). Thus, \(\leq \frac{1}{(R^{\prime}(\zeta)-\delta)} ,\forall k\geq N\). So for \(m>N\), we have, \(\overline{\Big|R^{\prime}(\phi_{\zeta}^{k}(\gamma_{m}))\Big|}\) + +1 + +\[\begin{aligned} \Big|\phi_{\zeta}^{m}(x_{1})-\phi_{\zeta}^{m}(x_{2})\Big|&\leq \frac{1}{\Big|R^{\prime}(\phi_{\zeta}^{m}(\gamma_{m}))\Big|} \ldots \frac{1}{\big|R^{\prime}(\phi_{\zeta}(\gamma_{m}))\big|} |x_{1}-x_{2}| \\ &\leq \frac{1}{(R^{\prime}(\zeta)-\delta)^{m-N}} \frac{1}{R^{\prime}(\phi_{\zeta}^{N}(\gamma_{m}))} \ldots \frac{1}{R^{\prime}(\phi_{\zeta}(\gamma_{m}))} |x_{1}-x_{2}|. \end{aligned}\] + +Now note that + +\[\Big|c_{\Delta}^{n+j+m}\phi_{0}^{j}\phi_{\zeta}^{m}(x_{1})-c_{\Delta}^{n+j+m}\phi_{0}^{j}\phi_{\zeta}^{m}(x_{2})\Big|=c_{\Delta}^{n}\cdot c_{\Delta}^{j}\Big|(\phi_{0}^{j})^{\prime}(\gamma_{j})\Big|\cdot c_{\Delta}^{m}\Big|\phi_{\zeta}^{m}(x_{1})-\phi_{\zeta}^{m}(x_{2})\Big|.\] + +for some \(\gamma_{j}\) lying between \(\phi_{\zeta}^{m}(x_{1})\text{and}\phi_{\zeta}^{m}(x_{2})\). Choose \(\delta\) so that \(R^{\prime}(0)< \big|R^{\prime}(\zeta)\big|-\delta\) and observe that \(c_{\Delta}^{m}\Big|\phi_{\zeta}^{m}(x_{1})-\phi_{\zeta}^{m}(x_{2})\Big|\rightarrow0\text{as}m\rightarrow\infty\). + +Hence by [Shi96, Proposition 3.1], it remains to show that \(c_{\Delta}^{j}\Big|(\phi_{0}^{j})^{\prime}(\gamma_{j})\Big|\) converges to a finite number as \(j\rightarrow\infty\). diff --git a/outputs/arxiv_math/2503.05614_pg12_pg1_repeat1.md b/outputs/arxiv_math/2503.05614_pg12_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..469333db090f662a882974b329c533d2803104cd --- /dev/null +++ b/outputs/arxiv_math/2503.05614_pg12_pg1_repeat1.md @@ -0,0 +1,49 @@ +### 5.2 Arithmetic Duality and Functional Equations + +The connection between the DACC framework and L-functions is further strengthened by the relationship between Poitou-Tate duality and the functional equation of L-functions. + +Proposition 5.5 (Arithmetic Duality and Height Pairing). For an elliptic curve \(E/\mathbb{Q}\), the pairing induced by Poitou-Tate duality on the adelic complex coincides with the Néron-Tate height pairing on \(E(\mathbb{Q})\). + +Proof Sketch. We outline this result in three steps: + +Step 1: Establishing Derived Duality. For each local component, we prove: + +\[R\Gamma(\mathbb{Q}_{v},D_{v})\simeq R\text{Hom}(R\Gamma(\mathbb{Q}_{v},D_{v}),\mathbb{Q}/\mathbb{Z}(1))[1]\] + +For the global component: + +\[R\Gamma_{\text{global}}(E,D)\simeq R\text{Hom}(R\Gamma_{\text{global}}(E,D),\mathbb{Q}/\mathbb{Z}(1))[1]\] + +Using the mapping cone construction, we derive the duality for the adelic complex. + +Step 2: Connection to Height Pairing. For \(P,Q\in E(\mathbb{Q})\), we show that the pairing \(\langle P,Q\rangle\) induced by duality equals the Néron-Tate height pairing \(\hat{h}(P,Q)\) by analyzing local and global components. + +For the archimedean component, we use the connection between periods and heights. + +For non-archimedean components, we use the theory of local heights and their relation to Galois cohomology. + +The global pairing emerges as the sum of these local contributions, matching the Néron-Tate height pairing. + +Step 3: Determinant Interpretation. The determinant of this pairing on a basis of \(E(\mathbb{Q})/E(\mathbb{Q})_{\text{tors}}\) gives the regulator \(R_{E}\), which explains its appearance in the formula for \(\det(d_{r})\). + +The proof establishes explicit isomorphisms in the derived category and verifies the duality relations at each place through detailed Galois cohomology calculations. + +Theorem 5.6 (Arithmetic Duality). The adelic complex \(C^{\bullet}(E)\) satisfies a derived version of Poitou-Tate duality: + +\[C^{\bullet}(E)\simeq RHom(C^{\bullet}(E),\mathbb{Q}/\mathbb{Z}(1))[1]\] + +This duality reflects the functional equation of the L-function: + +\[\Lambda(E,s)=\varepsilon_{E}\cdot\Lambda(E,2-s)\] + +Proof. Step 1: Local and global duality. + +For each local component, we establish: + +\[R\Gamma(\mathbb{Q}_{v},D_{v})\simeq R\text{Hom}(R\Gamma(\mathbb{Q}_{v},D_{v}),\mathbb{Q}/\mathbb{Z}(1))[1]\] + +This follows from local Tate duality in Galois cohomology. + +For the global component: + +\[R\Gamma_{\text{global}}(E,D)\simeq R\text{Hom}(R\Gamma_{\text{global}}(E,D),\mathbb{Q}/\mathbb{Z}(1))[1]\] diff --git a/outputs/arxiv_math/2503.05642_pg4_pg1_repeat1.md b/outputs/arxiv_math/2503.05642_pg4_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..3dd658ce3d8af96b32f65d914c358fab4a56094c --- /dev/null +++ b/outputs/arxiv_math/2503.05642_pg4_pg1_repeat1.md @@ -0,0 +1,40 @@ +For graph \(G\), denote \(l_{u}\) as the label of node \(u,e_{u,v}\) as the shortest path from \(u\) to \(v\) (which may not be unique), and \(d_{u,v}\) as the shortest distance from node \(u\) to \(v\) (which is unique). Borgwardt and Kriegel (2005) define an SP kernel between graphs \(G^{1}=(V^{1},E^{1})\text{and}G^{2}=(V^{2},E^{2})\) as: + +\[\begin{aligned} k_{SP}(G^{1},G^{2})&=_{1}\sum_{,v_{1}\in V^{1},u_{2},v_{2}\in V}k(e_{u_{1},v_{1}},e_{u_{2},v_{2}}) \\ u^{2} \end{aligned}\] + +\(k(\cdot,\cdot)\) compares the labels and lengths of two shortest paths: + +\[k(e_{u_{1},v_{1}},e_{u_{2},v_{2}})=k_{v}(l_{u_{1}},l_{u_{2}})\cdot k_{e}(d_{u_{1},v_{1}},d_{u_{2},v_{2}})\cdot k_{v}(l_{v_{1}},l_{v_{2}})\] + +where \(k_{v}\) is a kernel comparing node labels, \(k_{e}\) is a kernel comparing path lengths. Both \(k_{v}\text{and}k_{e}\) are usually chosen as Dirac kernels, giving the explicit representation of the SP kernel as: + +\[k_{SP}(G^{1},G^{2})= \frac{1}{n_{1}^{2}n_{2}^{2}}_{u_{1}} \sum_{,v_{1}\in V^{1},u_{2},v_{2}\in V^{2}}\mathbf{1}(l_{u_{1}}=l_{u_{2}})\cdot\mathbf{1}(d_{u_{1},v_{1}}=d_{u_{2},v_{2}})\cdot\mathbf{1}(l_{v_{1}}=l_{v_{2}})\] (SP) + +where \(\frac{1}{n_{1}^{2}n_{2}^{2}}\) is introduced as a normalizing coefficient with \(n_{1},n_{2}\) as the number of nodes in graph \(G^{1},G^{2}\), respectively. Note that each node may have more problem-specific features beyond a single label. From here on, we use \(X=(G,F)\) to denote an attributed graph with \(G\) as the underlying labeled graph and \(F\) as node features. Intuitively, we can compare the features of two nodes instead of labels in \(k_{v}\). However, this could unnecessarily reduce the number of matching paths between two graphs, as requiring identical node features is restrictive and may introduce additional subgraph information into path comparison. Another option is to use a more complicated kernel \(k_{v}\) that measures similarity between features of two nodes, which may significantly increase the computational cost of optimization (similarity is computed for all node pairings). Therefore, we borrow from Cui and Yang (2018) the idea to separate the implicit and explicit information of graphs, i.e., the kernel value between two attributed graphs \(X^{1},X^{2}\) becomes: + +\[k(X^{1},X^{2})=\alpha\cdot k_{G}(G^{1},G^{2})+\beta\cdot k_{F}(F^{1},F^{2})\] (3) + +where \(k_{G}\) is any graph kernel, \(k_{F}\) is any kernel over features, and \(\alpha,\beta\) are trainable parameters controlling the trade-off between graph similarity and feature similarity. Appendix A.4 describes an example of \(k_{F}\) . Since node label is usually included as a node feature and considered in \(k_{F}\) term, and comparing labels in Eq. (SP) increases the complexity of our optimization formulations, we propose a simplified shortest-path (SSP) kernel corresponding to an unlabeled SP kernel: + +\[k_{SSP}(G^{1},G^{2})= \frac{1}{n_{1}^{2}n_{2}^{2}}_{u_{1}} \sum_{,v_{1}\in V^{1},u_{2},v_{2}\in V^{2}}\mathbf{1}(d_{u_{1},v_{1}}=d_{u_{2},v_{2}})= \frac{1}{n_{1}^{2}n_{2}^{2}}_{0} \sum_{\leq s<\min(n_{1},n_{2})}D_{s}(G^{1})\cdot D_{s}(G^{2})_{(\text{SSP})}\] + +| kSSP(G1, G2) = 1 X | 1(du1,v1 = du2,v2) = | 1 | X | Ds(G1) · Ds(G2) | | +| --- | --- | --- | --- | --- | --- | +| n21n2 2 | | n21n2 2 | | | (SSP) | +| u1,v1∈V 1,u2,v2∈V 2 | | | 0≤s1\), we get + +\[\Vert u^{\mathcal{A}}\Vert_{L_{x}^{2}}\ll\Vert w\Vert_{L_{x}^{2}},\] + +and hence \(u^{\mathcal{A}}\) can be used as a temporal corrector to cancel the error \(f\). + +Notations. Let \(\mathbb{N}_{+}\) denote the set of positive integers. For \(p\in[1,\infty]\text{and}s\in\mathbb{R}\), we set + +\[L_{x}^{p}:=L^{p}(\mathbb{T}^{3}),W_{x}^{s,p}:=W^{s,p}(\mathbb{T}^{3}),H_{x}^{s}:=H^{s}(\mathbb{T}^{3}),\] + +where \(W_{x}^{s,p}\) is the Sobolev space and \(H_{x}^{s}=W_{x}^{s,2}\) . We also use \(L_{\sigma}^{2}\) for divergence free functions in \(L_{x}^{2}\). Given any Banach space \(\mathbb{X}\), we denote by \(C([0,T];\mathbb{X})\) the space of continuous functions from \([0,T]\) to \(\mathbb{X}\), equipped with the norm \(\Vert u\Vert_{C_{T}\mathbb{X}}:=\sup_{s\in[0,T]}\Vert u(s)\Vert_{\mathbb{X}}\). In particular, for \(N\in\mathbb{N}_{+}\) we set + +\[\Vert u\Vert_{C_{T,x}^{N}}:=\sum_{0\leq m+|\zeta|\leq N}\Vert\partial_{t}^{m}\nabla^{\zeta}u\Vert_{L^{\infty}([0,T];L_{x}^{\infty})},\] + +where \(\zeta=(\zeta_{1},\zeta_{2},\zeta_{3})\) denotes the multi-index and \(\nabla^{\zeta}:=\partial_{x_{1}}^{\zeta_{1}}\partial_{x_{2}}^{\zeta_{2}}\partial_{x_{3}}^{\zeta_{3}}\). We also use the product spaces + +\[\mathcal{L}_{x}^{p}:=L^{p}(\mathbb{T}^{3})\times L^{p}(\mathbb{T}^{3}),\mathcal{H}_{x}^{s}:=H_{x}^{s}\times H_{x}^{s},\mathcal{C}_{x}^{1}:=C_{x}^{1}\times C_{x}^{1}.\] + +Throughout this paper the notation \(a≲b\) means that \(a\leq Cb\) for some constant \(C>0\). diff --git a/outputs/arxiv_math/2503.05698_pg14_pg1_repeat1.md b/outputs/arxiv_math/2503.05698_pg14_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..6f38b736ecdee27ec0f4033caa3dec5f96b1d455 --- /dev/null +++ b/outputs/arxiv_math/2503.05698_pg14_pg1_repeat1.md @@ -0,0 +1,49 @@ +Moreover, we found that, in contrast to the random circuit setting, the \(k-\)design preparation in minimally random quantum circuits occurs in a two-step fashion — similarly to what happens for purity [30–32] and out-of-time-ordered correlators [33, 34] — and provided an analytic description of this phenomenon in the case of dual-unitary circuits. These findings suggest that that the two settings we studied here are more similar to each other than to Haar random circuits, meaning that the setting + +(b) represents a qualitatively more accurate modelling of non-random dynamics than the fully random circuit. Our work raises many interesting questions for future research. In particular, we can identify two compelling directions. One is to rigorously prove some of the statements that are argued or conjectured here. For example, that the design production speed achieved by perfect tensors is the theoretical maximum achievable in a brickwork circuit and whether one can achieve this speed considering perfect tensors in the setting (a). The other is to ask, more generally, whether the our setting (a) is the optimal one for producing \(k-\)designs with a brickwork quantum circuit (i.e. the one requiring the least amount of resources) or can be further improved. + +# ACKNOWLEDGMENTS + +We acknowledge financial support from the Royal Society through the University Research Fellowship No. 201101 (B. B. and J. R.) and the Leverhulme Trust through the Early Career Fellowship No. ECF-2022-324 (K. K.). + +# Appendix A: Proof of Property 1 + +Property 1 follows by straightforward adaptation of Ippoliti and Ho’s Theorem (cf. Sec. 3.C in Ref. [26]). Let us provide a brief proof considering separately the three Cases (a.1), (a.2), and (b.1). + +## 1. Proof for Case (a.1) + +In Case (a.1) the mixed state \(\rho_{t}^{(k)}\) evolves according to + +\[\rho_{t+1}^{(k)}=\mathcal{B}_{k}[\rho_{t}^{(k)}]=\mathcal{U}^{\otimes k}\mathcal{D}_{k}[\rho_{t}^{(k)}](\mathcal{U}^{\dagger})^{\otimes k}(\mathrm{A}1)\] + +where \(\mathcal{U}\) is written in terms of matrices \(U^{(n)}\in\mathcal{A}_{V}\) as in Eq. (1) and + +\[\mathcal{D}_{k}[\cdot]=4^{-1}\sum_{\alpha\in\{I,\sigma^{x},\sigma^{y}}(\alpha_{,\sigma^{2_{z}L}\}})^{\otimes k}\cdot(\alpha_{2L})^{\otimes k}.\] (A2) + +This is very similar to the setting considered in Ref. [26]: the only difference is that \(\mathcal{A}_{V}\) is an open subset of \(U(4)\) and not necessarily of the dual-unitary submanifold. Therefore, we need to show that the Lemma in Appendix E of Ref. [26], the one guaranteeing that one can approximate any unitary by a brickwork circuit of gates taken from \(\mathcal{A}_{V}\) , still holds. Repeating the arguments of Ref. [26] we then need to show that + +\[(UW^{\dagger})_{n,n+1},\] (A3) + +can generate any unitary operation by varying \(n\in\mathbb{Z}_{2L}\), and \(U,W\in\mathcal{A}_{V}\) . Considering \(W\) close to \(U\) we see that one can then generate any arbitrary infinitesimal unitary transformation on the qubits at position \(n\text{and}n+1\). Taking then sufficiently high powers one can generate any unitary transformation on the two qubits. Since \(n\) is arbitrary the statement immediately follows. Following Ref. [26] we then have that the limit + +\[\rho_{\infty}^{(k}=\lim_{t\rightarrow\infty}^{)}\rho_{t}^{(k)},\] (A4) + +exists and commutes with + +\[(\mathcal{U}\sigma^{\alpha}\mathcal{U}^{\dagger})^{\otimes k},\] (A5) + +where \(\mathcal{U}\) is an arbitrary unitary matrix in \(U(2^{2L})\). These two matrices can be chosen in such a way to generate a complete gate set for each pair of qubits on the chain. Therefore, we conclude that \(\rho_{\infty}^{(k)}\) commutes with any unitary matrix of the form \(\mathcal{U}^{\otimes k},\text{with}\mathcal{U}\text{in}U(2^{2L})\). The Schur–Weyl duality [64, 65] then implies that \(\rho_{\infty}^{(k)}=\rho_{H}^{(k)}\) . + +## 2. Proof for Case (a.2) + +In Case (a.2) the channel \(\mathcal{B}_{k}[\cdot]\) can again be written as in Eq. (A1). In this case, however, \(d\) is a generic integer \(\geq2\text{and}\mathcal{D}_{k}[\cdot]\) is modified to + +\[\mathcal{D}_{k}[\cdot]=\int_{U(d)}(\alpha)^{\otimes k}\cdot(\alpha_{2L}^{\dagger})^{\otimes k}\mu_{H}(\alpha_{2L}),\] (A6) + +Therefore the discussion in Ref. [26] does not directly apply. Specifically, we need to prove again the Lemmas in Appendix D of Ref. [26], those concerning the existence of the limit state and the fact that it commutes with the Krauss operators of the channel. By noting that, however, \(\mathcal{D}_{k}[\cdot]\) projects the state at site \(2L\) on the space spanned by \(\{P_{\sigma}\}_{\sigma\in S_{k}},\text{where}S_{k}\) is the symmetric group of of \(k\) elements and + +\[[P_{\sigma}]_{\boldsymbol{i},\boldsymbol{j}}=\prod_{p=1}^{k}\delta_{i_{p},j_{\sigma(p)}},\] (A7) + +one finds that the arguments of Ref. [26] can be straightforwardly repeated and both Lemmas continue to hold. Also the Lemma in Appendix E holds as the arguments of the previous subsection are not restricted to \(d=2\). Therefore, we again have that the limit + +\[\rho_{\infty}^{(k}=\lim_{t\rightarrow\infty}^{)}\rho_{t}^{(k)},\] (A8) diff --git a/outputs/arxiv_math/2503.05716_pg6_pg1_repeat1.md b/outputs/arxiv_math/2503.05716_pg6_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..da86df7bbf49a039b1cec1ee8e120a4f174e1386 --- /dev/null +++ b/outputs/arxiv_math/2503.05716_pg6_pg1_repeat1.md @@ -0,0 +1,149 @@ +between the approximated solution and the raw data, thereby enhancing the overall accuracy of the model. + +\[Loss_{D}= \frac{1}{N_{D}} \sum_{i=1}^{N_{D}}\bigg|u_{NN}(\boldsymbol{x}_{D}^{i},t_{D}^{i})-u_{Data}^{i}\bigg|^{2}\] (2.6) + + + + + +\(^{2}u(\) , ) \(t^{2} u_{NN}x (x\) + +2 + +x t + +\(1^{N_{R}}^{2}u(\) , ) \(t^{2}\) + +xi ti + +( + +) + + + +NN + +Loss\(=_{PDE}\) + +\[-a^{2}\] + +xt\(-fx_{R}^{i}\) , R i , + +a uNN ( R i , R i ) + +t uNN ( R i , R i ) + +NN + +R R x t + +## x + + + + + +N 1 N 1 N + +=1 + +R i + +2 + +uNN + +N + +B + + + +( ,t ) + +Loss\(=_{BC}\) + +\(x^{i}t^{i}-g(_{B}\) , B ) + +xi ti + +ΒuNN ( B , B ) + + + + + +t + +=1 + +B i N + +2 + +) + +I + + + +Loss\(=_{IC}\) + +x t − h x + +IuNN ( I i , 0 ) ( I ) + +f + +,t , uNN ( , ) + +x t + +=1 + +I i + + + + + +# update + +Loss + +# Done + +# N + +Loss + +# Y + += + +Loss Loss + +PDE + +\(+\text{Loss}_{BC}+\) Loss + +IC + +![](figure) + +Figure 2: FPINN framework to solve wave propagation + +## 3. Normalized Fourier induced PINN to solve the wave equation + +## 3.1. The analysis of general PINN and FPINN method to the wave equation in two different scale range + +Although various PINN models have been successfully applied to the study of ordinary and partial differential equations, particularly in the case of the wave equation, our investigation shows that their performance deteriorates in large scale domain and long time range, potentially leading to non-convergence. For example, let us consider two scenarios for two-dimensional wave propagation equation with Dirichlet boundary in \(\Omega_{1}=[0,2\pi]\times[0,2\pi],t\in(0,2).\Omega_{2}=[0,10\pi]\times[0,10\pi],t\in(0,10)\), respectively. The governed equation is + +\[\frac{\partial^{2}u}{\partial t^{2}} = \frac{1}{2} \left( \frac{\partial^{2}u}{\partial x_{1}^{2}} + \frac{\partial^{2}u}{\partial x_{2}^{2}} \right)+12t^{2}\] (3.1) + +An analytical solution is given by + +\[u(x_{1},x_{2},t)=t^{4}+\sin(x_{1})\cdot\sin(x_{2})\cdot\sin(t).\] + +Since the boundary and initial constraint functions can be directly derived from the exact solution, we will not explicitly state them here. In this experiment, the solvers for PINN and FPINN are configured as a DNN and a FFM-based DNN with \(N\) subnetworks, respectively, and the scale factors are set as (1, 2, 3, 4, 5, 6, 7, 8, 9, 10), and each subnetwork is configured with sizes of (20, 15, 15, 10). The first hidden layer of all subnetworks employs Fourier feature mapping as the activation function (see Eq.(2.3)), while the activation functions for the other layers (except for the output layer) are selected as \(\text{GELU}(x)=x\cdot \frac{1}{2} [1+erf( \frac{x}{\sqrt{2}} )],\text{where}erf(x)\) is Gaussian error function, and the output layers of all subnetworks are linear. We train the previously mentioned PINN and FPINN models for 30,000 epochs, performing testing every 1,000 epochs during the training process. diff --git a/outputs/arxiv_math/2503.05717_pg14_pg1_repeat1.md b/outputs/arxiv_math/2503.05717_pg14_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..dfcde7a3e392406da92ce0528e2dd8abecd476df --- /dev/null +++ b/outputs/arxiv_math/2503.05717_pg14_pg1_repeat1.md @@ -0,0 +1,25 @@ +the crack tip for \(\beta=0\). The greater the positive \(\beta\) is, the higher the reduction is. Similar to Fig. 4, all curves show slightly wavy patterns due to the impact of the density-dependent moduli. + +![](figure) + +(a) For negative \(\beta\) values. + +![](figure) + +(b) For positive \(\beta\) values. + +![](figure) + +Figure 6: \(T_{22}\) vs. \(r(T_{22}\) unit: 104 Pa). In the first panel, at \(r=0\) (the crack-tip), negative \(\beta\) does not reduce \(T_{22}\) compared with \(T_{22}\) at \(\beta=0\). Generally, a more negative value does not necessarily generate greater \(T_{22}\) at the crack tip. In the second panel for \(\beta>0\), positive \(\beta\) generates smaller \(T_{22}\) value compared with \(T_{22}\) at \(\beta=0\). Greater \(\beta\) creates smaller \(T_{22}\) at \(r=0\). Slightly wavy patterns also appear in all the curves due to the effect of the density-dependent moduli. + +### 5.3 SIF + +For the current problem, we do not have an asymptotic or analytical solution. Therefore, an explicit description of crack-tip SIF is not available. However, one can utilize the crack-tip SIF defined for the linear elastic fracture mechanics model and use the finite element solution for the nonlinear model to glean some vital physical insight into SIF. + +The SIF \(K_{I}\) in the context of linear elasticity defined as + +\[K_{I}=\lim_{r\rightarrow0^{+}} \overline{2\pi r} T_{22}.\] (33) + +It is challenging to analytically compute \(K_{I}\) for complicated models such as the ones in 3-D that we are investigating in this paper. We compute \(K_{I}\) numerically for illustrative purposes to + +understand its value as a function of the nonlinear modeling parameter \(\beta\). The function \(\sqrt{2\pi r} T_{22}\) vs. \(r\) is shown in Fig. 7. For \(\beta=0,-10,-20,-30\), the \(K_{I}\) values are, respectively, 0.0110, 0.0185, 0.0251, 0.0178; For \(\beta=0,10,20,30\), the \(K_{I}\) values are, respectively, 0.0110, 0.0060, 0.0037, 0.0026 (Unit: \(10^{4}\text{mm}^{1/2}\text{Pa})\). All the \(K_{I}\) values are very close, implying an identical theoretical \(K_{I}\) value for all \(\beta\) is highly likely. Such numerical approximation for \(K_{I}\) shows that the nonlinear model with density-dependent moduli is designed appropriately under the common crack criteria used for the linear-elasticity fracture model. Also, in the two panels of Fig. 7, the curve shapes for \(\beta<0\text{and}\beta>0\) are slightly distinct. The \(\beta<0\) curves show a greater curvature, while those for \(\beta>0\) show flatter behaviors. Such difference is caused by the sign of \(\beta\), disclosing how negative and positive \(\beta\) values impact the SIF differently. diff --git a/outputs/arxiv_math/2503.05736_pg17_pg1_repeat1.md b/outputs/arxiv_math/2503.05736_pg17_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..804988b3e6423c6e7d8a35f75f1de39e2c1a4e53 --- /dev/null +++ b/outputs/arxiv_math/2503.05736_pg17_pg1_repeat1.md @@ -0,0 +1,25 @@ +additional bisection refinements: we found that quadratic interpolation produces better results in our numerical case studies. + +We refer to Figure 2 for a practical implementation of the algorithm described above: the colored disks in the center panel correspond to the parameter values at which the pattern statistics was computed for predictor-corrector step as outlined above. + +## 4 Assessing dependence on algorithmic parameters + +We now outline the parameters that enter the algorithm and discuss how they affect accuracy and computational efficiency. The implementation of the continuation algorithm described above requires the following choices: + +1. Feature function: We choose a feature function \(f\) that can differentiate between the patterns we want to distinguish. + +2. Spatial discretization: We choose the number \(K\) of Fourier modes so that we can resolve the nonlinearity and the expected patterns at the expected wavelength. Alternatively, and this is how our numerical computations were conducted, we can use finite differences to solve the PDE model for a sufficiently small spatial stepsize that resolves the patterns we are interested in and use the resulting grid also for \(\alpha-\text{shapes}\). + +3. Initial data: We choose two fixed functions \(U_{\mathrm{b}}(x)\text{and}U_{\mathrm{r}}(x)\) to construct the initial data in (2.11). The deterministic part \(U_{\mathrm{b}}\) is selected to ensure that we reach the patterns we are interested in: for domain-filling patterns, \(U_{\mathrm{b}}\) is typically an unstable homogeneous rest state. The function \(U_{\mathrm{r}}\) that will be used for the randomized part is given by \(U_{\mathrm{r}}(x)=\sum_{|k|\leq K}a_{k}e_{k}(x)\) for a fixed nonzero choice of coefficients \((a_{k})_{|k|\leq K}\) (we set \(U_{\mathrm{r}}=0\) for deterministic initial data). + +4. Randomization: We select \(N\) samples of the random variables \((b_{k}(\omega))_{|k|\leq K}\) from a uniform distribution and form the ensemble of \(N\) randomized functions \(U_{\mathrm{r}}^{\omega}(x):=\sum_{|k|\leq K}a_{k}b_{k}(\omega)e_{k}(x)\). The resulting \(N\) functions \(U_{0}(x)=U_{\mathrm{b}}(x)+U_{\mathrm{r}}^{\omega}(x)\) are then used as initial data in the numerical solver. The number \(N\) of samples that are used to calculate the empirical measure \(\mu_{f}^{N,\omega}\) (or the empirical feature mean \(E_{f}^{N,\omega}\) if applicable) can be adjusted using, for instance, a small-sample paired t-test (which tests the null hypothesis that the mean of the difference of feature samples is zero) to ensure that there is a statistically significant difference between the two empirical measures in the argument of the bifurcation function \(g\). + +5. Integration time: The integration time \(T>0\) is chosen so that we reach the relevant pattern regime from the initial data \(U_{\mathrm{b}}+U_{\mathrm{r}}^{\omega}\) within the time interval \([0,T]\). It is possible to adapt \(T\) during continuation, for instance by choosing shorter or longer values and comparing the resulting feature values. + +6. Sublevel sets: We evaluate the feature function on the sublevel sets \(U_{j}^{-1}((-\infty,c])\) of the \(j\)th component of the solution \(U\) to (2.1). We choose the index \(1\leq j\leq d\) and the threshold \(c\in\mathbb{R}\) so that the corresponding sublevel sets best reflect the patterns we are interested in. + +7. Alpha-shapes: The evaluation of the feature functions we consider require the computation of the \(\alpha-\)shapes of the sublevel sets. We need to pick the radius \(\alpha\) of the \(\alpha-\)shape and the number \(M^{2}\) of lattice points on which we evaluate the solution to approximate the sublevel set. We usually choose \(M:=K\) and set \(\alpha=10/M\). We note that we can adapt \(M\text{and}\alpha\) by comparing the resulting pattern statistics using the Wasserstein metric to ensure that they do not change upon increasing \(M\) or varying \(\alpha\). + +8. Predictor-corrector steps: We need to choose the arclength stepsize \(s\) and the parameter offset \(h\). We normally pick \(h:=s\) and note that the stepsize \(s\) can be adapted based on the successive changes of the angle of the secants, which are indicative of the curvature of the curve \(\Gamma\). + +Our algorithm is robust with respect to these choices, and we never had to adjust them during continuation. Generally, increasing the number \(K\) of Fourier modes (or mesh points when using finite differences) for the PDE solver, the number \(M\) of lattice points on which we evaluate patterns and their features, and the ensemble size \(N\) of randomized initial data will provide smoother and more accurate continuation curves. The algorithm is diff --git a/outputs/arxiv_math/2503.05752_pg4_pg1_repeat1.md b/outputs/arxiv_math/2503.05752_pg4_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..0e183a7e8e48f712af2b4e9a677b617c55c83602 --- /dev/null +++ b/outputs/arxiv_math/2503.05752_pg4_pg1_repeat1.md @@ -0,0 +1,36 @@ +0.1 + +0.1 + +0.05 + +0.05 + +0 + +0 + +-0.05 + +-0.05 + +| -0.1 | | -0.1 | +| --- | --- | --- | +| -0.1 -0.05 0 0.05 | 0.1 | -0.1 -0.05 0 0.05 0.1 | +| (a) Interpolation nodes | | (b) Center evaluation nodes | + +Figure 1: Distribution of interpolation and evaluation nodes within a circle of radius 0.1. + +(a) Error in \(f\) vs. shape parameter + +(c) Error in \(f_{y}\) vs. shape parameter + +(b) Error in \(f_{x}\) vs. shape parameter + +(d) Condition number vs. shape parameter + +![](figure) + +Figure 2: Comparison of Accuracy and Numerical Stability for HRBF and MHRBF: The errors (function, \(f\), and its first derivatives, \(f_{x}\text{and}f_{y})\) and condition number computed in double precision for GA kernel, as functions of shape parameter \(\varepsilon\), with polynomial degree of 6 and \(n=4\) for MHRBF. + +The results in Figure 2 reveal significant accuracy improvements with MHRBF compared to HRBF. Since the error trends for \(\nabla f\) closely follow that of \(f\), the subsequent analysis focuses primarily on the function interpolation. Figure 2a demonstrates that the MHRBF achieves low error over a large range of \(\varepsilon\) values, with the error \(\sim10^{-14}\text{for all}\varepsilon≳0.5\), despite the fact that the condition number of the MHRBF being diff --git a/outputs/arxiv_math/2503.05829_pg8_pg1_repeat1.md b/outputs/arxiv_math/2503.05829_pg8_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..955e9219b4a5f9fe3de99ecb74595dc222bf6cd3 --- /dev/null +++ b/outputs/arxiv_math/2503.05829_pg8_pg1_repeat1.md @@ -0,0 +1,29 @@ +## 2 Results + +In this section, we introduce the definition of the model of interest along with the relevant notation. We then proceed to state the main results of this work. + +### 2.1 Definition of the model + +Throughout, we use the notation \(\mathbb{N}:=\{1,2,\ldots\},\mathbb{N}_{0}:=\mathbb{N}\cup\{0\}\text{and}[n]:=\{1,\ldots,n\}\) for each \(n\in\mathbb{N}\). We consider a labeled directed graph \((V,E)\) with vertex set \(V= [n]\text{for some}n\in\mathbb{N}\) and edge set \(E\subset\{(v,w):v,w\in V\}\). The natural number representing a vertex is then referred to as its label. In such a graph, a sequence of vertices \((v_{1},v_{2},\ldots,v_{k+1})\) is a directed path of length \(k\in\mathbb{N}\) if + +\[\forall i\in[k]:(v_{i},v_{i+1})\in E\] + +and an undirected path of length \(k\in\mathbb{N}\) if + +\[\forall i\in[k]:(v_{i},v_{i+1})\in E\vee(v_{i+1},v_{i})\in E.\] + +A labeled directed graph is called + +- connected if, for each pair of vertices, there is an undirected path joining those vertices, + +- acyclic if it does not contain any directed cycles, i.e. directed paths that visit the same vertex more than once, + +- increasing if the labels are decreasing along all directed paths. + +We observe that if a graph is increasing, then it is acyclic in particular. For \(v\in V\) , we call a vertex \(w\in V\) an out-neighbor if \((v,w)\in E\) and an in-neighbor if \((w,v)\in E\). Then, the number of out-neighbors of \(v\) is denoted by \(\mathbf{outdeg}(v)\). On the other hand, we call the number of in-neighbors of \(v\) the degree of vertex + +\(v\), denoted by \(d_{V}(v).\text{For all}n,m\in\mathbb{N}\) we define \(\mathcal{I}_{n}^{(m)}\) to be the set of increasing directed graphs on \([n]\) satisfying the condition + +\[\forall v\in[n]:\text{outdeg}(v)=m\wedge(v-1).\] (2.1) + +Now, we are ready to formally define RRDAGs. diff --git a/outputs/arxiv_math/2503.05844_pg8_pg1_repeat1.md b/outputs/arxiv_math/2503.05844_pg8_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..43923edb0eecd6516af532551e36af613536a1d6 --- /dev/null +++ b/outputs/arxiv_math/2503.05844_pg8_pg1_repeat1.md @@ -0,0 +1,63 @@ +\[\begin{bmatrix} 10 & 0 \\ 0 & 50 \end{bmatrix} ,R=0.1.\] + +simplifies the problem and focuses on the key states. At the same time, the computational cost of this method is comparable to that of standard linear MPC over the same prediction time domain. Based on the predictor in (24), design the objective function (29), The simulation lasts 300 s and the prediction step \(N_{t}\) is set to 20. Since we are concerned with the state of motion in the vertical direction, we set \(Q=\) As shown in Fig. 5, the simulation results demonstrate that the BE-MPC algorithm successfully achieves precise control of the DSRV at the target depth of \(z=50\) m. The figure illustrates the dynamic changes in the submersible’s diving depth, vertical velocity, and control inputs (rudder angle) throughout the simulation. The initial state was selected as \(x_{0}=[0.2;0;0;0.1;0]^{T}\) . The control input (rudder angle) was constrained within the range of \([-30,30]\). From the figure, it can be observed that the controller effectively adjusts the rudder angle to optimize the motion trajectory and attitude control. During the entire simulation process, the vertical position of the submersible steadily converges to the target depth of 50 meters. This verifies that the BE-MPC algorithm can successfully perform precise control tasks in a complex nonlinear dynamic system. + +## V. CONCLUSIONS + +In this paper, we propose a BLS-EDMD method for approximating the Koopman operator and use this method to design a new MPC. This method solves the problem of model prediction error caused by the randomness associated with the choice of basis functions in the traditional EDMD method. By leveraging the feature and enhancement layers of the BLS network, the BLS-EDMD method refines the generation of basis functions, thereby enhancing the system state representation and boosting the model’s overall prediction accuracy. In simulation experiments, we apply the proposed method to the target tracking control task of a classical van der Pol oscillator system and a DSRV. Among the prediction experiments, the results show that the BLS-EDMD-based Koopman predictor has significant advantages in terms of accuracy and stability. In addition, the control accuracy of the BLS-EDMD-based MPC controller is effectively validated in a complex DSRV scenario, highlighting its potential to effectively handle high-dimensional nonlinear systems. In future work we will further extend the current BE-MPC methodology for applications in more complex nonlinear dynamic systems, such as multi-degree-of-freedom UUV and underwater devices with more complex hydrodynamic properties. + +## REFERENCES + +[1] Zhang, W. and Zhu, J., “A spline kernel-based approach for nonlinear system identification with dimensionality reduction,” Electronics, vol. 9, Art. no. 940, 2020. + +[2] Wang, H. and Xu, K., and Zhang, H., “Adaptive finite-time tracking control of nonlinear systems with dynamics uncertainties,” IEEE Trans. Autom. Control, vol. 68, no. 9, pp. 5737–5744, 2022. + +[3] Meng, X., Yu, H., and Zhang, J., “An EPCH control strategy for complex nonlinear systems with actuator saturation and disturbances,” Inf. Sci., vol. 625, pp. 639–655, 2023. + +[4] Bwambale, E., Abagale, F. K., and Anornu, G. K., “Data-driven model predictive control for precision irrigation management,” Smart Agric. Technol., vol. 3, Art. no. 100074, 2023. + +[5] Coskun, M. Y., and Itik, M., “Intelligent PID control of an industrial electro-hydraulic system,” ISA Trans., vol. 139, pp. 484–498, 2023. + +[6] Muhssin, M. T., Ajaweed, M. N., and Khalaf, S. K., “Optimal control of underwater vehicle using LQR controller driven by new matrix decision control algorithm,” Int. J. Dyn. Control, vol. 11, no. 6, pp. 2911–2923, 2023. + +[7] Wang, Y., Li, S., Tian, Q., Li, Y., Sun, H., and Zhang, Y., “An adaptive control method with disturbance estimation for underwater manipulation,” J. Marine Sci. Eng., vol. 11, no. 6, Art. no. 1113, 2023. + +[8] Ławryńczuk, M., “Koopman operator-based multi-model for predictive control,” Nonlinear Dyn., pp. 1–28, 2024. + +[9] Nguyen, T. H., Bui, D. Q., and Dao, P. N., “An efficient Min/Max robust model predictive control for nonlinear discrete-time systems with dynamic disturbance,” Chaos Solitons Fractals, vol. 180, Art. no. 114551, 2024. + +[10] Wei, H. and Shi, Y., “MPC-based motion planning and control enables smarter and safer autonomous marine vehicles: Perspectives and a tutorial survey,” IEEE/CAA J. Autom. Sinica, vol. 10, no. 1, pp. 8–24, 2022. + +[11] Xu, J., Fan, Z., and Zhang, F., “Underwater maneuvering target tracking control based on model predictive control,” in Proc. Int. Conf. Autonomous Unmanned Syst., 2021, pp. 1036–1045. + +[12] Li, X., Yu, S., Gao, X., Yan, Y., and Zhao, Y., “Path planning and obstacle avoidance control of UUV based on an enhanced A* algorithm and MPC in dynamic environment,” Ocean Eng., vol. 302, Art. no. 117584, 2024. + +[13] Yuan, S., Liu, Z., Zheng, L., Sun, Y., and Wang, Z., “Event-based adaptive horizon nonlinear model predictive control for trajectory tracking of marine surface vessel,” Ocean Eng., vol. 258, Art. no. 111082, 2022. + +[14] Koopman, B. O., “Hamiltonian systems and transformation in Hilbert space,” Proc. Natl. Acad. Sci., vol. 17, no. 5, pp. 315–318, 1931. + +[15] Morgan, P., “An algebraic approach to Koopman classical mechanics,” Ann. Phys., vol. 414, Art. no. 168090, 2020. + +[16] Li, C. Y., Chen, Z., Zhang, X., Tim, K. T., and Lin, C., “Koopman analysis by the dynamic mode decomposition in wind engineering,” J. Wind Eng. Ind. Aerodyn., vol. 232, Art. no. 105295, 2023. DOI:10.1016/j.jweia.2022.105295 + +[17] Gong, X., and Wang, X., “A novel Koopman-inspired method for the secondary control of microgrids with grid-forming and grid-following sources,” Appl. Energy, vol. 333, Art. no. 120631, 2023. DOI:10.1016/j.apenergy.2022.120631 + +[18] Nandanoori, S. P., Guan, S., Kundu, S., Pal, S., Agarwal, K., Wu, Y., and Choudhury, S., “Graph neural network and Koopman models for learning networked dynamics: A comparative study on power grid transients prediction,” IEEE Access, vol. 10, pp. 32337–32349, 2022. + +[19] Liang, Z., Luo, Z., Liu, K., Qiu, J., and Liu, Q., “Online learning Koopman operator for closed-loop electrical neurostimulation in epilepsy,” IEEE J. Biomed. Health Inform., vol. 27, no. 1, pp. 492–503, 2022. + +[20] Golany, T., Radinsky, K., and Freedman, D., “12-lead ECG reconstruction via Koopman operators,” in Proc. Int. Conf. Mach. Learn., pp. 3745–3754, 2021. + +[21] Snyder, G., and Song, Z., “Koopman operator theory for nonlinear dynamic modeling using dynamic mode decomposition,” arXiv preprint, arXiv:2110.08442, 2021. + +[22] Williams, M. O., Kevrekidis, I. G., and Rowley, C. W., “A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition,” J. Nonlinear Sci., vol. 25, pp. 1307–1346, 2015. + +[23] Yang, Q., and Liu, Z., “Model-based versus model-free optimal tracking for soft robots: Analytical and data-driven Koopman modeling, control design and experimental validation,” Nonlinear Dyn., pp. 1–21, 2024. + +[24] Proctor, J. L., Brunton, S. L., and Kutz, J. N., “Generalizing Koopman theory to allow for inputs and control,” SIAM J. Appl. Dyn. Syst., vol. 17, no. 1, pp. 909–930, 2018. + +[25] Guo, W., Zhao, S., Cao, H., and Yi, B., “Koopman operator-based driver-vehicle dynamic model for shared control systems,” Appl. Math. Model., vol. 114, pp. 423–446, 2023. + +[26] Toro, V., Tellez-Castro, D., Mojica-Nava, E., and Rakoto-Ravalontsalama, N., “Data-driven distributed voltage control for microgrids: A Koopman-based approach,” Int. J. Electr. Power Energy Syst., vol. 145, Art. no. 108636, 2023. + +[27] Švec, M., Ileš, Š., and Matuško, J., “Model predictive control of vehicle dynamics based on the Koopman operator with extended dynamic mode decomposition,” in Proc. IEEE Int. Conf. Ind. Technol. (ICIT), 2021, pp. 68–73. diff --git a/outputs/arxiv_math/2503.05871_pg24_pg1_repeat1.md b/outputs/arxiv_math/2503.05871_pg24_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..6dacf78a189938d2bc2646139bba3fe1c2e9aa5f --- /dev/null +++ b/outputs/arxiv_math/2503.05871_pg24_pg1_repeat1.md @@ -0,0 +1,33 @@ +\(\Phi(t,x,y)\), and pressure perturbation \(\delta P(t,x,y)\), viz. + +\[u^{x}=U(y)+\partial_{y}\Phi(t,x,y),\] (A.2) + +\[\begin{aligned} u^{y}&=0-\partial_{x}\Phi(t,x,y), \\ u^{z}&=0, \\ P&=-2\nu\rho x+\delta P(t,x,y), \end{aligned}\] (A.3) (A.4) (A.5) + +⃗ + +where \(U(y)=1-y^{2}\). The perturbed flow is incompressible, \(\nabla\cdot\vec{u}=0\), and we impose no-slip boundary conditions \(\Phi(t,\pm1)=\partial_{y}\Phi(t,\pm1)=0\). In order to correctly account for energy in the system we work to second order in perturbation theory, + +\[\begin{aligned} \delta P&=\epsilon P^{(1)}(t,x,y)+\epsilon^{2}P^{(2)}(t,x,y)+O(\epsilon)^{3} \\ \Phi&=\epsilon\Phi^{(1)}(t,x,y)+\epsilon^{2}\Phi^{(2)}(t,x,y)+O(\epsilon)^{3}, \end{aligned}\] (A.6) (A.7) + +where \(\epsilon\) is a formal parameter counting orders in the expansion. Let us consider each order in turn. At \(O(\epsilon)\) we take a real perturbation formed as follows: + +\[\Phi^{(1)}(t,x,y)=\phi(t,y)e^{i\alpha x}+ \overline{\phi} (t,y)e^{-i\alpha x}\] (A.8) + +with wavenumber \(\alpha\neq 0.P^{(1)}\) is determined algebraically by the \(x-\)component of (A.1) at order \(\epsilon\). The \(y\) component of (A.1), after inserting \(P^{(1)}\), gives the Orr-Sommerfeld equation (2.1), where the Orr-Sommerfeld operator is + +\[O_{\text{OS}}=-\Delta_{2}^{-1}\left[(i\alpha Re)^{-1}\Delta_{2}^{2}-U(y)\Delta_{2}+U^{\prime\prime}(y)\right],\] (A.9) + +with Reynolds number \(Re=\nu^{-1},\text{and where}\Delta_{2}=\partial_{y}^{2}-\alpha^{2}\) is the spatial Laplacian for the \(x,y-\text{plane}\). At \(O(\epsilon)^{2}\) we have perturbations sourced by \(O(\epsilon)\) terms. These take the form, + +\[\Phi^{(2)}(t,x,y)=\delta\phi_{0}(t,y)+\sum_{\pm}\delta\phi_{\pm\alpha}(t,y)e^{\pm i2\alpha x}.