diff --git a/outputs/arxiv_math/2503.06804_pg14_pg1_repeat1.md b/outputs/arxiv_math/2503.06804_pg14_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..6df15f902281b21f7d0dfa1e2b157ac2f5cde861 --- /dev/null +++ b/outputs/arxiv_math/2503.06804_pg14_pg1_repeat1.md @@ -0,0 +1,15 @@ +1. Economic costs: At a given time t, individuals who are primarily concerned by social distancing measures are susceptible individuals. Besides asymptomatic people, namely non-detected infectious \((I^{-})\) and non-detected recovered \((R^{-})\) are concerned as well since their infection status is not known. Furthermore, since the social distancing measures usually apply even to recovered individuals, we define those who are directly affected by restrictive measures at time n as \(X_{n}^{\text{work}}=S_{n}+I_{n}^{-}+R_{n}^{-}+R_{n}^{+}\) . Using the normalization, \(X_{n}^{\text{work}}=N-I_{n}^{+}-H_{n}\) becomes observable and represents the labor force within the population which are not in quarantine (I+). When restrictive measures with strength \(u^{L}>0\) are implemented, the proportion uLXwork of individuals is unable to fulfill their role in the social economy as a labor force. Consequently, the loss of productivity due to restriction measures will depend on the number of people concerned by the lock-down. In the literature, as in Charpentier et al. [10], Federico et al. [13], Federico and Ferrari [12] a quadratic function or more generally, a convex function is commonly used to describe the socio-economic cost because of its mathematical tractability. Therefore, we define the social distancing running cost at time n by + +\[C_{L}\left(u_{n}^{L}X_{n}^{\text{Work}},0\right),\] + +where the threshold is taken to be 0 which allows to penalize any additional number of people affected by restrictive measures. + +2. Detection costs: Decision-makers must design detection policies for individuals who are infected and ignore their status, and eventually for those who have recovered from the disease but also ignore their status. In this regard, they must also plan a testing strategy that will provide more information about the current state of the epidemic and reduce some uncertainties about the current and future course of the epidemic. Those who can be tested at a given time n are susceptible S, non-detected infectious \(I^{-}\) and non-detected recovered \(R^{-}\). Then, the number of individuals who can be tested is \(X^{\text{Test}}=I^{-}+R^{-}+S\). Hence, the running cost for the testing control variable uT is as follows : + +\[C_{T}\left(u_{n}^{T}X_{n}^{\text{Test}}, \overline{\mathrm{x}}^{\text{Test}} \right),\] + +where xTest > 0 represents the maximum capacity for testing that can be reached. Note that in a model with constant population size, as in our case, it holds that \(X^{\text{Test}}=N-(I^{+}+R^{+}+H)\) which makes XTest observable. + +3. Vaccination costs: Large-scale vaccination programs require significant human and financial resources. We assume that this vaccination cost depends on the number of vaccinated persons per unit of time, which is a portion of XTest. Therefore, the running cost corresponding to the vaccination strategy uV is given as + +\[C_{V}\left(u_{n}^{V}X_{n}^{\text{Test}}, \overline{\mathrm{x}}^{\text{Vacc}} \right).\] diff --git a/outputs/arxiv_math/2503.06817_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.06817_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..2ff70e42a9da142eb5779adae03e63c70074e8b7 --- /dev/null +++ b/outputs/arxiv_math/2503.06817_pg3_pg1_repeat1.md @@ -0,0 +1,19 @@ +obtained from validating whether the LB method satisfies the stability structure [38, 50], but also the convergence proof of the nonlinear problems can be established [30, 29]. Nevertheless, such stability has only been demonstrated for the D2Q9 orthogonal MRT model and how to prove it for many other MRT models is still unclear. To bridge this knowledge gap, Yong et al. [48] put forward an automatic approach for the stability analysis of MRT-LB models and applied it to ten different MRT-LB models [20, 21, 41, 32, 35, 17, 23]. The key step in this approach is to decompose the Jacobian matrix of the collision term in the LB model into the product of a symmetric semi-negative definite matrix and a diagonally positive definite matrix, in particular, this decomposition process can be automatically verified by a simple computer code. With the aid of this work [48], in our current study, we will develop a high-order LB model for (1) and present its explicit stability structure preserving condition. The remainder of this paper is structured as follows. In Section 2, we present the MRT-LB model for the ddimensional diagonal-anisotropic diffusion equation. In Section 3, through the direct Taylor expansion, we first deduce the conditions that ensure the present MRT-LB model to be fourth-order consistent with the diagonal-anisotropic diffusion equation, and then present the fourth-order initialization scheme for the MRT-LB model. Thereafter, the condition which guarantees that the MRT-LB model can satisfy the stability structure is provided, and the relationship between the stability structure preserving condition and the L2 stability of the MRT-LB model is also discussed. Some numerical experiments are carried out in Section 4, and finally some conclusions are summarized in Section 5. + +## 2. The MRT-LB model for the diagonal-anisotropic diffusion equation + +In this section, we will present a unified MRT-LB model for the diagonal-anisotropic diffusion equation (1), where the transformation matrix constructed in a natural way and the DdQ(2d2 + 1) lattice structure are employed. In particular, for the isotropic diffusion equation, another MRT-LB model with the DdQ(2d + 1) lattice structure [fewer discrete velocities than the DdQ(2d2 + 1) lattice structure] is also provided. + +### 2.1. Spatial and temporal discretization + +For the sake of brevity and to facilitate the following analysis, the physical domain \(\Omega\) is first discretized into a uniform mesh \(L\) with a lattice spacing \(\Delta x>0,\text{where}L=\Delta x\mathbb{Z}^{d}\), and the corresponding lattice node is denoted by xi. The time is uniformly discretized as \(t_{n}=n\Delta t,\text{where}\Delta t>0\) represents the time step. The lattice velocity in the LB method can then be expressed as \(c=\Delta x/\Delta t\). For the diagonal-anisotropic diffusion equation (1), the diffusive scaling \((\Delta t\propto\Delta x^{2})\) is adopted in this work. Without loss of generality, we can also take \(\Delta t=\xi\Delta x^{2}\), where the parameter + +\(\xi\in\mathbb{R}^{+}\text{and}\xi=0\text{when}\Delta x,\Delta t\rightarrow0\). + +### 2.2. The MRT-LB model + +To develop a high-order LB method for the diagonal-anisotropic diffusion equation (1), we here adopt the general MRT-LB model [21, 6, 7] with some additional adjustable relaxation parameters, which can be used to eliminate some certain high-order truncation errors of the LB method, in which the evolution equation can be written as + +\[f_{k}(\mathbf{x}+\mathbf{c}_{k}\Delta t,t+\Delta t)=f_{k}(\mathbf{x},t)-\sum_{i=1}^{q}\left[\Lambda_{ki}\left(f_{i}-f_{i}^{eq}\right)\right](\mathbf{x},t)+\Delta t\sum_{i=1}^{q}\left[\left(\mathbf{I}- \frac{\Lambda}{2} \right)_{ki}R_{i}\right](\mathbf{x},t),k\in\heartsuit1,q\spadesuit,^{1}\] (2) + +where \(f_{k}(\mathbf{x},t),f_{k}^{eq}(\mathbf{x},t)\), and Rk represent the distribution function, equilibrium distribution function, and discrete source term at position x and time t, respectively. In terms of the present MRT-LB model (2) for (1), we consider the DdQq lattice structure with the number of the discrete velocities q = (2d2 + 1). To be specific, the velocity set c of the diff --git a/outputs/arxiv_math/2503.06825_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.06825_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..7d9ed9b7de1a663be0deffe1ce7b21c91066d148 --- /dev/null +++ b/outputs/arxiv_math/2503.06825_pg2_pg1_repeat1.md @@ -0,0 +1,9 @@ +In presence of outliers, traditional estimation algorithms based on quadratic cost function can be overly influenced by outliers resulting in overall poor performance. In such cases estimation algorithms with Huber type cost functions ([9], [6] and [5]) may be more appropriate as the cost function switches from quadratic to linear when errors become sufficiently large as is the case with outliers. The second cost function considered in this paper is a hybrid of Huber loss function and \(\epsilon-\)insensitive loss function as the one considered in [16] for ARMA system identification. The proposed cost function a) ignores small measurement errors (is \(\epsilon-\text{insensitive})\), and b) has linear instead of quadratic penalty for large errors making the estimates less sensitive to small noises as well as outliers. For linear dynamical systems, various other algorithms for handling outliers have also been proposed in [1], [3] and [8]. At a general level, Aravkin et al. [4] have considered estimation problems with such convex non-quadratic cost functions and drawn links to machine learning. There are instances when additional information is available about the system beyond the description of the the dynamical model. Examples of such additional information are maximum or minimum value of certain states (for example price of an asset can never be negative or physical constraints that limit movement of an object) or knowledge about the magnitude of disturbances and measurement noises such as an upper bound on measurement noise. Incorporating such additional information can be helpful not only in identification of outliers but can also lead to improved estimates as such information puts constraints on exogenous signals and possible trajectories of states. One approach for estimation under such constraints is to obtain sets constraining possible state values (see for example [12] and [8]). At a more general overarching level, convex constraints with convex cost functions for smoothing problem have been considered by Aravkin et al. [4]. In this paper we assume the additional information about the system can be described in terms of inequality constraints that are linear with respect to states and exogenous signals and develop estimation algorithms that satisfy the constraints as well as minimize the two cost functions considered in this paper \((\epsilon\) insensitive quadratic as well as Huber). In this paper we extend results from Nagpal [13] where optimal smoothing algorithms were developed for \(\epsilon-\)insensitive \(\epsilon\) insensitive Huber M loss function. The algorithms in this paper apply those results for one step horizon to develop recursive algorithms that are easily implemented and where complexity does not grow with the number of observations. Remarkably, the algorithms bear strong structural resemblance to Kalman Filter with the primary difference being that the update based on the new information ("innovation term") is based on solution of a quadratic optimization problem with linear constraints. This paper is organized as follows. In the next section we describe the the two objective functions and the background results from Nagpal [13] which form the basis of the results presented here. Main results are described in Section 3 and the last section concludes with a summary of the results. + +## 2 Problem Formulation and Background Results + +Throughout the paper, \(N\) represents a positive integer which will be used to describe the number of measurements available for estimation. For vectors \(v,w\in\mathbb{R}^{n},v\geq w\) implies that all the components of the vector \(v-w\) are non-negative. In particular for a real valued vector, \(v\geq0\) would imply that all elements of the vector \(v\) are non-negative. For a matrix \(C\in\mathbb{R}^{m\times n},C^{\prime}\) will indicate its transpose. For a vector \(x_{k}\in\mathbb{R}^{n},x_{k_{j}}\) denotes the \(j^{\prime}th\) element of \(x_{k}\). For all the estimation problems we will assume that the underlying system is known and finite dimensional linear system of the following form: + +\[\begin{aligned} x_{k+1}&=Ax_{k}+Bw_{k},\text{initial condition}x_{0}\text{is not known with}\bar{x}_{0}\text{itsbest estimate} \\ y_{k}&=Cx_{k}+v_{k} \end{aligned}\] (1) + +where \(x_{k}\in\mathbb{R}^{n}\) is the state, \(y_{k}\in\mathbb{R}^{m}\) are the noisy measurements, \(w_{k}\text{and}v_{k}\) are unknown exogenous signals and measurement noises respectively. Given measurements \(\{y_{1},\ldots,y_{N}\}\), the filtering problem involves estimating \(x_{N}\). The proposed algorithms described in this paper are applicable for linear time varying systems as well (when \(A,B\) and \(C\) depend on time index \(k\) in equation (1)) but for ease of transparency, we will assume the system parameters \(A\in\mathbb{R}^{n\times n},B\in\mathbb{R}^{n\times l}\text{and}C\in\mathbb{R}^{m\times n}\) are known constant matrices. For any \(k\geq0,\hat{x}_{k}\) will denote the estimate of \(x_{k}\) based on the given measurements \(\{y_{1},\ldots,y_{k}\}.I_{m}\) will denote identity matrix of dimension \(m\). Given a sequence of vectors \(x_{k}\) and matrices \(R_{k}\), we will use the following vector and diff --git a/outputs/arxiv_math/2503.06826_pg10_pg1_repeat1.md b/outputs/arxiv_math/2503.06826_pg10_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..2bebcb772cff6359a3d52b128fc1e0ac7e5f56de --- /dev/null +++ b/outputs/arxiv_math/2503.06826_pg10_pg1_repeat1.md @@ -0,0 +1,37 @@ +We are now ready to prove Theorem 1.3. Proof of Theorem 1.3. Let \(X\subseteq V(G)\) be a subset given by Lemma 2.4. We pass on to \(\Gamma=G[X]\). The graph \(\Gamma\) has \(N\geq\alpha n/128\) vertices, and for \(\beta=\alpha n/(8N)\text{and}d=t/4\), the following holds: + +(i) \(\Gamma\) is a \((\beta,d)-\text{expander}\), and + +(ii) for every partition \(V(\Gamma)=R\cup A\cup B\text{with}|R|\leq\alpha N/16\text{and}|A|,|B|\geq\beta N/4\), there is an edge in \(\Gamma\) between \(A\text{and}B\). + +Let \(K=25/\alpha^{2}\) (the constant in the upper bound on the size of \(T\) in Lemma 3.3). Consider the + +family \(P\) of all ordered partitions \(V(\Gamma)=D\cup\left(\bigcup_{i=1}^{k}W_{i}\right)\cup U,\text{where}k\geq0\) (this can be a different number for different partitions), such that the following holds: + +\[(\mathrm{P}1)\text{For every}i\in[k]:|W_{i}|= \sqrt{\frac{2KN\log N}{d}} =:\ell\text{and}\Gamma[W_{i}]\text{is connected},\] + +(P2) for every two distinct \(i,i^{\prime}\in[k]\), there is an edge in \(\Gamma\) between \(W_{i}\text{and}W_{i^{\prime}}\), and + +\[(\mathrm{P}3)|D|\leq\alpha N/(32d)\text{and}|N_{\Gamma}(D)\cap U|2|U|\text{and}|U|/s>\log|U|\), where the former + +\[|T|\leq K \frac{|U|}{s} \log\left( \frac{qs}{|U|} \right)_{\cdot} \frac{\log|U|}{\log d} \leq\ell,\] diff --git a/outputs/arxiv_math/2503.06838_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.06838_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..9b67e162d09655fdff7c83af78df7047bd377ceb --- /dev/null +++ b/outputs/arxiv_math/2503.06838_pg3_pg1_repeat1.md @@ -0,0 +1,5 @@ +The general idea is that we would like to find that transformation of \(\mu\text{on}\mathcal{X}\) that aligns most closely, in an optimal transport sense, to \(\nu\text{on}\mathcal{Z}\). See Figure 1 for an illustration when \(R\equiv0\). In practice, the penalization term \(R(\theta)\) can be thought of as penalizing the complexity of the transformation \(T_{\theta}\). Also see (1.5) in Example 3 below which shows that the inner product Gromov-Wasserstein distance can indeed be written in the form of (1.3) for a suitable \(R(\theta)\). Let us list a few examples where special cases of this, or similar, problems have been studied. + +Example 1 (Procrustes problem). The Procrustes problem seeks an optimal transformation—such as translation, scaling, rotation, or reflection—that best aligns one data matrix, \(\mathbf{X}\in\mathbb{R}^{N\times n}\), with another, \(\mathbf{Y}\in\mathbb{R}^{N\times d}[21]\). Specifically, the objective is to find a transformation matrix \(T\in\mathbb{R}^{n\times d}\) that solves \(\min_{T}\Vert\mathbf{X}T-\mathbf{Y}\Vert^{2}\), where \(\Vert\cdot\Vert\) denotes the Frobenius norm. The matrix \(T\) can be restricted to different subclasses of linear transformations. In essence, Procrustes analysis learns a linear transformation that maps one set of matched points—represented by the rows of X and Y—onto the other. A particularly well-studied case is the orthogonal Procrustes problem, where \(T\) is restricted to be an orthogonal matrix (i.e., \(n=d\) and \(T^{\top}T=I_{n})\). This variant has a closed form solution [35] and has gained renewed interest in machine learning [40]. The Procrustes problem has widespread applications, from multivariate statistics to machine learning, including shape analysis in 2D [17, 20] and learning a linear mapping between word embeddings in different languages using a bilingual lexicon [29]. Our problem (1.2) extends the Procrustes problem by allowing for an arbitrary family \(\mathcal{T}\) of transformations as opposed to just linear transformations. More importantly, we remove the assumption that the correspondences between the two sets are known—that is, we do not assume prior knowledge of which row in X matches with which row in Y. Instead, we leverage the Wasserstein distance in (1.2) to infer these correspondences. Our next example fits this setup more closely. + +Example 2 (Point-cloud registration or scan matching). Point-cloud registration is an important problem for aligning 2D/3D shapes with applications in shape analysis, computer vision, robotics, and medical imaging [5, 36, 17]. The problem is to find the optimal spatial transformation (scaling, rotation and translation) that minimizes the discrepancy between two point clouds. The Iterative Closest Point or ICP algorithm is a widely used method to solve this problem. The algorithm iteratively refines the alignment by: (1) finding correspondences by pairing each point in one data set with its “nearest” point (either in a greedy or an optimal matching sense) in the other, (2) estimating a transformation that minimizes the sum of squared distances between the matched points, and (3) applying the transformation and repeating until convergence. Despite its efficiency, ICP suffers from sensitivity to initialization, often converging to local minima if the initial alignment is poor. Additionally, it is vulnerable to noise and outliers. Many variants of the ICP algorithm have been developed over the years to improve its accuracy, robustness, and convergence properties [33, 32]. Our objective in (1.3) indeed tackles a problem similar to that of the ICP algorithm. However, our proposed convex diff --git a/outputs/arxiv_math/2503.06856_pg9_pg1_repeat1.md b/outputs/arxiv_math/2503.06856_pg9_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..1a4363589785937265dc0cbe765280b554d1f38c --- /dev/null +++ b/outputs/arxiv_math/2503.06856_pg9_pg1_repeat1.md @@ -0,0 +1,27 @@ +and this last expression is equal to the expectation in (3.4), from the definition of the process \(\Pi(\cdot)\) and the fact that \(\tau\) is \(\mathcal{F}_{\tau}\) –measurable. + +Remark 3.2. The expectation in (3.4) is still well-defined for the infinite time horizon \(T=\infty\). Indeed, on the event \(\{\tau<\infty\}\) this is obvious, while on the event \(\{\tau=\infty\}\) the expression is well-defined since the functions \(c_{i}(\cdot),i=0,1\) are defined on \([0,T]\), while the process \(\Pi(\cdot)\) of (3.2) has a limit at infinity (by the Martingale Convergence Theorem). + +The identity (3.4) and the fact that the processes \(X(\cdot)\text{and}\Pi(\cdot)\) generate the same filtrations enable us to reinterpret the optimal stopping problem (2.9) as maximizing the expression (3.4) over all stopping times \(\tau\) of the filtration generated by the process \(\Pi(\cdot)\). To solve this problem, we embed it into a general Markovian framework allowing for arbitrary initial data \((t,\pi)\). + +#### 3.1.1. General Markovian Framework + +We fix a probability space \((\Omega,\mathcal{F},\mathbb{P})\), supporting a standard Brownian motion \(B=\{B(t),0\leq t<\infty\}\). For each \(\pi\in[0,1]\), we consider a stochastic process \(\Pi^{\pi}=\{\Pi^{\pi}(t),0\leq t<\infty\}\) with state space \([0,1]\), which satisfies + +\[\Pi^{\pi}(t)=\pi+\int_{0}^{t}a\Pi^{\pi}(s)(1-\Pi^{\pi}(s))dB(s),0\leq t<\infty.\] (3.5) + +The processes \(\Pi^{\pi}(\cdot),\pi\in[0,1]\) mimic the conditional process (3.2) with arbitrary starting positions \(\pi\). For any \(\pi\in[0,1]\), the Lipschitz property of the coefficients on \([0,1]\) guarantees the existence and uniqueness of the strong solution to the equation (3.5) (see, e.g., Chapter 5.2.B in Karatzas & Shreve [30]). Moreover, the processes \(B(\cdot)\text{and}\Pi^{\pi}(\cdot),\pi\in(0,1)\), generate the same filtration, as in the proof of Lemma 3.1. Hence, regardless of \(\pi\), we let \(\mathbb{F}:=\{\mathcal{F}(t)\}_{t\geq0}\) be the augmentation of the filtration generated by the processes \(\Pi^{\pi}(\cdot)\pi\in(0,1)\), i.e., we set \(\mathcal{F}(t):= \overline{\sigma} (B(s),0\leq s\leq t)\). In addition, we denote by \(\mathcal{T}_{T}\) the collection of all \(\mathbb{F}-\)stopping times \(\tau\text{such that}\mathbb{P}(\tau\leq T)=1\). + +For fixed \(\pi\in[0,1]\), time horizon \(T\) , and initial time \(t\), we now consider the corresponding optimal stopping problem of maximizing the expected reward + +\[J_{T}(t,\pi,\tau):=\mathbb{E}^{(}c_{1}(t+\tau)\Pi^{\pi}(\tau)-c_{0}(t+\tau)(1-\Pi^{\pi}(\tau))^{)_{+}}\] (3.6) + +over all stopping times \(\tau\in\mathcal{T}_{T-t}\), where the functions \(c_{i}:[0,T]\rightarrow[0,1],i=0,1\), satisfy the assumptions (A1)–(A5). We denote the gain function of this problem by + +\[G(t,\pi):=^{(}c_{1}(t)\pi-c_{0}(t)(1-\pi)^{)_{+}},\] (3.7) + +and its value function by + +\[V_{T}(t,\pi):=\sup_{\tau\in\mathcal{T}_{T-t}}J_{T}(t,\pi,\tau)=\sup_{\tau\in\mathcal{T}_{T-t}}\mathbb{E}G^{(}t+\tau,\Pi^{\pi}(\tau)^{)}.\] (3.8) + +Clearly, the original optimal stopping problem (2.9) can be embedded into the one above by setting \(\pi=p,t=0\). Therefore, if we find the value function \(V_{T}(\cdot,\cdot)\) of (3.8) and the corresponding optimal stopping time, we will automatically solve the original optimal stopping problem as well. For some future arguments, it will be convenient to treat the processes \(\Pi^{\pi}(\cdot),\pi\in[0,1]\) of the new diff --git a/outputs/arxiv_math/2503.06865_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.06865_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..be31cfc1fb3efebaff7271aeed927739add63fb8 --- /dev/null +++ b/outputs/arxiv_math/2503.06865_pg3_pg1_repeat1.md @@ -0,0 +1,19 @@ +Let \(N\) be a compact connected oriented hypersurface in \(\mathbb{C}H^{2}\) which is quadratically convex, that is, the shape operator at each point is definite. In particular, by [2], it is the border of a convex body and diffeomorphic to a sphere. We call \(U\) the exterior of \(N\), that is, the connected component of \(\mathbb{C}H^{2}-N\) into which the opposite of the mean curvature field points. We call \(J\) the complex structure of \(\mathbb{C}H^{2}\). Given a unit tangent vector \(u\) we denote by \(\gamma_{u}\) the unique geodesic of \(\mathbb{C}H^{2}\) whose initial velocity is \(u\). Let \(F:N\times\mathbb{R}\rightarrow\mathbb{C}H^{2}\) be the smooth map defined by + +\[F(q,r)=\gamma_{J\nu(q)}(r),\] + +where \(\nu\) is the unit normal vector field pointing inwards, that is, in the direction of the mean curvature vector field of \(N\). This is our first result: + +Theorem 1 Let \(N\) be a quadratically convex compact connected oriented hypersurface of \(\mathbb{C}H^{2}\). Then, the restrictions of \(F\) to \(N\times(0,\infty)\text{and}N\times (-\infty,0)\) are diffeomorphisms onto the exterior \(U\) of \(N\). + +Thus, we have a double geodesic ray foliation of \(U\) and this allows us to define well an outer billiard map, as follows: For \(t>0\), + +\[B:U\rightarrow U,B(\gamma_{J\nu(p)}(-t))=\gamma_{J\nu(p)}(t).\] + +As a direct consequence of Theorem 1, we have: + +Theorem 2 The outer billiard map \(B\) is a diffeomorphism. + +Remark 3 In general, for a Hadamard manifold (a simply connected manifold with nonpositive sectional curvature), weaker hypotheses on the convexity of the hypersurface may allow the outer billiard map to be well defined without being smooth. For instance, Proposition 1.3 of [9] presents an example of a strictly convex surface \(Sin\mathbb{R}^{3}\) (that is, for each \(p\in S\), the affine plane tangent to \(S\) at \(p\) intersects \(S\) only at \(p\), near \(p)\), for which the outer billiard map does exist, but it is not smooth. + +The following proposition provides examples. Part (b) is a particular case of part (a). We recall that a complex line in \(\mathbb{C}H^{2}\) is a complete totally geodesic surface which is a complex submanifold (its tangent spaces are invariant by \(J)\). It is isometric to the real hyperbolic plane \(H^{2}(-4)\) of constant curvature −4. A complex line admits an isometric reflection in \(\mathbb{C}H^{2}\) with respect to it. diff --git a/outputs/arxiv_math/2503.06877_pg16_pg1_repeat1.md b/outputs/arxiv_math/2503.06877_pg16_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..e89d6c920657f98195cf271163f82e5cbb9a508c --- /dev/null +++ b/outputs/arxiv_math/2503.06877_pg16_pg1_repeat1.md @@ -0,0 +1,17 @@ +Proposition 5.10. Suppose that \((U_{*},\lambda_{*})\in\mathrm{V}(r,n_{1})\times\cdots \times\mathrm{V}(r,n_{s})\times\text{OB}(r,n_{s+1})\times\cdots \times\text{OB}(r,n_{k})\times\mathbb{R}_{*}^{r}\). + +Then \((U_{*},\lambda_{*})\) is a critical point of \(g\) if and only if it is a KKT point of (32). The following fact is also straightforward to verify. Lemma 5.11 (Local Diffeomorphism). [49] The set \(\mathrm{W}_{r}:=\mathrm{V}(r,n_{1})\times\cdots \times\mathrm{V}(r,n_{s})\times\text{OB}(r,n_{s+1})\times \cdots \times\text{OB}(r,n_{k})\times\mathbb{R}_{*}^{r}\) is a smooth manifold and is locally diffeomorphic to the manifold \(\psi(\mathrm{W}_{r})\), where \(\psi\) is the map defined in (23). Theorem 5.12 (Generic Linear Convergence Rate). Let \(s\geq1\) and let \(\{U_{[p]}\}\) be a sequence generated + +by the iAPD-ALS algorithm for a generic tensor \(\mathcal{A}\in\mathbb{R}^{n_{1}}\otimes\cdots \otimes\mathbb{R}^{n_{k}}.Ifs=1\), assume in addition that + +\[\sum_{i^{\prime}=2}^{k}n_{i^{\prime}}\geq n_{1}+k-1,\sum_{i^{\prime}=2}^{k}n_{i^{\prime}}\geq n_{i^{\prime\prime}}+k,2\leq i^{\prime\prime}\leq k.\] + +The sequence \(\{U_{[p]}\}\) converges \(R-\)linearly to a KKT point of (32). Proof. It suffices to verify conditions a)–d) in Theorem 5.5. It follows from Lemma 5.8 that there + +- is some \(t\leq r\text{such that}(U_{[p]},\lambda_{[p]})\in\mathrm{V}(t,n_{1})\times\cdots \times\mathrm{V}(t,n_{s})\times\mathrm{B}(t,n_{s+1})\times\cdots \times\mathrm{B}(t,n_{k})\times\mathbb{R}_{*}^{t}\) for sufficiently large \(p\). According to Lemma 5.11, \(\mathrm{W}_{t}\) is locally diffeomorphic to \(\psi(\mathrm{W}_{t})\). Hence a) is satisfied. Moreover, we already see that b) holds in the proof of Proposition 4.4. By Proposition 5.6, the sequence \(\{(U_{[p]},\lambda_{[p]})\}\) globally converges to a KKT point \((U_{*},\lambda_{*})\) of (32) and this verifies c). Lastly, Proposition 5.7 and Lemma 5.9 ensure that d) is satisfied. Note that by Corollary 4.5, the requirement for (25) and (26) can be removed if \(s=1,k\geq4\) and \(n_{1}=\cdots =n_{k}\). Hence we obtain the following. Theorem 5.13 (Square Tensors). Assume that \(k\geq4\text{and}n_{1}=\cdots =n_{k}\geq2\). For a generic \(\mathcal{A}\), the sequence \(\{U_{[p]}\}\) generated by Algorithm 1 converges \(R-\)linearly to a KKT point of (32). We remark that the sublinear convergence rate can be established for the general case [49], while in Theorem 5.12 linear convergence rate is given for a generic case. This can be interpreted via the fact that the local branches of the solution mapping for the best low rank partially orthogonal tensor approximation problem have local Lipschitz continuity whenever the nondegeneracy of the converged KKT point holds [12]. However, the nondegeneracy only holds generically. Equivalently, for general polynomial systems, only local Hölderian error bounds can hold [34]. The explicit examples given in [16, 17] present a witness for this phenomenon when \(r=1\). + +## 6. Conclusions + +In this paper, we study the non-linear least squares (NLS) problem associated with a smooth variety + +M and a polynomial map \(\psi\) : M \(\rightarrow\mathbb{R}^{n}\). This model encompasses a wide range of practical problems, including low rank (structured) matrix approximation problems and low rank (structured) tensor approximation problems. Under mild assumptions, we prove that critical points of the NLS problem are all mapped into the smooth locus of \(\psi(\mathrm{M})\). One significant implication of this result is that, when designing and analyzing algorithms for the NLS problem, misbehaved points are guaranteed to be avoided. As a consequence, we obtain a linear convergence rate of an iterative sequence generated by algorithms for the NLS problem. In particular, we apply our general framework to establish the generic linear convergence rate of the iAPD-ALS algorithm for the low rank partially orthogonal tensor approximation problem, without any further assumption. Given that numerous problems can be recast as the NLS problem for different choices of M and \(\psi\), our framework is potentially applicable to the convergence analysis of numerical algorithms for them. It is possible to establish an unconditional convergence result analogous to Theorem 5.13. We also want to mention that our framework heavily relies on the geometry of M and \(\psi\). Hence, the framework is applicable if the geometry of \((\mathrm{M},\psi)\) is well-understood. Of particular interest is the case where \(\psi(\mathrm{M})\) is a variety of structured matrices or tensors, since such varieties are already extensively studied in the context of computational algebraic geometry [13, 31, 35]. Acknowledgement. This work is partially supported by the Natural Science Foundation of Hunan Province of China (Grant No. 2025JJ20006), and the National Science Foundation of China (Grant No. 12171128). diff --git a/outputs/arxiv_math/2503.06880_pg36_pg1_repeat1.md b/outputs/arxiv_math/2503.06880_pg36_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..c60265e1fbcbf3dd918c7a346fd3e40984e629c2 --- /dev/null +++ b/outputs/arxiv_math/2503.06880_pg36_pg1_repeat1.md @@ -0,0 +1,43 @@ +Obviously \(W\) contains an open neighborhood of zero. Now let \(z=(r(f_{1}),r(f_{2}),\cdots ,r(f_{n}))\). By assumption and the fact that \(\overline{W}\) is compact, \(z\not\in \overline{W}\) . Since \(\overline{W}\) compact we can find a bounded linear functional that strictly separates \(\overline{W} \text{and}z_{0}\). Namely we can find \((\lambda_{1},\lambda_{2},\cdots ,\lambda_{n})\in\mathbb{C}^{n}\) such that: + +\[\sum_{i\leq n}\lambda_{i}r(f_{i})>1\mathbf{and}\forall(x_{1},\cdots ,x_{n})\in \overline{W} ,\sum_{i\leq n}\lambda_{i}x_{i}\leq1\] + +Therefore \(\|_{\|}\sum_{i\leq n}\lambda_{i}f_{i}^{\|_{\|}}\leq1\) and hence: + +\[10\). Below we will start constructing the desired bi-orthogonal system. + +Suppose \(\|r\|=1\). Fix \(\sigma\in(0,1).y_{1}\in X_{=1}^{*}\) so that \(\beta_{1}=r(y_{1})> \frac{1}{2}\). By Proposition 5.2 we can find \(b_{1}\in X_{\leq1+\sigma}\) so that \(y_{1}(b_{1})=\beta_{1}\). Define \(E_{1}=\text{Span}\{b_{1}\}\) and: + +\[\rho_{1}=\sup^{\{}r(y)|y\in X_{=1}^{*}\cap E_{1}^{\}_{\perp}}\] + +By the previous remark, we have \(\rho_{1}>0\). Next find \(y_{2}\in X_{\leq1}^{*}\cap E_{1}^{\perp}\) so that \(\beta_{2}=r(y_{2})> \frac{1}{2} \rho_{1}\). Again by Proposition 5.2, find \(b_{2}\in X_{\leq1+\sigma}\) so that \(y_{1}(b_{2})=\beta_{1},y_{2}(b_{2})=\beta_{2}\). Define \(E_{2}=\text{Span}\{b_{1},b_{2}\}\) and: + +\[\rho_{2}=\sup^{\{}r(y)|y\in X_{=1}^{*}\cap E_{2}^{\}_{\perp}}\] + +and similarly \(\rho_{2}>0\). By induction for each \(n\in\mathbb{N}\) we will have \(\{b_{i}\}_{i\leq n}\subseteq X_{\leq1+\sigma},E_{n}=\text{Span}\{b_{i}\}_{i\leq n}\), \(y_{i+1}\in X_{\leq1}^{*}\cap E_{i}^{\perp},\{\beta_{i}\}_{i\leq n}\) and: + +\[\begin{aligned} \rho_{n}&=\sup^{\{}r(y)|y\in X_{=1}^{*}\cap E_{n}^{\}_{\perp}} \\ \text{such that}\beta_{1}&=r(y_{1})> \frac{1}{2} ,\beta_{n+1}=r(y_{n+1})\in^{(} \frac{1}{2} \rho_{n},\rho_{n}\text{and for each}i,j\in\{1,2,\cdots ,n\}: \\ y_{i}(b_{j})&= \begin{cases} r(y_{i}), & 1\leq i\leq j\leq n \\ 0, & 1\leq j1, \\ \zeta(m\eta^{-1}|z-\xi_{j}|), & \text{if} & |z|\leq1. \end{cases}\] + +Here \(\zeta(s)\) is a smooth function such that \(\zeta(s)=1\text{for}s<1\text{and}\zeta(s)=0\text{for}s>2,\text{and}\eta\) is a small positive constant. It is not difficult to check that + +\[U(z)= \frac{1}{|z|} U\left( \frac{z}{|z|^{2}} \right),U_{j}(z)= \frac{1}{|z|} U_{j}\left( \frac{z}{|z|^{2}} \right),j=1,\cdots ,m,\] (2.8) + +and + +\[\zeta_{j}(z)=\zeta_{j}\left( \frac{z}{|z|^{2}} \right),j=1,\cdots ,m.\] (2.9) + +Together with the properties that the ansatz is \(2\pi/m-\)rotationally invariant in \(z_{1}z_{2}\) plane and even symmetry with respect to the \(z_{3}\) axis, one can solve \(\psi\) from (2.7) uniquely in a suitable Sobolev space, which is the orthogonal complement of the space spanned by the kernels of the linear operator \(\Delta+5U^{4}\). More precisely, the authors in [20] proved that + +\[\Vert\psi\Vert_{L_{w}^{\infty}}\leq C\left(\Vert V(z)\psi\Vert_{L_{w}^{q}}+5\Bigg\||U_{*}|^{4}\sum_{j}(1-\zeta_{j})\tilde{\phi}_{j}\Bigg\|_{L_{w}^{q}}+\Vert M(\psi)\Vert_{L_{w}^{q}}\right),\] + +where + +\[\Vert\psi\Vert_{L_{w}^{\infty}}:=\Vert(1+|z|)\psi\Vert_{L^{\infty}(\mathbb{R}^{3})}\text{and}\Vert h\Vert_{L_{w}^{q}}:=\Big\|(1+|z|)^{5- \frac{6}{q}}h\Big\|_{L^{q}(\mathbb{R}^{3})}.\] + +Particularly, for \(q> \frac{3}{2}\) it has been shown that + +\[\Bigg\|\left(1-\sum_{j=1}^{m}\zeta_{j}\right)N\left(\sum_{j=1}^{m}\tilde{\phi}_{j}+\psi\right)\Bigg\|_{L_{w}^{q}}+\Bigg\||U_{*}|^{4}\sum_{j}(1-\zeta_{j})\tilde{\phi}_{j}\Bigg\|_{L_{w}^{q}}+\Vert V(z)\psi\Vert_{L_{w}^{q}}\leq \frac{C}{(\log m)^{2}} .\] diff --git a/outputs/arxiv_math/2503.06908_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.06908_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..d8aa5f2e0000ec4fe4819e8a92388e0705142e58 --- /dev/null +++ b/outputs/arxiv_math/2503.06908_pg3_pg1_repeat1.md @@ -0,0 +1,63 @@ +We denote by \(\pi:G\rightarrow G/H\) the quotient map and let \(\pi^{*}:L(G/H)\rightarrow L(G)\) be the map defined + +\[\text{by}\pi^{*}(f)=f\circ\pi.\] + +\(\left(\cup\right)\) + +\(\subseteq H^{\perp}\) and the restriction map \(\pi^{*}:G/H\) + +Lemma 2.3 ([29], Lemma 4.5). We have \(\pi^{*}G/H^{\cup}\rightarrow H^{\perp}\) is an isomorphism of groups. + +2. 2. Weighted graphs. The primary goal of this subsection is to introduce weighted graphs and record an inequality involving their eigenvalues. Standard references are [28, 27, 22]. Let \(\mathcal{G}= (V,E,w)\) be an undirected, weighted and connected graph. Here \(V\) is the set of vertices, \(E\) is the + +set of edges and \(\omega:V\times V\rightarrow\mathbb{R}_{\geq0}\) satisfies \(\omega(x,y)=\omega(y,x)\) for any \((x,y)\in V\times V\) , and \(\omega(x,y)>0\) if and only if \(\{x,y\}\in E\). The function \(\omega\) is called the weight function of \(\mathcal{G}\). Given \(x,y\in V\) , we say that \(x\text{and}y\) are adjacent, denoted as \(x\sim y,\text{if}\{x,y\}\in E\). The degree of a vertex \(x\in V\) is \(d_{x}:=\sum_{y\in V,y\sim x}\omega(x,y)\). We denote \(d_{G}:=\max_{x\in V}d_{x}\). + +Let \(L(V,\mathbb{R})\) be the vector space consisting of all real-valued functions on \(V\) . The Laplacian + +operator of \(\mathcal{G}\) is the linear map \(\Delta:L(V,\mathbb{R})\rightarrow L(V,\mathbb{R})\) defined by + +\[(\Delta u)(x)=\sum_{y_{y}\in_{\sim}V_{x}}\omega(x,y)(u(y)-u(x)),u\in L(V,\mathbb{R}),x\in G.\] (2) + +We remark that \(\Delta=A-D\text{where}A:=(\omega(x,y))_{x,y\in V}\in\mathbb{R}^{|V|\times|V|}\text{and}D:=\text{diag}(d_{x})_{x\in V}\in\mathbb{R}^{|V|\times|V|}\). + +Moreover, we define two bilinear operators on \(L(V,\mathbb{R})\): + +\[(3)\Gamma(u,v)= \frac{1}{2} (\Delta(uv)-u\Delta v-(\Delta u)v),\Gamma_{2}(u,v)= \frac{1}{2} (\Delta\Gamma(u,v)-\Gamma(u,\Delta v)-\Gamma(\Delta u,v)),\] + +where \(u,v\in L(V,\mathbb{R})\). For simplicity, we denote \(\Gamma(u):=\Gamma(u,u)\text{and}\Gamma_{2}(u):=\Gamma_{2}(u,u)\). We say that + +\(\mathcal{G}\) is \(\text{CD}(0,\infty)\text{if}\Gamma_{2}(u)\geq0\text{for all}u\in L(G,\mathbb{R})\). Theorem 2.4 ([28], Theorem 1.2). Let \(\mathcal{G}=(V,E,\omega)\) be an undirected, weighted and connected graph. Suppose that \(\mathcal{G}\) is \(\text{CD}(0,\infty)\text{and}0=\lambda_{1}<\lambda_{2}\leq\cdots \leq\lambda_{n}\) are eigenvalues of \(-\Delta\). Then for + +2 + +any natural number \(k\geq2\), we have \(\lambda_{k}\leq Cd_{G}k^{2}\lambda_{2}\text{where}C:=\Big( \frac{20 \sqrt{2} e}{e-1}\) . + +## 3. Positive definite functions + +In this section, we investigate real-valued functions whose Fourier coefficients are all non-negative. These functions, known as positive definite functions, play a crucial role in harmonic analysis on groups [7, 8, 32]. To begin with, we summarize several basic properties of functions on finite abelian groups in the following lemma. + +Lemma 3.1. Given a function \(f:G\rightarrow\mathbb{C}\), we denote by \(H\) the subgroup of \(\widehat{G}\) generated by + +\(\text{supp}(\widehat{f}):=\{\chi\in\widehat{G}:\widehat{f}(\chi)\neq 0\}\). Then the followings hold: + +(a) There is a well-defined function \(\tilde{f}:G/H^{\perp}\rightarrow\mathbb{C}\text{such that}\tilde{f}(xH^{\perp})=f(x)\) for any \(x\in G\). + +\(\setminus\perp\rightarrow\widehat{G}\) be the induced map defined by + +(b) Let \(\pi:G\rightarrow G/H^{\perp}\) be the quotient map and let \(\pi^{*}:G/H\) + +\(\setminus\perp\) to \(H\). + +\(\pi^{*}(\rho):=\rho\circ\pi\) is an isomorphism from \(G/H\) + +(c) \(\tilde{f}=\sum_{\chi\in\text{supp}\widehat{f}}\widehat{f}(\chi)(\pi^{*})^{-1}(\chi)\). + +(d) Suppose that \(f\) is real-valued. For any \(\chi\in\widehat{G}\), we have \(\widehat{f}(\chi^{-1})= \overline{\widehat{f}(\chi)}\). If moreover \(\widehat{f}(\chi)\in\mathbb{R}\), + +then \(\widehat{f}(\chi^{-1})=\widehat{f}(\chi)\). + +Proof. To prove (a), it is sufficient to show \(f(y)=f(x)\text{for all}y\in xH^{\perp}\). By definition, there exists + +some \(z\in H^{\perp}\text{such that}y=xz\). Since \(H\) is generated by \(\text{supp}(\widehat{f}),H^{\perp}=\{z\in G:\chi(z)=1,\chi\in \text{supp}(\widehat{f})\}\). This implies + +\[\begin{aligned} \hat{}\hat{}\hat{} \\ f(y)&=\sum_{\chi\in\text{supp}(f)}f(\chi)\chi(y)=\sum_{\text{supp}}f(\chi)\chi(x)\chi(z)=\sum_{\text{supp}}f(\chi)\chi(x)=f(x). \\ \hat{}\hat{}\hat{} \\ \chi&\in(f)\chi\in(f) \end{aligned}\] diff --git a/outputs/arxiv_math/2503.06939_pg20_pg1_repeat1.md b/outputs/arxiv_math/2503.06939_pg20_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..1505c9a7bd460a20c3b5558836ba57944b45f095 --- /dev/null +++ b/outputs/arxiv_math/2503.06939_pg20_pg1_repeat1.md @@ -0,0 +1,29 @@ +\[\sigma=\left[ \frac{1}{N} \sum_{k=1}^{N}(z_{k}-\nu)^{2}\right]^{\frac{1}{2}},\nu= \frac{1}{N} \sum_{k=1}^{N}z_{k}.(79)\] + +The normalized standard deviation of the interspike interval is then simply + +\[\bar{\sigma}= \frac{\sigma}{\nu} .