\] (A.10) + +The zero-momentum piece \(\partial_{y}\delta\phi_{0}\) obeys the following diffusion equation, + +\[\left(\partial_{t}-\nu\partial_{y}^{2}\right)\partial_{y}\delta\phi_{0}=i\alpha\partial_{y}\left(\phi\partial_{y} \overline{\phi} - \overline{\phi} \partial_{y}\phi\right),\] (A.11) + +with viscosity \(\nu\) serving as the diffusivity, and a current source term coming from the \(O(\epsilon)\) perturbations. \(P^{(2)},\delta\phi_{\pm\alpha}(t,y)\) are determined by other differential equations but we will not need them here. To \(O(\epsilon)^{2}\) the energy of the perturbed flow evaluates to + +\[\begin{aligned} E&=\int_{-1}dy\int^{1}dx \overline{u} \cdot u=\text{vol}_{x}\bigg( \frac{16}{15} +2\epsilon^{2}\int^{1}\left(|\partial_{y}\phi|^{2}+\alpha^{2}|\phi|^{2}\right)dy \\ \int^{-1}\bigg) \end{aligned}\] (A.12) + +\[+2\epsilon^{2}\int_{-1}^{1}(1-y^{2})\partial_{y}\delta\phi_{0}dy+O(\epsilon)^{3}\bigg).(\mathrm{A}.13)\] diff --git a/outputs/arxiv_math/2503.05873_pg4_pg1_repeat1.md b/outputs/arxiv_math/2503.05873_pg4_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..0bdfb5437e80e3c23b75a0b59510cd71e27e3d2c --- /dev/null +++ b/outputs/arxiv_math/2503.05873_pg4_pg1_repeat1.md @@ -0,0 +1,29 @@ +scheme as well. After the encryption, the size of the seed used to compress each of files is \(3.27[\text{KB}]\). The encrypted seed is stored on the second server. The rest of the files are stored over the rest of the servers, s.t. each server stores one file. This hybrid scheme is secured against both IT and Crypto-Eve’s. The overall data rate of the proposed scheme is \(0.79\). 4) Finally, we assess the computational complexity of the proposed communication scheme. Complexity is measured in this paper, as the number of binary operations required to perform encoding and decoding of all the messages. We show that NU-HUNCC, which encrypts only a subset of links using McEliece, exhibits a more efficient run-time complexity compared to NUM. NU-HUNCC’s efficiency makes it a promising candidate for practical applications. The remainder of this paper is structured in the following manner. Sec. II presents NU-HUNCC setting, while Sec. III provides the comprehensive security definitions for IS and ISS-CCA1. In Sec. V, we introduce the encoding/decoding algorithm for NU-HUNCC. In Sec. VI, we provide the key findings and theorems of this paper. Sec. VII offers numerical results demonstrating the performance of NU-HUNCC. The proofs of the theorems provided in the paper are given in Sec. VIII, IX, X, and Appendixes A and B. We conclude the paper is Sec. XI. + +## II. SYSTEM MODEL + +We consider a communication system where Alice wishes to transmit \(\ell\) non-uniform confidential message3 over \(\ell\) noiseless links, \(\mathcal{L}=\{1,...,\ell\}\), to Bob, in the presence of an eavesdropper, Eve. The messages are taken from a DMS \((\mathcal{V},p_{V})\) s.t. \(\mathcal{V}\in\{0,1\}\). We denote the source message matrix by \(\underline{V}_{\mathcal{L}} \in\mathbb{F}_{2}^{\ell\times n}\text{when}n\) is the size of each source message. Bob’s observations are denoted by \(\underline{Y}_{\mathcal{L}}\). Those observations, provide Bob reliable decode \(\underline{V}_{\mathcal{L}}\) with high probability. That is, \(\mathbb{P}( \underline{\hat{V}}_{\mathcal{L}} ( \underline{Y}_{\mathcal{L}} )\neq \underline{V}_{\mathcal{L}} )\leq\epsilon_{e},\text{wh} \overline{\text{er}} \mathrm{e} \underline{\hat{V}}_{\mathcal{L}} ( \underline{Y}_{\mathcal{L}} )\) is the estimation of \(\underline{V}_{\mathcal{L}}\). We consider two types of Eve: 1) IT-Eve, which observes any subset \(\mathcal{W}\subset\mathcal{L}\) of the links s.t. \(|\mathcal{W}|≜w<\ell\), but is computationally unbounded, and 2) Crypto-Eve which observes all the links, but is bounded computationally. We denote IT-Eve’s observations by \(\underline{Z}_{\mathcal{W}}\) and Crypto-Eve’s observations by \(\underline{Z}_{\mathcal{L}}\). + +## III. SECURITY DEFINITIONS + +In this section, we provide the formal security definitions used throughout this paper. + +## A. Security against IT-Eve + +Against IT-Eve, we consider information-theoretic security. For any subset of \(k_{s}<\ell-w\) source messages, we use the notion of \(k_{s}\) individual security \((k_{s}-\text{IS})\). We measure the leakage of information to the eavesdropper using nonnormalized variational distance, denoted by \(\mathbb{V}(\cdot,\cdot)\). Additionally, we require the code to be reliable where the reliability is measured by the decoding error probability at Bob’s. For a + +3In this paper, we assume the messages are independent to focus on the key methods and results. However, our proposed solution can be easily shown to be valid for dependent sources by using joint source coding schemes [54]. + +code to be \(k_{s}-\text{IS}\) we require the information leakage and error probability to be negligible. We now formally define \(k_{s}-\text{IS}\): Definition \(\mathbf{1}.(k_{s}\) Individual Security) Let \(\underline{V}_{\mathcal{L}} \in\mathbb{F}_{q}^{\ell\times n}\) be a set of \(\ell\) confidential source messages Alice intends to send, \(\underline{Y}_{\mathcal{L}}\) be Bob’s observations of the encoded messages, and \(\underline{Z}_{\mathcal{W}}\) be IT-Eve’s observations of the encoded messages. We say that the coding scheme is \(k_{s}-IS\) if: + +1. Security: \(\forall\epsilon_{s}>0,\forall\mathcal{W}\subset\mathcal{L}s.t.|\mathcal{W}|=w<\ell\), and \(\overline{\forall \underline{V}_{\mathcal{K}_{s}}} \subset \underline{V}_{\mathcal{L}} s.t.|\mathcal{K}_{s}|=k_{s}<\ell-w\), it holds that \(\mathbb{V}(p_{\underline{Z}_{\mathcal{W}} | \underline{V}_{\mathcal{K}s} = \underline{v}_{\mathcal{K}s}},p_{\underline{Z}_{\mathcal{W}}})\leq\epsilon_{s}\). + +2. Reliability: \(\forall\epsilon_{e}>0\) it holds that \(\mathbb{P}( \underline{\hat{V}}_{\mathcal{L}} ( \underline{Y}_{\mathcal{L}} )\neq \underline{V}_{\mathcal{L}} )\leq \overline{\epsilon_{e},\text{where}} \underline{\hat{V}}_{\mathcal{L}} ( \underline{Y}_{\mathcal{L}} )\) is the decoding estimation of the message matrix from Bob’s observations. + +Thus, IT-Eve that observes any subset of \(w\) links in the network can’t obtain any information about any set of \(k_{s}<\ell- w\) individual messages, \(\underline{V}_{\mathcal{K}_{s}}\). However, IT-Eve might be able to obtain some insignificant information about the mixture of all the messages. Yet, this negligible information is controlled [12], [55], [56]. Bob can reliably decode the message matrix from his observations of the encoded messages. + +## B. Security against Crypto-Eve + +Crypto-Eve may perform passive/active attacks against Alice and the ciphertexts she produces, to obtain information about confidential messages. Two of the most common attacks given in the literature are the chosen-plaintext attack (CPA) and the chosen-ciphertext attack (CCA) [57]. In both attacks, Crypto-Eve is given a ciphertext from which she tries to obtain information about the plaintext. In CCA Crypto-Eve is active and prior to receiving the ciphertext, she can question a decryption oracle to provide her the plaintexts for a limited number of ciphertexts of her choice. However, she can’t use this decryption oracle after receiving the test ciphertext [57]. The information leakage is measured by Crypto-Eve’s ability to distinguish between plaintexts given the test ciphertext provided by Alice. There are two ways in which this leakage is measured: 1) indistinguishability (IND) - by observing a ciphertext created from one of two possible plaintexts, Crypto-Eve can’t distinguish between the two plaintexts better than a uniform coin-toss, 2) semantic security (SS) - by observing a ciphertext, Crypto-Eve can’t obtain any information about the original plaintext or some function applied on it, that is more significant than the information she obtains from the plaintexts original probability distribution. Those two definitions are equivalent, i.e. a cryptosystem that is IND is also SS and vice versa [1]. The security level of a cryptosystem is defined by the combination of Crpyto-Eve’s attack model and its information leakage. In this paper, we introduce a new notion of security, individual semantic security against a chosen ciphertext attack (ISS-CCA1). This notion of security is based on SS-CCA1 cryptographic security [1], [57], and usually requires the encryption scheme to be probabilistic[^4]. To properly define ISS- + +[^4]: To focus on our main contributions, we choose to work with public key encryption schemes, but any other PQ probabilistic encryption schemes would have achieved similar results. diff --git a/outputs/arxiv_math/2503.05874_pg8_pg1_repeat1.md b/outputs/arxiv_math/2503.05874_pg8_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..afb7e57484d695cd639545d5aeb15df9f85d6ec9 --- /dev/null +++ b/outputs/arxiv_math/2503.05874_pg8_pg1_repeat1.md @@ -0,0 +1,51 @@ +Remark 3. According to Remark 2, each \(I_{ij}\) can be written as an interval \(I_{ij}=[L_{ij},U_{ij}]\text{where}L_{ij}\text{and}U_{ij}\) are the lower and upper bounds of the interval, respectively. Also, each \(S_{ij}\) (if it is non-empty) is obtained as one of the forms of \(S_{ij}=\{L_{ij}\},S_{ij}=\{U_{ij}\},S_{ij}=\{L_{ij},U_{ij}\}\text{or}S_{ij}=[L_{ij},U_{ij}]\). + +Definition 4. For each \(j\in\mathscr{J}\) , we define \(I_{j}=\bigcap_{i\in\mathscr{I}}I_{ij}\). Also, let \(S_{ij}^{\prime}=S_{ij}\cap I_{j},\forall i\in\mathscr{I}\text{and}\forall j\in\mathscr{J}\) . + +Corollary 4. Let \(i\in\mathscr{I}\text{and}j\in\mathscr{J}\) . If \(S_{ij}^{\prime}\neq \emptyset,\text{then}S_{ij}^{\prime}\) is as one of the forms of \(S_{ij}^{\prime}=\{L_{j}\},S_{ij}^{\prime}=\{U_{j}\}\), + +\(S_{ij}^{\prime}=\{L_{j},U_{j}\}\text{or}S_{ij}^{\prime}=[L_{j},U_{j}],\text{where}L_{j}\text{and}U_{j}\) are the lower and upper bound of \(I_{j}=[L_{j},U_{j}]\), respectively. + +Proof. From Definition 4 and Remark 3, we have + +\[I_{j}=\bigcap_{i\in\mathscr{I}}I_{ij}=\bigcap_{i\in\mathscr{I}}[L_{ij},U_{ij}]=\left[\max_{i\in\mathscr{I}}\{L_{ij}\},\min_{i\in\mathscr{I}}\{U_{ij}\}\right]=[L_{j},U_{j}]\] + +Hence, for each \(i\in\mathscr{I}\) , if \(L_{ij}\in I_{j}(U_{ij}\in I_{j})\), then we must have \(L_{ij}=L_{j}(U_{ij}=U_{j})\). On the other hand, from Remark 3, a non-empty \(S_{ij}\) is as one of the forms of \(S_{ij}=\{L_{ij}\},S_{ij}=\{U_{ij}\},S_{ij}=\{L_{ij},U_{ij}\}\text{or}S_{ij}=[L_{ij},U_{ij}]\). So, + +if \(S_{ij}=\{L_{ij}\}\text{and}S_{ij}^{\prime}=S_{ij}\cap I_{j}\neq \emptyset,\text{then}L_{ij}\in I_{j}\); that is, \(L_{ij}=L_{j}\) and therefore \(S_{ij}^{\prime}=S_{ij}\cap I_{j}=\{L_{j}\}\). For the other cases, the results are attained by the similar argument. + +## Example 1. Consider Problem (1) includes Yager operator where + +\[z=c^{T}x=x_{1}+0.35x_{2}+0.93x_{3}+3.28x_{4}+5.03x_{5}+2.96x_{6}+x_{7}+2.75x_{8}+5.25x_{9}+6.39x_{10}\] + +\[AA_{-}^{+}=_{=} \begin{bmatrix} 0.25 & 0.32 & 0.41 & 0.19 & 0.70 & 0.13 & 0.44 & 0.37 & 0.28 & 0.50 \\ 0.80 & 0.73 & 0.64 & 0.79 & 0.80 & 0.22 & 0.80 & 0.56 & 0.10 & 0.28 \\ 0.11 & 0.20 & 0.12 & 0.13 & 0.05 & 0.25 & 0.40 & 0.25 & 0.20 & 0.18 \\ 0.10 & 0.23 & 0.25 & 0.15 & 0.12 & 0.05 & 0.02 & 0.01 & 0.15 & 0.15 \\ 0.45 & 0.35 & 0.70 & 0.50 & 0.41 & 0.27 & 0.39 & 0.48 & 0.17 & 0.39 \\ 0.60 & 0.70 & 0.25 & 0.38 & 0.63 & 0.58 & 0.46 & 0.47 & 0.85 & 0.33 \\ 0.01 & 0.02 & 0.15 & 0.09 & 0.12 & 0.10 & 0.15 & 0.15 & 0.04 & 0.25 \\ 0.75 & 0.64 & 0.32 & 0.29 & 0.39 & 0.61 & 0.57 & 0.34 & 1.00 & 0.46 \\ 0.22 & 0.20 & 0.35 & 0.23 & 0.30 & 0.18 & 0.29 & 0.25 & 0.35 & 0.10 \\ 0.41 & 0.25 & 0.50 & 0.20 & 0.56 & 0.60 & 0.59 & 0.60 & 0.47 & 0.31 \\ \big] \\ 0.70 & 0.70 & 0.32 & 0.44 & 0.00 & 0.16 & 0.20 & 0.50 & 0.40 & 0.39 \\ 0.70 & 0.65 & 0.14 & 0.12 & 0.80 & 0.76 & 0.00 & 1.00 & 0.15 & 0.79 \\ 0.17 & 0.24 & 0.20 & 0.20 & 0.06 & 0.25 & 0.13 & 0.19 & 0.22 & 0.02 \\ 0.14 & 0.10 & 0.04 & 0.00 & 0.10 & 0.00 & 0.14 & 0.02 & 0.15 & 0.08 \\ 0.70 & 0.04 & 0.27 & 0.36 & 0.60 & 0.40 & 0.48 & 0.50 & 0.50 & 0.50 \\ 0.66 & 0.63 & 0.14 & 0.73 & 0.53 & 0.46 & 0.61 & 0.85 & 0.85 & 0.39 \\ 0.00 & 0.15 & 0.15 & 0.05 & 0.02 & 0.03 & 0.10 & 0.12 & 0.18 & 0.09 \\ 0.63 & 0.03 & 0.55 & 0.77 & 0.79 & 0.49 & 0.21 & 0.32 & 0.80 & 0.71 \\ 0.27 & 0.30 & 0.35 & 0.24 & 0.35 & 0.07 & 0.29 & 0.35 & 0.20 & 0.75 \\ 0.59 & 0.34 & 0.26 & 0.38 & 0.02 & 0.60 & 0.52 & 0.43 & 0.27 & 0.44 \end{bmatrix}\] + +| | 0.25 | 0.32 | 0.41 | 0.19 | 0.70 | 0.13 | 0.44 | 0.37 | 0.28 | 0.50 | | +| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | +| | 0.80 | 0.73 | 0.64 | 0.79 | 0.80 | 0.22 | 0.80 | 0.56 | 0.10 | 0.28 | | +| | 0.11 | 0.20 | 0.12 | 0.13 | 0.05 | 0.25 | 0.40 | 0.25 | 0.20 | 0.18 | | +| | 0.10 | 0.23 | 0.25 | 0.15 | 0.12 | 0.05 | 0.02 | 0.01 | 0.15 | 0.15 | | +| | 0.45 | 0.35 | 0.70 | 0.50 | 0.41 | 0.27 | 0.39 | 0.48 | 0.17 | 0.39 | | +| A+ = | | | | | | | | | | | | +| | 0.60 | 0.70 | 0.25 | 0.38 | 0.63 | 0.58 | 0.46 | 0.47 | 0.85 | 0.33 | | +| | 0.01 | 0.02 | 0.15 | 0.09 | 0.12 | 0.10 | 0.15 | 0.15 | 0.04 | 0.25 | | +| | 0.75 | 0.64 | 0.32 | 0.29 | 0.39 | 0.61 | 0.57 | 0.34 | 1.00 | 0.46 | | +| | 0.22 | 0.20 | 0.35 | 0.23 | 0.30 | 0.18 | 0.29 | 0.25 | 0.35 | 0.10 | | +| | 0.41 | 0.25 | 0.50 | 0.20 | 0.56 | 0.60 | 0.59 | 0.60 | 0.47 | 0.31 | | +| | 0.70 | 0.70 | 0.32 | 0.44 | 0.00 | 0.16 | 0.20 | 0.50 | 0.40 | 0.39 | | +| | 0.70 | 0.65 | 0.14 | 0.12 | 0.80 | 0.76 | 0.00 | 1.00 | 0.15 | 0.79 | | +| | 0.17 | 0.24 | 0.20 | 0.20 | 0.06 | 0.25 | 0.13 | 0.19 | 0.22 | 0.02 | | +| | 0.14 | 0.10 | 0.04 | 0.00 | 0.10 | 0.00 | 0.14 | 0.02 | 0.15 | 0.08 | | +| | 0.70 | 0.04 | 0.27 | 0.36 | 0.60 | 0.40 | 0.48 | 0.50 | 0.50 | 0.50 | | +| A− = | | | | | | | | | | | | +| | 0.66 | 0.63 | 0.14 | 0.73 | 0.53 | 0.46 | 0.61 | 0.85 | 0.85 | 0.39 | | +| | 0.00 | 0.15 | 0.15 | 0.05 | 0.02 | 0.03 | 0.10 | 0.12 | 0.18 | 0.09 | | +| | 0.63 | 0.03 | 0.55 | 0.77 | 0.79 | 0.49 | 0.21 | 0.32 | 0.80 | 0.71 | | +| | 0.27 | 0.30 | 0.35 | 0.24 | 0.35 | 0.07 | 0.29 | 0.35 | 0.20 | 0.75 | | +| | 0.59 | 0.34 | 0.26 | 0.38 | 0.02 | 0.60 | 0.52 | 0.43 | 0.27 | 0.44 | | + +\[b^{T}=[0.50,0.80,0.25,0.15,0.50,0.75,0.15,0.80,0.35,0.60]\] + +In this example, \(\mathscr{I}=\{1,2,\ldots,10\},\mathscr{J}=\{1,2,\ldots,10\}\text{and}\varphi\) is the Yager t-norm (that is continuous and nilpotent) with \(p=2\). So, according to Table A1 we have + +\[\varphi(x,y)=T_{Y}^{2}(x,y)=\max\left\{0,1-\left((1-x)^{2}+(1-y)^{2}\right)^{1/2}\right\}\] diff --git a/outputs/arxiv_math/2503.05880_pg5_pg1_repeat1.md b/outputs/arxiv_math/2503.05880_pg5_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..ff62cf68614646f1bf1ff07a00e224141c400795 --- /dev/null +++ b/outputs/arxiv_math/2503.05880_pg5_pg1_repeat1.md @@ -0,0 +1,15 @@ +To derive the asymptotic properties of the MCLEs, we use results obtained in [8] and [9]. In [8] we considered sums of square increments of an isotropic fractional Brownian field based on the edges of the Delaunay triangles and provided asymptotic results using Malliavin calculus (see Theorem 1 in [8]). In [9] we considered sums of square increments of the pointwise maximum of two independent isotropic fractional Brownian fields and showed that the asymptotic behaviors of the sums depend on the local time at the level 0 of the difference between the two fractional Brownian fields and no more on the Gaussian limits of the sums of square increments of each isotropic fractional Brownian field (see Theorem 2 in [9]). In this paper we generalize this result to the max-stable Brown-Resnick random field which is built as the pointwise maximum of an infinite number of isotropic fractional Brownian fields (see Theorem 3). Using approximations of the pairwise and triplewise CL objective functions, we then derive the asymptotic properties of the MCLEs (see Theorem 6). The rates of convergence of the estimators as well as their limit distributions are not standard and are specific to the structure of the max-stable process. It is important to identify them in order to avoid using the Gaussian distributions typically obtained with data from multiple time observations. It is noteworthy that we only consider isotropic fractional Brownian fields with Hurst index in \((0,1/2)\) as in [8] and [9]. This is not a very restrictive constraint since almost all empirical studies that use the spatial Brown-Resnick random field obtain estimates in this interval (see e.g. [12, 13, 19, 20]). + +Our paper is organized as follows. In Section 2 we present the family of stationary Brown-Resnick random fields introduced in [27], then we review some established concepts related to the Delaunay triangulation, and we end with the definition of the local time between two independent and identically distributed fractional Brownian random fields. In section 3, we first study the asymptotic distributions of the “normalized” increments of the logarithm of the Brown-Resnick random field based on pairs and triples of sites as the distances between sites tend to zero. We then provide asymptotic results for squared increment sums of the max-stable Brown-Resnick random field. In Section 4, we introduce the randomized sampling scheme and define the CL estimators of the scale and Hurst parameters. The asymptotic properties of the MCLEs for these parameters are then given. The proofs and some intermediate results are deferred to Section 5 and Section 6. + +## 2 Preliminaries + +### 2.1 Max-stable Brown-Resnick random fields + +In this paper we are concerned with the class of max-stable random fields known as Brown-Resnick random fields introduced in [27]. This class of random fields is based on Gaussian random fields with linear stationary increments. Recall that a random process \((W(x))_{x\in\mathbf{R}^{d}}\) is said to have linear stationary + +increments if the law of \((W(x+x_{0})-W(x_{0}))_{x\in\mathbf{R}^{d}}\) does not depend on the choice of \(x_{0}\in\mathbf{R}^{d}\). A prominent example is the isotropic fractional Brownian field where \(W(0)=0\) a.s. and semi-variogram given by + +\[\gamma(x)= \frac{\text{var}(W(x))}{2} = \frac{\sigma^{2}\Vert x\Vert^{\alpha}}{2}\] (2.1) + +for some \(\alpha\in(0,2)\text{and}\sigma^{2}>0,\text{where}\Vert x\Vert\) is the Euclidean norm of \(x\). The parameter \(\sigma\) is called the scale parameter while \(\alpha\) is called the range parameter \((H=\alpha/2\) is also known as the Hurst parameter and relates to the Hölder continuity exponent of \(W)\). It is noteworthy that \(W\) is a self-similar random diff --git a/outputs/arxiv_math/2503.05885_pg8_pg1_repeat1.md b/outputs/arxiv_math/2503.05885_pg8_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..cdd8c72fbeb8d52ae67997abc2c050815a923d46 --- /dev/null +++ b/outputs/arxiv_math/2503.05885_pg8_pg1_repeat1.md @@ -0,0 +1,40 @@ +### 2.2 Upper bounds on the mass distribution + +Our main results give lower bounds on the mass measure \(m^{\nu}\) defined in (1.14). It is then natural to ask about matching upper bounds. Additionally, we only give lower bounds for \(|k|\ll\nu^{-1/2}\). For \(|k|\gg\nu^{-1/2}\), the dynamics should be dominated by the \(\nu\Delta\) dissipation, and so the lower bound should not generally hold. + +2.2.1. The \(|k|\ll\nu^{-1/2}\) range. First, when \(|k|\ll\nu^{-1/2}\), we note the following upper bound. This bound is essentially identical to the cumulative Batchelor spectrum upper bound obtained in [BBPS21b] and relies on the uniform-in-diffusivity exponential mixing. + +\[\begin{aligned} \mathbf{Proposition}\mathbf{2}.\mathbf{1}.\text{For all}R&\geq2, \\ m^{\nu}([1,R])&\leq C\log R. \end{aligned}\] (2.2) + +Proof. Using the exponential mixing, we note that for any \(\alpha>0\) + +\[\begin{aligned} m^{\nu}([1,R])&=\int_{\alpha\mathrm{l}}^{\infty}\mathbb{E}\|\Pi_{\leq R}\varphi_{t}^{\nu}\|_{L^{2}}^{2} \\ &\leq\int_{0}^{0\text{og}R}\mathbb{E}\|\varphi_{t}^{\nu}\|_{L^{2}}^{2}+CR^{2}\int_{\mathrm{g}R\alpha\log R}^{\infty}e^{-C^{-1}t}dt \\ &\leq\alpha\log R+CR^{2}e^{-C^{-1}\alpha\text{lo}}. \end{aligned}\] + +Choosing \(\alpha\) large enough, we conclude. + +One can then use Chebyshev’s inequality to obtain a log-density bound on the “overcharged” radii. We define + +\[\overline{B}_{h,\alpha}^{\nu} :=r\in[1,\infty):m^{\nu}([r,r+h])\geq \frac{\alpha h}{r} \}\] (2.3) + +\[\mu_{1,R}( \overline{B}_{h,\alpha}^{\nu} )= \frac{1}{\log R} \int_{\overline{B}_{h,\alpha}^{\nu} \cap[1,R]} \frac{1}{r} dr\leq \frac{1}{\alpha h\log R} \int_{1}^{R}m^{\nu}([r,r+h])dr\leq \frac{1}{\alpha\log R} m^{\nu}([1,R+h])\leq C\alpha^{-1}.\] + +| and compute | | | | | +| --- | --- | --- | --- | --- | +| 1 1 | | 1 | | 1 | +| µ1,RpBh,αqν “ r | log R żBh,αXr1,Rsν | dr ď αh log R | ż1 R mνprr, r`hsq dr ď | mνpr1, R`hsq ď Cα´1. α log R | + +We have proven the following density estimate. + +## Proposition 2.2. For all \(R\geq2\), + +\[\mu_{1,R}( \overline{B}_{h,\alpha}^{\nu} )\leq C\alpha^{-1}.\] + +We note that, in some sense, this upper bound matches the second term of the lower bound density estimate in (1.10) or the lower bound estimate in (1.12). It however does not match the estimates (1.11) or (1.13), which do not require taking a limit in \(\alpha\). + +2.2.2. The \(|k|\gg\nu^{-1/2}\) range. The bound given by Proposition 2.1 holds for all \(R\geq2\), however one can get a much sharper statement for \(R\gg\nu^{-1/2}\). Computing the Itô differential, we have the energy estimate + +\[\frac{d}{dt} \mathbb{E}\|\psi_{t}^{\nu}\|_{L^{2}}^{2}=-2\nu\mathbb{E}\|\nabla\psi_{t}^{\nu}\|_{L^{2}}^{2}+\|g\|_{L^{2}}^{2}.\] + +By the Itô isometry, Duhamel’s principle, and the stationarity of the \(u_{t}\) process, + +\[\frac{d}{dt} \mathbb{E}\|\psi_{t}^{\nu}\|_{L^{2}}^{2}=\mathbb{E}\|\varphi_{t}^{\nu}\|_{L^{2}}^{2}\leq e^{-C^{-1}\nu tt\rightarrow}\rightarrow^{\infty}0.\] diff --git a/outputs/arxiv_math/2503.05886_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.05886_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..8e09eca3cc11b12a71a7cb7a93375e19f95ecc68 --- /dev/null +++ b/outputs/arxiv_math/2503.05886_pg3_pg1_repeat1.md @@ -0,0 +1,43 @@ +![](figure) + +| reported initial state | pre-select | hidden realizations performed by the | assistant post-select | reported final state | +| --- | --- | --- | --- | --- | +| Po = Dai\|xi)(r8I | \|x- | Likj j)Lkj i,k,j | -\|yI> | P1= D 3j\|y}X(yI | +| F Po | | | | F P1 | +| | | Pij often | | | + +FIG. 1. Quantum analog of Schrödinger’s 1931 bridge problem. + +initial state \(|x_{0}^{i}\rangle\) is subjected to the prescribed Markovian dynamics (4). Finally, our assistant performs the final projective measurement with respect to states \(\{|y_{1}^{j}\rangle\}\), leading to a certain final state \(|y_{1}^{j}\rangle\). In other words, our assistant realizes an experiment in which the states are pre- and post-selected. After a large number of such experiments, say \(N\), our assistant reports to us the fraction \(\tilde{\alpha}_{i}\) of experiments that started with state \(|x_{0}^{i}\rangle\) and the fraction \(\tilde{\beta}_{j}\) of experiments that ended on state \(|y_{1}^{j}\rangle\), without telling us which combination of states was chosen in each realization. That is, without reporting the joint distribution \(\tilde{p}_{ij}\). Thus, from our point of view, the experiment starts and concludes with the system in the mixed states + +\[\tilde{\rho}_{0}=\sum_{i}\tilde{\alpha}_{i}|x_{0}^{i}\rangle\langle x_{0}^{i}|\text{and}\tilde{\rho}_{1}=\sum_{j}\tilde{\beta}_{j}|y_{1}^{j}\rangle\langle y_{1}^{j}|,(7)\] + +respectively, which do not align with our expectation \(\rho_{0}\) and \(\rho_{1}\). Moreover, the reported states are such that + +\[\tilde{\rho}_{1}\neq \sum_{i,k,j}L_{ikj}(1,0)\tilde{\rho}_{0}L_{ikj}(1,0)^{\dagger}.\] + +See Figure 1 for a schematic representation of the experiment. The discrepancy between the expected and the reported initial and final states may be due to large deviations, i.e., obtaining an unlikely ensemble from a collection of measurements due the finite size of the experimental record, or due to pre- and post-selection, where our assistant might have chosen a sub-ensemble of realizations to cook up some desired \(\tilde{\alpha}_{i}\text{and}\tilde{\beta}_{j}\) probabilities. Either way, in the spirit of Schr¨odinger’s original gedanken experiment, we pose the following question: what is the most likely joint probability \(\tilde{p}_{ij}\) that led to the outcomes \(\tilde{\rho}_{0}\text{and}\tilde{\rho}_{1}\)? In other words, if the outcomes were postselected, what is the most likely way this post-selection was achieved? + +# B. A large deviations solution + +The question raised above reduces to a classical large deviations problem of the same nature as the one that Schr¨odinger answered in [1]. In modern terms, we seek to quantify the likelihood of unexpected outcomes through Sanov’s theorem, which states that the probability \(P\) of drawing an atypical distribution from a finite collection of \(N\) realizations decays exponentially as \(N\rightarrow\infty\). Specifically, + +\[P\sim e^{-IN},\] + +where the decay rate is given by the large deviations rate function \(I\) that quantifies the distance between \(P\) and the typical probability distribution. Thereby, the most likely atypical distribution that is consistent with given outcomes is the one that minimizes the rate function. Although the experiment we consider involves a quantum evolution, the measurement at the two sites where pre- and post-selection takes place, render the probabilistic model of the experimental setting classical. As a result, the rate function is the relative entropy + +\[\sum_{i,j}\tilde{p}_{ij}\log \frac{\tilde{p}_{ij}}{p_{ij}} ,\] + +between the atypical observed distribution \(\tilde{p}_{ij}\), and the expected (i.e., typical) one \(p_{ij}\), which would have been obtained if no rare events or selection took place. Hence, we now seek the most likely (classical) joint probability distribution \(\tilde{p}_{ij}\) between initial and final states \(|x_{0}^{i}\rangle\text{and}|y_{1}^{j}\rangle\) that is in agreement with the observation record, i.e., + +\[\sum_{j}\tilde{p}_{ij}=\tilde{\alpha}_{i}\text{and}\sum_{i}\tilde{p}_{ij}=\tilde{\beta}_{j}.\] (8) + +This is nothing but a classical one-time-step Schr¨odinger bridge problem, to find the minimizer + +\[\operatorname*{arg\,min}_{\tilde{p}_{ij}}\sum_{i,j}\tilde{p}_{ij}\log \frac{\tilde{p}_{ij}}{p_{ij}} \text{so that}(8)\text{holds}.(9)\] + +To solve this problem, we consider the augmented Lagrangian + +\[\mathcal{L}=\sum_{i,j}\left(\tilde{p}_{ij}\log \frac{\tilde{p}_{ij}}{p_{ij}} +\lambda_{i}(\tilde{p}_{ij}-\tilde{\alpha}_{i})+\gamma_{j}(\tilde{p}_{ij}-\tilde{\beta}_{j})\right),\] + +where \(\lambda_{i}\text{and}\gamma_{j}\) are Lagrange multipliers. Setting its first variation with respect to \(\tilde{p}_{ij}\) to zero, we obtain that the diff --git a/outputs/arxiv_math/2503.05896_pg10_pg1_repeat1.md b/outputs/arxiv_math/2503.05896_pg10_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..dbb596cd53012f0835fa03753f5a64ca2ea39c51 --- /dev/null +++ b/outputs/arxiv_math/2503.05896_pg10_pg1_repeat1.md @@ -0,0 +1,39 @@ +Proof. We show that there exists \(00\text{such that}|\text{Sec}(g_{i}(t))|\leq\Delta\text{for all}i\in\mathbb{N}\text{and}t\in[0,b]\). + +Fix \(t_{0}\in(0,b)\), and consider a \(C^{1}-\)continuous curve \(\gamma_{i}:[0,1]\rightarrow M_{i}\). We have the following relation between the length of \(\gamma_{i}\) with respect to \(g_{i}(0)\), denoted by \(\ell(g_{i}(0))(\gamma_{i})\), and the length of \(\gamma_{i}\) with respect to \(g_{i}(0)\), denoted by \(\ell(g_{i}(t_{0}))(\gamma_{i})\): + +\[\begin{aligned} \ell(g_{i}(0))(\gamma_{i})&=\int_{0_{1}}^{1} \sqrt{g_{i}(0)(\gamma^{\prime}(s),\gamma^{\prime}(s))} ds \\ &\geq\int_{0} \sqrt{e^{-2Kt_{0}}g_{i}(t_{0})(\gamma^{\prime}(s),\gamma^{\prime}(s))} ds \\ &\geq e^{-Kt_{0}}\int_{Kt^{0_{0}}}^{1} \sqrt{g_{i}(t_{0})(\gamma^{\prime}(s),\gamma^{\prime}(s))} ds \\ &=e^{-}\ell(g_{i}(t_{0}))(\gamma_{i}). \end{aligned}\] + +From this, it follows that for all \(i\in\mathbb{N}\) and arbitrary \(x_{i},y_{i}\in M_{i}\) we have + +\[d_{g_{i}(0)}(x_{i},y_{i})\geq e^{-Kt_{0}}d_{g_{i}(t_{0})}(x_{i},y_{i}).\] + +For \(p_{i}\) fixed and \(r>0\), let \(q_{i}\in B_{e^{-Kt_{0}}r}^{g_{i}(0)}(p_{i}).\) Then since \(e^{-Kt_{0}}r>d_{g_{i}(0)}(p_{i},q_{i})\), it follows that + +\[r>e^{Kt_{0}}d_{g_{i}(0)}(p_{i},q_{i})\geq d_{g_{i}(t_{0})}(p_{i},q_{i}),\] + +That is the open ball \(B_{e^{-Kt_{0}}r}^{g_{i}(0)}(p_{i})\) of radius \(e^{-Kt_{0}}r\) centered at \(p_{i}\) with respect to \(g_{i}(0)\) is contained in the open ball of radius \(r\) centered at \(p_{i}\) with respect to \(g_{i}(t_{0})\), i.e. + +\[B_{e^{-Kt_{0}}r}^{g_{i}(0)}(p_{i})\subset B_{r}^{g_{i}(t_{0})}(p_{i}).\] + +\(e^{-Knb}\). Choose \(2r_{0}=\min\{\nu_{0},2\}\). Then we have that \(2e^{-Kt_{0}}r_{0}\leq2r_{0}\leq\nu_{0}\leq\text{Injrad}(g_{i}(0))\) for + +Observe that since \(t_{0}\in(0,b)\text{and}K>0\), then we have \(1\geq e^{-Kt_{0}}\geq e^{-Kb},\text{and}1\geq e^{-Knt_{0}}\geq\) all \(i\in\mathbb{N}\). Then by [12, Proposition 14], there exists a constant \(A(n)\), which depends only on the dimension \(n\), such that: + +\[\text{vol}(g_{i}(0))\left(B_{e^{-Kt_{0}}r_{0}}^{g_{i}(0)}(x_{i})\right)\geq \frac{A(n)}{n^{n}} r_{0}^{n}e^{-Knt_{0}}.\] + +for each \(i\in\mathbb{N}\text{and all}x_{i}\in M_{i}\). diff --git a/outputs/arxiv_math/2503.05901_pg4_pg1_repeat1.md b/outputs/arxiv_math/2503.05901_pg4_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..184de30fbdfce3bc40d471732bbca07988b5d3a0 --- /dev/null +++ b/outputs/arxiv_math/2503.05901_pg4_pg1_repeat1.md @@ -0,0 +1,47 @@ +but there are no equidistant points above the graph of the function \(f\). Therefore it is impossible to find different equidistant points along a vertical line. In particular, by choosing \(y_{1}=G(x)< y=y_{2}\), Lemma 1 shows that + +\[\begin{aligned} d((x,y_{2}),L)-d((x,y_{1}),L)&\min\{G_{i}(x)|i\in I\}.\] + +Then there exists at least one index \(i\in I\text{such that}y>G_{i}(x)\) and Lemma 2 implies that \(d((x,y),L_{i})G_{\text{min}}(x)\). On the other hand, suppose that + +\[y<\min\{G_{i}(x)|i\in I\}.\] + +Then \(y0:\int_{R}A\left( \frac{|f(x)|}{\lambda} \right)d\mu(x)\leq1\right\},f\in\mathscr{M}^{+}(R,\mu),\] + +where \(A:[0,\infty]\rightarrow[0,\infty]\) is a Young function. A function \(A:[0,\infty]\rightarrow[0,\infty]\) is called a Young function if it is convex, left-continuous, vanishing at 0, and not constant on the entire interval \((0,\infty)\). For example, when \(p\in[1,\infty)\), we have \(\|\cdot\|_{L^{p}(R,\mu)}=\|\cdot\|_{L^{A}(R,\mu)}\text{with}A(t)=t^{p},t\geq0\). We also have \(\|\cdot\|_{L^{\infty}(R,\mu)}=\|\cdot\|_{L^{A}(R,\mu)}\) with \(A(t)=\infty\cdot\chi_{(1,\infty]}(t),t\geq0\). Besides the classical textbooks [6, 53], the interested reader can find more information on the contemporary theory of Orlicz spaces and in particular Orlicz–Sobolev spaces in [16, 46]. An analogue of Fatou’s lemma is at our disposal in the framework of r.i. spaces. More precisely, if \(\mathscr{M}(R,\mu)\ni f_{k}\rightarrow f\) pointwise \(\mu-\text{a.e}\)., then + +\[\|f\|_{X(R,\mu)}\leq\liminf_{k\rightarrow\infty}\|f_{k}\|_{X(R,\mu)}.\] (2.1) + +With any r.i. function norm \(\|\cdot\|_{X(R,\mu)}\), there is associated another r.i. function norm, \(\|\cdot\|_{X^{\prime}(R,\mu)}\), defined for \(g\in\mathscr{M}^{+}(R,\mu)\) as + +\[\begin{aligned} \|g\|_{X^{\prime}(R,\mu)}&=\sup_{X(R,\mu}\int_{R}|f(x)||g(x)|d\mu(x),g\in\mathscr{M}^{+}(R,\mu). \\ \|f\|_{)}&\leq1 \end{aligned}\] (2.2) + +The r.i. function norm \(\|\cdot\|_{X^{\prime}(R,\mu)}\) is called the associate norm of \(\|\cdot\|_{X(R,\mu)}\). The resulting r.i. space \(X^{\prime}(R,\mu)\) is called the associate space. The definition of \(\|\cdot\|_{X^{\prime}(R,\mu)}\) immediately gives us that the Hölder inequality + +\[\int_{R}|f||g|d\mu\leq\|f\|_{X(R,\mu)}\|g\|_{X^{\prime}(R,\mu)}\text{for all}f,g\in\mathscr{M}(R,\mu)\] (2.3) diff --git a/outputs/arxiv_math/2503.05941_pg1_pg1_repeat1.md b/outputs/arxiv_math/2503.05941_pg1_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..d537678973b75d769c794dacce9a48d6880aa0a1 --- /dev/null +++ b/outputs/arxiv_math/2503.05941_pg1_pg1_repeat1.md @@ -0,0 +1,15 @@ +# Choosing Augmentation Parameters in OSQP-A New Approach based on Conjugate Directions + +Abstract— This work proposes a new method to select the augmentation parameters in the operator splitting quadratic program (OSQP) algorithm so as to reduce the computation time of overall algorithm. The selection is based upon the information of conjugate directions of the coefficient matrix of a linear system of equations present in the algorithm. This selection makes it possible to cache these conjugate directions, instead of computing them at each iteration, resulting in faster computation of the solution of the linear system thus reducing the overall computation time. This reduction is demonstrated by a numerical example. Notation:Throughout this manuscript, scalars and scalar-valued functions are denoted by small ordinary alphabets, vectors are denoted by bold ordinary alphabets, and matrices are denoted by capital alphabets. \(X\succ\mathbf{0}\) means that the matrix \(X\) is symmetric positive definite. \(diag(\bullet)\) represents a diagonal matrix. A \(‘T\) ’ in the superscript represents the transpose. \(‘^{k}’\) in the superscript of a quantity denotes the value of the quantity at the \(k^{\text{th}}\) iteration. + +## I. INTRODUCTION + +The analysis of convex optimization problems is of great importance in applied mathematics. Convex optimization problems arise in numerous fields including - machine learning, control engineering (model predictive control), Lasso and Hubber fitting and so on [1]. With the advent of big data and consequently higher dimensional data availability, the size of the convex optimization problems that are required to be solved continues to grow in size. Therefore, to handle real-time optimization-based tasks, it becomes necessary to develop algorithms that can work efficiently and effectively with the high-dimensional data. Furthermore, study of the solution algorithms for convex optimization problems turns out fruitful for the analysis of non-convex problems. In this work, we consider the convex optimization problem, called quadratic program, of the form as stated below. + +\[\begin{aligned} \text{minimize}\left( \frac{1}{2} \right)\mathbf{x}^{T}P\mathbf{x}+\mathbf{q}^{T}\mathbf{x} \\ \text{subject to}\mathbf{l}&\leq A\mathbf{x}\leq\mathbf{u} \end{aligned}\] (1) + +where \(\mathbf{x}\in\mathbb{R}^{n}\) is the decision variable, \(P\in\mathbb{R}^{n\times n}\) is a positive semi-definite matrix, \(\mathbf{q}\in\mathbb{R}^{n},A\in\mathbb{R}^{m\times n}\), and \(\mathbf{l}\in\mathbf{R}^{m}\text{and}\mathbf{u}\in\mathbb{R}^{m}\) define the lower and upper limits on the variable \(A\mathbf{x}\). This formulation allows handling equality constraints by setting \(\mathbf{l}_{i}=\mathbf{u}_{i}\). There exist a plethora of algorithms to solve the optimization problem 1 and can + +Acknowledgement: The author is thankful to Prof. Hoai Nam Nguyen (Télécom SudParis) for the fruitful discussions while working at Télécom SudParis, Institut Polytechnique de Paris. + +be broadly classified into three categories: (1) active-set methods [2], [3], (2) interior-point methods [4], and (3) first-order methods [5]. The major drawback of active set methods is that their worst-case complexity grows exponentially with the number of constraints [6] while the interior point methods are not scalable for very large scale problems because each iteration requires heavy computational tasks to be performed [7]. In recent years, the alternating direction method of multipliers (ADMM), which is an operator splitting based first-order method, has received a lot of attention by the researchers. This is because this method seems to be very well suited for the large scale problems because of the decomposability (inherited from the dual ascent algorithm) and superior convergence properties (inherited from the method of multipliers) [8]. Moreover, the steps involved in the ADMM algorithm are computationally cheap and simple to implement [1]. However, the optimal selection of the parameters, inherited from the augmented Lagrangian, involved in the ADMM algorithm is still an open problem [8]. Operator Splitting Quadratic Program (OSQP), proposed in [1], is one of the most popular general-purpose solvers for the convex quadratic programs. The algorithm is based on a novel splitting technique and subsequent usage of the auxiliary variables leading to a quasi-definite linear system which is always solvable. One of the most computationally expensive step in the OSQP algorithm involves the solution of a linear system-a common trait of the algorithms derived from ADMM. It is this linear system which we propose to solve in a more efficient manner. To do so we use the information about the conjugate directions of the coefficient matrix. The information about the conjugate directions is of grave importance in the study of convex optimization. To the best of author’s knowledge, this line of research, where the information of the conjugate directions is exploited to improve the characteristics of the algorithm, remains unexplored in the literature and the author believes that it holds promise. The information about the conjugate directions allows us to compute offline a set of parameters and cache them thus avoiding their computation at each iteration of the algorithm leading to reduction in the computation time of the overall algorithm which is demonstrated by a numerical example. The rest of the paper is organised as follows. Section II presents a review of the OSQP solver as reported in [1] followed by the background of conjugate direction and conjugate gradient methods. In Section III, we present the method to choose augmentation parameters by utilising the information of the conjugate directions of a specific coef- diff --git a/outputs/arxiv_math/2503.05960_pg4_pg1_repeat1.md b/outputs/arxiv_math/2503.05960_pg4_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..5a1a514e3e3448f93651dc2be54340e208304aee --- /dev/null +++ b/outputs/arxiv_math/2503.05960_pg4_pg1_repeat1.md @@ -0,0 +1,17 @@ +We thank Amol Aggarwal, Andrew Hardt and Travis Scrimshaw for helpful conversations about the groupoid. + +## 2. Groupoids + +A groupoid is a set \(G\) with a partially defined composition. This consists of a map \(\mu\) : \(S\longrightarrow G,\text{where}S\) is a subset of \(G\times G.\text{If}a,b\in G\) we say that the product \(a\star b\) is defined if \((a,b)\in S\), and then we write \(a\star b=\mu(a,b)\). The groupoid is also required to have an “inverse map” \(x\mapsto x^{\prime}\) from \(G\rightarrow G\). The inverse map is more commonly denoted as \(x\mapsto x^{-1}\), but we will be concerned with a groupoid whose elements are matrices, and we will reserve the notation \(x^{-1}\) for the matrix inverse. The following axioms are required. Axiom 1 (Associative Law). If \(a\star b\text{and}b\star c\) are defined then \((a\star b)\star c\text{and}a\star(b\star c)\) are defined, and they are equal. We say that \(a\star b\star c\) is defined if \(a\star b\text{and}b\star c\) are defined, and then we denote \((a\star b)\star c= a\star(b\star c)\text{as}a\star b\star c\). Axiom 2 (Inverse). The compositions \(a\star a^{\prime}\text{and}a^{\prime}\star a\) are always defined. Thus if \(a\star b\) is defined, then \(a\star b\star b^{\prime}\) is defined, and this is required to equal \(a\). Similarly \(a^{\prime}\star a\star b\) is defined, and this is required to equal \(b\). Example 2.1. A category \(\mathcal{C}\) is small if its class of objects is a set. A small category is a groupoid category if every morphism is an isomorphism. Assuming this, the disjoint union + +\[G=\bigsqcup_{A,B\in\mathcal{C}}\text{Hom}(A,B)\] + +is a groupoid, with the ⋆ operation being composition: thus if \(a\in\text{Hom}(A,B)\text{and}b\in \text{Hom}(C,D),\text{then}a\star b\) is defined if and only if \(B=C\). The groupoid axioms are clear. Lemma 2.2. In a groupoid, we have \((a^{\prime})^{\prime}=a\). Moreover if \(a\star b\) is defined then so is \(b^{\prime}\star a^{\prime}\) and \((a\star b)^{\prime}=b^{\prime}\star a^{\prime}\). Proof. Since \((a^{\prime})^{\prime}\star a^{\prime}\text{and}a^{\prime}\star a\) are both defined, by the Associative Law the product \((a^{\prime})^{\prime}\star a^{\prime}\star a\) is defined, and using the Inverse Axiom, this equals both \((a^{\prime})^{\prime}\text{and}a\). For the second assertion, assume \(a\star b\) is defined. It follows from the axioms that + +\[(a\star b)^{\prime}=(a\star b)^{\prime}\star a\star b\star b^{\prime}\star a^{\prime}=b^{\prime}\star a^{\prime}.□\] + +- + +Given a groupoid \(G\), let us say an element \(A\) is idempotent if \(A\star A\) is defined and \(A\star A=A\). Lemma 2.3. An element \(A\in G\) is an idempotent if and only if \(A=g\star g^{\prime}\text{for some}g\in G\). If \(A\) is idempotent then \(A=A^{\prime}\). Proof. It is easy to check that \(g\star g^{\prime}\) is idempotent. Conversely if \(A\) is idempotent, then \(A=A\star A^{\prime}\) since \(A=A\star A=A\star A\star A^{\prime}=A\star A^{\prime}\), and so \(A\) can be written \(g\star g^{\prime}\text{with}g=A\). Now if \(A=g\star g^{\prime}\text{then}A=A^{\prime}\) as a consequence of Lemma 2.2. Lemma 2.4. If \(g\in G\) then there are unique idempotents \(A\text{and}B\text{such that}g=g\star A\) and \(g=B\star g\). Proof. We can take \(A=g^{\prime}\star g\), and this is an idempotent such that \(g\star A=g\). Conversely if \(A^{\prime}\) is any other element such that \(g\star A^{\prime}=g,\text{then}g^{-1}\star g=g^{-1}\star g\star A^{\prime}=A^{\prime}\), so \(A^{\prime}=A\). The statements about \(B\) are proved similarly. + +- diff --git a/outputs/arxiv_math/2503.05976_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.05976_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..8beff022d95c5c6ad3ee7db7c673c9f45b1ca210 --- /dev/null +++ b/outputs/arxiv_math/2503.05976_pg3_pg1_repeat1.md @@ -0,0 +1,21 @@ +We note that the above is the most general statement of this result possible in the sense that no hypothesis can be dropped, and the bound is sharp. Firstly, the bound is sharp as equality occurs if \(Q\) is a nonzero constant, and it trivially fails if \(Q\equiv0\). Less trivially, the conclusion fails if \(P\) has no zero set: For \(d>0,P=1+\Vert z\Vert^{2}\), and \(Q= \frac{1}{P^{d}}\) , we get that \(P\) has no zero set, and rank of \(P^{d}\) is \(\left(_{d}^{n+d}\right)>1\), but rank of \(QP^{d}\) is + +> 1, but rank of \(QP^{d}\) is 1. + +\(d\) + +1. More generally, the conclusion fails simply if \(Q\) is not defined in a neighborhood of any point on the zero set of \(P:\text{For}d>0,P=1-\Vert z\Vert^{2},\text{and}Q= \frac{1}{P^{d}}\) , the rank of \(P^{d}\) is \(\left(^{n+d}\right)\) The conclusion √ may also fail if the bidegree of \(P\) is bigger than \((1,1):\text{For}d=1\), \(n=2,P=|z_{1}|^{4}-\) of bidegree \((2,2)\) and rank of \(P^{d}\) is 3, but rank of \(QP^{d}=|z_{1}|^{8}+|z_{2}|^{8}\) is 2. The proof of [4, Proposition 4.1] generalizes this to a family of examples with \(d\) a power of 2. The key idea in the proof is to reduce to the case when \(P\) is of the form + +\(\overline{2} |z_{1}|^{2}|z_{2}|^{2}+|z_{2}|^{4},\text{and}Q=|z_{1}|^{4}+ \overline{2} |z_{1}|^{2}|z_{2}|^{2}+|z_{2}|^{4}\), we get that \(P\) is + +\[(7)w+\bar{w}+\Vert z\Vert^{2}+\text{bidegree}-(1,1)\text{terms involving}w\text{or}\bar{w}\] + +where we split the variables to \(z\in C^{n-1}\text{and}w\in C\). The combinatorics of the bound on the rank in the case considered in [5] turns out to be somewhat straightforward once the problem is viewed in the correct context; one bounds the rank of the matrix by considering the number of nonzero entries on an extremal superdiagonal (or subdiagonal), and the count reduces to what could be termed a “monomial version” of the problem. In the presence of the linear terms \(w+\bar{w}\) , we can no longer reduce to a single superdiagonal (a monomial version), and the combinatorics required for the degree bound are significantly more difficult. If \(Q\) were a polynomial, then one could work in projective space and get rid of the linear terms by an automorphism of \(\in^{n}\). However, if \(Q\) is a real-analytic function, then such a change of coordinates is unavailable. The idea of the proof is that both the matrix of coefficients of \(P^{d}\) and a certain submatrix of the matrix of coefficients of \(QP^{d}\) in the reduced case have enough zero entries to allow row reduction preserving certain nonzero entries. These nonzero entries raise diagonal submatrices of full rank in the row echelon form. + +## 2. PRELIMINARIES + +This section establishes the fundamental definitions and notations for the (hermit-ian) rank of real-analytic functions. These concepts form the basis of our result. In this section, we write \(z=(z_{1},\ldots,z_{n})\text{and}\zeta=(\zeta_{1},\ldots,\zeta_{n})\) for the coordinates in \(C^{n}\), where \(\zeta\) is used for polarization. + +Notation 1. For any positive integer \(k\), denote by \([k]\) the set \(\{1,\ldots,k\}\) of integers from 1 to \(k\). For convenience, we will let \([0]\) denote the empty set, and \([\infty]\) denote the natural numbers \(N\). We will use this notation extensively to index terms in sums. + +Definition 2.1. Let \(U\subset C^{n}\) be a domain, and \(R:U\rightarrow C\) a real-analytic function. We define the rank of \(R\) at \(p\in U\) to be the smallest \(r\in\{0\}\cup N\cup\{\infty\}\) such that there exists diff --git a/outputs/arxiv_math/2503.05983_pg12_pg1_repeat1.md b/outputs/arxiv_math/2503.05983_pg12_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..843a8e0be597a61f1db8710c81a05922e527cb4a --- /dev/null +++ b/outputs/arxiv_math/2503.05983_pg12_pg1_repeat1.md @@ -0,0 +1,55 @@ +2. 2. The compact Kähler case. An important class of compact manifolds are those admitting a K¨ahler metric. This includes all complex submanifolds of projective space. + +In this case, the cohomological story greatly simplifies: Roughly speaking, all cohomologies are determined by Dolbeault cohomology. More precisely: + +Proposition 2.3 (The \(\partial\bar{\partial}-\text{Lemma}\), [DGMS75]). For any bicomplex \(A= (A,\partial,\bar{\partial})\), the following assertions are equivalent: + +1. For any \(a\in A\text{such that}\partial a=\bar{\partial}a=0\text{and}a=db\text{for some}b\in A\), there + +exists \(ac\in A\text{such that}a=\partial\bar{\partial}c\) . + +2. There is an isomorphism \(A\cong A^{sq}\oplus A^{dot},\text{where}A^{sq}\) is a direct sum + +of squares, i.e. bicomplexes of the form + +\[\mathbb{C}\mathbb{C}\] + +\[\mathbb{C}\mathbb{C},\] + +![](figure) + +where all arrows are ± id and all other maps vanish, and \(A^{dot}\) is a + +direct sum of dots, i.e. one-dimensional bicomplexes with all differentials being zero. + +## (3) All maps in the diagram + +(2.4) + +\[H_{\bar{\partial}}(A)\] + +![](figure) + +are isomorphisms. + +\[H_{BC}(A)\] + +\[H_{dR}(A)\] + +\[H_{A}(A)\] + +\[H_{\partial}(A)\] + +4. The spectral sequences in the previous diagram degenerate, and the + +filtrations on de Rham cohomology are \(n-\text{opposed}\), i.e. \(b_{k}^{p,q}=0\) unless + +\[\begin{aligned} k&=p+q. \\ er,fora \\ nditions. \end{aligned}\] + +Moreover, for a compact Kähler manifold \(X\), the bicomplex \(A_{X}\) satisfies + +A complex manifold \(X\) for which \(A_{X}\) satisfies the above conditions is called a \(\partial\bar{\partial}-\text{manifold}\). Compact K¨ahler manifolds are \(\partial\bar{\partial}-\text{manifolds}\), but the converse it not true. A broader class is for example given by those manifolds + +bimeromorphic to compact K¨ahler manifolds, i.e. Fujiki’s class \(C\), but also + +these do not exhaust all \(\partial\bar{\partial}-\text{manifolds}\), see e.g. [Fri19], [Li24], [KS23a]. One readily checks that all cohomologies introduced above are compatible with direct sums, vanish on squares and are one-dimensional on dots. Thus, as soon as a cohomology allows to reconstruct the information on the position of the dots, e.g. Dolbeault cohomology, or de Rham cohomology diff --git a/outputs/arxiv_math/2503.05989_pg1_pg1_repeat1.md b/outputs/arxiv_math/2503.05989_pg1_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..0498ecd56fcdf8e1a6bb2e0b73a3315f57826755 --- /dev/null +++ b/outputs/arxiv_math/2503.05989_pg1_pg1_repeat1.md @@ -0,0 +1,23 @@ +# Learning about passivity from data + +Alexandre Sanfelici Bazanella + +Abstract— This paper presents a data-driven methodology to estimate the storage function of a passive system. The methodology consists in parametrizing the storage function with a dictionary then running a linear program. Results on a benchmark are presented to illustrate its properties, including its robustness to noise. Various uses of the storage function that do not require knowledge of a model are also discussed. + +## I. INTRODUCTION + +Passivity is a fundamental property of dynamical systems that plays a major role in systems and control theory at least since the 1960’s. Determining whether a system is passive or not, and to which degree, revolves arounds finding an appropriate function describing its energy dissipation properties - the storage function. Passivity concerns the input-output relation of the system, and yet a storage function can also be used as a Lyapunov function, thus allowing to study the stability of the unforced behavior of the system as well. Once a storage function is known, various tasks can be performed using its knowledge: estimate the domain of attraction, estimate the convergence rate of autonomous trajectories, design a controller to improve this convergence rate, certify a previously designed controller, etc; this is all standard theory [6], [13]. What is much less recognized in the model-based tradition of systems and control theory is that these tasks require knowledge only of the storage function and not of the model. A storage function for a given system is usually obtained analytically from its model, but knowledge of a model is by no means a guarantee that a good storage function can be found. There are no general fail-proof methods for that, except for linear systems. Known methods are either for restricted classes of systems [6], rely on nonrobust numerical procedures [8], [14], and/or are constructive design methods that actually passivate a given system through control, instead of verifying the passivity of the uncontrolled system [7], [13]. So, from an applications perspective the relationship between models and storage functions is somewhat problematic: on one hand, in a variety of applications the model serves only as a means to find a storage function, becoming superfluous once it is found; but on the other hand the knowledge of a model is often not enough to find a useful storage function. Therefore, alternative ways of finding a storage function that do not rely on an analytical model are to be welcomed, even when such a model is available. + +A. S. Bazanella is with the Department of Automation and Energy, Universidade Federal do Rio Grande do Sul (DELAE/UFRGS), Porto Alegre-RS, Brazil. Email: bazanella@ufrgs.br. This study was financed by Conselho Nacional de Desenvolvimento Cientıfíco e Tecnológico (CNPq) + +This is what this paper is about: in it, a purely data-driven methodology to obtain a storage function is presented. The methodology consists in solving a linear optimization program (LP) formed with input/output/state data collected from the system, without the need for any priors. The LP yields the optimal parameter values of a dictionary parametrization of the storage function. A hyperparameter of the LP allows to make the solution robust to noise and to numerical errors. Recently, research on data-driven control has experimented a boost [1], [4] and a considerable amount of this work involves Lyapunov analysis. Among these works, the ones that deal with nonlinear systems usually consist of control synthesis [3] (as opposed to analysis), and rarely deal with passivity [9], [11]. Also, the vast majority of this work treats discrete-time systems, whereas this paper is devoted to continuous time systems. I start by reviewing in Section II the basic definitions of passivity. Then the proposed data-driven method for determination of passivity and identification of the storage function will be presented in Section III. Results for a classical benchmark - the pendulum - are presented in Section IV. Various uses of the storage function are discussed in Section + +V. Finally, lines of future and present work are outlined in Section VI. + +## II. PASSIVITY + +An overview of the theory on passivity of continuous time dynamical systems is given in this Section. The presentation follows the fundamental reference [13]. The class of systems considered are SISO (single-input-single-output) systems in the standard input-affine form: + +\[\begin{aligned} \dot{\mathbf{x}}&=f(\mathbf{x})+g(\mathbf{x}).u \\ y&=h(\mathbf{x}) \end{aligned}\] (1) (2) + +where \(\mathbf{x}\in\mathbb{R}^{n}\) is the state, \(u\in\mathbb{R}\text{and}y\in\mathbb{R}\) are the input and output respectively, \(f(\cdot):\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\text{and}g(\cdot):\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\) are vector fields, \(f(\mathbf{0})=\mathbf{0}-\) the origin is an equilibrium of the autonomous system - and \(h(\cdot):\mathbb{R}^{n}\rightarrow\mathbb{R}\) is the output function. Definition 1: Dissipativity and Passivity. Let \(\omega(\cdot,\cdot)\) : \(\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}\) be an integrable function, called the supply rate. A system in the form (1)-(2) is said to be dissipative with supply rate \(\omega(u(t),y(t))\) if there exists a function \(S(\cdot):\mathbb{R}^{n}\rightarrow\mathbb{R}\) satisfying, in some domain of the state diff --git a/outputs/arxiv_math/2503.05993_pg15_pg1_repeat1.md b/outputs/arxiv_math/2503.05993_pg15_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..3d431c1c80a03b439159f042ac46ce3ceaa6a56d --- /dev/null +++ b/outputs/arxiv_math/2503.05993_pg15_pg1_repeat1.md @@ -0,0 +1,35 @@ +![](figure) + +(a) Single Pendulum + +Double Pendulum + +(b) Pixel Data: Pendulum Experiment + +Pixel Data : Double Pendulum + +(c) Data Requirement + +(d) SVD Analysis for Double Pendulum[^inferred] + +![](figure) + +(a) Single Pendulum + +Double Pendulum + +(b) Pixel Data: Pendulum Experiment + +Pixel Data : Double Pendulum + +(c) Data Requirement + +(d) SVD Analysis for Double Pendulum[^inferred] + +Figure 4. Example 3: Application to non-linear pendulum. (a) Schematic diagram: single pendulum (top) and double pendulum (bottom). (b) Scaled pixel data: single pendulum experiments (left) and double pendulum animation (right). (c) Data requirement: comparison between damped single pendulum and double pendulum. (d) SVD analysis: determining the number of algebraic constraints in the case of the chaotic double pendulum with degree 5 library. + +in the absence of additional information about the system. Indeed, under the translation \(\phi=\omega+ \frac{3\pi}{2}\) , it can be easily verified that the equations we discovered after applying the transformation (35) are consistent with (30) and (28). To discover the dynamic equations in the polar coordinate system, we applied a Savitzky-Golay filter [48] with a window size of 12 to compute the first and second derivatives of the \(\phi\). The candidate library consisted of monomials of \(\phi,\dot{\phi},\cos\phi\text{and sin}\phi\) : \(\Theta_{*}=\{\phi,\dot{\phi},\cos\phi,\sin\phi,\phi^{2},\phi\dot{\phi},\ldots\}\). Using (35), we included x and y in the library instead of calculating \(\cos\phi\text{and sin}\phi\) directly from \(\phi\). Similar to the algebraic finder step, we restricted the monomial degree to 3 for Case 1 and varied it between 2 and 5 for Case 2 for a comparative study. After using sparse regression, where sequential threshold with inner iterations consisting of LASSO regularization was applied, we used a curve fit package to determine the correct parameters. In both case 1 and case 2, simulated solutions based on the discovered equations were able to predict the test pixel data accurately (see supplementary Section 3 for details on discovered equations and parameter estimation). For Case 3, SODAs robustly identified the constraints (33)–(34) from pixel data obtained from a single video segment, using various upper bounds on the degree of the monomial library. Note that the polar nature of the states \((x_{2},y_{2})\) is not immediately evident from the pixel data of the secondary pendulum in the double-pendulum (Figure 4b - right). This is because the secondary pendulum does not have a fixed pivot and sweeps trajectory in the 2D plane that doesn’t correspond to an identifiable pattern. Nevertheless, SODAs discovered an expanded version of (34). This algebraic discovery enables us to determine the transformation to the polar coordinate system: \(\phi_{1}=\arctan \frac{y_{1}}{x_{1}} \text{and}\phi_{2}=\arctan \frac{y_{1}-y_{2}}{x_{1}-x_{2}}\) , which can be further used for dynamic discovery. The dynamic discovery of equations governing the motion of the double pendulum similar to the form + +(31) –(32) based on polar coordinates, has been successfully demonstrated using SINDy-based methods [27]. We do not repeat this work here using our pixel data because it requires addressing noisy derivatives and may require advanced smoothing filters + +[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter. diff --git a/outputs/arxiv_math/2503.06000_pg17_pg1_repeat1.md b/outputs/arxiv_math/2503.06000_pg17_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..0cc40ae95973589c4cdd6b2d558621723b73a698 --- /dev/null +++ b/outputs/arxiv_math/2503.06000_pg17_pg1_repeat1.md @@ -0,0 +1,56 @@ +to compute dose volume histograms (DVHs) and associated dosimetric indices. To this end, we voxelize the basic structures using voxel with side length of \(1.5\text{mm}\), resulting in 21219 total voxels, of which 2140 correspond to the tumor and 19079 do not. Resulting DVHs are shown in Figure 3b. + +![](figure) + +100 + +RE + +| 90 | LE | +| --- | --- | +| | RB | +| 80 | RS LB | +| 70 | LS ST | +| 60 | SW data1 | + +50 + +40 + +30 + +20 + +10 + +0 + +3 + +6 + +9 + +12 + +15 + +18 + +0 + +Dose (Gy) + +(a) Needle configuration + +(b) DVHs + +Figure 3 + +Clinical results + +Figure 3b shows that doses among the various body structures are inhomogeneous. Specifically, the left side of the tumor surface receives higher doses compared to its right side, and there is a noticeable difference in the amount of radiation received by the boundary voxels on the left and right side of the nose. Further, the skin tissues exhibit long tails in their DVHs and most voxels on the left side of the tumor are exposed to a high dose. In the following section, we will compare the characteristics of this clinical plan with those obtained with the approaches we propose. To make the comparison easier, we normalize the treatment plans by uniformly scaling their dwell times so that their \(RB_{V_{100}}\) are approximately the same. In this process, \(RB_{V_{100}}\) describes the volume of target tissue \(RB\) that receives 100% of the prescribed radiation dose. The maximum normalization error is \(0.001\%\). This normalization error is \(\Big)*100,\text{where}RB_{V_{100}}^{1}\text{and}RB_{V_{100}}^{2}\) are the dosimetric values, after normalization, for the two treatment plans compared. In our models, we consider the five exiting planes shown in Figure 1b. To avoid negative bounds on some of the variables, which can affect the quality of relaxations of bilinear terms, we translate all of the body structures by the same vector, so that they all belong to the positive orthant. Then, all needle reference points can be chosen among the faces of the box \([1,173]\times[1,60]\times[95,200]\). For all optimization models that require dose computation, we use piecewise-linear penalty function \(f_{i}^{c}(s)=\max\{-5000s,0,5000(s-3)\}\text{with}\delta_{i}=6\) for each voxel \(i\) in a tumor region. We use + +calculated as + +\[\Big|RB_{V_{100}}^{1}-RB_{V}^{2}\Big|/\max_{100}\Big(RB_{V_{100}}^{1},RB_{V_{100}}^{2}\] diff --git a/outputs/arxiv_math/2503.06020_pg21_pg1_repeat1.md b/outputs/arxiv_math/2503.06020_pg21_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..e9eebd1babc33393cb3da16371623fc3ad4cdb01 --- /dev/null +++ b/outputs/arxiv_math/2503.06020_pg21_pg1_repeat1.md @@ -0,0 +1,59 @@ +### 3.3. Global existence. + +Theorem 3.10. Assuming that (3.2) and (3.3) are satisfied, then the conclusions in Theorem 3.4 hold for any \(T>0\). Proof. By Theorem 3.4, we may assume that (3.1) has a unique solution \((u,g,h)\) defined on some maximal time interval \((0,T_{m})\text{with}T_{m}\in(0,\infty]\), and + +\[h,g\in C^{1+ \frac{\beta}{2}}((0,T_{m})),u\in C^{1+ \frac{\beta}{2} ,2+\beta}(\Omega_{T_{m}}),\] + +where \(\Omega_{T_{m}}:=\{(t,x):t\in(0,T_{m}),x\in[g(t),h(t)]\}\). To complete the proof, we must demonstrate that \(T_{m}=\infty\). Suppose \(T_{m}<\infty\). Then, by the proof of Theorem 3.4, along with Lemmas 3.8 and 3.9, there exist positive constants \(C_{1},C_{2}=C_{2}(T_{m}),\text{such that for}t\in[0,T_{m})\text{and}x\in[g(t),h(t)]\), + +\[0\leq u(x,t)\leq C_{1},|h^{\prime}(t)|+|g^{\prime}(t)|\leq C_{2},|h(t)|,|g(t)|\leq C_{2}t+h_{0}.\] + +For any small constant \(\varepsilon>0\), it follows from the proof of Theorem 3.4 and Lemmas 3.8 and 3.9 that \(u,v\in C^{1+\beta, \frac{1+\beta}{2}}(\Omega_{T_{m}-\varepsilon})\). Thus, as in Step 4 of the proof of Theorem 3.4, applying Schauder’s estimates, and \(C_{i}\text{for}i=1,2\), but is independent of \(\varepsilon\). Since \(\varepsilon>0\) can be made arbitrarily small, it follows that for any \(t\in[T_{0},T_{m})\), + +\[C^{2+\beta,1+ \frac{\beta}{2}}(\Omega_{Tm-\varepsilon}\setminus\Omega_{T_{0}})\] + +\[\Vert u(t,\cdot)\Vert_{C^{2+\beta}([g(t),h(t)])}\leq Q^{*}.\] + +\(\underline{T}\) 2 + +By repeating the arguments used in the proof of Theorem 3.4, we can conclude that there exists \(T>0\) small, depending on \(Q^{*}\text{and}C_{i}(i=1,2)\), such that the solution to (3.1) with initial time \(T_{m}-\) can be extended uniquely to \(t=T_{m}- \frac{T}{2} +T>T_{m}\), a contradiction to the definition that \(T_{m}\) is the maximal time interval for the solution. Thus, we must have \(T_{m}=\infty\). + +- + +3. 4. Proof of Theorem 3.1. Let us first note that \(f(t,x,1)\equiv0\text{and}f(t,x,u)\leq\bar{f}(u)<0\text{for}u>1\) imply \(\bar{f}(1)=0\). Let \(M_{0}=\max\{\Vert u_{0}\Vert_{\infty},1\}\text{and}v(t)\) be the solution of + +\[v^{\prime}=\bar{f}(v),v(0)=M_{0}.\] + +Since \(\bar{f}(v)<0\text{for}v>1\), we clearly have \(1\leq v(t)\leq M_{0}\text{and}v(t)\rightarrow1\text{as}t\rightarrow\infty\). Since \(f(t,x,u)\leq\bar{f}(u)\) for \(u\geq1\), we obtain + +\[v_{t}-dv_{xx}=\bar{f}(v)\geq f(t,x,v)\text{for}t>0,x\in[g(t),h(t)].\] + +We also have \(v\geq1>\delta=u\text{for}x\in\{g(t),h(t)\}\text{and}v(0)=M_{0}\geq u(0,x)\text{for}x\in[-h_{0},h_{0}]\). Therefore the standard comparison principle over the region \(\{(t,x):t>0,x\in[g(t),h(t)]\}\) infers \(u(t,x)\leq v(t)\) in this region. It follows that + +\[\limsup_{t\rightarrow\infty}u(t,x)\leq\lim_{t\rightarrow\infty}v(t)=1\text{uniformly for}x\in[g(t),h(t)].\] (3.31) + +To bound \(u(t,x)\) from below, we first make use of (3.4) to show that there exists \(T_{0}>0\) such that + +\[u(t,x)\geq\delta\text{for}t\geq T_{0}\text{and}x\in[g(t),h(t)].\] + +(3.32) + +Similar to the proof of Theorem 1.1, since \(\underline{f}\) satisfies (fA), we are able to choose a function \(\hat{f}\in C^{1}\) sufficiently close to \(\underline{f} \text{in}L^{\infty}\text{such that}\hat{f}(s) \overline{\leq} f(s)\text{for}s\geq0,\text{and}\hat{f}\) satisfies (Fb) with \((P,Q)=(\hat{\theta},\delta)\) for some \(\hat{\theta}\in[\theta,\delta)\cap \overline{(} 0,\delta)\). Then, by Lemma 2.1, the traveling wave problem (2.4) has a solution pair \((c,q)=(c_{0},q_{0})\text{with}c_{0}>0\text{and}q_{0}(\cdot)\) strictly increasing. The same reasoning as in the proof of Theorem 1.1 shows that for some \(L>0\) sufficiently large, + +\[\underline{u} (t,x):=\max\{q_{0}(ct-x-L),q_{0}(ct+x-L)\}\] + +satisfies (in the weak sense) + +\[\underline{u}_{t} \leq d \underline{u}_{xx} + \underline{f} ( \underline{u} )\leq f(t,x, \underline{u} )\text{for}t>0,x\in\mathbb{R}.\] + +Additionally, + +and + +\[0\leq \underline{u} (t,x)\leq\delta\leq u(t,x)\text{for}t>0,x\in\{g(t),h(t)\},\] + +\[\underline{u} (0,x)\leq u_{0}(x)\text{for}x\in[-h_{0},h_{0}].\] + +Therefore we can apply the standard comparison principle over \(\{(t,x):t>0,x\in[g(t),h(t)]\}\) to deduce \(u(t,x)\geq \underline{u} (t,x)\) in this region. Moreover, using + +\[\lim_{t\rightarrow\infty}\Vert \underline{u} (t,\cdot)-\delta\Vert_{L^{\infty}(\mathbb{R})}=0,\] diff --git a/outputs/arxiv_math/2503.06022_pg26_pg1_repeat1.md b/outputs/arxiv_math/2503.06022_pg26_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..57b3ae1f16f8925035ce5832a26aabff9e2d4da9 --- /dev/null +++ b/outputs/arxiv_math/2503.06022_pg26_pg1_repeat1.md @@ -0,0 +1,33 @@ +\[\begin{aligned} ii.\text{For all}t&>0,\text{we have}\tilde{F}(tX,t^{\beta}Y)=t^{d-e}\tilde{F}(X,Y)\text{and}\tilde{G}(tX,t^{\beta}Y)=t^{d-e}\tilde{G}(X,Y),\text{where}e=e_{F}= \\ e_{G}. \end{aligned}\] + +Proof. Let \(\beta=r/s,\text{where}r>s>0\text{andgcd}(r,s)=1\). Since \(F\text{and}G\) are \(\beta-\)quasihomogeneous polynomials of degree \(d\), we have: + +\[\begin{aligned} mn \\ F(X,Y)&=a_{k}X^{d-rk}Y^{sk}\text{and}G(X,Y)=b_{k}X^{d-rk}Y^{sk}, \\ k&=0k=0 \end{aligned}\] + +where \(a_{m}\neq0,b_{n}\neq0,\text{and}0\leq m,n\leq\lfloor d/r\rfloor\). From these equations, it follows that \(e_{F}=d-rm\) and \(e_{G}=d-rn\). Thus, in order to prove that \(e_{F}=e_{G}\), it suffices to show that \(m=n\). Let us prove that \(m=n\). Since \(f_{+}(t)=a_{k}t^{sk}\text{and}f_{-}(t)=(-1)^{d-rk}a_{k}t^{sk}\), we have \(\deg f_{+}= \deg f_{-}=sm\); and since \(g_{+}(t)=b_{k}t^{sk}\text{and}g_{-}(t)=(-1)^{d-rk}b_{k}t^{sk}\), we have \(\deg g_{+}=\deg g_{-}=sn\). Also, by hypothesis, \(f_{+}\text{and}f_{-}\) are Lipschitz equivalent to \(g_{+}\text{and}g_{-}\) in some order. Hence, by Lemma 2.2, it follows that \(sm=sn\). Therefore, \(m=n\). From now on, we drop the subscript and denote simply by \(e\) the multiplicity of \(X\) as a factor of either \(F\text{or}G\). Let us prove the second part of the proposition. Since \(F\) is a \(\beta-\)quasihomogeneous polynomial of degree \(d\), we have: + +\[F(tX,t^{\beta}Y)=t^{d}F(X,Y)=t^{d}X^{e}\tilde{F}(X,Y).\] + +On the other hand, since the multiplicity of \(X\) as a factor of \(F\) is equal to \(e\), we have: + +\[F(tX,t^{\beta}Y)=(tX)^{e}\tilde{F}(tX,t^{\beta}Y)=t^{e}X^{e}\tilde{F}(tX,t^{\beta}Y).\] + +From these equalities, it follows that + +\[\tilde{F}(tX,t^{\beta}Y)=t^{d-e}\tilde{F}(X,Y).\] + + + +Clearly, the deduction above with \(F\) replaced with \(G\) yields the other identity. + +Corollary 3.5. Let \(F,G\in\mathbb{R}[X,Y]\) be \(\beta-\)quasihomogeneous polynomials of degree \(d\). If the height functions of \(F\text{and}G\) can be arranged into pairs of Lipschitz equivalent functions, then either both \(F\text{and}G\) are of the form \(cX^{d}\), or neither of them is. + +Proof. Denote by \(e_{F}\) the multiplicity of \(X\) as a factor of \(F\), and by \(e_{G}\) the multiplicity of \(X\) as a factor of \(G\). By Proposition 3.10, \(F\) is of the form \(cX^{d}\) if and only if \(e_{F}=d,\text{and}G\) is of the form \(cX^{d}\) if and only if \(e_{G}=d\). Suppose that the height functions of \(F\text{and}G\) can be arranged into pairs of Lipschitz equivalent functions. Then, by Proposition 3.11, \(e_{F}=e_{G}\), so that \(e_{F}=d\) if and only if \(e_{G}=d\). Therefore \(F\) is of the form \(cX^{d}\) if and only if \(G\) is of the form \(cX^{d}\). + + + +The next proposition shows how to determine whether any two polynomials of the form \(cX^{d},\text{with}c\neq0\) and \(d\geq1\), are \(R-\)semialgebraically Lipschitz equivalent. Proposition 3.12. Let \(F(X,Y)=aX^{d}\text{and}G(X,Y)=bX^{d},\text{where}a,b\in\mathbb{R}\setminus\{0\}\text{and}d\geq1\). + +i. If \(d\) is even, then \(F\text{and}G\) are \(R-\)semialgebraically Lipschitz equivalent if and only if \(a\text{and}b\) have the same sign. + +ii. If \(d\) is odd, then \(F\text{and}G\) are \(R-\)semialgebraically Lipschitz equivalent. diff --git a/outputs/arxiv_math/2503.06051_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.06051_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..1d58bfa9e43fc0dce0f73ea1380120e90ee360c2 --- /dev/null +++ b/outputs/arxiv_math/2503.06051_pg2_pg1_repeat1.md @@ -0,0 +1,15 @@ +Theorem 1.1 (equivalent to [Ale19, Theorem 22]). Let \(\alpha=\alpha_{1}\ldots\alpha_{n}\) be a composition with \(\alpha_{i}=\alpha_{i+1}\text{for some}i\in[n-1]\). Let \(\sigma\in\mathcal{S}_{n},\text{such that}\sigma_{i+1}=\sigma_{i}\pm1\). Then, + +\[E_{\alpha}^{\sigma}(\mathbf{x};q,t)=E_{\alpha}^{\sigma s_{i}}(\mathbf{x};q,t).\] + +Our main result is stated in the following theorem, which extends Theorem 1.1 by removing the assumption that \(\sigma_{i}\text{and}\sigma_{i+1}\) are consecutive. Theorem 1.2. Let \(\alpha=\alpha_{1}\ldots\alpha_{n}\) be a composition with \(\alpha_{i}=\alpha_{i+1}\text{for some}i\in[n-1]\), and let \(\sigma\in\mathcal{S}_{n}\). Then, + +\[E_{\alpha}^{\sigma}(\mathbf{x};q,t)=E_{\alpha}^{\sigma s_{i}}(\mathbf{x};q,t).\] + +Theorem 1.2 is proved algebraically from Theorem 1.1 in Section 2.3. However, in this article, our focus is on providing a fully combinatorial proof of Theorems 1.1 and 1.2 by constructing a probabilistic bijection on non-attacking fillings. The combinatorial interpretation of Theorem 1.2 is that swapping the basement entries in two adjacent columns of the same height does not change the generating function for non-attacking fillings. Our approach for proving this builds upon a (classical) bijection introduced in [CHMMW22] for unrestricted fillings of partition shape, in which two adjacent entries of the filling in the bottom row of columns of the same height are swapped while preserving the overall weight of the filling. In [Man24], the second author extended this technique by introducing a probabilistic bijection for partition-shaped non-attacking fillings. In this article, we generalize this probabilistic bijection to composition-shaped non-attacking fillings, thereby giving a bijective proof of Theorem 1.2. As implications of Theorem 1.2, certain assumptions can be removed from theorems in [Ale19] and [CMW22], which we summarize in Section 4. This article is organized as follows. Section 2 contains preliminaries on permuted-basement Macdonald polynomials and their tableaux formula in terms of non-attacking fillings. Section 3 defines a probabilistic bijection on non-attacking fillings, leading to a combinatorial proof of Theorem 1.2 in Section 3.1. Finally, Section 4 discusses some implications of Theorem 1.2. + +Acknowledgements. We would like to thank Per Alexandersson, Sarah Mason, and Arun Ram for helpful discussions. OM was supported by NSERC grant RGPIN-2021-02568. + +## 2. Preliminaries + +2.1. Permutations and compositions. Throughout the paper, let \(n\) be a nonnegative integer. The symmetric group \(\mathcal{S}_{n}\) is the group of all permutations of the set \([n]=\{1,2,\ldots,n\}\). We write permutations in one-line notation \(\sigma=[\sigma_{1},\sigma_{2},\ldots,\sigma_{n}]\), with brackets to distinguish them from compositions. Given two permutations \(\sigma,\pi\in\mathcal{S}_{n}\), their product is the permutation \(\sigma\pi\in\mathcal{S}_{n}\) defined by \((\sigma\pi)_{i}=\sigma_{\pi_{i}}\). The reverse \(\text{rev}(\sigma)\) of a permutation \(\sigma\in\mathcal{S}_{n}\) is the permutation \(\sigma w_{0}=[\sigma_{n},\sigma_{n-1},\ldots,\sigma_{1}],\text{where}w_{0}=[n,n-1,\cdots ,1]\). The symmetric group \(\mathcal{S}_{n}\) is generated by the simple transpositions \(s_{1},s_{2},\ldots,s_{n-1},\text{where}s_{i}\) swaps \(i\text{and}i+1\). The length \(\ell(\sigma)\) of a permutation \(\sigma\in\mathcal{S}_{n}\) is the smallest number of simple transpositions whose product equals \(\sigma\). A reduced expression of a permutation \(\sigma\) is a sequence \(s_{i_{1}},s_{i_{2}},\ldots,s_{i_{\ell(\sigma)}}\) of simple transpositions whose product \(s_{i_{1}}s_{i_{2}}\cdots s_{i_{\ell(\sigma)}}\) equals \(\sigma\). A composition of length \(n\) is a sequence \(\alpha=\alpha_{1}\alpha_{2}\ldots\alpha_{n}\) of nonnegative integers. A partition is a weakly decreasing composition, and an antipartition is a weakly increasing composition. Given a composition \(\alpha\), let \(\text{dec}(\alpha)\text{and inc}(\alpha)\) denote the partition and antipartition obtained diff --git a/outputs/arxiv_math/2503.06055_pg5_pg1_repeat1.md b/outputs/arxiv_math/2503.06055_pg5_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..aee2d8b0e6ee15eded91fa009734dff3cf675c50 --- /dev/null +++ b/outputs/arxiv_math/2503.06055_pg5_pg1_repeat1.md @@ -0,0 +1,25 @@ +2.3. Markov Dynamics. If \((\mathrm{X},\mathscr{B})\) is a measurable space and \(U\) is a topological space, a stochastic kernel \(P\) from \(U\) to X is a function \(P:U\times\mathscr{B}\rightarrow\) R+ with the property that \(u\mapsto P(u,B)\) is Borel measurable for each \(B\in\mathscr{B}\text{and}B\mapsto P(u,B)\) is a probability on \((\mathrm{X},\mathscr{B})\text{for all}u\in U.