\] (80) + +If a spike train is exactly periodic then \(\bar{\sigma}=0\). If it is instead a Poisson process, then \(\bar{\sigma}=1\). We have computed \(1/\bar{\sigma}\text{for}Y_{\star}(t)\) at various values of \(\kappa\). The results are shown in Fig. 8 as purple dots [joined for ease of visualization, and colored differently for (a), (b), and (c)]. The data indicates a resonance effect of \(\Upsilon(t)\text{on}Y_{\star}(t)\). This refers to the existence of an optimal value of \(\kappa\) at which \(Y_{\star}(t)\) oscillates most regularly. The same can also be seen in \(X_{\star}(t)\) (the results for which are not shown). Starting at point (a) \((\kappa=0.050)\), the spike train can be seen to be more and more periodic as \(\kappa\) increases, until we reach (b) \((\kappa=0.243)\) when the period is optimized. Adding more noise thereafter only reduces the regularity of the spike train, which can be seen to level off around point (c) \((\kappa=1.178)\). The manner in which \(Y_{\star}(t)\) changes with noise is reminiscent of coherence resonance in excitable systems [23]. The effect is well known for classical systems [164–168], but has appeared only relatively recently in quantum systems [121, 169]. A notable difference between our results in Fig. 8 and coherence resonance is that we have not operated the noise-free quantum Fitzhugh–Nagumo system in the excitable regime (defined classically by having a stable fixed point in the system while being close to the supercritical Hopf bifurcation). If we define quantum excitability based on the classical theory, then the excitable regime of the noise-free quantum Fitzhugh– Nagumo model should look something like Fig. 7(a). We have in fact calculated \(\bar{\sigma}(\kappa)\) using such a state for \(W_{\Upsilon}(x,y,0)\). The resonance effect of \(\Upsilon(t)\text{on}Y_{\star}(t)\) is retained, i.e. we get a similar \(\bar{\sigma}^{-1}(\kappa)\) as in Fig. 8, but the mode of \(W_{\Upsilon}(x,y,t)\) is also a lot noisier for such a \(W_{\Upsilon}(x,y,0)\), so the spikes are no longer apparent. A more careful study is required to understand the noise-driven quantum Fitzhugh–Nagumo model in this regime. This is beyond the scope of this paper as our goal here is simply to illustrate the range of systems amenable to cascade quantization. There is however a separate note to this end that is worth mentioning, and that is—excitable systems come in different flavors, depending on the bifurcation that gives rise to their excitability. In the case of the Fitzhugh–Nagumo model, it is a supercritical Hopf bifurcation that facilitates its excitability. Another commonly used mechanism is the infinite-period bifurcation, an example of which was considered in Sec. IV E (and recall that this is the same as a saddle-node bifurcation on an invariant circle). In this case, (59) and (60) become excitable for \(|\mu|≲1\), i.e. near the onset of the infinite-period bifurcation. In the classical literature, these two bifurcations—the Hopf and the infinite period—lead to substantially different interspike intervals and the two modes of excitability are classified as type I (saddle-node on invariant circle) and type II (Hopf) [23]. Therefore as a byproduct, the Lindbladian in (63) [and (64)] can also be said to quantize a type-I excitable system for an appropriate value of \(\mu\). This classification also puts the system in Ref. [121] as type I excitable, but in contrast to (63), Ref. [121] considers a bistable system instead of a monostable one. It may thus be interesting to also investigate how quantum versions of the different excitable systems respond to noise. + +## 2. Average effect of noise + +We can also get an understanding of how \(\Upsilon(t)\) affects the system by considering its effect on average. Suppose we are interested in how a function \(s(\hat{a},\hat{a}^{\dagger})\) might change. Then in principle we could use (75) to obtain many runs + +of \(\langle s(\hat{a},\hat{a}^{\dagger})\rangle_{\Upsilon}=\text{Tr}[s(\hat{a},\hat{a}^{\dagger})\rho_{\Upsilon}]\) and calculate their ensemble average \(\mathrm{E}[\langle s(\hat{a},\hat{a}^{\dagger})\rangle_{\Upsilon}]\). This approach works, but is rather indirect. It does not provide a simple way to capture the average effect of \(\Upsilon\) on the system. Here we derive an equation that propagates the quantum state of the noise-driven system while averaging over \(\Upsilon\). That is, we seek the evolution of a state \(\rho\) such that, for any \(s(\hat{a},\hat{a}^{\dagger})\) and any time, + +\[\text{Tr}\left\{s(\hat{a},\hat{a}^{\dagger})\rho\right\}=\mathrm{E}\left[\langle s(\hat{a},\hat{a}^{\dagger})\rangle_{\Upsilon}\right].\] (81) + +Such a state can be found by noting that + +\[\begin{aligned} \mathrm{E}\left[\langle s(\hat{a},\hat{a}^{\dagger})\rangle_{\Upsilon}\right]&\equiv\int_{-\infty}^{\infty}d\upsilon\wp_{\Upsilon}(\upsilon,t)\text{Tr}\left[s(\hat{a},\hat{a}^{\dagger})\rho_{\Upsilon}(t)\right] \\ &=\text{Tr}\left[s(\hat{a},\hat{a}^{\dagger})\int_{-\infty}^{\infty}d\upsilon\wp_{\Upsilon}(\upsilon,t)\rho_{\Upsilon}(t)\right], \end{aligned}\] + +where \(\wp_{\Upsilon}(\upsilon,t)\) is such that \(\wp_{\Upsilon}(\upsilon,t)d\upsilon\) gives the probability that \(\Upsilon(t)\in[\upsilon,\upsilon+d\upsilon]\). Comparing (81) to (82), we see that + +\[\rho(t)=\mathrm{E}[\rho_{\Upsilon}(t)]=\int_{-\infty}^{\infty}d\upsilon\rho_{\Upsilon}(t)\wp_{\Upsilon}(\upsilon,t),(83)\] + +Equation (83) implies that an equation of motion for \(\rho(t)\) can be obtained by averaging (75). One way to calculate the ensemble average of (75) is to convert it to its Itˆo equivalent first [107], given by + +\[d\rho_{\Upsilon}=\mathcal{L}_{+^{0}i}\rho \frac{\kappa^{\Upsilon}dt}{\sqrt{2}} [^{+}\hat{a} \frac{\kappa^{2}}{+^{2}\hat{a}} \mathcal{D}_{,^{\dagger}}[_{\rho}\hat{a}_{\Upsilon}+_{]d}\hat{a}_{W^{\dagger}}]\rho_{.^{\Upsilon}}dt\] (84) + +As (84) is now an Itˆo stochastic differential equation, it obeys Itˆo calculus for which \(dW\) has zero mean and diff --git a/outputs/arxiv_math/2503.06944_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.06944_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..f649f7668aab8a4a93bb4ed76e0865f4c5cb44c6 --- /dev/null +++ b/outputs/arxiv_math/2503.06944_pg2_pg1_repeat1.md @@ -0,0 +1,45 @@ +Hr + +Hd + +RIS N reflecting elements + +1 2 ... Mr + +Downlink + +User + +BS + +![](figure) + +Fig. 1. An RIS-assisted point-to-point MIMO communication system. predesigned codebook for RIS-assisted point-to-point MIMO communications. In contrast to the direct codeword selection from the predesigned codebook, the proposed scheme designs a set of weights for each codeword in the codebook according to the corresponding output results, yielding a new RIS RC vector. For illustration, we contrast the proposed scheme to its existing channel estimation, passive beamforming and predesigned codebook counterparts in Table I. The simulation results demonstrate that the proposed scheme has improved performance compared to existing passive beamforming and codebook schemes. + +## II. SYSTEM MODEL + +We consider an RIS-assisted point-to-point MIMO system in a single cell as shown in Fig. 1, where a base station (BS) with \(M_{t}\) transmit antennas sends \(M_{s}\) data streams to a user with \(M_{r}\) antennas, with \(M_{s}\leq\min\{M_{t},M_{r}\}\). The RIS consists of \(N\) reflecting elements and is equipped with a smart controller capable of adjusting the RCs according to instructions from the BS. The signals from both the cascaded BS-RIS-user link and the direct BS-user link are superimposed at the user. We assume that the frequency-flat baseband equivalent channels spanning from the BS to the RIS, from the RIS to the user, and from the BS to the user are denoted by \(\mathbf{H}_{t}\in\mathbb{C}^{N\times M_{t}},\mathbf{H}_{r}\in \mathbb{C}^{M_{r}\times N}\text{and}\mathbf{H}_{d}\in\mathbb{C}^{M_{r}\times M_{t}}\), respectively. Let \(\varphi=[\varphi_{1},\varphi_{2},\cdots ,\varphi_{N}]^{T}\) represent the RIS RC vector, where \(\varphi_{n}=e^{j\theta_{n}}\) denotes the RC of the \(n\)th RIS element with phase shift \(\theta_{n}\), satisfying \(\theta_{n}\in[0,2\pi)\text{for}n=1,2,\cdots ,N\). Thus the composite end-to-end channel \(\mathbf{H}_{e}\in\mathbb{C}^{M_{r}\times M_{t}}\) from the BS to the user can be expressed as + +\[\mathbf{H}_{e}=\mathbf{H}_{d}+\mathbf{H}_{r}\text{diag}(\varphi)\mathbf{H}_{t}.\] (1) + +During the channel training process in the uplink phase, the user sends the pilot signal \(\mathbf{X}=[\mathbf{x}_{1},\mathbf{x}_{2},\cdots ,\mathbf{x}_{M_{r}}]^{T}\in\mathbb{C}^{M_{r}\times\tau}\) to the BS, where \(\mathbf{x}_{m}^{T}\in\mathbb{C}^{1\times\tau},m=1,2,\cdots ,M_{r}\) is the pilot loaded on the \(m\)th antenna at the user. The pilot matrix satisfies \(\Vert\mathbf{X}\Vert_{F}^{2}=\tau p_{u},\text{where}p_{u}\) is the average pilot power. As we consider a time-division duplexing protocol for both uplink as well as downlink transmissions and assume the channel’s reciprocity, the pilot signal received at the BS is given by + +\[\mathbf{Y}=\mathbf{H}_{e}^{H}\mathbf{X}+\mathbf{N}_{\text{BS}},\] (2) + +where \(\mathbf{N}_{\text{BS}}\in\mathbb{C}^{M_{t}\times\tau}\) denotes the noise matrix at the BS with an average noise power of \(\sigma_{\mathrm{B}\mathrm{S}}^{2}\), whose \(i\)th column vector follows \(\mathbf{n}_{\text{BS},i}\sim\mathcal{C}\mathcal{N}\left(\mathbf{0}_{M_{t}},\sigma_{\mathrm{B}\mathrm{S}}^{2}\mathbf{I}_{M_{t}}\right),\text{for}i=1,2,\cdots ,\tau\). We employ mutually orthogonal pilots, and the length of the pilot signal is designed such that \(\tau\geq M_{r}[12]\). Next, we consider the downlink of data transmission where the BS applies a baseband precoder \(\mathbf{W}\in\mathbb{C}^{M_{t}\times M_{s}}\) to transmit symbol \(\mathbf{s}\in\mathbb{C}^{M_{s}\times1},\text{with}\mathbb{E}\left\{\mathbf{ss}^{H}\right\}=\mathbf{I}_{M_{s}}\). Furthermore, the precoder satisfies \(\Vert\mathbf{W}\Vert_{F}^{2}\leq p_{d}\text{and}p_{d}\) is the total transmit power at the BS. Thus the received signal at the user is obtained as + +\[\mathbf{r}=\mathbf{H}_{e}\mathbf{Ws}+\mathbf{n}_{\text{UE}},\] (3) + +where \(\mathbf{n}_{\text{UE}}\in\mathbb{C}^{M_{r}\times1}\) is the noise at the user with an average noise power of \(\sigma_{\mathrm{U}\mathrm{E}}^{2}\), satisfying \(\mathbf{n}_{\text{UE}}\sim\mathcal{C}\mathcal{N}\left(\mathbf{0}_{M_{r}},\sigma_{\mathrm{U}\mathrm{E}}^{2}\mathbf{I}_{M_{r}}\right)\). Meanwhile, we adopt the Rician channel in this paper. + +\[\mathbf{H}_{r}= \sqrt{\beta_{r}} \left( \sqrt{\frac{F_{r}}{F_{r}+1}} \mathbf{H}_{r}^{\mathrm{L}\text{oS}}+ \sqrt{\frac{1}{F_{r}+1}} \mathbf{H}_{r}^{\mathrm{N}\text{LoS}}\right),(4)\] + +HNLoS \(r\) + +\[,^{\frac{\pi}{\lambda} (m_{t}-1)d_{\text{BS}}\sin(\delta)}m_{t}=\] + +where \(\beta_{r}\text{and}F_{r}\) are the path loss and the Rician factor of RIS-user channel, respectively; \(\mathbf{H}_{r}^{\mathrm{L}\text{oS}}\in\mathbb{C}^{M_{r}\times N}\) and \(\in\mathbb{C}^{M_{r}\times N}\) represent the line-of-sight (LoS) and the non-line-of-sight (NLoS) components of the RIS-user channel, respectively. The element on the \(m_{r}\)th row and the \(n\)th column of the NLoS matrix is modeled by Rayleigh fading, which follows \(\mathbf{H}_{rm_{r},n}^{\mathrm{N}\text{LoS}}\sim\mathcal{C}\mathcal{N}(0,1)\). Similarly, the BS-user channel and BS-RIS channel can be modeled by using (4). Moreover, we consider a uniform linear array (ULA) at the BS, a ULA at the user, and a uniform planar array (UPA) at the RIS. Let \(\mathbf{a}_{\text{BS}}(\delta)\in\mathbb{C}^{M_{t}\times1},\mathbf{a}_{\text{UE}}(\delta)\in\mathbb{C}^{M_{r}\times1}\) and \(\mathbf{a}_{\mathrm{R}}(\zeta,\gamma)\in\mathbb{C}^{N\times1}\) denote the steering vector of the BS, the user and the RIS, respectively. Specifically, the \(m_{t}\)th entry of aBS is denoted as \(e^{j^{\underline{2\pi}}} 1,2,\cdots ,M_{t},\text{where}d_{\text{BS}}\) denotes the element spacing of the BS, \(\lambda\) denotes the signal wavelength, and \(\delta\in[-\pi/2,\pi/2)\) denotes the angle of departure (AoD) or the angle of arrival (AoA). Similarly, the \(m_{r}\)th entry of aUE is denoted as \(e^{j^{\underline{2\pi}}}\) the element spacing of the user. The \(n\)th entry of aR is denoted as \(e^{j2\pi d_{\mathrm{R}}\sin(\gamma)[\lfloor \frac{n-1}{Nx} \rfloor\sin(\zeta)+((n-1)-\lfloor \frac{n-1}{Nx} \rfloor N_{x})\cos(\zeta)]/\lambda},n 1,2,\cdots ,N,\text{where}d_{\mathrm{R}}\) denotes the element spacing of the RIS. \(N_{x}\) is the number of elements deployed at each row of the RIS. \(\zeta\in[0,\pi)\text{and}\gamma\in[-\pi/2,\pi/2)\) denote the azimuth and elevation AoA/AoD, respectively. Thus, the LoS component of the \(\mathbf{H}_{t},\mathbf{H}_{r}\text{and}\mathbf{H}_{d}\) are given by \(\mathbf{a}_{\mathrm{R}}\left(\zeta_{t}^{\text{AoA}},\gamma_{t}^{\text{AoA}}\right)\mathbf{a}_{\text{BS}}\left(\delta_{t}^{\text{AoD}}\right)^{H}\) , \(\mathbf{a}_{\text{UE}}\left(\delta_{r}^{\text{AoA}}\right)\mathbf{a}_{\mathrm{R}}\left(\zeta_{r}^{\text{AoD}},\gamma_{r}^{\text{AoD}}\right)^{H}\text{and}\mathbf{a}_{\text{UE}}\left(\delta_{d}^{\text{AoA}}\right)\mathbf{a}_{\text{BS}}\left(\delta_{d}^{\text{AoD}}\right)^{H}\) , respectively, where \(\delta_{t}^{\text{AoD}},\zeta_{t}^{\text{AoA}}\text{and}\gamma_{t}^{\text{AoA}}\) represent the AoD, the azimuth and elevation AoA from the BS to the RIS, respectively; \(\delta_{r}^{\text{AoA}},\zeta_{r}^{\text{AoD}}\text{and}\gamma_{r}^{\text{AoD}}\) represent the AoA, the azimuth and elevation AoD from the RIS to the user, respectively; \(\delta_{d}^{\text{AoA}}\) and \(\delta_{d}^{\text{AoD}}\) represent the AoA and the AoD from the BS to the user, respectively. In the next section, unlike traditional passive beamforming and codebook schemes, the proposed scheme maximizes the channel capacity of the point-to-point MIMO systems by + +\(,^{(_{\overline{\lambda}}m_{r}-1)d_{\text{UE}}\sin(\delta)}m_{r}=1,2,\cdots ,M_{r},\text{where}d_{\text{UE}}\) denotes + += diff --git a/outputs/arxiv_math/2503.06958_pg1_pg1_repeat1.md b/outputs/arxiv_math/2503.06958_pg1_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..ecd085fcdde86484d375d69c82101425f35ab4b2 --- /dev/null +++ b/outputs/arxiv_math/2503.06958_pg1_pg1_repeat1.md @@ -0,0 +1,15 @@ +# ON THE STABILITY OF THE PENALTY FUNCTION FOR THE \(\mathbb{Z}^{2}-\mathbf{HARD}\) SQUARE SHIFT + +CHIHIRO OGURI AND MAO SHINODA + +Abstract. We investigate the stability of maximizing measures for a penalty function of a two-dimensional subshift of finite type, building on the work of Gonschorowski et al. [GQS21]. In the one-dimensional case, such measures remain stable under Lipschitz perturbations for any subshift of finite type. However, instability arises for a penalty function of the Robinson tiling, which is a two-dimensional subshift of finite type with no periodic point and zero entropy. This raises the question of whether stability persists in two-dimensional subshifts of finite type with positive topological entropy. In this paper, we address this question by studying the \(\mathbb{Z}^{2}-\)hard square shift, a well-known example of a two-dimensional subshift with positive entropy. Our main theorem establishes that, in contrast to previous results, a penalty function of the hard square shift remains stable under Lipschitz perturbations. + +## 1. Introduction + +Ergodic optimization is the study of maximizing measures. In its most basic form, let \(T:X\rightarrow X\) be a continuous map on a compact metric space \(X\) and for a continuous function \(\varphi:X\rightarrow\mathbb{R}\) we consider the maximum ergodic average + +\[\begin{aligned} \beta(\varphi)&=\sup_{\mathcal{M}_{T}(}\int\varphi d\mu \\ \mu&\in X) \end{aligned}\] + +where \(\mathcal{M}_{T}(X)\) is the space of \(T\) -invariant Borel probability measures on \(X\) endowed with the weak*-topology. An invariant measure which attains the maximum is called a maximizing measure for \(\varphi\) and denote by \(\mathcal{M}_{\text{max}}(\varphi)\) the set of maximizing measures for \(\varphi\). The stability of maximizing measures for a penalty function of a subshift of finite type was established by Gonschorowski et al. [GQS21]. A penalty function is defined on the forbidden set of a subshift of finite type, assigning a value of 0 to admissible local configurations near the origin and −1 otherwise (see §2 for more details). It is straightforward to see that every maximizing measure of a penalty function is supported on the given subshift of finite type. In the one-dimensional case, maximizing measures remain supported on the given subshift under Lipschitz perturbations for any subshift of finite type. However, in the two-dimensional case, there exists a subshift of finite type where this stability fails. In [GQS21], the authors highlight the difference between one and two dimensions, demonstrating that instability arises in a penalty function of the Robinson tiling, which is a two-dimensional subshift of finite type with no + +2010 Mathematics Subject Classification. Primary 37B51, 37B10, 37B25. diff --git a/outputs/arxiv_math/2503.06967_pg8_pg1_repeat1.md b/outputs/arxiv_math/2503.06967_pg8_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..9d1e39be1f46f95f1d91c89e42d478a0f1e362b4 --- /dev/null +++ b/outputs/arxiv_math/2503.06967_pg8_pg1_repeat1.md @@ -0,0 +1,27 @@ +(A3) There exists a constant \(c_{L}>0\) such that for all \(t\in[0,T],x_{0}^{\prime},x_{0}\in\mathbb{R}^{d_{0}},\alpha_{0}^{\prime},\alpha_{0}\in A_{0}\), \(x^{\prime},x\in\mathbb{R}^{d},\alpha^{\prime},\alpha\in A,\mu^{\prime},\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})\), we have: + +\[\begin{aligned} |(f_{0},g_{0})(t,x_{0}^{\prime},\alpha_{0}^{\prime},\mu^{\prime})-(f_{0},g_{0})(t,x_{0},\alpha_{0},\mu)| \\ &\leq_{\times}c_{L}\left(_{|}1_{x_{0}^{\prime}}+_{-}|x_{x_{0}^{\prime_{0}}}|_{|}+_{+}|x_{|\alpha^{0}}|_{\prime_{0}}+_{-}|_{\alpha}\alpha_{|_{0}^{0}}^{\prime}|_{+}+_{W}|\alpha_{(_{2}^{0}}|_{\mu}+_{,^{\prime}}M_{\mu)^{2}}\right)(_{.}\mu^{\prime})+M_{2}(\mu)\big) \end{aligned}\] + +and + +\[\begin{aligned} |(f,g)(t,x_{0},x^{\prime},\alpha_{0},\alpha^{\prime},\mu^{\prime})-(f,g)(t,x_{0},x,\alpha_{0},\alpha,\mu)| \\ &\leq_{\times}c_{L}\left(_{|}1_{x_{0}^{\prime}}+_{-}|x_{x_{0}^{\prime_{0}}}|_{|}+_{+}|x_{|x^{0_{\prime}}}|_{-}+_{x}|x_{|^{\prime}+}|+_{|\alpha}|x_{\prime_{0}}|_{-}+_{\alpha}|_{0}\alpha_{|^{\prime_{0}}+}|+_{|\alpha}|_{\prime}\alpha_{-^{0}}|_{\alpha}+_{|}|_{+}\alpha_{W}^{\prime}|+_{(_{2}}|_{\mu}\alpha_{\prime}|_{,}+_{\mu)}\right)M_{.^{2}}(\mu^{\prime})+M_{2}(\mu)\big) \end{aligned}\] + +(A4) The maps \((x_{0},\alpha_{0})\mapsto b_{0},b,f_{0}\) are differentiable, and the maps \((x_{0},\alpha_{0},\mu)\mapsto\partial_{x_{0}}(b_{0},b,f_{0})\) and \((x_{0},\alpha_{0},\mu)\mapsto\partial_{\alpha_{0}}(b_{0},b,f_{0})\) are continuous. The maps \((x,\alpha)\mapsto b,\sigma,f\) are differentiable, and the maps \((x,\alpha,\mu)\mapsto\partial_{x}(b,\sigma,f)\text{and}(x,\alpha,\mu)\mapsto\partial_{\alpha}(b,f)\) are continuous. The maps \(\mu\mapsto b_{0},b,f_{0}\) are L differentiable, and the maps \((x_{0},X,\alpha_{0})\mapsto\partial_{\mu}(b_{0},f_{0})(t,x_{0},\alpha_{0},\mathcal{L}(X))(X)\) are continuous. + +The map \(x_{0}\mapsto g_{0}\) is differentiable, and the map \((x_{0},\mu)\mapsto\partial_{x_{0}}g_{0}\) is continuous. The map \(\mu\mapsto g_{0}\) is L differentiable, and the map \((x_{0},X)\mapsto\partial_{\mu}g_{0}(x_{0},\mathcal{L}(X))(X)\) is continuous. The map \(x\mapsto g\) is differentiable, and the mapping \((x_{0},x,\mu)\mapsto\partial_{x}g\) is continuous. + +2. 2. Finite player setting. The MmMFG strategy is meant to serve as a mean field description of the following finite player framework, which is described through a stochastic differential game. Suppose we have \(N+1\) players, where the 0-th player corresponds to the major player, and players \(1,\ldots,N\) are the minor players. Let \(\boldsymbol{B}=(B^{1},\ldots,B^{N})\) denote a \(Nd-\)dimensional Brownian motion. Denote by \(\mathbb{A}_{i}\) the space of admissible strategies for the \(i-\)th minor player, namely strategies \(\alpha^{i}:[0,T]\times\Omega\rightarrow A\) which are progressively measurable w.r.t. the augmented filtration generated by \(B^{i},\text{and}\mathbb{E}[\int_{0}^{T}|\alpha_{t}^{i}|^{2}dt]<\infty\). The major player’s admissible strategies remain elements of \(\mathbb{A}_{0}\). Suppose the players possess the following state processes: + +\[\begin{aligned} dX_{0t}&=b_{0}(t,X_{0t},\alpha_{0t},\bar{\mu}_{\boldsymbol{X}_{t}})dt, \\ dX_{t}^{i}&=b(t,X_{0t},X_{t}^{i},\alpha_{0t},\alpha_{t}^{i},\bar{\mu}_{\boldsymbol{X}_{t}^{-i}})dt+\sigma(t,X_{t}^{i})dB_{t}^{i},X_{0}^{i}=x_{0}^{i},i=1,\ldots,N, \end{aligned}\] + +where \(X_{0}\) refers to the state process of the major player, and \(X^{i}\) refers to the state process of the \(i-\)th minor player. Moreover, we write \(\boldsymbol{X}:=(X^{1},\ldots,X^{N})\text{and}\boldsymbol{\alpha}:=(\alpha^{1},\ldots,\alpha^{N})\). + +The utility function for the major player is given by: + +\[J_{0}(\alpha_{0},\boldsymbol{\alpha})=\mathbb{E}\left[\int_{0}^{T}f_{0}(t,X_{0t},\alpha_{0t},\bar{\mu}_{\boldsymbol{X}_{t}})dt+g_{0}(X_{0T},\bar{\mu}_{\boldsymbol{X}_{T}})\right],\] + +and for the \(i-\)th minor player: + +\[J^{i}(\alpha_{0},\alpha^{i},\boldsymbol{\alpha}^{-i})=\mathbb{E}\left[\int_{0}^{T}f(t,X_{0t},X_{t}^{i},\alpha_{0t},\alpha_{t}^{i},\bar{\mu}_{\boldsymbol{X}_{t}^{-i}})dt+g(X_{0T},X_{T}^{i},\bar{\mu}_{\boldsymbol{X}_{T}^{-i}})\right],i=1,\ldots,N.\] + +The objective of the above \(N+1\) player game is for the major player to minimise \(\alpha_{0}\mapsto J_{0}(\alpha,\boldsymbol{\alpha})\) and the minor players to minimise \(\alpha^{i}\mapsto J^{i}(\alpha_{0},\alpha^{i},\boldsymbol{\alpha}^{-i})\) in the sense of Nash equilibrium. However, finding true Nash equilibria to this stochastic differential game is difficult. Put simply, this diff --git a/outputs/arxiv_math/2503.06969_pg11_pg1_repeat1.md b/outputs/arxiv_math/2503.06969_pg11_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..e520e8ca0703bdc0630b379ff337c94608171b73 --- /dev/null +++ b/outputs/arxiv_math/2503.06969_pg11_pg1_repeat1.md @@ -0,0 +1,27 @@ +We point out that there are no conditions on spaces and maps, except normality, making the theorem as broad as possible. Proof. We keep the same data as in Lemma 34, so liftcat\((_{f}^{\mathrm{o}\mathrm{p}}h)=\text{liftcat}_{j\circ f}^{\mathrm{o}\mathrm{p}}(\mu)\). On the other hand, Proposition 12 shows that liftcat\((_{j\circ f}\mu)=\text{liftcat}_{f}(h)\). So we can work with \(\mu\text{and}f^{\prime}=j\circ f\) rather than with \(h\text{and}f\). By definition, liftcat\((_{f}^{\prime}\mu)\leqslant n\) if and only if there exists a map \(\Phi:X\rightarrow T^{n}(\mu)\) making the following diagram homotopy commutative: + +\[T^{n}(\mu)\] + +\(\Phi\) + +\(t_{n}\) + +| X | | Zh | Zn+1 . | +| --- | --- | --- | --- | +| | f′ | ∆n+1 | h | + +From Remark 6, we regard the cofibrations \(\mu\text{and}t_{n}\) as inclusions and consider that \(T^{n}(\mu)=\bigcup_{i=0}^{n}p_{i}^{-1}(A)\subseteq Z_{h}^{n+1}\text{where}p_{i}\) is the \(i-\)th projection \(p_{i}:Z_{h}^{n+1}\rightarrow Z_{h}\). Assume we have the map \(\Phi\) and the homotopy \(H:X\times I\rightarrow Z_{h}^{n+1}\) between \(\Delta_{n+1}\circ f^{\prime}\text{and}t_{n}\circ\Phi\). Consider the open neighborhood \(\mathcal{N}\) of \(A\text{in}Z_{h}\) which is deformable in \(Z_{h}\) into \(A\) with some homotopy \(G\) rel \(A.\text{For}0\leqslant i\leqslant n\), let \(h_{i}=p_{i}\circ t_{n}\circ\Phi\text{and}U_{i}=h_{i}^{-1}(\mathcal{N})\). As \((t_{n}\circ\Phi)(X)\subseteq T^{n}(\mu)=\bigcup_{i=0}^{n}p_{i}^{-1}(A)\), we have \(X=\bigcup_{i=0}^{n}U_{i}\). Now we can define homotopies \(L_{i}:U_{i}\times I\rightarrow Z_{h}\) by: + +\[L_{i}(u,t)= \begin{cases} p_{i}(H(u,2t)) & \text{if}0\leqslant t\leqslant \tfrac{1}{2} \\ G(h_{i}(u),2t-1) & \text{if} \tfrac{1}{2} \leqslant t\leqslant 1 . \end{cases}\] + +Each \(L_{i}\) is well defined because \(p_{i}(H(u,1))=p_{i}((t_{n}\circ\Phi)(u))=h_{i}(u)=G(h_{i}(u),0)\). We have \(L_{i}(u,0)=f^{\prime}(u)\text{and}L_{i}(u,1)\in A\), so \(L_{i}\) is a homotopy between \(f^{\prime}|_{U_{i}}\) and \(\mu\circ l_{i}\text{where}l_{i}=L_{i}(-,1)\). Conversely assume we have an open cover \((U_{i})_{0\leqslant i\leqslant n}\) of \(X\) and homotopies \(K_{i}:U_{i}\times I\rightarrow Z_{h}\text{with}K_{i}(u,0)=f^{\prime}(u)\text{and}K_{i}(u,1)\in A\). Choose a closed subset \(F_{i}\subset U_{i}\) (possibly the empty set) in each \(U_{i}.\text{If}X\) is normal, we have two more open covers \((V_{i})_{0\leqslant i\leqslant n}\text{and}(W_{i})_{0\leqslant i\leqslant n}\) of \(X\) such that + +\[F_{i}\subset W_{i}\subset \overline{W_{i}} \subset V_{i}\subset \overline{V_{i}} \subset U_{i}.\] + +Moreover, as \(\overline{W_{i}} \text{and}∁V_{i}=X\setminus V_{i}\) are disjoint closed subspaces of the normal space \(X\), we have an Urysohn map \(\varphi_{i}:X\rightarrow I\text{with}\varphi_{i}( \overline{W_{i}} )=1\text{and}\varphi_{i}(∁V_{i})=0\). Now we define \(H_{i}:X\times I\rightarrow Z_{h}\) by: + +\[H_{i}(x,t)= \begin{cases} f^{\prime}(x) & \text{if}x\in∁ \overline{V_{i}} \\ K_{i}(x,t\varphi_{i}(x)) & \text{if}x\in U_{i}. \end{cases}\] + +Each \(H_{i}\) is well defined because if \(x\in∁ \overline{V_{i}} \cap U_{i},\text{then}K_{i}(x,t\varphi_{i}(x))=K_{i}(x,0)= f^{\prime}(x)\). Let \(H=(H_{0},\ldots,H_{n}):X\times I\rightarrow Z_{h}^{n+1}\text{and}\Phi=H(-,1)\). We have \(H(-,0)= \Delta_{n+1}\circ f^{\prime}\). On the other hand, for any \(x\in X\), there is (at least) one i such that \(x\in W_{i}\), so \(\varphi_{i}(x)=1\text{and}H_{i}(x,1)=K_{i}(x,1)\in A\); this means that \(\Phi(X)\subseteq T^{n}(\mu)\). □ + +Thus \(H\) is a homotopy between \(\Delta_{n+1}\circ f^{\prime}\text{and}t_{n}\circ\Phi\). diff --git a/outputs/arxiv_math/2503.06979_pg8_pg1_repeat1.md b/outputs/arxiv_math/2503.06979_pg8_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..623fd603fa2594f9f8fed25b6eef0eeb0a57b780 --- /dev/null +++ b/outputs/arxiv_math/2503.06979_pg8_pg1_repeat1.md @@ -0,0 +1,23 @@ +Lemma 2.8, it follows that \(\tilde{K}\cap\widetilde{P}_{1}✂\widetilde{P}_{1}\). As shown in Equation 3.1, the projection of \(y\) is given by \(([\mu,\lambda][\mu,\lambda^{2}],1)\), which lies in \(\tilde{K}\cap\widetilde{P}_{1}\). By Lemma 3.3, we have \(P=\langle[\mu,\lambda],[\mu,\lambda][\mu,\lambda^{2}]\rangle\), implying that \(\widetilde{P}_{1}/(\tilde{K}\cap\widetilde{P}_{1})\) is cyclic. Since \(P\) is perfect, it follows that \(\tilde{K}\cap\widetilde{P}_{1}=\widetilde{P}_{1}\). Consequently, we conclude that \(\tilde{K}=\widetilde{P}_{1}\times\widetilde{P}_{3}\). + +To streamline the proofs of the subsequent lemmas, for any subgroup \(Q\) of \(P\) , we consistently denote by \(Q_{i}\) the corresponding subgroup in the \(i-\)th direct factor \(P_{i}\) and denote by \(Q^{5}\) their direct product \(Q_{1}\times\ldots Q_{5}\). + +## Lemma 3.5. The graph \(\Gamma\) is a connected graph. + +Proof. Let \(P\) be a minimal counterexample for the statement \(X\neq G\). Then for any nontrivial normal subgroup \(M\) of \(P\) , the quotient group \(P/M\) is perfect and generated by \(\mu M\text{and}\lambda M\). Let \(L=M^{5}.\text{Then}X/L\cong (P/M)≀\) A5, and by assumption, \(X/L=GL/L\). Therefore, \(X=GL\), and it follows that + +\[N=GL\cap N=(G\cap N)L=KL.\] + +We claim that \(Z=Z(P)\), the center of \(P\) , is trivial. If \(Z\neq 1\), From the discussion above, we have \(N=Z^{5}K\). Thus, \(N=N^{\prime}=(Z^{5}K)^{\prime}=K\), and hence \(N=K\). Therefore, \(X=G\), which contradicts that \(P\) is a counterexample. Hence, our claim holds. + +## Step 1: the projection of \(K\mathbf{on}P_{1}\times P_{2}\times P_{3}\) is surjective + +Assume the projection of \(K\text{on}P_{1}\times P_{2}\times P_{3}\) is not surjective. Let \(U=P_{1}\times P_{2}\times P_{3}\) and \(V=P_{4}\times P_{5}\) and let \(\tilde{K}\text{and}\tilde{P}_{i}\) be the projections of \(K\text{and}P_{i}\text{on}U,\text{where}1\leq i\leq3.\text{Then}\tilde{K}=KV/V,\tilde{P}_{i}=P_{i}V/V\text{and}KV0,\] (1) + +where the dot denotes the time derivative. The solution of this equation, + +\[n(t)= \frac{n_{0}}{1-\beta n_{0}t} ,\] (2) + +where \(n_{0}\equiv n(t=0)>0\) is the initial condition, blows up in a finite time \(T=(\beta n_{0})^{-1}\). In the presence of noise the blowup time becomes a random quantity, and it is interesting to determine its statistics. One way of addressing this class of problems is to interpret Eq. (1) as the equation of motion of an overdamped particle, with the coordinate \(n=n(t)\), in a repulsive potential \(V(n)\sim-n^{3}\). By adding a noise term to Eq. (1), one turns this mean-field equation into a Langevin equation: + +\[\dot{x}(t)=\beta x^{2}(t)+\eta(t),\] (3) + +where \(\eta(t)\) is white Gaussian noise, and we have replaced \(n\) by \(x\). The finite-time blowup statistics for this model and its extensions has been recently studied in Ref. [2], see also Refs. [3–5]. Here we take a different approach by observing that Eq. (1) also provides a mean-field description to the simple Markovian stochastic model of asexual binary reproduction \(2A\rightarrow3A\), where two identical \(A-\text{individuals}\), which we will call particles, give birth to a third one. The master equation for this stochastic process is + +\[\dot{P}_{n}(t)= \frac{\beta(n-1)(n-2)}{2} P_{n-1}(t)- \frac{\beta n(n-1)}{2} P_{n}(t),(4)\] + +where \(P_{n}(t)\) is the probability of observing \(n\) particles at time \(t\). We suppose that there are \(m\geq2\) particles at \(t=0\), so the initial condition is + +\[P_{n}(t=0)=\delta_{nm}.\] (5) + +By rescaling time, \(\beta t\rightarrow t\), we can get rid of the rate coefficient \(\beta\). Therefore, from now on we set \(\beta=1\). In Sec. II we determine exactly the average blowup time and the probability distribution of the blowup time for the \(2A\rightarrow3A\) model. We also determine several first moments of the distribution and provide an additional insight into the short-time tail of the probability distribution of the blowup time by employing the optimal fluctuation method (OFM). In Sec. III we briefly discuss our main results and compare them with those for the Langevin equation (3), obtained in Ref. [2]. + +### II. STATISTICS OF BLOWUP TIME + +### A. Average blowup time + +As a warmup, let us calculate the average blowup time \(\Theta(m)\) as a function of the initial number of particles \(m\). This quantity can be determined from the backward master equation, see, e.g., Ref. [6]. For the process \(2A\rightarrow3A\) the backward master equation is + +\[r(m)[\Theta(m+1)-\Theta(m)]=-1,\] (6) + +where \(r(m)=m(m-1)/2\). The blowup, which can be viewed as the first passage to infinity, is described by the absorbing boundary condition [6] + +\[\Theta(m\rightarrow+\infty)=0.\] (7) + +The solution of the problem (6) and (7) is remarkably simple: + +\[\Theta(m)= \frac{2}{m-1} .\] (8) + +In particular, for \(m=2\) (there are exactly two particles at \(t=0)\) we obtain \(\Theta(2)\equiv\langle T\rangle=2\). It is twice as large as the mean-field result \(t_{0}=1\) for the same initial number of particles. Notice that the average blowup time (8) falls off quite slowly as a function of \(m\). diff --git a/outputs/arxiv_math/2503.07088_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.07088_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..2003f109a5c59a5951c29ea07fbc298f9e7d9468 --- /dev/null +++ b/outputs/arxiv_math/2503.07088_pg3_pg1_repeat1.md @@ -0,0 +1,39 @@ +## 2. Preliminaries + +In this section, we briefly recall main definitions, notations of some basics of \(q-\)calculus that will be useful to us in this paper. These notions are presented in [1, 3, 6] among many others. + +Throughout this paper we suppose \(00\forall x\in\mathbb{R}^{n}\setminus\{0\},(12)\] + +for all systems \((A,B)\) compatible with the data. In the earlier work \([8],H(x)\) is taken to be equal to the identity matrix, which implies that \(Z(x)=F(x)\). In contrast, [9] considers more general \(Z(x)\) satisfying (10). This strategy is appealing because it leads to data-based linear matrix inequalities for control design. Unfortunately, however, the method also has some major limitations. + +1. The matrix \(\frac{\partial Z}{\partial x} (x)\) must have full row rank for all \(x\in\mathbb{R}^{n}\setminus\{0\}\). + +Indeed, suppose that there exists a nonzero \(x\) such that \(\frac{\partial Z}{\partial x} (x)\) does not have full row rank. Then \(\Theta(x)\) is singular, which implies that (12) does not hold. Note that the full row rank condition can only hold if \(p\leq n\), i.e., the number of polynomials in \(Z\) is less than or equal to the state-space dimension of the system. This limits the class of Lyapunov functions of the form (11) that can be considered by the diff --git a/outputs/arxiv_math/2503.07105_pg18_pg1_repeat1.md b/outputs/arxiv_math/2503.07105_pg18_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..4cc73a6b474c39993687b2ede67ca2d22c8f2ba4 --- /dev/null +++ b/outputs/arxiv_math/2503.07105_pg18_pg1_repeat1.md @@ -0,0 +1,27 @@ +By summing up these integrals, some terms cancel out and we get + +\[\begin{aligned} \int_{P}x^{\top}vdx&= \frac{v_{2}(q_{i,2}+q_{i-1,2})+v_{1}(q_{i,1}+q_{i-1,1})}{6} (q_{i-1,1}q_{i,2}-q_{i,1}q_{i-1,2}) \\ &=\sum_{i=^{m}1}^{i=^{m}1} \frac{v^{\top}(q_{i-1}+q_{i})}{3} \frac{q_{i-1,1}q_{i,2}-q_{i,1}q_{i-1,2}}{2} . \end{aligned}\] + +For the calculation of \(W\) , we sum up these integrals with \(P=D_{i}(w)\text{and}v=a_{i}\) for all \(i\in\{1,\ldots,n\}\). For practical reasons we write this in terms of the edges, where \(E\) denotes all edges. Let \(e\in E\) be an edge with vertices \(q_{1}^{e},q_{2}^{e}\). Let \(a_{l},a_{r}\in\mathbb{R}^{2}\) be the vectors \(a_{i}\) on the left/right side of \(e\). We set \(a_{l}=0\text{or}a_{r}=0\) for the non-existing side of boundary edges. Previously, every edge was considered twice, thus we can write + +\[\sum_{i=1}^{n}\int_{D_{i}}x^{\top}a_{i}dx= \frac{1}{3} \sum_{e\in E}\left((a_{l}-a_{r})^{\top}(q_{1}^{e}+q_{2}^{e})\right) \frac{q_{1,1}^{e}q_{2,2}^{e}-q_{2,1}^{e}q_{1,2}^{e}}{2} .\] + +For an inner edge, the addend can be rewritten as + +\[2(w_{l}-w_{r}) \frac{q_{1,1}^{e}q_{2,2}^{e}-q_{2,1}^{e}q_{1,2}^{e}}{2} .\] + +## 4 Numerical algorithms + +In this section, we address numerical algorithms for the solution of (Pw). + +### 4.1 Fixed-point iteration + +As an optimality condition for (Pw) we derived \(r(w)=0\), cf. Theorem 2.6. This can be transformed into the fixed-point equation \(w=w+\tau r(w)\text{with}\tau>0\). A possible algorithm for the solution is the fixed-point iteration \(w_{k+1}:=w_{k}+\tau_{k}r(w_{k})\) with appropriate step sizes \(\tau_{k}>0\). It is not clear, how to choose these step sizes. We will see in Lemma 4.2 below, that the direction \(r(w_{k})\) is a descent direction for \(J\). However, since \(J\) is not \(C^{1}\), Armijo step sizes could become arbitrarily small and the usual convergence proof does not work. Similarly, constant (but small) step sizes might not give a decrease of \(J\). Instead, we can view the above iteration scheme as a discretization of an ODE. Indeed, the above update formula yields \((w_{k+1}-w_{k})/\tau_{k}=r(w_{k})\). This is an explicit Euler scheme for the ODE + +\[w_{w^{\prime}(}(_{0}t_{)})_{=}=r_{w}(_{0}w_{.}(t)),_{(\text{ODE}(\mathrm{w}))}\] + +We state some simple properties for (ODE(w)). + +Lemma 4.1. The following properties hold. + +(i) The function \(r\) is globally Lipschitz continuous with constant \(L>0\). diff --git a/outputs/arxiv_math/2503.07122_pg4_pg1_repeat1.md b/outputs/arxiv_math/2503.07122_pg4_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..5b962a59f2bae97b6e7da7f0abc972a8285ef361 --- /dev/null +++ b/outputs/arxiv_math/2503.07122_pg4_pg1_repeat1.md @@ -0,0 +1,25 @@ +Assumption 1.5. The growth functions \(\Theta\) is such that the associated generalized modulus of continuity \(\varphi_{\Theta}:[0,+\infty)\rightarrow[0,+\infty)\) is continuous, where + +\[\varphi_{\Theta}(s):= \begin{cases} 0 & ifs=0, \\ s|\log s|\Theta(|\log s|) & if00\text{for}n\) sufficiently large and it follows from the definition of \(t_{n}\) that + +\[\begin{aligned} \tilde{} \\ \forall t&\in[0,t_{n}),\forall x\in tW,u_{n}(t,x)> \underline{u} (x), \end{aligned}\] (4.11) + +and, moreover, being \(t_{n}\tilde{W}\) compact, that there exists \(x_{n}\in t_{n}\tilde{W}\) such that + +\[u_{n}(t_{n},x_{n})= \underline{u} (x_{n}).\] + +Recall that \(\underline{u}\) is a strict subsolution, hence the parabolic strong maximum principle necessarily implies that \(x_{n}\in\partial \overline{(} t_{n}\tilde{W})\), that is, \(x_{n}/t_{n}\in\partial\tilde{W}\). Consider now \(h_{n}\in\mathbb{Z}^{N}\text{such that}\xi_{n}:=x_{n}-h_{n}\in[0,1)^{N}\). We define + +\[\tilde{u}_{n}(t,x):=u_{n}(t_{n}+t,h_{n}+x).\] + +Up to extraction of a subsequence, the following limits exist: + +\[\xi_{n}\rightarrow\xi_{\infty}\in[0,1]^{N},x_{n}/t_{n}\rightarrow\zeta\in\partial\tilde{W}.\] + +Also, always up to subsequences, by standard parabolic estimates and spatial periodicity of the equation, the functions \(\tilde{u}_{n}\) converge to \(\tilde{u}_{\infty}\), an entire solution of (1.1) which fulfils by construction (and by periodicity of \(\underline{u} )\) + +\[\tilde{u}_{\infty}(0,\xi_{\infty})= \underline{u} (\xi_{\infty}).\] (4.12) + +Moreover, (4.11) rewrites for the \(\tilde{u}_{n}\) as + +\[\begin{aligned} \tilde{} \\ \forall t&\in[-t_{n},0),\forall x\in(t_{n}+t)W-\{x_{n}\},\tilde{u}_{n}(t,x+\xi_{n})> \underline{u} (x+\xi_{n}). \end{aligned}\] (4.13) + +We assert that this entails + +\[\forall t\leq0,\forall x\cdot\nu\leq\left(1- \frac{\varepsilon}{3} \right)c_{1}(\nu)t,\tilde{u}_{\infty}(t,x+\xi_{\infty})\geq \underline{u} (x+\xi_{\infty}),\] (4.14) + +where \(\nu\) is the outward unit normal vector to \(\tilde{W}\) at the point \(\zeta\) and, we recall, \(c_{1}(\nu)\) is the speed of the uppermost front of the terrace \(\mathcal{T}^{\nu}\) in the direction \(\nu\). The first crucial observation to derive (4.14) is that the \(t-\)dependent sets \((t_{n}+t)\tilde{W}\) expand at a given boundary point \((t_{n}+t)\tilde{w}(e)e\) with the (positive) constant normal speed \(\tilde{w}(e)e\cdot\tilde{\nu},\text{where}\tilde{\nu}\) is the outward normal at that point, hence \(e\cdot\tilde{\nu}>0\). The second observation is that \(0\in\partial\left(t_{n}\tilde{W}-\{x_{n}\}\right)\), for any \(n\in\mathbb{N}\), and that the normal at that point converges to \(\nu\). The last one is that, because \(\tilde{W}\) is compact and smooth, it satisfies uniform interior and exterior sphere conditions of some radius \(\rho>0\) on the boundary, whence its dilation \((t_{n}+t)\tilde{W}\) fulfils these conditions with radius \((t_{n}+t)\rho\), which for any \(t\) tends to \(+\infty\text{as}n\rightarrow+\infty\). This means that \((t_{n}+t)\tilde{W}\) “flattens” to a half-space around each of its boundary points as \(n\rightarrow+\infty\). These geometric observations are made rigorous in diff --git a/outputs/arxiv_math/2503.07145_pg13_pg1_repeat1.md b/outputs/arxiv_math/2503.07145_pg13_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..d838620f53fd4444db36b67eb75e5b26f4ff6afb --- /dev/null +++ b/outputs/arxiv_math/2503.07145_pg13_pg1_repeat1.md @@ -0,0 +1,41 @@ +Theorem 3.6. Let \(\mathbf{x}≼(0,\ldots,n-1)\) be an irreducible score, so that \(C\cap(0,\infty)^{n\times n}\neq \emptyset\). Then + +\[\lim_{k\rightarrow\infty}M=\operatorname*{arg\,min}_{L\in C}^{k}D(L,M)=\operatorname*{arg\,min}_{L\in C}^{0}H(L).\] (3.9) + +Proof. Note that \(\text{dom}(h)=\mathbb{R}_{+},\text{idom}(h)=(0,\infty)\text{and bdom}(h)=\{0\}\). It is easy to check: + +\((1)h\) is a proper convex function. Moreover, \(h\) is closed because \(\{x\in\text{dom}(h):h(x)\leq\alpha\}\) is closed for all \(\alpha\in\mathbb{R}\). + +\((2)h\) is Legendre because + + \(\bullet h\) is differentiable on \(\text{idom}(h)\); + + \(\bullet\lim_{t\rightarrow0+}h^{\prime}(x+t(y-x))(y-x)=-\infty\text{for all}x\in\text{bdom}(f)\text{and}y\in\text{idom}(h)\); + + \(\bullet h\) is strictly convex on \(\text{idom}(h)\). + +\((3)h\) is co-finite because \(\lim_{t\rightarrow\infty} \frac{h(tx)}{t} =\infty\text{for all}x\neq 0\). + +\((4)h\) is very strictly convex because \(h^{\prime\prime}(x)>0\text{for all}x\in\text{idom}(h)\). + +- + +Since \(C_{1},C_{2},C_{3}\) are all affine subsets, it suffices to apply [7, Theorem 4.3] to conclude. + +See also [45] for recent development on the convergence rate of Bregman’s iteration under further technical assumptions, which we do not pursue here. Next we propose a computational scheme inspired by Theorem 3.6. The key is to compute numerically, for each \(M\in(0,\infty)^{n\times n}\), its Bregman projection on \(C_{k}\). We distinguish three cases: + +\(\bullet k=1\): We introduce the Lagrange multiplier \(\lambda=(\lambda_{1},\ldots,\lambda_{n})\in\mathbb{R}^{n}\), and set + +\(\Phi(L,\lambda):=D(L,M)+\sum_{i=1}^{n}\lambda_{i}\left(\sum_{j=1}^{n}l_{ij}-1\right)\). Differentiating \(\Phi\) with respect to \(m_{ij}\text{and}\lambda_{i}\) yields \(\partial_{l_{ij}}\Phi=\log(l_{ij})-\log(m_{ij})+\lambda_{i}\text{and}\partial_{\lambda_{i}}\Phi=\sum_{j=1}^{n}l_{ij}-1\). By setting these to zero, we get + +\[\operatorname*{arg\,min}_{L\in C_{1}}D(L,M)=\left( \frac{m_{ij}}{\sum_{k=1}^{n}m_{ik}} \right)_{1\leq i,j\leq n}.\] (3.10) + +\(\bullet k=2\): The same reasoning as in the previous case yields: + +\[\operatorname*{arg\,min}_{L\in C_{2}}D(L,M)=\left( \frac{m_{ij}}{\sum_{k=1}^{n}m_{kj}} \right)_{1\leq i,j\leq n}.\] (3.11) + +\(\bullet k=3\): Define \(\Phi(L,\lambda):=D(L,M)+\sum_{i=1}^{n}\lambda_{i}\left(\sum_{j=1}^{n}(j-1)l_{ij}-x_{i}\right)\), and differentiate \(\Phi\) with respect to \(m_{ij}\text{and}\lambda_{i}\) yields \(\partial_{l_{ij}}\Phi=\log(l_{ij})-\log(m_{ij})+\lambda_{i}(j-1)\) and \(\partial_{\lambda_{i}}\Phi=\sum_{j=1}^{n}(j-1)l_{ij}-x_{i}\). Setting these to zero yields + +\[\operatorname*{arg\,min}_{L\in C_{3}}D(L,M)=\left(m_{ij}r_{ij}^{j-1}\right)_{1\leq i,j\leq n},\] (3.12) + +where \(r_{ij}\) is the unique positive root to the polynomial equation \(\sum_{j=1}^{n}(j-1)m_{ij}r^{j-1}- x_{i}=0\). Let \(f(r):=\sum_{j=1}^{n}(j-1)m_{ij}r^{j-1}-x_{i}\). It is easy to see that \(f\) is strictly increasing on \([0,\infty)\text{with}f(0)\leq0\text{and}\lim_{r\rightarrow\infty}f(r)=+\infty\). Thus, it is easy (and quick) to find a numerical root of \(f\text{on}[0,\infty)\) by Newton’s method. diff --git a/outputs/arxiv_math/2503.07147_pg16_pg1_repeat1.md b/outputs/arxiv_math/2503.07147_pg16_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..399e73b87597cc9059eea2317d1f9876c72bf139 --- /dev/null +++ b/outputs/arxiv_math/2503.07147_pg16_pg1_repeat1.md @@ -0,0 +1,33 @@ +Lemma 5.3. There is an \(n_{0}\) such that the following holds for all \(n\geq n_{0},2^{-9}<\varepsilon<1,c>0,r,t\geq(\log n)^{2}\) and \(s\geq20rt\). Let \(G\) be an \(n-\text{vertex}(\varepsilon,c,s)-expander\), let \(U\subseteq V(G)\) satisfy \(|U|\leq2n/3.