\text{When}U=\) X, we say that \(P\) is a stochastic kernel on X. Given a stochastic kernel \(P\) from \(U\) to X, we set + +\[(Ph)(u):=\int h(x^{\prime})P(u,dx^{\prime})(h\in bm\mathrm{X},u\in U).\] + +When X is a topological space we use the following standard terminology: We call \(P\) Feller if \(Ph\) is continuous on \(U\text{whenever}h\in bm\mathrm{X}\) is continuous on X. We call \(P\) strong Feller if \(Ph\) is continuous on \(U\text{whenever}h\in bm\mathrm{X}\). + +## 3. Abstract Dynamic Programs + +This section is a brief review of the dynamic programming framework from Sargent and Stachurski (2025). We repeat the key definitions to make the paper partly self-contained. + +3.1. Definition and Properties. An abstract dynamic program (ADP) is a pair \((V,\) T), where + +(i) \(V=(V,\preceq)\) is a poset and + +(ii) T \(=\{T_{\sigma}:\sigma\in\Sigma\}\) is a nonempty family of order preserving self-maps on \(V\) . + +We call \(V\) the value space. Each operator \(T_{\sigma}\) in T is called a policy operator. \(\Sigma\) is an arbitrary index set and elements of \(\Sigma\) are called policies. In applications, we impose conditions under which each \(T_{\sigma}\) has a unique fixed point. In these settings, the significance of each policy operator \(T_{\sigma}\) is that its fixed point, denoted below by \(v_{\sigma}\), represents the lifetime value of following policy \(\sigma\). + +Let \((V,\) T) be an ADP with policy set \(\Sigma\). Given \(v\in V\) , a policy \(\sigma\in\Sigma\) is called \(v-greedy\text{if}T_{\tau}v\preceq T_{\sigma}v\text{for all}\tau\in\Sigma\). An element \(v\in V\) is said to satisfy the Bellman equation if + +\[v:=\bigvee_{\sigma}T_{\sigma}v(v\in V).\] (1) + +In (1), the supremum is taken over all \(\sigma\in\Sigma\). The Bellman operator generated by \((V,\) T) is defined via + +\[Tv:=\bigvee_{\sigma}T_{\sigma}v\text{whenever the supremum exists}.\] diff --git a/outputs/arxiv_math/2503.06070_pg1_pg1_repeat1.md b/outputs/arxiv_math/2503.06070_pg1_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..311234e66381f9b42dcdfc134ab82ac832a4678b --- /dev/null +++ b/outputs/arxiv_math/2503.06070_pg1_pg1_repeat1.md @@ -0,0 +1,25 @@ +# Natural Gradient Descent for Control + +#### Ramin Esmzad ∗ Farnaz Adib Yaghmaie ∗∗ Hamidreza Modares ∗ + +∗ Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824, USA (e-mails: \(\{esmzadra\), modaresh\(\}@msu.edu)\). ∗∗ Department of Electrical Engineering, Linköping University, 58183 Linköping, Sweden (e-mail: farnaz.adib.yaghmaie@liu.se) + +Abstract: This paper bridges optimization and control, and presents a novel closed-loop control framework based on natural gradient descent, offering a trajectory-oriented alternative to traditional cost-function tuning. By leveraging the Fisher Information Matrix, we formulate a preconditioned gradient descent update that explicitly shapes system trajectories. We show that, in sharp contrast to traditional controllers, our approach provides flexibility to shape the system’s low-level behavior. To this end, the proposed method parameterizes closed-loop dynamics in terms of stationary covariance and an unknown cost function, providing a geometric interpretation of control adjustments. We establish theoretical stability conditions. The simulation results on a rotary inverted pendulum benchmark highlight the advantages of natural gradient descent in trajectory shaping. + +Keywords: Natural Gradient Descent, Trajectory Optimization, Stationary Covariance, Fisher Information Matrix (FIM). + +## 1. INTRODUCTION + +Optimization techniques, particularly gradient descent (GD) and its numerous variants (Ruder, 2017; Laborde and Oberman, 2020; Rattray et al., 1998; Martens, 2020), have become fundamental in modern control and machine learning. These methods are broadly classified into two categories when applied to control systems: GD-based control, where gradient methods optimize controller parameters, and controlled GD, where control-theoretic tools improve the convergence properties of a gradient-based optimizer (Lessard et al., 2016; Padmanabhan and Seiler, 2024; Nayyer et al., 2022). GD-based methods have demonstrated significant success in learning controllers for uncertain environments (Narendra and Parthasarathy, 1990; Sutton, 2018; Kiumarsi et al., 2018; Luo et al., 2017), as well as in system identification (Ljung, 1998; Hardt et al., 2018) and adaptive control (Gaudio et al., 2019; Ioannou and Sun, 2012; Cheng et al., 2024; Landau et al., 2011). Despite their effectiveness, a key challenge persists: the closed-loop trajectory behavior—how system states evolve over time—is typically an indirect outcome of weight tuning or cost-function optimization. Traditional methods + +⋆ This work is supported in part by the National Science Foundation under award ECCS-2227311. Farnaz Adib Yaghmaie is supported by the Excellence Center at Link¨oping–Lund in Information Technology (ELLIIT), ZENITH, and partially by Sensor informatics and Decision-making for the Digital Transformation (SEDDIT). This work was partly performed within the Competence Center SEDDIT-Sensor Informatics and Decision making for the Digital Transformation, supported by Sweden’s Innovation Agency within the research and innovation program Advanced digitalization. + +rely on defining an objective function, applying an optimization algorithm, and then observing the resulting state trajectories (Cothren et al., 2021). If these trajectories do not meet performance expectations, the objective function often requires manual adjustments, leading to a cumbersome iterative tuning process. Furthermore, this process is susceptible to “reward hacking,” where the optimized policy achieves a low-cost function value but deviates from the designer’s intended behavior (Skalse et al., 2022). In our previous work (Esmzad and Modares, 2024), we proposed a fundamentally different perspective by directly shaping system trajectories through a gradient-descent-like closed-loop approach. We introduced a novel parameterization of the stable closed-loop dynamics + +\[A+BK=I-2\Gamma P,\] (1) + +where \(A,B\) are the system matrices, \(K\) is a feedback gain, and \(\Gamma,P\succ0\) represent a GD step and a quadratic Lyapunov cost matrix, respectively. This formulation ensures that the closed-loop system behaves analogously to a GD update applied to the Lyapunov function \(V(x_{k})=x_{k}^{\top}Px_{k}\), leading to the explicit trajectory update dynamic + +\[x_{k+1}=x_{k}-\Gamma\nabla_{x_{k}}V(x_{k}).\] (2) + +Rather than iteratively adjusting an unknown cost function to produce desired state trajectories, we directly impose a controlled gradient flow on the system states. This trajectory-oriented perspective offers several advantages. Firstly, it provides explicit trajectory control. The gradient formulation provides direct control over the trajectory shape, eliminating the need for trial-and-error cost function tuning. Secondly, it quantifies robustness by the step size. For a small step size, closed-loop eigenvalues have small variations. Lastly, it unifies with a Linear Quadratic diff --git a/outputs/arxiv_math/2503.06086_pg17_pg1_repeat1.md b/outputs/arxiv_math/2503.06086_pg17_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..136a3cc056848285660d64f60a813e36ee5c8706 --- /dev/null +++ b/outputs/arxiv_math/2503.06086_pg17_pg1_repeat1.md @@ -0,0 +1,41 @@ +\[v_{i_{l}}\] + +\[v_{i_{l}}\] + +\(v\) + +\(xv\) + +\(x\) + +\[v_{1}^{\prime}C_{x_{1}^{\prime}v_{1}^{\prime}y}\] + +\(P\) + +\(v^{\prime}\) + +(a) Another shortest path \(P\) from \(v^{\prime}\) to \(x^{\prime}\) + +\(x^{\prime}\) + +1 + +\[x^{\prime}v^{\prime}\] + +\[x^{\prime}\] + +(b) \(v_{i_{l}}\) is incident to exactly one chord in \(C\) + +![](figure) + +Figure 12: The cycle \(C\) created by two distinct paths from \(v^{\prime}\) to \(x^{\prime}\) + +to it in \(C\). Let \(v_{i_{l}}y\) be one such chord; then we show that \(y\) lies in \(P(v_{1}^{\prime},x_{1}^{\prime})\) in the next claim. + +## Claim 6.6. \(y\) lies in \(P(v_{1}^{\prime},x_{1}^{\prime})\). + +Proof. If \(y\) does not lie in \(P(v_{1}^{\prime},x_{1}^{\prime}),\text{then}y\) must lie in the path between \(v_{i_{l}}\text{and}x_{1}^{\prime}\) (or the path between \(v_{i_{l}}\text{and}v_{1}^{\prime})\). If it lies on the path between \(v_{i_{l}}\text{and}x_{1}^{\prime}\) (resp. between \(v_{i_{l}}\) and \(v_{1}^{\prime})\), then we have a path from \(v_{i_{l}}\) to \(x^{\prime}\) (resp. between \(v_{i_{l}}\text{and}v^{\prime})\) which has length smaller than \(d(v_{i_{l}},x^{\prime})\) (resp. \(d(v_{i_{l}},v^{\prime}))\), which is a contradiction. Hence, \(y\) lies in \(P(v_{1}^{\prime},x^{\prime} \underline{)_{1}}\) (refer to Figure 12b). Note that \(d(x_{1}^{\prime},y)\geq d(x_{1}^{\prime},v_{i_{l}})\text{and}d(v_{1}^{\prime},y)\geq d(v_{1}^{\prime},v_{i_{l}})\). If not, then there exists a shorter or equal length path from \(v^{\prime}\) to \(v_{i_{l}}\) (from \(x^{\prime}\) to \(v_{i_{l}})\) that bypasses the edge \(v_{i_{l}}v\) (resp. \(v_{i_{l}}x)\), which is a contradiction to the fact that the pair \(v_{i_{l}},v^{\prime}\) (resp. \(v_{i_{l}},x^{\prime})\) monitors the edge \(v_{i_{l}}v\) (resp. \(v_{i_{l}}x)\). Hence \(d_{P}(x_{1}^{\prime},v_{1}^{\prime})=d(x_{1}^{\prime},y)+d(y,v_{1}^{\prime})\geq d(v_{1}^{\prime},v_{i_{l}})+d(x_{1}^{\prime},v_{i_{l}})\), implying \(d_{P}(x_{1}^{\prime},v_{1}^{\prime})=d(v_{1}^{\prime},v_{i_{l}})+d(x_{1}^{\prime},v_{i_{l}})\). This implies that \(d(x_{1}^{\prime},y)=d(x_{1}^{\prime},v_{i_{l}})\) and \(d(v_{1}^{\prime},y)=d(v_{1}^{\prime},v_{i_{l}})\), which implies that the vertex \(y\) is unique in \(P(v_{1}^{\prime},x_{1}^{\prime})\). This fact together with Claim 6.6 implies that in \(C,v_{i_{l}}\) is incident to exactly one chord. Now note that if \(v\text{and}y\) are not adjacent, then the induced 2-path \(yv_{i_{l}}v\) is part of a chordless cycle of length at least 4, which is a contradiction. Hence \(vy\in E(G_{k+1})\). Similarly, it can be shown that \(xy\in E(G_{k+1})\). Hence \(vv_{i_{l}}x\) is part of a 4-cycle \(vv_{i_{l}}xyv\), which leads to a contradiction. Hence \(v^{\prime}\text{and}x^{\prime}\) monitor \(vv_{i_{l}}\). Hence, until now, we showed that all edges in \(E_{l}\) can be monitored by the vertices of \(Man(G_{k})\setminus\{v_{i_{l}}\}\). Hence, combining this fact with Claim 6.1 and 6.3, we can conclude that \(Man(G_{k+1})\) forms an optimal MEG set of \(G_{k+1}\). Hence, \(meg(G)=|Man(G)|\) for every strongly chordal graph \(G\). + +## 7. Conclusion and future aspects + +In this paper, we solved the complexity status of the MIN-MEG problem for some well-known graph classes. Next, it will be interesting to address the following questions: diff --git a/outputs/arxiv_math/2503.06102_pg16_pg1_repeat1.md b/outputs/arxiv_math/2503.06102_pg16_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..6f151516b8cd87446f07b2bc9d029fc13cd3dcc1 --- /dev/null +++ b/outputs/arxiv_math/2503.06102_pg16_pg1_repeat1.md @@ -0,0 +1,23 @@ +\[Da2k\] + +\[a2k+2\] + +\[a \frac{D_{2k+1}2k}{kk+12h-k2h+1-k}\] + +\[Da2k\] + +\[a2k+2\] + +\[ta_{2k+1}\] + +\[\begin{aligned} \frac{ta_{2k+1}(D2k)}{kk+12h-k2h+1-k} \\ \text{isotopic} \end{aligned}\] + +\[Da2k\] + +\[a2k+2\] + +\[a \frac{t_{2k+1}a_{2k+1}(D2k)}{kk+12h-k2h+1-k}\] + +![](figure) + +Figure 11. The simple closed curves \(D_{2k}\text{and}\Phi_{K}(D_{2k})= \Phi_{T_{2,2h+1}}(D_{2k})=t_{a_{2k+1}(D_{2k})}\). diff --git a/outputs/arxiv_math/2503.06110_pg13_pg1_repeat1.md b/outputs/arxiv_math/2503.06110_pg13_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..85fcb8f749548c668984d09d6c815b63b7a08625 --- /dev/null +++ b/outputs/arxiv_math/2503.06110_pg13_pg1_repeat1.md @@ -0,0 +1,43 @@ +We finally show that the set \(A_{C}\) is contained within a neighborhood of size \(N^{-l}e^{-M}\) around a hyperplane in \(\mathbb{F}_{q}((X^{-1}))^{n}\). For any \(x\in C\), define \(y=e^{(n+1)lM}(x-x_{0})\text{in}\mathbb{Z}_{\mathrm{O}}^{n}\), so that + +\[g_{t}u_{x}=u_{y}g_{t}u_{x_{0}}.\] + +Consider a linear functional \(\phi_{t,C}\) of norm 1 that is zero on the hyperplane \(H_{t,C}=g_{t}u_{x_{0}}H_{C}\). Then, we have + +\[d([e_{1}],g_{t}u_{x}H_{C})=d([e_{1}],u_{y}H_{t,C})≍d([u_{-y}e_{1}],H_{t,C})≍\phi_{t,C}(u_{-y}e_{1}).\] + +Thus, if \(x\in A_{C}\), we obtain \(\phi_{t,C}(u_{-y}e_{1})≲e^{-M}\) , implying that \(y\) lies within a neighborhood of size \(O(e^{- \frac{M}{n}})\) of the affine hyperplane in \(\mathbb{F}_{q}((X^{-1}))^{n}\) defined by + +\[\phi_{t,C}(e_{1}-y_{1}e_{2}-\cdots -y_{n}e_{n+1})=0.\] + +This means that \(x\) lies in a neighborhood of size \(O(N^{-l}e^{-M})\) around an affine hyperplane in \(\mathbb{F}_{q}((X^{-1}))^{n}\). The number of subcubes of \(C\) that intersect this neighborhood is bounded by + +and thus we obtain + +\[≲_{n}N^{n}e^{-M},\] + +\[\begin{aligned} \text{card}K_{l}(C)&\geq N^{n}(1-O_{n}(e^{-M}))=N^{n}-O_{n}(N^{n- \frac{1}{n+1}}). \\ □ \end{aligned}\] + +Now, we proceed to the proof of Theorem 1.1. Proof of Theorem 1.1. To obtain a lower bound for the Hausdorff dimension of \(K_{\infty}\), we apply the Mass Distribution Principle. This argument follows identically from the work of [BdS23], but we include the proof here for the sake of completeness. For \(l\geq1\), define + +\[b_{l}= \begin{cases} \lfloor N^{n}(1-R_{3}e^{- \frac{M}{n}})\rfloor & \text{if}l_{k-1}^{+}0\), + +\[\mu(B(x,r))\leq Cr^{\alpha}.\] + +To justify this, choose \(l\text{such that}N^{-l}\tau_{n})\leq\bar{L}(|x-x_{0}|),x\in B_{r_{0}}^{c}(x_{0}).\] + +Hence, by letting \(n\rightarrow\infty\), we conclude that + +\[\mathbb{P}^{x}(\tau_{B_{r_{0}}(x_{0})}=\infty)\leq \frac{\bar{L}(|x-x_{0}|)}{\bar{L}(\infty)} <1,x\in B_{r_{0}}^{c}(x_{0}),\] + +i.e., \(\mathbb{P}^{x}(\tau_{B_{r_{0}}(x_{0})}<\infty)>0forallx\in\mathbb{R}^{d}.\) We next show that \(\{X(t)\}_{t\geq0}\) is open-set irreducible. According to [21, Theorems 7.3.6 and 7.3.7], there exists a strictly positive function \(q(t,x,y)on(0,\infty)\times\bar{B}_{r_{0}}(x_{0})\times\bar{B}_{r_{0}}(x_{0})\), jointly continuous in \(t,x\text{and}y\), and twice continuously differentiable in \(x\in B_{r_{0}}(x_{0})\), satisfying + +\[\mathbb{E}^{x}[f(X(t))1_{\{\tau_{\bar{B}_{r}}c_{0(x_{0})}>t\}}]=\int_{B_{r_{0}}(x_{0})}q(t,x,y)f(y)dy,\] + +for any \(t>0,x\in B_{r_{0}}(x_{0})\) and any continuous, bounded function \(f:\mathbb{R}^{d}\rightarrow\mathbb{R}.\) By employing the dominated convergence theorem, the above relation holds also for \(1_{O},\text{where}O\subseteq B_{r_{0}}(x_{0})\) is + +an open set. Denote by the class of all \(B\in\mathcal{B}(B_{r_{0}}(x_{0}))\) (the Borel \(\sigma-algebraonB_{r_{0}}(x_{0}))\) such that + +\[\mathbb{P}^{x}(X_{t}\in B,\tau_{\bar{B}_{r_{0}}^{c}(x_{0})}>t)=\int_{B}q(t,x,y)dy,t>0,x\in B_{r_{0}}(x_{0}).\] + +Clearly, contains the \(\pi-\)system of open rectangles in \(B_{r_{0}}(x_{0})\), and forms a \(\lambda-\text{system}\). By + +Dynkin’s \(\pi-\lambda\) theorem, we conclude that \(\mathcal{D}=\mathcal{B}(B_{r_{0}}(x_{0})).\) Consequently, for any \(t>0,x\in B_{r_{0}}(x_{0})\text{and}B\in\mathcal{B}(\mathbb{R}^{d})\), we have + +\[p(t,x,B)\geq\int_{B\cap B_{r_{0}}(x_{0})}q(t,x,y)dy.\] + +Define \(\psi(\cdot):=\lambda(\cdot\cap B_{r_{0}}(x_{0})),\text{where}\lambda(dx)\) denotes the Lebesgue measure on \(\mathbb{R}^{d}\). Clearly, \(\psi(dx)\) is a \(\sigma-\)finite measure whose support has a non-empty interior. It remains to prove that + +\[\int_{0}^{\infty}p(t,x,B)dt>0,x\in\mathbb{R}^{d},\] + +whenever \(\psi(B)>0,B\in\mathcal{B}(\mathbb{R}^{d}).\text{For}x\in B_{r_{0}}(x_{0})\) the implication holds trivially. Let \(x\in B_{r_{0}}^{c}(x_{0})\text{and}B\in\mathcal{B}(\mathbb{R}^{d}),\psi(B)>0\), be arbitrary. For all \(s>0\) we have + +\[\begin{aligned} \int_{0}^{\infty}p(t,x,B)dt&\geq\int_{\infty}^{\infty}p(t,x,B)dt \\ &=\int^{s}\int_{\infty}p(t-s,x,dy)p(s,y,B)dt \\ &\geq\int^{s}\int_{B_{r_{0}}(x_{0})}^{\mathbb{R}^{d}}p(t-s,x,dy)p(s,y,B)dt \\ &=\int_{B_{r_{0}}(x_{0})}^{s}p(s,y,B)\int_{s}^{\infty}p(t-s,x,dy)dt. \end{aligned}\] + +The assertion follows from the fact that \(p(s,y,B)>0\text{for}y\in B_{r_{0}}(x_{0})\), and + +\[\int_{s}^{\infty}p(t-s,x,B_{r_{0}}(x_{0}))dt=\int_{0}^{\infty}p(t,x,B_{r_{0}}(x_{0}))dt=\mathbb{E}^{[_{x}}\int_{0}^{\infty}1_{\{X(t)\in B_{r_{0}}(x_{0})\}}dt^{]}>0,\] diff --git a/outputs/arxiv_math/2503.06126_pg25_pg1_repeat1.md b/outputs/arxiv_math/2503.06126_pg25_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..2f2a4fce8187b60bacac359a21b71d96469fa1e1 --- /dev/null +++ b/outputs/arxiv_math/2503.06126_pg25_pg1_repeat1.md @@ -0,0 +1,31 @@ +One implication of this Gamma convergence is that: if \(u_{n}\) minimizes \(\tilde{\mathcal{E}}_{n}\) over \(g+ W^{1,n}(\Omega)\), then any cluster point, say \(u\), of \(u_{n}\) should minnimize \(\tilde{\mathcal{E}}_{\infty},\) i.e. + +\[u=\arg\min_{w\in g+W_{0}^{1,\infty}(\Omega)}\Vert\nabla w\Vert_{L^{\infty}}.\] + +Hence \(u\) is infinity harmonic, i.e. the Gamma convergence of \(\tilde{\mathcal{E}}_{n}\) echoes the convergence of \(u_{n}\). This can be regarded as an alternative proof for the classical result that \(-\Delta_{p}\) is approximately an infinity Laplacian when the constant exponent \(p\) is large. Coming back to the Orlicz setting, in general, we cannot simply replace \(\Vert\nabla u\Vert_{L^{n}}\) with \(\Vert\nabla u\Vert_{L^{\Phi n(\cdot)}}\) and expect a clean analogy for the above to hold. One reason is that the Luxemburg norm of an Orlicz space is not a complete parallel for the usual \(L^{p}\) space. For example, \(\Phi(x,s)=s^{p}(p\geqslant1)\) a generalized Φ-function, its Luxemburg norm \(\Vert\cdot\Vert_{L^{\Phi(\cdot)}}\) is equivalent but not equal to \(\Vert\cdot\Vert_{L^{p}}\). Some scalings are needed to make them agree. For detailed dicussions of the scalings of generalized Orlicz spaces and its consequences on Luxemburg norms, we refer to [HH19, Section 3.4]. That being said, there are some situations where such analogy exists. Theorem 11. Assume that \(\Phi_{n}\) satisfies \((4.3)\) and the unit normalization condition \((6.4)\). Assume in addition that (a stronger version of \((4.4)\) holds, i.e.) + +\[\lim_{n\rightarrow\infty}p_{n}^{-}=\infty\text{and}\lim_{n\rightarrow\infty} \frac{p_{n}^{+}}{p_{n}^{-}} =1.\] + +Let + +\[\tilde{E}_{n}:L^{1}(\Omega)\mapsto \begin{cases} |\Vert\nabla u\Vert|_{L^{\Phi n(\cdot)}}, & u\in W^{1,\Phi_{n}(\cdot)}(\Omega); \\ \infty, & \text{otherwise}, \end{cases}\] + +and + +\[\tilde{E}_{\infty}:L^{1}(\Omega)\mapsto \begin{cases} \Vert\nabla u\Vert_{L^{\infty}}, & u\in W^{1,\infty}(\Omega); \\ \infty, & \text{otherwise}. \end{cases}\] + +\[\text{Then}\Gamma\left(L^{1}(\Omega)\right)-\lim_{n\rightarrow\infty}\tilde{E}_{n}=\tilde{E}_{\infty}.\] + +This can be proved by similar arguments as [BM15, Theorem 2]. We omit the details. + +## 7. Comparison Principle and Uniqueness of the Limit Equation + +Let \(u_{n}\) be the sequence of the unique continuous weak solution for equation \((5.3)\). In Theorem 8 or Theorem 10, we have established that \(\{\Vert\nabla u_{n}\Vert_{L^{m}}:p_{n}^{-}\geqslant m\}\) is bounded for all \(m>d\) and there is a subsequence of \(u_{n}\) which converges uniformly to the following limit equation + +\[\begin{cases} -\Delta_{\infty,\Lambda}u=0, & x\in\Omega; \\ u=g, & x\in\partial\Omega, \end{cases}\] (7.1) + +where + +\[-\Delta_{\infty,\Lambda}u=-\Delta_{\infty}u-|\nabla u|\langle\Lambda(x,|\nabla u|),\nabla u\rangle.\] + +Remark 7. We shall show that the equation \((7.1)\) has a unique continuous viscosity solution. Since \(u_{n}\) is continuous and \(\{\Vert\nabla u_{n}\Vert_{L^{m}}:p_{n}^{-}\geqslant m\}\) is bounded for all \(m>d,\) the uniform convergence limit of every subsequence of \(u_{n}\) is again continuous. Hence the uniqueness result indicates that the whole sequence \(u_{n}\), not just a subsequence, converges uniformly to the unique continuous viscosity solution of \((7.1).\) diff --git a/outputs/arxiv_math/2503.06153_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.06153_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..ed6c8bbfecceddc77971cb4cf9e40de4e0d94c1a --- /dev/null +++ b/outputs/arxiv_math/2503.06153_pg2_pg1_repeat1.md @@ -0,0 +1,23 @@ +When \(r=1,M(1,L,c_{2})\) is isomorphic to \(S^{[c_{2}]}\), the Hilbert scheme of 0-dimensional closed subschemes in \(S\) of length \(c_{2}\). Fogarty showed that \(S^{[n]}(n\in\mathbb{Z}_{\geq0})\) is a smooth projective variety of dimension \(2n\) [Fog68] and computed its Picard group [Fog73]. The Betti numbers of \(S^{[n]}\) were completely determined by Göttsche [G¨90]. When \(r\geq2\), \(M(r,L,c_{2})\) can be reducible or non-reduced. But starting from Donaldson’s generic smoothness result [Don90], the study of \(M(r,L,c_{2})\) has revealed that \(M(r,L,c_{2})\) has better geometric properties as we fix \(r\geq2,L\), and let \(c_{2}\) increase. For example, when \(c_{2}\) is large enough, Gieseker–Li [GL94] [GL96] and O’Grady [O’G96] showed that \(M(r,L,c_{2})\) is irreducible, normal, and generically smooth of expected dimension. Some asymptotic phenomena also occur in the Picard group and Betti numbers of \(M(r,L,c_{2})(r\geq2)\) [Qin92][Li96][Li97][CW22]. This raises a natural question: + +## Question 1.1. Are there asymptotic phenomena similar to those above in the geometry of \(M(r,L,c_{2})\text{when}r=\) 0? + +Sheaves on \(S\) of rank zero are supported on points or curves. In this paper, we focus on those pure sheaves supported on curves, which are called one-dimensional sheaves. In contrast to the positive rank case, there are few general results for the moduli space of one-dimensional sheaves. An essential difficulty is that the supports of one-dimensional sheaves can be singular, even non-reduced. To distinguish this case, we use the following notation. Given a nonzero effective divisor \(\beta\text{on}S\) and an integer \(\chi\), denote by \(M_{\beta,\chi}\) the moduli space parametrizing polystable one-dimensional sheaves \(F\text{on}S\), with determinant \(\det(F)\cong \mathcal{O}_{S}(\beta)\) and Euler characteristic \(\chi(F)=\chi\). We will make the following assumption on \(\beta\), unless otherwise stated. + +## Assumption 1.2. The following conditions hold: + +\((1)\beta\) is base-point-free. + +2. A general curve in \(|\beta|\) is smooth and connected. + +A conjecture [SZ24, Conjecture 1.3] was proposed on the stabilization of (intersection cohomology) Betti numbers of \(M_{\beta,\chi}\text{when}\chi\) is fixed and \(\beta\) is sufficiently positive, as an analogue of [CW22, Conjecture 1.1]. When \(S\) is a del Pezzo surface, this conjecture was resolved in [PSSZ24] by proving an asymptotic formula of the refined BPS invariants which also refine the Betti numbers. The primary motivation of this paper is to explore the stabilization of the Picard number \(\rho(M_{\beta,\chi})\) of \(M_{\beta,\chi}\text{when}\beta\) is sufficiently positive. Here, the Picard number \(\rho(X)\) of a proper scheme \(X\) is defined as the dimension of the \(\mathbb{Q}-\)vector space Num\((_{\mathbb{Q}}X)\) of the numerically equivalence classes of line bundles on \(X\) with rational coefficients, in the sense of [Kle66]. Fix a basis \([L_{i}](i=1,\cdots ,\rho(S))\text{for Num}_{\mathbb{Q}}(S)\), where each \(L_{i}\) is a smooth, connected effective divisor. We specify a positivity condition for \(\beta\) as follows. + +## Definition 1.3. We say a divisor \(\beta onS\) satisfies (P) if + +\((1)\beta\) satisfies Assumption 1.2. + +2. The arithmetic genus \(p_{a}(\beta)>0\). + +3. For each \(1\leq i\leq\rho(S)\), there exists a smooth and connected curve \(L_{i}^{\prime}\in|\beta-L_{i}|\) such that \(L_{i}\) meets \(L_{i}^{\prime}\) transversally with intersection number \(L_{i}\cdot L_{i}^{\prime}\geq|\chi|+1\). + +Note that for an ample divisor \(\beta\), the multiple \(n\beta\) satisfies (P) for all \(n\gg0\). Our first main result is the following. We obtain an asymptotic lower bound of the Picard number of \(M_{\beta,\chi}\). diff --git a/outputs/arxiv_math/2503.06167_pg12_pg1_repeat1.md b/outputs/arxiv_math/2503.06167_pg12_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..989dac8c51d7ea513e24876e0d2506ae9d8ff8f1 --- /dev/null +++ b/outputs/arxiv_math/2503.06167_pg12_pg1_repeat1.md @@ -0,0 +1,19 @@ +where \(\eta>0\) is the step-rate, \(k\) is the time step, \(\mathcal{N}_{i}\) denotes the set of neighbours of agent \(i,W_{ij}\) is the consensus weighting factor on the shared information over link \((j,i),0\leq\mu<1\) is the momentum-rate, and variable \(y_{i}(k)\) denotes the momentum term of node \(i\) at time \(k\). The momentum term \(y_{i}\) is added to improve the convergence rate and accelerate the solution. For initialization, the agents set their initial state values as \(x_{i}(0)=b_{i}\) and \(y_{i}(0)=0\). + +### 3.2. Solution Subject to Sector-Bound Nonlinearity + +The data transmission channels (or the communication links) between agents might be subject to nonlinear constraints. This implies that the sent data \(\partial_{x_{j}}f_{j}(k)\) over a link \((j,i)\) is delivered to agent \(i\text{as}g_{l}(\partial_{x_{j}}f_{j}(k))\), where \(g_{l}:\mathbb{R}\mapsto\mathbb{R}\) denotes the nonlinear mapping. Then, the decentralized solution is updated as, + +\[\begin{aligned} x_{i}(k+1)&=x_{i}(k)+\mu y_{i}(k) \\ +\eta\sum_{j\in\mathcal{N}_{i}}W_{ij}(g_{l}(\partial_{x_{j}}f_{j}(k))-g_{l}(\partial_{x_{i}}f_{i}(k))), \end{aligned}\] (12) + +\[y_{i}(k+1)=x_{i}(k+1)-x_{i}(k),\] (13) + +where the weight matrix \(W\) is only weight-balanced and not necessarily stochastic. + +## Assumption 3. The nonlinear function \(g_{l}(\cdot)\) is assumed to be odd, strongly sign-preserving, and sector-bound. + +One example of such link nonlinearity is saturation or clipping. Another example is log-scale quantization [55, 56] illustrated in Fig. 2 with its formula as follows: + +\[g_{l}(u):=\text{sgn}(u)\exp(g_{u}(\log(|u|))),\] (14) + +with \(g_{u}(u):=\rho\left[ \frac{u}{\rho} \right]\) (the operator \([\cdot]\) as rounding to the nearest integer) and \(\rho>0\) as the quantization level. diff --git a/outputs/arxiv_math/2503.06185_pg6_pg1_repeat1.md b/outputs/arxiv_math/2503.06185_pg6_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..d71333b98bd18b1e9fce6ed31acc140e672c4b55 --- /dev/null +++ b/outputs/arxiv_math/2503.06185_pg6_pg1_repeat1.md @@ -0,0 +1,33 @@ +\[\alpha_{k}^{RBB}=\operatorname*{arg\,min}_{\alpha\in\mathbb{R}}\left\{\Vert\alpha\Delta y^{k-1}-\Delta\Psi^{k-1}\Vert_{2}^{2}+\tau_{k}\Vert\alpha \sqrt{H_{k}} \Delta y^{k-1}- \sqrt{H_{k}} \Delta\Psi^{k-1}\Vert_{2}^{2}\right\},\] + +where regularized parameter \(\tau_{k}\in[0,\infty)\). + +Property 1. [1, Them.2.2] Assume that \((\Delta y^{k-1})^{\mathrm{T}}\Delta\Psi^{k-1}>0,\tau_{k}\geq0.\text{Then}\alpha_{k}^{RBB}\in[\alpha_{k}^{BB1},\alpha_{k}^{BB2}]\) and it is monotonically increasing with respect to the parameter \(\tau_{k}\). Using a similar process, we obtain \(\beta_{k}^{RBB}\) as + +\[\beta_{k}^{RBB}= \frac{(\Delta y^{k-1})^{\mathrm{T}}\Delta\Phi^{k-1}+\tau_{k}\Vert\Delta\Phi^{k-1}\Vert_{2}^{2}}{\Vert\Delta y^{k-1}\Vert_{2}^{2}+\tau_{k}(\Delta y^{k-1})^{\mathrm{T}}\Delta\Phi^{k-1}} .\] (29) + +Combining the rule of penalty parameter (19) with (28) and (29), we get the RBB step size penalty parameters in ADMM as + +\[\rho_{k}^{RBB}=1/ \sqrt{\alpha_{k}^{RBB}\beta_{k}^{RBB}} .\] (30) + +In the spectral gradient step size methods, alternating strategy between long and short step sizes can improve the performance of algorithm. By adaptively adjusting the parameter \(\tau_{k}\), RBB method can be viewed as a continuous alternating step size method. One of its advantages is that if the regularization parameter is appropriately selected according to the nature of problem, then RBB generates reasonable step sizes, avoiding manual setting of alternating thresholds. Based on the analysis in the preceding section, the penalty parameter selection problem in ADMM is now transformed to the regularization parameter \(\tau_{k}\) selection problem in RBB. We now consider how to select appropriate parameter \(\tau_{k}\) in the RBB step sizes (28) and (29). The RB strategy (12) is an effective method for adjusting the penalty parameter, which describes the convergence properties of ADMM [22]. Combined with the spectral penalty parameter scheme (30), the RB strategy + +inspires us that if \(\Vert r^{k}\Vert_{2}>\Vert d^{k}\Vert_{2}\), then in the next iteration, a small spectral gradient step size is reasonable (corresponding to a large regularization parameter \(\tau_{k+1})\); otherwise, a large spectral gradient step size + +(corresponding to a small regularization parameter \(\tau_{k+1})\) is selected when \(\Vert r^{k}\Vert_{2}<\Vert d^{k}\Vert_{2}\). Based on these analyses, a natural conclusion is that the size of regularization parameter should be proportional to the + +ratio of \(\Vert r^{k}\Vert_{2}\text{and}\Vert d^{k}\Vert_{2}\). Therefore, we obtain the regularization parameter as follows + +\[\tau_{k+1}=\left( \frac{\Vert r^{k}\Vert_{2}}{\Vert d^{k}\Vert_{2}} \right)^{q},\] (31) + +where \(q>0\) is a constant that acts as a scaling factor. Remark 1. Similar to the procedure in [26], the penalty parameter generated by the RBB step size is scaling covariant and translation invariant. + +### 3.2 Selecting regularization parameter of regularized MV model + +In this subsection, we first give the initial regularization parameter \(\lambda_{0}\) based on the sample size and the number of assets, and then consider adaptively adjusting the regularization parameters based on the financial goal: controlling the number of short sales. For analysis, the \(\ell_{1}\) regularized model (3) can be rewritten as + +\[\min_{x\in\mathbb{R}^{n}} \frac{1}{2m} \Vert e\mathbf{1}_{m}-Rx\Vert_{2}^{2}+\lambda\Vert x\Vert_{1},\text{subject to}x^{\mathrm{T}}\mu=e,x^{\mathrm{T}}\mathbf{1}_{n}=1,\] (32) + +where matrix \(\mathbb{R}^{m\times n}\) is the historical returns of asset \(i\) on its \(i-\)th column over \(m\) observation periods, see [7]. The observation data contained in this matrix is used to estimate two important parameters in MV model: expected return and covariance matrix. If the number of the chosen observations \(m\) is small compared to the number of assets, then the sample covariance matrix becomes ill-conditioned, one suffers from the over-fitting problem [12, 3]. In this case, a high regularization promotes the sparsity of the solutions and mitigates over-fitting. On the other hand, if the number of samples is large enough, then the model obtained is truth and a small amount of regularization is reasonable. Based on these, we regard \(\frac{1}{m}\) as a factor of the initial regularization parameter \(\lambda_{0}\). The second factor that affects regularization is the number of assets \(n\). For any \(x\neq 0\), the following inequalities hold + +\[\frac{1}{\sqrt{n}} \leq \frac{\Vert x\Vert_{2}}{\Vert x\Vert_{1}} \leq1.\] diff --git a/outputs/arxiv_math/2503.06191_pg26_pg1_repeat1.md b/outputs/arxiv_math/2503.06191_pg26_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..148963216ff1ba81abd3af44014986eda3dd3b27 --- /dev/null +++ b/outputs/arxiv_math/2503.06191_pg26_pg1_repeat1.md @@ -0,0 +1,33 @@ +Corollary 31. Fix \(m,n \underline{\in} \mathbb{N}\). Let \(\psi:\mathbb{R}^{n}\rightarrow\mathbb{R}^{+}\) be an even \(C^{1}\) function + +\[m+1\] + +with \(\psi,|\nabla\psi|^{2}\in L^{1}(\mathbb{R}^{n},\gamma \frac{m}{n} )\text{and}D^{m}\psi,|\nabla D^{m}\psi|^{2}\in L^{1}(\mathbb{R}^{nm},\gamma_{n,m})\). Then, + +\[\begin{aligned} \left( \frac{m+1}{m} \right)\text{Var}_{\gamma} \tfrac{m}{nm+1} \psi+ \frac{1}{m+1} \text{Var}_{\gamma_{n,m}}D^{m}\psi \\ &\leq \frac{1}{2} \left( \frac{1}{m+1} \int_{\mathbb{R}^{nm}}|\nabla D^{m}\psi(x)|^{2}d\gamma_{n,m}(x)+\int_{\mathbb{R}^{n}}|\nabla\psi(x)|^{2}d\gamma^{\frac{m}{n^{m+1}}}(x)\right). \end{aligned}\] + +Observe that \(\nabla\psi\) refers to taking the gradient of \(\psi\text{on}\mathbb{R}^{n}\) while \(\nabla D^{m}\psi\) + +contains the gradient on \(\mathbb{R}^{nm}\). The quantity \(|\nabla D^{m}\psi|^{2}\) seems a bit mysterious, but in fact it is not so complicated. Since \(\nabla\) respects the product + +structure of \(\mathbb{R}^{nm}\), one has that + +\[\nabla D^{m}\psi(x_{1},\ldots,x_{m})=\left\{\nabla\psi(x_{i})-\nabla\psi\left(-\sum_{i=1}^{m}x_{i}\right)\right\}_{i=1}^{m}.\] + +Thus, + +\[|\nabla D^{m}\psi(x_{1},\ldots,x_{m})|^{2}=\sum_{i=1}^{m}\Bigg|\nabla\psi(x_{i})-\nabla\psi\left(-\sum_{i=1}^{m}x_{i}\right)\Bigg|^{2}.\] + +\[m+1\] + +Proof. Using standard approximation techniques, and the fast decay of the densities of the measures \(\gamma \frac{m}{n} \text{and}\gamma_{n,m}\), we may assume that \(\psi\) is compactly supported. In Theorem 4, write \(f=e^{-V}\) and switch the roles of \(V\text{and}V^{\star}\). Then, one can take logarithm to obtain + +\[(45)_{\leq}^{m\log}\left(\int_{m\log^{\mathbb{R}}}\left(\int^{e_{\mathbb{R}^{n_{n}}}^{-}} \frac{m+1}{e^{-^{m}}}^{V_{m}} \frac{(_{+1}^{\star}}{m} \frac{d_{|x|}^{x)_{2}}}{2} xd\right)x\right)^{+}+^{\log}\log\left(\left(\int\int_{\mathbb{R}^{nm}}^{e_{\mathbb{R}^{nm}}^{-D}}e^{d_{-D^{m} \frac{|\cdot|^{2}}{2} (x)}^{(^{m}V(x))}x}d\right)x\right).\] + +We consider the case when \(V(x)=V_{\varepsilon}(x)= \frac{|x|^{2}}{2} +\varepsilon\psi(x)\). Focusing on the second integral in (45), we have + +\[\begin{aligned} \log\left(\int_{\mathbb{R}}e^{-D^{m}(V_{\varepsilon}(x))}dx\right)-\log\left(\int_{\mathbb{R}^{nm}}e^{-D^{m} \frac{|\cdot|^{2}}{2} (x)}dx\right) \\ &=\log\left(\int_{\mathbb{R}^{nm}}^{nm}\exp(-\varepsilon D^{m}(\psi(x)))d\gamma_{n,m}(x)\right) \\ &=\log\left(1-\varepsilon\int_{\mathbb{R}^{nm}}\left(D^{m}\psi(x)+ \frac{\varepsilon^{2}}{2} (D^{m}\psi(x))^{2}\right)d\gamma_{n,m}(x)+o(\varepsilon^{3})\right) \\ &=-\varepsilon\int_{\mathbb{R}^{nm}}D^{m}\psi(x)d\gamma_{n,m}(x)+ \frac{\varepsilon^{2}}{2} \left(\text{Var}_{\gamma_{n,m}}D^{m}\psi\right)+o(\varepsilon^{3}). \end{aligned}\] + +We next need the well-known fact that if \(\psi\in C^{1}(\mathbb{R}^{n})\) is compactly supported, then + +\[V_{\varepsilon}^{\star}= \frac{|x|^{2}}{2} -\varepsilon\psi(x)+ \frac{\varepsilon^{2}}{2} |\nabla\psi|^{2}+o(\varepsilon^{3}),\] diff --git a/outputs/arxiv_math/2503.06194_pg10_pg1_repeat1.md b/outputs/arxiv_math/2503.06194_pg10_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..999a994c7bd097347f3826b240edd1744baae510 --- /dev/null +++ b/outputs/arxiv_math/2503.06194_pg10_pg1_repeat1.md @@ -0,0 +1,33 @@ +4. 1. The formula of the first homology groups. We recall the construction + +of abelian branched coverings of \((M,L)\). Let \(G\) be a finite abelian group and \(\pi\) : \(\pi_{1}(M-L)\rightarrow G\) be a surjective homomorphism. Let \(M_{\pi}\rightarrow M\) denote the branched cover corresponding to \(\pi\), that is, the Fox completion of the unbranched covering corresponding to \(\ker\pi\). Then the transformation group of \(M_{\pi}\rightarrow M\) is \(G\). Note that any finite abelian cover of \(M\) branched along \(L\) is obtained in this way. + +We also recall the formula for the size of the first homology groups of abelian branched coverings given by Mayberry–Murasugi and Porti. Let \(\hat{G}\) denote the set of homomorphisms from \(G\) to \(\mathbb{C}^{*}\). We choose meridians \(m_{1},...,m_{d}\in\pi_{1}(M-L;\mathbb{Z})\). For \(\xi\in\hat{G}\), we define a sublink \(L_{\xi}\) of \(L\) as + +\[L_{\xi}=\cup_{\xi(\pi(m_{i}))\neq 1}l_{i}.\] + +We write \(L_{\xi}=l_{i_{1}}\cup\cdots l_{i_{k}}\). For the trivial homomorphism \(1\in\hat{G}\), we define \(L_{1}=\emptyset\) and \(\Delta_{L_{1}}=1\). We also set + +\[\hat{G}^{(1)}=\{\xi\in\hat{G}|L_{\xi}\text{has a single component}\}.\] + +We write \(i(\xi)\) for the index of the meridian corresponding to \(\xi\in\hat{G}^{(1)}\). For a group \(H\), let \(|H|\) be the size of \(H\text{if}H\) is finite and 0 if \(H\) is infinite. Then Mayberry–Murasugi and Porti gave the following theorem. Theorem 4.1 ([8, Theorem 10.1], [4, Theorem 1.1]). + +\[|H_{1}(M_{\pi};\mathbb{Z})|= \frac{|G|}{\prod_{\xi\in\hat{G}^{(1)}}|1-\xi(\pi(m_{i(\xi)}))|} \prod_{\xi\in\hat{G}}|\Delta_{L_{\xi}}(\xi(\pi(m_{i_{1}})),...,\xi(\pi(m_{i_{k}})))|.(4.1)\] + +4. 2. \(\mathbb{Z}_{p}^{d}-\mathbf{coverings}\). Let \(\pi:\pi_{1}(M-L)\rightarrow\mathbb{Z}_{p}^{d}\) be the composition map of the abelianization \(\pi_{1}(M-L)\rightarrow\pi_{1}(M-L)^{ab}(\cong \mathbb{Z}^{d})\) and the inclusion \(\mathbb{Z}^{d}\hookrightarrow\mathbb{Z}_{p}^{d}\). For a finite index subgroup \(\Gamma\subset\mathbb{Z}_{p}^{d}\) let \(\pi_{\Gamma}:\pi_{1}(M-L)\rightarrow\mathbb{Z}_{p}^{d}/\Gamma\) denote the composition of \(\pi\) and the canonical surjective map \(\mathbb{Z}_{p}^{d}\rightarrow\mathbb{Z}_{p}^{d}/\Gamma\). Let \(M_{\Gamma}\rightarrow M\) denote the branched covering of \((M,L)\) corresponding to \(\pi_{\Gamma}\). + +Definition 4.2. A branched \(\mathbb{Z}_{p}^{d}-\)covering of \((M,L)\) is the inverse system of branched coverings \(M_{\Gamma}\rightarrow M\text{where}\Gamma\) runs finite index open subgroups in \(\mathbb{Z}_{p}^{d}\) . + +We obtain the \(p-\)adic convergence of the non\(-p-\)parts of the sizes of the first homology groups in branched \(\mathbb{Z}_{p}^{d}-\)covering of \((M,L)\). For a \(\mathbb{Z}_{p}-\text{module}A\) let \(\text{rank}A= \dim_{\mathbb{F}_{p}}A\otimes\mathbb{F}_{p}\). + +Lemma 4.3. Let \(\Gamma_{1}\supset\Gamma_{2}\supset\cdots\) be a descending sequence of finite open subgroups such that \(\cap_{n}\Gamma_{n}=\{0\}\text{in}\mathbb{Z}_{p}^{d}\) . Then we have + +1. for all sufficiently large \(n,\text{rank}\mathbb{Z}_{p}^{d}/\Gamma_{n}=d\). + +2. for each meridian \(m\) of \(L,\lim_{n\rightarrow\infty}\text{ord}(\pi_{\Gamma_{n}}(m))=\infty\). + +Proof. We put \(G_{n}=\mathbb{Z}_{p}^{d}/\Gamma_{n}\). + +1. We note that lim + +\(\longleftarrow nG_{n}\cong \mathbb{Z}_{p}^{d}\) by the assumption \(\cap_{n}\Gamma_{n}=\{0\}\). Since the canonical homomorphisms \(F_{n}:\mathbb{Z}_{p}^{d}\rightarrow G_{n}\text{and}f_{n/n-1}:G_{n}\rightarrow G_{n-1}\) are surjective the induced homomorphisms \(\tilde{F}_{n}:\mathbb{Z}_{p}^{d}\otimes\mathbb{F}_{p}\rightarrow G_{n}\otimes\mathbb{F}_{p}\text{and}\tilde{f}_{n/n-1}:G_{n}\otimes\mathbb{F}_{p}\rightarrow G_{n-1}\otimes\mathbb{F}_{p}\) are also surjective. Thus we have that the sequence \((\text{rank}G_{n})\) is monotonically increasing. We suppose that \(\text{rank}G_{n}0\). Then we have that \(\dim_{\mathbb{F}_{p}}\ker\tilde{F}_{n}\geq1\) for all \(n\). Since \(\ker\tilde{F}_{n}\subset\ker\tilde{F}_{n-1}\text{forall}n>1\), we see that \(\dim_{\mathbb{F}_{n}}\cap_{n}\ker\tilde{F}_{n}\geq1\). Thus diff --git a/outputs/arxiv_math/2503.06205_pg1_pg1_repeat1.md b/outputs/arxiv_math/2503.06205_pg1_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..30329bb2647eafce0f908314e46c785f2187981b --- /dev/null +++ b/outputs/arxiv_math/2503.06205_pg1_pg1_repeat1.md @@ -0,0 +1,23 @@ +# THE INITIAL-TO-FINAL-STATE INVERSE PROBLEM WITH TIME-INDEPENDENT POTENTIALS + +MANUEL CAÑIZARES, PEDRO CARO, IOANNIS PARISSIS, AND THANASIS ZACHAROPOULOS + +Abstract. The initial-to-final-state inverse problem consists in determining a quantum Hamiltonian assuming the knowledge of the state of the system at some fixed time, for every initial state. This problem was formulated by Caro and Ruiz and motivated by the data-driven prediction problem in quantum mechanics. Caro and Ruiz analysed the question of uniqueness for Hamiltonians of the form \(-\Delta+V\) with an electric potential \(V=V(\mathrm{t},\mathrm{x})\) that depends on the time and space variables. In this context, they proved that uniqueness holds in dimension \(n\geq2\) whenever the potentials are bounded and have super-exponential decay at infinity. Although their result does not seem to be optimal, one would expect at least some degree of exponential decay to be necessary for the potentials. However, in this paper, we show that by restricting the analysis to Hamiltonians with time-independent electric potentials, namely \(V=V(\mathrm{x})\), uniqueness can be established for bounded integrable potentials exhibiting only super-linear decay at infinity, in any dimension \(n\geq2\). This surprising improvement is possible because, unlike Caro and Ruiz’s approach, our argument avoids the use of complex geometrical optics (CGO). Instead, we rely on the construction of stationary states at different energies—this is possible because the potential does not depend on time. These states will have an explicit leading term, given by a Herglotz wave, plus a correction term that will vanish as the energy grows. Besides the significant relaxation of decay assumptions on the potential, the avoidance of CGO solutions is important in its own right, since such solutions are not readily available in more complicated geometric settings. + +## 1. Introduction + +In quantum mechanics, the family of wave functions \(\{u(t,\bullet):t\in[0,T]\}\) describes the state of the system during an interval of time \([0,T]\). If the motion takes place only under the influence of an electric potential \(V\) and the initial state \(u(0,\bullet)\) is prescribed by \(f\), then states + +\[u:(t,x)\in[0,T]\times\mathbb{R}^{n}\mapsto u(t,x)\in\mathbb{C}\] + +are solutions of the initial-value problem for the Schrödinger equation + +\[\begin{cases} i\partial_{\mathrm{t}}u=-\Delta u+Vu & \text{in}(0,T)\times\mathbb{R}^{n}, \\ u(0,\bullet)=f & \text{in}\mathbb{R}^{n}. \end{cases}\] (1) + +It is a well known fact that, if \(V\in L^{1}((0,T);L^{\infty}(\mathbb{R}^{n})),\text{then for every}f\in L^{2}(\mathbb{R}^{n})\) there exists a unique \(u\in C([0,T];L^{2}(\mathbb{R}^{n}))\) solving (1). Moreover, the linear map \(f\in L^{2}(\mathbb{R}^{n})\mapsto u\in C([0,T];L^{2}(\mathbb{R}^{n}))\) is bounded. The solutions with the previous properties will be referred to as physical. Using the physical solutions associated to a potential \(V\) , Caro and Ruiz formulated in [9] an inverse problem consisting in determining the electric potential from data measured only at the initial and final times. These data were modelled by the initial-to-final-state map, which is defined by + +\[\mathcal{U}_{T}:f\in L^{2}(\mathbb{R}^{n})\mapsto u(T,\bullet)\in L^{2}(\mathbb{R}^{n}),\] + +where \(u\) is the solution of the problem (1). This mapping is bounded in \(L^{2}(\mathbb{R}^{n})\). In [9], Caro and Ruiz proved that \(\mathcal{U}_{T}\) uniquely determines the potential \(V=V(\mathrm{t},\mathrm{x})\) in dimension \(n\geq2\) whenever \(V\in L^{1}((0,T);L^{\infty}(\mathbb{R}^{n}))\) has super-exponential decay, that is, \(e^{\rho|\mathrm{x}|}V\in L^{\infty}((0,T)\times\mathbb{R}^{n})\text{for all}\rho>0\). The reason that explains why the super-exponential decay was needed in [9] is the use of a family of solutions, usually called complex geometrical optics (CGO) solutions, that grow exponentially at infinity. In light + +Date: March 11, 2025. 2010 Mathematics Subject Classification. 35R30, 35J10, 81U40. Key words and phrases. Initial-to-final-state map, Schrödinger equation, uniqueness, inverse problem, inverse problems. diff --git a/outputs/arxiv_math/2503.06226_pg4_pg1_repeat1.md b/outputs/arxiv_math/2503.06226_pg4_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..1a9a73c65c556bae44eef26930ad73d588168404 --- /dev/null +++ b/outputs/arxiv_math/2503.06226_pg4_pg1_repeat1.md @@ -0,0 +1,41 @@ +Different from the state reconstruction given in (8) which is specific to the discrete-time systems, an alternative way to obtain the state information is to use a state observer to estimate the real state. Concretely, the state of system (1) can be estimated by using the following Luenberger observer: + +\[\hat{x}(k+1)=(A-LC)\hat{x}(k)+Bu(k)+Ly(k).\] (10) + +It is known that the Luenberger observer (10) has the property that \(\lim_{k\rightarrow\infty}\) xˆ(k) \(=x(k)\) when all the eigenvalues of \((A-LC)\) are set within the unit circle under the observability condition. As given in Rizvi & Lin (2019), by setting ˆx(k) = 0, the estimated state ˆx(k) can be parametrized by + +\[\hat{x}(k)=W_{y}\omega(k)+W_{u}\sigma(k)=\left[W_{y},W_{u}\right] \begin{bmatrix} \omega(k) \\ \sigma(k) \end{bmatrix} :=\bar{W}(k)(11)\] + +where the parameterization matrices \(W_{y}\in\mathbb{R}^{n\times np}\text{and}W_{u}\in \mathbb{R}^{n\times nm}\) are determined by the coefficients of the numerators in the transfer function matrix of the Luenberger observer system + +10. with y(k) and u(k) being the inputs to the observer system, and the dynamics of \(\omega(k)\text{and}\sigma(k)\) are given as + +\[\begin{aligned} \omega(k+1)&=\bar{A}_{y}\omega(k)+\bar{B}_{y}y(k),\omega(0)=\mathbf{0} \\ \sigma(k+1)&=\bar{A}_{u}\sigma(k)+\bar{B}_{u}u(k),\sigma(0)=\mathbf{0} \end{aligned}\] + +where \(\bar{A}_{y}\in\mathbb{R}^{np\times np}\text{and}\bar{A}_{u}\in\mathbb{R}^{nm\times nm}\) are user-defined matrices determined by the eigenvalues of \((A-\) LC), and \(\bar{B}_{y}\in\mathbb{R}^{np\times p}\) and \(\bar{B}_{u}\in\mathbb{R}^{nm\times m}\) are also user-defined matrices which can make \((\bar{A}_{y},\bar{B}_{y})\text{and}(\bar{A}_{u},\bar{B}_{u})\) controllable. It is easy to see that the data of \(\omega(k)\text{and}\sigma(k)\) are available since their system matrices \((\bar{A}_{y},\bar{B}_{y})\text{and}(\bar{A}_{u},\bar{B}_{u})\) are user-defined. Using the state parameterization in (11), the LQR problem can be solved by + +\[u(k)=-K^{*}\bar{W}(k):=-K_{\bar{W}}(k)\] (12) + +when the state of the observer system converges to the real state. It is worth mentioning that, if all the eigenvalues of \((A-LC)\) are chosen as 0, the observer state ˆx(k) converges to the real state x(k) when \(k\geq n\). Thus, by using the state reconstruction in (8) or the state parameterization in (11), the LQR problem can be solved by the output feedback controller (9) or (12) when the accurate knowledge of system matrices are known. + +### 2.3. Problem Formulation + +In this paper, we aim to solve the LQR problem of linear discrete-time systems with completely unknown system matrices and unmeasurable state, which can be formulated as follows: + +Problem 1. For system (1) where the system matrices A, B, and C are unknown, and the state x is unmeasurable, find an optimal control policy sequence u(k) to satisfy + +\[V(x(0))=\sum_{k=0}^{\infty}y^{T}(k)Q_{y}y(k)+u^{T}(k)Ru(k)=x^{T}(0)P^{*}x(0)\] + +where \(P^{*}\) is the solution to ARE (6). + +The challenges of Problem 1 arise from the solution of ARE + +(6) and the design of the parameterization matrix M¯ or W¯ , which becomes difficult in the absence of knowledge of the system matrices A, B, and C. By combining the ADP method with a series of historical data of input and output, some ADP-based output feedback learning approaches focused on estimating the optimal control gain KM¯ and KW¯ directly, for instance, Chen et al. (2023); Gao et al. (2016); Lewis & Vamvoudakis (2011); Rizvi & Lin (2019, 2020, 2023). Particularly, it is proven in Rizvi & Lin (2023) that the requirement on the full row rank of W¯ is necessary to guarantee the convergence performance of the aforementioned ADP-based output feedback learning approaches, so is the same requirement on M¯ along the similar proof. However, as we can see from (8), under Assumption 1, the parameterization matrix M¯ may not be of full row rank when the system matrix A has the eigenvalue 0. It follows from Rizvi & Lin (2023, Theorem 4) that the parameterization matrix W¯ may not be of full row rank when the matrices \((A-LC)\) and A have common eigenvalues. On the other hand, as stated in Postoyan et al. (2016), the stability of the closed-loop system under the learning control approach proposed in Lewis & Vamvoudakis (2011) cannot be guaranteed. Although this issue was eliminated in Chen et al. (2023); Rizvi & Lin (2019, 2023), the requirement of the convergence of state observer is needed to ensure the equivalence of the dynamic output feedback controller with the state feedback controller. This implies that the observer error will influence the convergence and optimality performance of the ADP-based output feedback learning control approaches in Chen et al. (2023); Rizvi & Lin (2019, 2023). To deal with the above issues, the objective of this paper is to provide a generalized learning based output feedback solution to Problem 1 and a detailed analysis of convergence, stability, and optimality for the proposed output feedback learning control approach. + +## 3. Main Results + +In this section, we propose a novel output feedback learning control approach to solve the LQR problem. In particular, a new dynamic output feedback controller is designed that is equal to a static state feedback controller. Then, a data-driven learning algorithm is established to estimate the optimal control gain without prior knowledge of system matrices. Finally, the convergence, stability, and optimality analyses of the proposed learning control approach are given. + +### 3.1. Dynamic Output Feedback Controller Design + +Now we present a generalized dynamic output feedback controller design method, where the internal model is established without using any prior knowledge of the system dynamics. Moreover, the equivalence relationship between the proposed output feedback controller and the state feedback controller always holds even in the presence of observer error. diff --git a/outputs/arxiv_math/2503.06228_pg1_pg1_repeat1.md b/outputs/arxiv_math/2503.06228_pg1_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..336de241c78db464a2de1e73cb10a927bfc3a6d7 --- /dev/null +++ b/outputs/arxiv_math/2503.06228_pg1_pg1_repeat1.md @@ -0,0 +1,31 @@ +# CONGRUENCES AND DENSITY RESULTS FOR PARTITIONS INTO DISTINCT EVEN PARTS + +HEMJYOTI NATH AND ABHISHEK SARMA + +Abstract. In this paper, we consider the set of partitions \(ped(n)\) which counts the number of partitions of \(n\) wherein the even parts are distinct (and the odd parts are unrestricted). Using an algorithm developed by Radu, we prove congruences modulo 192 which were conjectured by Nath [Nat24]. Further, we prove a few infinite families of congruences modulo 24 by using a result of Newman. Also, we prove that \(ped(9n+7)\) is lacunary modulo \(2^{k+2}\cdot3\text{and}3^{k+1}\cdot4\) for all positive integers \(k\geq0\). We further prove an infinite family of congruences for \(ped(n)\) modulo arbitrary powers of 2 by employing a result of Ono and Taguchi on the nilpotency of Hecke operators. + +## 1. Introduction + +The partition of a positive integer \(n\) is a non-increasing sequence of positive integers whose sum is equal to \(n\). For example, 5+3+2 is a partition of 10. If \(p(n)\) denotes the number of partitions of a positive integer \(n\), then with the convention that \(p(0)=1\), the generating function of \(p(n)\) (due to Euler) is given by + +\[\sum_{n=0}^{\infty}p(n)q^{n}= \frac{1}{(q;q)_{\infty}} ,\] + +where + +\[(a;q)_{\infty}:=\prod_{n=0}^{\infty}(1-aq^{n}),\] + +where \(a\text{and}q\) are complex numbers with \(|q|<1\). Throughout this paper, we set + +\[f_{k}:=(q^{k};q^{k})_{\infty},\text{for any integer}k\geq1.\] + +Ramanujan [Ram19], [Ram20] and [Ram21] discovered three beautiful congruences for the partition function, namely + +\[\begin{aligned} p(5n+4)&\equiv0\pmod{5}, \\ p(7n+5)&\equiv0\pmod{7}, \end{aligned}\] + +and + +\[p(11n+7)\equiv0\pmod{11}.\] + +The subject then got a lot of interest and many mathematicians over the years have found many more interesting results. Many Ramanujan-Type congruences for other classes of partitions such as \(l-\)regular partitions, etc (See for example, [CG13]) have also been found by many mathematicians. In this paper, the object of interest is the class of partitions where even parts are distinct and odd parts are unrestricted. + +Date: March 11, 2025. 2020 Mathematics Subject Classification. 11P81, 11P83, 05A17, 11F11. Key words and phrases. Integer partitions, Ramanujan-type congruences, Radu’s algorithm, arithmetic density. diff --git a/outputs/arxiv_math/2503.06230_pg7_pg1_repeat1.md b/outputs/arxiv_math/2503.06230_pg7_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..0bf83d38f75f6aa3a2c0a986630dcc3f5e7265cd --- /dev/null +++ b/outputs/arxiv_math/2503.06230_pg7_pg1_repeat1.md @@ -0,0 +1,15 @@ +Proposition 3.0.8. [8, Proposition 6] Soit \(L\) un anneau de Lie \(M_{c}\) tel que pour tout \(x\in L\), il existe un \(n\) tel que \([x+y,_{n}x]=[y,_{n}x]=0\) pour tout \(y\in L^{\prime}\). Alors \(L\) est nilpotent. + +Démonstration. On procède par récurrence sur la classe de résolubilité, le cas abélien étant trivial. On peut donc supposer que \(L^{\prime}\) est nilpotent. Il suffit de montrer par récurrence que tout centre itéré de \(L^{\prime}\) est contenu dans un centre itéré de \(L\) : il existera alors un indice \(i\) tel que \(L^{\prime}=Z_{n}(L^{\prime})\leq Z_{i}(L)\) et donc \(L/Z_{i}(L)\) est abélien et \(L=Z_{i+1}(L)\). Evidemment, \(\{0\}=Z_{0}(L^{\prime})\) est contenu dans \(Z_{0}(L)\). Supposons que \(Z_{i}(L^{\prime})\leq Z_{k}(L)\) ; les sous-anneaux \(Z_{i+1}(L^{\prime})\) et \(Z_{i}(L^{\prime})\) sont des idéaux de \(L\). On considère \(A=Z_{i+1}(L^{\prime})/Z_{i}(L^{\prime})\) ; c’est un anneau de Lie abélien sur lequel \(L\) agit de façon adjointe, de sorte que \(A\) est un \(L-\text{module}\). L’anneau de Lie \(A\) est centralisé par \(L^{\prime}\) et on obtient donc une structure de \(L/L^{\prime}-\text{module}\). Comme \(L\) est \(Z_{f}\), il existe \(x_{1},...,x_{n}\) dans \(L\) tels que \(C_{A}(L)=C_{A}(x_{1},..,x_{n})\). Pour tout \(x_{i}\), il existe un entier \(n_{i}\) tel que \([x_{i}+y,_{n_{i}}x_{i}]=[y,_{n_{i}}x_{i}]=0\) ; ainsi, pour tout \(y\in L^{\prime}\), l’action de \(x_{i}+y\) sur \(A\) coïncide avec celle de \(x_{i}\). D’après le Lemme 3.0.7, on obtient donc pour tout \(y\in Z_{i+1}(L^{\prime}),[y,_{m}L]\leq Z_{i}(L^{\prime})\leq Z_{k}(L)\) pour \(m=1+\sum_{i=1}^{k}(n_{i}-1)\). Par conséquent, d’après la Proposition 2.0.6, \(y\in Z_{k+m}(L)\). Finalement, \(Z_{i+1}(L^{\prime})\leq Z_{k+m}(L)\). + +Théorème 3.0.9. [8, Theorem 8] Soit \(L\) un anneau de Lie \(M_{c}\). Alors \(F(L)\) est un idéal nilpotent. + +Démonstration. L’idéal \(F=F(L)\) est localement nilpotent et donc résoluble d’après la Proposition 3.0.3. Soit \(x\in F\) ; cet élément appartient à un idéal nilpotent \(I\) de \(L\), de classe de nilpotence \(r\). Par conséquent, pour un élément \(y\) de \(F^{\prime},[x,y]\in I\) et donc \(ad_{x}^{r}(x+y)=ad_{x}^{r}(y)=0\). On conclut d’après la Proposition 3.0.8. + +## 4 Théorème de Engel + +Dans cette section, nous allons établir un analogue du théorème de Engel. Sauf mention explicite du contraire, \(L\) est une algèbre de Lie sur un corps de caractéristique nulle. D’après les travaux de Hartley, il est possible de définir un analogue du radical de Baer dans ce contexte. Définition 4.0.1. Soit \(L\) une algèbre de Lie. On dit que \(H\) est un sous-idéal de \(L\) s’il existe une suite finie de sous-algèbres telle que : + +\[H_{0}=H\rhd H_{1}\rhd...\rhd H_{n}=L.\] + +L’indice d’un sous-idéal désigne la longueur minimale d’une suite de la forme précédente. Fait 4.0.2. [9] Le radical de Baer \(B(L)\) est la sous-algèbre engendrée par les sous-idéaux nilpotents et de dimension finie. On a : diff --git a/outputs/arxiv_math/2503.06256_pg5_pg1_repeat1.md b/outputs/arxiv_math/2503.06256_pg5_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..f272ae3b14a6b198a5b19d54b39a24a466f91a3b --- /dev/null +++ b/outputs/arxiv_math/2503.06256_pg5_pg1_repeat1.md @@ -0,0 +1,33 @@ +Previously it was only known (using arguments from [15]) that the low moments of these quantities should be of the same order of magnitude. Now, having shown that our variances concentrate around the mean square of an Euler product, we are in a hopeful position, since previous work of Saksman and Webb [30, Theorem 1.9] tells us that, when \(t\) is restricted to a bounded interval, we have convergence in distribution to an integral with respect to a random measure known as a critical Gaussian multiplicative chaos measure. In the following theorem, we extend their result to give convergence over the full range of integration. To state our result, we redefine the Euler product over primes \(p\leq x\) by + +\[\big)^{-1}\] + +\(F_{(x)}(s):=\prod_{p\leq x}\big(1- \frac{f(p)}{p^{s}}\) in the Steinhaus case. This is a notational convenience that makes the statement of our theorem more natural. + +Theorem 3 (Extension of Theorem 1.9 of Saksman and Webb [30]). For \(g\in C^{l}[a,b]\), + +denote \(\Vert g\Vert_{C^{l}[a,b]}=\sum_{j=0}^{l}\Vert g^{(j)}\Vert_{L^{\infty}[a,b]}.\text{For}f\) a Steinhaus random multiplicative function, we have + +\[\frac{(\log\log x)^{1/2}}{\log x} \int_{\mathbb{R}}\bigg| \frac{F_{(x)}(1/2+it)}{1/2+it} \bigg|^{2}dt\longrightarrow ^{d}V_{\text{crit}}asx\rightarrow\infty,\] + +where \(\mathbb{E}[V_{\mathrm{c}\text{rit}}^{q}]\) exists for each \(00\), all norms \(\Vert g\Vert_{C^{l}[-n,n]}\text{and}\Vert1/g\Vert_{C^{l}[-n,n]}\) possess moments of all orders, and \(\lambda(dt)\) is a critical Gaussian multiplicative chaos measure. Note that \(g\text{and}\lambda\) may not be independent. The random variable \(V_{\text{crit}}\) is precisely the one that appears in Theorem 1. For details of this weak convergence, see Section 7, and more specifically Section 7.2. An analogous theorem is not known in the Rademacher case (and would involve non-trivial adjustments to the argument). For this reason, we do not give a proof of Theorem 1 in the Rademacher case. + +1. 4. Connection to Gaussian multiplicative chaos. We now explain the connection + +between our variance (which we think of as mean squares of random Euler products like (1.2)) and the theory of Gaussian multiplicative chaos, restricting ourselves to the Steinhaus case. For an introduction to the theory, see Rhodes and Vargas [29]. A similar discussion can be found in the introduction of Harper [15]. Here, we put a particular emphasis on understanding the limiting distribution of our variances. + +We begin by noting that, for fixed \(t\), the quantity \(\log|F(1/2+it)|\approx\Re\sum\) + +\[\sum^{p\leq \sqrt{x}} \frac{f(p)}{p_{1^{/2+it}}^{1}}\] + +is distributed like a Gaussian with mean 0 and variance approximately \(\frac{1}{2} \sum_{p\leq \sqrt{x}} \frac{1}{p} = \frac{1}{2} \log\log x+O(1)\). Consequently, we think of \(X(t)=\log|F(1/2+it)|\) as a Gaussian field. The reader is reminded that a key property of Gaussian fields is that their distribution is completely determined by their means, \(\mathbb{E}[X(t)]\) (which will be zero in the Steinhaus case), and their covariances, \(\mathbb{E}[X(t_{1})X(t_{2})]\). With this in mind, we rewrite our variances in (1.2) as \(\frac{(\log\log x)^{1/2}}{\log x} \int_{0}^{1}e^{2X(t)}dt\). For simplicity, we have restricted ourselves to a compact interval and ignored the denominator. It is natural to normalise our integral so that the integrand has expectation one, and, if \(X(t)\) were actually Gaussian for each \(t\in[0,1]\), the + +correct2 renormalisation would be to multiply the integrand by \(e^{-2\mathbb{E}X(t)^{2}}\). This accounts for the factor of \(\frac{1}{\log x}\) in (1.1), so our variances are roughly + +\[(\log\log x)^{1/2}\int_{0}^{1}\exp\left(2X(t)-2\mathbb{E}\left(X(t)^{2}\right)\right)dt,\] (1.3) + +2This follows from the moment generating function for Gaussian random variables. diff --git a/outputs/arxiv_math/2503.06267_pg10_pg1_repeat1.md b/outputs/arxiv_math/2503.06267_pg10_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..403fc1ecf82118a406c2972786eaa68cefb17ac0 --- /dev/null +++ b/outputs/arxiv_math/2503.06267_pg10_pg1_repeat1.md @@ -0,0 +1,37 @@ +A subrepresentation of a representation V of the magnetic point group \((G,\phi)\) is a subspace \(W\subset\) V such that the restriction of the G-action to W is also a representation of \((G,\phi)\). A representation V of \((G,\phi)\) is irreducible if its only representations are V and 0. Now, the category of representations of finite magnetic groups is semisimple. + +Proposition 1.9. Every representation of a finite magnetic group is a sum of irreducible representations. + +Which follows from Maschke’s lemma applied to representations of magnetic groups. + +Lemma 1.10 (Maschke). If W is a subrepresentation of V of a finite magnetic group, then there exists a subrepresentation \(W^{\prime}\) of V such that \(V=W\oplus W^{\prime}\). + +Proof. Write V as a direct sum \(V=W\oplus U\) (with U not necessary a subrepresentation) and consider the projection \(\pi_{1}:V\longrightarrow W\) given by \(\pi_{1}(w+u)=w\). Define + +\[\begin{aligned} \pi:V\longrightarrow W \\ v\longmapsto\sum_{g\in G}g\cdot\pi_{1}(g^{-1}\cdot v). \end{aligned}\] (1.10) + +The morphism \(\pi\) is in \(\text{Hom}_{\mathbf{Rep}(G,\phi)}(V,W)\) because if \(h\in G\text{and}v\in V\), then + +\[\pi(h\cdot v)=\sum_{g\in G}g\cdot\pi_{1}(g^{-1}\cdot(h\cdot v))\] (1.11) + +\[=\sum_{g\in G}g\cdot\pi_{1}((g^{-1}h)\cdot v)\] (1.12) + +\[=\sum_{g\in G}(hh^{-1}g)\cdot\pi_{1}((h^{-1}g)^{-1}\cdot v)\] (1.13) + +\[=h\cdot\sum_{g\in G}(h^{-1}g)\cdot\pi_{1}((h^{-1}g)^{-1}\cdot v)\] (1.14) + +\[=h\cdot\sum_{l\in G}l\cdot\pi_{1}(l^{-1}\cdot v)\] (1.15) + +\[=h\cdot\pi(v)\] (1.16) + +The morphism \(\pi\) is surjective, this follows since \(\pi_{1}(w)=w\text{for}w\in W\), so \(\pi(w)=|G|w\). If we denote by \(W^{\prime}=\ker\pi\), then we have the decomposition as a direct sum of representations of \((G,\phi)\) + +\[V\supset\text{Im}\pi\oplus\ker\pi=W\oplus W^{\prime}.\] (1.17) + +□ + +One key step required to classify the irreducible representations of magnetic groups \((G,\phi)\) is to construct the induced representations of the core \(G_{0}\leq G\). + +Definition 1.11. Let \((G,\phi)\) be a finite magnetic point group and V a representation of the core group G0. The induced representation of V is the complex vector space + +\[\text{Ind}_{G_{0}}^{(G,\phi)}V:=\mathbb{C}[G,\phi]\otimes_{\mathbb{C}[G_{0}]}V\] (1.18) diff --git a/outputs/arxiv_math/2503.06285_pg1_pg1_repeat1.md b/outputs/arxiv_math/2503.06285_pg1_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..bcd85c103676054e2d55a291ff0788bc1d7a8b19 --- /dev/null +++ b/outputs/arxiv_math/2503.06285_pg1_pg1_repeat1.md @@ -0,0 +1,27 @@ +# Modified Bregman Golden Ratio Algorithm for Mixed Variational Inequality Problems + +Gourav Kumara and V. Vetrivela + +aDepartment of Mathematics, Indian Institute of Technology Madras, Chennai, 600036, India + +## ARTICLE HISTORY + +Compiled March 11, 2025 + +#### ABSTRACT + +In this article, we provide a modification to the Bregman Golden Ratio Algorithm (B-GRAAL). We analyze the B-GRAAL algorithm with a new step size rule, where the step size increases after a certain number of iterations and does not require prior knowledge of the global Lipschitz constant of the cost operator. Under suitable assumptions, we establish the global iterate convergence as well as the R-linear rate of convergence of the modified algorithm. The numerical performance of the proposed approach is validated for the matrix game problem and the sparse logistic regression problem in machine learning. + +#### KEYWORDS + +Variational inequality; Bregman distance; KL divergence; Sparse logistic regression; Matrix games + +MSC 47J20, 49J40, 65K15, 65Y20 + +## 1. Introduction + +Let \(\mathcal{H}\) be a real finite dimentional Hilbert space. A mixed variational inequality problem (MVIP) is a problem to find \(w^{*}\in\mathcal{H}\) such that + +\[\langle A(w^{*}),w-w^{*}\rangle+g(w)-g(w^{*})\geq0\forall w\in\mathcal{H},\] (1) + +where \(A:\mathcal{H}\rightarrow\mathcal{H}\) is the cost operator, and \(g:\mathcal{H}\rightarrow\mathbb{R}\cup\{+\infty\}\) is an extended real-valued function. We denote the set of solutions of (1) by \(S\). Mixed variational inequality problems (MVIPs) of the form (1) generalize various optimization problems encountered in nonlinear programming and variational analysis, including minimization problems, linear complementarity problems, and variational inequalities, see [1,3,11,33]. These problems have widespread applications in diverse fields, such as data science, image processing, mechanics, control theory, economics, structural engineering, and many more; see [1,9,14,15,21,23–25], and the references therein. Many algorithms solving the variational inequality (1) (for example; see [11,29,31]), need the exact value of the global Lipschitz constant of the cost operator \(A\). This diff --git a/outputs/arxiv_math/2503.06293_pg24_pg1_repeat1.md b/outputs/arxiv_math/2503.06293_pg24_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..8d9b09f5f56f23f5199c0602ce82ef80fb0646d3 --- /dev/null +++ b/outputs/arxiv_math/2503.06293_pg24_pg1_repeat1.md @@ -0,0 +1,40 @@ +with the contours \(\Gamma_{\pm}\). It is found that, when grade restricting the branes of Eq. (4.7) to a specific window, the open Witten index was consistent when calculated on the positive phase and on the negative phase, i.e. \(\chi_{k,N}^{+}=\chi_{k,N}^{-}\), suggesting strongly that successful brane transport has taken place. The results of a few examples are presented below. For \(K_{Gr(2,4)}\), the basis of branes used was \(\{(0,0),(1,0),(1,1),(2,0),(2,1),(2,2)\}\) and the open Witten indices was calculated to be + +\[\chi_{2,4}^{+}=\chi_{2,4}^{-}= \begin{pmatrix} 0 & 4 & 6 & 10 & 20 & 20 \\ -4 & 0 & 4 & 4 & 16 & 20 \\ -6 & -4 & 0 & 0 & 4 & 6 \\ -10 & -4 & 0 & 0 & 4 & 10 \\ -20 & -16 & -4 & -4 & 0 & 4 \\ -20 & -20 & -6 & -10 & -4 & 0 \end{pmatrix} .\] (4.14) + +| χ+2,4 = χ−2,4 = | | | | | 6 . (4.14) | +| --- | --- | --- | --- | --- | --- | +| −10 | −4 | 0 | 0 | 4 | 10 | +| −20 | −16 | −4 | −4 | 0 | 4 | + +A special note is needed here: this was found to be consistent for a set of branes all grade restricted to one of the charge window types in Fig. 4. On the other hand, if a set of branes grade restricted to different window types were mixed together, so that the entire set was not grade restricted to the same charge window type, then the open Witten indices was found to be inconsistent. Thus it is believed that branes must be grade restricted in exactly the same way when computing the Witten index, despite multiple ways of charge restriction being valid for a single choice of theta-angle. For \(K_{Gr(2,5)}\), the basis of branes used was \(\{(0,0),(1,0),(1,1),(2,0),(2,1),(2,2),(3,0),(3,1),(3,2),(3,3)\}\). Then we found14 + +\[\chi_{2,5}^{+}=\chi_{2,5}^{-}= \begin{pmatrix} 0 & -5 & -10 & -15 & -40 & -50 & -35 & -105 & -175 & -175 \\ 5 & 0 & -5 & -5 & -25 & -40 & -15 & -75 & -155 & -175 \\ 10 & 5 & 0 & 0 & -5 & -10 & 0 & -15 & -40 & -50 \\ 15 & 5 & 0 & 0 & -5 & -15 & -5 & -25 & -75 & -105 \\ 40 & 25 & 5 & 5 & 0 & -5 & 0 & -5 & -25 & -40 \\ 50 & 40 & 10 & 15 & 5 & 0 & 0 & 0 & -5 & -10 \\ 35 & 15 & 0 & 5 & 0 & 0 & 0 & -5 & -15 & -35 \\ 105 & 75 & 15 & 25 & 5 & 0 & 5 & 0 & -5 & -15 \\ 175 & 155 & 40 & 75 & 25 & 5 & 15 & 5 & 0 & -5 \\ 175 & 175 & 50 & 105 & 40 & 10 & 35 & 15 & 5 & 0 \end{pmatrix} .(4.15)\] + +| 0 | −5 | −10 | −15 | −40 | −50 | −35 | −105 | −175 | −175 | +| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | +| 5 | 0 | −5 | −5 | −25 | −40 | −15 | −75 | −155 | −175 | +| 10 | 5 | 0 | 0 | −5 | −10 | 0 | −15 | −40 | −50 | +| 15 | 5 | 0 | 0 | −5 | −15 | −5 | −25 | −75 | −105 | + +| −5 | −15 | +| --- | --- | +| 0 | −5 | +| 5 | 0 | + +For \(K_{Gr(2,6)}\), we have 15 generators so, we just present the computation for the subset \(\{(0,0),(1,0),(1,1),(2,1),(2,2),(3,2),(3,3),(4,3),(4,4)\}\): + +\[\chi_{2,6}^{+}=\chi_{2,6}^{-}= \begin{pmatrix} 0 & 6 & 15 & 70 & 105 & 420 & 490 & 1764 & 1764 \\ -6 & 0 & 6 & 36 & 70 & 315 & 420 & 1624 & 1764 \\ -15 & -6 & 0 & 6 & 15 & 70 & 105 & 420 & 490 \\ -70 & -36 & -6 & 0 & 6 & 36 & 70 & 315 & 420 \\ -105 & -70 & -15 & -6 & 0 & 6 & 15 & 70 & 105 \\ -420 & -315 & -70 & -36 & -6 & 0 & 6 & 36 & 70 \\ -490 & -420 & -105 & -70 & -15 & -6 & 0 & 6 & 15 \\ -1764 & -1624 & -420 & -315 & -70 & -36 & -6 & 0 & 6 \\ -1764 & -1764 & -490 & -420 & -105 & -70 & -15 & -6 & 0 \end{pmatrix} .(4.16)\] + +| 0 | 6 | 15 | 70 | | | | | | +| --- | --- | --- | --- | --- | --- | --- | --- | --- | +| | | | | 105 | 420 | 490 | 1764 | 1764 | +| −6 | 0 | 6 | 36 | 70 | 315 | 420 | 1624 | 1764 | +| −15 | −6 | 0 | 6 | 15 | 70 | 105 | 420 | 490 | +| −70 | −36 | −6 | 0 | 6 | 36 | 70 | | | +| χ+2,6 = χ−2,6 = −420 −105 | −315 −70 | −70 −15 | −36 −6 | −6 0 | 6 0 | 15 6 | 315 70 36 | 420 105 70 . (4.16) | +| −490 | −420 | −105 | −70 | −15 | −6 | 0 | 6 | 15 | +| −1764 | −1624 | −420 | −315 | −70 | −36 | −6 | 0 | 6 | +| −1764 | −1764 | −490 | −420 | −105 | −70 | −15 | −6 | 0 | + +14We remark that (4.15) was computed in [15]. It corresponds to the open Witten index computed in what is called the ’S-basis’ in [15]. More precisely, using the notation of [15]: \((\chi_{-}^{-T}-\chi_{-}^{-1})=\chi_{2,5}^{+}=\chi_{2,5}^{-}\). diff --git a/outputs/arxiv_math/2503.06329_pg24_pg1_repeat1.md b/outputs/arxiv_math/2503.06329_pg24_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..5249cb0686ea830e11a2d5adfa2e032f1d69fcdd --- /dev/null +++ b/outputs/arxiv_math/2503.06329_pg24_pg1_repeat1.md @@ -0,0 +1,50 @@ +Theorem 4.2. We have \(\theta_{LC_{n}}(X)\neq0\) if and only if \(n<8\). + +Proof. By Lemma 3.7, we have + +\[\tilde{L}_{a_{1}a_{5}}=\{a_{1}a_{5},a_{1}a_{4}a_{5},a_{n}a_{1}a_{5},a_{n}a_{1}a_{4}a_{5}\}\] + +and + +\[\tilde{R}_{a_{1}a_{5}}=\{a_{1}a_{5},a_{1}a_{2}a_{5},a_{1}a_{5}a_{6},a_{1}a_{2}a_{5}a_{6}\}.\] + +For \(n\geq8\), we have \(∣\tilde{L}_{a_{1}a_{5}}∣=∣\tilde{R}_{a_{1}a_{5}}∣=4\). By Lemma 5.8 in [12], we + +compute the submatrix \(\tilde{L}_{a_{1}a_{5}}\times\tilde{R}_{a_{1}a_{5}}\), extracted from the Cayley table \((S,*)\), as follows: + +\[\frac{a_{1}a_{5}a_{1}a_{2}a_{5}a_{1}a_{5}a_{6}a_{1}a_{2}a_{5}a_{6}}{a_{1}a_{5}a_{1}a_{5}a_{1}a_{2}a_{5}a_{1}a_{5}a_{6}a_{1}a_{2}a_{5}a_{6}}\] + +| a1a4a5 | a1a4a5 | ⋅ | ⋅ | ⋅ | +| --- | --- | --- | --- | --- | +| a5ana1 | a5ana1 | ⋅ | ana1a5a6 | ⋅ | +| a4a5ana1 | a4a5ana1 | ⋅ | ⋅ | ⋅ | + +where the zero is represented with a dot. To compute the entries the table, + +at position \((s,t)\), we only need to verify that + +\[(st)^{+}=s^{+}\text{and}(st)^{*}=t^{*}.\] + +Since its determinant is zero, this is sufficient to show that the determinant + +of \(\theta_{LC_{n}}(X)\) is zero, for every \(n\geq8\). It is evident that the determinant of the monoid \(LC_{1}\) is non-zero. The Cayley table of \(LC_{2}\) is as follows: + +\[\begin{aligned} \frac{a_{1}a_{2}1}{a_{1}a_{1}\cdot a_{1}} \\ a_{2}\cdot a_{2}a_{2} \\ 1a_{1}a_{2}1 \end{aligned}\] + +| a1 a2 | a1 a1 ⋅ | a2 ⋅ a2 | 1 a1 a2 | +| --- | --- | --- | --- | +| 1 | a1 | a2 | 1 | + +which indicates that the determinant is non-zero. Now, suppose that \(3\leq n\leq7\). Let \(e\) be an idempotent of \(LC_{n}\) and let \(W_{1}W_{2}\cdots W_{m_{e}}\) denote the canonical form of \(e\) over \(A.\text{As}n\leq7\) and there is a two-letter distance between every neighboring segments of \(W\) , we have \(m_{e}\leq2\). By Lemma 4.1, since \(n\leq7\), we have + +\[l_{1}=∣\{W_{i}∣∣W_{i}∣=1,\text{and}d(W_{i-1},W_{i})\geq3\}∣\leq1.\] + +Thus, it follows that \(1\leq∣\tilde{L}_{e}∣\leq2.\) + +\[\begin{aligned} \mathrm{T}_{∣\tilde{L}}\mathrm{h}_{e}\mathrm{e}_{∣}\mathrm{n}_{=},_{1}\mathrm{w}_{\mathrm{a}}\mathrm{e}_{\mathrm{n}}\mathrm{h}_{d}\text{av}_{m}\mathrm{e}_{e}\mathrm{t}_{=}\text{he}_{1:} \\ \text{In this case},e\text{is a} \end{aligned}\] + +In this case, \(e\) is an idempotent with a single segment of length greater + +than one. Thus, the submatrix \(\tilde{L}_{e}\times\tilde{R}_{e}\) contains only a single entity, which + +\[e_{∣\tilde{L}},_{e}\mathrm{a}_{∣}\mathrm{n}_{=}d_{1}\text{it}_{\mathrm{a}}\mathrm{s}_{\mathrm{n}}d_{d}\text{et}_{m}\text{er}_{e}\mathrm{m}_{=}\text{in}_{2}\mathrm{a}_{:}\] diff --git a/outputs/arxiv_math/2503.06334_pg4_pg1_repeat1.md b/outputs/arxiv_math/2503.06334_pg4_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..2c33e14bcd1ef72d4e3e3e577c72d940b4dd0e2a --- /dev/null +++ b/outputs/arxiv_math/2503.06334_pg4_pg1_repeat1.md @@ -0,0 +1,9 @@ +The packing \(P\) can also be viewed as a collection of interconnected flowers: a flower consists of a central circle \(C_{v}\) and the chain of successively tangent petal circles, \(\{C_{v_{0}},\cdots ,C_{v_{n-1}}\}\), all tangent to \(C_{v}\). A flower is closed if \(C_{v_{n-1}}\) is tangent to \(C_{v_{0}}\), in which case \(v\) is an interior vertex of \(K\), whereas a flower is open if and only if \(v\) is a boundary vertex. (To avoid pathologies, we require of \(K\) that every boundary vertex has at least one interior neighbor.) There are three classes of closed flowers: A univalent flower is one whose petals have mutually disjoint interiors. An un-branched flower is one whose petals wrap once around the center, possibly with overlaps between non-contiguous petals. Finally, a branched flower is one whose petals wrap more than once about the center and its degree \(d\) is number of times it wraps. A circle packing \(P\) is univalent if its circles have mutually disjoint interior. It is necessary (but not sufficient that the flowers for interior vertices are univalent). If an interior circle \(C_{v}\) has a branched flower, then we say that \(P\) has a branch point at \(v\). The surprising richness of the topic is seen in the foundational existence and uniqueness result; namely, the Koebe-Andreev-Thurston (KAT) Theorem, which states that for any triangulation \(K\) of a topological sphere, there exists an associated univalent circle packing \(\mathcal{P}_{K}\) of the Riemann sphere \(\mathbb{P},\text{and that}\mathcal{P}_{K}\) is unique up to M¨obius transformations (and inversions) of \(\mathbb{P}\). Thurston also proposed in [24] a clever algorithm for actually computing such packings, allowing us today to treat circle packing as a verb: to ”circle pack” a complex \(K\) is to create and manipulate associated circle packings. + +1. 