\text{Then},inG\) we can find either + +(a) \(\frac{|U|}{10r}\) pairwise vertex-disjoint stars of size \(t\), whose centers are in \(U\) and whose leaves are in \(V(G)-U\), or + +(b) a bipartite subgraph \(H\) with vertex classes \(U\text{and}X\subseteq V(G)-U\) such that + +\(\varepsilon|U|\) + + - \(|X|\geq\) + +2(log \(n)^{c}\) and + + - every vertex in \(X\) has degree at least \(rinH\) and every vertex in \(U\) has degree at most \(2tinH\). + +Proof. Take a maximal collection \(\mathcal{C}\) of pairwise vertex-disjoint stars in \(G\text{with}t\) leaves, centres in \(U\) and leaves outside of \(U\). Let \(C\subseteq U\) be the set of centres of these stars and \(L\subseteq V(G)-U\) be the set consisting of all their leaves. Suppose a) does not hold. Then we can assume that \(|C|\leq \frac{|U|}{10r}\) and thus \(|L|=|C|\cdot t\leq \frac{|U|}{10r} \cdot t\), and, by the maximality of \(\mathcal{C}\), that there is no vertex in \(U-C\) with at least \(t\) neighbours in \(G\text{in}V(G)-(U\cup L)\). Thus, + +\[|N_{G}(U-C)|\leq|C|+|L|+|U-C|\cdot t\leq \frac{|U|}{10r} +|C|\cdot t+|U-C|\cdot t<2|U|\cdot t.\] (6) + +We now construct a set \(X\subseteq V(G)-U\) and a bipartite subgraph \(H\) with vertex classes \(U\text{and}X\) using the following process, starting with \(X_{0}=\emptyset\) and setting \(H_{0}\) to be the graph with vertex set \(U\cup X_{0}\) and no edges. Let \(k=|V(G)-U|\) and label the vertices of \(V(G)-U\) arbitrarily as \(v_{1},\ldots,v_{k}\). For each \(i\geq1\), if possible, pick a star \(S_{i}\text{in}G\) with centre \(v_{i}\text{and}r\) leaves in \(U\) such that the vertices in \(U\) in the graph \(H_{i-1}\cup S_{i}\) have degree at most \(2t\), and let \(H_{i}=H_{i-1}\cup S_{i}\text{and}X_{i}=X_{i-1}\cup\{v_{i}\}\), while otherwise we set \(H_{i}=H_{i-1}\text{and}X_{i}=X_{i-1}\). Finally, let \(H=H_{k}\) and \(X=X_{k}=V(H_{k})-U\). We will now show that b) holds for this choice of \(H\) (with vertex classes \(U\text{and}X)\). Firstly, observe that every vertex of \(U\) has degree at most \(2t\text{in}H_{i}\) for each \(i\in[k]\) by construction, and that every vertex \(v_{i}\text{in}X\) has degree exactly \(r\text{in}H\), so the second condition in b) holds. Thus, we only need to show that \(|X|\geq \frac{\varepsilon|U|}{2(\log n)^{c}}\) holds. To see this, let \(U^{\prime}\) be the set of vertices in \(U-C\) with degree exactly \(2t\text{in}H\). As each vertex in \(U-C\) has fewer than \(t\) neighbours in \(G\text{in}X-L\) (due to the maximality of the collection of stars \(\mathcal{C})\), the vertices in \(U^{\prime}\) must have at least \(t\) neighbours in \(H\text{in}X\cap L\). As each vertex in \(X\cap L\) has \(r\) neighbours in \(H\), we have + +Let \(B=C\cup U^{\prime}\), so that + +\[|U^{\prime}|\leq \frac{r|X\cap L|}{t} \leq \frac{r}{t} \cdot|L|\leq \frac{r}{t} \cdot \frac{|U|\cdot t}{10r} = \frac{|U|}{10} .\] + +\[|B|\leq \frac{|U|}{10r} + \frac{|U|}{10} \leq \frac{|U|}{2} ,\] + +and, thus, \(|U-B|\geq \frac{|U|}{2}\) . Then, by Proposition 5.2 applied to \(U-B\text{with}d=r\), we have either \(|N_{G}(U-B)|\geq \frac{s|U-B|}{2r} \text{or}|N_{G,r}(U-B)|\geq \frac{\varepsilon|U-B|}{(\log n)^{c}}\) . As + +\[\frac{s|U-B|}{2r} \geq \frac{s|U|}{4r} \geq5t|U|,\] + +the former inequality contradicts (6), so we have that \(|N_{G,r}(U-B)|\geq \frac{\varepsilon|U-B|}{(\log n)^{c}}\) . Every vertex \(v_{i}\text{in}N_{G,r}(U-B)\) has at least \(r\) neighbours in \(G\text{in}U-B\), and vertices of \(U-B\) must all have degree strictly less than \(2t\text{in}H\) (as they are not in \(U^{\prime})\). This implies that every \(v_{i}\text{in}N_{G,r}(U-B)\), satisfies \(v_{i}\in X\), since we could add it along with some \(r\) of its neighbours while constructing \(H\). Hence, \(N_{G,r}(U-B)\subseteq X\), and + +\[|X|\geq|N_{G,r}(U-B)|\geq \frac{\varepsilon|U-B|}{(\log n)^{c}} \geq \frac{\varepsilon|U|}{2(\log n)^{c}} ,\] diff --git a/outputs/arxiv_math/2503.07149_pg18_pg1_repeat1.md b/outputs/arxiv_math/2503.07149_pg18_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..2a3ee138b31926fe2fe0338f591fd808cf7724c3 --- /dev/null +++ b/outputs/arxiv_math/2503.07149_pg18_pg1_repeat1.md @@ -0,0 +1 @@ +REC, both entities stop charging and start exporting their energy production once the first DR request horizon begins (see Fig. 3). During this period, Entity 1 exports 294 kWh, while Entity 2 exports 126 kWh. After the first DR request ends, entities resume charging due to low energy price and later export to the grid during the period with highest energy price. Between 15:15 and 17:00 both entities start charging again, so that their storage systems can be discharged during the second DR request, as expected. During the second DR request, this strategy enables Entities 1 and 2 to provide 729 kWh and 318 kWh, respectively. From Fig. 4 it is apparent that both \(\delta_{1}\text{and}\delta_{2}\) are higher when the objective function is set to \(H^{E}\) compared to \(H^{M}\). This result is expected, as \(H^{E}\) is designed to maximize the total revenue for the entities, whereas \(H^{M}\) focuses on maximizing the revenue of the REC manager, which naturally leads to lower extra profits for the entities. Fig. 4 also demonstrates a fair redistribution of rewards among the entities, such that the additional profit be proportional to their respective individual revenuse when operating independently, i.e., outside the REC. This ensures that the additional profits \(\delta_{1}\text{and}\delta_{2}\) are allocated equitably, reflecting the contribution provided by each entity to the REC. Notice that, when the objective function \(H^{M}\) is used, there are some days such that \(\delta_{1}=\delta_{2}=0\), that is, the profit of the entities joining the REC equals the profit they would gain if acting individually outside the REC. Regarding the DR rewards assigned to the entities, Fig. 5 shows that when optimizing \(H^{M}\) the total community reward is no less than that obtained by using \(H^{E}\), as expected. However, even if optimizing \(H^{M}\), the REC is in general unable to provide the upper DR request energy bounds (i.e., \(\gamma_{1}^{*}+\gamma_{2}^{*}< \overline{\gamma}_{1} + \overline{\gamma}_{2} )\). Clearly, the sum of the rewards assigned to the two entities is less than the total reward received by the community, since a fraction of it is retained by the REC manager according to (30). The sensitivity of the proposed method with respect to the DR reward bound \(\overline{\gamma}_{j}\) is explored in Table 4. When the DR reward bound \(\overline{\gamma}_{j}\) is increased from 40 to 100e, the achieved DR rewards \(\gamma_{1}^{*}+\gamma_{2}^{*}\) under both objectives \(H^{E}\text{and}H^{M}\) are the same. This basically means that the DR reward is large enough to make fulfillment of DR requests always advantageous regardless of all costs and energy losses arising when operating storage. So, any further increase in \(\overline{\gamma}_{j}\) will provide the same BESS control commands and consequently the same amount of energy injected into the grid. Concerning Example 2, in Fig. 7 one can observe that during the DR periods entities discharge their storage systems to increase the injected energy into the grid. Thanks to this operation, the REC receives a monetary reward which can be shared among entities, allowing them to substantially increase their profit compared to their baseline, see Tables 6 and 7. Regarding the two considered objective functions \(H^{E}\text{and}H^{M}\), they yield behaviors similar to those in Example 1, favouring the entity profit and the total DR reward, respectively. diff --git a/outputs/arxiv_math/2503.07156_pg35_pg1_repeat1.md b/outputs/arxiv_math/2503.07156_pg35_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..4a6f00d4f497e95b14fce9c5595fa76ebb80632f --- /dev/null +++ b/outputs/arxiv_math/2503.07156_pg35_pg1_repeat1.md @@ -0,0 +1,35 @@ +Finally, we define the nonlinear mapping \(F^{M}:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}\) as + +\[F^{M}(U):=(f_{a}^{M}(U)+\varepsilon^{-1}Q^{M}(U),f_{b}^{M}(U)-\varepsilon^{-1}Q^{M}(U),f_{v}^{M}).\] + +Note that the functions \(f_{\nu}^{M},\psi^{M},\phi^{M}\) are globally Lipschitz and bounded. Therefore, \(F^{M}\) is globally Lipschitz, bounded and + +\[F^{M}(U):=(f_{a}(U)+\varepsilon^{-1}Q(U),f_{b}(U)-\varepsilon^{-1}Q(U),f_{v}(U)),\text{if}|U|\leq M.\] + +Given \(p\in(1,+\infty)\), we consider the operator \(A_{p}\text{on}X_{p}:=(L^{p}(\Omega))^{3}\) defined by + +\[\begin{cases} D(A_{p})=D_{p}^{3} & (\text{see}(2.1)) \\ A_{p}U=(d_{a}\Delta u_{a},d_{b}\Delta u_{b},d_{v}\Delta v) & \text{for}U\in D(A_{p}), \end{cases}\] (A.1) + +and the abstract initial value problem + +\[U^{\prime}(t)=A_{p}U(t)+F^{M}(U(t)),t>0,U(0)=U^{\text{in}}:=(u_{a}^{\mathrm{i}\mathrm{n}},u_{b}^{\mathrm{i}\mathrm{n}},v^{\text{in}}).(\mathrm{A}.2)\] + +We will solve (A.1),(A.2) and then we will get rid of the truncation. The main ingredient is that \(A_{p}:D(A_{p})\subset X_{p}\rightarrow X_{p}\) is a sectorial operator ([21], Theorem 3.1.3). Hence it generates in \(X_{p}\) an analytic semigroup denoted \((e^{tA_{p}})_{t\geq0}([21]\), Chapter 2). Moreover, \(A_{p}\) is closed so that \(D(A_{p})\), endowed with the graph norm, is a Banach space. \(D(A_{p})\) being also dense in \(X_{p}\), the semigroup is strongly continuous, i.e. \(\lim_{t\rightarrow0}e^{tA_{p}}U=U,\text{for all}U\in X_{p}\). Furthermore, there exists \(K_{p}>0\text{and}\omega_{p}\in\mathbb{R}\) such that (see [21], Proposition 2.1.1) + +\[\Vert e^{tA}\Vert_{L(X_{p})}\leq K_{p}e^{\omega_{p}t},\forall t\geq0.\] (A.3) + +First step: well-posedness of (A.1),(A.2). Let \(\Vert\cdot\Vert_{p}\) denote the usual norm in \(X_{p}\). We start proving that (A.1),(A.2) has a unique mild solution, i.e. a unique function \(U\in C([0,\infty),X_{p})\) such that + +\[U(t)=e^{tA}U^{\text{in}}+\int_{0}^{t}e^{(t-s)A}F^{M}(U(s))ds,\forall t\geq0.\] (A.4) + +It is easily seen that \(F^{M}\) maps \(X_{p}\) into \(X_{p}\) and + +\[\Vert F^{M}(U)\Vert_{p}\leq\Vert F^{M}(0)\Vert_{p}+L_{M}\Vert U\Vert_{p}=L_{M}\Vert U\Vert_{p},\forall U\in X_{p}.(\mathrm{A}.5)\] + +Therefore, (A.4) makes sense since, by assumption \((2.3),U^{\text{in}}\in X_{p}\) and, for all \(U\in C([0,\infty),X_{p})\text{andall}t>0,F^{M}(U(\cdot))\in L^{1}((0,t);X_{p})\). Moreover, the Lipschitz property of \(F^{M}\) together with (A.3) and Gronwall’s Lemma gives us the uniqueness of (A.4). The same ingredients give us the continuous dependence of \(U\) with respect to \(U\) in. Therefore, it remains to prove the existence of \(U\text{and that}U\) belongs to \(C^{1}([0,\infty);X_{p})\cap C([0,\infty);D_{p}(A))\text{for all}p\in(1,+\infty)\). Let \(\theta>0\) be such that \(\omega_{p}+\theta>0\). The existence is proved using the contraction mapping principle in the space + +\[E:=\{U\in C([0,\infty),X_{p}):\Vert U\Vert_{E}=\sup_{t\geq0}e^{-(\omega_{p}+\theta)t}\Vert U(t)\Vert_{p}<\infty\},\] + +that is a Banach space when endowed with the norm \(\Vert U\Vert_{E}\). Hence, given \(U\in E\), we set + +\[\Phi(U)(t)=e^{tA}U^{\text{in}}+\int_{0}^{t}e^{(t-s)A}F^{M}(U(s))ds,\forall t\geq0.\] diff --git a/outputs/arxiv_math/2503.07166_pg4_pg1_repeat1.md b/outputs/arxiv_math/2503.07166_pg4_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..13194db037ca6f654601d7c0dfab35b423da56f9 --- /dev/null +++ b/outputs/arxiv_math/2503.07166_pg4_pg1_repeat1.md @@ -0,0 +1,47 @@ +Lemma 2.3. For a binary (near) equireplicate \(r\times c\) row-column design on \(v\) symbols, + +\[\begin{aligned} (a)\lambda_{rc}&=e^{-}+e^{+}- \tfrac{e^{-}e^{+}}{e} =e+ \tfrac{(e^{+}-e)(e-e^{-})}{e} , \\ (b)\lambda_{rc}^{-}&=e^{-}\text{and}\lambda_{rc}^{+}=e^{+}, \end{aligned}\] + +\[(c)\lambda_{rr}= \tfrac{c(\lambda_{rc}-1)}{r-1} ,\] + +\[(d)\lambda_{cc}= \tfrac{r(\lambda_{rc}-1)}{c-1} .\] + +Remark 2.4. Lemma 2.3 (a) implies that \(\lambda_{rc}\geq e\), and that the equality holds if and only if \(e\) is an integer, i.e. the row-column design is equireplicate. + +Proof of Lemma 2.3. For an equireplicate design all claims hold due to Lemma 2.2, so from now on we assume that it is near equireplicate, that is, \(e^{-}0\) fixed, + +\[S(\lambda):D(S(\lambda))\rightarrow L^{\infty}[-\tau,\tau],[S(\lambda)y](t):=\dot{y}(t)-A(t,\lambda)y(t)\] + +for a.a. \(t\in[-\tau,\tau]\), which due to Lemma 2.1(a) is well-defined on the domain + +\[D(S(\lambda)):=\left\{u\in W^{1,\infty}[-\tau,\tau]|u(-\tau)\in N(P_{\lambda}^{-}(-\tau)),u(\tau)\in R(P_{\lambda}^{+}(\tau))\right\}.\] + +(IV.1) Claim: \(\dim N(S(\lambda))=\dim N(T(\lambda))<\infty\). + +Consider the commutative diagram + +\[\begin{aligned} W^{1,\infty}(\mathbb{R})^{\underline{T(\lambda)}}L^{\infty}(\mathbb{R}) \\ i_{\lambda}p \\ D(S(\lambda))^{\underline{S(\lambda)}}L^{\infty}[-\tau,\tau], \end{aligned}\] (3.2) + +where \(p\) is the restriction of functions in \(L^{\infty}(\mathbb{R})\) to \(L^{\infty}[-\tau,\tau]\) given by \(p(u):=u|_{[-\tau,\tau]}\) + +and a canonical map \(i_{\lambda}:D(S(\lambda))\rightarrow W^{1,\infty}(\mathbb{R})\) defined by extending a given function + +\(u\in D(S(\lambda))\) to the intervals \((-\infty,-\tau)\text{and}(\tau,\infty)\) as solution of \((V_{\lambda})\). Observe that + +\(i_{\lambda}\) is injective and \(i_{\lambda}\left(N(S(\lambda))\right)=N(T(\lambda))\) holds, where \(i_{\lambda}\left(N(S(\lambda))\right)\subseteq N(T(\lambda))\) + +results directly due to the construction of \(i_{\lambda}\), while \(i_{\lambda}\left(N(S(\lambda))\right)\supseteq N(T(\lambda))\) as converse + +inclusion follows from the fact that privided \(u\in N(T(\lambda)),\text{then}u(\tau)\in R(P_{\lambda}^{+}(\tau))\) and + +\(u(-\tau)\in N(P_{\lambda}^{-}(-\tau))\) for any \(\tau>0\) (recall (2.3)). Finally, this yields the assertion. + +(IV.2) We decompose \([-\tau,\tau]=[-\tau,0]\cup[0,\tau]\) and define the spaces + +\[\begin{aligned} X_{+}:&=\{u\in W^{1,\infty}[0,\tau]|u(\tau)\in R(P_{\lambda}^{+}(\tau))\},Y_{+}:=L^{\infty}[0,\tau], \\ X_{-}:&=\{u\in W^{1,\infty}[-\tau,0]|u(-\tau)\in N(P_{\lambda}^{-}(-\tau))\},Y_{-}:=L^{\infty}[-\tau,0]. \end{aligned}\] + +Consider the following linear operators \(S^{\pm}(\lambda):X_{\pm}\rightarrow Y_{\pm}\) pointwise defined as + +\[[S^{\pm}(\lambda)y](t)=\dot{y}(t)-D_{2}f(t,\phi_{\lambda}(t),\lambda)y(t)\text{for a.e.}t\in\mathbb{R}_{\pm}.\] + +Next consider the following commutative diagram + +\[\begin{aligned} X_{-}\oplus X_{+}^{\underline{S^{-}(\lambda)\oplus S^{+}(\lambda)}}Y_{-}\oplus Y_{+} \\ J_{\lambda}J \\ D(S(\lambda))^{\underline{S(\lambda)}}L^{\infty}[-\tau,\tau], \end{aligned}\] (3.3) + + + + + + + + + + + + + + + + + + + + + + +
S−(λ)⊕S+(λ)
(3.3)X− ⊕ X+Y− ⊕ Y+
JλJ
S(λ)
D(S(λ))L∞[−τ, τ ],
+ +where \(J:L^{\infty}[-\tau,\tau]\rightarrow Y_{-}\oplus Y_{+}\text{and}J_{\lambda}:D(S(\lambda))\rightarrow X_{-}\oplus X_{+}\) are defined by + +\[Ju:=(u_{-},u_{+})\text{and}J_{\lambda}v:=(v_{-},v_{+})\] + +with \(u_{+}\text{and}v_{+}\) (resp. \(u_{-}\text{and}v_{-})\) being the corresponding restrictions to \([0,\tau]\) (resp. to + +the interval \([-\tau,0])\). It is clear that \(J\) is an isomorphism, while \(J_{\lambda}\) is injective with range + +\(R(J_{\lambda})=\{(v_{-},v_{+})|v_{-}(0)=v_{+}(0)\}\). Defining the mapping \(\Sigma:X_{-}\oplus X_{+}\rightarrow\mathbb{R}^{d}\) by diff --git a/outputs/arxiv_math/2503.07226_pg7_pg1_repeat1.md b/outputs/arxiv_math/2503.07226_pg7_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..a4d7aa853b3d1a5a4cddd28c1e500287844c3051 --- /dev/null +++ b/outputs/arxiv_math/2503.07226_pg7_pg1_repeat1.md @@ -0,0 +1,27 @@ +## 3. Analytical solutions + +Our objective is to build exact solutions that fit, in particular, the physiological problem under study, + +\[\tau\partial_{t}u-\alpha\Delta u+Bu=f,\] (12) + +via the separable variable method, in cylindrical coordinates i.e. at the position \((r,z)\) and the time \(t\). Using Bernoulli–Fourier technique, we have + +\[\left(\tau X^{\prime}(t)+[-\alpha(\beta+\eta^{2})+B]X(t)\right)R(r)Z(z)=f(r,z,t),\] (13) + +where the constants \(\beta,\eta\in\mathbb{R}\) are arbitrary, Radiative transfer: \(\tau=1/\nu,\alpha=D,B=\mu_{\mathrm{a}}\text{and}f=S\) in Subsection 2.1; Heat transfer: \(\tau=\rho c_{\mathrm{p}},\alpha=k,B=c_{\mathrm{b}}\omega_{\mathrm{b}}(t)\text{and}f=q\) in Subsection 2.2. The functions \(X,Z\text{and}R\) are elementary solutions to the system of ordinary differential equations (ODE) + +\[\begin{cases} X^{\prime}(t)=\zeta X(t) \\ Z^{\prime\prime}(z)=\eta^{2}Z(z) \\ \big( & \big)^{\prime} \\ rR^{\prime}(r) & =\beta rR(r) \end{cases}\] (14) + +for some time parameter \(\zeta\in\mathbb{R}\). The nonconstant behavior of \(\omega_{\mathrm{b}}\) (cf. (11)) does not invalidate the application of this technique in determining an analytical solution for the heat transfer. Firstly, a particular solution is available due to the Duhamel principle (for details, see [9] and the references therein). Secondly, let us seek for elementary solutions of the ODE system (14). The first-order ODE in (14) admits the elementary solution + +\[X(t)=\exp[\int_{0}^{t}\zeta(s)ds],t>0.\] (15) + +The second-order ODE in (14), with constant coefficients, admits the elementary solutions + +\[Z(z)=\exp[\pm\eta z],z\in\mathbb{R}.\] (16) + +The second-order ODE in (14), with nonconstant coefficients, admits the Bessel functions of first and second kind and order 0, respectively, \(J_{0}( \sqrt{|\beta|} r)\text{and}Y_{0}( \sqrt{|\beta|} r)\) if \(\beta<0;\mathrm{o}\mathrm{r}\) the modified√ Bessel functions of first and second kind and order 0, respectively, \(I_{0}( \overline{\beta} r)\text{and}K_{0}( \overline{\beta} r)\text{if}\beta>0[33].\text{If}\beta=0\), the elementary solutions reduce to \(R(r)=\log[r]\) and the unity function. Then, a general solution solving the homogeneous equation \((13)(f=0)\) is available by the above elementary solutions, if + +\[\tau\zeta+B=\alpha(\beta+\eta^{2}).\] (17) + +Finally, we analyze the PDE (12) at the period of time \(02\). In this case, \(g\) has exactly two fixed points on the real axis. + +- Case 2: \(\det g=-1\) (Orientation-reversing isometries) + + - Reflection if \(\text{Tr}g=0\). In this case, \(g\) is an involution with a geodesic of fixed points. + + - Glide-reflection if \(\text{Tr}g\neq 0\). In this case, \(g\) has two fixed points on the real axis. + +For each \(g\in\text{PSL}_{2}^{\pm}(\mathbb{R})\), the translation length \(t_{g}\) is defined as: + +\[t_{g}=\inf_{z\in\mathbb{H}}d(z,gz).\] + +If \(t_{g}\) is positive, we call \(g\) a positive translation isometry. For a positive translation isometry \(g\), the absolute value of the trace of \(g\) satisfies the following: + +\[|\text{Tr}g|= \begin{cases} \underline{t_{g}} \\ 2\cosh & , & \text{if}g\text{is hyperbolic}, \\ \overline{2} \\ \underline{t_{g}} \\ 2\sinh & , & \text{if}g\text{is a glide}-\text{reflection}. \\ \overline{2} \end{cases}\] (1) + +For a positive translation isometry \(g\in\text{PSL}_{2}^{\pm}(\mathbb{R})\), we define \(A_{g}\) as the axis of \(g\), i.e., the geodesic joining the two fixed points of \(g\), and let \(\rho_{A_{g}}\) be the reflection with respect to \(A_{g}\). We state the following result from [1, Theorem 7.38.6] because it will be used later. + +Theorem 2.1. [1, Theorem 7.38.6] Let \(g\text{and}h\) be hyperbolic transformations of the hyperbolic plane and suppose that \(A_{g}\text{and}A_{h}\) intersect at a point \(P\) . Denote by \(\theta_{P}\) the angle at P between forward direction of \(A_{g}\text{and}A_{h}\). Then the composition \(g\circ h\) is hyperbolic and + +\[\frac{1}{2} |\text{Tr}gh|=\cosh\left( \frac{t_{g}}{2} \right)\cosh\left( \frac{t_{h}}{2} \right)+\sinh\left( \frac{t_{g}}{2} \right)\sinh\left( \frac{t_{h}}{2} \right)\cos(\theta_{P}).\] (2) + +| 1 | tg | | tg | | th | | +| --- | --- | --- | --- | --- | --- | --- | +| \|Tr gh\| = cosh | | th cosh + sinh | | sinh | cos(θP ). | (2) | +| 2 | 2 | 2 | 2 | | 2 | | + +Now, we generalize the above theorem to positive translation isometries. The proof of [1, Theorem 7.38.6] uses the law of cosines for the triangle formed by the three axes \(A_{g},A_{h},\text{and}A_{gh}\) to compute the absolute value of the trace. In this paper we compute the absolute value of the trace by matrix computations as follows. + +Theorem 2.2. Let \(g\text{and}h\) be positive translation isometries and suppose that \(A_{g}\text{and}A_{h}\) intersect at a point \(P\) . Denote by \(\theta_{P}\) the angle at P between forward direction of \(A_{g}\text{and}A_{h}\). Then, the following hold. Case 1: If \(g\) is a glide-reflection and \(h\) is a hyperbolic element, + +\[\frac{1}{2} |\text{Tr}gh|=\bigg|\sinh\left( \frac{t_{g}}{2} \right)\cosh\left( \frac{t_{h}}{2} \right)+\cosh\left( \frac{t_{g}}{2} \right)\sinh\left( \frac{t_{h}}{2} \right)\cos\theta_{P}\bigg|.\] (3) + +Case 2: If both \(g\text{and}h\) are glide-reflections, + +\[\frac{1}{2} |\text{Tr}gh|=\bigg|\sinh\left( \frac{t_{g}}{2} \right)\sinh\left( \frac{t_{h}}{2} \right)+\cosh\left( \frac{t_{g}}{2} \right)\cosh\left( \frac{t_{h}}{2} \right)\cos\theta_{P}\bigg|.\] (4) + +Proof. Without loss of generality, we may assume the axis \(A_{g}\) equals the imaginary axis with the intersection point \(P\in A_{g}\cap A_{h}\) located at \(i\in\mathbb{H}\) . See Figure 1. diff --git a/outputs/arxiv_math/2503.07251_pg28_pg1_repeat1.md b/outputs/arxiv_math/2503.07251_pg28_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..adace82876aa603e144ece718094a3da699ecd1b --- /dev/null +++ b/outputs/arxiv_math/2503.07251_pg28_pg1_repeat1.md @@ -0,0 +1,13 @@ +vaccinated compartment, we can document individuals receiving the vaccine, but the timing of immunity loss remains uncertain. To make better use of the available information, we propose to divide each compartment into two distinct groups: an observable part and an unobservable part. The unobservable part, called the hidden compartment with waning immunity, represents those individuals within the recovered and vaccinated compartments whose status cannot be directly observed. On the other hand, for the observable part, we want to divide it into LG sequential or cascade compartments. We assume perfect immunity for a known period of time, which is divided into LG sub-periods. These cascading compartments are used to model the progressive loss of immunity over time for both recovered and vaccinated individuals. In certain epidemic models with hidden states, some compartments may have one or more observable incoming transitions, while their outgoing transitions may be partially or completely unobservable. As a result, these compartments are treated as hidden. To extract useful information from the observed incoming transitions, the compartment can be divided into a sequence of generic sub-compartments, denoted as Gi, for \(i=1,\ldots,L^{G}\). Each subcompartment Gi groups individuals based on their time since entering the compartment, effectively tracking their Gi age. The number of sub-compartments LG is typically chosen to match the period during which no outgoing transitions occur, such as the duration of full immunity following recovery or vaccination. + +#### 4.5.1 Continuous-time Dynamics + +Consider the diagram in Figure 4.4 where we assume a random observable inflow into the first generic compartment, G1, and a non-observable outflow from the last compartment, denoted by \(G^{-}\). Between these two compartments, we introduce \(L^{G}-1\) additional compartments, representing different G-ages within the overall compartment G. Additionally, except for the random inflow and outflow, all transitions between compartments are deterministic. Let \(\tau_{1},\ldots,\tau_{L^{G}}\) represent the accounting dates, times at which individuals in each Gi compartment transition to the next. If the time spent in each compartment Gi is \(\Delta\tau=\tau_{i+1}-\tau_{i}\), the dynamics from G1 to G can be described as in Figure 4.4. Now, we assume that the state process X can be decomposed into \((X^{e},G)\), where Xe con- + +![](figure) + +![](figure) + +Fig. 4.4: Cascade state illustration with 8 compartment \((G_{l},l=1\ldots,L^{G}=8)\); observed inflow : random inflow in the compartment 1 (G1) with G-age, \(1\Delta\tau\), non-observed outflow : random outflow from the compartment \(G^{-}\) with G-age, \(\geq9\Delta\tau\) Between G1 and \(G^{-}\), only deterministic transition. + +tains the \(d^{e}0\) such that for all \((\lambda,x)\in[ \underline{H} , \overline{H} ]\times\mathbb{R},\) + +\[\big|h^{[\lambda]}(x)\big|\leq c(3+|x|)^{\lambda- \frac{3}{2}}.\] (8) diff --git a/outputs/arxiv_math/2503.07310_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.07310_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..d3d6b5f5e716242dcc11ef2fd848c320935e9d46 --- /dev/null +++ b/outputs/arxiv_math/2503.07310_pg2_pg1_repeat1.md @@ -0,0 +1,5 @@ +uncertainty. There are two key drivers for the selection of the appropriate optimisation method: statistical data availability and desired degree of robustness. Robust optimisation has emerged as a leading method for problems with limited uncertain parameter data and requirements for a risk-averse solution. In a robust optimisation context, solutions that lead to attainable decisions under all parameter realisations are referred to as robust feasible, while the solution that also leads to the best objective value is called robust optimal. A general form of the robust optimisation problem is given by \((P_{rob})\). + +\[\mathrm{s}\min_{x\in\mathcal{X}}.\mathrm{t}.g^{f}(x^{(x},^{)}u)\leq0,\forall u\in\mathcal{U}\] (Prob) + +where \(x\in\mathcal{X}\subseteq\mathbb{R}^{n_{x}}\) denotes continuous decisions, and \(u\) is the vector of uncertain parameters that resides within an uncertainty set \(\mathcal{U}\). Despite the presence of an embedded optimisation problem within \((P_{rob})\), robust optimisation problems can yield tractable reformulations under restrictive assumptions about the problem structure. That is, if Problem \((P_{rob})\) is convex with respect to the decisions \(x\) and concave in \(u\), and \(\mathcal{U}\subset\) is a convex set, then using duality theory one can derive a tractable robust counterpart reformulation of Problem \((P_{rob})\). Because of this, the field of convex robust optimisation is well-established, whereas the field of non-convex robust optimisation is still at its infancy [1]. State-of-the-art robust optimisation algorithms for non-convex problems typically rely on global solvers to obtain robust optimal solutions. The prevailing approach involves deriving the dual reformulation and utilising a global solver, either directly or adaptively. However, the dual reformulation can increase the problem complexity, and in the case of challenging non-convex problems may only yield a robust feasible solution. An alternative approach is based on an iterative robust cutting plane algorithm. In this case, significant computational time can be spent searching for a global solution that may be deemed robust infeasible. Contributions of this work: Our research hypothesis is that integrated exploration of global and robust optimality can yield computational benefits. To this end, in this article we introduce a novel algorithm that conducts a concurrent global optimality and robustness search for continuous non-convex problems. The proposed approach integrates spatial branch-and-bound with robust cutting plane notions. The key idea is that, while exploring the branch-and-bound tree, we assess the robustness of nodes entailing the best-found solutions. We illustrate the performance and benefits of our proposed approach through benchmark Quadratically Constrained Quadratic Programs (QCQPs) of pooling problems. At each node, the non-convex problem is solved via a local solver. If the computed solution is as good as the best-found so far, an infeasibility test is performed to evaluate the robustness of the obtained solution. If not, then the corresponding cutting planes are added both to the non-convex and convex problems. The algorithm proceeds to the next step once no more violations are detected. The convex problem is solved next, and the solution of this problem is used to decide the most promising variable for branching into two child nodes. With the use of appropriate fathoming criteria, the tree nodes are exhausted, and the robust optimal solution is obtained. diff --git a/outputs/arxiv_math/2503.07322_pg9_pg1_repeat1.md b/outputs/arxiv_math/2503.07322_pg9_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..f98361da2c82fca1726820297451bae59b7e70f9 --- /dev/null +++ b/outputs/arxiv_math/2503.07322_pg9_pg1_repeat1.md @@ -0,0 +1,29 @@ +is a subcomplex since + +\[\overline{\frac{D}{D}} (_{(\bar{z}}z_{)_{i}^{i}})_{=}=_{-}m_{\Theta^{i}}-_{(m}e\bar{z}_{)_{i}^{i}=}\in B_{0,^{d+1}}\otimes\Lambda(V_{1}, \overline{V}_{1} ), \overline{D} ),\] + +by Corollary 2.15, where zi is a basis element of V1 with D(zi) = mi and the second equality holds as \(\Theta\) is Bd+1-linear. Denote this subcomplex by C1 and the cokernel by C2 and consider the short exact sequences + +\[0\rightarrow C_{1}\rightarrow(B_{d+1}\otimes\mathrm{C}_{CE}(H^{*}(\mathrm{S}^{d})\otimes L_{d}), \overline{D} )\rightarrow C_{2}\rightarrow0.\] (2.21) + +The subcomplex C1 is a sub-cdga which is a pure Sullivan algebra [FHT01, Ch. 32], i.e. the differential is only non-trivial for odd degree generators and lies in the algebra of even degree generators. Hence, it has an additional homological grading \(F_{k}C_{1}:=B_{d+1}\otimes\Lambda \overline{V}_{1} \otimes\Lambda^{k}V_{1}\) and + +\[H_{0}(C_{1})=\mathrm{Q}[e,p_{1},\ldots,p_{n},\{\bar{z}_{i}\}]/(\{m_{i}-e\bar{z}_{i}\}).\] + +The map \(B_{d+1}\rightarrow(B_{d+1}\otimes\mathrm{C}_{CE}(H^{*}(\mathrm{S}^{d})\otimes L_{d}), \overline{D} )\) factors through F0C1 and hence we need to understand the image of the connecting homomorphism of the above short exact sequence. For d = 3 the vector space V1 is 1-dimensional with generator z1 and D(z1) = p21 by Corollary 2.19. Then the sequence (2.21) actually splits as a sequence of cochain complexes so that + +This is because + +\[H(C_{1})֒\rightarrow H^{*}((B_{d+1}\otimes\mathrm{C}_{CE}(H^{*}(\mathrm{S}^{d})\otimes L_{d}), \overline{D} )).\] + +\[\tilde{C}_{2}:=B_{d+1}\otimes\Lambda(V_{1}\oplus \overline{V}_{1} )\otimes\Lambda^{+}(V_{2}\oplus \overline{V}_{2} )\] + +is a subcomplex isomorphic to C2 as the only way D|C˜2 can have image in C1 is if there is \(w\in V_{2}\) so that \(D(w)=fz_{1}+\chi\text{for}f\in B_{d+1}\text{and}\chi\in B_{d+1}\otimes\Lambda V_{1}\otimes\Lambda^{+}V_{2}\). Suppose w has the minimal degree where this happens, then + +\[0=D^{2}(w)=fp_{1}^{2}+D(\chi)\] + +and since \(D(\chi)\) cannot be contained in \(B_{d+1}\otimes1\) by construction it follows that f = 0. Hence, D1|V2 has image in \(B_{d+1}\otimes V_{2}\) and consequently (2.21) splits as as cochain complexes. This proves the second part the theorem as + +\[H(C_{1})\mathrm{Q}[e,p_{1},\bar{z}_{1}]/(p_{1}^{2}-e\bar{z}_{1})\] + +- injects into \(H^{*}((B_{d+1}\otimes\mathrm{C}_{CE}(H^{*}(\mathrm{S}^{d})\otimes L_{d}), \overline{D} ))H^{*}(\Gamma_{\mathrm{S}^{3}}//\text{SO}(4);\mathrm{Q})\). The first part then follows as the map \(B_{4}\rightarrow H(C_{1})\) is injective. Remark 2.22. The formulation of Corollary 2.19 for all dimensions d is due to the author’s attempt to prove a general version of Theorem 2.20, namely that H∗(BSO(d \(+1))\rightarrow H^{*}(\Gamma_{\mathrm{S}^{d}}\) //SO(d + 1) is injective for all odd d. However, for \(d\geq5\) the free resolution of the kernel of H∗(BSO(d)) \(\rightarrow H^{*}(F_{d}//\text{SO}(d))\), i.e. the ideal generated by monomials in the Pontrjagin classes of degree > 2d, has higher syzygies and the proof for d = 3 does not generalize to show that \(D^{1}(V_{2})\subset B_{d+1}\otimes V_{2}\). In fact, one would need to show a stronger version of Corollary 2.19 to get a splitting of (2.21) for odd d > 3. Nonetheless, it follows from Nariman’s results that the monomials in the Euler and Pontrjagin classes do not vanish and I expect that one can improve the statement of Corollary 2.19 or control the image of the connecting homomorphism of (2.21) in order to find a purely algebraic proof of Nariman’s result. For this reason, I have kept a general version of Corollary 2.19 in this article in the hope that it can serve as a starting point to prove a more general version of Theorem 2.20 – it is hard to imagine that the statement doesn’t generalize to all odd d. diff --git a/outputs/arxiv_math/2503.07335_pg33_pg1_repeat1.md b/outputs/arxiv_math/2503.07335_pg33_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..6db4e9420335b6bfd3289a1cfc5a1956fb33202e --- /dev/null +++ b/outputs/arxiv_math/2503.07335_pg33_pg1_repeat1.md @@ -0,0 +1,17 @@ +The sum \(\sum_{j=i_{0}+1}^{i_{1}}11\{j\in\mathcal{B}_{C}\}\) is thus stochastically dominated by a sum of independent \(\{0,1\}-\) random variables + +\[Z=\sum_{i=0}^{i_{\text{max}}-1}\sum_{h=1}^{\omega}Z_{h}^{(i)},\] + +where \(\Pr(Z_{h}^{(i)}=1)=(1+o(1))\left(1- \frac{2t_{i}}{3} \right)\text{for all}h=1,2,\ldots,\omega\), uniformly in \(i\). We compute + +\[\begin{aligned} \mathrm{E}[Z]&=(1+o(1))\omega\sum_{i=0}^{i_{\text{max}}-1}\left(1- \frac{2t_{i}}{3} \right) \\ &=(1+o(1))\omega\sum_{i=0}^{i_{\text{max}}-1}\left(1- \frac{2}{3} \cdot \frac{i_{0}+i\omega}{n} \right) \\ &=(1+o(1))\omega i_{\text{max}}\left(1- \frac{2i_{0}}{3n} \right)- \frac{2\omega}{3n} \sum_{i=0}^{i_{\text{max}}-1}i \\ &=(1+o(1))\left(n- \frac{2\omega}{3n} \frac{i_{\text{max}}^{2}}{2} \right) \\ &=(1+o(1)) \frac{2n}{3} . \end{aligned}\] + +It follows from Proposition 1.5 that \(Z\leq(1+o(1))\mathrm{E}[Z]=(1+o(1)) \frac{2n}{3}\) a.a.s. Since \(Z\) stochastically dominates \(|\mathcal{B}_{C}|\), we may conclude \(\frac{1}{2} |\mathcal{B}_{C}|\leq(1+o(1)) \frac{n}{3}\) a.a.s., completing the proof. + +## 7. Conclusion + +In this paper, we have shown that \(s(\mathcal{G}_{n,3})\leq(1+o(1)) \frac{n}{3}\) a.a.s., and have additionally made the conjecture that \(s(\mathcal{G}_{n,3})=(1+o(1)) \frac{n}{4}\) a.a.s., which is (asymptotically) smallest possible for cubic graphs on \(n\) vertices. Resolving this conjecture is an interesting direction for future research. Another one is to study \(s(\mathcal{G}_{n,d})\text{for}d\geq\) 4—a problem which at present appears challenging. We considered the following crude technique for upper bounding the Sudoku number of a graph \(G\) with chromatic number \(k\). Start with any \(k-\)coloring of \(G\) using colour classes \(\{1,2,\ldots,k\}\). Iteratively move vertices from class \(k\) to another available class until every remaining vertex has a neighbour in each class \(i\text{for}i=1,\ldots,k-1\). Then the union of classes 1 through \(k-1\) is a Sudoku set for \(G\), and we have the bound + +\[s(G)\leq(\chi(G)-1)\alpha(G),\] (38) + +where \(\alpha(G)\) is the size of the largest independent set in \(G\). Using the best available bound \(\alpha(\mathcal{G}_{n,4})\leq0.41635\) a.a.s. (see [23]) and the fact that \(\chi(\mathcal{G}_{n,4})=3\) a.a.s. ([26]), (38) gives \(s(\mathcal{G}_{n,4})\leq0.8327n\) a.a.s. The next value of \(d\) for which \(\chi(\mathcal{G}_{n,d})\) is known explicitly is \(d=6\), where we have \(\chi(\mathcal{G}_{n,d})=4\) a.a.s. [27]. However, it has been shown that \(\mathcal{G}_{n,6}\) has an independent set of size at least \(0.33296n\) a.a.s. [8], meaning that any bound using (38) for \(d=6\) will be very close to \(n\). In fact, currently available upper bounds, such as \(0.35799n\), are well above \(n/3\) (see for instance [8]) and thus, are far from providing any useful bound on \(s(\mathcal{G}_{n,6})\). For large \(d,(38)\) is too weak to say anything nontrivial about \(s(\mathcal{G}_{n,d})\). Indeed, for \(d\rightarrow\infty\) sufficiently slowly with respect to \(n\), it is known that \(\chi(\mathcal{G}_{n,d})=(1+o(1)) \frac{n}{\alpha(\mathcal{G}_{n,d})} =(1+o(1)) \frac{d}{2\log d}\) a.a.s. [10]. diff --git a/outputs/arxiv_math/2503.07337_pg40_pg1_repeat1.md b/outputs/arxiv_math/2503.07337_pg40_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..8278f93a511348b84e4f447ffb25511b3608cf6f --- /dev/null +++ b/outputs/arxiv_math/2503.07337_pg40_pg1_repeat1.md @@ -0,0 +1,47 @@ +Proposition A.2. Let \(q>n\), let \(\Omega\subset\mathbb{R}^{n}\) be an open bounded connected set, and let \(u\in W^{3,q}(\Omega)\) be a function such that for some positive constant \(k\) + +\[|\nabla u|(x)\geq k,\forall x\in u^{-1}(0).\] + +Assume in addition that \(d(u^{-1}(0),\partial\Omega)>0\). Let \(u_{j}\in C^{2,\alpha}(\Omega)\) be a sequence of functions such that + +\[u_{j}-^{W}--^{2,}-^{(^{q}}-^{\Omega}\rightarrow^{)}u.\] + +Then for every \(j\) large enough \(u_{j}^{-1}(0)\) is a \(C^{1,s}\) hypersurface, with \(s=1-n/q\), and there exist deformations \(\Phi_{j}\in W^{2,q}(\Omega;\mathbb{R}^{n})\) such that + +(i) \(\Phi_{j}\) are orthogonal to \(u^{-1}(0)\); + +\[(ii)u_{j}^{-1}(0)=(\text{Id}+\Phi_{j})(u^{-1}(0));\] + +\[(\text{iii})\lim_{j}\Vert\Phi_{j}\Vert_{2,q}=0\] + +Proof. • Step 1: we first notice that for every \(\varepsilon>0\) there exists \(t_{0}\text{such that if}|t|\leq t_{0}\) then + +\[u^{-1}(t)\subseteq(u^{-1}(0))^{\varepsilon},\] + +where we use the notation \((K)^{t}=\{p\in\mathbb{R}^{n}|d(p,K)\leq t\}\), for the outer parallel set. Indeed, let us assume by contradiction that there exist \(\varepsilon_{0}>0\) and points \(y_{k}\in u^{-1}(t_{k})\) with \(\lim_{k}t_{k}=0\) such that + +\[\forall k\in\mathbb{N},d(y_{k},u^{-1}(0))>\varepsilon_{0}.\] + +As \(\Omega\) is bounded, up to a subsequence, we may assume that \(y_{k}\) converge to some point \(\bar{y}\in \overline{\Omega}\). The continuity of \(u\) gives \(u(\bar{y})=0\), so that \(\bar{y}\in u^{-1}(0)\). On the other hand, + +which is a contradiction. + +\[d(\bar{y},u^{-1}(0))\geq\varepsilon_{0},\] + +- Step 2: We now want to extend the non-degeneracy property of the gradient of \(u\) to the functions \(u_{j}\) on their level sets \(u_{j}^{-1}(0)\). We notice that by uniform convergence and by the previous step, for every \(\varepsilon>0\) there exist \(t_{0}>0\text{and}j_{0}\) such that + +\[\forall j\geq j_{0},u_{j}^{-1}(0)\subset u^{-1}((-t_{0},t_{0}))\subset(u^{-1}(0))^{\varepsilon}.\] (A.2) + +On the other hand, \(\nabla u\) is uniformly continuous in \(\Omega\) so there exists \(\alpha_{0}>0\) such that + +\[\forall(x,z)\in\Omega\text{s.t.}|z-x|\leq\alpha_{0},|\nabla u(z)-\nabla u(x)|\leq \frac{k}{4} .\] + +In particular, + +\[\forall x\in(u^{-1}(0))^{\alpha_{0}},|\nabla u(x)|\geq \frac{k}{2} .\] + +Also, \(\nabla u_{j}\) uniformly converge to \(\nabla u\), so for \(j\) large enough + +\[|\nabla u_{j}(x)-\nabla u(x)|\leq \frac{k}{8} \forall x\in\Omega.\] + +From the previous estimates, we get \(|\nabla u_{j}|(x)>0\text{for every}x\in u_{j}^{-1}(0)\text{and}j\) large enough, which implies that the sets \(u_{j}^{-1}(0)\) are \(C^{1,s}\) hypersurfaces. diff --git a/outputs/arxiv_math/2503.07345_pg12_pg1_repeat1.md b/outputs/arxiv_math/2503.07345_pg12_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..603f5af5a20039806ed37e18f6ec3404445dc913 --- /dev/null +++ b/outputs/arxiv_math/2503.07345_pg12_pg1_repeat1.md @@ -0,0 +1,27 @@ +Proof. These operators are limit point at both end points \(r=0,\text{and}r=\infty\). Indeed, any solution of \(Lf=0\text{or}L_{j}f=0\) is asymptotic to a linear combination of \(r^{\frac{3}{2}},r^{- \frac{1}{2}}\text{as}r\rightarrow0+\), so no boundary condition is needed at \(r=0\). The other endpoint \(r=\infty\) is standard, see [53, Theorem X.8]. The claim about essential self-adjointness is [53, Theorem X.7]. See [30, Section 4] for the Bessel operator \(L_{0}\) and its domain. The Kato-Rellich theorem applies to \(L,L_{1},L_{2}\), which are relatively bounded perturbations of \(L_{0}\), see [17, Section 1.4] (in fact, these operators are perturbations of \(L_{0}\) by bounded operators). The Weyl criterium, see [54, Theorem XIII.14], implies that \(\text{spec}_{\text{ess}}(L_{0})= \text{spec}_{\text{ess}}(L_{1})=[0,\infty).\text{If}\lambda\in\text{spec}(L_{1})\text{for some}\lambda<0\), then there exists a ground state of negative energy, i.e., \(L_{1}\phi=\lambda_{0}\phi\text{for some}\lambda_{0}<0\text{and}\phi\in\mathcal{D}\text{with}\phi>0\) (in fact, \(\phi\) is smooth and \(\phi(r)\sim cr^{\underline{3}}\) as \(r\rightarrow0+\) for some constant \(c)\). Let \(\chi(r)=1\text{for}0\leq r\leq1,\chi\in C^{\infty}([0,\infty))\) with compact support, and set \(\chi_{b}(r)=\chi(r/b)\text{with}b\geq1\). Then + +2 + +\[\langle L_{1}\phi,\chi_{b}(r)r^{\frac{1}{2}}\rho_{1}\rangle=\lambda_{0}\langle\phi,\chi_{b}(r)r^{\frac{1}{2}}\rho_{1}\rangle\] + +- + +Integrating by parts on the left-hand side and sending \(b\rightarrow\infty\) now leads to a contradiction because of the vanishing \(L_{1}(r^{\frac{1}{2}}\rho_{1})=0\). Hence \(L_{1}>0\) as stated (note that \(L_{1}\) cannot have a zero energy eigenfunction because the unique 0-energy solution is not in \(X_{0})\). Pure a.c. spectrum is a consequence of the construction of the Weyl, Titchmarsh \(m-\)function for these operators, see [30,40]. The essential spectrum of \(L_{2}\) follows from the Weyl criterium as before. Since \(L_{2}=L_{1}+2\rho_{1}^{2}>0\), we conclude that the discrete spectrum of \(L_{2}\) – if it exists – is strictly positive. Remark 2.3. The exact value of \(c_{0}>0\) is not known, but the approximate value \(c_{0}\approx1.3326\) is obtained in [51] via a numerically assisted argument. Moreover, [51] shows that \(L_{2}\) has infinitely many eigenvalues in \((c_{0},2)\) and that 2 is a resonance. Next, we determine the spectrum of \(i\mathcal{L}\) and define the evolution \(e^{t\mathcal{L}}\) using the Hille-Yosida theorem. As we do not have a selfadjoint reference operator available as required for the standard version of Weyl’s theorem as in [54, Theorem XIII.14], we need to proceed differently. To this end we first obtain a proper understanding of the resolvent \((i\mathcal{L}_{0}-z)^{-1}\) of the free operator + +\[\mathcal{L}_{0}:=\left[-\left(-\partial_{r}^{2}+^{0} \tfrac{3}{4r^{2}} +2\right)-\partial_{0^{r}}^{2}+ \tfrac{3}{4r^{2}} \right].\] + +Let \(\tilde{p}_{+}(\zeta)\) denote the modified Hankel function + +\[\tilde{p}_{+}(\zeta):= \sqrt{\zeta} H_{1}^{(1)}(\zeta)= \sqrt{\zeta} \left(J_{1}(\zeta)+iY_{1}(\zeta)\right),\] + +and let \(\tilde{q}_{+}(\zeta)\) denote the modified Bessel function + +\[\tilde{q}_{+}(\zeta):= \sqrt{\zeta} J_{1}(\zeta).\] + +Here \(H_{1}^{(1)},J_{1},\text{and}Y_{1}\) denote the order one Hankel function, Bessel function of the first kind, and Bessel function of the second kind respectively. \(\tilde{p}_{+}\text{and}\tilde{q}_{+}\) satisfy the ODE + +\[- \frac{d^{2}}{d\zeta^{2}} \tilde{p}_{+}(\zeta)+ \frac{3}{4\zeta^{2}} \tilde{p}_{+}(\zeta)=\tilde{p}_{+}(\zeta),- \frac{d^{2}}{d\zeta^{2}} \tilde{q}_{+}(\zeta)+ \frac{3}{4\zeta^{2}} \tilde{q}_{+}(\zeta)=\tilde{q}_{+}(\zeta).\] + +Then by direct inspection, the vectors + +\[\tilde{\psi}_{\tilde{\psi}_{3}^{1}}(_{(}r_{r},_{,}z_{z})_{)}:_{:}=_{=} \begin{bmatrix} \tfrac{ik_{1}(z)^{2}}{z} \tilde{p}_{+}(k_{1}(z)r) \\ \tilde{p}_{+} (k_{1}(z)r) \end{bmatrix} \begin{bmatrix} \tfrac{ik_{1}(z)^{2}}{z} \tilde{q}_{+}(k_{1}(z)r)_{,} & \tilde{\psi}_{4}(r,z):= & \tfrac{ik_{2}(z)^{2}}{z} \tilde{q}_{+}(k_{2}(z)r) \\ \tilde{q}_{+} (k_{1}(z)r) & \tilde{q}_{+} (k_{2}(z)r) \end{bmatrix} ,\tilde{\psi}_{2}(r,z):=\bigg[ \frac{ik_{2}(z)^{2}}{\tilde{p}_{+}^{z}} (p_{\tilde{k}_{2}^{+}}(_{(}k_{z^{2})r}(z_{)})r)_{,,}\] diff --git a/outputs/arxiv_math/2503.07351_pg7_pg1_repeat1.md b/outputs/arxiv_math/2503.07351_pg7_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..acbed8dea9fff5f483b9c828cf8b82be3e256c38 --- /dev/null +++ b/outputs/arxiv_math/2503.07351_pg7_pg1_repeat1.md @@ -0,0 +1,35 @@ +Remark 2.14. In \(PL_{[0,1]}s\), the evaluation of formulas with connective & is traditionally extended as \(\Vert a\&b\Vert=T(\Vert a\Vert,\Vert b\Vert)=\Vert a\Vert*\Vert b\Vert\) and the evaluation of formulas with connective \(\wedge\) is extended as \(\Vert a\wedge b\Vert=\min\{\Vert a\Vert,\Vert b\Vert\}\). For keeping the unification of form of the normal encoding function, we extend the evaluation of formulas with connective \(\wedge\text{as}\Vert a\wedge b\Vert=T(\Vert a\Vert,\Vert b\Vert)=\Vert a\Vert*\Vert b\Vert\), which will not cause any confusion in this paper. + +### 2.3. Model Checking + +Besnard and Doutre [12] give a method of model checking. The method aims to translate an \(AF\) as a propositional formula in the \(PL_{2}\), such that the models of the propositional formula are corresponding to extensions of the \(AF\) under a kind of semantics. In fact, the core of each model is an extension of the \(AF\) and vice versa. The translating functions depend on various semantics. The mainly results in the chapter “model checking” in [12] are listed as the following, where each \(\Phi\) is a translating function. Let \(AF=(A,R)\) be an argumentation framework. + +1. Translating the \(AF\) to the \(PL_{2}\) for conflict-free semantics. + +A set \(S\subseteq A\) is conflict-free iff \(S\) is the core of a model of any formula below: + +\[\begin{aligned} \bigwedge_{a\in A}(a&\rightarrow\bigwedge_{b,a)}\neg b); \\ b:(&\in R \end{aligned}\] (2.1) + +\[\begin{aligned} \bigwedge_{a\in A}(a&\rightarrow\bigwedge_{a,b)}\neg b); \\ b:(&\in R \end{aligned}\] (2.2) + +\[\bigwedge_{(a,b)\in R}(\neg a\vee\neg b).\] (2.3) + +2. Translating the \(AF\) to the \(PL_{2}\) for stable semantics. + +A set \(S\subseteq A\) is a stable extension iff \(S\) is the core of a model of the formula below: + +\[\begin{aligned} \bigwedge_{a\in A}(a\leftrightarrow\bigwedge_{b,a)}\neg b). \\ b:(&\in R \end{aligned}\] (2.4) + +This formula is come up with by Creignou in [29] to characterize the kernels of a graph, and is applied in [12] to characterize the stable extensions of an \(AF\) . + +3. Translating the \(AF\) to the \(PL_{2}\) for admissible semantics. + +A set \(S\subseteq A\) is a admissible set iff \(S\) is the core of a model of the formula below: + +\[\begin{aligned} \bigwedge_{a\in A}((a&\rightarrow\bigwedge_{b,a)}\neg b)\wedge(a\rightarrow\bigwedge_{b,a)}(\bigvee_{c,b)}c))). \\ b:(&\in Rb:(\in Rc:(\in R \end{aligned}\] (2.5) + +4. Translating the \(AF\) to the \(PL_{2}\) for complete semantics. + +A set \(S\subseteq A\) is a complete extension iff \(S\) is the core of a model of the formula below: + +\[\begin{aligned} \bigwedge_{a\in A}((a&\rightarrow\bigwedge_{b,a)}\neg b)\wedge(a\leftrightarrow\bigwedge_{b,a)}(\bigvee_{c,b)}c))). \\ b:(&\in Rb:(\in Rc:(\in R \end{aligned}\] (2.6) diff --git a/outputs/arxiv_math/2503.07355_pg11_pg1_repeat1.md b/outputs/arxiv_math/2503.07355_pg11_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..5f61a849cb70cc7c70154ae0d7c662e9fb14e952 --- /dev/null +++ b/outputs/arxiv_math/2503.07355_pg11_pg1_repeat1.md @@ -0,0 +1,38 @@ +Table 2.2 also tells us whether the representation is real, complex or quaternionic. For the spinor representations, since Spin is a subspace of the even Clifford algebra, we only need to look at table 2.2, which implies that the number of irreducible inequivalent spinor representations is + +\[s_{r,s}= \begin{cases} 2\text{if}r-s=2,4\mod4, \\ 1\text{if}r-s=1,3\mod4. \end{cases}\] + +Notice that in the even dimensional case \(D=2k\) there are two inequivalent irreducible spinor representations (known as Weyl representations), which correspond to the Weyl spinors and can be understood by looking at the volume element \(v_{*}=v_{1}\cdots v_{D}\). Being \(v_{*}\) the product of an even number of generators, it anticommutes with them, i.e. \(\{v_{*},v_{a}\}=0\text{for all}a=1,\cdots ,D\), but it commutes with all the elements in the even Clifford subalgebra, hence with all the group elements in \(\text{Spin}(V)\), which implies that it must act as a scalar in the spinor representations. This means that the inequivalent Weyl representations can be labelled by the eigenvalues of \(v_{*}\). Furthermore, a straightforward computation gives, for even \(D=r+s=2k\) + +\[v_{*}^{2}=(-1)^{\frac{r-s}{2}}1.\] (15) + +One can classify the inequivalent spinor representation as follows + +- \(r-s=0\) mod 8. There are two inequivalent real spinor representations, of real dimension \(2^{\frac{D-2}{2}}\) , labeled by the eigenvalue of \(v_{*}\) being 1 or −1; + +- \(r-s=1,7\) mod 8. There is a unique spinor representation, which is real and of real dimension \(2^{\frac{D-1}{2}}\) ; + +- \(r-s=2,6\) mod 8. There are two inequivalent complex spinor representations, of complex dimension \(2^{\frac{D-2}{2}}\) , labeled by the eigenvalue of \(v_{*}\) being \(i\text{or}-i\); + +- \(r-s=3,5\) mod 8. There is a unique spinor representation, which is quaternionic and of quaternionic dimension \(2^{\frac{D-3}{2}}\) ; + +- \(r-s=4\) mod 8. There are two inequivalent quaternionic spinor representations, of quaternionic dimension \(2^{\frac{D-4}{2}}\) , labeled by the eigenvalue of \(v_{*}\) being 1 or −1; + +#### 2.4.1 Complex representations and the Lorentzian signature case + +As it is significantly easier to deal with complex Clifford algebras, we turn our attention to complex representations of \(\mathcal{C}(D)\). + +\[\mathcal{C}(2k)\simeq\mathbb{C}(2^{k})\text{and}\mathcal{C}(2k+1)\simeq\mathbb{C}(2^{k})\oplus\mathbb{C}(2^{k}),\] (16) + +\[\Gamma_{(2k)}:\mathcal{C}(2k)\rightarrow\text{End}(\mathbb{C}^{2^{k}})\] (17) + +\[\Gamma_{(2k+1)}:\mathcal{C}(2k+1)\rightarrow\text{End}(\mathbb{C}^{2^{k}})\oplus\text{End}(\mathbb{C}^{2^{k}}),\] (18) + +| We recall | | +| --- | --- | +| C(2k) ≃ C(2k) and C(2k + 1) ≃ C(2k) ⊕ C(2k), | (16) | +| which implies there are faithful representations | | +| Γ(2k) : C(2k) → End(C2k) | (17) | +| Γ(2k+1) : C(2k + 1) → End(C2k) ⊕ End(C2k), | (18) | + +where \(\Gamma_{2k}\) is irreducible and \(\Gamma_{2k+1}\) splits into two irreducible representations. These are precisely the pinor representations of the complex Clifford algebra. Remark 18. The above irreducible representations are unique up to conjugacy with unitary matrices. From now on, we drop the subscript and denote such representations just by \(\Gamma\). diff --git a/outputs/arxiv_math/2503.07361_pg12_pg1_repeat1.md b/outputs/arxiv_math/2503.07361_pg12_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..b316de4a65a27e6f2fb39d6c6b4d695c1f658094 --- /dev/null +++ b/outputs/arxiv_math/2503.07361_pg12_pg1_repeat1.md @@ -0,0 +1,25 @@ +respective circles intersect on \(\ell_{k+1}\). Observe that with this operation, the minimum distance between any two vertices is still \(d\). Next, we process first pairs. Observe that no vertex can be involved in several first pairs. Otherwise it is not on the outer face. However, a vertex can be contained in a first and another pair and it is also possible that a vertex closing another face is contained in a first pair. So some of the vertices might already have a fixed position. However, it is not possible that both vertices of a first pair are contained in another pair or close a face. Otherwise, the predecessor of the first pair was an internal vertex. So, we first process the first pairs that share a vertex with another pair or where one of its vertices close a face and reduce their distance to \(d\). Then we can also process all the other first pairs. In the end all pairs have distance \(d\text{and}d\) is still the minimum distance between any two vertices on \(\ell_{k+1}\). This concludes the construction for Items 1 and 2. The strips are defined as follows: Let \(u\in V_{k-1}\) and let \(v_{1},\ldots,v_{h}\) be the children of \(u\) from left to right. We partition the strip \(S_{u}\) into the strips \(S_{v_{1}},\ldots,S_{v_{h}}\) by splitting \(S_{u}\) at an arbitrary spot between \(x_{v_{i-1}}^{r}\text{and}x_{v_{i}}^{\ell},\text{for}i=2,\ldots,h\). If a vertex has two parents then its strip is the union of the two portions obtained from its parents. By construction, all edges of \(G_{s}\) have length at most one. It remains to show that edges in \(E_{\ell}\) have length greater than one. So let \(w_{j}w_{k}\in E_{\ell}\text{with}w_{j}\in V_{j},w_{k}\in V_{k}\), and \(j\leq k\). Since \(G\) is bipartite, we can exclude \(k=j\text{and}k=j+2.\text{If}k-j\geq3\) then \(y_{k}-y_{j}>k-1-j\geq2\). Hence, the distance between \(w_{j}\text{and}w_{k}\) is greater than one. If \(k-j=1\) then the parent \(w\text{if}w_{k}\) is in \(V_{j}\). By construction, we know that the distance between \(w_{j}\) and any child of \(w\) is greater than one. + +◀ + +longest possible short edge + +\[3n^{2}\] + +\[2n^{2}\] + +\(n^{2}\) + +\(n^{2}\) + +\[3n^{2}\] + +\(2n^{2}\) shortest possible long edge + +![](figure) + +● Figure 8 For each grid point \((i,j),1\leq i\leq n,1\leq j\leq n\) there are four possible points. If \(i>1\), the x-coordinate is \(in^{2}\) if the edge between \((i-1,j)\text{and}(i,j)\) is short and \(in^{2}+i\) otherwise. If \(j>1\), the y-coordinate is \(jn^{2}\) if the edge between \((i,j-1)\text{and}(i,j)\) is short and \(jn^{2}+j\) otherwise. + +## ▶ Theorem 9. A dichotomous ordinal graph \(G=(V,E_{s}\cup E_{\ell})\) admits a geometric realization if the set of short edges induces a subgraph of the grid. + +Proof. Extend \(G_{s}=(V,E_{s})\) by the remaining grid edges and require the new edges to be long. We now perturb the grid. For each grid point \((i,j),1\leq i\leq n,1\leq j\leq n\) of the original grid, there are four possible choices with the x-coordinates \(in^{2}\text{or}in^{2}+i\) and the y-coordinates \(jn^{2}\text{or}jn^{2}+j\). See Fig. 8. In the case where \(i=1\) choose as x-coordinate \(n^{2}\text{or}n^{2}+1\). Similarly, if \(j=1\) choose as y-coordinate \(n^{2}\text{or}n^{2}+1.\text{If}i>1\), we choose the x-coordinate \(in^{2}\) if the edge between \((i-1,j)\text{and}(i,j)\) is short and \(in^{2}+i\) otherwise. diff --git a/outputs/arxiv_math/2503.07368_pg8_pg1_repeat1.md b/outputs/arxiv_math/2503.07368_pg8_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..c4073f524e896f9631f8b6ee1ff687512c717ef7 --- /dev/null +++ b/outputs/arxiv_math/2503.07368_pg8_pg1_repeat1.md @@ -0,0 +1,23 @@ +If \(M\) is an interval module, then, when viewed as a persistence module of one-parameter persistence modules, every slice \(M_{(\exists,i)}\) is either zero or a single one-parameter interval (see Figure 2). This is a direct consequence of the definition of an interval module as an indecomposable thin persistence module supported on a convex connected subset of \(G(m,n)\). Since there is at most one non-zero morphism between two one-parameter interval modules, \(\text{id}=(\text{id},\ldots,\) id) is the only barcode basis for an interval module. + +▶ Proposition 7. For an interval module \(M,GC_{id}(M)\) is a directed path. + +Proof. By construction of \(GC_{\text{id}}(M)=GC(M)\), the vertices at every height of \(GC(M)\) correspond to the intervals in \(M_{(\exists,i)}\). Since \(M\) is an interval module, there is at most one vertex at every height. Since \(M\) is indecomposable all these vertices have to be connected. Otherwise Proposition 2 and 4 would imply that \(M\) is decomposable. Since there are only edges between vertices at consecutive height levels in a graphcode, \(GC(M)\) is a directed path. + +◀ + +Compressed graphcodes. A disadvantage of graphcodes is that every bar of every barcode is represented by a vertex, leading to a high number of vertices and therefore a large graph. However, in practical situations, the barcodes of consecutive slices often do not differ from each other, and the question is how to compress the information of a graphcode without losing any information. The following definition prepares our approach by relaxing the condition that graphcodes only connect bars on consecutive slices. + +▶ Definition 8 (generalized Graphcode). A directed graph \(G=(V,E,c)\) with vertex labels + +\[c:V\rightarrow\{1,\ldots,m\}\times\{1,\ldots,m+1\}\times\{1,\ldots,n\}\] + +is called a generalized graphcode if: + +1. for all \(v\in V\text{with}c(v)=(b,d,h)\), we have \(b)+ \frac{1}{8} \bar{\chi}\iota_{\hat{\gamma}}\iota_{\hat{\gamma}}( \underline{\gamma} d_{\omega}\psi)\right) \end{aligned}\] + +For \(c,\chi\text{and}\xi\), we can do the computations of \(Q_{0}^{2}\) right away, obtaining + +\[\begin{aligned} Q_{0}^{2}c&=Q_{PC}(\iota_{\xi}\delta_{\chi}\omega)+\delta_{\chi}\left( \frac{1}{2} \iota_{\xi}\iota_{\xi}F_{\omega}- \frac{1}{2} [c,c]+\iota_{\xi}\delta_{\chi}\omega\right) \\ &= \frac{1}{2} \iota_{[\xi,\xi]}\delta_{\chi}\omega+\iota_{\xi}\mathrm{L}_{\xi}^{\omega}\delta_{\chi}\omega-\iota_{\xi}[c,\delta_{\chi}\omega]+ \frac{1}{2} \iota_{\xi}\iota_{\varphi}F_{\omega}- \frac{1}{2} \iota_{\xi}\iota_{\xi}d_{\omega}\delta_{\chi}\omega \\ -[\iota_{\xi}\delta_{\chi}\omega,c]+ \frac{1}{2} \iota_{\varphi}\delta_{\chi}\omega+\iota_{\xi}\delta_{\chi}^{2}\omega \\ &= \frac{1}{2} \iota_{\varphi}\delta_{\chi}\omega+\iota_{\xi}Q_{0}^{2}\omega, \end{aligned}\] + +having used the fact that \(\frac{1}{2} \iota_{[\xi,\xi]}\delta_{\chi}\omega+\iota_{\xi}\mathrm{L}_{\xi}^{\omega}\delta_{\chi}\omega- \frac{1}{2} \iota_{\xi}\iota_{\xi}d_{\omega}\delta_{\chi}\omega=0\). For \(Q_{0}^{2}\xi\), we see + +\[\begin{aligned} Q_{0}^{2}(\xi)&= \frac{1}{2} Q_{0}([\xi,\xi]+\varphi)= \frac{1}{2} [\xi,\varphi]+ \frac{1}{2} Q_{0}(\bar{\chi}\gamma^{a}\chi e_{a}^{\mu})\partial_{\mu} \\ &= \frac{1}{2} [\xi,\varphi]+\mathrm{L}_{\xi}^{\omega}(\bar{\chi})\gamma^{\mu}\chi\partial_{\mu}-[c,\bar{\chi}]\gamma^{\mu}\chi\partial_{\mu}- \frac{1}{2} \iota_{\varphi}\bar{\psi}\gamma^{\mu}\chi\partial_{\mu}+ \frac{1}{2} \bar{\chi}\gamma^{a}\chi Q_{0}(e_{a}^{\mu})\partial_{\mu}. \end{aligned}\] + +Now, since \(e_{a}^{\mu}e_{\mu}^{b}=\delta_{a}^{b}\), we have \(Q_{0}(e_{a}^{\mu})=-e_{a}^{\nu}e_{b}^{\mu}Q_{0}(e_{\nu}^{b})\), obtaining + +\[\begin{aligned} Q_{0}^{2}(\xi)&= \frac{1}{2} [\varphi,\xi]+\mathrm{L}_{\xi}^{\omega}(\bar{\chi})\gamma^{\mu}\chi\partial_{\mu}-[c,\bar{\chi}]\gamma^{\mu}\chi\partial_{\mu}- \frac{1}{2} \iota_{\varphi}\bar{\psi}\gamma^{\mu}\chi\partial_{\mu} \\ - \frac{1}{2} e_{b}^{\mu}\bar{\chi}\gamma^{\nu}\chi\left((\mathrm{L}_{\xi}^{\omega}e)_{\nu}^{b}-[c,e_{\nu}]^{b}+\bar{\chi}\gamma^{b}\psi_{\nu}\right)\partial_{\mu} \\ &=- \frac{1}{2} [\xi,\varphi]+\mathrm{L}_{\xi}^{\omega}(\bar{\chi})\gamma^{\mu}\chi\partial_{\mu}-[c,\bar{\chi}]\gamma^{\mu}\chi\partial_{\mu}- \frac{1}{2} \iota_{\varphi}\bar{\psi}\gamma^{\mu}\chi\partial_{\mu} \\ + \frac{1}{2} \bar{\chi}[c,\gamma]_{V}^{\mu}\chi\partial_{\mu}- \frac{1}{2} \bar{\chi}\gamma^{\mu}\iota_{\varphi}\psi\partial_{\mu}+ \frac{1}{2} \bar{\chi}\mathrm{L}_{\xi}^{\omega}(\gamma^{\mu}\partial_{\mu})\chi \\ &=- \frac{1}{2} \mathrm{L}_{\xi}^{\omega}(\phi)+ \frac{1}{2} \mathrm{L}_{\xi}^{\omega}(\bar{\chi}\gamma^{\mu}\chi\partial_{\mu})=0. \end{aligned}\] diff --git a/outputs/arxiv_math/2503.07398_pg15_pg1_repeat1.md b/outputs/arxiv_math/2503.07398_pg15_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..993b4ef6655c04387a0d592ba69c3d58d2975eb0 --- /dev/null +++ b/outputs/arxiv_math/2503.07398_pg15_pg1_repeat1.md @@ -0,0 +1,37 @@ +\[\text{rk}(1_{A}^{C_{0}\oplus C_{1}})=\text{rk}(1_{A}^{C_{0}}\oplus1_{A}^{C_{1}})\geq\text{rk}(1_{A}^{C_{0}}),\] + +- for any measurable subset \(A\) of \(X\). Observe that \(\mathcal{A}_{\kappa}(\mathcal{X})\) does not necessarily contains a faithful \(\mathcal{X}\) -module. For instance, if \(\mathcal{X}\) has \(\aleph_{1}\) coarsely connected components, no coarse \(\mathcal{X}\) -module of \(\text{rank}\aleph_{0}\) can be faithful. However, it is possible to fully characterise the cardinals for which the approximable category contains a faithful module. Lemma 2.10. Let \(\mathcal{X}\) be a LFCM space with an infinite discrete partition. For any two discrete partitions \(\{A_{i}\}_{i\in I}\text{and}\{B_{j}\}_{j\in J}\), the cardinalities of \(I\text{and}J\) are equal. Moreover, for an infinite cardinal \(\kappa\), the following statements are equivalent: + +\((1)\kappa\) is greater than or equal to the cardinality of a discrete partition of \(\mathcal{X}\) ; + +\((2)\mathcal{A}_{\kappa}(\mathcal{X})\) contains a faithful \(\mathcal{X}\) -module; + +\((3)\mathcal{A}_{\kappa}(\mathcal{X})\) contains an ample \(\mathcal{X}\) -module. + +Proof. Since \(\{A_{i}\}_{i\in I}\) is a discrete partition of \(\mathcal{X}\) , for each \(j\in J\), there exist only finitely many \(i\in I\) such that \(B_{j}\cap A_{i}\neq \emptyset\). Consequently, \(|J|\leq|I|\times\aleph_{0}\). Similarly, one has \(|I|\leq|J|\times\aleph_{0}\). Note that if \(\mathcal{X}\) admits an infinite discrete partition, then any discrete partition of \(\mathcal{X}\) must also be infinite. It follows that + +\[|I|=|I|\times\aleph_{0}=|J|\times\aleph_{0}=|J|.\] + +For the second part of the statement, note that every ample \(\mathcal{X}\) -module is necessarily faithful. Conversely, given a faithful \(\mathcal{X}\) -module \(C\), let \(H\) be a separable, infinite-dimensional Hilbert space. Define the \(\mathcal{X}\) -module \(C\otimes H\), where the underlying Hilbert space is \(H_{C}\otimes H\) and the representation \(\bullet^{1}C\otimes H\) is given by \(=_{A^{1}A}^{C_{1}\otimes HC}\otimes\text{id}_{H}\) for all subsets \(A\). It is straightforward to verify that \(C\otimes H\) is an ample \(\mathcal{X}\) -module with the same rank as \(C\). It remains to establish the equivalence of (1) and (2). If \(\kappa\) exceeds the cardinality of the discrete partition, then the module constructed in Example 2.1 is a faithful \(\mathcal{X}\) -module of \(\text{rank}|I|\). Conversely, suppose \(C\) is a faithful \(\mathcal{X}\) -module of \(\text{rank}\kappa\). Define \(I_{C}=\{i\in I|A_{i}\cap\text{dom}_{1}(C)\neq \emptyset\}\subseteq I\). By discretising \(\mathcal{X}\text{and dom}_{1}(C)\), there exists a gauge \(E\in\mathcal{E}^{I}\) such that + +\[I=E[I_{C}]=\bigcup_{i\in I_{C}}E[\{i\}].\] + +- + +It follows that the cardinality of \(I\) is given by \(|I_{C}|\times\sup_{i\in I_{C}}|E[\{i\}]|\). Since \(|E[\{i\}]|\) is finite for all \(i\in I\), we deduce \(|I|=|I_{C}|\). We shall focus on the case of countably generated LFCM spaces. Accordingly, we aim to determine the cardinals \(\kappa\) for which the approximable category \(\mathcal{A}_{\kappa}(\mathcal{X})\) contains a faithful \(\mathcal{X}\) -module. The following lemma addresses this question in the connected case. Lemma 2.11. If \(\mathcal{X}\) is a countably generated LFCM space with a countable number of connected components, then \(\mathcal{A}_{\aleph_{0}}(\mathcal{X})\) contains a faithful \(\mathcal{X}\) -module. Proof. Let \(\{A_{i}\}_{i\in I}\) be a discrete partition of \(\mathcal{X}\) . By considering the discretisation \(\mathcal{I}=(I,\mathcal{E}^{I},\mathcal{P}(I))\) of \(\mathcal{X}\) , as in Proposition 1.9, we obtain a locally finite, countably generated coarse space. We may select a generating set \(\{E_{n}\}_{n\in\mathbb{N}}\text{for}\mathcal{I}\) satisfying the following conditions: + +1. For every \(n\in\mathbb{N}\), the inclusion \(E_{n}\subset E_{n+1}\) holds; + +2. For every \(n\in\mathbb{N}\), the entourage \(E_{n}\) is a gauge; + +3. For every entourage \(F\in\mathcal{E}^{I}\), there exists \(n\in\mathbb{N}\text{such that}F\subset E_{n}\). + +Since \(\mathcal{I}\) has a countable amount of connected components, there exists a sequence \(\{i_{k}\}_{k\in\mathbb{N}}\subset I\) such that for every \(j\in J\) there are \(n,k\in\mathbb{N}\text{such that}(i_{k},j)\in E_{n}.\text{As}\mathcal{I}\) is locally finite, the sets \(E_{n}[i_{k}]\) are finite for all \(n,k\in\mathbb{N}\). Hence, + +\[I=\bigcup_{k\in\mathbb{N}}\bigcup_{n\in\mathbb{N}}E_{n}[i_{k}]\] + +- is at most countable. Consequently, \(\mathcal{X}\) admits a countable discrete partition. By Lemma 2.10, the approximable category \(\mathcal{A}_{\aleph_{0}}(\mathcal{X})\) contains a faithful \(\mathcal{X}\) -module. In the case where \(\mathcal{X}\) has an uncountable number of connected components, the theorem above does not hold. For instance, consider an uncountable disjoint union of copies of \(\mathbb{N}\), equipped with + +an extended metric ( + +\[d(n,m)= \begin{cases} |n-m|, & \text{if}n,m\text{belong to the same copy of}\mathbb{N}; \\ \infty, & \text{otherwise}. \end{cases}\] diff --git a/outputs/arxiv_math/2503.07400_pg11_pg1_repeat1.md b/outputs/arxiv_math/2503.07400_pg11_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..e4222a6e3bd9f798f3ca691675ef5df57e45d1d7 --- /dev/null +++ b/outputs/arxiv_math/2503.07400_pg11_pg1_repeat1.md @@ -0,0 +1,27 @@ +Finally, we proceed to the main result of this section. We show that if \(G\) is a nontrivial \((k,g,s)-\text{multipole with}s\leq(k-2)g\), then it cannot happen that \(|G|\leq b_{1}(k,g,s)\). Thus, we obtain a nontrivial lower bound on the size of \(G\). + +Theorem 9. Let \(G\) be a nontrivial \((k,g,s)-\text{multipole for some}k\geq3,g\geq5\text{and}s\leq(k-2)g\). Then + +\[|G|\geq b_{2}(k,g,s)\geq \frac{1}{2} M(k,g).\] + +Proof. Note that \(b_{2}(k,g,s)\geq \frac{1}{2} M(k,g)\). It is easy to check that the statement is true for \(k=3\) and \(g\leq8\) (see Table 1), for other values we can use Lemma 7 and it is sufficient to prove \(|G|\geq b_{2}(k,g,(k-2)g)\) which is equivalent to \(|G|>b_{1}(k,g,(k-2)g)\). This is clearly true for \(s=0\). We will proceed by induction as follows. Let \(G\) be a nontrivial \((k,g,s)-\)multipole of order \(n\text{with}0b_{1}(3,g,g)\] + +We manually checked that this is true for all \(g<14.\text{For}g\geq14\), by Proposition 8 it is sufficient to show + +\[M(3,g-1)- \frac{(g-1)^{2}}{2} > \frac{g^{2}}{2} ,\text{or equivalently},M(3,g-1)>g^{2}-g+ \frac{1}{2} .(8)\] + +For \(g=2d+1\), Inequality (8) is equivalent to + +\[M(k,2d)=2\cdot2^{d}-2>4d^{2}+2d+ \frac{1}{2} ,\] (9) + +which is true for \(g\geq7\). For \(g=2d+2\), Inequality (8) is equivalent to + +\[M(3,2d+1)=3\cdot2^{d}-2>4d^{2}+6d+ \frac{11}{2} ,\] (10) + +which holds for \(d\geq6\), thus \(g\geq14\). diff --git a/outputs/arxiv_math/2503.07410_pg12_pg1_repeat1.md b/outputs/arxiv_math/2503.07410_pg12_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..48b59bc9e4e48d3b4596638c0b0393f1db69b464 --- /dev/null +++ b/outputs/arxiv_math/2503.07410_pg12_pg1_repeat1.md @@ -0,0 +1 @@ +accurately a polynomial time algorithm can approximate \(LV_{M,\ell^{p}}(\lambda)\). The known polynomial time algorithms give upper and lower bounds that differ by a power of \(N\). It is a closely related problem to approximate \(\Vert M\Vert_{p\rightarrow q}\) in polynomial time. In the regime \(21\), the known polynomial time algorithms give upper and lower bounds that differ by a power of \(N\) (cf. [GMU] and references therein). For some values of \(p,q\) in this range, it is known to be \(NP\) -hard to approximate \(\Vert M\Vert_{p\rightarrow q}\) within a constant factor – see [BV], [BBHKSZ], and [BGGLT]. The methods in these papers may lead to similar results for the large value problem. Because of these computational difficulties, there is no meaningful numerical evidence supporting the main conjectures about large values of Dirichlet polynomials. This is different from the situation with the Riemann hypothesis itself. There is a lot of numerical evidence supporting the Riemann hypothesis – for instance, we know that the first 109 zeroes of the Riemann zeta function have real part equal to 1/2 (EDIT: Reference). In contrast, we are not able to check the large value conjectures numerically even for \(N=200\). There is not currently a conjectural picture of how well polynomial time algorithms can approximate \(LV_{M,\ell^{p}}(\lambda).\text{If}M\) is allowed to vary among all \(T\times T\) matrices, then it looks reasonable to conjecture that it is \(NP\) -hard to approximate the function \(LV_{M,\ell^{p}}(N^{\sigma})\) to within a factor \(T^{\gamma}\), for some \(\gamma>0\). But there is no clear conjecture about the sharp value of the exponent \(\gamma\). Proving bounds for \(LV_{M,\ell^{p}}(\lambda)\) is not just hard in the worst case, it is also hard in the average case. If \(M\) is a random \(T\times N\) matrix, then Proposition 3.2 shows that with high probability, \(M\) obeys an essentially optimal large value estimate. But if we are given a particular matrix \(M\) that was sampled from the random distribution in Proposition 3.2, it is still hard to prove that \(M\) obeys strong large value estimates. In computer science, this is called the problem of certifying large value estimates for \(M\). It is closely related to the problem of distinguishing an honest random matrix \(M\) from a matrix that has a planted structure which causes \(LV_{M,\ell^{p}}(\lambda)\) to be much larger. I think the problem of certifying large value estimates for random matrices is closely parallel to the large value problem for Dirichlet polynomials. The two problems have been studied independently, but the methods and bounds that are known for the two problems are closely parallel, as we will see in the next few sections. The problem of certifying large value estimates for random matrices is part of the field of average case computational complexity. For a broad class of related problems in the field, the best current method is called the sum of squares hierarchy. The limits of this method are well understood in a number of cases. It is not fully understood what large value estimates for a random matrix \(M\) can be certified by the sum of squares method, but there was a lot of recent progress in [DHPT]. Based on recent developments in the field, there is good evidence that the sum of squares method cannot certify that a random matrix obeys the large value estimates in Proposition 3.2 or Conjecture 3.3 or 3.4. Experts in the field consider it to be plausible that the sum of squares method is the best polynomial time algorithm for a broad class of problems in average case computational complexity, including certifying large value estimates for random matrices. So there is a plausible conjectural picture about what large value estimates can be certified in polynomial time. In this picture, almost every random matrix obeys very strong large value estimates, but these strong large value estimates cannot be certified by a polynomial time algorithm. After this brief discussion of computational complexity, let us return to the question why the large value problem for Dirichlet polynomials is hard. Recall that \(M_{Dir}\) is the \(T\times N\) matrix that encodes Dirichlet polynomials. We would like to prove bounds for \(LV_{M_{Dir},\ell^{\infty}}(\lambda)\text{for all}N\) and all \(T=N^{O(1)}\). However, let us set a more modest goal and try to prove bounds when \(N=10^{3}\) and diff --git a/outputs/arxiv_math/2503.07421_pg26_pg1_repeat1.md b/outputs/arxiv_math/2503.07421_pg26_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..05de1ba8b9b936c1625d917be6016957229549a4 --- /dev/null +++ b/outputs/arxiv_math/2503.07421_pg26_pg1_repeat1.md @@ -0,0 +1,79 @@ +Recall the functions \(cov(l)\text{and}H(l)\) are defined in (4.3 and 4.4) respectively. For the sake of subsequent elaboration, we rewrite the functions \(cov\text{and}H\) as + +\[cov^{h}(l^{h})=cov(l^{*},l^{h})=cov(l)=\sum_{\sigma\in\mathcal{T}_{3,1}}c\widetilde{o}v_{\sigma}(l_{\sigma}^{h})+\sum_{\sigma\in\mathcal{T}_{4,0}}cov_{\sigma}(l_{\sigma})\] (5.3) + +and + +\[H^{h}(l^{h})=H(l^{*},l^{h})=H(l)=cov(l)-2\pi\sum_{i\in E^{i}}l,\] (5.4) + +where \(\mathcal{T}_{3,1}(\mathcal{T}_{4,0}\), resp.) represents the set of 3-1 type (4-0 type, resp.) tetrahedra in \(\mathcal{T}\) . + +Lemma 5.3. For \((M,\mathcal{T})\), let \(l^{h}\in\mathcal{L}^{h}(M,\mathcal{T})\) be a hyper-ideal edge length vector, then + +\[\frac{\partial H^{h}}{\partial l_{i}} =-K_{i},e_{i}\in E^{h}.\] + +Proof. For any \(e_{i}\in E^{h}\), if the hyper-ideal edge \(e_{i}\) is not a side of any 3-1 type tetrahedron in \(\mathcal{T}\) , then (4-1) in [9] had proved the conclusion. On the contrary, \(e_{i}\) is a side of some 3-1 type hyperbolic tetrahedra, take such a 3-1 tetrahedron \((\sigma,l_{\sigma})\), with + +\[l_{\sigma}=(l_{12}(l_{23},l_{24},l_{34}),l_{13}(l_{23},l_{24},l_{34}),l_{14}(l_{23},l_{24},l_{34}),l_{23},l_{24},l_{34}),\] + +where \((l_{23},l_{24},l_{34})=l_{\sigma}^{h}\in\mathbb{R}^{3}\), then + +\[\begin{aligned} \frac{\partial c\widetilde{o}v_{\sigma}(l_{\sigma}^{h})}{\partial l_{23}} &= \frac{\partial cov(l_{\sigma})}{\partial l_{23}} \\ &= \frac{\partial cov}{\partial l_{23}} + \frac{\partial cov}{\partial l_{12}} \frac{\partial l_{12}}{\partial l_{23}} + \frac{\partial cov}{\partial l_{13}} \frac{\partial l_{13}}{\partial l_{23}} + \frac{\partial cov}{\partial l_{14}} \frac{\partial l_{14}}{\partial l_{23}} \\ &=\alpha_{23}+\alpha_{12} \frac{\partial l_{12}}{\partial l_{23}} +\alpha_{13} \frac{\partial l_{13}}{\partial l_{23}} +\alpha_{14} \frac{\partial l_{14}}{\partial l_{23}} \\ &=\alpha_{23}+ \frac{\pi}{3} \frac{\partial}{\partial l_{23}} \left(l_{12}+l_{13}+l_{14}\right). \end{aligned}\] (5.5) + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
∂covf σ(lh)σ∂cov(lσ) ∂l23
=
∂l23
∂cov ∂cov ∂l12 ∂cov ∂l13 ∂cov ∂l14
(5.5)=+++
+ +Similarly, we also have + +\[\frac{\partial c\widetilde{o}v_{\sigma}(l_{\sigma}^{h})}{\partial l_{24}} =\alpha_{24}+ \frac{\pi}{3} \frac{\partial}{\partial l_{24}} \left(l_{12}+l_{13}+l_{14}\right).\] (5.6) + +\[\frac{\partial c\widetilde{o}v_{\sigma}(l_{\sigma}^{h})}{\partial l_{34}} =\alpha_{34}+ \frac{\pi}{3} \frac{\partial}{\partial l_{34}} \left(l_{12}+l_{13}+l_{14}\right).\] (5.7) + +| | ∂covf σ(lh) σ | π ∂ | +| --- | --- | --- | +| (5.6) | = α24 + | l12 + l13 + l14. | +| | ∂l24 | 3 ∂l24 | +| and | | | +| | ∂covf σ(lh) σ | π ∂ | +| (5.7) | = α34 + | l12 + l13 + l14. | +| | ∂l34 | 3 ∂l34 | + +Substitute (5.5), (5.6) and (5.7) into the following calculation of \(\partial H^{h}/\partial l_{i}\), we have diff --git a/outputs/arxiv_math/2503.07441_pg11_pg1_repeat1.md b/outputs/arxiv_math/2503.07441_pg11_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..c4ed546ab02970a0ace39798306d8f46f4797027 --- /dev/null +++ b/outputs/arxiv_math/2503.07441_pg11_pg1_repeat1.md @@ -0,0 +1,47 @@ +after a fixed choice of parametrization of boundary of \(\mathcal{D}\) by \(\mathbb{R}\). (In the case the end of \(\mathcal{D}\) is negative, the above is meant to be analogous.) + +A Hamiltonian connection \(\mathcal{A}\text{on}[0,1]\times M\) uniquely corresponds to a choice of a smooth function \(H:[0,1]\times M\rightarrow\mathbb{R}\), normalized to have mean zero at each moment. For the holonomy path of \(\mathcal{A}\) over \([0,1]\) is a path \(\phi_{\mathcal{A}}:[0,1]\rightarrow Ham(M,\omega)\), generated by a Hamiltonian \(H:[0,1]\times M\rightarrow\mathbb{R}\), and this uniquely determines the connection. Conversely, \(H\) uniquely determines a Hamiltonian connection with holonomy path generated by \(H\). We can say that \(H\) generates \(\mathcal{A}\). + +Lemma 2.26. Let \(p\text{and}\mathcal{L}_{p}\subset\partial\mathcal{D}\times M\) be as in definition above with \(L^{\pm}(\tilde{p})=\rho,\text{where}\tilde{p}\) is some lift of \(p\) to \(\text{Ham}(M,\omega)\), that is \(p(t)=\tilde{p}(t)(p(0))\). Let \(\mathcal{A}_{0}\) be a Hamiltonian connection on \([0,1]\times M\), generated by a Hamiltonian \(H:[0,1]\times M\rightarrow\mathbb{R}\text{with}L^{\pm}\) length \(\kappa\), constant for \(t\) near \(0,1\). Then there is a Hamiltonian connection \(\tilde{\mathcal{A}}_{0}^{p}on\mathcal{D}\times M\), preserving \(\mathcal{L}_{p}\), compatible with respect to \(\mathcal{A}_{0}\), and satisfying + +\[area(\tilde{\mathcal{A}}_{0}^{p})\leq\kappa+\rho.\] + +The construction is natural in the sense that \((\tilde{p},\mathcal{A}_{0})\mapsto\tilde{\mathcal{A}}_{0}^{p}\) can be made into a smooth map (of Frechet manifolds). + +Proof. Let \(q:[0,1]\rightarrow\text{Ham}(M,\omega)\) be the holonomy path of \(\mathcal{A}_{0},q(0)=id\), generated by \(H\). Let \(\tilde{p}\cdot q\) be the usual path concatenation in diagrammatic order, and \(H^{\prime}\) be its generating Hamiltonian. Define a coupling form \(\Omega^{\prime}\text{on}D^{2}\times M\): + +\[\Omega^{\prime}=\omega-d(\eta(rad)\cdot H^{\prime}d\theta),\] + +for \((rad,\theta)\) the modified angular coordinates on \(D^{2},\theta\in[0,1],0\leq rad\leq1,\text{and}\eta:[0,1]\rightarrow[0,1]\) is a smooth function satisfying + +\[0\leq\eta^{\prime}(rad),\] + +and + +\[\eta(rad)= \begin{cases} 1 & \text{if}1-\delta\leq rad\leq1, \\ rad^{2} & \text{if}rad\leq1-2\delta, \end{cases}\] (2.27) + +for a small \(\delta>0\). By an elementary calculation + +\[\text{area}(\mathcal{A}^{\prime})=L^{+}(p\cdot q)=L^{+}(p)+L^{+}(q),\] + +where \(\mathcal{A}^{\prime}\) is the connection induced by \(\Omega^{\prime}\). Set + +\[arc=\{(1,\theta)\in D^{2}|0\leq\theta\leq1/2\}.\] + +Let \(arc^{c}\) denote the complement of \(arc\text{in}\partial D^{2}\). Fix a smooth embedding \(i:D^{2}\hookrightarrow\mathcal{D}\) such that the following is satisfied (see Figure 1): + +- The image of the embedding contains \(\partial\mathcal{D}-end,\text{where}end\) is the image of the distinguished (say positive) strip end chart + +\(\bullet i(arc)\subset end^{c}\), + +\(\bullet i(arc^{c})\subset end\). + +\[[0,1]\times(0,\infty)\rightarrow\mathcal{D}.\] + +Next fix a deformation retraction \(ret\) of \(\mathcal{D}\) onto \(i(D^{2})\), so that in the strip end chart above, for \(r\geq1 ret\) is the composition \(i\circ param\circ pr\), where + +the projection and where + +\[pr:[0,1]\times(0,\infty)\rightarrow[0,1]\] + +\[param:[0,1]\rightarrow arc^{c}\subset D^{2}\] diff --git a/outputs/arxiv_math/2503.07447_pg1_pg1_repeat1.md b/outputs/arxiv_math/2503.07447_pg1_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..34298ab7bd601f6f369848c07a3f689be817e022 --- /dev/null +++ b/outputs/arxiv_math/2503.07447_pg1_pg1_repeat1.md @@ -0,0 +1,25 @@ +# A new density limit for unanimity in majority dynamics on random graphs + +##### Jeong Han Kim and BaoLinh Tran + +##### June 2022 + +###### Abstract + +Majority dynamics is a process on a simple, undirected graph \(G\) with an initial Red/Blue color for every vertex of \(G\). Each day, each vertex updates its color following the majority among its neighbors, using its previous color for tie-breaking. The dynamics achieves unanimity if every vertex has the same color after finitely many days, and such color is said to win. When \(G\) is a \(G(n,p)\) random graph, L. Tran and Vu (2019) found a codition in terms of \(p\) and the initial difference \(2\Delta\) beteween the sizes of the Red and Blue camps, such that unanimity + +is achieved with probability arbitrarily close to 1. They showed that if \(p\Delta^{2}\gg1,p\Delta\geq100\), and + +\(p\geq(1+\varepsilon)n^{-1}\log n\) for a positive constant \(\varepsilon\), then unanimity occurs with probability \(1-o(1)\). If \(p\) is not extremely small, namely \(p>\log^{-1/16}n\), then Sah and Sawhney (2022) showed that + +the condition \(p\Delta^{2}\gg1\) is sufficient. + +If \(n^{-1}\log^{2}n\ll p\ll n^{-1/2}\log^{1/4}n,\) we show that \(p^{3/2}\Delta\gg n^{-1/2}\log n\) is enough. Since this condition holds if \(p\Delta\geq100\text{for}p\) in this range, this is an improvement of Tran’s and Vu’s result. For the closely related problem of finding the optimal condition for \(p\) to achieve unanimity when the initial coloring is chosen uniformly at random among all possible Red/Blue assignments, our + +result implies a new lower bound \(p\gg n^{-2/3}\log^{2/3}n\), which improves upon the previous bound of \(n^{-3/5}\log n\) by Chakraborti, Kim, Lee and T. Tran (2021). + +## 1 Introduction + +### 1.1 Majority Dynamics + +Consider a parliamentary election with two parties. The election season begins with Day 0, where each voter supports one side. At the end of every day, each voter polls their friends, then supports the majority side the next morning, or keeps their affiliation in case of a tie. On election day, everyone votes for the party they currently supports and the parliament will be divided proportionally with each side’s number of votes. This process can be modeled with a graph process, where vertices represent voters and edges represent friendships. We let Red and Blue, or R and B, stand for the two parties. The state of affiliations at any time is simply a coloring on the graph. This process on a graph is called Majority Dynamics in literature, due to the majority-based updating rule. diff --git a/outputs/arxiv_math/2503.07448_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.07448_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..d5c4ed0ed79a1872ab43d0381b10a24ec0a3af16 --- /dev/null +++ b/outputs/arxiv_math/2503.07448_pg3_pg1_repeat1.md @@ -0,0 +1,11 @@ +Conjecture 3 ([6]). For all \(L,C\in\mathbb{N}\), there exists \(C^{\prime}\in\mathbb{N}\text{such that if}\varphi\) is an \((L,C)-\) quasi-isometry from a graph \(G\) to a graph \(H\), then there is an edge-weighting function \(w:E(H)\rightarrow\mathbb{N}\) such that the same function \(\varphi\) is a \((1,C^{\prime})-\text{quasi}-\)isometry from \(G\) to the weighted graph \((H,w)\). + +This conjecture, if true, would imply that for any class of graphs \(H\) closed under contracting edges and taking subdivisions, if a graph \(G\) is \((L,C)-\text{quasi}-\)isometric to a graph \(H\in H,\text{then}G\) is \((1,C^{\prime})-\text{quasi}-\)isometric to a graph \(H^{\prime}\in H\text{where}C^{\prime}\) depends only on \(L\) and \(C\). This would be a remarkably powerful result if true. In this paper, we disprove this conjecture by constructing explicit counterexamples demonstrating that multiplicative distortion for weighted graphs is, in fact, necessary. + +Theorem 4. For every \(C\in\mathbb{N}\), there exist graphs \(G\text{and}H\) and a \((2,\) 1)-quasi-isometry \(\varphi:V(G)\rightarrow V(H)\) such that, for every edge weighting \(w:E(H)\rightarrow\mathbb{R}^{+}\) of \(H\), the map \(\varphi\) is not a \((1,C)-\text{quasi}-\)isometry from \(G\) to \((H,w)\). + +Our proof for Theorem 4 is based on orientated graphs \(H\) with large girth and chromatic number. We use the orientation of the edges to split each vertex into a new edge to define our graph \(G\) (see Figure 1). Since \(H\) can be obtained from \(G\) by contracting disjoint edges, the natural map \(\varphi\) between \(G\text{and}H\) is a \((2,\) 1)-quasi-isometry. Next, we assume that \(H\) is given an edge-weighting function \(w:E(H)\rightarrow\mathbb{R}^{+}\). We then separate the light-weight edges from the heavy-weight edges and then find different types of long orientated paths within the graph. This allows us to find in \(H\) either a long path of light-weight edges that traverse many new edges, or a long path of heavy-weight edges that avoids the new edges. Since \(H\) has large girth, such paths are geodesic which allows us to contradict \(\varphi\) being a \((1,C)-\text{quasi}-\)isometry from \(G\) to \((H,w)\). + +### 1.1 Preliminaries + +Let \(G\) be a graph. The girth of \(G\) is the length of a shortest cycle in \(G.