2. On Discrete Analytic Functions. Intriguing as circle packings + +were in their own right, it was a conjecture in Thurston’s talk that really fired up the topic. An example in Figure 1.2 will set the stage. Let \(P\) be a univalent circle packing filling a simply connected region \(\Omega\) of the plane, as on the right in the figure, and let \(K\) be the underlying complex. Thurston proved using KAT that there exists a univalent circle packing for \(K\) in the unit disc \(\mathbb{D}\) whose boundary circles are all horocycles, as on the left in the figure. With two circle packings for the same complex, one may define \(F:\mathcal{P}_{K}\longrightarrow P\) by identifying corresponding circles. Essentially a mapping from the unit disc to \(\Omega\), \(F\) is roughly analogous to the classical Riemann Mapping. + +It is the conjecture Thurston made about such discrete conformal mappings that kicked off the nearly 40 years of development efforts in circle packing. He suggested that if one were to refine this construction + +- used circle packings \(P\) with ever more and smaller circles — that the diff --git a/outputs/arxiv_math/2503.06344_pg15_pg1_repeat1.md b/outputs/arxiv_math/2503.06344_pg15_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..3fa89c62d6009521ffb1aeea553f2e1f20104363 --- /dev/null +++ b/outputs/arxiv_math/2503.06344_pg15_pg1_repeat1.md @@ -0,0 +1,19 @@ +Remark 3.15. Another interesting situation is when we have two foliated manifolds \((M,\mathcal{F}^{M})\) and \((N,\mathcal{F}^{N})\), and a local diffeomorphism — say, a covering map — \(F:M\rightarrow N\) which sends leaves of \(\mathcal{F}^{M}\) into leaves of \(\mathcal{F}^{N}\), and we want to move affine information between them. Note that \(F_{*}\) maps \(\mathcal{F}^{M}-\)vertical vectors into \(\mathcal{F}^{N}-\)vertical vectors, but not necessarily surjectively. It will become clear that it is necessary that only \(\mathcal{F}^{M}-\)vertical vectors are mapped into \(\mathcal{F}^{N}-\)vertical vectors. For example, consider \(id:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}\), where the first \(\mathbb{R}^{3}\) is foliated by \(T\mathcal{F}^{1}=\mathbb{R} \frac{\partial}{\partial y}\) and the second one is foliated by \(T\mathcal{F}^{2}=\mathbb{R} \frac{\partial}{\partial x} \oplus\mathbb{R} \frac{\partial}{\partial y} .\) It is clear that \(id\) is a foliated diffeomorphism. The usual flat connection \(\nabla^{\text{flat}}\) of \(\mathbb{R}^{3}\) is a transverse affine connection to both foliations, albeit associated with different partner connections \(\omega^{1}\text{and}\omega^{2}\) (each of them is just \(\nabla^{\text{flat}}\) restricted to the respective vertical distributions). However, \(id^{*}\nabla^{\text{flat}}=\nabla^{\text{flat}}\text{and}id^{*}\omega^{2}=\omega^{2}\), which does not result in a transverse affine connection in the source foliation. To achieve our goal, we must require furthermore that \(\mathcal{F}^{M}\text{and}\mathcal{F}^{N}\) are of the same dimension, so that vertical vectors on \(TM\) are in one-to-one correspondence with the vertical vectors in \(TN\). Indeed, if \((N,\mathcal{F}^{N})\) is endowed with a transverse affine connection \(\hat{\nabla}^{N}\), we can define the pullback \(F^{*}\hat{\nabla}^{N}\) as follows: for \(X,Y\in X(M)\), we set \((F^{*}\hat{\nabla}^{N})_{X}Y=Z,\text{where}Z\) is given by: For \(p\in M\), let \(\mathcal{U}\) be any neighborhood of \(p\) for which \(\Phi:=F|_{\mathcal{U}}\) is a diffeomorphism onto its (open) + +image. Then, \(Z(p):=d\Phi_{p}^{-1}\left(\hat{\nabla}_{\Phi_{*}X}^{N}\Phi_{*}Y\right)_{\Phi(p)}.\text{If}\omega^{N}\) is the partner connection associated with \(\hat{\nabla}^{N}\), then we can check that, for any \(V\in X(\mathcal{F}^{M})\) and any \(X\in X(M)\) we have + +\[\begin{aligned} (F^{*}\hat{\nabla}^{N})_{X}V&=(F^{*}\omega^{N})_{X}V, \\ (F^{*}\hat{\nabla}^{N})_{V}X&=[V,X]+(F^{*}\omega^{N})_{X}V. \end{aligned}\] + +As for the holonomy invariance condition, given \(V\in X(\mathcal{F}^{M}),\text{and}X,Y\in L(\mathcal{F}^{M})\) we have + +\[(\mathcal{L}_{V}(F^{*}\hat{\nabla}^{N}))(X,Y)=[V,(F^{*}\hat{\nabla}^{N})_{X}Y]-(F^{*}\hat{\nabla}^{N})_{[V,X]}Y-(F^{*}\hat{\nabla}^{N})_{X}[V,Y].\] + +Note that the two properties that we already established imply that the two last terms above are vertical. As for the first one, note that since \(F\) is locally a diffeomorphism, on a neighborhood of each point we can write \(V=\Phi_{*}^{-1}(W)\text{for some}W\in X(\mathcal{F}^{N})\). Then we are left with + +\[[V,(F^{*}\hat{\nabla}^{N})_{X}Y]=\left[\Phi_{*}^{-1}(W),\Phi_{*}^{-1}\left(\hat{\nabla}_{\Phi_{*}X}^{N}\Phi_{*}Y\right)\right]=\Phi_{*}^{-1}\left[W,\hat{\nabla}_{\Phi_{*}X}^{N}\Phi_{*}Y\right],\] + +which is vertical because \(\hat{\nabla}^{N}\) is a transverse affine connection and \(\Phi_{*}\) stablishes a one-to-one correspondence between vertical vectors. In the converse direction, if now \((M,\mathcal{F}^{M})\) is given a transverse affine connection \(\hat{\nabla}^{M}\), the necessary and sufficient condition for the existence of the pushforward via \(F\) is that \(F\) is oneto-one, (that is, a full diffeomorphism). When this is the case, we can set the pushforward \((F_{*}\hat{\nabla}^{M})_{X}Y=W\) , for \(X,Y\in X(N),\text{where}W\) is constructed as: given \(q\in N\), let \(\mathcal{U}\) be any neighborhood of \(p=F^{-1}(q)\) for which \(\Phi:=F|_{\mathcal{U}}\) is a diffeomorphism. Put \(W(q):= d\Phi_{p}\left(\hat{\nabla}_{\Phi^{*}X}\Phi^{*}Y\right)_{p}\). As before, since vertical vectors in \(TM\text{and in}TN\) are in bijection, we obtain that \(F_{*}\hat{\nabla}^{M}\) is a transverse affine connection with partner connection given by \(F_{*}\omega^{M}\). The next important consistency check arises by considering semi-Riemannian foliations. We show that any one such gives rise to a unique transverse affine structure — a result that works as an analogue of the fundamental theorem of semi-Riemannian geometry. Theorem 3.16. Let \(g_{⊺}\) be a transverse semi-Riemannian metric on \((M,\mathcal{F})\). There exists a unique transverse affine structure \([ \overline{\nabla} ]on(M,\mathcal{F})\) such that, for any \(\hat{\nabla}\in[ \overline{\nabla} ]\) + +(i) the torsion tensor \(\text{Tor}(\hat{\nabla})\) takes values in \(\Gamma(T\mathcal{F})\), and + +\[(ii)Xg_{⊺}(Y,Z)=g_{⊺}(\hat{\nabla}_{X}Y,Z)+g_{⊺}(Y,\hat{\nabla}_{X}Z),\forall X,Y,Z\in X(M).\] diff --git a/outputs/arxiv_math/2503.06350_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.06350_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..67bd7112f4db3c727cf21b8e626daa08460b22cf --- /dev/null +++ b/outputs/arxiv_math/2503.06350_pg3_pg1_repeat1.md @@ -0,0 +1,23 @@ +If \(f:[0,\infty)\longrightarrow \mathbb{R}\) is l`adl`ag, we will denote by \(f_{-}\) the left limit of \(f\) (with the convention that \(f_{0-}=f_{0})\), and by \(f_{+}\) the right limit of \(f\). The left jump of \(f\) is denoted by \(\Delta f=f-f_{-}\). For later use, we will state Froda’s theorem (in a slightly more general form, applying to finite variation functions of time). + +## Lemma 1. Suppose \(f\) has locally finite variation. Then \(f\) is l`adl`ag and + +\[\{t\in[0,\infty):f\text{is discontinuous at}t\}=\{f\neq f_{-}\}\cup\{f\neq f_{+}\}.\] + +Furthermore, the above sets are all at most countable. + +## 3. Main results + +The main theorem of this article, Theorem 1, can be stated as follows. Theorem 1. Let \((X^{n})_{n=1}^{\infty}\) be a sequence of semimartingales such that, for each \(t\geq0\), the set + +\[\text{co}\left\{\bigg|\int_{0}^{t}\xi dX^{n}\bigg|:n\in\mathbb{N},\xi\text{is predictable and}|\xi|\leq1\right\}\] + +is bounded in probability, and \((X_{0}^{n})_{n=1}^{\infty}\) is bounded. Then there exists \(\tilde{X}^{n}\in\text{co}\{X^{m}\) : \(m\geq n\}\) and a semimartingale \(\tilde{X}\text{such that}\tilde{X}=\lim_{s\downarrow\cdot,s\in\mathbb{Q}_{+}}\lim_{n\rightarrow\infty}\tilde{X}_{s}^{n}\) and + +\[\lim_{s\downarrow\cdot,s\in\mathbb{Q}_{+}}\mathbb{P}-\lim_{n\rightarrow\infty}\int_{0}^{s}Yd\tilde{X}^{n}=\int_{0}^{\cdot}Yd\tilde{X}\] + +for each continuous semimartingale \(Y\) . + +The continuity condition on \(Y\) situates Theorem 1 as a stochastic counterpart to the following well-known version of Prokhorov’s theorem: if \((g_{n})_{n=1}^{\infty}\) is a sequence of finite variation functions on \([0,1]\text{with}\sup_{n}\sup_{t\in[0,1]}\text{var}(g_{n})_{t}\leq1\), there exists a subsequence \((n_{k})_{k=1}^{\infty}\) and a finite variation \(g\text{such that}\lim_{k\rightarrow\infty}\int_{0}^{1}fdg_{n_{k}}=\int_{0}^{1}fdg\) for all continuous \(f\). In Section 5, we show the continuity condition in Theorem 1 cannot be weakened—mirroring the deterministic case. The imposition of a boundedness condition on Theorem 1 is natural, and not a significant restriction. Indeed, the boundedness conditions assumed in the literature are often stronger (see, for example, [KP91; DS99]). Furthermore, since the boundedness condition from Theorem 1 is based on boundedness in probability, Theorem 1 is not chained to any particular choice of measure, an important property when working with multiple equivalent probability measures. Let us note the following corollary of Theorem 1 for \(H^{1}-\)bounded sequences of martingales. Corollary 1. Let \((M^{n})_{n=1}^{\infty}\) be a sequence of martingales such that + +\[\sup_{n}\int_{\Omega}(M^{n})_{t}^{*}d\mathbb{P}<\infty\] diff --git a/outputs/arxiv_math/2503.06354_pg6_pg1_repeat1.md b/outputs/arxiv_math/2503.06354_pg6_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..9e9d9f2aefc2b2f565820b77c58587a54741dd19 --- /dev/null +++ b/outputs/arxiv_math/2503.06354_pg6_pg1_repeat1.md @@ -0,0 +1,51 @@ +Lemma 3.2. Let \(M\) be an \(R-\text{module}\). Suppose that \(I\) is an ideal of \(R\text{and}N\in\mod R\) + +such that \(\text{Supp}N\subset V(I)\). Then the following assertions hold for any non-negative integer \(n\). + +1. If Ext\((_{R}^{i}R/I,M)\in\mathscr{S}\text{for}0\leqslant i\leqslant n,\text{then}\text{Ext}_{R}^{i}(N,M)\in\mathscr{S}\text{for}0\leqslant i\leqslant n\). + +2. If Tor\((_{i}^{R}R/I,M)\in\mathscr{S}\text{for}0\leqslant i\leqslant n,\text{then}\text{Tor}_{i}^{R}(N,M)\in\mathscr{S}\text{for}0\leqslant i\leqslant n\). + +Proof. Since \(N\) is finitely generated, it follows from [13, Theorem 6.4] that there exists a chain \(0=N_{0}\subset N_{1}\subset\cdots \subset N_{k}=N\) of submodules of \(N\) such that for each \(i\) we have + +\[\begin{aligned} N_{j}/N_{j-1}&\cong R/p_{j}\text{with}p_{j}\in\text{Supp}N. \\ (1)\text{Note that Hom}(R/I,E)\text{is} \end{aligned}\] + +1. Note that \(\text{Hom}_{R}(R/I,E)\) is an injective \(R/I-\)module for any injective \(R-\)module \(E\) by [12, Lemma 3.5]. For each \(p_{j}\), by [15, Theorem 10.64], there is a spectral sequence + +\[E_{2}^{p,q}=\text{Ext}_{R/I}^{p}(R/p_{j},\text{Ext}_{R}^{q}(R/I,M))\Rightarrow\text{Ext}_{R}^{n}(R/p_{j},M).\] + +Being a subquotient of a finite direct sum of copies of Ext\((_{R}^{q}R/I,M)\in\mathscr{S}\) , we have + +that \(E_{2}^{p,q}\in\mathscr{S}\text{for}0\leqslant q\leqslant n\) and any \(p\). Since \(E_{\infty}^{p,q}\) is isomorphic to a subquotient + +of \(E_{2}^{p,q}\) , we have \(E_{\infty}^{p,q}\in\mathscr{S}\text{for}0\leqslant q\leqslant n\) and any \(p\). Hence Ext\((_{R}^{i}R/p_{j},M)\in\mathscr{S}\) + +for \(0\leqslant i\leqslant n\). Consequently, applying the functor Ext\((_{R}^{i}-,M)\) to exact sequences + +\(0\rightarrow N_{j-1}\rightarrow N_{j}\rightarrow N_{i}/N_{j-1}\rightarrow0\) yields that Ext\((_{R}^{i}N,M)\in\mathscr{S}\text{for}0\leqslant i\leqslant n\). + +2. can be proved similarly by using [15, Theorem 10.60]. + +3. 3) Let \(I=(x_{1},x_{2},\cdots ,x_{n})\) be an ideal of \(R\text{and}M\) a complex of \(R-\text{modules}\). In the following, we use \(K(I)\) to denote the Koszul complex with respect to \(I= (x_{1},x_{2},\cdots ,x_{n})\). The Koszul complex \(K(I)\) is a bounded complex of finite free modules. Ext\((_{R}^{i}K(I),M)=H^{i}(\text{Hom}_{R}(K(I),M))\). + +Then we define \(H_{i}(K(I),M):=\text{Tor}_{i}^{R}(K(I),M)=H_{i}(K(I)\otimes_{R}M)\text{and}H^{i}(K(I),M):=\) + +The preceding lemma allows us to extend some parts of [3, Theorem 2.8] to + +3. 4) modules which are not necessarily finitely generated. + +Proposition 3.5. Let \(M\) be an \(R-\text{module}.IfI\text{and}J\) are ideals of \(R\text{such that} \sqrt{I} = \sqrt{J}\), then the following assertions are equivalent for any non-negative integer \(n\). + +\((1)\text{Ext}_{R}^{i}(R/I,M)\in\mathscr{S}\text{for}0\leqslant i\leqslant n\). + +\((2)\text{Ext}_{R}^{i}(R/J,M)\in\mathscr{S}\text{for}0\leqslant i\leqslant n\). + +\((3)H^{i}(K(I),M)\in\mathscr{S}\text{for}0\leqslant i\leqslant n\). + +\((4)H^{i}(K(J),M)\in\mathscr{S}\text{for}0\leqslant i\leqslant n\). + +Proof. \((1)\Leftrightarrow(2)\) follows from Lemma 3.2(1). + +\((1)\Rightarrow(3)\) There is a spectral sequence + +\[E_{2}^{p,q}=\text{Ext}_{R}^{q}(\mathrm{H}_{p}(K(I)),M)\Rightarrow\mathrm{H}^{p+q}(K(I),M).\] diff --git a/outputs/arxiv_math/2503.06377_pg8_pg1_repeat1.md b/outputs/arxiv_math/2503.06377_pg8_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..e5f35e956d7a3b44933c9d0e916117d34f5f0d34 --- /dev/null +++ b/outputs/arxiv_math/2503.06377_pg8_pg1_repeat1.md @@ -0,0 +1,49 @@ +Hence, \(X_{\pm3}=\emptyset\), and (4.2) follows. Furthermore, we see that for every vector \(\gamma=(\gamma_{1},\gamma_{2},\gamma_{3})\in X\), □ + +\(\gamma_{1},\gamma_{2}\text{and}\gamma_{3}\) are minimal representatives in \(m\alpha_{9}+A_{9},2m\alpha_{9}+A_{9}\text{and}\alpha_{1}+A_{1}\), respectively. By Lemma 3.1 again, we have (4.3) and (4.4). By direct calculation, we have the following. + +Lemma 4.3. Let \(i,j,k\in[10]\text{with}j\neq k,\text{and}i^{\prime},j^{\prime},k^{\prime}\in[10]\text{with}j^{\prime}\neq k^{\prime}\). Then, + +\[(v_{\varepsilon}(i,j,k),v_{\varepsilon}(i^{\prime},j^{\prime},k^{\prime}))=|\{i\}\cap\{i^{\prime}\}|+|\{j,k\}\cap\{j^{\prime},k^{\prime}\}|(\varepsilon=\pm1).\] (4.5) + +Let \(I,J\in\left(_{5}^{[10]}\right)\). Then, + +\[(v_{1}(i,j,k),v_{5}(I))=|\{i\}\cap I|,\] (4.6) + +\[(v_{5}(I),v_{5}(J))=|I\cap J|-2.\] (4.7) + +Definition 4.4. Let \(\mathcal{X},\mathcal{X}_{1}\text{and}\mathcal{X}_{5}\) be the sets of maximum affine equiangular sets in \(X,X_{1}\), and + +\(X_{5}\), respectively, with respect to \(r\). Our aim is to investigate maximum affine equiangular sets \(\omega\text{in}X\) with norm 3 with respect to + +\(r\) up to switching equivalence, that is, replacing some of the vectors \(u\in\omega\) by \(r-u\). From the + +following lemma, we may restrict our attention to affine equiangular sets in \(X_{1}\) and ones in \(X_{5}\) + +\[\begin{aligned} \mathbf{Lemma}\mathbf{4}.\mathbf{5}.\text{We have}\mathcal{X}/&\sim_{\text{sw}}=\{[\psi\cup\phi]:[\psi]\in\mathcal{X}_{5}/\sim_{\text{sw}},\phi\in\mathcal{X}_{1}\}.\text{In particular},|\mathcal{X}/\sim_{\text{sw}}|= \\ |\mathcal{X}_{5}/&\sim_{\text{sw}}||\mathcal{X}_{1}|. \end{aligned}\] + +Proof. Let \(\mathcal{X}_{\pm1}\) be the set of maximum affine equiangular sets in \(X_{1}\cup X_{-1}\). By (4.6), we have + +\((u,v)\in\{0,1\}\) for any \(u\in X_{1}\cup X_{-1}\text{and}v\in X_{5}\). Hence, + +\[\mathcal{X}=\{\psi\cup\phi:\psi\in\mathcal{X}_{5},\phi\in\mathcal{X}_{\pm1}\}.\] + +Since \(X_{-1}=\{r-u:u\in X_{1}\}\), we see that for each \(\phi\in\mathcal{X}_{\pm1}\), there exists (a unique) \(\phi^{\prime}\in\mathcal{X}_{1}\) with + +\(\phi^{\prime}\sim_{\text{sw}}\phi.\) Noting that \(X_{5}\text{and}X_{-1}\cup X_{1}\) are closed under switching equivalence \(\sim_{\text{sw}}\), we have + +\[\begin{aligned} \mathcal{X}/&\sim_{\text{sw}}=\{[\psi\cup\phi]:[\psi]\in\mathcal{X}_{5}/\sim_{\text{sw}},[\phi]\in\mathcal{X}_{\pm1}/\sim_{\text{sw}}\} \\ &=\{[\psi\cup\phi]:[\psi]\in\mathcal{X}_{5}/\sim_{\text{sw}},\phi\in\mathcal{X}_{1}\}. \end{aligned}\] + +Next, we verify \(|\mathcal{X}/\sim_{\text{sw}}|=|\mathcal{X}_{5}/\sim_{\text{sw}}||\mathcal{X}_{1}|.\) Let \(\psi,\psi^{\prime}\in\mathcal{X}_{5}\text{and}\phi,\phi^{\prime}\in\mathcal{X}_{1}\) be such that \([\psi\cup\phi]= [\psi^{\prime}\cup\phi^{\prime}]\). Then, it suffices to show \([\psi]=[\psi^{\prime}]\text{and}\phi=\phi^{\prime}\). Indeed, we obtain \([\psi]=[\psi^{\prime}]\) and of \([\phi]\), we conclude \(\phi=\phi^{\prime}\). + +\([\phi]=[\phi^{\prime}]\). Moreover, as discussed earlier, since an element of \(\mathcal{X}_{1}\) uniquely exists as a representative □ + +## 5. MAXIMUM AFFINE EQUIANGULAR SETS IN \(X_{5}\) + +In this section, we determine a unique maximum affine equiangular set in \(X_{5}\) up to \(\text{Aut}(\Lambda)_{r}\) with a computer. Lemma 5.1. Every affine equiangular set in \(X_{5}\) with respect to \(r\) is of cardinality at most 12. Proof. Recall that the vectors in \(X_{5}\) are + +\[\left( \frac{1}{2} \mathbf{1}-\sum_{i\in I}e_{i},0,\alpha_{1}\right)\text{for}I\in \begin{pmatrix} [10] \\ 5 \end{pmatrix} .\] + +Hence, the dimension of the linear space generated by \(X_{5}\) together with \(r\) is at most \(\dim(\mathbb{R}^{10}\perp\) + +1) + 1 = 10. Thus, an arbitrary affine equiangular set in \(X_{5}\) with respect to \(r\) induces a set of equiangular lines with common angle \(\arccos(1/5)\) in dimension 9. The cardinality \(N_{1/5}(9)\) of a maximum set of equiangular lines with common angle \(\arccos(1/5)\) in dimension 9 equals 12 [11, diff --git a/outputs/arxiv_math/2503.06379_pg4_pg1_repeat1.md b/outputs/arxiv_math/2503.06379_pg4_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..5a7ee98c3babd4f9ad5c53b1bceca27236b1217b --- /dev/null +++ b/outputs/arxiv_math/2503.06379_pg4_pg1_repeat1.md @@ -0,0 +1,19 @@ +## 2 Homotopy of \(\mathscr{C}_{p}(G)\) + +In this section, we review some of the standard facts of order complexes of posets. See the first two parts of Smith’s book[9] for more details. Let \((\mathcal{P},\leq)\) be a poset (short for partially ordered set). The order complex \(\Delta\mathcal{P}\) of \(\mathcal{P}\) consists of all finite chains in \(\mathcal{P}\), including the empty set. The chain of length \(i\text{in}\mathcal{P}\) consisting of \(i+1\) elements is called an \(i-\text{simplex in}\Delta\mathcal{P}\). In particular, the empty set \(\emptyset\) is the \((-1)-\text{simplex}\). It is well-known that such (abstract) simplicial complex \(\Delta\mathcal{P}\) can be embedded into a Euclidean topological space as a subspace denoted by \(|\Delta\mathcal{P}|\), so-called the geometric realization of \(\Delta\mathcal{P}\). We abbreviate \(|\Delta\mathcal{P}|\text{as}|\mathcal{P}|\) here. A poset map \(f:(\mathcal{P},\leq_{\mathcal{P}})\rightarrow(\mathcal{Q},\leq_{\mathcal{Q}})\) of posets \(\mathcal{P},\mathcal{Q}\), is a set-theoretical map such that \(x\leq_{\mathcal{P}}y\) implies \(f(x)\leq_{\mathcal{Q}}f(y).\) It is routine to check that a poset map \(f:\mathcal{P}\rightarrow\mathcal{Q}\) naturally induces a simplicial map \(\Delta f\) between order complexes \(\Delta\mathcal{P}\text{and}\Delta\mathcal{Q},\text{and}f\) also induces a continuous map \(|f|:|\mathcal{P}|\rightarrow|\mathcal{Q}|.\) The following lemma gives a sufficient condition for the homotopy between two continuous maps induced by poset maps. + +Lemma 2. [9, Lemma 3.1.7] Suppose that \(f,g:P\rightarrow Q\) satisfy \(f,g\) are poset maps with \(f\leq g\), that is, \(f(x)\leq g(x)\) for each \(x\in P\) . Then \(f\text{and}g\) are homotopic, that is, \(|f|,|g|:|\mathcal{P}|\rightarrow|\mathcal{Q}|\) are homotopic. + +Recall that topological spaces \(X\text{and}Y\) are homotopy equivalent if there are continuous maps \(f:X\rightarrow Y\text{and}g:Y\rightarrow X\) whose compositions satisfy the homotopies \(g\circ f\simeq\text{Id}_{X}\text{and}f\circ g\simeq\text{Id}_{Y}\) . For posets \(\mathcal{P},\mathcal{Q},\mathcal{P}\text{and}\mathcal{Q}\) are said to be homotopy-equivalent if \(|\mathcal{P}|\text{and}|\mathcal{Q}|\) are homotopy-equivalent, simply denoted by \(\mathcal{P}\simeq\mathcal{Q}\). For a prime \(p\) and a group \(G\), we denote + +\[\mathcal{I}_{p}(G)=\left\{P_{1}\cap P_{2}\cdots \cap P_{s}|P_{i}\in\text{Syl}_{p}(G)\text{for all}1\leq i\leq s,\text{and}s\geq1\right\}\] + +by the set of all intersections of some Sylow \(p-\)subgroups of \(G\). + +Lemma 3. Let \(G\) be a group and \(p\) be a prime. Write + +\[\mathscr{C}(\mathcal{I}_{p}(G))=\{Hx|H\in\mathcal{I}_{p}(G),x\in G\}.\] + +Then \(\mathscr{C}_{p}(G)\) is homotopy-equivalent to \(\mathscr{C}(\mathcal{I}_{p}(G))\). Proof. For each \(Hx\in\mathscr{C}_{p}(G)\), we define a map \(f\) from \(\mathscr{C}_{p}(G)\) to \(\mathscr{C}(\mathcal{I}_{p}(G))\) by + +\[f(Hx):=\bigcap_{H\subseteq P\in\text{Syl}_{p}(G)}Px=\left(_{H\subseteq P}\bigcap_{\in\text{Syl}_{p}(G)}P\right)x.\] diff --git a/outputs/arxiv_math/2503.06384_pg7_pg1_repeat1.md b/outputs/arxiv_math/2503.06384_pg7_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..23676a3c82b8ab4edf01a5dad79c736883c801c7 --- /dev/null +++ b/outputs/arxiv_math/2503.06384_pg7_pg1_repeat1.md @@ -0,0 +1,31 @@ +\[H(x,p,t)=p\dot{x}-L= \frac{1}{2m(t)} p^{2}+ \frac{1}{2} m(t)\omega^{2}(t)x^{2}.\] (24) + +Let us consider now a point transformation \(x\rightarrow\xi(x,t)=x/\rho(t)\), which represents a time-dependent scaling transformation with the scaling function \(\rho(t)\) satisfying the auxiliary inhomogeneous equation + +\[\ddot{\rho}+\gamma(t)\dot{\rho}+\omega^{2}(t)\rho= \frac{\Omega^{2}}{m^{2}(t)\rho^{3}} ,\] (25) + +for \(\Omega\) constant. Without loss of generality, we will assume \(\rho(t)>0[33]\). Under this point transformation, we have: + +\[L(x,\dot{x},t)=L(\xi,\dot{\xi},t)+ \frac{d}{dt} \left( \frac{1}{2} m(t)\rho\dot{\rho}\xi^{2}\right),\] (26) + +where + +\[\begin{aligned} L(\xi,\dot{\xi},t)&= \frac{m(t)}{2} \left[\rho^{2}\dot{\xi}^{2}-\rho(\ddot{\rho}+\gamma(t)\dot{\rho}+\omega^{2}(t)\rho)\xi^{2}\right] \\ &= \frac{m(t)}{2} \left[\rho^{2}\dot{\xi}^{2}-\left( \frac{\Omega^{2}}{m^{2}(t)\rho^{2}} \right)\xi^{2}\right], \end{aligned}\] (27) + +where the last line was obtained in view of (25). As it is well-knwon, the total derivative term in (26) does not affect the classical equation of motion; however, it plays a fundamental role as a generating function for canonical transformations. Further, in quantum mechanics, such a total derivative term manifests as a nontrivial time-dependent phase of the wave function [34]. Now, by transforming the time variable \(t\rightarrow\tau(t)\), the Lagrangian (26) is mapped to + +\[L\left(\xi, \frac{d\xi}{d\tau} ,\tau\right)= \frac{m(\tau)}{2\dot{\tau}} \left[\rho^{2}\dot{\tau}^{2}\left( \frac{d\xi}{d\tau} \right)^{2}-\left( \frac{\Omega^{2}}{m^{2}(\tau)\rho^{2}} \right)\xi^{2}\right],\] (28) + +where the global parameter \(\dot{\tau}\) in the denominator emerges to compensate the change of measure in the action induced by the transformation \(t\rightarrow\tau\) , and both \(\rho\text{and}\xi\) in this expression must be thought of as functions of the parameter \(\tau\) . In general, the new time variable \(\tau(t)\) may be considered as a nonlinear function in \(t\). In particular, we can redefine the time variable \(\tau(t)\) such that + +\[\dot{\tau}= \frac{1}{m(t)\rho^{2}(t)} .\] (29) + +This transformation implies that the coefficient of the kinetic energy term in equation + +28. becomes constant. Consequently, we obtain + +\[L\left(\xi, \frac{d\xi}{d\tau} ,\tau\right)= \frac{1}{2} \left(\left( \frac{d\xi}{d\tau} \right)^{2}-\Omega^{2}\xi^{2}\right).\] (30) + +This Lagrangian describes the motion of a simple harmonic oscillator with constant frequency \(\Omega\). The Euler-Lagrange equation for the dynamical variable \(\xi(\tau)\) is given by + +\[\frac{d^{2}\xi}{d\tau^{2}} +\Omega^{2}\xi=0.\] (31) diff --git a/outputs/arxiv_math/2503.06459_pg7_pg1_repeat1.md b/outputs/arxiv_math/2503.06459_pg7_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..1283f8dfa44c816038cc3cb465df4b5c7552c3ed --- /dev/null +++ b/outputs/arxiv_math/2503.06459_pg7_pg1_repeat1.md @@ -0,0 +1,21 @@ +Proof of Proposition 2.3. The coefficients of the variables in the defining constraints (eqs. (2.11) to (2.15)) of pGT \(\tilde{}(\lambda,\mu)\) do not depend upon \(\mu\). It thus follows that in modifying \(\mu\) to \(\mu^{\prime}\), each of the defining hyperplanes is translated parallel to itself. Further, hyperplanes corresponding to eqs. \((2.12)\) and (2.13) do not change. The right hand sides of the inequalities in eq. (2.11) change by at most \(\delta \overline{n-1}\), and the Euclidean norm of the coefficient vector for the corresponding hyperplanes + +is \(\sqrt{n-2}_{} ,\) so that the corresponding hyperplanes translate along their normal by a distance of + +- at √ most \(\frac{\overline{n-1}}{\sqrt{n-2}} \delta\). A similar argument applied to eq. (2.14) and eq. (2.15) gives upper bounds of \(\delta/ \overline{2i-1}\) (for \(1\leq i\leq n-2)\text{and}\delta\) respectively on the magnitude of the translation\(,\) so that the largest possible magnitude of the translation over all the defining hyperplanes is \(\frac{\overline{n-1}}{\sqrt{n-2}} \delta\). The volume of \(GT(\lambda,\mu)\) according to the \(\frac{(n-1)(n-2)}{2}\) -dimensional Hausdorff measure in \(\mathbb{R}^{n(n-1)/2}\) is denoted by \(V_{\lambda,\mu}\), while that of pGT \(\tilde{}(\lambda,\mu)\) with respect to the Lebesgue measure on \(\mathbb{R}^{(n-1)(n-2)/2}\) is denoted by \(\tilde{V}_{\lambda,\mu}\). It is then a simple consequence of the area formula [Fed96, Theorem 3.2.3] that + +\[\sqrt{(n-1)!} \cdot\tilde{V}_{\lambda,\mu}=V_{\lambda,\mu}.\] (2.16) + +(See Appendix A.1 for details). Note that for these volumes to be non-zero, it is necessary that (1) \(|\lambda|=|\mu|\), and (2) the entries of \(\lambda\) must be distinct. When both \(\lambda\text{and}\mu\) have only non-negative integer entries, integer points in \(GT(\lambda,\mu)\) are in one-to-one correspondence with semi-standard Young tableaux with shape \(\lambda\) and content \(\mu\). (This correspondence is obtained by interpreting the \(i\)th row of a pattern with integral entries as the shape of the sub-tableaux consisting only of entries in \(\{1,2,\ldots,i\}.)\) In particular, the number of integral points in \(GT(\lambda,\mu)\) is then equal to the Kostka number \(K_{\lambda,\mu}\). We thus refer to the polytope \(GT(\lambda,\mu)\) as a Kostka polytope. Definition 2.5 (The Schur-Horn polytope). Let \(\lambda\) be a partition with \(n\) parts. For \(\sigma\) a permutation of \(\{1,\ldots,n\}\), we denote its action on \(\lambda\) by + +\[\sigma(\lambda_{1},\ldots,\lambda_{n}):=(\lambda_{\sigma(1)},\ldots,\lambda_{\sigma(n)}).\] (2.17) + +We refer to the convex hull of all vectors \(\sigma(\lambda)\) (as \(\sigma\) ranges over all \(n-permutations)\) as the Schur-Horn polytope \(SH(\lambda).SH(\lambda)\) is also referred to as the permutohedron of \(\lambda\) [Pos09]. It is well known that \(SH(\lambda)\) has the following equivalent representation [Mar64, Theorem 1.1 and Remark 1.1]. + +\[SH(\lambda)=\left\{\mu\in\mathbb{R}_{\geq0}^{n}:\lambda\succeq\mu\right\},\] (2.18) + +where \(\lambda\succeq\mu\) means that \(\lambda\) majorizes \(\mu\) in the sense that for every permutation2 \(\tau\) of \(\{1,2,\ldots,n\}\), + +\[\begin{aligned} \sum_{j=1}^{i}(\lambda_{j}-\mu_{\tau(j)})&\geq0\text{for}1\leq i\leq n-1,\text{and} \\ \sum_{j=1}^{n}(\lambda_{j}-\mu_{j})&=0. \end{aligned}\] + +2It is enough to restrict to the permutation that puts the entries of \(\mu\) in non-increasing order. diff --git a/outputs/arxiv_math/2503.06464_pg8_pg1_repeat1.md b/outputs/arxiv_math/2503.06464_pg8_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..9630c83aab540b3ef118f1ff7e4cc417ecc53ac5 --- /dev/null +++ b/outputs/arxiv_math/2503.06464_pg8_pg1_repeat1.md @@ -0,0 +1,17 @@ +with \(n\) trials and success probability \(p\). Recall that \(\mathbb{P}=\mathbb{P}_{n},\mathbb{Q}=\mathbb{Q}_{n}\) are two probability measures on a pair of random graphs on \([n]=\{1,\ldots,n\}\). Denote \(S_{n}\) the set of permutations in \([n]\) and \(\mu=\mu_{n}\) the uniform distribution on \(S_{n}\). In addition, denote by \(\nu=\nu_{n}\) the uniform distribution on \(\{-1,+1\}^{n}\). We will use the following notation conventions for graphs. + +- Labeled graphs. Denote by \(\mathcal{K}_{n}\) the complete graph with vertex set \([n]\) and edge set \(\mathrm{U}_{n}\). For any graph \(H\), let \(V(H)\) denote the vertex set of \(H\) and let \(E(H)\) denote the edge set of \(H\). We say \(H\) is a subgraph of \(G\), denoted by \(H\subset G,\text{if}V(H)\subset V(G)\text{and}E(H)\subset E(G)\). We say \(\varphi:V(H)\rightarrow V(S)\) is an injection, if for all \((i,j)\in E(H)\) we have \((\varphi(i),\varphi(j))\in E(S)\). For \(H,S\subset\mathcal{K}_{n}\), denote by \(H\cap S\) the graph with vertex set given by \(V(H)\cap V(S)\) and edge set given by \(E(H)\cap E(S)\), and denote by \(S\cup H\) the graph with vertex set given by \(V(H)\cup V(S)\) and edge set given by \(E(H)\cup E(S)\). For any graph \(H\), denote the excess of \(H\) by \(\tau(H)=|E(H)|-|V(H)|\). Given \(u\in V(H)\), define Nei\((_{H}u)\) to be the set of neighbors of \(u\) in \(H\). For two vertices \(u,v\in V(H)\), we define \(\text{Dist}_{H}(u,v)\) to be their graph distance. Denote the diameter of a connected graph \(H\) by \(\text{Diam}(H)=\max_{u,v\in V(H)}\text{Dist}_{H}(u,v)\). + +- Graph isomorphisms and unlabeled graphs. Two graphs \(H\text{and}H^{\prime}\) are isomorphic, denoted by \(H\cong H^{\prime}\), if there exists a bijection \(\pi:V(H)\rightarrow V(H^{\prime})\text{such that}(\pi(u),\pi(v))\in E(H^{\prime})\) if and only if \((u,v)\in E(H)\). Denote by \(\mathcal{H}\) the isomorphism class of graphs; it is customary to refer to these isomorphic classes as unlabeled graphs. Let \(\text{Aut}(H)\) be the number of automorphisms of \(H\) (graph isomorphisms to itself). For any graph \(H\), define \(\text{Fix}(H)=\{u\in V(H):\varphi(u)= u,\forall\varphi\in\text{Aut}(H)\}\). + +- Induced subgraphs. For a graph \(H=(V,E)\) and a subset \(A\subset V\) , define \(H_{A}=(A,E_{A})\) to be the induced subgraph of \(H\text{in}A,\text{where}E_{A}=\{(u,v)\in E:u,v\in A\}\). Also, define \(H_{\setminus A}=(V,E_{\setminus A})\) to be the subgraph of \(H\) obtained by deleting all edges with both endpoints in \(A\). Note that \(E_{A}\cup E_{\setminus A}=E\). + +- Isolated vertices. For \(u\in V(H)\), we say \(u\) is an isolated vertex of \(H\) if there is no edge in \(E(H)\) incident to \(u\). Denote \(\mathcal{I}(H)\) as the set of isolated vertices of \(H\). + +- Paths, self-avoiding paths and non-backtracking paths. We say a subgraph \(H\subset\mathcal{K}_{n}\) is a path with endpoints \(u,v\) (possibly with \(u=v)\), if there exist \(w_{1},\ldots,w_{m}\in[n]\neq u,v\) such that \(V(H)=\{u,v,w_{1},\ldots,w_{m}\}\text{and}E(H)=\{(u,w_{1}),(w_{1},w_{2})\ldots,(w_{m},v)\}\) (we allow the occurrence of multiple vertices or edges). We say \(H\) is a self-avoiding path if \(w_{0},w_{1},\ldots,w_{m},w_{m+1}\) are distinct (where we denote \(w_{0}=u\text{and}w_{m+1}=v)\), and we say \(H\) is a non-backtracking path if \(w_{i+1}\neq w_{i-1}\text{for}1\leq i\leq m\). Denote \(\text{EndP}(P)\) as the set of endpoints of a path \(P\) . + +- Cycles and independent cycles. We say a subgraph \(H\) is an \(m-\text{cycle if}V(H)=\{v_{1},\ldots,v_{m}\}\) and \(E(H)=\{(v_{1},v_{2}),\ldots,(v_{m-1},v_{m}),(v_{m},v_{1})\}\). For a subgraph \(K\subset H\), we say \(K\) is an independent \(m-\)cycle of \(H,\text{if}K\) is an \(m-\)cycle and no edge in \(E(H)\setminus E(K)\) is incident to \(V(K)\). Denote \(\mathcal{C}_{m}(H)\) as the set of independent \(m-\)cycles of \(H.\text{For}H\subset S\), we define \(C_{m}(S,H)\) to be the set of independent \(m-\text{cycles in}S\) whose vertex set is disjoint from \(V(H)\). + +- Leaves. A vertex \(u\in V(H)\) is called a leaf of \(H\), if the degree of \(u\text{in}H\) is 1; denote \(\mathcal{L}(H)\) as the set of leaves of \(H\). + +- Trees and rooted trees. We say a graph \(T=(V(T),E(T))\) is a tree, if \(T\) is connected and has no cycles. We say a pair \((T,R(T))\) is a rooted tree with root \(R(T),\text{if}T\) is a tree and \(R(T)\in V(T)\). For a rooted tree \(T\text{and}u\in V(T)\), we define Dep\((_{T}u)=\text{Dist}_{T}(R(T),u)\) to be diff --git a/outputs/arxiv_math/2503.06466_pg22_pg1_repeat1.md b/outputs/arxiv_math/2503.06466_pg22_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..f2e0a6ff96801c24b7d5c33875d76b787b1be41b --- /dev/null +++ b/outputs/arxiv_math/2503.06466_pg22_pg1_repeat1.md @@ -0,0 +1,13 @@ +### 5.8 Canonical Double Cover Construction + +The following is a well-known recursive construction that preserves the degree of the original graph, and in the special case of the canonical double cover, it doubles its order and its cycles of odd lengths. This means that it produces a \((k,g^{\prime})-\)graph from a \((k,g)-\)graph of odd girth \(g\), with \(g^{\prime}>g\) and even. For the sake of completeness, we present the construction, but only for the case when the starting graph is already a \((k,g)-\text{graph}\). Let \(\Gamma\) be a finite \((k,g)-\text{graph}\), and let \(D(\Gamma)\) denote the set of darts of \(\Gamma\), obtained by replacing each edge \(e\) of \(\Gamma\) with a pair of opposing darts (or arcs) \(e\text{and}e^{-1}\). A mapping \(\alpha:D(\Gamma)\rightarrow G\) is called a voltage assignment on \(\Gamma\) if it satisfies the condition \(\alpha(e^{-1})=(\alpha(e))^{-1}\) for all \(e\in D(\Gamma),\text{where}G\) is a group referred to as the voltage group. The voltage graph (also known as the derived graph or the lift) of \(\Gamma\) with respect to \(\alpha\), denoted by \(\Gamma^{\alpha}\), is a new graph with vertex set \(V(\Gamma^{\alpha})=V(\Gamma)\times G\) and edge set \(E(\Gamma^{\alpha})\), where vertices \(u_{a}\text{and}v_{b}\) are adjacent in \(\Gamma^{\alpha}\text{if}e=(u,v)\in D(\Gamma)\text{and}b=a\alpha(e)\). A voltage graph \(\Gamma^{\alpha}\) is a canonical double cover of \(\Gamma\) if the voltage group is \(\mathbb{Z}_{2}\) and each edge of \(\Gamma\) is assigned the non-zero voltage \(1\in\mathbb{Z}_{2}\). The canonical double cover construction is a particular type of voltage graph construction, which has been extensively studied by various authors (of the large number of articles considering the canonical double cover, consult, for example, [9, 12]). In our work, we applied the canonical double \(Graph(3,12,228)\), \(Graph(6,4,16)\), \(Graph(7,4,16)\). As is well-known, a canonical double cover of a bipartite graph consists of two disconnected copies of the original graph. Thus, using the canonical double cover construction systematically, we also obtained the following graphs, which are all disconnected but were used as starting graphs with respect to erasing vertices and adding edges as described in Subsection 5.3: \((3,12)-\) graph of order \(252,(5,\) 4)-graph of order \(20,(5,\) 8)-graph of order 304, and \((7,\) 4)-graph of order 28 (which consists of two disconnected copies of \(K_{7,7})\). + +\[\begin{aligned} \text{to obtain the following graphs}:Graph(3,8,48;1),Graph(3,12,224;1), \\ Graph(4,8,134),Graph(4,8,146),Graph(5,6,60),Graph(6,4,14), \\ Graph(6,4,18),Graph(6,4,20),Graph(6,4,22),Graph(6,4,24),\text{and} \end{aligned}\] + +## 6 Concluding Remarks + +As already stated in the introduction, the aim of the research and results presented in our paper is to gain insights into the structure of \((k,g)-\text{graphs}\); with the graphs of small orders being of particular interest. We conclude by outlining possible directions for future research or applications. Of the several open questions highlighted in [7], there is one that is attributed to several authors and which is based on the simple observation that all known cages as well as record graphs of even girth happen to be bipartite. This lead to the repeatedly stated conjecture that all even-girth cages must be bipartite. Whether the conjecture holds true or not, it still leads to the following open problem we find both interesting and related to \((k,g)-\text{spectra}\): + +For any given \(k\geqslant3\) and even \(g\geqslant4\), determine the smallest order \(n\) such that there exists a \((k,g)-\)graph of order \(n\) which is not bipartite. + +Our findings not only expand the current state of affairs with regard to the spectra of orders of \((k,g)-\text{graphs}\), but also highlight the challenges in determining complete spectra for larger values of \(k\text{and}g\). The gaps in our results, particularly for higher girths, underscore the complexity of diff --git a/outputs/arxiv_math/2503.06476_pg11_pg1_repeat1.md b/outputs/arxiv_math/2503.06476_pg11_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..988dab7d8471c0ed24cbb82860b356bb75baa086 --- /dev/null +++ b/outputs/arxiv_math/2503.06476_pg11_pg1_repeat1.md @@ -0,0 +1,33 @@ +which implies + +\[\sum_{k=0}^{N}\alpha_{k}\left(|\Theta(x^{k})|+ \frac{\Vert t^{k}\Vert^{2}}{2} \right)\leq \frac{1}{\beta} (\Phi_{j}(x^{0})-\hat{y}_{j}),∵\theta(x^{k})<0.\] + +Since the right-hand side of the above inequality is finite and inequality holds for any positive integer \(N\), then we get + +\[\sum_{k=0}^{\infty}\alpha_{k}\left(|\Theta(x^{k})|+ \frac{\Vert t^{k}\Vert^{2}}{2} \right)<\infty.\] + +The above inequality implies the result of this lemma. + +Theorem 4.1. Suppose that \(\Phi\) is convex in component-wise sense (i.e., \(\Phi\) is \(\mathbb{R}^{m}-\) convex) and the Assumption 1 holds. Then any sequence produced by Algorithm 3.1 converges to a WPOS \(x^{*}\in\mathbb{R}^{n}.\) + +Proof. Since by Algorithm 3.1, \(\{\Phi(x^{k})\}\) is a component-wise decreasing sequence, then by assumption, there exists \(\tilde{x}\in\mathbb{R}^{n}\) such that + +\[\Phi(\tilde{x})\leq\Phi(x^{k})\text{for all}k=0,1,2.\ldots\] (4.18) + +It is observed that \(0<\alpha_{k}\leq1\text{for all}k,\) so + +\[\begin{aligned} \Vert x^{k+1}-x^{k}\Vert^{2}&\leq \frac{1}{\alpha_{k}} \Vert x^{k+1}-x^{k}\Vert^{2}\text{for all}k=0,1,2,\ldots \\ &\leq \frac{1}{\alpha_{k}} \Vert\alpha_{k}t^{k}\Vert^{2}=\alpha_{k}\Vert t^{k}\Vert^{2}\text{for all}k=0,1,2,\ldots(∵x^{k+1}=x^{k}+\alpha_{k}t^{k}). \end{aligned}\] + +Therefore, by above inequality, (4.18), and Lemma (4.5) we obtained + +\[\sum_{k=0}^{\infty}\Vert x^{k+1}-x^{k}\Vert^{2}\leq\sum_{k=0}^{\infty}\alpha_{k}\Vert t^{k}\Vert^{2}<\infty.