\text{For}k\in\mathbb{N}\), a proper \(k-\)colouring of a \(G\) is a function \(c:V(G)\rightarrow\{1,\ldots,k\}\text{such that}c(u)/=c(v)\) whenever \(uv\in E(G)\). The chromatic number \(\chi(G)\) is the minimum \(k\in\mathbb{N}\) for which \(G\) has a proper \(k-\text{colouring}\). Let \((H,w)\) be a weighted graph. For a path \(P=(v_{0},v_{1},\ldots,v_{n})\text{in}(H,w)\), we say that the length of \(P\) is \(\sum_{i=1}^{n}w(v_{i-1}v_{i})\) and the hop-length of \(P\) is \(n\). The path \(P\) is geodesic if it is a path of minimum length in \((H,w)\) between \(v_{0}\text{and}v_{n}\). An oriented graph \(\overline{H}\) is a graph where each edge has a direction. The chromatic number of \(\overline{H}\) is the chromatic number of \(H\). Let \(\overline{P_{n}} =(v_{0},v_{1},v_{2},\ldots,v_{n})\) be an oriented path. We diff --git a/outputs/arxiv_math/2503.07449_pg17_pg1_repeat1.md b/outputs/arxiv_math/2503.07449_pg17_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..8e819fe311759f5da2e784eef44bce106fb60c5a --- /dev/null +++ b/outputs/arxiv_math/2503.07449_pg17_pg1_repeat1.md @@ -0,0 +1,27 @@ +![A plot showing L_2 error (hat rho) versus Delta hat t, with data points and a fitted line indicating an exponent of 2. The y-axis is labeled 'L_2 error, hat rho' and the x-axis is labeled 'Delta hat t'.](figure)[^inferred] + +![A plot showing L_2 error, \( \hat{q} \), versus \( \Delta \hat{t} \) on a log-log scale. The y-axis is labeled 'L_2 error, \( \hat{q} \)' and ranges from \( 10^{-6} \) to \( 10^{-5} \). The x-axis is labeled \( \Delta \hat{t} \) and ranges from \( 10^{-4} \) to 1. Several data points are shown as blue dots, and a black line fits through them, indicating a power law relationship. The exponent of the fit is marked as 2.](figure)[^inferred] + +![](figure) + +FIG. 9. Second-order convergence of the numerical solution of the fields \(\hat{\varrho}\text{and}\hat{q}˙\). + +![A plot showing the relative error (\( \hat{T}_{\text{rel. err.}} \)) versus time (\( \hat{t} \)) for two curves labeled 'rear side (20)' in blue and 'rear side (50)' in orange. The y-axis is labeled \( \hat{T}_{\text{rel. err.}} \) ranging from -0.01 to 0.01, and the x-axis is labeled \( \hat{t} \) ranging from 0 to 30. A legend indicates the color coding for each curve.](figure)[^inferred] + +![](figure) + +![A line plot showing the relative error (\( \hat{\rho} \) rel. err.) versus time (\( \hat{t} \)) for two curves labeled 'rear side (20)' in blue and 'rear side (50)' in orange. The y-axis is labeled 'rel. err.' and ranges from -0.025 to 0.025. The x-axis is labeled 't' and ranges from 0 to 30. A legend indicates the color coding for each curve.](figure)[^inferred] + +FIG. 10. Grid dependence of time evolution of temperature and density at the rear side of the sample. + +![A line plot showing temperature difference \( \hat{T} - \hat{T}_0 \) versus time \( \hat{t} \) for four cases: front side, nonviscous; front side, viscous; rear side, nonviscous; rear side, viscous. The y-axis is labeled \( \hat{T} - \hat{T}_0 \) ranging from 0.00 to 0.04, and the x-axis is labeled \( \hat{t} \) ranging from 0 to 20. Four colored lines represent different cases, with a legend indicating which line corresponds to each case.](figure)[^inferred] + +![A line plot showing temperature difference \( \hat{T} - \hat{T}_0 \) versus time \( \hat{t} \) for four cases: front side, nonviscous; front side, viscous; rear side, nonviscous; rear side, viscous. The y-axis is labeled \( \hat{T} - \hat{T}_0 \) ranging from 0.00 to 0.04, and the x-axis is labeled \( \hat{t} \) ranging from 0 to 20. Four colored lines represent different cases, with a legend indicating which line corresponds to each case.](figure)[^inferred] + +FIG. 11. The effect of viscous momentum transport on the temperature field. + +# B. The thermoacoustic approximation of the piston effect + +This section is devoted to analyze the piston effect in supercritical fluids via the developed scheme. In this case the duration of excitation is much longer than the acoustic time scale. Therefore, wave propagation occurs only at the beginning of the process until the initial pressure disturbance reaches the other end of the pipe, correspondingly, no reflection takes place. In order to prove the intensive effect of thermal expansion near the critical point, we present the same phenomenon with same dimensional excitation parameters for an ideal gas state, too. Based on [19], we have defined a cross-section specific heat of \(q=30 \frac{\mathrm{J}}{\mathrm{m}^{2}}\) , to which in the above defined supercritical state the dimensionless value \(\hat{q}=0.00035\) corresponds, the non-dimensional pulse duration \(\hat{t}_{\mathrm{P}}=50\) and Péclet number Pea = 107 are chosen, which latter implies \(Re_{\mathrm{a}}=1722695.711\). Now, viscous momentum transport is also taken into + +[^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/2503.07452_pg8_pg1_repeat1.md b/outputs/arxiv_math/2503.07452_pg8_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..e7d3c3b0f118c6034753b7ba7d21135463d2f965 --- /dev/null +++ b/outputs/arxiv_math/2503.07452_pg8_pg1_repeat1.md @@ -0,0 +1,39 @@ +### 3.2. Constraints + +To obtain feasible solutions within this simulation-based optimization framework, it is crucial to establish suitable constraints. Constraints are tied to the simulation residuals, which guide the optimization process towards achieving the desired fitness and ensuring numerically stable results. When the residuals are below the specified thresholds, the constraints are deemed satisfied. If the residuals exceed the limits, a penalty is imposed for failing to meet the criteria. The pressure residual constraint, given by: + +\[r_{p}(\mathbf{b})\leq1e^{-3}\] (11) + +helps ensure consistent pressure values during the optimization process, preventing pressure imbalances that could result in unrealistic behavior. The velocity residuals ensure controlled and physically realistic fluid motion and are defined for both velocity components: + +\[r_{u_{x}}(\mathbf{b})\leq1e^{-4}\] (12) + +\[r_{u_{y}}(\mathbf{b})\leq1e^{-4}\] (13) + +\[r_{k}(\mathbf{b})\leq1e^{-4}\] (14) + +| rux(b) ≤ 1e−4 | (12) | +| --- | --- | +| ruy(b) ≤ 1e−4 | (13) | +| The k residual constraint, defined as: | | +| rk(b) ≤ 1e−4 | (14) | + +ensures that the turbulent kinetic energy remains within acceptable bounds. The \(\omega\) residual constraint, set as: + +\[r_{\omega}(\mathbf{b})\leq1e^{-4}\] (15) + +limits the specific dissipation rate of turbulence, maintaining alignment with the fundamental physics. These constraints collectively guide the optimization process, ensuring physical realism, stability, and relevant fluid dynamics representation. All five constraints are assessed across all test cases. + +## 4. Optimization procedure + +The selection of the most suitable optimization algorithm for similar flow field reconstruction method was previously addressed in [47], where the performance, efficiency, robustness, and scalability of various methods were evaluated, leading to the adoption of PSO as the preferred approach for this type of modeling. Although the authors noted that optimization outcomes are case-dependent, they observed similar results across different test cases due to the synthetic nature of the simulations, despite variations in data sources. As outlined in [47], the mean square difference of the velocities at measurement points, denoted as \(\epsilon_{d}\), is used as the fitness function in all optimization tests. The optimization is thus defined as follows: + +\[\text{mini}_{\mathbf{b}}\text{mize}\epsilon_{d}(\mathbf{b})= \frac{1}{n_{MP}} \sum_{i=1}^{n_{MP}}(\mathbf{u}_{r_{i}}-\mathbf{u}_{s_{i}}(\mathbf{b}))^{2}\] (16) + +\[\text{subject to}\mathbf{b}_{l}\leq\mathbf{b}\leq\mathbf{b}_{u}\] + +| minimize | ϵd(b) = 1 XMP (uri − usi(b))2 | | +| --- | --- | --- | +| b subject to | nMP i=1 bl≤ b ≤ bu | (16) | + +Convergence of the optimization process is considered to be achieved when the drifter error threshold, \(\epsilon_{d}=1e^{-4}\), is reached, corresponding to a drifter velocity error in \(m/s\). All reconstruction results for which these threshold is met are deemed satisfactory. diff --git a/outputs/arxiv_math/2503.07453_pg68_pg1_repeat1.md b/outputs/arxiv_math/2503.07453_pg68_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..cc9f14b2b1344e972069293a9cab714b0050e08c --- /dev/null +++ b/outputs/arxiv_math/2503.07453_pg68_pg1_repeat1.md @@ -0,0 +1,35 @@ +# J Guarantee for UncertainStateAction + +In this section, we present the main guarantee of UncertainStateAction (Algorithm 6) as a standalone algorithm; see Lemma J.1. Then, in Lemma J.2, we provide its guarantee when used as a subroutine within MTSS (Algorithm 4). For a discussion of the motivation for these results, we refer back to Section I.1.3. Lemma J.1. Consider a call to UncertainStateAction\((_{h}\mathcal{C}_{0:h-1},\widehat{\pi}_{1:h},\Sigma_{h};\mathrm{a},N, \overline{N} )\) (Algorithm 6) for some given \(h,\mathcal{C}_{0:h-1},\widehat{\pi}_{1:h},\Sigma_{h},\mathrm{a}\in\mathcal{A},N,\text{and} \overline{N} \text{such that}\sigma_{\text{min}}(\Sigma_{h})\geq\lambda,\text{for some}\lambda\in(0,1)\). Then, for any \(\delta^{\prime}\in(0,1)\text{and}\zeta\in(0,1/2)\), with probability at least \(1-\delta^{\prime}\), the output \((\hat{x}_{h},\hat{a}_{h})\) of UncertainStateAction satisfies: + +- For all \(\ell\in[0..h-1]\text{and}(x_{\ell},a_{\ell})\in\mathcal{C}_{\ell}\), + +\[\mathbb{P}_{\widehat{\pi}_{\ell+1:h}}\left[\Vert\varphi_{h}(\boldsymbol{x}_{h},\boldsymbol{a}_{h})\Vert_{\Sigma_{h}^{-1}}^{2}>2\left(\zeta\vee\Vert\varphi_{h}(\hat{x}_{h},\hat{a}_{h})\Vert_{\Sigma_{h}^{-1}}^{2}\right)|\boldsymbol{x}_{\ell}=x_{\ell},\boldsymbol{a}_{\ell}=a_{\ell}\right]\leq\max_{h\in[H]} \frac{4\log\left( \tfrac{16H|\mathcal{C}_{h}|}{\lambda\delta^{\prime}\zeta} \right)}{N} ,(90)\] + +\[\text{where}\varphi_{h}(\cdot,\cdot):=\phi_{h}(\cdot,\cdot)-\phi_{h}(\cdot,\mathrm{a}).\] + +- Furthermore, there exists \(\mathcal{X}_{h,\text{span}}\subseteq\mathcal{X}\) such that for all \(\ell\in[0..h-1]\text{and}(x_{\ell},a_{\ell})\in\mathcal{C}_{\ell},\mathbb{P}_{\widehat{\pi}\ell+1:h-1}[\boldsymbol{x}_{h}\in \frac{4}{N} \log^{\underline{32HN|\mathcal{C}_{h}|}}\) + +and + +\(\mathcal{X}_{h,\text{span}}|\boldsymbol{x}_{\ell}=x_{\ell},\boldsymbol{a}_{\ell}=a_{\ell}]\geq1-\max_{h\in[H]}\) + +\(\lambda\delta^{\prime}\zeta\) + +\[\forall x_{h}\in\mathcal{X}_{h,\text{span}},\mathbb{P}_{\boldsymbol{a}\sim\pi_{h,\text{ref}}(\cdot|x_{h})}\left[\Vert\varphi_{h}(x_{h},\boldsymbol{a})\Vert_{\Sigma_{h}^{-1}}^{2}>2\left(\zeta\vee\Vert\varphi_{h}(\hat{x}_{h},\hat{a}_{h})\Vert_{\Sigma_{h}^{-1}}^{2}\right)\right]\leq\max_{h\in[H]} \frac{4\log\left( \tfrac{16H|\mathcal{C}_{h}|}{\lambda\delta^{\prime}\zeta} \right)}{\overline{N}} .(91)\] + +## Proof of Lemma J.1. + +Fix \(\delta^{\prime}\in(0,1)\text{and}\zeta\in(0,1/2)\), and let \(\Gamma:=\{\zeta,2\zeta,\ldots,\lceil \frac{4}{\zeta\lambda} \rceil\zeta\}.\) Further, for \(\ell\in[0..h-1]\text{and}(x_{\ell},a_{\ell})\in\mathcal{C}_{\ell}\), let \(\mathcal{D}_{\ell}(x_{\ell},a_{\ell})\) be the dataset in Algorithm 6 when the algorithm returns. Note that \(\mathcal{D}_{\ell}(x_{\ell},a_{\ell})\) consists of \(N\) i.i.d. pairs sampled from \(\mathbb{P}_{\widehat{\pi}_{\ell+1:h}}[(\boldsymbol{x}_{h},\boldsymbol{a}_{h})=\cdot|\boldsymbol{x}_{\ell}=x_{\ell},\boldsymbol{a}_{\ell}=a_{\ell}]\). Thus, by Freedman’s inequality (Lemma C.2) and the union bound over \(\ell\in[0..h-1],(x_{\ell},a_{\ell})\in\mathcal{C}_{\ell},\text{and}\gamma\in\Gamma\), there is an event \(\mathcal{E}\) of probability at least \(1-\delta^{\prime}/2\) under which, + +\[\begin{aligned} \forall\ell&\in[0..h-1],\forall(x_{\ell},a_{\ell})\in\mathcal{C}_{\ell},\forall\gamma\in\Gamma,\mathbb{P}_{\widehat{\pi}_{\ell+1:h}}\left[\Vert\varphi_{h}(\boldsymbol{x}_{h},\boldsymbol{a}_{h})\Vert_{\Sigma_{h}^{-1}}^{2}\geq\gamma|\boldsymbol{x}_{\ell}=x_{\ell},\boldsymbol{a}_{\ell}=a_{\ell}\right] \\ &\leq \frac{2}{N} \sum_{(x,a)\in\mathcal{D}_{\ell}(x_{\ell},a_{\ell})}\mathbb{I}\left\{\Vert\varphi_{h}(x,a)\Vert_{\Sigma_{h}^{-1}}^{2}>\gamma\right\}+ \frac{4\log(2H|\mathcal{C}_{\ell}||\Gamma|/\delta^{\prime})}{N} , \\ &\leq \frac{2}{N} \sum_{(x,a)\in\mathcal{D}_{\ell}(x_{\ell},a_{\ell})}\mathbb{I}\left\{\Vert\varphi_{h}(x,a)\Vert_{\Sigma_{h}^{-1}}^{2}>\gamma\right\}+ \frac{4\log\left( \tfrac{16H|\mathcal{C}_{\ell}|}{\lambda\delta^{\prime}\zeta} \right)}{N} ,(92) \end{aligned}\] + +where the last step follows by the facts that \(|\Gamma|\leq\lceil \frac{4}{\zeta\lambda} \rceil,\lambda\in(0,1),\text{and}\zeta\in(0,1/2)\). Now, since \(\sigma_{\text{min}}(\Sigma_{h})\geq\lambda\) and \(\sup_{(x,a)\in\mathcal{X}\times\mathcal{A}}\Vert\phi_{h}(\cdot,\cdot)\Vert\leq1\) (Assumption H.1), we have that \(\sup_{(x,a)\in\mathcal{X}\times\mathcal{A}}\Vert\varphi_{h}(x,a)\Vert_{\Sigma^{-1}}^{2}\leq \frac{4}{\lambda} .\) Therefore, by the definition of \(\Gamma\), we have that for all \(\ell\in[0..h-1]\text{and}(x_{\ell},a_{\ell})\in\mathcal{C}_{\ell}\), there exists \(\gamma_{\ell}(x_{\ell},a_{\ell})\in\Gamma\) such that + +\(h\) + +\[\begin{aligned} \max_{(x,a)\in\mathcal{D}_{\ell}(x_{\ell},a_{\ell})}\Vert\varphi_{h}(x,a)\Vert_{\Sigma_{h}^{-1}}^{2}&\leq\gamma_{\ell}(x_{\ell},a_{\ell}), \\ &\leq\max_{(x,a)\in\mathcal{D}_{\ell}(x_{\ell},a_{\ell})}\Vert\varphi_{h}(x,a)\Vert_{\Sigma_{h}^{-1}}^{2}+\zeta, \\ &\leq2\left(\zeta\vee\max_{(x,a)\in\mathcal{D}_{\ell}(x_{\ell},a_{\ell})}\Vert\varphi_{h}(x,a)\Vert_{\Sigma_{h}^{-1}}^{2}\right), \\ &\leq2\left(\zeta\vee\Vert\varphi_{h}(\hat{x}_{h},\hat{a}_{h})\Vert_{\Sigma_{h}^{-1}}^{2}\right), \end{aligned}\] (93) + +where the last inequality follows by the fact that + +\[\max_{\ell\in[0..h-1](x,a)}\max_{\in\mathcal{D}_{\ell}(x_{\ell},a_{\ell})}\Vert\varphi_{h}(x,a)\Vert_{\Sigma_{h}^{-1}}^{2}\leq\Vert\varphi_{h}(\hat{x}_{h},\hat{a}_{h})\Vert_{\Sigma_{h}^{-1}}^{2}\] diff --git a/outputs/arxiv_math/2503.07467_pg6_pg1_repeat1.md b/outputs/arxiv_math/2503.07467_pg6_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..8c9188477367afcb552747e397f1ae2afb154725 --- /dev/null +++ b/outputs/arxiv_math/2503.07467_pg6_pg1_repeat1.md @@ -0,0 +1,43 @@ +Based on the above notions, we get the existence of a \(N-\)bubble solution. + +Theorem 1.1 Suppose that \((A_{1})\) holds and either of the following holds + +\(e_{t=1}^{(\hat{m}^{*}-2)H_{i,t}(p^{*})}h_{i}(p^{*}\) + +\(\sum^{N}\) + +\((1)\hat{m}^{*}<4\text{and}\sum\) + +\()_{t}D_{i,t}\neq 0\); + +\(i\in\hat{I}\) + +\((2)\hat{m}^{*}=4\text{and}\sum_{i=1}^{n}\) + +\(\sum^{N}\) + +\(e_{t=1}^{2H_{i,t}(p^{*})}h_{i}(p^{*}\) + +\()_{t}L_{i,t}\neq 0\); + +\[(3)\hat{m}^{*}=4,\sum_{i=1}^{n}\sum_{t=1}^{N}e^{2H_{i,t}(p^{*})}h_{i}(p_{t}^{*})L_{i,t}=0\text{and}\sum_{i=1}^{N}\sum_{t=1}^{N}e^{2H_{i,t}(p^{*})}h_{i}(p_{t}^{*})D_{i,t}\neq 0.\] + +Then, there exist \(\rho_{\varepsilon}:=(\rho_{1,\varepsilon},\cdots ,\rho_{n,\varepsilon})\) and a family of solutions \(\{u_{\varepsilon}:=(u_{1,\varepsilon},\cdots ,u_{n,\varepsilon})\}_{\varepsilon}\) to Problem (1.1) corresponding to \(\rho_{\epsilon}\text{such that}\rho_{i,\varepsilon}\rightarrow\rho_{i}^{*}(i=1,...,n)\) and + +\[\frac{h_{i}(x)e^{u_{i,\varepsilon}}}{\int_{\Omega}h_{i}(x)e^{u_{i,\varepsilon}}} \rightarrow\sum_{t=1}^{N}2\pi m_{i}^{*}\delta_{p_{t}^{*}}\text{in the sense of measure}\] + +as \(\varepsilon\rightarrow0+\). Here, \(\delta_{p}\) is the Dirac measure supported at \(p\in\Omega\) and the numbers \(D_{i,s}\text{and}L_{i,s}\) are defined as in (1.8) and (1.9). Moreover, the solution \(u_{\varepsilon}\) satisfies the sharp estimates \((SE_{1})\) and \((SE_{3})\). Furthermore, \((SE_{2})\) holds if Cases (1) or (2) occur, we get + +\[\Lambda_{I,N}(\rho_{\varepsilon})=-\sum_{i=1}^{n} \frac{2}{N} \left(D_{i,t}+\sum_{s\neq t} \frac{e^{2H_{i,s}(p^{*})}h_{i}(p_{s}^{*})}{e^{2H_{i,t}(p^{*})}h_{i}(p_{t}^{*})} D_{i,s}+o(1)\right)\varepsilon_{t}^{2}\] (1.11) + +if Case (3) occurs. + +Remark 1.2 \((A1)\) and Assumptions \((1-3)\) in Theorem 1.1 are based on the blowup analysis of Liouville systems in [31, 23, 20]. In particular, the surprising (1.10) plays a crucial role. + +Remark 1.3 The subtlety of our construction can be observed from Assumption (3) in Theorem 1.1 because it is new even for the blowup analysis in [20]. + +Remark 1.4 \(D_{i,t}\) is a global quantity because it is involved with global integration, \(L_{i,t}\) is a local quantity because its definition only needs information at blowup point. This difference is consistent with the blowup analysis not only for Liouville systems [31], but also for the singular Liouville equations [5] + +Remark 1.5 Because of the difficulty on the structure of solutions, there has not been too many works on the construction of blowup solutions of Liouville systems. In [22] Huang constructed a sequence of one-bubble blowup solutions under further restrictions. In this article we removed all the extra assumptions and constructed the multi-bubble blowup solutions without additional restrictions. + +Remark 1.6 Strictly speaking, the family of solutions \(\{u_{\varepsilon}=(u_{1,\varepsilon},\cdots ,u_{n,\varepsilon})\}_{\varepsilon}\) depends on \(N\) parameters \(\varepsilon_{1},\cdots ,\varepsilon_{N}\), which also depend on each other. For the sake of simplicity, we will still use \(\varepsilon\) to parameterize the solutions. diff --git a/outputs/arxiv_math/2503.07488_pg6_pg1_repeat1.md b/outputs/arxiv_math/2503.07488_pg6_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..7f97bf8284dc05b6628ec56d338fb4432a618f1d --- /dev/null +++ b/outputs/arxiv_math/2503.07488_pg6_pg1_repeat1.md @@ -0,0 +1,31 @@ +![Diagram showing an ellipse with axes labeled \( \lambda_1 \), \( l_1 \), \( \lambda \), \( l \), and angles \( \varphi_1 \), \( \theta \), \( \psi \), and \( \varphi \). The point O is marked at the center of the ellipse.](figure)[^inferred] + +![Diagram showing an ellipse with axes labeled \( \lambda_1 \), \( l_1 \), \( \lambda \), \( l \), and angles \( \varphi_1 \), \( \theta \), \( \psi \), and \( \varphi \). The point O is marked at the center of the ellipse.](figure)[^inferred] + +FIGURE 1. The billiard map \(f(\varphi,\lambda)=(\varphi_{1},\lambda_{1})\), the normal angle \(\psi=(\varphi_{1}+\varphi)/2\) and the incidence-reflection angle \(\theta=(\varphi_{1}-\varphi)/2\). + +Let \(p/q\in(0,1)\) be a rational number with \(\gcd(p,q)=1\). We call the elements of the space + +\[X=\left\{\{\varphi_{j}\}\in\mathbb{R}^{\mathbb{Z}}:\varphi_{j}<\varphi_{j+1}<\varphi_{j}+2\pi,\varphi_{j+q}=\varphi_{j}+2\pi p,\forall j\in\mathbb{Z}\right\}\] + +\(p/q-\)periodic sequences. We define the \(p/q-\)periodic action \(A:X\rightarrow\mathbb{R}\) as + +\[A\{\varphi_{j}\}=\sum_{j=0}^{q-1}S(\varphi_{j},\varphi_{j+1})=2\sum_{j=0}^{q-1}h(\psi_{j})\sin\theta_{j},\] + +where \(\psi_{j}=\sigma\{\varphi_{j}\}/2\text{and}\theta_{j}=\delta\{\varphi_{j}\}/2\). Periodicities \(\varphi_{j+q}=\varphi_{j}+2\pi p,\psi_{j+q}=\psi_{j}+2\pi p\) and \(\theta_{j+q}=\theta_{j}\) imply that \(\mu\{S(\varphi_{j_{0}+j},\varphi_{j_{0}+j+1})\}=2\mu\{h(\psi_{j_{0}+j})\sin\theta_{j_{0}+j}\}=A\{\varphi_{j}\}/q\) for all \(j_{0}\in\mathbb{Z}\). Critical points of the action, which we call \(p/q-\)periodic configurations, can be lifted to full \(p/q-\)periodic orbits of the billiard map \(f\) by taking \(\lambda_{j}=-\partial_{1}S(\varphi_{j},\varphi_{j+1})=\partial_{2}S(\varphi_{j-1},\varphi_{j})\). Thus, any \(p/q-\)periodic configuration defines a \(p/q-\)periodic billiard trajectory inside \(\Gamma\) with side angles \(\varphi_{j}\), normal angles \(\psi_{j}=\sigma\{\varphi_{j}\}/2\) and incidence-reflection angles \(\theta_{j}=\delta\{\varphi_{j}\}/2\). We characterize such configurations in the next proposition, where we also recall a formula for the length of their corresponding periodic billiard trajectories. + +## Proposition \(\mathbf{3}.Ap/q-\)periodic sequence \(\{\varphi_{j}\}\) is \(ap/q-\)periodic configuration if and only if + +\[\sigma\{h^{\prime}(\psi_{j})\sin\theta_{j}\}-\delta\{h(\psi_{j})\cos\theta_{j}\}=0,\forall j\in\mathbb{Z},\] + +where \(\psi_{j}=\sigma\{\varphi_{j}\}/2\text{and}\theta_{j}=\delta\{\varphi_{j}\}/2\), in which case + +\[2q\mu\left\{h(\psi_{j})\sin\theta_{j}\right\}=A\{\varphi_{j}\}=L,\mu\{h^{\prime}(\psi_{j})\sin\theta_{j}\}=0,\mu\{\theta_{j}\}=\pi p/q,\] + +where \(L\) is the length of the \(p/q-\)periodic configuration \(\{\varphi_{j}\}\). Proof. A \(p/q-\)periodic sequence \(\{\varphi_{j}\}\) is a critical point of the action if and only if + +\[\sigma\{h^{\prime}(\psi_{j})\sin\theta_{j}\}-\delta\{h(\psi_{j})\cos\theta_{j}\}=\partial_{2}S(\varphi_{j},\varphi_{j+1})+\partial_{1}S(\varphi_{j+1},\varphi_{j+2})=0\] + +for all \(j\in\mathbb{Z}\). The first equality above follows from \(S(\varphi_{j},\varphi_{j+1})=2h(\psi_{j})\sin\theta_{j}\) and relations \(\psi_{j}=\sigma\{\varphi_{j}\}/2=(\varphi_{j+1}+\varphi_{j})/2\text{and}\theta_{j}=\delta\{\varphi_{j}\}/2=(\varphi_{j+1}-\varphi_{j})/2\). + +[^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/2503.07508_pg14_pg1_repeat1.md b/outputs/arxiv_math/2503.07508_pg14_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..7d6fe386bd5054890a2eb36d97dae1633ee61c95 --- /dev/null +++ b/outputs/arxiv_math/2503.07508_pg14_pg1_repeat1.md @@ -0,0 +1,21 @@ +very similar. Lemma 3.2 will be applied for self-similar measures \(\mu\), but we do not yet need to assume that \(\mu\) is self-similar. + +Lemma 3.2. Let \(\mu\) be a compactly supported Borel probability measure on \(\mathbb{R}^{k}\) and let \(f:\mathbb{R}^{k}\rightarrow\mathbb{R}^{d}\) be continuous. Consider the map \(T:\mathbb{R}^{k}\rightarrow\mathbb{R}^{k+d}\), \(\mathbf{x}\mapsto(\mathbf{x},f(\mathbf{x}))\) and the image measure \(\mu_{T}\) . Then for all \(\boldsymbol{\xi}\in\mathbb{R}^{d}\), letting 0 be the zero vector with \(k\) entries, + +\[\hat{\mu}_{T}(\mathbf{0},\boldsymbol{\xi})=\hat{\mu}_{f}(\boldsymbol{\xi}).\] + +- + +Proof. This is simply a matter of unpacking the definition of the Fourier transform from (1.1). Now \(\mu\) be a self-similar measure on \(\mathbb{R}^{k}\). Without loss of generality, we can assume that the support of \(\mu\) is contained in a unit cube. Let \(\Lambda\) be an IFS for \(\mu\). For each \(\omega\in\Lambda^{*}\) (the space of finite-length sequences with symbols in \(\Lambda)\), following the composition of maps \(\omega_{0}\omega_{1}\cdots\) , there is a least \(l\geq0\) such that the contraction ratio corresponding to \(\omega_{0}^{l}=\omega_{0}\cdots \omega_{l}\) is at most \(1/2^{n}\). We write \(l=l_{\omega,n}\). Notice that the finite collection of paths + +\[\{\omega_{0}^{l\omega,n}\}_{\omega\in\Lambda}\] + +corresponds to a covering of \(\text{supp}(\mu)\). Such a covering uses similar copies of \(\text{supp}(\mu)\) of sizes at most \(1/2^{n}\) and at least \(\rho_{m}2^{-n}\text{where}\rho_{m}\) is smallest contraction ratio for \(\mu\). Let \(\mu_{\omega,n}\) be the corresponding similar copy of \(\mu\) with probability weight \(|\mu_{\omega,n}|\). Clearly, each such branch \(\text{supp}(\mu_{\omega,n})\) intersects at least one and at most \(2^{k}\) many dyadic cubes in \(\mathcal{D}_{n}\). We choose only one such dyadic cube and associate it with \(\mu_{\omega,n}\). As a result, for each dyadic cube \(D_{n}\in\mathcal{D}_{n}\), there is a collection of similar copies \(\mu_{\omega,n}\) that are associated with this cube. We write \(\mu_{\omega,n}\sim D_{n}\) for this association. We will call such a collection of \(\omega\) to be \(\Lambda_{D_{n}}\). We then obtain the following decomposition of \(\Lambda\) + +\[\{\Lambda_{D_{n}}\}_{D_{n}\in\mathcal{D}_{n}}\] + +which induces a decomposition (or disintegration) of \(\mu\). For each \(\omega\in\Lambda_{D_{n}}\), the copy \(\mu_{\omega,n}\) is supported in \(3D_{n}\), the tripling of \(D_{n}\) with the same centre. Given a \(C^{2}\) map \(f:\mathbb{R}^{k}\rightarrow\mathbb{R}^{d}\), write \(\Gamma_{f}=\{(\mathbf{x},f(\mathbf{x})):\mathbf{x}\in\mathbb{R}^{k}\}\subset\mathbb{R}^{k+d}\) for the graph of \(f\). Let \(T_{\omega,n}\) be the \((k-\text{dimensional})\) tangent plane of \(\Gamma_{f}\) at \(\mathbf{y}_{0}=(\mathbf{x}_{0},f(\mathbf{x}_{0}))\) for x0 being the centre of \(D_{n}\). Given \(\mu_{\omega,n}\sim D_{n}\subset\mathbb{R}^{k}\) and \(\boldsymbol{\xi}\in\mathbb{R}^{k+d}\) with first \(k\) coordinates 0, let \(\boldsymbol{\xi}_{\omega,n}\in\mathbb{R}^{k}\) be such that \((\boldsymbol{\xi}_{\omega,n},\mathbf{0})\) is the projection of \(\boldsymbol{\xi}\) to \(\mathbb{R}^{k}\times\mathbf{0}\subset\mathbb{R}^{k+d}\) along the direction that is orthogonal to \(T_{\omega,n}\), i.e. + +\[((\boldsymbol{\xi}_{\omega,n},\mathbf{0})-\boldsymbol{\xi})\perp T_{\omega,n}.\] (3.1) + +Note that in the following statement we do not yet assume that \(f\) has positive Gaussian curvature. Lemma 3.3. Let \(\mu\) be a self-similar measure on \(\mathbb{R}^{k}\), let \(f:\mathbb{R}^{k}\rightarrow\mathbb{R}^{d}\) be \(C^{2}\), and let \(T(\mathbf{x})=(\mathbf{x},f(\mathbf{x}))\). Then for sufficiently large \(n\in\mathbb{N}\text{and}\boldsymbol{\xi}\in\mathbb{R}^{k+d}\) diff --git a/outputs/arxiv_math/2503.07509_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.07509_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..85d740cd968b840527f0c738456284cebb610e3b --- /dev/null +++ b/outputs/arxiv_math/2503.07509_pg2_pg1_repeat1.md @@ -0,0 +1,23 @@ +![Block diagram of a neural network system with Tx Autoencoder, Rx1 Autodecoder, Rx2 Autodecoder, and noise addition blocks. The Tx Autoencoder takes input vectors \( s_1 \) and \( s_2 \), processes through FCNN layers, and outputs \( h_1 \) and \( h_2 \). These are then fed into the Rx1 and Rx2 Autodecoders, which also include batch normalization and FCNN layers, and produce outputs \( \hat{s}_1 \) and \( \hat{s}_2 \) respectively.](figure)[^inferred] + +![Block diagram of a neural network system with Tx Autoencoder, Rx1 Autodecoder, Rx2 Autodecoder, and noise addition blocks. The Tx Autoencoder takes input vectors \( s_1 \) and \( s_2 \), processes through FCNN layers, and outputs \( h_1 \) and \( h_2 \). These are then fed into the Rx1 and Rx2 Autodecoders, which also include batch normalization and FCNN layers, and produce outputs \( \hat{s}_1 \) and \( \hat{s}_2 \) respectively.](figure)[^inferred] + +Fig. 1: The architecture of the implemented AE-NOMA for two-users. Each \(\text{FCNN}[\ell]\) represents a fully connected layer with \(\ell\) nodes. The main structure which consists five hidden layers each with 32 neurons is replicated at Tx, Rx1 and Rx2. The Tx, however, has a side block (Sub-Network 2) which is used to adjust power allocation for each symbol. + +## II. SYSTEM MODEL + +Consider a downlink NOMA system involving a base station (BS) communicating with two users. All nodes are equipped with single antennas. The user with weaker (stronger) channel is referred to as the weak (strong) user. The channels between the BS and the weak and strong users are respectively denoted by \(h_{1}\text{and}h_{2}\) satisfying \(|h_{1}|^{2}\leq|h_{2}|^{2}\). Both channel gains are known at the BS, but each user only knows its own channel gain. All channel gains are complex-valued and independently drawn from a continuous distribution. In a NOMA system, the BS employs superposition coding to concurrently transmit messages√ to both√ users. The transmitted signal \(x\) is defined by \(x= \overline{\alpha P} s_{1}+ \overline{\bar{\alpha}P} s_{2},\text{where}s_{1}\text{and}s_{2}\) are independent and identically distributed complex Gaussian signals with zero mean and unit variance, \(\mathcal{C}\mathcal{N}(0,1)\). Here, \(P\) represents the BS’s transmit power budget, \(\alpha\in[0,1]\) is the fraction of power allocated to the weak user, and \(\bar{\alpha}≜1-\alpha\). The received signals at the users are expressed as + +\[y_{k}=h_{k}x+n_{k},k\in\{1,2\}\] (1) + +where the noises \(n_{1}\text{and}n_{2}\) are independent and identically distributed circularly-symmetric complex Gaussian random variables with zero mean and unit variance, \(\mathcal{C}\mathcal{N}(0,1)\). The weak user decodes its message by treating the interfering signal from the other user as noise. In contrast, the strong user first decodes the weak user’s message, treating its own interfering signal as noise, and then applies SIC to decode its own message. The combined use of superposition coding and SIC decoding with Gaussian codebooks results in achieving the capacity region of this channel. + +## III. NETWORK STRUCTURE + +## A. Autoencoder’s Structure + +An autoencoder consists of two parts, an encoder and a decoder. The AE processes a binary input vector \(\mathbf{s}\in[0,1]^{d}\) + +by first mapping it to a hidden representation \(\mathbf{u}\in[0,1]^{d^{\prime}}\) through a deterministic mapping \(\mathbf{u}=f_{\theta}(\mathbf{s})=\sigma(W\mathbf{s}+\mathbf{b})\), where \(\theta=\{W,\mathbf{b}\},W\) is a \(d^{\prime}\times d\) weight matrix, and b is a bias vector. Here, \(\sigma(x)= \frac{1}{1+e^{x}}\) is the sigmoid function and \(\sigma(\mathbf{x})= [\sigma(\mathbf{x}_{1}),\ldots,\sigma(\mathbf{x}_{d})]^{T}\) . The resulting latent representation u is then transformed back to a reconstructed vector ˆs \(\in[0,1]^{d}\) in the input space, given by ˆs \(=g_{\theta^{\prime}}(\mathbf{u})=\sigma(W^{\prime}\mathbf{u}+\mathbf{b}^{\prime})\) with \(\theta^{\prime}=\{W^{\prime},\mathbf{b}^{\prime}\}\). For each training sample \(\mathbf{s}^{(i)}\), the AE maps it to a corresponding \(\mathbf{u}^{(i)}\) and a reconstruction \(\hat{\mathbf{s}}^{(i)}\). The proposed AE-NOMA network, illustrated in Fig. 1, consists of one transmitter (Tx) encoder and two decoders each at one of the receivers, i.e., Rx1 and Rx2. Each encoder and decoder, include a fully connected neural network (FCNN) composed of a sequence of fully connected layers and some residual connections. As seen in Fig. 1, the encoder (Tx) comprises a main block and a secondary network, referred to as Sub-Network 1 and Sub-Network 2, respectively. This main block, replicated at Tx, Rx1 and Rx2, consists of an input layer, five hidden layers, and an output layer. The hidden layers contain 32 neurons each, while the final layer (output layer) has two neurons representing the in-phase (I) and quadrature-phase (Q) components of the symbol designed by the autoencoder for transmission. In addition to the main block, the Tx has a side block (Sub-network 2) which helps to better adjust the I and Q components under a given average power constraint, thereby optimizing the use of the I/Q plane in a way similar to a quadrature amplitude modulation (QAM). In the main block, we have also implemented residual connections to enhance learning capacity and improves performance without the need for additional parameters or a wider network. These shortcuts also play a crucial role in preserving gradients within the network. We have used the activation function exponential linear unit which provides a smooth, nonzero output for negative inputs, and can help improve learning stability and performance. + +[^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/2503.07532_pg14_pg1_repeat1.md b/outputs/arxiv_math/2503.07532_pg14_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..4edffe1c3b998322e8b0e8319393cc8bc250adf3 --- /dev/null +++ b/outputs/arxiv_math/2503.07532_pg14_pg1_repeat1.md @@ -0,0 +1,9 @@ +In particular \(F_{\text{min}}(\alpha)\) determines \(\mathcal{F}[\alpha]\), and furthermore \(\mathcal{F}[\alpha]\) determines \(F_{\text{min}}(\alpha)\) up to conjugacy except in the case that \(\mathcal{F}[\alpha]=\mathcal{A}\): in that case, \(F_{\text{min}}(\alpha)\) can be either trivial or atomic. Despite this ambiguity we will refer to either the free factor \(F_{\text{min}}(\alpha)\) or the free factor system \(\mathcal{F}[\alpha]\) as the filling support of \(\alpha\); in context the exact referent should be clear. We sometimes use the notation \(\mathcal{F}[\alpha;T]\) to highlight the dependence on the free splitting \(T\) that contains \(\alpha\). Also, we note that \(\mathcal{F}[\alpha]\) is determined by the connected subgraph \(\beta_{\mathrm{I}d}^{\mathrm{o}}(\alpha)\), in that it is the unique smallest free factor system that “carries” the subgroup \(\text{Stab}(\beta_{\mathrm{I}d}^{\mathrm{o}}(\alpha))\). For this reason we sometimes denote \(\mathcal{F}[\alpha]=\mathcal{F}[\beta^{\mathrm{o}}(\alpha;T)]\text{and KR}(\alpha)=\text{KR}(\beta^{\mathrm{o}}(\alpha;T))\). Remark. In the statement of Definition 2.8, Case (1) holds if and only if \(\beta_{\mathrm{I}d}^{\mathrm{o}}(\alpha)=\alpha\) if and only if for all \(g\neq h\in\Gamma\) the translated paths \(g\cdot\alpha\text{and}h\cdot\alpha\) do not overlap; in this case the protocomponents of \(\beta^{\mathrm{o}}(\alpha)\) are precisely the translates of \(\alpha\). Also, Case (2) holds if and only if there exists a vertex \(V\in\beta_{\mathrm{I}d}^{\mathrm{o}}(\alpha)\text{such that Stab}(\beta_{\mathrm{I}d}^{\mathrm{o}}(\alpha))=\text{Stab}(V)\text{and}[\text{Stab}(V)]\in\mathcal{A}\); it follows in this case that \(V\in\alpha,\text{and that}\beta_{\mathrm{I}d}^{\mathrm{o}}(\alpha)=\bigcup_{g\in\text{Stab}(V)}g\cdot\alpha\). + +A continued example. In the example we gave preceding the statement of Theorem 2.4, the protocomponents of \(\beta_{\mathrm{I}d}^{\mathrm{o}}(\alpha)\) are the axes in \(T\) of the infinite cyclic free factor \(\langle a\rangle\) of \(\Gamma=\langle a,b\rangle\), and so the filling support is \(\mathcal{F}[\alpha;T]=\{[\langle a\rangle]\}\) which is a nonfull free factor system, despite the fact that the path \(\alpha\subset T\) does indeed have an interior crossing of a translate of every natural edge orbit of \(T\) : the tree \(T\) has only one natural edge orbit, and \(\alpha\) crosses three different natural edges of that orbit, the middle one of those three being an interior crossing. + +A new example. This example shows that the subgroup \(\text{Stab}(\beta_{\mathrm{I}d}^{\mathrm{o}}(\alpha))\) need not be a free factor of \(\Gamma\) rel \(\mathcal{A}\), in contrast to the fact that the stabilizer of every component of the Γ-invariant subforest \(\beta(\alpha)\subset T\) is a free factor of \(\Gamma\) rel \(\mathcal{A}\). Consider the rank 4 free group \(\Gamma=\langle a,b,c,d\rangle\). Let \(G\) be a marked graph with two vertices \(p,q\), with the \(\langle a,b\rangle\) rose attached to \(p\), the \(\langle c,d\rangle\) rose attached to \(q\), and an edge \(e\) from \(p\) to \(q\); we identify \(\pi_{1}(G,p)\approx\langle a,b\rangle*\langle c,d\rangle\approx\langle a,b,c,d\rangle\) by using \(e\) as a maximal subtree of \(G\). Let \(U=\tilde{G}\) be the universal covering with edge labels lifted from \(G\). Choosing a lift \(\tilde{p}\in U\) of \(p\) determines the deck transformation action \(\Gamma↷U\). Let \(\tilde{\alpha}\subset U\) be the path with initial vertex \(\tilde{p}\) that is labelled by the word \(ecd\bar{c}\bar{d}\bar{e}ab\bar{a}\bar{b}e\). Note that \(\tilde{\alpha}\) is contained in the \(U-\)axis of the infinite cyclic subgroup + +\[C=\langle cdc^{-1}d^{-1}aba^{-1}b^{-1}\rangle<\langle a,b,c,d\rangle\] + +Also, \(\tilde{\alpha}\) consists of one entire fundamental domain for the action of \(C\) on its \(U-\text{axis}\), followed by the first \(e\) edge of the next fundamental domain. Let \(T\) be the free splitting obtained from \(U\) by collapsing all edges labelled \(a,b,c,d\), so \(T\) has a single natural edge orbit, represented by the \(T\) -image of any \(e-\)edge of \(U\). Let \(\alpha\subset T\) be the image of \(\tilde{\alpha}\). Note that \(\alpha\) crosses the unique natural edge orbit three times, the middle crossing being an interior crossing. Again \(\alpha\) is contained in the axis of \(C\text{in}T\) , and \(\alpha\) consists of one entire fundamental domain of that axis followed by the first edge of the next fundamental domain. Note that no two distinct translates of the axis of \(C\) have a common edge in \(T\) . It follows that the protocomponent \(\beta_{\mathrm{I}d}^{\mathrm{o}}(\alpha)\) equals the axis of \(C\text{in}T\) , and that \(C=\text{Stab}(\beta_{\mathrm{I}d}^{\mathrm{o}}(\alpha))\). But \(C\) has trivial image under abelianization of \(\langle a,b,c,d\rangle\) and hence \(C\) is not a free factor of \(\langle a,b,c,d\rangle\). In fact we have the equation \(\mathcal{F}[\alpha;T]=\{[\langle a,b,c,d\rangle]\}\), in other words \(C\) fills the group \(\langle a,b,c,d\rangle\), and hence \(\alpha\) fills \(T\) by Proposition 2.9. The fact that \(C\) fills \(\langle a,b,c,d\rangle\) can be proved using a beautiful transversality argument that we learned from a paper of Stallings [Sta00]. Here are some details of this argument. Consider \(S\) diff --git a/outputs/arxiv_math/2503.07533_pg5_pg1_repeat1.md b/outputs/arxiv_math/2503.07533_pg5_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..c221567ba92efe082a22f6ac825e4dfd6d032e0a --- /dev/null +++ b/outputs/arxiv_math/2503.07533_pg5_pg1_repeat1.md @@ -0,0 +1,13 @@ +Remark 1. Convergence of the Markov process to the deterministic gradient flow describing trait dynamics occurs in the limit of small jumps \(\sigma^{2}\rightarrow0[9]\). In system \((1),\varepsilon\) therefore acts as a small timescale separation parameter \(0<\varepsilon\ll1\). On the slow timescale \(\tau=\varepsilon t\) we recover the canonical equation of adaptive dynamics in the limit \(\varepsilon\rightarrow0\). Methods from geometric singular perturbation theory will be useful in the study of (1) for fixed \(a\in A\) later. + +![](figure) + +Figure 2. Examples of growth landscape \(h(u,a)=h_{0}(u)-ah_{1}(u)\) + +(2) for treatment values \(a\in\{0,a_{M}\}\). (a) Simple cost of resistance landscape. (b) An example without cost of resistance, where the growth landscape takes a unique maximum for high trait-values, remaining nearly unaffected by the treatment. (c) A more complex cost of resistance example. Under the effect of treatment, a single global maximum of the landscape is deformed into a “double well”type potential at \(a=a_{M}\). (d) A landscape with partial cost of resistance. It has a global maximum at high resistance for \(a=0\), shielded by a local maximum near \(u=0\). Parameter values and functions used can be found in Appendix B. + +## 3. Preliminary results + +We write the system (1) more compactly as an affine control or switching system + +\[\dot{x}=f(x,\alpha(t))=f_{0}(x)+\alpha(t)f_{1}(x).\] (3) diff --git a/outputs/arxiv_math/2503.07537_pg4_pg1_repeat1.md b/outputs/arxiv_math/2503.07537_pg4_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..202b60b9c175e77fa3fc2caa97acb33e5b664b35 --- /dev/null +++ b/outputs/arxiv_math/2503.07537_pg4_pg1_repeat1.md @@ -0,0 +1,27 @@ +Note that the level lines \(\{(x_{1},x_{2})\in\mathbb{R}^{2}:2U(x_{1})+x_{2}^{2}=r^{2}\}\text{for all}0<|r|<(2\vartheta)^{-1/2}\) correspond to \(T(r)-\)periodic solutions + +\[\begin{aligned} \hat{x}_{1,0}(t,r)&\equiv r\text{sn}\left( \frac{t}{\sqrt{k_{r}^{2}+1}} ,k_{r}\right) \sqrt{k_{r}^{2}+1} , \\ \hat{x}_{2,0}(t,r)&\equiv r\text{cn}\left( \frac{t}{\sqrt{k_{r}^{2}+1}} ,k_{r}\right)\text{dn}\left( \frac{t}{\sqrt{k_{r}^{2}+1}} ,k_{r}\right), \\ T(r)&\equiv4\mathcal{K}(k_{r}) \sqrt{k_{r}^{2}+1} ,\nu(r)\equiv \frac{2\pi}{T(r)} , \end{aligned}\] (8) + +where \(\text{sn}(u,k),\text{cn}(u,k),\text{dn}(u,k)\) are the Jacoby elliptic functions, \(\mathcal{K}(k)\) is the complete elliptic integral + +of the first kind, and \(k_{r}\in(0,1)\) is a root of the equation \((k_{r}+k_{r}^{-1})^{-2}=\vartheta r^{2}/2\). Define auxiliary \(2\pi-\) periodic functions + +\[X_{1}(\varphi,r)\equiv\hat{x}_{1,0}\left( \frac{\varphi}{\nu(r)} ,r\right),X_{2}(\varphi,r)\equiv\hat{x}_{2,0}\left( \frac{\varphi}{\nu(r)} ,r\right).