\] + +Thus, + +\[\sum_{k=0}^{\infty}\Vert x^{k+1}-x^{k}\Vert^{2}<\infty.\] (4.19) + +Let us define \(\tilde{L}=\{x\in\mathbb{R}^{n}:\Phi(x)\leq\Phi(x^{k}),k=0,1,2,\ldots\}.\) By the component-wise convexity of \(\Phi\) and Lemma 4.3, for any \(x\in\tilde{L}\) we have + +\[\Vert x-x^{k+1}\Vert^{2}\leq\Vert x-x^{k}\Vert^{2}+\Vert x^{k}-x^{k+1}\Vert^{2}\text{for all}k=0,1,2.\ldots\] + +As \(\tilde{L}\) is non empty because \(\tilde{x}\in\tilde{L}\), by (4.19) and the above inequality, it follows that \(\{x^{k}\}\) is quasi-Fejer convergent to the set \(\tilde{L}.\) Then by Theorem 2.4, \(\{x^{k}\}\) is bounded and hence \(\{x^{k}\}\) has an accumulation point. Let \(x^{*}\) be one of them. Then by Lemma 4.4, \(x^{*}\in\tilde{L}.\) Then by Theorem 2.4, we observe that \(\{x^{k}\}\) converges to \(x^{*}\). Therefore, by Theorem 3.3, \(x^{*}\) is a critical point, and hence \(\mathbb{R}^{m}-\) convexity implies that \(x^{*}\) is a weak Pareto optimal solution for \(\Phi.\) diff --git a/outputs/arxiv_math/2503.06509_pg11_pg1_repeat1.md b/outputs/arxiv_math/2503.06509_pg11_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..3bc331c30e520f3e2e6b5b8501ac934c4c218301 --- /dev/null +++ b/outputs/arxiv_math/2503.06509_pg11_pg1_repeat1.md @@ -0,0 +1,45 @@ +On contrary, assume that \(00\) sufficiently small such that \(H_{j}(x^{*}+\eta s)r_{s}\), which can be written using the relation between the dilaton and the electromagnetic charge as + +\[\sqrt{(e^{2}-2\mu^{2})^{2}-4a^{2}\mu^{2}} >e^{2}-2m\mu.\] (103) + +If we consider the case \(e^{2}>2m\mu\) than the square of this inequality leads to + +\[(\mu-m)e^{2}+(a^{2}-n^{2})\mu<0,\] (104) + +which cannot be satisfied due to \(|n|0\), which leads to + +\[m> \overline{a^{2}-n^{2}} .\] (106) + +So in the case \(|n|0\) because \(f_{t}^{*}(\sum_{i}C_{i})\cdot D_{t}=-K_{X}\cdot D>0\). Furthermore, \(D_{t}\) and \(E_{t}\) intersect at smooth points of \(Y_{t}\) because every singular point of \(Y_{t}\) lies on the support of \(\sum_{i}C_{i,t}\). Thus, \(D_{t}\cdot E_{t}=1\). This implies that + +\[-K_{X}\cdot D=f_{t}^{(_{*}}\sum_{i}C_{i}^{)}\cdot D_{t}=^{(}(p+q)E_{t}+\sum_{i}C_{i,t}^{)}\cdot D_{t}=p+q\] + +and + +\[f_{t}^{*}D=D_{t}+pqE_{t}\] + +since \(E_{t}^{2}=- \frac{1}{pq}\) . Thus, \(D_{t}^{2}=D^{2}-pq\) is an integer. Since \(-2\leq D_{t}^{2}-E_{t}\cdot D_{t}\) (by adjunction) and \(D_{t}^{2}<0\), + +we have \(D_{t}^{2}=-1\text{and}D^{2}=pq-1\). The equality + +\[(K_{Y_{t}}+D_{t})\cdot D_{t}=-2\] + +also implies that \(D_{t}\) is a smooth rational curve on \(Y_{t}\). In particular, \(D\) is smooth or has a unique singular point at \(x\). Next, we show that \(D\) is a \((p,q)-\)unicuspidal curve well-formed with respect to \((C,x)\). Let \(x_{1},x_{2}\) be a local (analytic) coordinate near \(x\) such that the local equation of \(C\) is \(x_{1}x_{2}=0\). Since \(D\) is irreducible, we can write the local equation of \(D\) as + +\[c_{1}x_{1}^{\alpha}+c_{2}x_{2}^{\beta}+g(x_{1},x_{2})\] + +for some \(c_{1},c_{2}\neq0\) and a power series \(g\), such that the coefficients of \(x_{1}^{i}\text{and}x_{2}^{j}\text{in}g\) vanish whenever \(i\leq\alpha\) and \(j\leq\beta\). Then the local intersection product at \(x\) satisfies + +Let \(v=\text{ord}_{E_{t}}\in\text{QM}(X,C)\). Then + +\[(D\cdot C)_{x}=\alpha+\beta.\] + +\[v(D)=v(f_{t}^{*}D)=pq.\] + +Since \(v(D)\leq p\alpha,q\beta\) by the definition of a quasi-monomial valuation, we have \(\alpha\geq p\text{and}\beta\geq q\). However, + +\[p+q=D\cdot C\geq(D\cdot C)_{x}=\alpha+\beta\geq p+q,\] + +- so \(\alpha=q\text{and}\beta=p\). This implies that the Newton polygon of \(D\text{in}x_{1},x_{2}\) coordinate is precisely the region above the line connecting \((q,0)\text{and}(0,p)\), so \(D\) is a \((p,q)-\)unicuspidal curve which is well-formed with respect to \((C,x)\). We need the following property about the set of unicuspidal curves on \(X\) well-formed with respect to \((C,x)\). + +## Lemma 3.9. Let \(T\) denote the set of \(t= \frac{q}{p}\) such that + +‚ There exists a \((p,q)-\)unicuspidal curve \(DonX\) which is well-formed with respect to \((C,x)\), + +\(\bullet-K_{X}\cdot D=p+q\), + +\(\bullet D^{2}=pq-1\), and + +\(\bullet E_{t}\) is a special divisor over \(X\). diff --git a/outputs/arxiv_math/2503.06622_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.06622_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..5e07d23194682ef238bb59b972180d73b256a9d8 --- /dev/null +++ b/outputs/arxiv_math/2503.06622_pg3_pg1_repeat1.md @@ -0,0 +1,27 @@ +In Section 2, we give motivation from non-linear stochastic filtering, volatility modelling in finance, pathwise stochastic optimal control and mean-fields SDE with common noise system; topics also closely related to [11, 12]. We do not strive for precise statements, but try to explain how rough stochastic processes naturally emerge, together with motivation for what is to come: Section 3 deals with the mathematics of randomisation of rough Itô processes, one obstacle being the potentially uncountable number of rough paths that figure as parameter in (1.1). The remaining section discuss applications to RSDE and then also mean-field RSDEs. The body of work surrounding RSDEs, control and common noise also resonates with early remarks made by the authors of [10], conjecturing the use of rough paths for conditioned forward stochastic dynamics (in their context of mean field games with common noise). Acknowledgement: PKF and HZ acknowledge support from DFG CRC/TRR 388 “Rough Analysis, Stochastic Dynamics and Related Fields”, Projects A07, B04 and B05. Part of this work was carried out during a visit of the first author to Shandong University. KL acknowledges supports from EPSRC [grant number EP/Y016955/1]. HZ is partially supported by the Fundamental Research Funds for the Central Universities, NSF of China and Shandong (Grant Numbers 12031009, ZR2023MA026), Young Research Project of Tai-Shan (No.tsqn202306054). + +## 2 Motivating examples + +### 2.1 Non-linear stochastic filtering + +We follow [16, 8]. One has an observation process \(Y_{t}=\int_{0}^{t}h(X_{s},Y_{s})ds+B_{t}^{\perp}\) where the signal process \(X\) is assumed to have “correlated” dynamics, + +\[\begin{aligned} dX_{t}&=\tilde{b}(t,X_{t},Y_{t})dt+\sigma(t,X_{t},Y_{t})dB_{t}(\omega)+f(t,X_{t},Y_{t})dB_{t}^{\perp}(\omega) \\ &=b(t,X_{t},Y_{t})dt+\sigma(t,X_{t},Y_{t})dB_{t}(\omega)+f(t,X_{t},Y_{t})dY_{t}(\omega). \end{aligned}\] (2.1) + +Here, \((B,B^{\perp})\text{and}(B,Y)\) are independent Brownians under some original measure \(\mathbb{P}^{o}\text{and}\mathbb{P}\), respectively, related by the Girsanov formula + +\[\frac{d\mathbb{P}^{o}}{d\mathbb{P}} \bigg|_{\mathcal{F}_{t}}=\exp\left(\int_{0}^{t}h(s,X_{s},Y_{s})dY_{s}- \frac{1}{2} \int_{0}^{t}|h(s,X_{s},Y_{s})|^{2}ds\right)=:\exp(I_{t}).\] + +Formally at least, the conditional dynamics of \(X_{t}\) given the observation \(\{Y_{t}:0\leqslant t\leqslant T\}\) should be captured by the solution \(X^{\mathbf{Y}}\) to a rough SDE of the form (1.5)2 + +\[dX_{t}^{\mathbf{Y}}=b(t,X_{t}^{\mathbf{Y}},Y_{t})dt+\sigma(t,X_{t}^{\mathbf{Y}},Y_{t})dB_{t}(\omega)+(f,D_{y}f)(t,X_{t}^{\mathbf{Y}},Y_{t})d\mathbf{Y}_{t},\] (2.2) + +together with the “rough stochastic” Girsanov exponent + +\[I_{t}^{\mathbf{Y}}=\int_{0}^{t}(h,(D_{x}hf+D_{y}h))(s,X_{s}^{\mathbf{Y}},Y_{s})d\mathbf{Y}_{s}- \frac{1}{2} \int_{0}^{t}|h(s,X_{s}^{\mathbf{Y}},Y_{s})|^{2}ds.\] + +Bearing exponential integrability, one defines a flow of measures + +\[\langle\mu_{t}^{\mathbf{Y}},\varphi\rangle:=\mathbb{E}[\varphi(X_{t}^{\mathbf{Y}})\exp(I_{t}^{\mathbf{Y}})],\] + +2Assuming regular \(t\) dependence in \(f\). diff --git a/outputs/arxiv_math/2503.06630_pg14_pg1_repeat1.md b/outputs/arxiv_math/2503.06630_pg14_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..69486edb3de23be6c2ac43f49d5add288b64e5fa --- /dev/null +++ b/outputs/arxiv_math/2503.06630_pg14_pg1_repeat1.md @@ -0,0 +1,61 @@ +differ on a set of null measure). So we have that \(\nabla w_{1}(x)=\nabla w_{2}(x)=0\) for a.a. \(x\in\Omega\setminus\tilde{\Omega}\). We may also write + +\[\text{that} \frac{\nabla w_{1}(x)}{w_{1}(x)} = \frac{\nabla w_{2}(x)}{w_{2}(x)} =0\text{for a.a.}x\in\Omega\setminus\tilde{\Omega}.\] + +\[\mathbf{FactII}:\text{For a.a.}x\in\tilde{\Omega}\text{we have that} \frac{\nabla w_{1}(x)}{w_{1}(x)} = \frac{\nabla w_{2}(x)}{w_{2}(x)} .\] + +\(|\nabla w_{2}(x)|=\nabla w_{1}(x)\cdot\nabla w_{2}(x)\). This means that there is some \(\lambda(x)>0\text{such that}\nabla w_{2}(x)=\lambda(x)\nabla w_{1}(x)\). + +Since \(x\in\tilde{\Omega}\) we have that \(\nabla w_{1}(x),\nabla w_{2}(x)\neq 0\). But to have equalities in (43) we deduce that \(|\nabla w_{1}(x)|\cdot\) Following the cases in which equality can occur in Lemma 4.1, having \(r>1\text{and}|\nabla w_{1}(x)|,|\nabla w_{2}(x)|>0\), we \(\underline{|\nabla w_{2}(x)|} \underline{|\nabla w_{1}(x)|} \underline{\lambda(x)|\nabla w_{1}(x)|}\) + +are in the case (iii) from which we can write that: \(\frac{|\nabla w_{1}(x)|}{w_{1}(x)} = \overline{w_{2}(x)} \Longrightarrow \overline{w_{1}(x)} = \overline{w_{2}(x)} \Longrightarrow\) + +\(\lambda(x)= \frac{w_{2}(x)}{w_{1}(x)}\) . So the claim is proved, since \(\nabla w_{2}(x)= \frac{w_{2}(x)}{w_{1}(x)} \nabla w_{1}(x)\). + +Fact III: The function \(\frac{w_{2}}{w_{1}}\) is constant. + +\[\underline{\nabla w_{2}(x)}\] + +At this moment we have that \(\frac{\nabla w_{1}(x)}{w_{1}(x)} =\) a.e. on the entire domain \(\Omega\). Using now Lemma 5.1 proved = 0 a.e. on \(\Omega\). + +\[\begin{aligned} w_{2}(x) \\ \tfrac{2}{1} &\in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,1}(\Omega)\text{and}:\nabla\left( \frac{w_{2}}{w_{1}} \right)= \frac{w_{2}}{w_{1}} \left( \frac{\nabla w_{2}}{w_{2}} - \frac{\nabla w_{1}}{w_{1}} \right) \end{aligned}\] + +at the end of this proof, we deduce that \(\frac{w_{2}}{w_{1}} \in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,1}(\Omega)\text{and}:\nabla \underline{w_{2}} =\) + +Since \(\Omega\) is a connected domain we infer that \(\frac{w_{2}}{w_{1}}\) is a constant function. Let \(w_{2}=\lambda w_{1}\) a.e. on \(\Omega\text{for some}\lambda>0\). Substituting this in (54) we obtain a.e. on \(\Omega\) that: + +\[\begin{aligned} \Longleftrightarrow ^{\mathbf{a}(}\left(^{x}1^{,}-^{\nabla w}\lambda^{r1}\right)^{)}\mathbf{a}^{\cdot}(^{\nabla}x,\left(\nabla^{w_{1}}w^{-_{1}})^{\lambda}\cdot^{r}\nabla^{w}w^{1}\right)_{1}&== \frac{\mathbf{a}_{1}(_{-}x,_{\lambda}\lambda_{r}}{\lambda^{r-1}} \nabla\mathbf{a}^{w}(x^{1},^{)}\lambda^{\cdot}\nabla^{\nabla}w\left(_{1} \frac{w_{1}}{\lambda)^{r}\cdot^{-}\nabla^{1}} w-_{1}\lambda w_{1}\right) \\ &\Leftarrow\Leftarrow_{\Rightarrow}\Rightarrow \begin{pmatrix} 1-\lambda^{r} \\ 1-\lambda^{r} \end{pmatrix} \lambda\lambda_{\Phi^{r-1}}^{r-1}\Phi_{(}(_{x}x_{,},_{|}|_{\nabla}\nabla_{w}w_{|_{1}^{1}}|_{)})_{|}|_{\nabla}\nabla_{w}w_{|_{1}^{1}}|_{=}= \begin{pmatrix} 1-\lambda^{r} \\ 1-\lambda^{r} \end{pmatrix} \Phi\Psi_{(}(_{x}x_{,},_{\lambda}\lambda_{|}|_{\nabla}\nabla_{w}w_{|_{1}^{1}}|_{)})_{|}\lambda_{\nabla}\nabla_{w}w_{|_{1}^{1}}\cdot\nabla w_{1} \\ \Longleftrightarrow \left(1-\lambda^{r}\right)|\nabla w_{1}|\left[\Phi(x,\lambda|\nabla w_{1}|)-\lambda^{r-1}\Phi(x,|\nabla w_{1}|)\right]&=0. \end{aligned}\] + +Therefore we may have \(\lambda=1\) and the equality holds. If \(\lambda\neq 1\text{then}\nabla w_{1}(x)=0\text{or}\Phi(x,\lambda|\nabla w_{1}|)= \lambda^{r-1}\Phi(x,|\nabla w_{1}|)\) for a.a. \(x\in\Omega\). This disjunction is equivalent to the second relation since \(\Phi(x,0)=0\). + +If we know that the function \((0,\infty)\ni s\mapsto \frac{\Phi(x,s)}{s^{r-1}}\) is strictly increasing for a.a. \(x\in\Omega\) (i.e. (H7’) holds), + +then equality can occur when \(\lambda=1\) (i.e. \(w_{1}\equiv w_{2}).\text{If}\lambda\neq 1\text{then for}x\in\tilde{\Omega}\) (i.e. \(\nabla w_{1}(x)\neq 0)\) we have that + +\[\frac{\Phi(x,\lambda|\nabla w_{1}(x)|)}{\left(\right)^{r-1}}\] + +\(\Phi(x,\lambda|\nabla w_{1}(x)|)=\lambda^{r-1}\Phi(x,|\nabla w_{1}(x)|)\) which rewrites as \(\frac{\Phi(x,|\nabla w_{1}(x)|)}{|w_{1}(x)|^{r-1}} =\big)^{r-1}\) . This equality + +\[\begin{aligned} \big(\lambda|w_{1}(x)| \\ ˜ \end{aligned}\] + +is impossible to hold from the strict monotony of the function involved there. So \(|\tilde{\Omega}|=0\) which means that □ + +\(\nabla w_{1}=0\) a.e. on \(\Omega\). Therefore \(\nabla w_{2}=\lambda\nabla w_{1}=0\) a.e. on \(\Omega\). Finally, since \(\Omega\) is connected we have that \(w_{1}\) and \(w_{2}\) are different constant functions. The theorem is now completely proved. + +Now we shall state and prove the lemma we have used above. + +Lemma 5.1. Let \(\Omega\subset\mathbb{R}^{N}\) be an open, bounded and connected domain. If \(w_{1},w_{2}\in W^{1,p(x)}(\Omega)\) with: + + (i) \(w_{1},w_{2}>0\) a.e. on \(\Omega\); + + (ii) \(\frac{w_{1}}{w_{2}} , \frac{w_{2}}{w_{1}} \in L^{\infty}(\Omega)\); + +(iii) \(\underline{\nabla w_{1}} ,\in L_{\text{loc}}^{1}(\Omega)^{N}\). + +\(\underline{\nabla w_{2}} w_{1}w_{2}\) + +Then \(\underline{w_{2}} , \underline{w_{1}} \in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,1}(\Omega)\) and moreover: + +\[w_{1}w_{2}\] + +\[\nabla\left( \frac{w_{2}}{w_{1}} \right)= \frac{w_{1}\nabla w_{2}-w_{2}\nabla w_{1}}{w_{1}^{2}} = \frac{w_{2}}{w_{1}} \left( \frac{\nabla w_{2}}{w_{2}} - \frac{\nabla w_{1}}{w_{1}} \right).\] (55) diff --git a/outputs/arxiv_math/2503.06651_pg13_pg1_repeat1.md b/outputs/arxiv_math/2503.06651_pg13_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..3d286d378c65be7d413e44477b87af045c5e572b --- /dev/null +++ b/outputs/arxiv_math/2503.06651_pg13_pg1_repeat1.md @@ -0,0 +1,73 @@ +![](figure) + +BS sends CSI-RS +UE feedback PMI and \( \delta \) + +BS obtains the full channel information by SRS and PMI + +strong ports weak ports +strong ports weak ports +strong ports weak ports[^inferred] + +![](figure) + +BS sends CSI-RS +UE feedback PMI and \( \delta \) + +BS obtains the full channel information by SRS and PMI + +strong ports weak ports +strong ports weak ports +strong ports weak ports[^inferred] + +Fig. 12. Illustration of integrated channel estimation method for tri-polarized MIMO. + +![](figure) + +Tri-polarized antenna[^inferred] + +(a) + +![](figure) + +![](figure) + +(b) + +(c) + +![](figure) + +Fig. 13. Illustration of the antenna radiation patterns at UEs. (a) Test scenario. + +(b) The first two polarizations. (c) The third polarization. + +estimation accuracy for some antenna ports deteriorates due to variations in antenna gain among different polarizations. To mitigate this issue, a joint uplink-downlink channel estimation method is proposed. As shown in Fig. 12, the main idea of the proposed method involves measuring the channel for port groups with high antenna gain using the uplink channel estimation method, while measuring the channel for port groups with low antenna gain through downlink channel estimation. The specific steps are as follows, illustrated with single-user MIMO system as an example. + +- Step 1: Port grouping: Based on the receive power on Rx’s each antenna port, the antenna ports are categorized into two distinct groups: one for ports with high antenna gain (marked as G1), and another for those with low antenna gain (marked as G2). + +- Step 2: Uplink channel estimation for G1: The Rx sends pilots through the ports in G1, such that Tx can estimate the channel for corresponding ports, denoted by \(\mathbf{H}_{1,\mathrm{U}}\), by leveraging the channel reciprocity. + +- Step 3: Downlink channel measurement for G2: The Tx sends pilots and then the Rx measures the full-dimension channel HD, which consists of two components corresponding G1 and G2, respectively, denoted by \(\mathbf{H}_{1,\mathrm{D}}\text{and}\mathbf{H}_{2,\mathrm{D}}\) with \(\mathbf{H}_{\mathrm{D}}=[\mathbf{H}_{1,\mathrm{D}},\mathbf{H}_{2,\mathrm{D}}]\). Next, the estimated channel for G2 is + +normalized and then fed back to Tx, given by + +\[\mathbf{H}_{2,\mathrm{D}}^{\prime}= \frac{\mathbf{H}_{2,\mathrm{D}}}{\rho_{2}e^{j\omega_{2}}} ,\] (47) + +where \(\rho_{2}\text{and}\omega_{2}\) represents the normalization amplitude and phase, respectively. Similarly, estimated channel for G1 can be normalized with a normalization amplitude and phase of \(\rho_{1}\text{and}\omega_{1}\). + +- Step 4: Port combining reference value feedback: The estimated channel of antenna ports in G2 by downlink measurement cannot be directly combined with that in G1 due to the unknown combining reference value. Therefore, the Rx needs to feedback the channel combining reference value, denoted by \(\delta\), from the full channel information HD, and is given by + +\[\delta= \frac{\rho_{1}e^{j\omega_{1}}}{\rho_{2}e^{j\omega_{2}}} .\] (48) + +- Step 5: Joint channel estimation: The estimated channel of antenna ports in G1, \(\mathbf{H}_{1,\mathrm{U}}\), is first normalized as \(\bar{\mathbf{H}}_{1,\mathrm{U}}\) and further adjusted with the channel combining reference value for channel combining as + +\[\mathbf{H}_{1,\mathrm{U}}^{\prime}= \frac{\bar{\mathbf{H}}_{1,U}}{\delta} .\] (49) + +As such, the channel for all antenna ports can be obtained as + +\[\mathbf{H}=[\mathbf{H}_{1,\mathrm{U}}^{\prime},\mathbf{H}_{2,\mathrm{D}}^{\prime}].\] (50) + +3. Performance Evaluation: We conduct system-level simulation to evaluate the efficiency of our proposed joint uplinkdownlink channel estimation method by taking into account real-world scenarios with tri-polarized MIMO. It is assumed that 50 UEs are randomly distributed in a standard cell with three sectors. The BS and each UE is equipped with 256 and 12 antenna ports respectively. The directional radiation patterns of the tri-polarized antennas used are illustrated in Fig. 13. Other simulation assumptions are detailed in Table I. Fig. 14 illustrates the cumulative distribution function (CDF) curves of the capacity, comparing a benchmark utilizing uplink channel estimation with the proposed method. It is observable that the capacity of the proposed method consistently surpasses the benchmark across all intervals. Furthermore, it is demonstrated that the average capacity attained by the proposed method is 29% higher than that of the benchmark. This is expected because our proposed method efficiently compensate for the imbalance in receive power among different antenna ports. + +[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter. diff --git a/outputs/arxiv_math/2503.06657_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.06657_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..01433377efb48a04902f63990c8c419bc874428c --- /dev/null +++ b/outputs/arxiv_math/2503.06657_pg2_pg1_repeat1.md @@ -0,0 +1,9 @@ +residuated lattice L is inserted into a residuated lattice K where it replaces the unit of the monoid operation. The requirements placed on L and K are discussed in Section 2. We essentially extend the conditions for residuated lattices to the additional unary operations in the signature of qRAs. A consequence of the conditions on K is that its FL-algebra reduct will be odd (i.e. 0 = 1). Having shown that the K[L] construction gives us a method for constructing new qRAs from existing ones, we then turn to the question of representability. Abstract relation algebras, first described by Tarski in 1941 [17], were designed to provide an abstract algebraic setting for studying binary relations. Jónsson and Tarski [12] asked whether all algebras were isomorphic to algebras of binary relations (with the monoid operation given by relational composition). This question was answered in the negative by Lyndon [13]. The task of representing relation algebras as algebras of binary relations has been an intriguing area of research ever since. In [3], two of the current authors gave a definition of representability of distributive quasi relation algebras (see Section 3). Partially ordered sets equipped with an equivalence relation (and satisfying certain symmetry requirements) can be used to build distributive quasi relation algebras (DqRAs) as algebras of binary relations. This method was further exploited to give relational representations of Sugihara monoids [4]. In Section 4 we show that if K = S3, and if L is representable, then their generalised ordinal sum K[L] will again be representable (Theorem 4.1). This is extended in Corollary 5.4 to the case of K being any finite odd Sugihara chain. These results are similar in spirit to the result for integral relation algebras where the relation algebra obtained from A and B by Comer’s construction [1] is representable if and only if A and B are representable. A further application of Theorem 4.1 is given in Section 5, where we show that all finite Sugihara chains can be represented using a set of binary relations on a finite poset (Theorem 5.1). This improves the results in [15] and [4] where infinite posets (using copies of \(\mathbb{Q}\), the rational numbers) were required to obtain a relational representation of \(\mathbf{S}_{n}\text{for}n\geqslant4\). In Problem 1 we pose an intriguing question about the structures of possible algebras K for which we can guarantee that K[L] will be representable. Specifically we ask whether K must be a finite Sugihara chain. + +## 1 Preliminaries + +In this section we recall the basic definitions of the algebras used in this paper. The reader is referred to [9] for more information. + +### 1.1 Quasi relation algebras + +A quasi relation algebra is an expansion of a residuated lattice [6]. Recall that a residuated lattice (RL) is an algebra \(\mathbf{A}=\langle A,\wedge,\vee,\cdot,\setminus,/,1\rangle\text{such that}\langle A,\cdot,1\rangle\) is a monoid, \(\langle A,\wedge,\vee\rangle\) is a lattice and the monoid operation · is residuated with residuals diff --git a/outputs/arxiv_math/2503.06658_pg1_pg1_repeat1.md b/outputs/arxiv_math/2503.06658_pg1_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..6588af43b04f5d570cde900f9a8b5b58403d4ae2 --- /dev/null +++ b/outputs/arxiv_math/2503.06658_pg1_pg1_repeat1.md @@ -0,0 +1,37 @@ +# First- and Half-order Schemes for Regime Switching Stochastic Differential Equation with Non-differentiable Drift Coefficient + +#### Abstract + +Divyanshu Vashistha, Chaman Kumar∗ + +Indian Institute of Technology Roorkee, Haridwar, 247667, India + +An explicit first-order drift-randomized Milstein scheme for a regime switching stochastic differential equation is proposed and its bi-stability and rate of strong convergence are investigated for a non-differentiable drift coefficient. Precisely, drift is Lipschitz continuous while diffusion along with its derivative is Lipschitz continuous. Further, we explore the significance of evaluating Brownian trajectories at every switching time of the underlying Markov chain in achieving the convergence rate \(1.0\) of the proposed scheme. In this context, possible variants of the scheme, namely modified randomized and reduced randomized schemes, are considered and their convergence rates are shown to be \(1/2\). Numerical experiments are performed to illustrate the convergence rates of these schemes along with their corresponding non-randomized versions. Further, it is illustrated that the half-order non-randomized reduced and modified schemes outperforms the classical Euler scheme. Keywords: Randomized Milstein scheme, SDEs with Markovian switching, Bi-stability, First- and Half-order schemes, Rate of Convergence. 2020 MSC: 60H35, 65L20, 60H10, 65C30, 60J60. + +## 1. Introduction + +Let \((\tilde{\Omega},\tilde{\mathscr{F}},\tilde{\mathbb{P}})\) be a complete probability space. Consider a \(\tilde{d}-\)dimensional standard Brownian motion + +\(B:=\{B(t)\}_{t\geq0}\) with natural filtration \(\tilde{\mathbb{F}}^{B}:=\{\tilde{\mathscr{F}}_{t}^{B}\}_{t\geq0}\). Further, assume that \(r:=\{r(t)\}_{t\geq0}\) is a Markov + +chain with finite state space \(S:=\{1,\ldots,m^{\prime}\}\) and generator \(\mathcal{Q}:=(q_{j_{0}k_{0}})_{j_{0},k_{0}\in S}\text{where}q_{j_{0}k_{0}}\geq0\text{for}j_{0}\neq k_{0}\) + +and \(q_{j_{0}j_{0}}=-\sum q_{j_{0}k_{0}}\) which implies that the transition probability matrix of \(r\) is given by, + +\[k_{0}\neq j_{0}\] + +\[\tilde{\mathbb{P}}(r(t+\Delta)=k_{0}|r(t)=j_{0})= \begin{cases} q_{j_{0}k_{0}}\Delta+\mathrm{o}(\Delta), & k_{0}\neq j_{0}, \\ 1+q_{j_{0}j_{0}}\Delta+\mathrm{o}(\Delta), & k_{0}=j_{0}, \end{cases}\] + +for any \(t\geq0,j_{0},k_{0}\in S\text{and}\Delta>0\). The natural filtration of \(r\) is denoted by \(\tilde{\mathbb{F}}^{r}:=\{\tilde{\mathscr{F}}_{t}^{r}\}_{t\geq0}\). Also, + +consider an \(\tilde{\mathscr{F}}_{0}^{B}\) -measurable random variable \(X_{0}\) and assume that \(r,B\text{and}X_{0}\) are independent. Define + +\(\tilde{\mathbb{F}}:=\{\tilde{\mathscr{F}}_{t}\}_{t\geq0}\text{where}\tilde{\mathscr{F}}_{t}:=\tilde{\mathscr{F}}_{t}^{B}\vee\tilde{\mathscr{F}}_{t}^{r}\text{for}t\geq0\). For a fixed \(T>0\), let \(b:[0,T]\times\mathbb{R}^{d}\times S\mapsto\mathbb{R}^{d}\) and + +\(\sigma:[0,T]\times\mathbb{R}^{d}\times S\mapsto\mathbb{R}^{d\times\tilde{d}}\) be Borel measurable functions. + +Consider the following \(d-\)dimensional regime switching stochastic differential equation, also referred to as stochastic differential equation with Markovian switching (SDEwMS), + +\[X(t)=X_{0}+\int_{0}^{t}b(s,X(s),r(s))ds+\sum_{\ell=1}^{\tilde{d}}\int_{0}^{t}\sigma_{\ell}(s,X(s),r(s))dB_{\ell}(s)\] (1) + +∗Corresponding author Email addresses: dvashistha@ma.iitr.ac.in (Divyanshu Vashistha), chaman.kumar@ma.iitr.ac.in (Chaman Kumar) diff --git a/outputs/arxiv_math/2503.06662_pg5_pg1_repeat1.md b/outputs/arxiv_math/2503.06662_pg5_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..347717be4ef4fe1c4686f6e754c05556000a2406 --- /dev/null +++ b/outputs/arxiv_math/2503.06662_pg5_pg1_repeat1.md @@ -0,0 +1,29 @@ +stable. The robustness properties of exponential stability are exploited here to deal with the fast network dynamics, which is therefore treated as a perturbation of the centralized algorithm that can be overcome by a suitably small stepsize. While specialized to handle such a network consensus phenomena, the underlying rationale in which perturbations are handled thanks to the robustness margin of exponential stability can be easily extended to cover other problems of interest, such as time-varying or time-delayed communications, whenever such phenomena can be reduced to fast parasite dynamics that become negligible when the stepsize is sufficiently small. This analysis approach is enabled by the fact that Algorithm (2) employs a non-diminishing stepsize. As a byproduct of the stability analysis, an explicit theoretical upper bound for the stepsize to guarantee semiglobal linear convergence is provided. Such a bound highlights how the network size, structure, and connectivity, the Lipschitz constant, and the convexity properties of the involved functions influence the stepsize and the convergence rate. Finally, we remark that, for simplicity of exposition, a single step-size for both the primal and the dual updates is assumed. While this is generally a more difficult case to analyze, as it prevents a further time-scale separation between the primal and the dual dynamics (see, e.g., [2]), there are cases where different step-size values are preferable (for instance, if one wants to embed the multiplication by n of \(\nabla\ell_{i}(x_{i}^{t},\lambda_{i}^{t})\) in (2) within the stepsize bound). The analysis carried out here can be easily extended to cover such a case. + +### 1.4 Notation + +If \(\sim\) is a binary relation on a set S and \(z\in S\), we let \(S_{\sim z}:=\{s\in S:s\sim z\}\). If not otherwise specified, binary relations are applied component-wise to vectors. We denote by \(\nabla f\) the gradient of a differentiable function f ; \(\sigma(A)\) denotes the spectrum of a matrix + +\(A\in\mathbb{R}^{n\times n};A\) is Schur if \(\sigma(A)\) is contained in the open unit disk in the complex plane; \(A>0\) means that A is positive definite. If \(\mathscr{I}\) is an ordered set of finite cardinality \(n,(x_{i})_{i\in\mathscr{I}}\) denotes the n-tuple of \(\mathbb{R}^{n}\) ordered by \(\mathscr{I}\) in the obvious way. Set inclusion + +(either strict or not) is denoted by \(\subset.\text{With}n\in\mathbb{N}\), we let \(1:=(1,\ldots,1)\in\mathbb{R}^{n},I\in\mathbb{R}^{n\times n}\) + +be the identity matrix, and \(\langle x,y\rangle:=\sum_{h=1}^{n}x_{h}y_{h}\) denote the standard inner product of + +two vectors \(x,y\in\mathbb{R}^{n}\). Moreover, \(|\cdot|\) denotes the Euclidean norm of a vector or the + +matrix-induced 2-norm, and the closed ball of radius r around point ¯x \(\in\mathbb{R}^{n}\) is denoted + +by \(\overline{\mathbb{B}}_{r} (\bar{x}):=\{x\in\mathbb{R}^{n}:|x-\bar{x}|\leq r\}\). + +We let \(S\in\mathbb{R}^{n\times(n-1)}\) be a matrix satisfying + +\[S^{\top}1=0,S^{\top}S=I_{n-1}.\] (3) + +Moreover, we define the matrix \(T\in\mathbb{R}^{n\times n}\) and its inverse as + +\[T:=\left[ \frac{1^{\top}}{S^{n_{\top}}} \right],T^{-1}:=\left[1S\right].\] (4) + +From (3)-(4), we deduce that + +\[I_{n}= \tfrac{1}{n} 11^{\top}+SS^{\top},|S|=1.\] (5) diff --git a/outputs/arxiv_math/2503.06701_pg5_pg1_repeat1.md b/outputs/arxiv_math/2503.06701_pg5_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..c81ba8aa43dd0d97bed4e56687b8f44071843c8d --- /dev/null +++ b/outputs/arxiv_math/2503.06701_pg5_pg1_repeat1.md @@ -0,0 +1,33 @@ +1. Observations: The observations that were fed to the RL agent from the environment included current blood glucose levels and their rate of change. It’s worth noting that these observations undergo a normalization process. Normalization is employed to prevent the emergence of excessively large or excessively small parameters, thereby enhancing numerical stability within the system. + +2. The reward function: The reward function is designed based on the error \(e(t)\) we carefully considered key criteria derived from the glucose levels of a type 1 diabetes patient. The normal glucose range, defined as 80 to 100 mg/dL, serves as a crucial reference point. Within this range, the RL agent receives a significantly higher reward to reinforce behaviors that maintain glucose levels within the healthy boundaries. We also designated an acceptable range, consisting of two sub-ranges (100 to 160 mg/dL and 70 to 80 mg/dL), where the agent receives a moderate reward. This encourages the agent to navigate glucose levels within an acceptable but broader spectrum. Additionally, we identified two distinct bad ranges (160 or higher and 0 to 70 mg/dL) and assigned a negative reward within these intervals.we also considered the derivative \(de(t)\) to guide the reinforcement learning agent toward desired performance. Additionally, \(i(t)\) the integral of the absolute value of the normalized error, aimed at mitigating large overshooting and oscillations. Furthermore, \(c(t)\) denotes the integral of the control signal, serving to penalize the agent for excessive insulin usage. The reward is given to the agent at each step. And the formulation is as follows: + + - If the error is less than 90 (the glucose is less than the reference), the reward will be given by the linear function: \((1-|e|)/20\) + + - If the error is greater than 90 (the glucose is greater than the reference), the reward will be given by the linear function: \((1-|e|)/70\) + + - If the error is less than 10 (the glucose is close to our reference), the reward will be given by the function, which at \(e=0\) gives the value 20, and at \(e=10\) gives 0: \(1.262\cdot(|e|)^{\frac{1}{5}}+2\) + + - We then apply penalties to our agent as follows: \(-2\times 10^{-6}\cdot i-10^{-6}\cdot c\), where the first term is introduced to mitigate oscillations in the blood glucose signal, ensuring stability, and the second term is designed to optimize the injected insulin, discouraging excessive usage. These penalties contribute to fine-tuning the agent’s behavior for more effective and stable blood glucose regulation. + +3. Stop of episode: In our specific implementation, episodes conclude under three conditions: after reaching the predefined simulation time of 24 hours, or when the glucose level exceeds 300 (indicating hyperglycemia), or falls below 50 (indicating hypoglycemia). These extreme glucose levels are considered hazardous situations for the patient, demanding immediate intervention. In such instances, we assert that the agent has failed to maintain glucose within the desired range, prompting a termination of the episode. This safety-centric approach aligns with our commitment to avoiding potentially + +harmful glucose extremes and ensures that the reinforcement learning agent operates within clinically acceptable bounds during the artificial pancreas control simulations. + +## IV. EXPERIMENTATION AND RESULTS + +We present a thorough and comprehensive comparative analysis of three distinct and well-defined methods for controlling blood glucose levels, specifically tailored to address the challenges associated with Type 1 diabetes management. The methods evaluated in this study include a direct reinforcement learning agent, which autonomously learns optimal control strategies; a non-adaptive fuzzy logic controller with finely tuned and optimized parameters; and an adaptive fuzzy logic controller, where the parameters are continuously and dynamically adjusted by the RL agent at each simulation timestep. It is important to explicitly highlight that all simulations, as well as the tuning processes for each of the proposed methods, were systematically carried out using MATLAB2022b and Simulink. + +## A. Direct insulin control through Reinforcement Learning + +The results of the Direct insulin control through Reinforcement Learning in the nominal case are presented below: + +![Three line plots showing glucose-level nominal, glucose-intake nominal, and insulin-injection nominal over time.](figure)[^inferred] + +![Three line plots showing glucose-level nominal, glucose-intake nominal, and insulin-injection nominal over time.](figure)[^inferred] + +Fig. 1. Glucose level (mg/dL) response of the Direct insulin control through Reinforcement Learning, along with corresponding meals intake (mg/dl) and insulin injections in units in the nominal case. + +In the Glucose-level subplot, the two black dotted lines clearly indicate the boundaries of the safe range for blood glucose levels, spanning from 70 to 180 units, while the green dotted line represents our chosen reference value, set at 90 units. It is important to emphasize that this visual representation, including both the safe range and the reference line, will consistently appear in all the glucose-level subplots presented throughout this article to maintain clarity and uniformity. In order to thoroughly examine the robustness of the proposed approach, the Reinforcement Learning Controller was subjected to a comprehensive series of tests conducted under various sets of randomly generated time spans and differing quantities of meal intakes, allowing us to assess its performance across a wide range of possible scenarios. + +[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter. diff --git a/outputs/arxiv_math/2503.06703_pg4_pg1_repeat1.md b/outputs/arxiv_math/2503.06703_pg4_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..c8de4de103192ea611e3d34a7eceb415e4515df3 --- /dev/null +++ b/outputs/arxiv_math/2503.06703_pg4_pg1_repeat1.md @@ -0,0 +1,85 @@ +being inspected at the same time. We then denote by \(\mathcal{S}_{m}\subset\mathcal{S}\) + +the subset of sensing regions inspected by transmit AP \(m\), + +with \(\mathcal{S}\) the complete set of all clusters such that \(|\mathcal{S}|=S\). On + +the contrary, receive APs are responsible for only detecting a single target within their sensing region. During the tracking phase, the goal is to confirm the presence of already detected targets and update their parameters. In this case, transmit AP \(m\) tracks the detected targets within the + +subset \(\mathcal{L}_{m}\), and the receive AP focuses on one or more targets. + +Obviously, the network has no control over the position of the detected targets, and it can happen that close targets are to be tracked: mutual interference effects are thus to be properly taken into account. This is discussed further in Section IV. + +## III. SYSTEM MODEL + +We now describe the proposed scalable ISAC-enabled CFmMIMO system model, focusing mainly on the sensing tasks: + +\(\mathbf{p}_{l},\text{with}l\in\{1,\ldots,L\}\). Assuming block fading channel and + +detection at positions \(\mathbf{p}_{i},\text{with}i\in\mathcal{S}\), and tracking at positions OFDM modulation, each coherence block consists of \(\tau_{c}\) time-frequency samples during which channels and reflections are + +constant and flat. This way, \(t\in\{1,\ldots,\tau_{c}\}\) will denote the + +index for the symbols transmitted within a single coherence block to sense the positions \(\mathbf{p}_{i}\) (detection) or \(\mathbf{p}_{l}\) (tracking). In that sense, \(\mathbf{h}_{k,m}(\mathbf{h}_{k,m}^{\mathrm{H}})\) denotes the \(N-\)dimensional UL + +(DL)1 channel from UE \(k\) to AP \(m;\mathbf{G}_{m,m^{\prime}}\) is the \((N\times N)- \mathcal{M}^{\text{tx}}\) to AP \(m\in\mathcal{M}^{\text{rx}};\text{and}\mathbf{H}_{\mathbf{p},m,m^{\prime}}\) refers to the \((N\times N)-\) + +dimensional matrix representing the channel from AP \(m^{\prime}\in\) dimensional matrix for the composite channel linking transmit AP \(m^{\prime}\) to receive AP \(m\) through the reflection from the target located in position p (either \(\mathbf{p}_{i}\text{or}\mathbf{p}_{l})\). + +## A. Propagation Channels + +Under the above assumptions, the \(N-\)dimensional UL channel from UE \(k\) to AP \(m\) is [23] + +\[\mathbf{h}_{k,m}\sim\mathcal{C}\mathcal{N}(\mathbf{0},\mathbf{C}_{k,m}),\] (1) + +where \(\mathbf{C}_{k,m}\in\mathbb{C}^{N\times N}\) refers to the spatial correlation matrix + +of the Rayleigh-distributed non-line-of-sight (NLoS) components. The corresponding large-scale fading coefficient (LSF), which includes path loss, is \(\text{tr}(\mathbf{C}_{k,m})/N\). For the AP-AP link, we adopt a Rician modeling: + +\[\mathbf{G}_{m,m^{\prime}}= \frac{bm,m^{\prime}}{1+c_{m,m^{\prime}}} \left(\bar{\mathbf{G}}_{m,m^{\prime}}+ \sqrt{cm,m^{\prime}} e^{j\psi_{m,m}\prime}\mathbf{V}_{m,m^{\prime}}\right),\] + +where \(bm,m^{\prime}\) is the LSF coefficient, \(cm,m^{\prime}\) is the Rician factor, + +G¯ \(\in_{m,m}^{\prime}\mathbb{C}^{N\times N}\) contains the correlated2 NLoS components such that \(\bar{\mathbf{g}}_{m,m^{\prime}}=\) vec(G¯ \()_{m,m}^{\prime}\sim\mathcal{C}\mathcal{N}(\mathbf{0}_{N^{2}},\mathbf{Q}_{m,m^{\prime}})\), the equivalent array response at the LoS direction: + +\(\psi_{m,m^{\prime}}\sim\mathcal{U}[0,2\pi]\) is the phase offset, and \(\mathbf{V}_{m,m^{\prime}}\in\mathbb{C}^{N\times N}\) is + +\[\mathbf{V}_{m,m^{\prime}}=\mathbf{a}_{m}(\varphi_{m,m^{\prime}},\theta_{m,m^{\prime}})\mathbf{a}_{m^{\prime}}^{\mathrm{H}}(\varphi_{m^{\prime},m},\theta_{m^{\prime},m}),(3)\] + +1We assume that system operations happen within one channel coherence interval, and that time-division-duplex (TDD) protocol is used, to ensure equivalence (reciprocity) of UL and DL channels. 2In line with [11], a popular choice for the channel covariance matrix \(\mathbf{Q}_{m,m^{\prime}}\) is the well-known Kronecker model. However, we leave this unspecified until the simulations section to keep our analysis general. + +with \(\mathbf{a}_{m}(\varphi_{m,m^{\prime}},\theta_{m,m^{\prime}})\) the response of the array (or steering vector) for the azimuth \(\varphi_{m,m^{\prime}}(\varphi_{m^{\prime},m})\) and elevation \(\theta_{m,m^{\prime}} (\theta_{m^{\prime},m})\) angles of arrival (departure) from AP \(m\) to AP \(m^{\prime}\). According to [24], the Rician factors can be defined as + +\[c_{m,m^{\prime}}= \frac{p_{\text{LoS}}(d_{m,m^{\prime}})}{1-p_{\text{LoS}}(d_{m,m^{\prime}})} ,\] (4) + +where \(p_{\text{LoS}}(d_{m,m^{\prime}})\) is the LoS probability that depends on the distance \(d_{m,m^{\prime}}\) between AP \(m\text{and}m^{\prime}[25\), Table B.1.2.1-2]. Finally, assuming LoS propagation between the targets and the involved APs, a convenient expression for \(\mathbf{H}_{\mathbf{p},m,m^{\prime}}\) is + +\[\mathbf{H}_{\mathbf{p},m,m^{\prime}}=\tilde{\alpha}_{\mathbf{p},m,m^{\prime}}\mathbf{a} \frac{(_{m}\varphi_{m,\mathbf{p}},\theta_{m,\mathbf{p}})}{\mathbf{A}_{\mathbf{p}}}_{,} \mathbf{a}_{m} \frac{(_{m}^{\mathrm{H}_{\prime}}\varphi_{m^{\prime},\mathbf{p}},\theta_{m^{\prime},\mathbf{p}}}{,m^{\prime}} ),\] + +where \(\tilde{\alpha}_{\mathbf{p},m,m^{\prime}}=\alpha_{\mathbf{p},m,m^{\prime}} \sqrt{\beta\mathbf{p},m,m^{\prime}}\) is a complex scalar coefficient, with \(\alpha_{\mathbf{p},m,m^{\prime}}\) the target reflectivity, or radar cross-section (RCS), and \(\beta\mathbf{p},m,m^{\prime}\) the product of the path loss from transmit AP \(m^{\prime}\) to the target at position p and that from the target to receive AP \(m\). As previously mentioned, we follow the Swerling-I model for the RCS, in which \(\alpha\mathbf{p},m,m^{\prime}\) is kept constant over consecutive symbols within the coherence block [11]. Moreover, \(\varphi_{m,\mathbf{p}},\theta_{m,\mathbf{p}}\) are the azimuth and elevation angle of the position p with respect to (w.r.t.) the antenna array of AP \(m\). A similar meaning have the quantities \(\varphi_{m^{\prime},\mathbf{p}}\) and \(\theta_{m^{\prime},\mathbf{p}}\) w.r.t. AP \(m^{\prime}\). + +## B. Uplink Channel Estimation + +Assuming perfect CSI can be overly optimistic in many applications. In practice, obtaining such channel knowledge locally at the APs via UL orthogonal pilots is more realistic. This approach enables the characterization of the sufficient statistics of the channels [26]. A feasible option could be the minimum mean-square error (MMSE) estimation [27, Subsection V-B]: + +\[\hat{\mathbf{h}}_{k,m}= \frac{1}{\sqrt{\tau_{p}\bar{\eta}_{k}}} \boldsymbol{\Lambda}_{k,m}\varphi_{k,m},\] (6) + +where \(\boldsymbol{\Lambda}_{k,m}=\tau_{p}\bar{\eta}_{k}\mathbf{C}_{k,m}\boldsymbol{\Gamma}_{k,m}^{-1}\) is the MMSE matrix, \(\tau_{p}\) is the number of pilots, \(\bar{\eta}_{k}\) is the UL training power, and + +\[\boldsymbol{\Gamma}_{k,m}=\sum_{j=1}^{K}\tau_{p}\bar{\eta}_{j}\mathbf{C}_{j,m}\big|\boldsymbol{\pi}_{j}^{\mathrm{H}}\boldsymbol{\pi}_{k}\big|^{2}+\sigma_{m}^{2}\mathbf{I}_{L},(7)\] + +denotes the covariance matrix of the least-squares observations + +\(\varphi_{k,m}\in\mathbb{C}^{N}\) providing sufficient statistics, i.e., + +\[\varphi_{k,m}=\sum_{j=1}^{K} \sqrt{\tau_{p}\bar{\eta}_{j}} \mathbf{h}_{j,m}\boldsymbol{\pi}_{j}^{\mathrm{H}}\boldsymbol{\pi}_{k}+\boldsymbol{\omega}_{k,m},\] (8) + +with \(\boldsymbol{\pi}_{k}\in\mathbb{C}^{\tau_{p}}\) the sequence of pilots sent by UE \(k\) to estimate variance \(\sigma_{\omega}^{2}[23\), Subsection II-B]. + +the channel, and \(\boldsymbol{\omega}_{k,m}\in\mathbb{C}^{N}\) the equalized ambient noise with + +## C. Downlink Transmission + +Assuming to be in the DL transmission and sensing phase, + +AP \(m\in\mathcal{M}^{\text{tx}}\) sends data to the UEs in the set \(\mathcal{K}_{m}\) plus additional beams to first, detect the presence of potential targets within the surveillance area and later, track their location. diff --git a/outputs/arxiv_math/2503.06710_pg5_pg1_repeat1.md b/outputs/arxiv_math/2503.06710_pg5_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..c4363ec607576fdf277e999b2281099a18ea1b97 --- /dev/null +++ b/outputs/arxiv_math/2503.06710_pg5_pg1_repeat1.md @@ -0,0 +1,19 @@ +4. This is the first time a DTMP is showed for (GE)CMV matrices. Some results in Baire category for certain classes of almost periodic extended CMV matrices were previously obtained by [27, 41]. It is an open question whether it holds for other CMV matrices with, for example, subshift or Sturmian Verblunsky coefficients. + +We base our proof on the recent understanding of Anderson localization for Diophantine frequencies obtained in [20] and techniques developed therein: the Anderson localization for the UAMO in the supercritical setting \(\lambda_{1}<\lambda_{2}\) proved in [21] is a full measure result. In [20], an arithmetic version of Anderson localization is proved, albeit for a mosaic model where every other local coin in (2.1) is trivial. However, the proof of [20] works in a straightforward way for UAMO as well, compare also [52]. Theorem 2.4. Let \(\Phi\in\text{DC}(\kappa,\tau)\text{and}\lambda_{1}<\lambda_{2}\). Then for each “ \(\Phi-nonresonant”\theta\), i.e., each \(\theta\) such that + +\[|\sin2\pi(\theta+n\Phi)|<\exp(-|n|^{\frac{1}{2\tau}})\] + +does not hold for infinitely many \(n,W_{\lambda_{1},\lambda_{2},\Phi,\theta}\) admits Anderson localization. Proof. In the case \(\Phi\in\mathbb{R}\setminus\mathbb{Q}\text{and}\lambda_{1}<\lambda_{2}\), according to [21, Theorem 2.9], the Lyapunov exponent characterizing the (typical) decay of generalized eigenfunctions is positive: + +\[L_{\lambda_{1},\lambda_{2},\Phi}(z)\geq\log\left[ \frac{\lambda_{2}(1+\lambda_{1}^{\prime})}{\lambda_{1}(1+\lambda_{2}^{\prime})} \right]>0,\] (2.6) + +- with equality if and only if \(z\in\Sigma_{\lambda_{1},\lambda_{2},\Phi}.\) The rest of the proof follows the same outline as the proof of [20, Theorem 6.3]. Remark 2.5. This result is a full measure result in \(\theta\). It is sharp in the sense that it cannot be strengthened to all \(\theta[18]\). We shall also need the following dynamical duality formulation of Autry-André duality for the UAMO, which can be seen as the reverse statement to [21, Theorem 2.4]. As such, we expect it to be of interest beyond this paper. Theorem 2.6 (Aubry-André Duality). Let \(\varphi=\varphi^{\xi}=\left[\varphi^{\xi,+},\varphi^{\xi,-}\right]^{\top},\xi\in\mathbb{T}\) be a solution to the generalized eigenvalue equation \(W_{\lambda_{1},\lambda_{2},\xi,\Phi}^{♯}\varphi=z\varphi\) which has the following form + +\[\begin{aligned} \begin{bmatrix} \varphi_{n}^{\xi,+} \\ \varphi_{n}^{\xi,-} \end{bmatrix} &=e^{2\pi in\theta} \begin{bmatrix} \check{\phi}^{+}(\xi+n\Phi) \\ \check{\phi}^{-}(\xi+n\Phi) \end{bmatrix} = \frac{1}{\sqrt{2}} e^{2\pi in\theta} \begin{bmatrix} \check{\psi}^{+}(\xi+n\Phi)+i\check{\psi}^{-}(\xi+n\Phi) \\ i\check{\psi}^{+}(\xi+n\Phi)+\check{\psi}^{-}(\xi+n\Phi) \end{bmatrix} . \\ \text{Let} \begin{bmatrix} \check{\psi}^{+} \\ \check{\psi}^{-} \end{bmatrix} &= \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & -i \\ -i & 1 \end{bmatrix} \begin{bmatrix} \check{\phi}^{+} \\ \check{\phi}^{-} \end{bmatrix} \end{aligned}\] (2.7) + +with \(n-\)th Fourier coefficients \(\psi_{n}^{+}\text{and}\psi_{n}^{-}\) , respectively. Then \(\psi=[\psi^{+},\psi^{-}]^{\top}\) solves the eigenvalue equation \(W_{\lambda_{1},\lambda_{2},\Phi,\theta}\psi=z\psi\). + +## 3. Preliminaries + +Our proof of Theorem 2.1 utilizes techniques from the theory of one-frequency cocycles of CMV matrices, which we hence review in this section to keep the present treatise as self-contained as possible. We first review the construction of so-called Cantero-Moral-Vel´azquez matrices (CMV matrices), whose intimate connection with quantum walks on diff --git a/outputs/arxiv_math/2503.06715_pg4_pg1_repeat1.md b/outputs/arxiv_math/2503.06715_pg4_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..70b3c25391374232fe86758970613d68abf30b52 --- /dev/null +++ b/outputs/arxiv_math/2503.06715_pg4_pg1_repeat1.md @@ -0,0 +1,11 @@ +Definition 2.6 ([23, Section 4.3]). Fix a strongly \(E^{*}-\)unitary inverse semigroup \((S,\varphi)\) with group \(G\) and morphism \(\varphi:S^{\times}\rightarrow G.\text{For all}g\in G\), define the sets \(E_{g}=\{x\in E:x\leq ss^{*}\text{for some}s\in S\text{with}\varphi(s)=g\}\subseteq E\) which are all semilattices as well. We define morphisms \(\phi_{g}:E_{g^{-1}}\rightarrow E_{g}\text{and that for}x\in E_{g^{-1}}\) map to \(\phi_{g}(x)=sxs^{*}\) where \(s\in S\) is any element where \(\varphi(s)=g\text{and}x\leq s^{*}s\). The inverse semigroup \((S,\varphi)\) induces a partial action of \(G\) on the generalized Boolean algebra \(\mathcal{T}_{c}(E)\) with ideals identified with \(\mathcal{T}_{c}(E_{g})\) and actions on ideals \(\tilde{\phi}_{g}:\mathcal{T}_{c}(E_{g^{-1}})\rightarrow\mathcal{T}_{c}(E_{g})\) generated by \(V_{x}\mapsto V_{\phi_{g}(x)}\). We denote \(\Phi=(\{\mathcal{T}_{c}(E_{g})\}_{g\in G},\{\tilde{\phi}_{g}\}_{g\in G}\}\) define \(L_{R}(S,\varphi)=\text{Lc}(R,\mathcal{T}_{c}(E))\rtimes_{\Phi}G\). The \(R-\)algebra \(L_{R}(S,\varphi)\) is generated by elements of the form \(\{x\delta_{g}\}_{g\in G,x\in E_{g}}\) and calculations involving these elements are described in [23, Lemma 4.15]. As long as \(S\) is strongly \(E^{*}-\text{unitary}\), the resulting algebra \(L_{R}(S,\varphi)\) is isomorphic irrespective of the chosen group \(G\) and map \(\varphi:S^{\times}\rightarrow G\), hence we unambiguously write \(L_{R}(S)\) whenever we don’t need to keep a specific grading in mind. For two strongly \(E^{*}-\)unitary inverse semigroups \(S_{1}\text{and}S_{2}\), we say that \(S_{1}\subseteq_{c}S_{2}\text{if}S_{1}\subseteq S_{2}\text{and}E_{1}\subseteq E_{2}\) preserves finite covers. If \(S_{1}\subseteq_{c}S_{2}\), then for any group \(G\) and grading \(\varphi:S_{2}^{\times}\rightarrow G\text{on}S_{2}\) there is an induced grading \(\varphi_{|S_{1}^{\times}}\text{on}S_{1}\). Furthermore, the induced partial actions from these gradings satisfy \((\mathcal{T}_{c}(E_{1}),\Phi_{1})\subseteq(\mathcal{T}_{c}(E_{2}),\Phi_{2})\) and hence \(L_{R}(S_{1})\subseteq L_{R}(S_{2})\). + +## 3. Morita Equivalence of Subrings for Rings with Local units + +When we refer to Morita equivalence in this paper, we are referring to the notion of Morita equivalence in rings with local units first developed by Abrams in [4] and further explored by Anh and Marki in [24]. We will denote Morita equivalent rings with local units \(R\text{and}S\text{as}R\sim_{M}S\). In this section, we present a technique to prove that a subring \(S\subseteq R\) of a ring with local units is Morita equivalent. We will later apply this to the partial skew group rings and derive various conditions to make the rings Morita equivalent. Most of this section is a generalization of the ideas presented in [3] and [10] to prove a desingularization result for graph and ultragraph algebras respectively. A significant generalization of this section is removing the countability assumption used previously by being slightly more careful with our directed limits. This lets our results apply to all rings with local units instead of just rings with \(\sigma-\text{units}\), which will be useful later when we apply our results to objects that aren’t necessarily countable. + +Definition 3.1 ([24, Definition 1]). A ring \(R\) (not necessarily unital) is called a ring with local units if there exists some set \(E\subseteq R\) such that elements of \(E\) are idempotents, commute with each other, and \(\bigcup_{e\in E}eRe=R\). For a ring \(R\) with local units, the category of (left) \(R-\)modules is defined to be the modules \(M\), with the additional requirement \(RM=M\). These are often called unitary modules. Morphisms of \(R-\)modules are defined in the usual way. Rings with local units \(R\text{and}S\) are said to be Morita equivalent if their respective categories \(R-\text{Mod and}S-\text{Mod}\) of (unitary) modules are equivalent. + +All modules we consider in this section will be left modules, so we omit this from our notation. + +Example 3.2. For a ring \(R\) with local units, we present examples of \(R-\)modules and morphisms that are used throughout this section. diff --git a/outputs/arxiv_math/2503.06716_pg12_pg1_repeat1.md b/outputs/arxiv_math/2503.06716_pg12_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..b433daae76729dbd3c32d53178d4e8b6ce753c25 --- /dev/null +++ b/outputs/arxiv_math/2503.06716_pg12_pg1_repeat1.md @@ -0,0 +1,43 @@ +We are now ready to present the proof of the Bianchi-Egnell type stability for the fractional Hardy-Sobolev inequality (1.1). + +Proof of Theorem 1.3. Clearly the sharpness of the result follows from the Lemma 3.2 above. Suppose that the theorem is not true. Then we can find a sequence \((u_{k})_{k\geq1}\text{in}\dot{H}^{s}\left(\mathbb{R}^{N}\right)\) such that + +\[\lim_{k\rightarrow\infty} \frac{\Vert u_{k}\Vert_{\dot{H}^{s}}^{2}-\mu_{s,t}\Vert|x|^{\frac{-t}{2_{s}^{*}(t)}}u_{k}\Vert_{2_{s}^{*}(t)}^{2}}{d(u_{k},\mathcal{M})^{2}} =0\] (3.8) + +Since the ratio in (3.8) is homogeneous of degree 2 in \(u_{k}\), we may also suppose that \(\Vert u_{k}\Vert_{\dot{H}^{s}}^{2}=\mu_{s,t}\) for all + +\(k\geq1\). Moreover, \(0\in\mathcal{M}\) implies that \(d(u_{k},\mathcal{M})\leq d(u_{k} ,0) =\Vert u_{k}\Vert_{\dot{H}^{s}}= \sqrt{\mu_{s,t}} <\infty\). Thus, up to a subsequence + +still denoted by \(u_{k}\), we see that \(d(u_{k},\mathcal{M})\rightarrow L\in\left[0, \sqrt{\mu_{s,t}} \right]\). We now consider two cases, specifically \(L=0\text{and}L>0\). When \(L=0< \sqrt{\mu_{s,t}}\) ; we can assume (if required up to a further subsequence still denoted by \(u_{k})\) that for + +all \(k\geq1\) one has \(d(u_{k},\mathcal{M})< \sqrt{\mu_{s,t}} =\Vert u_{k}\Vert_{\dot{H}^{s}}\). Lemma 3.1 now gives that the above ratio is greater than or equal to \(\alpha+\mathcal{O}(1)\rightarrow\alpha>0\text{as}k\rightarrow\infty\) and it’s a contradiction to (3.8). Thus \(L>0\) and then we must have that the numerator of the above ratio goes to 0 as \(k\rightarrow\infty\), this in turn satisfies the hypothesis of Lemma 3.3. Consequently, we obtain + +\[L=\lim_{k\rightarrow\infty}d(u_{k},\mathcal{M})\leq\lim_{k\rightarrow\infty}\Vert u_{k}-U_{s,t}^{\lambda_{k}}\Vert_{\dot{H}^{s}}=0\] + +and this gives us the desired contradiction. + +- + +## 4. Palais-Smale decomposition + +Although the energy functional associated with our problem (1.3) is given as in (3.7), for purely technical reasons, we consider the following normalized functional + +\[I_{s,t}(u):= \frac{1}{2} \Vert u\Vert_{\dot{H}^{s}}^{2}- \frac{1}{2_{s}^{*}(t)} \Big\||x|^{- \frac{t}{2_{s}^{*}(t)}}u\Big\|_{2_{s}^{*}(t)}^{2_{s}^{*}(t)}.\] (4.1) + +There is a one-to-one correspondence between the critical points of (4.1) and the critical points of (3.7). + +Specifically, one has that \(u\in\dot{H}^{s}\left(\mathbb{R}^{N}\right)\) is a weak solution to (1.3) (i.e., a critical point of (3.7)) if and only if \(\mu_{s,t}u\) is a weak solution to the normalized Euler-Lagrange equation + +\[\frac{1}{2_{s}^{*}(t)-2}\] + +\[\begin{cases} \underline{|u|^{2_{s}^{*}(t)-2}u} & \left(^{sN}\right) \\ (-\Delta)^{s}u= & \overline{|x|^{t}} & ;u\in\dot{H} & \mathbb{R} \\ u>0\text{in}\mathbb{R}^{N} \end{cases}\] (4.2) + +(i.e., a critical point of (4.1)). Due to this one-one correspondence, all the crucial properties of the energy functional are unchanged, therefore we only analyze the normalized functional moving onward. + +Definition 4.1. We say that the sequence \((u_{n})_{n\geq1}\subset\dot{H}^{s}(\mathbb{R}^{N})\) is a Palais-Smale sequence for \(I_{s,t}\) at level \(\beta\) + +\(((PS)_{\beta}\) condition in-short ), if \(I_{s,t}(u_{n})\rightarrow\beta\text{and}(I_{s,t})^{\prime}(u_{n})\rightarrow0in\dot{H}^{s}\left(\mathbb{R}^{N}\right)asn\rightarrow\infty\). The functional \(I_{s,t}\) is said to satisfy \((PS)_{\beta}\) condition if every \((PS)_{\beta}\) sequence has a convergent subsequence in \(\dot{H}^{s}\left(\mathbb{R}^{N}\right)\). + +It is easy to see that the weak limit of a \((PS)\) sequence solves (4.2) except for the positivity. However, the main difficulty is that the \((PS)\) sequence may not converge strongly and the weak limit can be zero even if \(\beta>0.\) The content of this section is the classification of \((PS)\) sequences of the functional \(I_{s,t}\) which is given in the next proposition and the proof follows by arguments analogous to [4, Theorem 2.1]. + +Proposition 4.2. Let \((u_{n})_{n\geq1}\subset\dot{H}^{s}\left(\mathbb{R}^{N}\right)\) be a Palais-Smale sequence for \(I_{s,t}\) at level \(\beta\). Then up to a subsequence (still denoted by \(u_{n})\) the following properties hold: diff --git a/outputs/arxiv_math/2503.06722_pg17_pg1_repeat1.md b/outputs/arxiv_math/2503.06722_pg17_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..7d02536299d8370cdaf3538eafb87c918ce0ced2 --- /dev/null +++ b/outputs/arxiv_math/2503.06722_pg17_pg1_repeat1.md @@ -0,0 +1,35 @@ +For any simplicial complex \(\Sigma\) we can consider its face poset \(F(\Sigma)\), whose nerve is (the barycentric subdivision of) \(\Sigma.\text{Then},\text{Inj}(F(\Sigma))\) is homotopy equivalent to \(\Sigma\), and the complex of injective words on directed graphs has any homotopy type. + +## Example 4.8. Consider the following digraphs with reciprocal edges: + +![](figure) + +0 + +0 + +\(12\mathcal{S}_{2}=1\) + +\[\mathcal{S}_{1}=\] + +2 + +3 + +3 + +![](figure) + +Note that the associated complexes of injective words are isomorphic (and homeomorphic to the 2-sphere). By Theorem 3.9, the graph \(\mathcal{S}_{2}\) is regularly diagonal. Also the graph \(\mathcal{S}_{1}\) is regularly diagonal. However, the associated eulerian magnitude homology groups are not isomorphic, such as in bidegree \((3,3)\). The last example shows that, although the complexes of injective words on two graphs are homotopy equivalent, we can have different eulerian magnitude homology groups. In the next section we aim to investigate this property more accurately. + +## 5. The injective nerve of categories and the regular magnitude-path spectral sequence + +The goal of this section is construct the regular magnitude-path spectral sequence, in analogy with the magnitude-path spectral sequence developed in [Asa23]. To do so, it shall be more convenient to use the categorical language, and to generalize the framework of eulerian magnitude homology to quivers. In the first subsection we introduce the tool of injective nerves of categories, and then we shall apply it to the reachable categories, getting the regular magnitude-path spectral sequence. As an application of the construction, we shall provide some computations of regular path homology of regularly diagonal graphs. + +5. 1. The injective nerve. Let 2 denote the category consisting of the objects \(E\) and \(V\) , with two non-identity morphisms \(s,t:E\rightarrow V\) called the source and the target. We shall also represent, and identify, a quiver \(Q\) by the corresponding directed graph with set of vertices \(V\) , and set of edges \(E\). Let Fin be the full subcategory of Set of finite sets. By a (finite) quiver we shall mean a functor \(Q:\mathbf{2}\rightarrow\) Fin. Morphisms of quivers are natural transformations of functors. The category Quiver of finite quivers and morphisms of quivers is the functor category \(\mathbf{Fun}(\mathbf{2},\) Fin). + +There is an adjunction + +\[\begin{aligned} \text{Free} \\ \mathbf{Quiver}_{U}\mathbf{Cat} \end{aligned}\] (6) + +between the category of quivers and the 1-category Cat of small categories and functors. In this adjunction, the functor Free is the free functor that turns a quiver into a category, and \(U\) is the corresponding forgetful functor. Recall that an ordered \(k-\)clique of a directed graph \(G\) is a totally ordered \(k-\)tuple \((v_{1},...,v_{k})\) of vertices of \(G\) with the property that, for every \(i0,Q(\alpha,r)=\alpha+(-r/2,r/2)^{d}\) (if \(\alpha=0\), simply \(Q_{r})\) is the open cube in \(\mathbb{R}^{d}\) of center \(\alpha\) and side length \(r\). We denote by \(\mathbb{S}^{d-1}\) the unit sphere in \(\mathbb{R}^{d}.\text{If}A\) is a subset of \(\mathbb{R}^{d}\text{then}\text{dist}(x,A)=\inf\{|y-x|:y\in A\};\mathcal{A}^{\text{reg}}(A)\) is the family of open subsets with Lipschitz boundary. We use standard notation for Γ-convergence [10]. Unless otherwise stated, \(C\) will always denote a generic strictly positive constant that may change from line to line. + +## 3 Setting of the problem and the main result + +We fix \(p\in(1,d)\) and we let \(\Omega\subset\mathbb{R}^{d}\) be a bounded open set with Lipschitz boundary. For fixed \(\varepsilon>0\), we denote by \(\Omega_{\varepsilon}\) the lattice \(\Omega_{\varepsilon}:=\varepsilon\mathbb{Z}^{d}\cap\Omega\) and later for a given infinitesimal sequence \(\varepsilon_{j}\) we will use the notation \(\Omega_{j}:=\Omega_{\varepsilon_{j}}\). We denote by \(\mathcal{A}_{\varepsilon}(\Omega;\mathbb{R}^{m})\) the set of functions + +\[\mathcal{A}_{\varepsilon}(\Omega;\mathbb{R}^{m}):=\{u:\Omega_{\varepsilon}\rightarrow\mathbb{R}^{m}\}.\] + +We will identify the functions in \(\mathcal{A}_{\varepsilon}(\Omega;\mathbb{R}^{m})\) by their piecewise constant interpolation on the cells of the lattice \(\varepsilon\mathbb{Z}^{d}\) that is + +\[\mathcal{A}_{\varepsilon}(\Omega;\mathbb{R}^{m})=\{u:\mathbb{R}^{d}\rightarrow\mathbb{R}^{m}:u\text{constant on}\alpha+[0,\varepsilon)^{d}\text{for any}\alpha\in\Omega_{\varepsilon}\}.\] + +Given \(\xi\in\mathbb{Z}^{d}\text{and}E\subset\Omega\) we define + +\[E_{\varepsilon}(\xi):=\{\alpha\in E|\alpha+\varepsilon\xi\in E\}\cap\varepsilon\mathbb{Z}^{d}.\] (3.1) + +Given a function \(v\in\mathcal{A}_{\varepsilon}(\Omega;\mathbb{R}^{m})\), we denote by \(D_{\varepsilon}^{\xi}v\) the different quotient along the direction \(\xi\); i.e. for \(\alpha\in\Omega_{\varepsilon}(\xi)\) + +\[D_{\varepsilon}^{\xi}v(\alpha):= \frac{v(\alpha+\varepsilon\xi)-v(\alpha)}{\varepsilon|\xi|}\] (3.2) diff --git a/outputs/arxiv_math/2503.06725_pg7_pg1_repeat1.md b/outputs/arxiv_math/2503.06725_pg7_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..9e7c1e5b1f63ffa038493c2c542580bbe5bf6b0e --- /dev/null +++ b/outputs/arxiv_math/2503.06725_pg7_pg1_repeat1.md @@ -0,0 +1,51 @@ +by performing actions, making observations, and receiving rewards, all without prior knowledge of the source’s dynamics, the potential usefulness of the updates to be queried, or the status of the update channels. Within this framework, the hub takes an action \(a(t)\in\mathcal{A}\) while the environment is in state \(s(t)\in\mathcal{S}\) during the \(t-\)th slot. The action is passed to the environment, and after consulting the updated knowledge base, the hub observes the new state of the environment, \(s(t+1)\). Based on this transition, the immediate net reward \(r(t)\) resulting from the chosen action is computed. The recurring interaction between the hub and the environment over \(T_{\mathrm{e}}\) steps is represented as a sequence \(\langle s(t),a(t),r(t),s(t+1)\rangle_{t=0}^{T_{\mathrm{e}}}\). Specifically, the reward \(r(t)\) at the \(t-\)th slot is given by + +\[\begin{aligned} r(t)&=r_{\mu}(s(t),a(t),s(t+1)) \\ &=v_{\text{cpt}}(\text{GoE}(t+1))-\mu v_{\mathrm{c}\text{pt}}^{+}(f_{c}(a(t)))(18) \end{aligned}\] + +where \(\mu\) is the Lagrange multiplier (see Section IV). This formalism defines the state space \(\mathcal{S}\), the action space \(\mathcal{A}\), and the net rewards of the modeled CMDP in Section III-B. Leveraging the model-free approach, we develop a learning-based iterative algorithm to derive policies for solving the dual scheduling problem, i.e., \(\widehat{\mathcal{P}}\), from Section IV-A. + +## B. Learning-Based Iterative Algorithm + +To derive the scheduling policies, we + +We adopt a similar iterative process to that outlined in Algorithm 1. The main distinction here is that we use model-free, learning-based solutions to find the class of effect-aware scheduling policies, \(\pi\), within the inner loop. 1) Computing \(\pi\): adapt two prominent on-policy DRL algorithms, advantage actor-critic (A2C) [35] and proximal policy optimization (PPO) [36], along with an off-policy algorithm, Deep Qnetwork (DQN) [37]. These algorithms are particularly well-suited for decision-making in high-dimensional spaces. In each algorithm, the Q-function or the corresponding value function is computed based on the net reward defined in (18). 2) Computing \(\mu^{*}\): Algorithm 1 to determine the optimal Lagrange multiplier for a given policy from the inner loop. In this context, the bisection search method is employed in the outer loop to iteratively and gradually identify the minimum Lagrange multiplier that meets the cost constraint of the scheduling problem \(\mathcal{P}\), as formulated in (6). + +We employ the same method as in + +## VI. SIMULATION RESULTS + +In this section, we evaluate the performance of the proposed model-based and model-free solutions outlined in Sections IV and V, respectively, within the context of effect-aware query scheduling. To assess their effectiveness, we compare these solutions against several well-established benchmark approaches. + +## A. Setup and Assumptions + +We consider a system with \(N=4\) SAs observing a source characterized by \(M=2\) attributes and \(K=4\) AAs performing actions over \(T=1,000\) slots. The usefulness of an update on + +TABLE I PARAMETERS FOR SIMULATION RESULTS. + +| Parameter | Symbol | Value | Parameter | Symbol | Value | +| --- | --- | --- | --- | --- | --- | +| Number of slots | T | 103 | | αcpt | 0.5 | +| Number of SAs | N | 4 | Parameters | βcpt | | +| Number of attributes | M | 2 | shaping vcpt(·) | λcpt | 2 | +| Number of AAs | K | 4 | Refs. point | GoEref | 0.2 | +| Corr. observation probability | po,nm, ∀n, m | 0.8 | Cost per query | fc(1) | 0.5 | +| Required attributes | \|Mk\|, ∀k | 2 | Cost flex. index | Cflex | 0.75 | +| Erasure probability | pe,n, ∀n | 0.2 | Maximum AoI | ∆max | 4 | +| Discount factor | γ | 0.9 | Converg. sens. | ϵπ | 10−6 | +| Shape parameters | {α1, α2} | {0.5, 2} | Tolerance sens. | ϵµ | | +| for gm(·; ·), ∀m | {β1, β2} | {0.5, 5} | Mixing factor | η | 0.5 | +| Size of usefulness | \|U\| | 4 | – | – | – | + +the \(m-\)th attribute is mapped to a value within the \(\text{range}[0,1]\), determined by applying + +\[g_{m}(t;y_{m}(t))=\min\left\{1, \frac{y_{m}^{\alpha_{m}-1}(t)(1-y_{m}(t))^{\beta_{m}-1}}{\mathrm{B}(\alpha_{m},\beta_{m})} \right\}(19)\] + +at the \(t-\)th slot, where \(\alpha_{m},\beta_{m}>0,\forall m\), are shape parameters, and \(\mathrm{B}(\cdot,\cdot)\) is Beta function. Besides, we consider GoE\((_{m}t)= \frac{u_{m}(t)}{\Delta_{m}(t)} ,\forall m\), and the CPT-based value function for an arbitrary \(x\in\mathbb{R}\) is given by [12] + +\[v_{\text{cpt}}(x)= \begin{cases} v_{\mathrm{c}\text{pt}}^{+}(x)=(x-x_{\text{ref}})^{\alpha_{\text{cpt}}},x\geq x_{\text{ref}}; \\ v_{\mathrm{c}\text{pt}}^{-}(x)=-\lambda_{\text{cpt}}(x_{\text{ref}}-x)^{\beta_{\text{cpt}}},xb\text{and}x_{2}>b\) such that + +\((x_{1}\wedge x_{2})>b\text{and}((b\vee c)\wedge x_{k})>b\), for each k. The latter condition is equivalent to \(b\vee(c\wedge x_{k})>b\), which + +is equivalent to \(c\wedge x_{k}\not\leqslant b\). We need to show that + +\[(b\vee c)\wedge x_{1}\wedge x_{2}>b\] + +which is equivalent to \(c\wedge x_{1}\wedge x_{2}\not\leqslant b\). Since b is a pseudo-complement of a in \(L,c\wedge x_{1}\wedge x_{2}\not\leqslant b\) holds if and only if + +\[a\wedge c\wedge x_{1}\wedge x_{2}\neq0.\] + +Now, consider the elements \(a,x_{1},x_{2}\in L.a\wedge x_{1}\wedge x_{2}\neq0\) because otherwise we would have \(x_{1}\wedge x_{2}\leqslant b\), + +as b is a pseudo-complement of a. Moreover \(c\wedge x_{k}\neq0\) for each k, because otherwise we would have + +\(c\wedge x_{k}\leqslant b\). Finally \(c\wedge a\neq0\), because otherwise \(c\leqslant b\), contradicting our assumption. As c is a \(\mu-\)element + +in L, we conclude that \(c\wedge a\wedge x_{1}\wedge x_{2}\neq0\), which proves the claim. + +Remark 3.16. — An analogue of part (2) of Proposition 3.15 does not hold for upsets. Consider + +the frame of power set \(P(X)\) of the set \(X=\{1,2,3,4,5\}\). We have \(\{1,2\}↝_{\mu}\{1\}^{\uparrow}\text{and}\{3,4\}↝_{\mu}\{3\}^{\uparrow}\), + +by Remark 3.12. However, it is easy to see that \(\{1,2\}\vee\{3,4\}=\{1,2,3,4\}\) is not a \(\mu-\)element in + +\(\{1,3\}^{\uparrow}=\{1\}\vee\{3\}^{\uparrow}\). Moreover, it is also not a \(\mu-\text{elementin}P(X)\). + +Proposition 3.17. — Let L be a frame, \(a\in L\), and b a pseudo-complement of a in L. If \(c\in L\) is maximal + +relative to the properties \(a\leqslant candb\wedge c=0,thena↝_{\mu}c^{\downarrow}\). diff --git a/outputs/arxiv_math/2503.06758_pg6_pg1_repeat1.md b/outputs/arxiv_math/2503.06758_pg6_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..c86f94f4ad337554e7f5efe90cd77c3b61cf1279 --- /dev/null +++ b/outputs/arxiv_math/2503.06758_pg6_pg1_repeat1.md @@ -0,0 +1,21 @@ +verified that \(g\) is an eub for \(\tilde{f}↾\alpha_{\xi}\). But then \(g\) witnesses that \(\alpha_{\xi}\) is good for \(\tilde{f}\). Moreover, by construction, \(\alpha_{\xi}\in C\). Since \(C\) was arbitrary, we have shown that there are stationarily many elements of \(S_{\lambda}^{\lambda^{+3}}\) that are good for \(\tilde{f}\). By Theorem 12, it follows that there is an eub \(h\text{for}\tilde{f}\text{such that cf}(h(i))>\lambda\) for all \(i<\omega\). + +\[\mathbf{Claim}\mathbf{1}\mathbf{4}.\text{cf}(h(i))\geq\lambda^{+3}\text{for all but fni itely many}i<\omega.\] + +Proof. If not, then there exist \(k\in\{1,2\}\) and an unbounded \(A\subseteq\omega\) such that, for all \(i\in A\), we have \(\text{cf}(h(i))=\lambda^{+k}\). For each \(i\in A\), let \(\{\delta_{\eta}^{i}:\eta<\lambda^{+k}\}\) enumerate, in increasing fashion, a set of ordinals cofinal in \(h(i)\). For each \(\eta<\lambda^{+k}\), define a function \(h_{\eta}\) from \(\omega\) to the ordinals by letting \(h_{\eta}(i)=\delta_{\eta}^{i}\text{if}i\in A\text{and}h_{\eta}(i)=0\) otherwise. For each \(\eta<\lambda^{+k}\), we have \(h_{\eta}<^{*}h\), so, since \(h\) is an eub for \(\tilde{f}\), there is \(\beta_{\eta}<\lambda^{+3}\text{such that}h_{\eta}<^{*}f_{\beta_{\eta}}\). Let \(\gamma=\sup\{\beta_{\eta}:\eta<\lambda^{+k}\}\). Since \(k<3\), we have \(\gamma<\lambda^{+3}\). Therefore, for all \(\eta<\lambda^{+k}\), we have \(h_{\eta}<^{*}f_{\gamma}\). Fix an unbounded \(B\subseteq\lambda^{+k}\) and an \(n<\omega\) such that, for all \(\eta\in B\), we have \(h_{\eta}<_{n}f_{\gamma}\). But then, for all \(i\in A\setminus n\), we must have \(f_{\gamma}(i)\geq\sup\{\delta_{\eta}^{i}:\eta\in B\}=h(i)\), contradicting the fact that \(h\) is an upper bound for \(\tilde{f}\). But this claim immediately contradicts the fact that \(\tilde{f}\) is a sequence of functions from \(\omega\) to \(\epsilon\text{and}\epsilon<\lambda^{+3}\). This is because, by the claim, we must have \(h(i)>\epsilon\) for all but finitely many \(i<\omega\). But then the constant function, taking value \(\epsilon\), witnesses that \(h\) fails to be an eub. + +The results in this section lead to the following corollary. + +## Corollary 15. Suppose that \(3\leq n<\omega\). + +1. If \(\eta<\omega_{n+1}\), then there is no strongly increasing sequence \(\langle f_{\alpha}:\alpha<\omega_{n+1}\rangle\) of functions from \(\omega\) to \(\eta\). + +\(2.(\aleph_{\omega+1},\aleph_{\omega})։(\aleph_{n+1},\aleph_{n})\). + +3. There are no inner models \(V\subseteq W\) of ZFC such that \((\aleph_{\omega+1})^{V}=(\aleph_{n+1})^{W}\) . + +It also follows that the only regular cardinals that can possibly be lengths of strongly increasing sequences from \(\omega^{\omega}\) are \(\aleph_{n}\text{for}0\leq n\leq3\). We have seen that there are always such sequences of length \(\aleph_{0}\text{and}\aleph_{1}\). We will prove, in Section 3, the consistency of the existence of a strongly increasing sequence of length \(\aleph_{2}\). The question about the consistency of the existence of a strongly increasing sequence of length \(\aleph_{3}\) remains open. + +## 3 Consistency via \(\mathbf{a}\mathbb{P}_{\text{max}}\) variation + +In this section we use a natural variation of Woodin’s partial order \(\mathbb{P}_{\text{max}}\) to produce a very strongly increasing sequence in \(\omega^{\omega}\) of length \(\omega_{2}\). We refer the reader to [14] for background on \(\mathbb{P}_{\text{max}}\), especially Chapter 4 and Section 9.2. The article [8] may also be helpful. Conditions in our partial order \(\mathbb{P}\) are triples \((M,F,a)\) such that diff --git a/outputs/arxiv_math/2503.06780_pg7_pg1_repeat1.md b/outputs/arxiv_math/2503.06780_pg7_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..e38500cefdc9671971a8bfea9b6cbdf07591b74b --- /dev/null +++ b/outputs/arxiv_math/2503.06780_pg7_pg1_repeat1.md @@ -0,0 +1,17 @@ +eigenvalues of the spheroidal equation), to the Nekrasov-Shatashvili instanton counting function. More details on the connection between exact WKB and \(\mathcal{N}=2\) supersymmetric field theories is available in [57]. WKB periods can also be calculated with thermodynamic-Bethe-ansatz (TBA) equations [58–62]. The TBA approach is perhaps the most efficient approach for calculating quantum periods and solving exact quantization conditions numerically. The TBA approach however does not produce asymptotic series with full parameter dependencies included. As described in [61], the seminal work of Voros [51] provided a method for computing spectral data from exact quantization conditions using what was termed the “analytic bootstrap.” In this approach the WKB periods play a fundamental role and are Borel resummed to form exact quantization conditions. The discontinuity structure encoded by the Stokes automorphism of the WKB periods determines a Riemann-Hilbert problem which has a solution in terms of a TBA-like system [61]. The connection between TBA and differential equations, the ODE/IM correspondence [63], has been used for the calculation of quasinormal modes [37] of D3 branes and the Reissner-Nordström geometry. However, to the knowledge of the author it has not been applied to the case of Kerr black holes. As a final comment we mention that exact quantization conditions can also be solved analytically in terms of transseries [54]. Here we lay the foundation for such computations. For this reason we focus on the computation of the quantum actions exactly and with full parameter dependence included. + +### 2.1 Quantum actions and the Bohr-Sommerfeld condition + +\(\hbar\) + +Applying a WKB ansatz \(\psi(x)=e^{\underline{i}}\int^{x}\phi dx\) to the Schrödinger equation (1.1) gives the following Riccati equation + +\[\phi^{2}=2(E-\cos^{2}x-\mu\csc^{2}x)-i\hbar \frac{d}{dx} \phi.\] (2.1) + +Setting \(\hbar=0\), this equation is algebraic and coincides with the classical equation of motion with the action density \(\phi\) playing the role of momentum. In the complex domain the classical equation of motion defines a Riemann surface which happens to be a torus2, call it \(\Sigma\). The equation (2.1) has an asymptotic series solution [50] for the action density \(\phi\). Let + +\[\phi(\mu,E,\hbar,x)=\sum_{k=0}^{\infty}\phi_{k}(\mu,E,x)\hbar^{k}.\] (2.2) + +Each of the \(\phi_{k}dx\) are differentials on the Riemann surface \(\Sigma\). More precisely they are differentials on \(\Sigma\) where the points \(x=\ldots,-2\pi,-\pi,0,\pi,\ldots\) (which correspond to simple poles of the \(\phi_{k})\) are removed.3 This means that we can integrate them along paths on \(\Sigma\) and define quantum actions. + +2The classical equation defines a two parameter family of tori. 3It is important to point out that the series has the property that the odd terms are all total derivatives with the exception of \(\phi_{1}\).