\] (9) + +It can easily be checked that + +\[\begin{aligned} \nu(r)\partial_{\varphi}X_{1}&=X_{2},\nu(r)\partial_{\varphi}X_{2}=-U(X_{1}), \\ 2U(X_{1})+X_{2}^{2}&\equiv r^{2},\det \frac{\partial(X_{1},X_{2})}{\partial(\varphi,r)} \equiv \begin{vmatrix} \partial_{\varphi}X_{1} & \partial_{\varphi}X_{2} \\ \partial_{r}X_{1} & \partial_{r}X_{2} \end{vmatrix} \equiv \frac{r}{\nu(r)} . \end{aligned}\] + +Hence, the transformation (9) is invertible for all \(0<|r|<(2\vartheta)^{-1/2}\text{and}\varphi\in[0,2\pi)\). Then, by applying Itô’s formula [39, §4.2], it can be shown that in the variables \((r,\varphi)\) system (7) takes the form (1) with + +\[\mu(t)\equiv t^{-1/4},s_{0}=3/2,s_{i}=0,\] + +\[\alpha_{i}a_{,j^{i}}(_{(}r_{r},_{,}\varphi_{\varphi},_{,}S_{S},_{,}t_{t})_{)}\equiv_{\equiv}t_{t^{-_{-} \frac{n}{\frac{p4}{4}}}\alpha}a_{ii^{,},^{n}j,}(_{p}r_{(},_{r}\varphi_{,\varphi},S_{,S})_{)}+_{,}t_{i}^{-} \frac{p}{,^{2}} ja_{\in^{i,2p}\{}^{\varepsilon}(_{1}r_{,},_{2}\varphi_{\}},_{,}S),i\in\{1,2\},\] (10) + +where + +\[\begin{aligned} a_{1,n}(r,\varphi,S)&\equiv \frac{X_{2}(\varphi,r)}{r} (\mathcal{Q}(S)X_{1}(\varphi,r)+\mathcal{P}(S)X_{2}(\varphi,r)), \\ a_{1,2p}^{\varepsilon}(r,\varphi,S)&\equiv \frac{\varepsilon^{2}\mathcal{B}^{2}(S)U(X_{1}(\varphi,r))}{r^{3}} , \\ a_{2,n}(r,\varphi,S)&\equiv- \frac{\nu(r)\partial_{r}X_{1}(\varphi,r)}{r} (\mathcal{Q}(S)X_{1}(\varphi,r)+\mathcal{P}(S)X_{2}(\varphi,r)), \\ a_{2,2p}^{\varepsilon}(r,\varphi,S)&\equiv- \frac{\varepsilon^{2}\mathcal{B}^{2}(S)\nu(r)}{2r} \partial_{r}\left( \frac{\partial_{r}X_{1}(\varphi,r)X_{2}(\varphi,r)}{r} \right), \\ \alpha_{1,1,p}(r,\varphi,S)&\equiv \frac{\mathcal{B}(S)X_{2}(\varphi,r)}{r} , \\ \alpha_{2,1,p}(r,\varphi,S)&\equiv- \frac{\mathcal{B}(S)\nu(r)\partial_{r}X_{1}(\varphi,r)}{r} , \end{aligned}\] + +and \(\alpha_{1,2,p}(r,\varphi,S)\equiv\alpha_{2,2,p}(r,\varphi,S)\equiv0\). Note that \(0<\nu(r)<1\text{for all}0<|r|<(2\vartheta)^{-1/2}\). Hence, If \(\mathcal{G}(x,y,S)\equiv0\text{and}\varepsilon=0,\text{then}r(t)\equiv_{0}\text{and}\varphi(t)\equiv\nu(_{0})t+\phi_{0}\) with arbitrary constants ̺ 0 and \(\phi_{0}\). Numerical analysis of the system with \(\mathcal{G}(x,y,S)\not\equiv 0\text{and}\varepsilon=0\) shows that, depending on the values of the perturbation parameters, two cases are possible. In the first case, the amplitude of the solutions either tends to zero or grows until it reaches the boundary of the considered domain, depending on the sign of \(\mathcal{Q}_{0}\) (see Fig. 1, a). In the second case, the solutions with an asymptotically constant amplitude \(r(t)\approx r_{0}\) may appear (see Fig. 1, b). The emergence of such solutions is associated with the capture of the system into resonance. If \(\varepsilon\neq 0\), the stochastic perturbations may disrupt such a behaviour (see Fig. 1, c). + +there exist \(\kappa,\varkappa\in\mathbb{Z}_{+}\text{and}0<|r_{0}|<(2\vartheta)^{-1/2}\) such that the condition (6) holds. diff --git a/outputs/arxiv_math/2503.07542_pg14_pg1_repeat1.md b/outputs/arxiv_math/2503.07542_pg14_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..575f60bf5acb1bbe057a1731bdc0f19605df0999 --- /dev/null +++ b/outputs/arxiv_math/2503.07542_pg14_pg1_repeat1.md @@ -0,0 +1,53 @@ +Note that \(X\) is Panchishkin ordinary if there exist a Greenberg local condition \(\Delta\text{such that}(X,\Delta)\) is 0-Panchiskin and all Hodge–Tate weights of \(\mathscr{F}_{p_{i}}^{+}X\) (resp. of \(X/\mathscr{F}_{p_{i}}^{+}X)\) are positive (resp. non-positive) for \(i=1,2\). + +## Example 3.12. — + +(i) Let \(x\in\mathcal{W}_{\mathrm{c}\mathrm{l}}^{(\mathbf{f},\text{bal})}(O)\) be an \(O-\)valued arithmetic point and let us denote \(X(x):=X\otimes_{x}O\text{for}X=\mathbf{T}_{1},\mathbf{T}_{2},\) and T3. Then + +\[\begin{aligned} \delta_{\text{Pan}}(\mathbf{T}_{3}(x),\Delta_{(\mathbf{g},\text{bal})})&=0; \\ \delta_{\text{Pan}}(\mathbf{T}_{2}(x),\text{tr}^{*}\Delta_{(\mathbf{g},\text{bal})})&=1=\delta_{\text{Pan}}(\mathbf{T}_{1}(x),\text{tr}^{*}\Delta_{(\mathbf{g},\text{bal})}). \end{aligned}\] + +(ii) Let \(x\in\mathcal{W}_{\mathrm{c}\mathrm{l}}^{\mathbf{f}}(O)\) be an \(O-\)valued arithmetic point and let us denote \(X(x):=X\otimes_{x}O\text{for}X=\mathbf{T}_{1},\mathbf{T}_{2},\) and T3. Then + +\[\begin{aligned} \delta_{\text{Pan}}(\mathbf{T}_{3}(x),\Delta_{\mathbf{g}})&=0=\delta_{\text{Pan}}(\mathbf{T}_{2}(x),\Delta_{\text{bal}}); \\ \delta_{\text{Pan}}(\mathbf{T}_{2}(x),\text{tr}^{*}\Delta_{\mathbf{g}})&=2=\delta_{\text{Pan}}(\mathbf{T}_{1}(x),\text{tr}^{*}\Delta_{\mathbf{g}}). \end{aligned}\] + +Remark 3.13. Using the Panchiskin defect we can indicate why the factorization problem for algebraic \(p-\)adic \(L-\)functions we consider in this article is considerably more challenging than those problems considered earlier in [Gre82, Pal18, BCS23]. All the factorizations result in op. cite are obtained when the associated pairs are either 0-Panchiskin or 1-Panchiskin while we also consider the 2-Panchiskin pairs (cf. [BCS23, §4.3.2]). This discussion also gives evidence towards the existence of higher rank Euler systems assumed in Section 6 (cf. [LZ20b]). + +3. 3. Remarks on Tamagawa Factors. Let \(v∤p\infty\) be a place in \(F\) . Suppose \(\varphi:G_{F_{v}}\rightarrow\text{GL}_{R}(V)\) be a continuous \(G_{F_{v}}-\text{representation},\text{where}V\) is a free module of finite rank over a complete Noetherian local ring \(R\) with finite residue field of characteristic \(p\). + +Remark 3.14. When \(R=\mathbb{Z}_{p},F=\mathbb{Q}\), the \(p-\)adic valuation of the order of \(H^{1}(I_{v},V)_{R-\text{to}\text{rs}}^{\mathrm{F}\mathrm{r}_{v}=1}\) is the local Tamagawa factor at \(v\) (see [FPR94, §I.4.2.2]). So it vanishes if \(H^{1}(I_{v},V)\) is a free \(R-\text{module}\). In this philosophy, we would concentrate on the following conditions: + +- The \(R-\text{module}H^{1}(I_{v},V)\) is free. + +This discussion about the Tamagawa factor is needed for verification that our Selmer complexes are perfect. 3.3.1. Let \(I_{v}^{(p)}\rhd I_{v}\) be the unique subgroup satisfying \(I_{v}/I_{v}^{\sim^{(p)}}=\mathbb{Z}_{p}\). Let us fix a topological generator \(t\in I_{v}/I_{v}^{(p)}\) . (cf. [NSW08, §7.5.2]) + +## Lemma 3.15. — + +1. The \(R-\text{module}V^{I_{v}^{(p)}}\) is free of finite rank. + +2. The complex \(C^{\bullet}(I_{v},V)\) is quasi-isomorphic to the perfect complex + +\[\cdots \rightarrow0\rightarrow V^{I_{v}^{(p)}}\xrightarrow{t-1} V^{I_{v}^{(p)}}\rightarrow0\rightarrow\cdots\] + +concentrated in degrees 0 and 1. + +3. For any ring homomorphism \(R\rightarrow S\), the induced map \(V^{I_{v}^{(p)}}\otimes_{R}S\rightarrow(V\otimes_{R}S)^{I_{v}^{(p)}}\) is an isomorphism. Therefore in the derived category + +\[R\Gamma(I_{v},V)\otimes_{R}^{\mathbb{L}}S-^{\sim}\rightarrow R\Gamma(I_{v},V\otimes_{R}S)\] + +Proof. The statement (1) and (2) follows from [Nek06, §7.5.8]. To prove (3), [Nek06, §7.5.8] gives the inclusion \(V^{I_{v}^{(p)}}\rightarrow V\) is split, the \(R-\text{module}V/V^{I_{v}^{(p)}}\) is flat. hence we have an exact sequence of \(S-\)modules + +\[0\longrightarrow V^{I_{v}^{(p)}}\otimes_{R}S\longrightarrow V\otimes_{R}S\longrightarrow V/V^{I_{v}^{(p)}}\otimes_{R}S\longrightarrow 0.\] + +Moreover, the group \(I_{v}^{(p)}\) acts trivially on the ring \(S\). Since \(\varphi(I_{v}^{(p)})\) is finite and \(p∤\#\varphi(I_{v}^{(p)})\), we have + +Therefore we have + +\[e:= \frac{1}{\#\varphi(I_{v}^{(p)})} \sum_{g\in\varphi(I_{v}^{(p)})}g\in O[\varphi(I_{v}^{(p)})]\] + +\[(V/V^{I_{v}^{(p)}}\otimes_{R}S)^{I_{v}^{(p)}}=e(V/V^{I_{v}^{(p)}}\otimes_{R}S)=e(V/V^{I_{v}^{(p)}})\otimes_{R}S=0.\] + +- + +This shows that the canonical homomorphism \(V^{I_{v}^{(p)}}\otimes_{R}S\rightarrow(V\otimes_{R}S)^{I_{v}^{(p)}}\) is an isomorphism. Corollary 3.16. Suppose \(\varphi(I_{v})\) is finite and \(p∤\#\varphi(I_{v})\text{then}R-\text{module}H^{1}(I_{v},V)\) is free. Proof. Follows from Lemma 3.15 since in this scenario we have isomorphism of \(R-\text{modules}H^{1}(I_{v},V)\cong V^{I_{v}^{(p)}}\) . + +□ diff --git a/outputs/arxiv_math/2503.07543_pg42_pg1_repeat1.md b/outputs/arxiv_math/2503.07543_pg42_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..4f5e140173783292f86e1b540b416c33f4a942e7 --- /dev/null +++ b/outputs/arxiv_math/2503.07543_pg42_pg1_repeat1.md @@ -0,0 +1,21 @@ +- + +Proof. Observe that the flow line \(\boldsymbol{\Gamma}\) ends at \(\boldsymbol{\gamma}\). Along \(\boldsymbol{\Gamma}\) the braid representative goes from the one on the top left to the one on the top right of Figure 4.4, implying a crossing. This crossing is derived from \(\tilde{q}\) and hence the claim for \((z_{1},z_{2})=(0,0)\in[0,1]_{z_{1}}\times[0,1]_{x_{2}}\) holds. By continuity there exists a crossing derived from \(\tilde{q}\)for each \((z_{1},z_{2})\) in a small neighborhood of \((0,0)\in[0,1]\times[0,1]\). Let \(U_{\tilde{q}}\) be the connected subset consisting of all \((z_{1},z_{2})\in[0,1]\times[0,1]\) satisfying Claim 4.6. For each \((z_{1},z_{2})\in U_{\tilde{q}}\) there exists an MFTS with 1 switching derived from \(\tilde{q}\). Let \(\boldsymbol{\Gamma}(z_{1},z_{2})\) be the flow line starting at \((z_{1},z_{2})\) and let \(s=s_{0}\) be the moment given by Claim 4.6 when the crossing derived from \(\tilde{q}\text{occurs}\). We write \(\boldsymbol{\Gamma}(z_{1},z_{2})(s_{0})=(\tilde{\gamma}_{1}^{\prime}(z_{1},z_{2}),\tilde{\gamma}_{2}^{\prime}(z_{1},z_{2}))\). Let \((\tilde{\gamma}_{1}^{\prime\prime}(z_{1},z_{2}),\tilde{\gamma}_{2}^{\prime\prime}(z_{1},z_{2}))\) be the result of applying the switching map; then \(\tilde{\gamma}_{1}^{\prime\prime}(z_{1},z_{2})\) (resp. \(\tilde{\gamma}_{2}^{\prime\prime}(z_{1},z_{2}))\) is approximately the concatenation \(a_{1}(z_{1},z_{2})♯b_{2}(z_{1},z_{2})\) (resp. \(a_{2}(z_{1},z_{2})♯b_{1}(z_{1},z_{2}))\), where we write \(\tilde{\gamma}_{i}^{\prime}(z_{1},z_{2})=a_{i}(z_{1},z_{2})♯b_{i}(z_{1},z_{2})\). + +\[\tilde{\gamma}_{2}^{\succ}\] + +\[\tilde{\gamma}_{1}^{\succ}\] + +\[q_{2}^{\succ\succ}\] + +\[q_{1}^{\succ\succ}q^{q_{q_{1\succ1}^{2}}^{\succ}}\] + +\(q_{2}\) + +![](figure) + +FIGURE 4.6. The curves \(\tilde{\gamma}_{1}^{\prime}\) (in blue) and \(\tilde{\gamma}_{2}^{\prime}\) (in green) intersect at the red dot, which represents the crossing derived from \(\tilde{q}\). The curves in gray show various stages of the flow line starting at \(\tilde{\gamma}_{1}^{\prime\prime}\). + +We also claim that there exists a unique value \(z_{1}^{\prime}\text{such that for}z_{1}>z_{1}^{\prime}\) there is no such MFTS with + +1 switching starting at \((z_{1},0)\in[0,1]\times[0,1]\). Similarly there is a unique \(z_{2}^{\prime}\) with analogous property for the second coordinate. Moreover, on each ray starting from \((0,0)\) there is a unique point satisfying analogous property and \(U_{\tilde{q}}\) has the triangular shape as in Figure 4.5. The sides of this triangle along the \(z_{1}-\text{and}z_{2}-\)axes are denoted \(I_{1}\text{and}I_{2}\) and the hypotenuse is denoted \(I_{3}\). Following the proof strategy of \(\mu_{M}^{2}(x,1)=x\) (see the paragraph on the verification of (1.6) above), we now analyze how the various MFTS approach the \(\text{generator}(1,1)\), represented by geodesics \(g(q_{1},q_{2}^{\prime\prime})\) and \(g(q_{2},q_{1}^{\prime\prime})\). For each of these generators there are two options: either a flow line approaches it from the right, or from the left. Hence there are 4 possible types of flow lines from \((x_{1},x_{2})\) which we denote by \(rr,rl,lr,ll\), where the first (resp. second) letter corresponds to the geodesic \(g(q_{1},q_{2}^{\prime\prime})\) (resp. \(g(q_{2},q_{1}^{\prime\prime}))\). Claim 4.7. The pair \(((1,\tilde{\gamma}_{2}),(\tilde{\gamma}_{1},1))\) corresponding to \((z_{1},z_{2})=(0,0)\) is of type \(lr\). Proof. Let \((\tilde{\gamma}_{1}^{\prime},\tilde{\gamma}_{2}^{\prime})=(\tilde{\gamma}_{1}^{\prime}(0,0),\tilde{\gamma}_{2}^{\prime}(0,0))\); see Figure 4.6. Note that \(\tilde{\gamma}_{1}^{\prime}\text{and}\tilde{\gamma}_{2}^{\prime}\) are very close to the generators \(x_{1}\text{and}x_{2}\). Applying the switching map, \(\tilde{\gamma}_{1}^{\prime\prime}=\tilde{\gamma}_{1}^{\prime\prime}(0,0)\text{and}\tilde{\gamma}_{2}^{\prime\prime}=\tilde{\gamma}_{2}^{\prime\prime}(0,0)\) are approximately concatenations \(a_{1}(0,0)♯b_{2}(0,0),a_{2}(0,0)♯b_{1}(0,0)\) which are very close to two geodesics that intersect at an angle slightly less than \(\pi\). The gradient vector at \(\tilde{\gamma}_{1}^{\prime}\), regarded as a vector field along \(\tilde{\gamma}_{1}^{\prime}\), has a relatively small norm away from this angle, whereas near the angle its norm is significantly greater than 0 and the vector points into the interior of this angle; see [Mil63, Chapter 12] for a more precise description. Hence \(\tilde{\gamma}_{1}^{\prime\prime}\) is pushed away to the right along the flow line as shown in Figure 4.6 and the flow line diff --git a/outputs/arxiv_math/2503.07567_pg6_pg1_repeat1.md b/outputs/arxiv_math/2503.07567_pg6_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..28e37b30981602636b4a65dd05505ea655c523ee --- /dev/null +++ b/outputs/arxiv_math/2503.07567_pg6_pg1_repeat1.md @@ -0,0 +1,21 @@ +## C. Orthogonality Condition for Base Matrices + +A necessary and sufficient condition for a pair of QC-LDPC codes to satisfy the symplectic inner product condition is given in [29]. We revisit the even multiplicity condition described therein first. Definition 1 (Even multiplicity): A vector \(\mu=(\mu_{1},\mu_{2},\ldots,\mu_{n})\) is said to have even multiplicity if, for every integer \(k\in\mathbb{Z}\), the number of indices \(i\in\{1,2,\ldots,n\}\text{such that}\mu_{i}=k\) is even. By definition, the vector has even multiplicity if each integer entry in \(\mu\) appears an even number of times while discounting the \(-\infty\) entries or the empty cells if any. Even multiplicity and orthogonal base matrices: Let \(B\) be a base matrix in exponent form of size \(m\times n\) with circulant size \(L\), where the entries \(b_{i,j}\in\{-\infty,0,1,\ldots,L-1\}\). Define the \(i-\)th row vector \(\mathbf{r}_{i}\text{for}i=1,2,\ldots,m\text{as}\mathbf{r}_{i}=\left[b_{i,1},b_{i,2},\cdots ,b_{i,n}\right].\) If the modulo\(-L\) difference of any two distinct rows \(\mathbf{r}_{i}\text{and}\mathbf{r}_{j}\): i.e., \(\mathbf{r}=\mathbf{r}_{i}-\mathbf{r}_{j},\text{modulo}-L\), has even multiplicity, then the corresponding binary matrices after lifting \(\mathrm{L}(\mathbf{r}_{i})\text{and L}(\mathbf{r}_{j})\) are orthogonal [29], [30]. The minus − operation is performed modulo\(-L\) and for the \(\infty\) entries is as follows: \(b_{i,j}-\infty=\infty-b_{i,j}=\infty-\infty:=-\infty.\) Theorem 1: LP-QLDPC code PCMs HX and HZ are orthogonal if the rows of quasi-cyclic base matrices \(B_{\mathrm{X}}\text{and}B_{\mathrm{Z}}\) have even multiplicity. Proof: To prove the orthogonality property for quasi-cyclic LP-QLDPC code parity check matrices, one can check the even multiplicity property for the base matrices \(B_{\mathrm{X}}\text{and}B_{\mathrm{Z}}\). Their rows obey the condition for orthogonality - the even multiplicity of integers in the modulo\(-L\) differences of the rows, where \(L\) is the circulant size. Take any pair of rows such that \(\mathbf{r}_{i}\) is taken from \(B_{\mathrm{X}},\text{and}\mathbf{r}_{j}\) from \(B_{\mathrm{Z}}\). These rows can be split based on the left part and right part - with exactly one overlap in each of the parts. Thus, the difference \(\mathbf{r}=\mathbf{r}_{i}-\mathbf{r}_{j},\text{modulo}-L\) will only have two integer terms. Suppose that the two overlapping terms are equal; then, the resulting difference vector has a pair of zeros. If the overlapping terms \(p\text{and}q\) are not equal, \(p\neq q\), then the corresponding terms in the difference appear as \(p-q\text{and}q^{*}-p^{*}\), respectively. Since the conjugate terms are \(q^{*}=L-q\) and \(p^{*}=L-p\), the overlapping terms are indeed equal, satisfying the even multiplicity condition. Therefore, the two parity □ + +check matrices are orthogonal to each other. + +## IV. MINIMUM DISTANCE OF LP-QLDPC CODES + +In the following, we examine the symmetric LP-QLDPC codes constructed from type-1 quasi-cyclic base matrices to understand how certain combinations of CPM values reduce the quantum minimum distance \(d_{\text{min}}^{\mathcal{Q}}\) from the minimum distance of the base code \(d_{\text{min}}^{\mathcal{C}}\). We first show how the minimum distance is limited to the Hamming weight of the stabilizer generators. We also prove that using a base matrix with only two rows \((m=2\) as in Eq. (11)) always limits the LP-QLDPC code to have a minimum distance \(d_{\text{min}}^{\mathcal{Q}}\leq n+2\). In general, we are interested in the following question: What constraints on the base code of an LP-QLDPC code reduce its minimum distance? This will guide us toward code construction recipes guarantee degeneracy for the obtained LP-QLDPC code. First, we provide examples of the choices of base codes of different sizes: \(m\text{and}n\) to demonstrate the reduction of the minimum distance of LP-QLDPC codes. Example 1 Continued: Consider the LP-QLDPC code constructed from the example base matrix we saw earlier with \(m= 2,n=3,L=7\), + +\[B= \begin{bmatrix} 1 & 2 & 4 \\ 6 & 5 & 3 \end{bmatrix} .\] (23) + +We start with a classical LDPC code \(\mathcal{C}:[21,8,6]\). Hence, the desired minimum distance of LP code is \(d_{\text{min}}^{\mathcal{C}}=6\). However, we verified that the \(d_{\text{min}}^{\mathcal{Q}}=m+n=5\). Note that for quasi-cyclic base matrices, it is enough to look at equivalent base matrices expressed in canonical form. Two base matrices are equivalent if their respective Tanner graphs are identical upon variable and/or check node permutations. Given a base matrix, \(B\) exchanging rows/columns, or adding a fixed integer to each element in row/column maintains equivalence. The rows and columns of the quasi-cyclic code can be reordered to put 0s in the first row and column to make an equivalent parity check matrix making the analysis easier. For instance, we have the equivalent matrix + +\[B= \begin{bmatrix} 0 & 0 & 0 \\ 0 & 1 & 3 \end{bmatrix} .\] (24) + +Example 2: Classical code \(\mathcal{C}:[104,30,14]\) with the base matrix given in Example 3 of [18]. We obtain the base matrix for the classical LDPC code \(\mathcal{C}:[104,30,14]\). The base matrix \(B\) in Eq. (25) corresponds to a \((3,4)\) regular LDPC code with circulant size \(L=26\). The minimum distance of the base code is \(d_{\text{min}}^{\mathcal{C}}=14\). + +\[B= \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 6 & 4 & 10 \\ 0 & 8 & 14 & 22 \end{bmatrix} .\] (25) + +We now construct a symmetric LP-QLDPC code from Eq. (25) to get a [[650, 50, 7]] code. Here, we have \(N=26\times(3^{2}+4^{2})\) physical qubits encoding \(K=50\) logical qubits with a minimum distance \(d_{\text{min}}^{\mathcal{Q}}=7\) instead of 14. Note that the quantum minimum distance is again equal to \(m+n\). diff --git a/outputs/arxiv_math/2503.07569_pg19_pg1_repeat1.md b/outputs/arxiv_math/2503.07569_pg19_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..bb97871f821fdb57dd5c2f28c11d8eb5b58b15ee --- /dev/null +++ b/outputs/arxiv_math/2503.07569_pg19_pg1_repeat1.md @@ -0,0 +1,45 @@ +Step 1: regularizing coefficients of \(A\): We follow [AN24]. Let \(\theta\in\mathcal{D}(\mathbb{R})\) a nonnegative function + +with \(´_{\mathbb{R}}\theta(t)dt=1.\text{For all}p\geq1\), let \(\theta_{p}(t)=p\theta(pt)\) be the associated mollifying sequence. We set For all \(p\geq1\text{and}t\in\mathbb{R}\), we set + +\(A_{p}(t,x):=(\theta_{p}\star A(\cdot,x))(t)\), i.e., we mollify the matrix-valued function A in the time variable only. + +\[B_{t}^{p}(u,v):=\int_{\mathbb{R}^{n}}\omega^{-1}A_{p}(t,\cdot)\nabla_{x}u\cdot \overline{\nabla_{x}v} d\omega+ \frac{1}{p} \langle u,v\rangle_{2,\omega}.\] + +We check easily that \(\min(1/l,\nu)\Vert u\Vert_{H_{\omega}^{1}(\mathbb{R}^{n})}^{2}\leq\text{Re}(B_{t}^{p}(u,u))\text{and Im}(B_{t}^{p}(u,u))\leq \frac{M}{\nu} \text{Re}(B_{t}^{p}(u,u))\). In + +particular, the quadratic form of \(\text{Re}(B_{t}^{p}(\cdot,\cdot))\) is closed. Moreover, we have + +\[|B_{t}^{p}(u,u)-B_{s}^{p}(u,u)|\leq M\bigg\| \frac{d\theta_{p}}{dt} \bigg\|_{L^{1}(\mathbb{R})}|t-s|\Vert\nabla_{x}u\Vert_{2,\omega}^{2}\leq \frac{pM\Vert\dot{\theta}\Vert_{L^{1}(\mathbb{R})}}{\nu} |t-s|\text{Re}(B_{t}^{p}(u,u)),\] + +where \(\dot{\theta}\) is the derivative of \(\theta.\text{For all}p\geq1\), we set \(U_{p}(t):=\Gamma_{p}(t,s)f\text{where}\Gamma_{p}\) is the fundamental + +solution of the parabolic operator associated to the family \((B_{t}^{p})_{t\in\mathbb{R}}\). Combining [Kat61, Theorem III] + +with uniqueness in in \(L^{2}((s,T);H_{\omega}^{1}(\mathbb{R}^{n}))\) for any \(T>s\), we have for all \(p\geq1,U_{p}:(s,\infty)\rightarrow L_{\omega}^{2}(\mathbb{R}^{n})\) is strongly differentiable. Note that \(U_{p}\) is a real-valued function by the same argument as we did + +for \(U\). Since \(\nabla_{x}U_{p}(t)\in L_{\omega}^{2}(\mathbb{R}^{n})\), we have \(\partial_{t}|U_{p}(t)|,\nabla_{x}|U_{p}(t)|\in L_{\omega}^{2}(\mathbb{R}^{n})\) with + +\[\partial_{t}|U_{p}(t) \begin{vmatrix} \bigg\{ \\ \partial_{t}U_{p}(t)\text{if}U_{p}(t)\geq0, \\ = & \text{and}\nabla_{x} \\ -\partial_{t}U_{p}(t)\text{if}U_{p}(t)<0, \end{vmatrix} U_{p}(t)|= \begin{cases} \nabla_{x}U_{p}(t)\text{if}U_{p}(t)\geq0, \\ -\nabla_{x}U_{p}(t)\text{if}U_{p}(t)<0. \end{cases}\] + +Using this, we have + +\[\begin{aligned} - \frac{d}{dt} \langle U_{p}(t)-|U_{p}(t)|,U_{p}(t)-|U_{p}(t)|\rangle_{2,\omega}&=-2\langle\partial_{t}(U_{p}(t)-|U_{p}(t)|),U_{p}(t)-|U_{p}(t)|\rangle_{2,\omega} \\ &=-4\langle\partial U_{p}(t),U_{p}(t)-|U_{p}(t)|\rangle_{2,\omega} \\ &=4\int_{\mathbb{R}^{n}}^{t}\omega^{-1}A(t,\cdot)\nabla_{x}U_{p}(t)\cdot\nabla_{x}(U_{p}(t)-|U_{p}(t)|)d\omega\geq0. \end{aligned}\] + +Integrating from \(s\) to \(t\) in this inequality, we see that \(t\mapsto\Vert U_{p}(t)-|U_{p}(t)|\Vert_{2,\omega}^{2}\) is a non-increasing function. Since it vanishes at \(t=s\), we have for all \(t>s,U_{p}(t)=|U_{p}(t)|\), that is \(\Gamma_{p}(t,s)f= |\Gamma_{p}(t,s)f|\), hence \(\Gamma_{p}(t,s)\) is a nonnegative operator. the boundedness of \((U_{p})_{p\geq1}\text{in}L^{2}((s,T);H_{\omega}^{1}(\mathbb{R}^{n}))\) provided by the energy equality, it is easy to check that, up to extracting a sub-sequence, \((U_{p})_{p\geq1}\) converges weakly to \(U\text{when}p\rightarrow\infty\) in □ + +Step 2: passing to the limit: using uniqueness in \(L^{2}((s,T);H_{\omega}^{1}(\mathbb{R}^{n}))\) for any \(T>s\) combined with + +\(L^{2}((s,T);L_{\omega}^{2}(\mathbb{R}^{n}))\) for any \(T>s\), and therefore \(U(t)\) is nonnegative for all \(t\geq s\). Combining Caccioppoli inequality in Lemma 4.4, a weighted Sobolev inequality [HKM18, Theorem 15.26] and the Moser’s iteration principle, we have the following \(L^{\infty}-\)estimate on nonnegative local weak solutions. For a proof, one can follow the classical scheme or see [Ish99, Proposition 2.1] with lower order coefficients equal to zero. + +Lemma 6.2. Let \((t_{0},x_{0})\in\mathbb{R}^{1+n}\text{and}R>0.Ifu\) is is a nonnegative local weak solution of \(\mathcal{H}u=0\) in a neighborhood of \(Q_{2R}(t_{0},x_{0})\), then + +\[\mathrm{e}_{Q_{R}}\text{ss}_{(t}\mathrm{s}_{0}\mathrm{u}_{,x}\mathrm{p}_{)_{0}}u=\Vert u\Vert_{L^{\infty}(Q_{R}(t_{0},x_{0}))}\leq B\left( \frac{1}{\mu(Q_{2R}(t_{0},x_{0}))} \int_{Q_{2R}(t_{0},x_{0})}u^{2}d\mu\right)^{1/2}\] + +where \(B=B(n,D,M,\nu)>0\) is a constant. The same estimate holds for nonnegative local weak + +solution of \(\mathcal{H}^{\star}v=0\). By combining Lemma 6.2 above, Lemma 6.1 and Proposition 5.12, we obtain the following result. Proposition 6.3. The operator \(\mathcal{H}\) admits a nonnegative generalized fundamental solution \(\Gamma(t,x;s,y)\) with, for all \(t>s\), almost everywhere pointwise Gaussian upper bound, that is, + +\[0\leq\Gamma(t,x;s,y)\leq \frac{K_{0}}{\sqrt{\omega_{t-s}(x)} \sqrt{\omega_{t-s}(y)}} e^{-k_{0} \frac{|x-y|^{2}}{t-s}},\] (6.4) + +for almost every \((x,y)\in\mathbb{R}^{2n},\text{where}K_{0}=K_{0}(n,D,M,\nu)>0\text{and}k_{0}=k_{0}(M,\nu)>0\) are constants. diff --git a/outputs/arxiv_math/2503.07573_pg1_pg1_repeat1.md b/outputs/arxiv_math/2503.07573_pg1_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..9a86714438f0166f7455632888c6befbf405d094 --- /dev/null +++ b/outputs/arxiv_math/2503.07573_pg1_pg1_repeat1.md @@ -0,0 +1,21 @@ +# RECONSTRUCTING CURRENTS FROM THEIR PROJECTIONS + +## AIDAN BACKUS + +Abstract. We prove an inversion formula for the exterior \(k-\)plane transform. As a consequence, we show that if \(m0\) (which depends on \(k)\text{for every}A,B,A\leq CB\). The Schwartz space is the Fréchet space \(\mathscr{S}\) of \(m-\text{forms}\alpha\) such that for every \(N\in\mathbf{N},|\alpha(x)|≲_{N}\langle x\rangle^{-N}\) [Hör15, §7.1]. A density argument shows that a compactly supported \(m-\)current is determined by its action on \(\mathscr{S}\) . Our main theorem establishes [Sol11, Conjectures 6.5 and 6.8]: + +Theorem 1. For every \(\alpha\in\mathscr{S}\text{and}P\in G\), there exists a smooth \(m-\text{form}\alpha_{P}onP\) such that for every \(x\in\mathbf{R}^{n}\), + +\[\alpha(x)=\int_{G}P^{*}(\alpha_{P})(x)dP,\] (1) + +Date: March 11, 2025. 2020 Mathematics Subject Classification. 44A12. This research was supported by the National Science Foundation’s Graduate Research Fellowship Program under Grant No. DGE-2040433. diff --git a/outputs/arxiv_math/2503.07583_pg22_pg1_repeat1.md b/outputs/arxiv_math/2503.07583_pg22_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..c96b002635e50d8cc1e8e93dad7a47a434d5ddd5 --- /dev/null +++ b/outputs/arxiv_math/2503.07583_pg22_pg1_repeat1.md @@ -0,0 +1,27 @@ +These values of coefficients are enough to reconstruct the full dependence: + +\[\begin{aligned} A_{a,b,c}&=2(a+b+c)+3 \\ C_{a,b|c,d}&=3(2a+1)(2b+1) \\ G_{a}&=(2a+1)(2a^{2}+2a+1) \end{aligned}\] (132) + +Now we are able to write the full Hamiltonian for SIB polynomials: + +\[\begin{aligned} \hat{\mathcal{H}}&=\sum_{a,b,c=0}^{\infty}(2(a+b+c)+3)\tau_{a+b+c+1} \frac{\partial^{3}}{\partial\tau_{a}\partial\tau_{b}\partial\tau_{c}} +\sum_{a+b=c+d}3(2a+1)(2b+1)\tau_{a}\tau_{b} \frac{\partial^{2}}{\partial\tau_{c}\partial\tau_{d}} \\ \sum_{a,b,c=0}^{\infty}4(2a+1)(2b+1)(2c+1)\tau_{a}\tau_{b}\tau_{c} \frac{\partial}{\partial\tau_{a+b+c+1}} +\sum_{a=0}^{\infty}(2a+1)\left(2a^{2}+2a+1\right)\tau_{a} \frac{\partial}{\partial\tau_{a}} \end{aligned}\] + +One can check that eigenvalues of the above operator are positive integers and obey formula (130). This Hamiltonian itself does not define SIB basis due to degenerate spectrum, for example \(\varepsilon_{[8,7]}=8^{3}+7^{3}=855=9^{3}+5^{3}+1^{3}=\varepsilon_{[9,5,1]}\). However, with adding/removing rules the basis is fixed completely. + +### 3.5 Action of Virasoro operators on \(\mathbf{SIB}_{\lambda}\) polynomials + +Motivated by the action of time-variables and Virasoro operators on SIB polynomials that we discuss in this section it is instructive to slightly change notation of diagrams DP - distinct partitions. For this purpose we represent a Young diagram (turned upside down) as a way of tight packing of square boxes in a corner of a room. In this setting distinct partitions (DP) are in one-to-one correspondence with possible ways of tight packing of square boxes under a ladder. We demonstrate examples of these diagrams, left picture correspond to usual Young diagram, right one correspond to distinct partition: + +![](figure) + +(a) Usual partition + +(a) Distinct partition + +(134) + +![](figure) + +Form now on we represent distinct partitions as Young diagrams under the ladder. With these notations the action of \(\tau_{0}\text{and} \frac{\partial}{\partial\tau_{0}}\) have the following from: + +\[\tau_{0}\cdot\text{SIB}_{\lambda}=\sum_{2\in\mathrm{A}\text{ddDP}(\lambda)}\text{SIB}_{\lambda+2}\] (135) diff --git a/outputs/arxiv_math/2503.07611_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.07611_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..3d5a18f3833af3f9ef0c93547a172871d6707dde --- /dev/null +++ b/outputs/arxiv_math/2503.07611_pg3_pg1_repeat1.md @@ -0,0 +1,23 @@ +at the center of another, passing through the centers of the cells on its way horizontally, vertically, or turning 90 degrees. Two arrows cannot occupy the same cell. Question. Is there at least one solution to the puzzle, i.e., a mapping from the set of white cells to the set \(\{\emptyset,□\}\) that satisfies the rules of the puzzle (each block contains exactly one square placed on the arrow, each arrow passes through at least two blocks, the sizes of the blocks increase in the direction of the arrow, maintaining the shape)? We also consider a problem of Counting Evolomino, which has the same instance but asks how many distinct solutions the puzzle has. + +## 3 Evolomino \(\in\) NP + +## Lemma 1. Evolomino \(\in\) NP. + +Proof. Evolomino is a decision problem, which means that to prove that it belongs to the class NP, it is sufficient to show that verifying that a given solution satisfies all the rules of the puzzle is performed in polynomial time. Let us be given an Evolomino puzzle on a rectangular board of size \(p\times q\). If we guess some solution to the puzzle, then checking its correctness will require: + +- Rule: “Each block must contain exactly one square placed on a pre-drawn arrow.” + +Since each board cell can belong to no more than one block, we have at most \(O(pq)\) blocks. The complexity of traversing one block is linear in its size, which does not exceed the size of \(p\times q\) of the entire board. Thus, the total complexity of verifying this rule is \(O(p^{2}q^{2})\). + +- Rule: “Each arrow must pass through at least two blocks”. + +Checking this rule requires a single pass through each pre-drawn arrow. Since the total length of all arrows does not exceed the size of the entire board, the complexity of verifying the rule is \(O(pq)\). + +- Rule: “The second and later blocks on the route of an arrow from start to finish must progress by adding one square to the previous block without rotating or flipping”. + +Let us consider two consecutive blocks in the direction of the arrow, both of size no greater than \(O(pq)\). Let’s try to exclude each square of the second block one by one, after which it remains to go through both blocks and check that they coincide. Therefore, the complexity of verifying the rule for two consecutive blocks does not exceed \(O(p^{2}q^{2})\). + +It remains to note that the number of pairs of consecutive blocks does not exceed the total length of all pre-drawn arrows, i.e., \(O(pq)\). Which makes the overall complexity of verifying the rule equal to \(O(p^{3}q^{3})\). + +Summing up the complexity of all the rules, we find that a given puzzle solution can be verified in polynomial time \(O(p^{3}q^{3})\). Note that this is a fairly rough estimate, but to prove that Evolomino \(\in\) NP, it is sufficient to be polynomial in the size of the board \(p\times q\). diff --git a/outputs/arxiv_math/2503.07617_pg27_pg1_repeat1.md b/outputs/arxiv_math/2503.07617_pg27_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..d9b08b83d4b160877fd917a72cb7290ae1b47794 --- /dev/null +++ b/outputs/arxiv_math/2503.07617_pg27_pg1_repeat1.md @@ -0,0 +1,57 @@ +| 1 | 1 | +| --- | --- | +| 0.8 | 0.8 | +| 0.6 | 0.6 | +| 0.4 | 0.4 ✓ | +| 0.2 | 0.2 ✓ ↑ | +| 0 | 0 | +| 0 0.5 1 | 1.5 2 0 0.5 1 1.5 2 | + +Figure 29: [Test Case 3] (Left) Reference Darcy Velocity Field. (Right) Perturbed Velocity Field. + +2.5 + +![](figure) + +| 0.8 | 2.5 | 0.8 | 2 | +| --- | --- | --- | --- | +| | 2 | | | +| 0.6 | | 0.6 | 1.5 | +| | 1.5 | | | +| 0.4 | | 0.4 | 1 | +| | 1 | | | +| 0.2 | 0.5 | 0.2 | 0.5 | + +0.5 + +1 + +1.5 + +0.5 + +1 + +1.5 + +Figure 30: [Test Case 3] Reference Concentration Field. (Left) Heat Map. (Right) Contour Map. + +mesh size \(h=1/40\) and on a very fine temporal mesh \(\Delta t=T/800\text{with}T=5\). The data assimilation algorithm is performed over the same spatial mesh size and on a coarser temporal filtering steps \(\Delta t_{\text{Filter}}=T/Nt_{\text{Filter}}\text{where}Nt_{\text{Filter}}=50\), with the same assumption on the observational operator as in Test Case 2. Since concentration transport is driven by the Darcy velocity, we assume that there are some uncertainties in the process of solving the Darcy’s flow system to increase the complexity of state estimation. Thus, we add a small perturbation \(\xi_{\text{Darcy}}=0.0001\epsilon_{D}\) to the reference Darcy velocities with \(\epsilon_{\text{Darcy}}\sim N(0,I)\) and input the disturbed velocity to the forward solver. The reference Darcy velocity field and one example of the perturbed velocity field are shown in Figure 29 on a very coarse mesh for the purpose of illustration. We can see that even with a small amount of noise, the perturbed Darcy field is significantly chaotic, which makes the task of state estimation challenging. Similar to Test Case 2, we aim to investigate the performance of the United Filter under various levels of perturbation. Thus, we consider two types of disturbed noise \(\omega\) in (14): \(\tilde{\omega}^{1}= 0.001 \sqrt{\Delta t_{\text{Filter}}} \tilde{\varepsilon}^{1}\text{and}\tilde{\omega}^{2}=0.1 \sqrt{\Delta t_{\text{Filter}}} \tilde{\varepsilon}^{2},\text{where}\tilde{\varepsilon}^{i}\sim\mathcal{N}(0,I_{l})\text{for}i=1,2\). Regarding the setting for the parameter estimation, as the exact values are small, we apply the assimilation algorithm to approximate \(\rho_{i}=1/d_{i},i=1,2\text{and}\rho_{\gamma}=1/\alpha_{\gamma}\). We choose the number of Direct Filter particles to be \(M=40\), and the number of iterations in the United Filter to be \(R=4\). + +![](figure) + +![](figure) + +| 0.9 | | 0.9 | | +| --- | --- | --- | --- | +| 0.8 | 2.5 | 0.8 | 2.5 | +| 0.7 | 2 | 0.7 | 2 | +| 0.6 | | 0.6 | | +| 0.5 | 1.5 | 0.5 | 1.5 | +| 0.4 | | 0.4 | | +| 0.3 | 1 | 0.3 | 1 | +| 0.2 | 0.5 | 0.2 | 0.5 | +| 0.1 | | 0.1 | | +| 0.5 1 | 1.5 | 0.5 1 | 1.5 | + +Figure 31: [Test Case 3] Heat map of estimated concentration field by the United Filter: (Left) With noise \(\tilde{\omega}^{1}\). (Right) With noise \(\tilde{\omega}^{2}\). diff --git a/outputs/arxiv_math/2503.07623_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.07623_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..3f21b6235d518ef97d3d4f225bd6363d8c04a1e1 --- /dev/null +++ b/outputs/arxiv_math/2503.07623_pg2_pg1_repeat1.md @@ -0,0 +1,33 @@ +In this manuscript, we investigate Finslerian exponentially harmonic functions. Analogous to the Riemannian case, a function \(u\) defined on a Finsler metric measure space \((M,F,\mu)\) is said to be exponentially harmonic if it satisfies + +\[\tilde{\Delta}_{\mu}u:=\text{div}_{\mu}(V(u)Du)=0,\] (1.1) + +where \(V(u)=\exp( \frac{1}{2} F^{*2}(Du))\) represents the energy density of \(u\). Such a function arises as a critical point of the exponential energy functional (see Theorem 3.1). Finsler metric measure spaces encompass a richer set of geometric tensors compared to Riemannian manifolds (cf. [11]). It is important to emphasize that (1.1) is not merely a quasilinear elliptic equation, as the underlying space is an anisotropic and asymmetric manifold. Within this framework, we establish the following theorem. Theorem 1.1. Let \((M,F,\mu)\) be a forward complete \(n-\)dimensional Finsler metric measure space with finite misalignment \(\alpha\). Assume that the mixed weighted Ricci curvature \(R^{m}ic^{\infty}\) of \(M\) is nonnegative, and that the \(S-\)curvature as well as the non- + +Riemannian curvatures \(U,\mathcal{T}\text{and}divC=FC_{jk|i}^{i}dx^{j}\otimes dx^{k}\) satisfy the norm bound + +\(F^{-1}|S|+F(U)+F(\mathcal{T})+\Vert divC\Vert_{HS(V)}\leq K_{0},\) for any \((x,V)\in SM\). Then, any bounded exponentially harmonic function \(uonM\) is constant. + +## 2 Related concepts and notations of the Finsler metric measure spaces + +A Finsler metric measure space is a triple \((M,F,\mu),\text{where}M\) is a differentiable manifold equipped with a Finsler metric \(F\) and a Borel measure \(\mu\). The Cartan tensor is defined by + +\[C(X,Y,Z):=C_{ijk}X^{i}Y^{j}Z^{k}= \frac{1}{4} \frac{\partial^{3}F^{2}(x,y)}{\partial y^{i}\partial y^{j}\partial y^{k}} X^{i}Y^{j}Z^{k},\] + +for any local vector fields \(X,Y,Z\). We always adopt the Chern connection, which is uniquely determined by + +\[\begin{aligned} \nabla_{X}Y-\nabla_{Y}X&=[X,Y]; \\ Z(g_{y}(X,Y))-g_{y}(\nabla_{Z}X,Y)-g_{y}(X,\nabla_{Z}Y)&=2C_{y}(\nabla_{Z}y,X,Y), \end{aligned}\] + +for any \(X,Y,Z\in TM\setminus\{0\},\text{where}C_{y}\) is the Cartan tensor. The coefficients of the + +Chern connection are locally expressed as \(\Gamma_{jk}^{i}(x,y)\) in natural coordinates. These coefficients induce the spray coefficients as \(G^{i}= \frac{1}{2} \Gamma_{jk}^{i}y^{j}y^{k}\). The spray is given by + +\[G=y^{i} \frac{\delta}{\delta x^{i}} =y^{i} \frac{\partial}{\partial x^{i}} -2G^{i} \frac{\partial}{\partial y^{i}} ,\] (2.1) + +where \(\frac{\delta}{\delta x^{i}} = \frac{\partial}{\partial x^{i}} -N_{i}^{j} \frac{\partial}{\partial y^{j}}\) , and the nonlinear connection coefficients \(N_{j}^{i}\) are locally \(=_{j}^{\underline{\partial G^{i}}}\) + +derived from the spray coefficients as \(N^{i}\) + +\(\overline{\partial y^{j}}\) . By convention, the horizontal Chern + +derivative is denoted by \(“|”\) and the vertical Chern derivative by “; ”. Let \(\hat{D}\) denote the Levi-Civita connection of the induced Riemannian metric \(\hat{g}=g_{Y}\) , and let \(\{e_{i}\}\) diff --git a/outputs/arxiv_math/2503.07624_pg4_pg1_repeat1.md b/outputs/arxiv_math/2503.07624_pg4_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..da6941b903f3d8fa01ef0608a52bb26aea658541 --- /dev/null +++ b/outputs/arxiv_math/2503.07624_pg4_pg1_repeat1.md @@ -0,0 +1,25 @@ +with the homogeneous Neumann boundary condition, it is known that \(u\) has exactly a single peak (i.e. local maximum of \(u)\). Moreover, if the single peak is on the boundary of \(\Omega\), then it is located at the point where the mean curvature of the boundary \(\partial\Omega\) reaches its maximum value. These results have not been numerically verified, which limits the further application of multiple-solution analysis. The rest of this paper is organized as follows. In section 2, we describe the spectral Legendre–Fourier scheme we used to discretize equation (1.1) defined in an elliptic geometry. The IAOBDM is designed and presented in section 3. In section 4, ample numerical experiments are carried out to demonstrate the efficiency of this method, and to show the effect of varying geometry \(\Omega\) on multiple solutions of (1.1). Finally, we end the paper with some remarks in section 5. + +## 2. A Legendre–Fourier scheme for elliptic equations in an elliptic domain + +To numerically study the effect of varying geometry \(\Omega\) on multiple solutions of (1.1), the first step is to provide an efficient discretization scheme. Here, we adopt a Legendre–Fourier scheme for (1.1) in an elliptic domain. Let + +\[\Omega=\left\{(x,y): \tfrac{x^{2}}{a^{2}} + \tfrac{y^{2}}{b^{2}} \leq1\right\}.\] (2.1) + +The weak formulation of (1.1) with homogeneous Dirichlet boundary condition and \(\varepsilon=1\) is to find \(u\in H_{0}^{1}(\Omega)\) such that + +\[\int_{\Omega}\nabla u\nabla vdxdy=\int_{\Omega}f(\boldsymbol{x},u)vdxdy,\forall v\in H_{0}^{1}(\Omega),\] (2.2) + +where \(\boldsymbol{x}=(x,y)\). We use polar transformation \(x=ar\cos\theta,y=br\sin\theta\) to transform the Dirichlet problem into polar coordinate form: + +\[\begin{cases} \underline{1} & \underline{1} \\ \omega_{1}u_{rr}+ & (\omega_{2}u_{r}-\omega_{3}u_{r\theta})+ & (\omega_{3}u_{\theta}+\omega_{2}u_{\theta\theta})+f(r,\theta,u)=0, & \text{in}\Omega=(0,1)\times[0,2\pi), \\ \overline{r} & \overline{r^{2}} \\ u(1,\theta)=0, & u\text{periodic in}\theta, \end{cases}\] + +where + +\[\omega_{1}= \frac{\cos^{2}\theta}{a^{2}} + \frac{\sin^{2}\theta}{b^{2}} ,\omega_{2}= \frac{\cos^{2}\theta}{b^{2}} + \frac{\sin^{2}\theta}{a^{2}} ,\omega_{3}=\sin2\theta( \frac{1}{a^{2}} - \frac{1}{b^{2}} ).\] (2.4) + +The corresponding weak formulation (2.2) becomes + +\[\begin{aligned} \mathcal{L}(u,v):&=\int_{\Omega}(\omega_{3}u_{\theta}v_{r}-\omega_{2}u_{r}v-\omega_{1}(u_{r}v+ru_{r}v_{r}))drd\theta \\ +\int_{\Omega} \frac{1}{r} u_{\theta}(\omega_{3}v-\omega_{2}v_{\theta})drd\theta&=\int_{\Omega}fvdrd\theta,\forall v\in H_{0}^{1}(\Omega). \end{aligned}\] (2.5) + +Obviously, from (2.3) or (2.5), we can find that a singularity point appears at \(r=0\), indicating that additional polar conditions should be proposed to obtain the desired solution regularity at diff --git a/outputs/arxiv_math/2503.07626_pg12_pg1_repeat1.md b/outputs/arxiv_math/2503.07626_pg12_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..1dfe65cdd636db55266104f7bbd7d242c6f743a9 --- /dev/null +++ b/outputs/arxiv_math/2503.07626_pg12_pg1_repeat1.md @@ -0,0 +1,42 @@ +the Cartan involution \(\theta\). Furthermore as part of the same theorem one has that \(\theta\) can always be chosen in such a way that it maps the compact Cartan subalgebra into itself: + +\[\theta:\mathcal{H}_{c}\rightarrow\mathcal{H}_{c}\] (1.20) + +This short discussion reveals that the classification of real sections of a complex Lie Algebra \(g\) is in one-to-one correspondence with the classification of symmetric spaces, the complexification of whose Lie algebra of isometries is + +\(g\). + +Let us now consider the action of the Cartan involution \(\theta\) on the Cartan subalgebra: + +\[\mathcal{H}_{c}\equiv\text{span}\{iH_{j}\}\] (1.21) + +of the maximal compact section \(\mathbb{G}_{\mathbf{comp}}\). The subspace of \(\mathcal{H}_{c}\) belonging to the eigenspace of \(\theta\) corresponding to eigenvalue 1 is, according to eq.(1.15), part of the subalgebra \(\mathbb{H}_{\theta}=\mathbb{H}_{c}\), while the subspace of \(\mathcal{H}_{c}\) belonging to the eigenspace corresponding to the eigenvalue −1 is, according once again to eq. (1.15), part of the subspace \(\mathbb{P}\). It follows that the Cartan subalgebra of the real section \(\mathbb{G}_{\theta}\) can be split in two subalgebras as follows: + +\[\begin{aligned} \mathbb{G}_{\theta}&\supset\mathcal{H}=\mathcal{H}_{\theta}^{comp}\oplus\mathcal{H}_{\theta}^{n.c.} \\ \text{CSA of}\mathbb{G}_{\theta} \\ \mathcal{H}_{\theta}^{comp}&\equiv\mathcal{H}_{c}\bigcap\mathbb{H}_{\theta} \\ \mathcal{H}_{\theta}^{n.c.}&=i\mathcal{H}_{c}\bigcap\mathbb{P}_{\theta}\subset\mathbb{K}_{\theta} \end{aligned}\] (1.22) + +| Gθ ⊃ | \|{z}Hθ CSA of Gθ | = | Hcompθ ⊕ Hn.c.θ | | +| --- | --- | --- | --- | --- | +| | Hcomp θ | ≡ | Hc \ Hθ | | +| | Hn.c. θ | = | i Hc \ Pθ ⊂ Kθ | (1.22) | + +Since, as stated by the quoted theorems, any real section \(\mathbb{G}_{\mathrm{R}}\) of a complex \(g\) is isomorphic to \(\mathbb{G}_{\theta}\) for a suitable \(\theta\), it follows that the splitting of the Cartan subalgebra of any \(\mathbb{G}_{\mathrm{R}}\), into a compact and non-compact subalgebra has an intrinsic meaning. Hence for every real section \(\mathbb{G}_{\mathrm{R}}\) we can define the compact and non-compact rank as follows: + +\[r_{n.c.}=\dim\mathcal{H}_{\theta}^{n.c.}\leq r;r_{c}=\dim\mathcal{H}_{\theta}^{comp}=r-r_{n.c.}\] (1.23) + +Having clarified the relation between real sections \(\mathbb{G}_{\mathrm{R}}\) of a complex Lie algebra \(g\), Cartan involutions \(\theta\) and non-compact Riemannian symmetric spaces \(\frac{\mathrm{G}_{\theta}}{\mathrm{H}_{\theta}}\) , let us then reconsider a symmetric coset manifold: + +\[\mathcal{M}_{\mathbb{G}_{\mathrm{R}}}= \frac{\mathrm{G}_{\mathrm{R}}}{\mathrm{H}_{c}}\] (1.24) + +where GR is a simple Lie group and \(\mathrm{H}_{c}\subset\) GR is its maximal compact subgroup. The Lie algebra \(\mathbb{H}_{c}\) of the denominator \(\mathrm{H}_{c}\) is the maximal compact subalgebra \(\mathbb{H}_{c}\subset\mathbb{G}_{\mathbb{R}}\). Denoting, as usual, by \(\mathbb{K}\) the orthogonal complement of \(\mathbb{H}_{c}\text{in}\mathbb{G}_{\mathrm{R}}\): + +\[\mathbb{G}_{\mathrm{R}}=\mathbb{H}_{c}\oplus\mathbb{K}\] (1.25) + +we define rank of the coset \(\mathrm{G}_{\mathrm{R}}/\mathrm{H}_{\mathrm{c}}\) the dimension of the non-compact Cartan subalgebra: + +\[\text{rank}(\mathrm{G}_{\mathrm{R}}/\mathrm{H}_{\mathrm{c}})=r_{nc}\equiv\dim\mathcal{H}^{n.c.};\mathcal{H}^{n.c.}\equiv\text{CSA}_{\mathbb{G}_{\mathrm{R}}}\bigcap\mathbb{K};\mathcal{H}^{comp}\equiv\text{CSA}_{\mathbb{G}_{\mathrm{R}}}\bigcap\mathbb{H}_{c}\] (1.26) + +Next, choosing a basis of \(\mathcal{H}_{c}\) aligned with the simple roots: + +\[\mathcal{H}_{c}=\text{span}\{iH_{\alpha_{i}}\}\] (1.27) + +we see that by duality of vector spaces the action of the Cartan involution \(\theta\) is transferred from the Cartan subalgebra to the simple roots \(\alpha_{i}\) and hence to the entire root lattice. As a consequence we can introduce the notion of real and diff --git a/outputs/arxiv_math/2503.07629_pg32_pg1_repeat1.md b/outputs/arxiv_math/2503.07629_pg32_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..f3ef2c89df80b9dc7bd93de84efb9133abd451c4 --- /dev/null +++ b/outputs/arxiv_math/2503.07629_pg32_pg1_repeat1.md @@ -0,0 +1,27 @@ +### 5.4. Orthonormal Bases and Particulate, Integral, and Prime Wave Numbers + +One may extend the \(n\) orthogonal bases \(\boldsymbol{u}( \frac{1}{n} ,j),j=1,n\) of additive wave numbers defined in Section 4.3 to bases of invertible wave numbers that are analogous to the standard bases of Euclidean geometry. Theorem 5.24. For every \(n\epsilon\mathbb{N}^{+}\), there exist \(n\) invertible wave numbers + +\[\{\boldsymbol{e}( \tfrac{1}{n} ,\xi)=(0,...,0,1,0,...,0)|\xi=1,n^{\}},\] (93) + +in which the 1 appears in the \(\xi\)th place, that form an orthonormal basis for representing an invertible wave number Aw\((f,g)\). + +## Proof Using Equation 67 define + +\[\boldsymbol{e}( \tfrac{1}{n} ,j)= \tfrac{1}{\mathbf{n}} \boldsymbol{u}( \tfrac{1}{n} ,j),j=1,n\] (94) + +from which it follows that the jth term of the principal sequence is + +\[\boldsymbol{e}_{1}( \tfrac{1}{n} ,j)=(0,...,1,...,0)\] (95) + +with the 1 occurring in the jth position. Clearly the \(n\) numbers form an orthonormal set of invertible wave number under \(\otimes\) and it follows from Definition 5.1 for the invertible wave numbers that the expression \(\bigoplus_{j\epsilon S}\boldsymbol{e}( \frac{1}{n} ,j)(\mathbf{Aw}(f,g))\), in which Aw\((f,g)\) an invertible wave number of period \(n\text{and}S\) is a subset of\(n\) the first \(n\) natural numbers, is an invertible wave number of period \(n\) with \(\bigoplus_{j=1}\mathbf{Aw}(f,g)\mathbf{e}( \frac{1}{n} ,\xi)=\mathbf{Aw}(f,g).\) + +\(\square\) + +under \(\otimes\) that select the phases that comprise a wave number. One may + +The wave numbers of the basis \(\boldsymbol{e}( \frac{1}{n} ,j),j=1,n^{\}}\) may be viewed as operators therefore employ these operators in constructing wave numbers that contain arbitrary subsets of the phases of an invertible wave number Aw\((f,g)\) and zeros in place of the non-included phases. Three interesting examples of such classes are the integral, particulate, and prime wave numbers. + +#### 5.4.1. The Integral Wave Numbers + +Integral wave numbers are sequences of numbers whose phases are the cumulative sums of the phases of a wave number. The integral wave numbers may be constructed by applying the orthonormal basis to previously-defined wave numbers in a manner specified in diff --git a/outputs/arxiv_math/2503.07632_pg11_pg1_repeat1.md b/outputs/arxiv_math/2503.07632_pg11_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..171bd269b999d4f3fa61f3954a517601d1808922 --- /dev/null +++ b/outputs/arxiv_math/2503.07632_pg11_pg1_repeat1.md @@ -0,0 +1,21 @@ +![](figure) + +![](figure) + +Figure 6. Experimentally motivated Plomp-Levelt consonance curve for \(f=440\) Hz, \(N=M=6\). + +\[w_{c}(f,N,A,t)=\sum_{n=1}^{N}a_{n}w(nf,t),\] + +The next lemma shows that the denominators present in the cosine similarity formula play little role in our considerations under appropriate assumptions on \(f\text{and}N\). + +Lemma 2. Let \(a_{n}\in[0,1]\text{for}n=1,\ldots,N\). We have + +\[\bigg|\int_{0}^{T}w_{c}(f,N,A,t)^{2}dt- \frac{NT}{2} \bigg|\leq \frac{N^{2}}{2\pi f}\] + +Proof. We are looking for a bound on the integral of \(w_{c}^{2}\), that is the following expression + +\[\sum_{n=1}^{N}\sum_{m=1}^{N}\int_{0}^{T}a_{n}a_{m}w(nf,t)w(mf,t)dt\] + +We obtain the desired bound by looking at the diagonal (i.e. those, where \(n=m)\) and the non-diagonal entries in the summation, and applying the first and second part of the earlier lemma. For the diagonal entries, we have + +\[\frac{NT}{2} - \frac{N}{8\pi f} \leq\sum_{n=1}^{N}\int_{0}^{T}a_{n}^{2}w^{2}(nf,t)dt\leq \frac{NT}{2} + \frac{N}{8\pi f}\] diff --git a/outputs/arxiv_math/2503.07692_pg35_pg1_repeat1.md b/outputs/arxiv_math/2503.07692_pg35_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..c9b9474b458a9233b2b2f93170f1f5f7a85029ad --- /dev/null +++ b/outputs/arxiv_math/2503.07692_pg35_pg1_repeat1.md @@ -0,0 +1,7 @@ +## 7 Numerical results + +In this section, we present a few numerical experiment results to validate the theoretical analysis, including the numerical tests for convergence rate, energy stability, mass conservation and the concentration positivity. Since the proposed numerical scheme (3.13) is nonlinear and coupled, its implementation turns out to be quite technical. A linearized iteration solver is applied to implement the numerical algorithm. In more details, the nonlinear parts are evaluated in terms of the numerical solution at the previous stage, while the linear diffusion and temporal derivative parts are implicitly computed at each iteration stage. In turn, only a linear numerical solver is needed at each iteration stage, although the numerical scheme (3.13) is nonlinear. Such a linearized iteration solver has been widely reported for various nonlinear numerical schemes; in particular, a geometric iteration convergence rate has been theoretically justified for the Poisson-Nernst-Planck (PNP) system [20], a highly nonlinear and singular gradient flow model. A similar theoretical analysis is expected for the linearized iteration approach to the numerical scheme (3.13), while the technical details will be left in the future works. Such a linearized iteration method is highly efficient; the theoretical analysis in [20] indicates a geometric iteration convergence rate, while the practical computations have revealed an even better iteration convergence rate in the implementation process. Only five to ten linear solvers are needed in the iteration process for most computational examples reported in this article, and the computational cost of the linear solver is comparable with a standard Poisson solver. Moreover, other than the linearized linear solver, some other alternate iteration approaches, such as preconditioned steepest descent (PSD) solver [9, 14], could be chosen, and a comparison between difference iteration methods will be considered in the future works. A two dimensional domain is set as \(\Omega=(-2,2)^{2}\). At the initial time step, a first-order scheme is used to obtain the numerical solution. In the subsequent time steps, an iterative algorithm (similar to the one in [21]) is used to implement the fully nonlinear scheme (3.13). The initial data is chosen as + +\[\begin{aligned} p_{0}(x,y)&=0.6+0.2\cos(\pi x)\cos(0.5\pi y), \\ n_{0}(x,y)&=0.6+0.2\cos(0.5\pi x)\cos(\pi y), \\ u_{0}(x,y)&=-0.25\sin^{2}(\pi x)\sin(2\pi y), \\ v_{0}(x,y)&=0.25\sin(2\pi x)\sin^{2}(\pi y), \\ \psi_{0}(x,y)&=\cos(0.5\pi x)\cos(0.5\pi y), \end{aligned}\] (7.1) + +where periodic boundary condition is used. The computation is performed with a sequence of uniform mesh resolutions, and the time step size is taken as \(\tau=0.1h\). Since the exact solution could not be explicitly represented, we measure the Cauchy error to test the convergence rate, a similar approach to that of [34]. In particular, the error between coarse and fine grid spacings \(h\) and \(h/2\) is recorded by \(\Vert e_{\zeta}\Vert=\Vert\zeta_{h}-\zeta_{h/2}\Vert\). We present the \(\ell^{2}\text{and}\ell^{\infty}\) errors of all the physical variables at a final time \(T=0.1\). An almost perfect second order accuracy, in both time and space, has been observed in this numerical experiment, which agrees with the theoretical analysis. In addition, the simulation results are used to demonstrate the numerical performance to preserve certain physical properties. The total mass conservation of the ion concentration variables (over the computational domain) has been perfectly confirmed in the upper panel of the Figure 1. Moreover, in the same figure, a monotone dissipation property of the discrete total energy \(E_{h}\) is also clearly observed, which confirms the theoretical analysis. To explore the positivity-preserving property, we focus on the evolution of the minimum concentration value, i.e., \(C_{\text{min}}:=\min_{i,j}(\min_{i,j}n^{m},\min_{i,ji,j}p_{i,j}^{m})\). As displayed in Figure 2, the numerical solutions of ion concentration variables remain positive all the time, even though their values could become very low. Overall, these numerical evidences have diff --git a/outputs/arxiv_math/2503.07696_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.07696_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..54f1da1ee49c2281a11214d0f2522aaeeb1f4f46 --- /dev/null +++ b/outputs/arxiv_math/2503.07696_pg3_pg1_repeat1.md @@ -0,0 +1,27 @@ +Throughout we assume the Riemann Hypothesis, in the strong form that the nontrivial zeros are also simple. We also need to assume the following: + +Hypothesis D. The level curves \(\text{Re}(\eta(s))=0\) are differentiable. This is automatic except at isolated points where \(\eta^{\prime}(s)=0\), so we are really assuming that when \(\eta^{\prime}(s)=0,\arg(\eta(s))\neq \pm\pi/2\). This prevents the level curves from branching. Hypothesis D is plausible because \(\pm\pi/2\) are only two points on the unit circle, while the zeros of \(\eta^{\prime}(s)\) form a countable set. + +Here’s a summary of the sections of the paper: + +§1 Classification of the zeros of \(\zeta(s)\text{and}\zeta^{\prime}(s)\) into different types by means of the level curves, and results on the asymptotics of the types. + +§2 Computation and classification of 106 zeros of \(\zeta^{\prime}(s)\) near T = 1010. + +§3 Two lemmas. + +§4 A closer look at the type 2 zeros, and the curvature of the level curve. + +§5 A canonical bijection between the complex zeros \(\rho^{\prime}\) of \(\zeta^{\prime}(s)\), and the complex zeros \(\rho^{\prime\prime}\) of \(\zeta^{\prime\prime}(s)\). + +§6 Adaptation of a theorem of Marden, and the location of \(\rho^{\prime\prime}\) relative to \(\rho^{\prime}\). + +§7 Curvature of the level curve at \(\rho^{\prime}\) in terms of all the other zeros \(\lambda^{\prime}\neq \rho^{\prime}\). + +§8 Appendix: With pA(z) the characteristic polynomial of a unitary matrix A, we give a classification of the zeros of pA(z) and \(p_{A}^{\prime}(z)\) analogous to that in §1. + +## 1. CLASSIFICATION OF ZEROS + +Proposition 1. With the usual indexing \(\gamma_{1}<\gamma_{2}<\) . . . of the imaginary parts of the zeros of \(\zeta(s)\), every odd indexed zero lies on a contour \(Im(\eta(s))<0\). Every even indexed zero lies on a contour \(Im(\eta(s))>0\). + +Proof. This follows from Lemma 2 below, which says that as t increases, the argument of \(\eta(1/2+\) it) decreases by exactly \(\pi\) between diff --git a/outputs/arxiv_math/2503.07705_pg1_pg1_repeat1.md b/outputs/arxiv_math/2503.07705_pg1_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..5d8d47ce7d3e5fefe2cb388de4ba02f8845abbe6 --- /dev/null +++ b/outputs/arxiv_math/2503.07705_pg1_pg1_repeat1.md @@ -0,0 +1,29 @@ +# Geometric Delocalization in Two Dimensions + +Laura Shou,1 Alireza Parhizkar,1 and Victor Galitski1 + +1Joint Quantum Institute, Department of Physics, University of Maryland, College Park 20742 + +We demonstrate the existence of transient two-dimensional surfaces where a random-walking particle escapes to infinity in contrast to localization in standard flat 2D space. We first prove that any rotationally symmetric 2D membrane embedded in flat 3D space cannot be transient. Then we formulate a criterion for the transience of a general asymmetric 2D membrane. We use it to explicitly construct a class of transient 2D manifolds with a non-trivial metric and height function but “zero average curvature,” which we dub tablecloth manifolds. The absence of the logarithmic infrared divergence of the Laplace–Beltrami operator in turn implies the absence of weak localization, non-existence of bound states in shallow potentials, and breakdown of the Mermin–Wagner theorem and Kosterlitz–Thouless transition on the tablecloth manifolds, which may be realizable in both quantum simulators and corrugated two-dimensional materials. + +Introduction— There are many seemingly disconnected physical phenomena which are related to the properties of the Laplace–Beltrami operator. They include random walk or diffusion, the standard Schrödinger equation, the properties of fluctuations in symmetry broken phases, interactions between topological excitations, and many others. In particular, if the heat kernel \(p(x,y;t)\) integrated over time — the Green’s function of the Laplace–Beltrami operator on a manifold — is infinite it implies automatically the following properties of this space: a random-walking particle is guaranteed to return to its starting region infinitely often (recurrence), there is Anderson localization in an arbitrarily weak disorder potential, any shallow quantum potential well hosts a bound state, no long-range order with spontaneously broken continuous symmetry can exist at finite temperature in this space (Mermin–Wagner theorem), to name a few. Specifically in flat Euclidean space, the two-dimensional case is critical as the corresponding heat kernel integral diverges logarithmically: + +\[\begin{aligned} \int_{\text{min}}^{\tau_{\text{max}}}p(x,y;t)dt&\propto\ln(\tau_{\text{min}}/\tau_{\text{max}}), \\ \tau \end{aligned}\] (1) + +which means that the two-dimensional flat space is recurrent, while higher dimensional flat spaces, which have finite integrals, are transient (a random-walking particle always escapes its starting region). An additional physical phenomenon tied to the Green’s function of the Laplace–Beltrami operator specific to \(O(2)\) models on two-dimensional manifolds is the behavior of topological excitations there. In the conventional flat space, the logarithmic divergence (1) is tied to logarithmic vortex-vortex interactions that in turn lead to a finite-temperature Berezinskii–Kosterlitz–Thouless transition due to the competition with entropic effects which also scale logarithmically. Due to the abundance of fundamental physical phenomena tied to the Laplace–Beltrami operator, it is reasonable to ask if there exist two-dimensional curved transient manifolds. Recent works [1, 2] have studied some of these phenomena on hyperbolic manifolds and lattices, which are known to be transient [3–5]. Here we ask: is it possible to have a two-dimensional membrane (i.e., a two-dimensional smooth manifold embedded in flat Euclidean three-dimensional space described by a height function) that is transient, \(\int_{\tau_{\text{min}}}^{\infty}p(x,y;t)dt<\infty\)? We answer this question in the affirmative and explicitly construct 2D transient “tablecloth manifolds” (Fig. 1), which hence lead to the breakdown of standard two-dimensional physics. + +![](figure) + +![](figure) + +FIG. 1. Two examples of tablecloth manifolds. On the right, a generic tablecloth manifold, and on the left the simplified example constructed in Eq. (12). The colors demonstrate the ratio of the volume element at any point on the manifold, \(\sqrt{g(r,\theta)} drd\theta\), to the regular flat volume element at the same point, \(rdrd\theta\). Therefore, the colors encode the value of \(\sqrt{g(r,\theta)} /r\). This allows us to compare the volume growth to the regular flat one as we go away from the origin with increasing \(r\). Purple corresponds to the regular \(\pi r^{2}\) volume growth of a flat disk, while other colors designate faster growths. + +Transience on a 2-dimensional membrane— The general form of a two dimensional Riemannian metric in polar-like coordinates, with no isometries assumed, is given by, + +\[ds^{2}=A^{2}(r,\theta)dr^{2}+2B(r,\theta)drd\theta+C^{2}(r,\theta)d\theta^{2}.(2)\] + +For rotationally symmetric manifolds with metric of the form \(ds^{2}=dr^{2}+f(r)^{2}d\theta^{2}\), a well-known result [6, 7] is that transience is equivalent to the condition + +\[\int_{1}^{\infty} \frac{1}{f(r)} dr<\infty.\] (3) + +Using this, we can find rotationally symmetric metrics \(ds^{2}= dr^{2}+f(r)^{2}d\theta^{2}\) corresponding to transient Brownian motion, and which also satisfy an average zero curvature condition, as diff --git a/outputs/arxiv_math/2503.07708_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.07708_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..bd1a52b7f35182b5383d32d87ae823cd95af33b5 --- /dev/null +++ b/outputs/arxiv_math/2503.07708_pg3_pg1_repeat1.md @@ -0,0 +1,29 @@ +\[\begin{aligned} \hat{S}_{0}(\mathbf{k})&=\tau^{z}\hat{Q}_{0}=\sum_{\mathbf{k}}d_{\mathbf{k}}^{\dagger}\hat{S}_{0}(\mathbf{k})d_{\mathbf{k}} \\ \hat{S}_{1}(\mathbf{k})&=\cos k_{z}\tau^{z}+\sin k_{z}\tau^{x}\hat{Q}_{1}=\sum_{\mathbf{k}}d_{\mathbf{k}}^{\dagger}\hat{S}_{1}(\mathbf{k})d_{\mathbf{k}}. \end{aligned}\] + +The first is the usual \(U(1)\) symmetry, while the second is composed of Kitaev Majorana chains [16] along \(z-\)axis wires. These generators do not commute, instead they generate an infinite-dimensional Lie algebra known as the Onsager algebra, introduced in [17]. This algebra has recently appeared in the study of the 1+1D chiral anomaly on the lattice [18, 19], and offers another route to defining exact symmetries on the lattice giving anomalous symmetries in the IR. We can actually write a Hamiltonian that has this symmetry and two Weyl nodes, which in the BdG formalism above (using the \(d_{\mathbf{k}}^{\dagger}\) basis) is + +\[\begin{aligned} h_{\text{do}\mathrm{u}\text{ble Weyl}}^{\mathrm{B}d\mathrm{G}}(\mathbf{k})&=1_{\tau}\otimes \frac{1}{2} \bigg[\sin k_{x}\sigma^{x}+\sin k_{y}\sigma^{z} \\ +[\cos k_{z}-\cos K+m(\mathbf{k})]\sigma^{y}\bigg], \end{aligned}\] + +where the identity in the \(\tau\) basis ensures it has both \(U(1)\) symmetries, \(K\) is a parameter, and \(m(\mathbf{k})\) is the same as in (3). This model is a magnetic Weyl semimetal model that has the two Weyl nodes at \(\mathbf{k}=\pm\mathbf{K}\text{where}\mathbf{K}=(0,0,K)\). Let’s linearize around the Weyl nodes. We get + +\[h_{l}^{\mathrm{B}\text{dG}}=1_{\tau}\otimes \frac{1}{2} (k_{x}\sigma^{x}+k_{y}\sigma^{z}-\sin Kk_{z}\sigma^{y}).(12)\] + +This shows that for \(K\neq 0,\pi\), the two Weyl nodes have an opposite handedness. To figure out the effect of \(\hat{S}_{1}(\mathbf{k})\) at low energy, we can also linearize it, and obtain + +\[\hat{S}_{1,K}(\mathbf{k})=\cos K\tau^{z}+\sin K\tau^{x}.\] (13) + +The important feature for \(K\neq 0,\pi\) is that the second term is non-zero. Thus, together with \(\hat{S}_{0}=\tau^{z}\), these generate an \(su(2)\) algebra acting on the low energy theory. Note that \(\tau^{x}\) acts by exchanging particles at K with holes at \(-\mathbf{K}\). Thus, it is convenient to apply a charge conjugation the right-handed Weyl fermion, to give a low energy theory in terms of two left-handed Weyl fermions, now with opposite charge w.r.t. \(\tau^{z}\) rotations. \(\tau^{x}\) rotations meanwhile act by a flavor rotation exchanging the two Weyl fermions. Thus, our symmetry generators \(\hat{Q}_{0},\hat{Q}_{1}\) correspond to two \(su(2)\) generators in the flavor symmetry of the low energy, at an angle of \(K.\text{For}K\neq 0,\pi\), they thus generate the whole chiral symmetry. We can demonstrate that this symmetry protects the gapless Weyl points. To do so, we must break translation symmetry, since otherwise \(z-\)axis translations also act as a discrete axial symmetry and help to stabilize the Weyl nodes [12, 13]. To analyze translation-symmetry breaking, we consider an extended basis \(e_{\mathbf{k}}\equiv(c_{\mathbf{k}-\mathbf{K}},c_{-\mathbf{k}+\mathbf{K}}^{\dagger},c_{\mathbf{k}+\mathbf{K}},c_{-\mathbf{k}-\mathbf{K}}^{\dagger})^{T}\) (we suppress the spin component). Hamiltonians in this basis may couple states at k with k + 2K but are automatically \(\hat{Q}_{0}\) preserving. In this basis the symmetry action of \(\hat{Q}_{1}\) becomes + +\[\begin{aligned} U(1)_{\hat{Q}_{1}}:\delta e_{\mathbf{k}}&=i(\cos k_{z}\cos K\tau^{z}+\sin k_{z}\sin K\eta^{z}\tau^{z} \\ +\sin k_{z}\cos K\tau^{x}-\cos k_{z}\sin K\eta^{z}\tau^{x})e_{\mathbf{k}}, \end{aligned}\] + +which prohibits all mass terms except \(m_{j}(\mathbf{k})\eta^{z}\sigma^{j}\). However, these terms always commute with at least one term in the original Hamiltonian, so the result is a shift in the gapless modes rather than a gap. At a large enough pertubation, we can move the modes until \(K=0\text{or}\pi\), where the symmetry generators are aligned and no longer generate the whole \(SU(2)\) symmetry. At these special points, we will be able to open a symmetric gap. + +# IV. TIME REVERSAL SYMMETRIC SINGLE WEYL FERMION IN 3+1D + +So far, we have considered time-reversal breaking models. We can also construct time-reversal invariant models, at the cost of making the symmetry generator slightly more not-on-site. As long as \(S(\mathbf{k})\) is a smooth function of the momentum, then the charge density in real space will be a sum of terms with faster-than-polynomial decay. Such “almost-local” operators share many properties with local operators, while being closed under Hamiltonian evolution generated by such terms [20, 21]. This will allow us to employ bump functions and partitions of unity in momentum space. To construct a time-reversal invariant model with a single protected Weyl fermion, we begin with a model on a cubic lattice with eight Weyl nodes. We use the BdG formalism with the basis \(d_{\mathbf{k}}^{\dagger}\equiv(c_{\mathbf{k}\uparrow}^{\dagger},c_{\mathbf{k}\downarrow}^{\dagger},c_{-\mathbf{k}\uparrow},c_{-\mathbf{k}\downarrow})\) used above, giving the Hamiltonian + +\[h_{8}^{\mathrm{B}\text{dG}}(\mathbf{k})= \frac{1}{2} [\sin k_{x}\sigma^{x}+\sin k_{y}\sigma^{z}+\sin k_{z}\tau^{z}\sigma^{y}].(15)\] + +This model has Weyl nodes at all eight time-reversal-invariant-momentum (TRIM) points of the Brillouin zone, as well as a time-reversal symmetry \(\Theta=i\sigma^{y}\mathcal{K}\), where \(\mathcal{K}\) is complex conjugation, satisfying \(\Theta^{2}=-1\). We will now add a \(U(1)\) symmetry-breaking term that will gap out all Weyl nodes except the one at k = 0. In order to facilitate our discussion, let us first define a bump function \(B(k)\) given by + +\[B(\mathbf{k},w)= \begin{cases} e^{- \frac{w2}{|\mathbf{k}|^{2}-w^{2}}} & \text{for}|\mathbf{k}|\)cy for some large constant \(c>1\). By “almost all” basis states, we mean that the fraction of basis states to which the above statement applies approaches 1 as our experimental resolution improves (i.e., \(\epsilon\rightarrow0,T\Delta E\rightarrow\infty)\). We describe this result as applying to “almost all” basis states as this is the statement we expect to be accessible in experiments for large systems; however, with sufficient resolution, our results actually allow us to constrain thermalization in every single state in the Hilbert space as well (which we expect may be possible for sufficiently small systems). Eq. (9) applies independently of whether the energy levels are degenerate or show any kind of statistical correlations. Most significantly, limits such as \(T\rightarrow\infty\) or \(\Delta E\rightarrow0\) are not required for this result, but may be taken if desired and allowed by the specific context. Our main physical emphasis here is that knowing the properties of observables even at a low resolution corresponding to an energy band \(\Delta E\) is sufficient to establish thermalization over finite or infinite time averages in arbitrary physical states, without requiring any higher resolution knowledge of the energy levels such as in Eq. (2). In addition to time averages, one is also interested in showing that the observable attains thermal equilibrium at almost all times in a finite time ranget\(\in[-T,T]\). Unlike the distinction between eigenstate thermalization in Eq. (2) and the full statement of ETH in Eq. (3), where ETH assumes a separate off-diagonal structure for thermal equilibrium, we will show that the same notion of energy-band thermalization is also sufficient to describe thermal equilibrium. Specifically, we will require (in Sec. 7) that a cloned version of Aˆ in a doubled Hilbert space (each with dynamics generated by Hˆ) also satisfies energy-band thermalization for the doubled Hamiltonian: + +\[[\hat{A}\otimes\hat{A}]_{\Delta E}\approx A_{\text{th}}^{2}\hat{1}\otimes\hat{1}.\] (10) + +We note that this cloning of the operator is a mere formal device, and such properties can be experimentally established entirely within a single copy of the system. Eq. (10) is sufficient for the observable to attain thermal equilibrium in the following sense: + +\[〈\psi_{k}(t)|\hat{A}|\psi_{k}(t)〉\approx A_{\text{th}}\text{for almost all}t\in[-T,T],\text{for any}T\gg2\pi/\Delta E.\] (11) + +Here, “almost all t” should be interpreted as “all t except a set whose length is smaller than \(T/c_{2}\) for some large constant \(c_{2}>\) 1”. This “cloning” strategy may be regarded as a quantum version of a theorem in classical ergodic theory \([6]\), which states that a dynamical system is weakly mixing (which loosely corresponds to thermal equilibrium) if and only if its “cloned” version with two copies of the system is ergodic (which loosely corresponds to time-averaged thermalization). This is especially convenient for us, as it allows us to use the same conceptual notion of energy-band thermalization (but applied to the observable vs. its cloned version) to address both time-averaged thermalization and instantaneous thermal equilibrium over finite timescales. Collectively, Eqs. (8), (9), (10) and (11) establish a fully quantum connection between properties of observables with a finite energy resolution and their thermalization over finite time scales. + +#### 1.2.2 Bypassing energy levels with finite-time autocorrelators + +We will now describe how energy-band thermalization \([\text{Eqs}.(8),(10)]\) is advantageous compared to conventional statements of ETH \([\text{Eqs}.(2),(3)]\) not only in the ability to access finite time scales, diff --git a/outputs/arxiv_math/2503.07741_pg8_pg1_repeat1.md b/outputs/arxiv_math/2503.07741_pg8_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..f6d9377f83ceea9c9d9a08a096d4436d361902d4 --- /dev/null +++ b/outputs/arxiv_math/2503.07741_pg8_pg1_repeat1.md @@ -0,0 +1,53 @@ +1 + +\[0.2\] + +\[0.1\] + +\[0.6\] + +\[0.5\] + +\[0.4\] + +−4 + +0 + +4 + +\[(\lambda/\lambda_{\mathrm{c}}^{(1)}-1)L^{1/\nu}\] + +−4 + +0 + +4 + +\[(\lambda/\lambda_{\mathrm{c}}^{(2)}-1)L^{1/\nu}\] + +![](figure) + +FIG. 5. Data collapses around the critical region. + +● N=480 N=960 ✗ N=1920 ↓ N=2880 + +−4 + +0 + +4 + +\[(\lambda/\lambda_{\mathrm{c}}^{(3)}-1)L^{1/\nu}\] + +Scaling plots for the order parameter \(\langle|z_{m}|\rangle\) (top row), the susceptibility \(\langle\Delta|z_{m}|^{2}\rangle\) (middle row), and the Binder cumulant \(U_{4}\) (bottom row), using all data shown in Fig. 3 (corresponding to \(\rho_{0}=1/3,L\in[480,2880],m=1,2, \beta=1/2\) corresponding to the (mean field) Kuramoto universality class. Vertical dashed lines signal the critical point. + +and \(\epsilon(m)=2.5m)\) and the critical exponents \(\nu=2,\gamma=1\), + +3 + +should cross at an \(L-\)independent value of \(\lambda\) that identifies the infinite-size critical point \(\lambda_{c}^{(m)}\). This is indeed clearly observed in Figs. 3.(g)-(i), where we can also verify that these measured critical packing field couplings agree remarkably well with the hydrodynamic predictions for \(\lambda_{c}^{(m)}\forall m\) (vertical dashed lines), obtained from a local stability analysis of the homogeneous density solution, see Eq. (7). + +Interestingly, if we monitor the average density at a given point along the ring, e.g. \(\langle\rho(x_{0},t)\rangle\), we find persistent oscillations as a function of time for any packing order \(m\), see Fig. 4, a universal signature of time-crystalline order [1–7, 9–11, 13, 19– 23, 25, 27, 28, 38, 84–91]. These oscillations are characterized by a period \(2\pi\omega_{m}^{-1}\) in the diffusive scale, with \(\omega_{m}\) the condensates average velocity. However, for any finite system size \(L\), the oscillations amplitude is clearly damped in time, see Fig. 4, with a damping rate \(\gamma_{L}\) that can be obtained from an exponential fit to the envelope of \(\langle\rho(x_{0},t)\rangle\) for each \(L\), see dot-dashed lines in Figs. 4.(a)-(c). In any case, the measured \(\gamma_{L}\) decays to zero for each \(m\) in the thermodynamic \(L\rightarrow\infty\) limit, see Fig. 4.(d), meaning that the observed local density oscillations converge toward the hydrodynamic (undamped) periodic prediction in this limit (dashed line in Figs. 4.(a)-(c)). This observation clearly signals the rigidity of the long-range spatiotemporal order emerging in the observed programmable time-crystal phases. We also measured the condensate speed \(\omega_{m}\) for different packing orders \(m\) as a function of \(1/L\), see Fig. 4.(e), finding a remarkable agreement with the hydrodynamic predictions in the large system size limit \(L\rightarrow\infty\). + +In order to determine the critical exponents characterizing the universality class of the observed phase transitions to these programmable time-crystal phases, we performed extensive measurements around the critical region in our Monte Carlo simulations. Due to the scale invariance underlying any continuous phase transitions, the relevant observables near the critical point are expected to be homogeneous functions of the two length scales characterizing the problem at hand, namely the correlation length \(\xi\) and the system size \(L[8,79]\). This immediately implies the following scaling relations diff --git a/outputs/arxiv_math/2503.07765_pg8_pg1_repeat1.md b/outputs/arxiv_math/2503.07765_pg8_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..70a73bcd066646d8f58c1eb7103888b87c96d3a3 --- /dev/null +++ b/outputs/arxiv_math/2503.07765_pg8_pg1_repeat1.md @@ -0,0 +1,59 @@ +\[0.5\] + +\[0.0\] + +\[-0.5\] + +\[-1.0\] + +\[7.2^{\circ}\] + +\[45^{\circ}\] + +✗ Orthogonal ● Non-orthogonal + +\[-1.0-0.50.00.5\] + +![](figure) + +Fig. 2. Constellation in terms of the basis formed by the decision boundaries [d0 d1] when the signal vectors are orthogonal and not orthogonal. + +## B. Upper Bound for Quasi-Biorthogonal Signaling + +Here, we divert our attention to the case where \(M>2\). Let \(A_{i+}\) be the event of detecting \(+\mathbf{s}_{i},\text{and}A_{i-}\) be the event of detecting \(-\mathbf{s}_{i}\). Then, the probability of a symbol error given \(+\boldsymbol{s}_{i}\) was transmitted is given by + +\[P_{e|\mathbf{s}_{i}}=\Pr\left[\bigcup_{jj^{=}\neq ^{0}i}^{M-1}(A_{j+}\cup A_{j-})\cup A_{i-}\Bigg|+\boldsymbol{s}_{i}\text{sent}\right].(75)\] + +The right-hand side of (75) should be stated as the probability of one or more the indicated events happens. Then, taking note that for any pair of events \(A\text{and}B,\Pr[A\cup B]\leq \Pr[A]+\Pr[B],(75)\) implies + +\[\begin{aligned} P_{e|\mathbf{s}_{i}}&\leq\sum_{j=0}^{M-1}\Pr[A_{j+}\cup A_{j-}\cup A_{i-}|+\boldsymbol{s}_{i}\text{sent}].(76) \\ j&\neq i \end{aligned}\] + +Making use of the results in Section V-A, this result can be written as + +\[\begin{aligned} P_{e|\mathbf{s}_{i}}&\leq\sum_{j=0}^{M-1}\left(1-\prod_{k=0}^{1}\left[1-Q\left( \sqrt{\frac{2\mathcal{E}\rho_{i,j,k}^{2}}{N_{0}}} \right)\right]\right)(77) \\ j&\neq i \end{aligned}\] + +where + +\[\rho\rho_{=_{i,j,1}^{i,j,0}}= \frac{(\mathbf{s}_{i}-\mathbf{s}_{j})^{\mathrm{T}}\mathbf{s}_{j}}{\frac{(\mathbf{s}_{i}^{|\mathbf{s}}+^{-_{i}}\mathbf{s}_{j}^{\mathbf{s}})^{|_{j_{\mathrm{T}}}}\mathbf{s}_{j}}{|\mathbf{s}_{i}+\mathbf{s}_{j}|}} .\] (78) + +Using (77) and assuming that the data symbols \(\pm\mathbf{s}_{i}\), for \(i=0,1,\cdots ,N-1\), are equally likely to be transmitted, one will find that the probability of a symbol error, \(P_{e}\), is upper bounded by + +\[\begin{aligned} P_{e}&\leq \frac{1}{M} \sum_{i=0}^{M-1}\sum_{j=0}^{M-1}\left(1-\prod_{k=0}^{1}\left[1-Q\left( \sqrt{\frac{2\mathcal{E}\rho_{i,j,k}^{2}}{N_{0}}} \right)\right]\right). \\ j&\neq i \end{aligned}\] (79) + +The simulation results presented in the next section reveal that this upper bound becomes tight as the SNR increases. This observation may be explained as follows. As the SNR increases, the events referred to on the right-hand side of + +(75) become nearly non-overlapping, equivalently, it will be unlikely that any pair of these events happen simultaneously. In that case, the inequality in (76) will become closer to an equality. + +## C. Upper Bound for Quasi-Orthogonal Signaling + +The desired bound here can be derived following those of the biorthogonal case with some minor modifications. First, for the case where \(M=2\), the decision boundary that separates s0 and s1 is the bisect line between them. This is the line in the direction d1 in Fig. 1(a). With this observation, the probability of a symbol error here is found to be + +\[P_{e}=Q\left( \sqrt{\frac{2\mathcal{E}\rho_{0}^{2}}{N_{0}}} \right).\] (80) + +Next, following the same line of thoughts as those in Section V-B, and removing \(A_{j-}\text{and}A_{i-}\) from the right-hand side (75) and subsequent equations, it is not hard to arrive at the following upper bound for probability of symbol error for the present case. + +\[\begin{aligned} P_{e}&\leq \frac{1}{M} \sum_{i=0}^{M-1}\sum_{j=0}^{M-1}Q\left( \sqrt{\frac{2\mathcal{E}\rho_{i,j,0}^{2}}{N_{0}}} \right) \\ j&\neq i \end{aligned}\] (81) + +## VI. NUMERICAL RESULTS + +In this section, to get insight to the theoretical findings of the previous sections, we present a set of numerical results that compare the symbol error rates (SERs) of the quasi-orthogonal and quasi-biorthogonal signaling methods against their respective orthogonal and biorthogonal counterparts. The goal here is to explore the amount of performance loss incurred as the non-orthogonality of the symbol vectors increases. Also, through numerical results, we examine the tightness of the upper bounds that were derived in Section V. Here, to generate a set of non-orthogonal codes, we begin with generating a random correlation matrix W with diagonal elements of unity. For W to be a valid correlation matrix, it should be symmetric, i.e., WT = W, and all of its eigenvalues should be non-negative. To satisfy these conditions, first, the off diagonal elements in the upper triangular part of R are chosen to be random and independent taken from a uniform distribution in the interval \([-\rho_{\text{max}},+\rho_{\text{max}}]\). These elements are then copied to the lower triangular part of W so that to satisfy the symmetry condition WT = W. Finally, to make diff --git a/outputs/arxiv_math/2503.07785_pg20_pg1_repeat1.md b/outputs/arxiv_math/2503.07785_pg20_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..db7e193437edda01a3b25d34c1e09e2892f61a4b --- /dev/null +++ b/outputs/arxiv_math/2503.07785_pg20_pg1_repeat1.md @@ -0,0 +1,189 @@ +0 + +-2 + +-4 + +-6 + +-8 + +0 + +10 + +30 + +40 + +50 + +60 + +70 + +80 + +\(t\) + +(a) \(\mathbb{Q}_{1}\) Galerkin FE solution + +0 + +-2 + +-4 + +-6 + +-8 + +0 + +10 + +30 + +40 + +50 + +60 + +70 + +80 + +\(t\) + +(c) \(\mathbb{Q}_{2}\) Galerkin FE solution + +0 + +-2 + +-4 + +-6 + +-8 + +0 + +10 + +30 + +40 + +50 + +60 + +70 + +80 + +\(t\) + +(e) \(\mathbb{Q}_{3}\) Galerkin FE solution + +0 + +-2 + +-4 + +-6 + +-8 + +0 + +10 + +30 + +40 + +50 + +60 + +70 + +80 + +\(t\) + +(b) \(\mathbb{Q}_{1}\) RV solution + +0 + +-2 + +-4 + +-6 + +-8 + +0 + +10 + +30 + +40 + +50 + +60 + +70 + +80 + +\(t\) + +(d) \(\mathbb{Q}_{2}\) RV solution + +0 + +-2 + +-4 + +-6 + +-8 + +0 + +10 + +30 + +40 + +50 + +60 + +70 + +80 + +\(t\) + +(f) \(\mathbb{Q}_{3}\) RV solution + +![](figure) + +Figure 4: Landau damping. Strong damping, the evolution of \(E_{e}\text{for}\alpha=0.5\). The distribution of nodes is \(N_{x}\times N_{v}=\) stabilized RV solutions for different polynomial spaces. + +Comparison of the Galerkin solution and + +49 + +\(\times97\). diff --git a/outputs/arxiv_math/2503.07794_pg18_pg1_repeat1.md b/outputs/arxiv_math/2503.07794_pg18_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..7a7935089ee93f0881b5c30bb921813bfe1ecea9 --- /dev/null +++ b/outputs/arxiv_math/2503.07794_pg18_pg1_repeat1.md @@ -0,0 +1,54 @@ +\[0.5\] + +\[0.0\] + + + + + + + + + + + + + + + + + + + + +
−0.5
−1.0● Eq. (50)5
○ ● Eq. (3) ■2 ●
∂t(Eq. (31))0
+ +\[-1.5\] + +Numerics + +−4 + +−2 + +## \(\Delta[\partial_{t}\langle p^{2}\rangle]1\) + +\(\overline{\ell}\) + +0 + +## \(x\) + +\(|\text{Th} \underline{\text{eory}-\text{Numerics}|/}\)Error + +\[-4-2024\] + +2 + +4 + +Figure 5: We show the \(O(1/\ell)\) correction to time derivative of the second moment of hard rods velocity distribution from the initial state (59), as a function of space at \(t=1\). The black line represent the numerical simulation of the hard rods gas, for which the time derivative is evaluated through the difference quotient (61) with \(\Delta t=0.05\). Then, the \(O(1/\ell)\) correction is estimated as the f2 parameter of a fit with model \(f(\ell)=f_{1}+f_{2}/\ell\) to \(\partial_{t}〈p^{2}〉(x,t;\ell),\text{for}\ell\in\{500,600\), . . . , \(1000\}\). The fit is performed independently for each point in space and the associated error bar is the standard deviation of the fit parameter. The numerical data are averaged over an ensemble of \(8\times10^{10}\) different realizations. The red dots represent the theoretical prediction of the new theory Eq. (50), while green dots are the prediction due to the Navier-Stokes-like theory Eq. (3). Finally, blue circles represent the time derivative of the solution (31), computed through difference quotient with \(\Delta t=0.01\). The inset shows the distance between theoretical predictions and data points in error bars units. One can clearly see that the new theory fits the numerical simulations much better, while the old Navier-Stokes-like theory Eq. (3) differs significantly. For numerical simulation we use the same data as in \([1]\). + +## 5 Conclusion + +In this manuscript we derived the equation that governs the diffusive correction to the Euler (generalized) hydrodynamic description of the integrable hard rods model. The starting point for this is the exact microscopic solution formula (22). Assuming the usual thermodynamic/hydrodynamic large deviation scaling of correlation functions, we perform the large-scale \(\ell\rightarrow\infty\) limit of (22), keeping the Euler \(\sim1/\ell^{0}\) and the diffusive contributions \(\sim1/\ell\). Then, taking a \(t\rightarrow0^{+}\) limit we finally obtain the equation governing the diffusive dynamics. We find that, unlike the previous Navier-Stokes like equation \((3)[14]\), the equation for the one-point function up to, including, the diffusive scale, depends on the two-point function at the Euler scale. The equation for the Euler-scale two-point function depends on the Euler-scale one-point function, but not on higher order correlation functions (those can be neglected due to the assumed large deviation scaling). Therefore, the diffusive dynamics are described by a closed system of two coupled equations, instead of one, which can be solved diff --git a/outputs/arxiv_math/2503.07801_pg15_pg1_repeat1.md b/outputs/arxiv_math/2503.07801_pg15_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..04e6ce2bc45f89093711927cbc5f3e7c1d66862a --- /dev/null +++ b/outputs/arxiv_math/2503.07801_pg15_pg1_repeat1.md @@ -0,0 +1,45 @@ +By (13), we have + +\[\begin{aligned} g_{i}&\geq\prod_{p\text{ine}\text{rt in}K}2>\exp\left((\log2)\exp\left(C \frac{\log x_{i}}{\log\log x_{i}} \right)\right)>i \\ (p+1)|M_{x_{i}} \end{aligned}\] + +for sufficiently large \(i\). Moreover, we have \(L(g_{i})|M_{x_{i}}\), which implies that + +\[L(gi)≤Mxi≤x2i=(logi)(4/C)logloglogi<(loggi)c0loglogloggi\] + +for sufficiently large \(i,\text{where}c_{0}:=4/C\). The asserted upper bound follows by extracting a strictly monotonic subsequence \(\{f_{i}\}_{i\geq1}\) from \(\{g_{i}\}_{i\geq1}\). + +## 5. Minimal order in the real case: Proof of Theorem 1.4 + +Let \(K\) be a real quadratic field. We remind the reader that \(\epsilon\) denotes the fundamental unit of \(K\), and we set + +\[\delta= \begin{cases} 1 & \text{if}N_{K/\mathbb{Q}}(\epsilon)=1, \\ 2 & \text{if}N_{K/\mathbb{Q}}(\epsilon)=-1. \end{cases}\] + +If \(p\) is a prime inert in \(K\), then its associated Frobenius element is conjugation on \(K\) (the nontrivial element of \(\text{Gal}(K/\mathbb{Q}))\). Hence, + +\[\epsilon^{p+1}\equiv\epsilon^{p}\epsilon\equiv N_{K/\mathbb{Q}}(\epsilon)\pmod{p\mathcal{O}_{K}},\] + +and + +\[\epsilon^{\delta(p+1)}\equiv N_{K/\mathbb{Q}}(\epsilon)^{\delta}\equiv1\pmod{p\mathcal{O}_{K}}.\] + +Thus, the order of \(\epsilon\text{in}(\mathcal{O}_{K}/p\mathcal{O}_{K})^{\times}\) is a divisor of \(\delta(p+1)\). We will base our proof of Theorem 1.4 on the following result of Roskam [Ros00]. + +Proposition 5.1 (conditional on GRH). There are infinitely many primes \(p\), inert in \(K\), for + +which the order of \(\epsilon in(\mathcal{O}_{K}/p\mathcal{O}_{K})^{\times}\) is precisely \(\delta(p+1)\). (In fact, Roskam shows that the order is \(\delta(p+1)\) not only for infinitely many inert primes, but for a positive proportion of all inert primes. The weaker version here is sufficient for our purposes.) Theorem 1.4 is an immediate consequence of Proposition 5.1 in conjunction with the next assertion. + +Proposition 5.2. If \(p\) is a prime inert in \(K\) for which \(\epsilon\) has order \(\delta(p+1)in(\mathcal{O}_{K}/p\mathcal{O}_{K})^{\times}\), then + +\[\rho(\mathcal{O}_{p})\leq h_{K}+ \frac{3}{2} .\] + +Proof. Let \(p\) be as in the proposition. Then \(\epsilon^{\ell(p)}\equiv n\pmod{p\mathcal{O}_{K}}\) for some rational integer \(n\) prime to \(p\). By Fermat’s little theorem, + +\[\epsilon^{(p-1)\ell(p)}\equiv1\pmod{p\mathcal{O}_{K}}.\] + +We are assuming that \(\epsilon\) has order \(\delta(p+1)\text{in}(\mathcal{O}_{K}/p\mathcal{O}_{K})^{\times}\). Hence, the displayed congruence forces + +\[p+1|\delta(p+1)|(p-1)\ell(p).\] + +Writing \((p-1)\ell(p)=(p+1)\ell(p)-2\ell(p)\), we deduce that \(p+1|2\ell(p)\). In particular, + +\[\ell(p)\geq \frac{p+1}{2} .\] diff --git a/outputs/arxiv_math/2503.07802_pg90_pg1_repeat1.md b/outputs/arxiv_math/2503.07802_pg90_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..211f68eee9ffd1d8b693a3902429f7e81f13913f --- /dev/null +++ b/outputs/arxiv_math/2503.07802_pg90_pg1_repeat1.md @@ -0,0 +1,33 @@ +so that, by the assumption in (5.26b), + +\[\limsup_{n}\bigg|\int_{t_{0}}^{t_{n}}u_{n}^{\prime}(t)dt\bigg|^{2}\leq\limsup_{n}\Gamma(\theta-1)t_{0}^{1-\theta}E^{\theta}(u_{n})=0,\] + +- which contradicts (A.10) and concludes the assertion. + +## Appendix B. Measure-preserving diffeomorphisms + +Let \((M,g)\) be a smooth connected, orientable Riemannian manifold with Riemannian volume measure \(\text{vol}_{g}\). We collect here some auxiliary results about the group Diff\((_{0}^{+}g)\) of all compactly non-identical, orientation-preserving, \(\text{vol}_{g}-\)preserving diffeomorphisms. Firstly, let us recall that Diff\((_{0}^{+}g)\) is the (infinite-dimensional) Lie group corresponding to the Lie algebra of div\(-_{g}\)free vector fields on \(M\) with the Lie derivative as its Lie bracket. Let us further recall some virtually well-known results about the natural action of Diff\((_{0}^{+}g)\text{on}M\). The following may be easily inferred from the arguments in [13, §3]. + +Lemma B.1 (Extension lemma). Let \((M,g)\) be in addition open or boundaryless, and \(K\subset M\) be any contractible compact subset. Then, every smooth vector field on \(K\) has a compactly supported, div\(-_{g}\)free extension to the whole of \(M\). + +As an immediate consequence, we see that the Lie algebra of div\(-_{g}\)free vector fields is infinite-dimensional, thus so is Diff\((_{0}^{+}g)\). + +Proposition B.2 (Transitivity, [13, Thm. A, §3, p. \(98]).\text{Diff}_{0}^{+}(g)\) acts \(k-\)transitively on \(M\text{for every}k\in\mathbb{N}_{1}\). In particular, it acts transitively on \(M\). + +B.1. Actions on measures. In the following, let \(B_{1}\subset\mathbb{R}^{d}\) be the open unit ball equipped with the standard Euclidean metric \(g_{e}\). + +Lemma B.3. Let \(\nu\) be any finite measure on \(B_{1}\) invariant for the \((⟲_{♯})-\)action + +of Diff\((_{0}^{+}g_{e}).\text{Then}\nu\propto\mathscr{L}^{d}\big|_{B_{1}}\). In particular, \(\nu\) has no singular continuous part. + +Proof. Let \(Q_{1},Q_{2}\subset B_{1}\) be arbitrary closed cubes with equal volume, define \(K\) as the closed convex hull of \(Q_{1}\cup Q_{2}\), and let \(w\) be the vector field on \(K\) defining the translation of \(Q_{1}\) to \(Q_{2}\). By Lemma B.1 applied to \(K\text{and}w\), there exists a compactly supported div\(-_{g_{e}}\)free vector field \(w^{\prime}\text{on}M\) extending \(w\text{on}K\). Then, the flow of \(w^{\prime}\) at time 1 is a compactly non-identical orientation-preserving, \(\mathscr{L}^{d}-\) preserving diffeomorphism on \(M\) mapping \(Q_{1}\) to \(Q_{2}\). In other words, Diff\((_{0}^{+}g_{e})\) acts transitively on all closed cubes in \(B_{1}\), hence, by continuity, on the family \(\mathcal{C}\) of all + +semi-closed cubes \(Q\) of the form \(Q:=x+q[0,1)^{d}\text{with}x\in\mathbb{R}^{d},q\in\mathbb{Q}^{+}\) and such that \(\overline{Q} \subset B_{1}\). Since \(\nu\) is invariant, by a standard decomposition argument, for every rational \(q\in \mathbb{Q}^{+}\text{and every}Q\in\mathcal{C}\text{with}qQ\in\mathcal{C}\) we have \(\nu(qQ)=q^{d}\nu Q\). Now, let \(Q_{0}\in\mathcal{C}\) be that \(Q_{1}=x+qQ_{0}.\text{For such}q\), by invariance of \(\nu\) and of \(\mathscr{L}^{d}\) (also under rescaling), + +fixed, and set \(c:=\nu Q_{0}/\mathscr{L}^{d}Q_{0}.\text{For every}Q_{1}\in\mathcal{C}\) there exists \(q\in\mathbb{Q}^{+}\text{and}x\in\mathbb{R}^{d}\) so + +\[\nu Q_{1}=\nu(qQ_{0})=q^{d}\nu Q_{0}=q^{d}c\mathscr{L}^{d}Q_{0}=c\mathscr{L}^{d}(qQ_{0})=c\mathscr{L}^{d}Q_{1}.\] + +Since \(\mathcal{C}\) is a \(\pi-\)system generating the Borel \(\sigma-\)algebra of \(B_{1}\), it follows by a standard □ + +monotone class argument that \(\nu=c\mathscr{L}^{d}\) as Borel measures on \(B_{1}\), which concludes the proof. diff --git a/outputs/arxiv_math/2503.07803_pg5_pg1_repeat1.md b/outputs/arxiv_math/2503.07803_pg5_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..381dda20b7aca04856e6860bfec4b3409c4c1307 --- /dev/null +++ b/outputs/arxiv_math/2503.07803_pg5_pg1_repeat1.md @@ -0,0 +1,15 @@ +value of each sampled Feynman graph to about \(10^{-3}\) relative accuracy. The resulting number provides one data point for an evaluated period Feynman integral. We repeat these steps a large number of times. For example, Figure 1 summarizes all 44027 data points we obtained at 17 loops by running Algorithm 1 and the tropical sampling algorithm the same number of times. + +## 3 Results + +### 3.1 Histograms of Feynman integrals at large loop order + +We used the methods described in the last section to generate representative samples of primitive divergent \(\phi^{4}-\)theory graphs and evaluate their period Feynman integrals at loop orders 8 to 17. The number of graphs we sampled at each loop order is listed in Table 1. We ran the computation in bunches at low priority on the ETH Euler computing cluster. Due to a maintenance event, our computation was interrupted, and some data points were lost. Hence, the number of samples differs slightly at each loop order, but there is no correlation between the probability of a data point being lost and its value. At 17 loops, we took fewer samples because we only had limited access to the required large-memory nodes. Figure 1 and Figure 2 depict our results as histograms. All our obtained data points of randomly sampled Feynman graphs are also available as machine-readable tables in the ancillary material to the arXiv version of this article. We evaluated each Feynman graph to \(10^{-3}\) relative accuracy using the tropical sampling approach. As this uncertainty is small compared to the statistical uncertainty that stems from the variance of the different Feynman graphs, we can neglect this uncertainty. We confirmed this explicitly by performing our analysis with the uncertainty included and obtaining identical results. Our data suggests that the distribution of period Feynman integrals is modelled well by the distribution (2) for \(L\rightarrow\infty\). At each loop order, we fitted the parameters \(\alpha,\lambda,\text{and}P_{0}\) by maximizing the logarithmic likelihood function + +\[\log L=N(\alpha\log\lambda-\log\Gamma(\alpha))+(\alpha-1)\sum_{i}\log\log \frac{P_{i}}{P_{0}} -\lambda\sum_{i}\log \frac{P_{i}}{P_{0}} ,\] + +where we sum over all period samples \(P_{1},\ldots,P_{N}\) at a specific loop order. As the number of samples \(N\) is large, we can estimate the uncertainties of these parameters by approximating the prior distribution using a Gaussian. The resulting parameters with uncertainties are listed in Table 1. The uncertainties of the fit parameters were extremely large for \(L\in\{8,9\}\). So, we discarded these fits. The fitted probability distributions are depicted as red lines in Figure 1 and Figure 2. We checked Conjecture 1 quantitatively using Pearson’s \(\chi^{2}\) test: Let \(O_{i}\) be the number of evaluated period Feynman integrals that fall into the \(i-\)th percentile of the distribution (2) with the fitted maximum likelihood parameters at the respective loop order. The expectation value of the random variable \(O_{i}\) is obviously \(N/100\). So, under the hypothesis that our data follows (2), the quantity \(\chi^{2}= \frac{100}{N} \sum_{i}(O_{i}- \frac{N}{100} )^{2}\) is expected to follow a \(\chi^{2}-\)distribution with mean \(100-3=97\), as three parameters are fitted. Table 1 shows that the ratio \(\chi^{2}/97\) approaches 1 with increasing loop order, consistent with Conjecture 1. Figure 2 illustrates how the distribution is approached with increasing loop number, providing further evidence for Conjecture 1. + +### 3.2 Extrapolating \(\beta_{L}^{\text{prim}}\) to all loop orders using instanton input + +The results of [30] suggest that the moments \(\langle P(G)^{k}\rangle_{L}\) diverge for \(L\rightarrow\infty\text{if}k\geq2\) (see Table 8 and the discussion before eq. (4.8) of [30]). Hence, for sufficiently large loop order, the central diff --git a/outputs/arxiv_math/2503.07814_pg1_pg1_repeat1.md b/outputs/arxiv_math/2503.07814_pg1_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..d286bc573dc78568bed87f45410c0be00c4a4ef4 --- /dev/null +++ b/outputs/arxiv_math/2503.07814_pg1_pg1_repeat1.md @@ -0,0 +1,23 @@ +# Whiteness-based bilevel estimation of weighted TV parameter maps for image denoising + +Monica Pragliola[^1], Luca Calatroni[^2], and Alessandro Lanza3 + +[^1]: Department of Mathematics and Applications, University of Naples Federico II, Naples, Italy monica.pragliola@unina.it + +[^2]: MaLGa Center, DIBRIS, Universit`a di Genova, Istituto Italiano di Tecnologia, Genoa, Italy luca.calatroni@unige.it + +## 3 Department of Mathematics, University of Bologna, Italy + +alessandro.lanza2@unibo.it + +Abstract. We consider a bilevel optimisation strategy based on normalised residual whiteness loss for estimating the weighted total variation parameter maps for denoising images corrupted by additive white Gaussian noise. Compared to supervised and semi-supervised approaches relying on prior knowledge of (approximate) reference data and/or information on the noise magnitude, the proposal is fully unsupervised. To avoid noise overfitting an early stopping strategy is used, relying on simple statistics of optimal performances on a set of natural images. Numerical results comparing the supervised/unsupervised procedures for scalar/pixel-dependent parameter maps are shown. + +Keywords: Adaptive regularisation · Total variation image denoising · Unsupervised bilevel optimisation · Residual whiteness. + +## 1 Introduction + +Variational methods are a popular paradigm to overcome the difficulties related to inverse imaging problems: they seek an estimate of the unknown image by minimizing a functional encoding information on the noise and prior beliefs on the image. For image denoising, denoting by \(\mathbf{y}\in\mathbb{R}^{n}\) the (vectorised) data corrupted by Additive White Gaussian Noise (AWGN), variational methods solve + +\[\mathbf{x}^{*}(\boldsymbol{\lambda})\in\operatorname*{arg\,min}_{\mathbf{x}\in\mathbb{R}^{n}}\left\{ \frac{1}{2} \Vert\mathbf{x}-\mathbf{y}\Vert_{2}^{2}+R(\mathbf{x};\boldsymbol{\lambda})\right\},\] + +where \(\mathbf{x}^{*}\) depends on a certain number of parameters \(\boldsymbol{\lambda}\in\Lambda\) possibly involved in the so-called regularisation term \(R\). In general, flexible models are characterised by a large number of parameters for which robust selection strategies need to be introduced. How to properly select parameters in variational models is an issue that has attracted the interest of many researchers working on inverse problems. diff --git a/outputs/arxiv_math/2503.07843_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.07843_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..cec6370ddad61908228d71e062f6501c9b67bb86 --- /dev/null +++ b/outputs/arxiv_math/2503.07843_pg3_pg1_repeat1.md @@ -0,0 +1,29 @@ +particular with semisimple isometry group \(G\). The holonomy group at a point \(p\) coincides with the isotropy \(K\) of the factor \(Y\) (see [2, Proposition 10.79]). On the other hand, the fixed space of the \(K-\)action at \(p\) is exactly \(X^{L}\). Therefore, the stabilizer of \(p\) preserves the factor \(X^{L}\). The rest follows similarly as above. + +- + +## 3. Injectivity of the developing map + +In this section we assume that \(X=X^{L}\times Y\) is a simply connected Lorentz symmetric space whose Lorentz factor \(X^{L}=X_{\kappa}\) is the universal model of constant curvature \(\kappa\). A compact manifold \(M\) locally isometric to \(X\) is an \((\text{Isom}(X),X)-\)manifold in the sense of \((G,X)-\)structures (see [15, 38]). That is, we have a local diffeomorphism \(D:\tilde{M}\rightarrow X\) and a representation \(\rho:\pi_{1}(M)\rightarrow\text{Isom}(X)\) satisfying the equivariance property \(D\gamma=\rho(\gamma)D\text{for all}\gamma\in\pi_{1}(M)\). The local diffeomorphism \(D\) is called the developing map and \(\rho\) is called the holonomy representation. + +3. 1. Natural foliations. Proposition 2.5 implies that a compact manifold \(M\) locally isometric to \(X=X^{L}\times Y\) inherits two transverse foliations \(\mathcal{F}_{1}\text{and}\mathcal{F}_{2}\) where the leaves of \(\mathcal{F}_{1}\) are Lorentz of constant curvature locally modeled on \(X_{\kappa}\) and the leaves of \(\mathcal{F}_{2}\) are Riemannian modeled on \(Y\) . The same applies to \(\tilde{M}\), we have two foliations \(\tilde{\mathcal{F}}_{1}\text{and}\tilde{\mathcal{F}}_{2}\). + +Lemma 3.1. Let \(M\) be a compact manifold locally isometric to \(X\). Each leaf of \(\tilde{\mathcal{F}}_{2}\) is mapped, under the developing map, bijectively and isometrically onto a vertical Riemannian fiber \(\{x_{0}\}\times Y\) . + +Proof. The compactness of the manifold \(M\) implies that each leaf of the foliation \(\mathcal{F}_{2}\) is complete (with respect to its Riemannian structure locally modeled on \(Y)\). Hence each leaf \(\tilde{\mathcal{F}}_{2}(p)\) of the foliation \(\tilde{\mathcal{F}}_{2}\) is also complete. The developing map \(D\), restricted to \(\tilde{\mathcal{F}}_{2}(p)\), is a local isometry between \(\tilde{\mathcal{F}}_{2}(p)\) and a corresponding fiber \(\{x_{0}\}\times Y\) . Since \(\tilde{\mathcal{F}}_{2}(p)\) is complete, then \(D:\tilde{\mathcal{F}}_{2}(p)\rightarrow \{x_{0}\}\times Y\) is a covering map. But \(Y\) being simply connected implies that \(D:\tilde{\mathcal{F}}_{2}(p)\rightarrow\{x_{0}\}\times Y\) is in fact an isometric diffeomorphism. + +- + +Corollary 3.2. The universal cover \(\tilde{M}\) is mapped, under the the developing map, onto \(\Omega\times Y\) where \(\Omega\subset X_{\kappa}\) is an open subset. + +3. 2. A natural action on \(\tilde{M}\). Since each leaf of \(\tilde{\mathcal{F}}_{2}\) is identified canonically under the developing map to \(Y=G/K\), we obtain a well-defined \(G-\text{action on}\tilde{M}\) for which the developing map is \(G-\text{equivariant}\). Since the developing map is isometric, we have + +Corollary 3.3. The \(G-\text{action on}\tilde{M}\) is isometric, proper, and with orbits the \(\tilde{\mathcal{F}}_{2}-\text{leaves}\). + +3. 3. Product structure of the universal cover. We have seen in Corollary 3.2 that \(D(\tilde{M})= \Omega\times Y\text{where}\Omega\subset X_{\kappa}\) is an open subset. The subset \(\widehat{\Omega}=D^{-1}(\Omega\times\{y\})\) is a global cross section of the \(\tilde{\mathcal{F}}_{2}-\text{foliation}\). Indeed, consider the map \(\sigma:\tilde{M}\rightarrow Y\text{where}\sigma=\pi\circ D\text{and}\pi:X_{\kappa}\times Y\rightarrow Y\) is the natural projection. We have \(\sigma^{-1}(y)=\widehat{\Omega}\) and, by construction, \(\sigma\) is \(G-\text{equivariant}\). Since each leaf \(\tilde{\mathcal{F}}_{2}(p)\) is identified under the map \(\sigma\text{with}Y\) , then \(\tilde{\mathcal{F}}_{2}(p)\) intersects \(\sigma^{-1}(y)=\widehat{\Omega}\) exactly in one point. In other words, \(\widehat{\Omega}\) is a leaf of \(\tilde{\mathcal{F}}_{1}\) which is a global cross section of the foliation \(\tilde{\mathcal{F}}_{2}\). + +Corollary 3.4. The universal cover \(\tilde{M}\) is globally isometric to \(\widehat{\Omega}\times Y\) . + +In particular, \(\widehat{\Omega}\) is a connected and simply connected Lorentz manifold of constant curvature \(\kappa\). The injectivity of the developing map \(D:\tilde{M}\rightarrow X_{\kappa}\times Y\) is then equivalent to the fact that \(\widehat{\Omega}\) isometrically embeds in \(X_{\kappa}\). + +3. 4. The injectivity. We have seen in Corollary 3.4 that \(\tilde{M}=\widehat{\Omega}\times Y\text{and}\pi_{1}(M)=\Gamma\subset \text{Isom}(\widehat{\Omega})\times G\) acts freely, properly, and cocompactly on \(\tilde{M}\text{where}G\) is the isometry group of \(Y\) . Frances in [10, Sect. 6] considers, among many other things, a similar situation (in fact a more general setting of warped products). He observed [10, Proposition 6.9] that the injectivity result in the works of Carri`ere and Klingler [7, 24] can be adapted to the product setting. That is, diff --git a/outputs/arxiv_math/2503.07868_pg30_pg1_repeat1.md b/outputs/arxiv_math/2503.07868_pg30_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..6bcf3f37e72db86ce6709481ff685307dd7237c6 --- /dev/null +++ b/outputs/arxiv_math/2503.07868_pg30_pg1_repeat1.md @@ -0,0 +1,31 @@ +[13] M. Ledoux, Spectral gap, logarithmic Sobolev constant, and geometric bounds, Surv. Differ. Geom., 9, Int. Press, Somerville, MA, 2004. 5 + +[14] M. Ledoux, A simple analytic proof of an inequality by P. Buser, Proc. Amer. Math. Soc., 121 (1994), 951–959. 5 + +[15] V. Maz’ya, Sobolev spaces with applications to elliptic partial differential equations. Second, revised and augmented edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 342. Springer, Heidelberg, 2011. 2, 3, 4, 5, 6, 7, 11, 12, 15, 26 + +[16] V. Maz’ya, Sobolev spaces. Translated from the Russian by T. O. Shaposhnikova. Springer Ser. Soviet Math. Springer-Verlag, Berlin, 1985. 3, 25 + +[17] V. Maz’ya, M. Shubin, Can one see the fundamental frequency of a drum?, Lett. Math. Phys., 74 (2005), 135–151. 2, 3, 4, 5, 11, 14 + +[18] E. Parini, Reverse Cheeger inequality for planar convex sets, J. Convex Anal., 24 (2017), 107–122. 5 + +[19] G. Poliquin, Principal frequency of the \(p-\)Laplacian and the inradius of Euclidean domains, J. Topol. Anal., 7 (2015), 505–511. 25 + +[20] P. Souplet, Geometry of unbounded domains, Poincaré inequalities and stability in semilinear parabolic equations, Comm. Partial Differential Equations, 24 (1999), 951–973. 2, 16, 25 + +[21] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4), 110 (1976), 353–372. 13 + +[22] A. Vitolo, \(H^{1,p}-\text{eigenvalues and}L^{\infty}-\)estimates in quasicylindrical domains, Commun. Pure Appl. Anal., 10 (2011), 1315–1329. 25 + +(F. Bozzola) DIME Dipartimento di ingegneria meccanica, energetica, gestionale e dei trasporti Universita di Genova + +via alla Opera ` Pia 15, 16145 Genova, Italy + +Email address: francesco.bozzola@edu.unige.it + +(L. Brasco) Dipartimento di Matematica e Informatica Universita degli Studi di Ferrara + +Via Machiavelli ` 35, 44121 Ferrara, Italy + +Email address: lorenzo.brasco@unife.it diff --git a/outputs/arxiv_math/2503.07875_pg21_pg1_repeat1.md b/outputs/arxiv_math/2503.07875_pg21_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..a9fa703e86ffa6b487d7c5d987090f329686d7b5 --- /dev/null +++ b/outputs/arxiv_math/2503.07875_pg21_pg1_repeat1.md @@ -0,0 +1,92 @@ +1.3 + +1.4 + +1.2 + +1.2 + +1.1 + +1 + +1 + +0.9 + +0.8 + +0.8 + +0 + +0.5 + +1 + +1.5 + +0 + +0.5 + +1 + +1.5 + +2 + +2.5 + +3 + +3.5 + +## (a) + +## (b) + +1.2 + +1.4 + +1.1 + +1.2 + +1 + +0.9 + +1 + +0.8 + +0.8 + +0.7 + +| 0.6 | | | | | | 0.6 | | | | | +| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | +| 0 | 0.5 | 1 | 1.5 | 2 | 2.5 | 1 | 1.5 | | 2 | 2.5 | +| | | (c) | | | | | | (d) | | | + +Fig. 5 Single parameter numerical continuation of susceptible human population with respect to parameters \(\alpha_{1}\) (a), \(b_{1h}\) (b), \(b_{1v}\) (c) and \(c_{1v}\) (d) with \(\tau_{h}=\tau_{v}=\tau=0\text{and}\tau_{r}=2.4\). The colored lines represent the number of eigenvalues with positive real parts. The dashed lines stand for the branches emanating from Hopf bifurcation and represent the maximum of the periodic orbits. Purple, cyan and green squares respectively represent Hopf, Period doubling and transcritical bifurcations. + +### 5.4 Disease free equilibrium (DFE) + +Disease free equilibrium is found as \(\Sigma_{2}^{*}=\left(1,0,0,0, \frac{1}{c_{1v}} ,0,0\right)\). Using the linearization argument the explicit form of the matrix given in Eq. (25) is given + +\[J_{\tau}= \begin{pmatrix} -1 & 0 & 0 & \omega & 0 & 0 & -f\mathrm{e}^{-\lambda\tau_{h}} \\ 0-\sigma_{h} & 0 & 0 & 0 & 0 & \alpha_{1}f\mathrm{e}^{-\lambda\tau_{h}} \\ 0 & 1 & -\xi_{h}\mathrm{e}^{-\lambda\tau_{r}} & 0 & 0 & 0 & 0 \\ 0 & 0 & \mathrm{e}^{-\lambda\tau_{r}} & -\eta_{h} & 0 & 0 & 0 \\ 0 & 0 & -h\mathrm{e}^{-\lambda\tau_{v}} & 0 & -c_{1v} & 0 & 0 \\ 0 & 0 & \alpha_{2}h\mathrm{e}^{-\lambda\tau_{v}} & 0 & 0 & -c_{2v} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & -c_{1v} \end{pmatrix} .\] + +| as follows | | | | | | | +| --- | --- | --- | --- | --- | --- | --- | +| −1 | 0 | 0 | ω | 0 | 0 | −fe−λτh | +| 0 | −σh | 0 | 0 | 0 | 0 | α1fe−λτh | +| 0 | 1 | −ξhe−λτr | 0 | 0 | 0 | 0 | + +| 0 0 | α2he−λτv | 0 0 −c2v 0 | +| --- | --- | --- | +| 0 0 | 0 | 0 0 1 −c1v | + +The calculations performed in Sec. 5.1 are valid here with the exception of Eq. (31). Since only susceptible human and susceptible vector population exist for diff --git a/outputs/arxiv_math/2503.07897_pg25_pg1_repeat1.md b/outputs/arxiv_math/2503.07897_pg25_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..25659f2587a4cceb704ef9f1ef40e1c8a6e3c711 --- /dev/null +++ b/outputs/arxiv_math/2503.07897_pg25_pg1_repeat1.md @@ -0,0 +1,23 @@ +Let the function \(\mu_{r}(t)\) be: + +\[\mu_{r}(t)=\int_{0}^{t}\lambda_{r}(v)dv=(1+r \frac{\gamma}{\rho} )\ln(1+\rho t).\] (A.3) + +It can be seen from Eq. (A.3) that \(\mu_{r}^{\prime}(t)=\lambda_{r}(t)\text{and}\mu_{r}(0)=0\forall r\geq0\). We now define the auxiliary function + +\[\mu_{\Delta}(t)=\mu_{r+1}(t)-\mu_{r}(t)= \frac{\gamma}{\rho} \ln(1+\rho t),\] (A.4) + +which does not depend on \(r\). Note that \(\mu_{\Delta}(0)=0\). It is a fact that the following equality holds: + +\[\int_{0}^{t}\lambda_{r}(x)e^{\mu_{\Delta}(x)}(e^{\mu_{\Delta}(x)}-1)^{r}dx= \frac{( \tfrac{\rho}{\gamma} +r)}{r+1} (e^{\mu_{\Delta}(t)}-1)^{r+1}.\] (A.5) + +The proof can be done by differentiating the right side of Eq. (A.5) to show it’s indeed a primitive of the left side’s integrand. Since the functions are also equal on \(t=0\), fundamental theorem of calculus yields that the equalty is valid for every \(t\). Considering the initial condition \(P_{0}(0)=1\) (the beginning population is 0), the probability mass functions are given by: + +\[P_{r}(t)= \frac{\Gamma( \tfrac{\rho}{\gamma} +r)}{r!\Gamma( \tfrac{\rho}{\gamma} )} e^{-(\mu_{r}(t)-\mu_{0}(0))}(e^{\mu_{\Delta}(t)}-1)^{r}.\] (A.6) + +This is demonstrated by induction over \(r\), using Eq. (A.6) as inductive hypothesis and \(P_{0}(t)\) from Eq. (A.1). In fact, + +\[\begin{aligned} P_{r+1}(t)&=\int_{0_{\rho}}^{t}e^{-(\mu_{r+1}(t)-\mu_{r+1}(x))}\lambda_{r}(x)P_{r}(x)dx \\ &= \frac{\Gamma(_{\overline{\gamma}}+r)}{r!\Gamma( \tfrac{\rho}{\gamma} )} e^{-(\mu_{r+1}(t)-\mu_{0}(0))}\int_{\rho0}^{t}\lambda_{r}(x)e^{\mu_{\Delta}(x)}(e^{\mu_{\Delta}(x)}-1)^{r}(x)dx \\ &= \frac{\Gamma( \tfrac{\rho}{\gamma} +r)}{r!\Gamma( \tfrac{\rho}{\gamma} )} e^{-(\mu_{r+1}(t)-\mu_{0}(0))} \frac{(_{\overline{\gamma}}+r)}{r+1} (e^{\mu_{\Delta}(t)}-1)^{r+1}. \end{aligned}\] + +We can rewrite Eq. (A.6) as: + +\[P_{r}(t)= \frac{\Gamma( \tfrac{\rho}{\gamma} +r)}{r!\Gamma( \tfrac{\rho}{\gamma} )} e^{-(\mu_{r}(t)-\mu_{0}(0)-r\mu_{\Delta}(t))}(1-e^{-\mu_{\Delta}(t)})^{r}.\] (A.7) diff --git a/outputs/arxiv_math/2503.07910_pg7_pg1_repeat1.md b/outputs/arxiv_math/2503.07910_pg7_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..5909a7d41b556c28ed6e7a861a2cbe538ec42e3a --- /dev/null +++ b/outputs/arxiv_math/2503.07910_pg7_pg1_repeat1.md @@ -0,0 +1,37 @@ +where \(F_{h}(x)\) enumerates independent sets fixed by non-identity elements \(h\in H_{p}\). This proves part (ii)(c) of Theorem 3.1. + +Proof. Each term \(b_{j}\) in the polynomial modulo \(p\) counts exactly those independent sets with nontrivial stabilizers. By definition, these are precisely sets that are fixed by some non-identity element \(h\in H_{p}\). Thus, summing over non-identity elements gives the generating function for fixed independent sets, proving the equivalence stated in (ii)(c) of Theorem 3.1. + +The orbit structure established in Lemma 3.4 implies the modular collapse behavior described in Theorem 3.1. + +### 3.3 Algebraic Completion via Frobenius + +Building upon the results established in Lemmas 3.3 and 3.1, we now give an algebraic characterization involving polynomial congruences modulo prime divisors of \(n\). This analysis directly ties into the orbit-stabilizer theorem and the modular collapse phenomenon, demonstrating how independent sets are constrained under the action of \(H_{p}\). For prime \(n\), we have established that \(I(C_{n}^{⊠d},x)\equiv1\pmod{n}\). For composite \(n\) with prime divisor \(p\), the subgroup \(H_{p}\) partitions independent sets into orbits of size exactly \(p\), ensuring that the number of such sets satisfies congruences modulo \(p\). This leads to specific conditions under which \(I(C_{n}^{⊠d},x)\) exhibits structured congruences. + +## Frobenius Automorphism and Its Role + +The Frobenius automorphism, denoted Frob\(p\), is a key tool in understanding the structure of field extensions in characteristic \(p\). It is defined as the mapping: + +\[\text{Frob}_{p}:a\mapsto a^{p}\] + +for any element \(a\) in a field of characteristic \(p\). This automorphism preserves the algebraic structure of the field while raising each element to the power of \(p\), which naturally extends to polynomials and their roots. Applying this iteratively, we obtain: + +\[(x+1)^{p^{d}}\equiv x^{p^{d}}+1\pmod{p}.\] + +## Polynomial Congruences and Frobenius Completion + +\((x+1)^{m}\pmod{p}\) holds for some \(m\leq p^{d}\): + +Let us examine when the congruence \(I(C_{n}^{⊠d},x)\equiv\) + +Proposition 3.5. Let \(n\) be composite with prime divisor \(p.\text{Then}I(C_{n}^{⊠d},x)\equiv c\cdot(x+1)^{m}\pmod{p}\) for some constant \(c\in\mathbb{F}_{p}^{\times}\text{and}m\leq p^{d}\) if and only if the only \(H_{p}-\)invariant independent sets are those consisting entirely of vertices fixed point wise by the action of \(H_{p}\). + +Proof. Let \(\mathcal{F}_{p}\) be the set of all independent sets that are fixed by at least one non-identity element of \(H_{p}\). The generating function for these sets is: + +\[F(x)=\sum_{S\in\mathcal{F}_{p}}x^{|S|}\] + +By the orbit-stabilizer theorem, each \(H_{p}-\)invariant independent set belongs to an orbit whose size divides \(p\), leading to constraints on \(I(C_{n}^{⊠d},x).\text{When}\mathcal{F}_{p}\) contains only independent sets consisting of vertices fixed point wise by \(H_{p},F(x)\) takes the form \((x+1)^{m}-1\text{for some}m\leq p^{d}\), representing all possible subsets of the fixed vertex set minus the empty set. Under these conditions, we have: + +\[I(C_{n}^{⊠d},x)\equiv1+F(x)\equiv(x+1)^{m}\pmod{p}\] + +Conversely, if \(I(C_{n}^{⊠d},x)\equiv c\cdot(x+1)^{m}\pmod{p},\text{then}\mathcal{F}_{p}\) must correspond to the structure \((x+1)^{m}-1\), implying that the only invariant sets are those built from fixed vertices. Applying the Frobenius automorphism, we recall: diff --git a/outputs/arxiv_math/2503.07916_pg19_pg1_repeat1.md b/outputs/arxiv_math/2503.07916_pg19_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..e5a05bb08b85072e8db7785067302081e567eaa4 --- /dev/null +++ b/outputs/arxiv_math/2503.07916_pg19_pg1_repeat1.md @@ -0,0 +1,13 @@ +are met. An explanation of the stopping criterion (8.4) is provided below. We introduce the random noise in the observation data in (2.13) as follows: + +\[g_{g_{1}^{0\xi_{\xi_{1}^{0}}}}(_{(}\mathbf{x}_{\mathbf{x}},_{,}\mathbf{x}_{\mathbf{x}_{0}^{0}})_{)}=_{=}g_{g_{1}^{0}}(_{(}\mathbf{x}_{\mathbf{x}},_{,}\mathbf{x}_{\mathbf{x}_{0}^{0}})_{)}(_{(}1_{1}+_{+}\delta_{\delta}\xi_{\xi_{1}^{0}}(_{(}\mathbf{x}_{\mathbf{x}})_{)})_{)},_{,}\mathbf{x}_{\mathbf{x}}\in_{\in}\partial_{\Gamma}\Omega_{,},\] (8.5) + +where \(\xi_{0}\) is the uniformly distributed random variable in the interval \([-1,1]\) depending on the point \(\mathbf{x}\in\partial\Omega\). Also, \(\xi_{1}\) is the uniformly distributed random variable in the interval \([-1,1]\) depending on the point \(\mathbf{x}\in\Gamma,\text{and}\delta=0.03\) corresponds to the 3% noise level. Since we deal with the first \(\varphi-\)derivatives of the noisy functions \(g_{0}^{\xi_{0}}(\mathbf{x},\mathbf{x}_{0})\) and \(g_{1}^{\xi_{1}}(\mathbf{x},\mathbf{x}_{0})\), we have to design a numerical method to differentiate the noisy data. First, we use the natural cubic splines to approximate the noisy input data (8.5). Next, we use the derivatives of those splines to approximate the derivatives of corresponding noisy observation data. We generate the corresponding cubic splines in \((0,2\pi)\) with the mesh grid size \(h_{\varphi}=\pi/100\), and then we calculate their derivatives to approximate the first derivatives with respect to \(\varphi\). We choose the optimal pair of parameters \((\alpha,\varepsilon)\) by the trial and error procedure for the reference Test 1. For each considered pair \((\alpha,\varepsilon),\) we test different values of the parameter \(\lambda\) to obtain its optimal value \(\lambda_{\text{opt}}(\alpha,\varepsilon)\) for this pair. Once the so chosen triple \((\alpha,\varepsilon,\lambda)\) of parameters is selected, we consider it as the optimal choice of parameters. An important point to make here is that exactly the same triple of optimal parameters is used for all follow up tests when imaging letters below. However, when using the CT scan of the abdomen below, we deal with a different medium. This means that we repeat the procedure of our choice of parameters again for this case. + +## Remarks 8.1: + + (i) As the test media, we intentionally choose letter-like shapes of inclusions in the first series of numerical experiments and the CT scans of the abdomen in the second series. This is done to demonstrate that our technique works well for complicated media. + + (ii) The above procedure of the choice of an optimal triple \((\alpha,\varepsilon,\lambda)\) of parameters is similar to the conventional calibration procedure, which is often used in many real World applications. Furthermore, quite similar procedures were used in all above cited works [18, 19, 20, 22, 23, 24] on the numerical studies of the convexification method for CIPs. + +(iii) Even though theorems of our convergence analysis are valid only for sufficiently large values of the parameter \(\lambda,\) we have discovered in all our works on the convexification listed in item 2 that actually optimal values of \(\lambda\) belong to the interval \(\lambda\in[1,5].\) In fact, this is similar with many asymptotic theories. Indeed, it is typically established in such a theory that if a certain parameter \(X\) is sufficiently large, then a certain formula \(Y\) provides a good approximation for a process. However, it is also typical that in a computational practice only numerical experiments can tell one which exactly values of \(X\) are appropriate ones. diff --git a/outputs/arxiv_math/2503.07924_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.07924_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..21e35df0df726ddec8a7f14c762f6d2294d823fc --- /dev/null +++ b/outputs/arxiv_math/2503.07924_pg3_pg1_repeat1.md @@ -0,0 +1,33 @@ +The Quadratic Unconstrained Binary Optimization (QUBO) model provides a framework for solving combinatorial optimization problems, where the objective is to determine the optimal configuration of binary variables that minimizes a quadratic objective function. Formulated as an unconstrained optimization problem, it encodes the constraints within the objective function. The model is defined over binary variables \(x_{i}\in\{0,1\}\), and uses the quadratic function \(H(x)\) to represent pairwise interactions: + +\[H(x)=\sum_{i\neq j}Q_{ij}x_{i}x_{j}+\sum_{i}q_{i}x_{i}\] (12) + +where \(Q_{ij}\) denotes the interaction coefficients between \(x_{i}\) and \(x_{j},\text{and}q_{i}\) represents the linear coefficients for \(x_{i}\). The objective function \(H(x)\) can be interpreted as a Hamiltonian, analogous to the energy function in Ising model. To incorporate constraints from (3), we add penalty terms: + +\[\begin{aligned} H(x)&=f(x)+P_{1}\left(\sum_{i}x_{S,i}-1\right)^{2}+P_{2}\left(\sum_{i}x_{i,D}-1\right)^{2} \\ +P_{3}\sum_{i\not\in \{S,D\}}\left(\sum_{j}x_{i,j}-\sum_{k}x_{k,i}\right)^{2} \end{aligned}\] (13) + +where \(P_{1},P_{2},\text{and}P_{3}\) are positive penalty coefficients that must be set large enough to ensure that any violation of the constraints is penalized heavily. Typically, \(P_{3}\) is chosen to be larger than \(P_{1}\text{and}P_{2}\) to enforce constraints more strictly. Expanding (13) yields the quadratic terms \(Q_{ij}\) and linear terms \(q_{i}\) as follows: + +\[\begin{aligned} Q_{ij}&=P_{1}\sum_{i\neq j}x_{S,i}x_{S,j}+P_{2}\sum_{i\neq j}x_{i,D}x_{j,D} \\ +P_{3}\sum_{i\not\in \{S,D\}}\big(_{(}\sum_{i,j)\in E(i}\sum_{,_{k}k_{\neq })\in_{j}E}x_{i,j}x_{i,k} \\ +\sum_{(j,i)\in E(k}\sum_{k,i_{\neq })\in_{j}E}x_{j,i}x_{k,i}-2\sum_{(i,j)\in E(k}\sum_{,i)\in E}x_{i,j}x_{(^{k,}1^{i}4}\big)_{)} \\ q_{i}&=-P_{1}\sum_{i}x_{S,i}-P_{2}\sum_{i}x_{i,D} \\ +P_{3}\sum_{i\not\in \{S,D\}}\left(_{(}\sum_{i,j)\in E}x_{i,j}+\sum_{(k,i)\in E}x_{k,i}\right)+f(x)^{(15)} \end{aligned}\] + +## C. Ising Model + +The Ising model is a mathematical representation in statistical physics used to describe magnetic materials, where the system’s energy depends on the configuration of spin variables. + +In the Ising model, each spin variable \(\sigma_{i}\in\{-1,1\}\) represents a binary state. The model’s Hamiltonian is given by: + +\[H(\sigma)=-\sum_{i,j}J_{ij}\sigma_{i}\sigma_{j}-\sum_{i}h_{i}\sigma_{i}\] (16) + +where \(J_{ij}\) denotes the coupling strength between spins \(i\) and \(j,\text{and}h_{i}\) represents the influence of an external magnetic field on spin \(i\). To convert the QUBO model to the Ising model, we use the transformation \(\sigma_{i}=2x_{i}-1\), which maps the binary variables \(x_{i}\in\{0,1\}\) to Ising spins \(\sigma_{i}\in\{-1,1\}\). The QUBO expression can be rewritten in terms of Ising variables: + +\[\begin{aligned} H(\sigma)&=\sum_{i\neq j}Q_{ij}\cdot \frac{(\sigma_{i}+1)(\sigma_{j}+1)}{4} +\sum_{i}q_{i}\cdot \frac{\sigma_{i}+1}{2} \\ &=\sum_{i\neq j} \frac{Q_{ij}}{4} \sigma_{i}\sigma_{j}+\sum_{i}\left( \frac{q_{i}}{2} + \frac{\sum_{i\neq j}(Q_{ij}+Q_{ji})}{4} \right)\sigma_{i} \\ +\left( \frac{\sum_{i}q_{i}}{2} + \frac{\sum_{i\neq j}Q_{ij}}{4} \right) \end{aligned}\] (17) + +Thus, the coefficients in the Ising model are: + +\[\begin{aligned} J_{ij}&=- \frac{Q_{ij}}{4} \\ h_{i}&=- \frac{q_{i}}{2} - \frac{\sum_{i\neq j}(Q_{ij}+Q_{ji})}{4} \end{aligned}\] (18) (19) + +We use this mapping to transform the original optimization problem into the Ising model, allowing it to be solved using CIM. + +## III. COHERENT ISING MACHINE + +In this section, we introduce Coherent Ising Machine (CIM) and explain its application to solving our multi-objective routing model. CIM is a quantum-inspired optimization technique that leverages principles from the Ising model. The Ising model represents optimization problems as energy minimization tasks, where the goal is to find the ground state of a system by minimizing its energy. In the context of combinatorial optimization, this ground state corresponds to an optimal or near-optimal solution. CIM utilizes an optical network of degenerate optical parametric oscillators (DOPOs) to represent spins in the Ising model. Each DOPO encodes the spin states using phase-coherent laser pulses, with the two possible phases corresponding to spin values of +1 or −1. The system’s energy is minimized by continuously adjusting the pump power, allowing CIM to evolve the spin states toward lower energy configurations and find solutions to the optimization problem. Theoretically, CIM operates by solving an Ising problem defined by a Hamiltonian that represents the energy of the spin system. The objective is to find a spin configuration that minimizes this Hamiltonian, which corresponds to solving a combinatorial optimization problem. In practical implementations, the dynamic behavior of CIM can be modeled diff --git a/outputs/arxiv_math/2503.07925_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.07925_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..bf668e52e4ef5254677a5078ccb5c501ba0e157f --- /dev/null +++ b/outputs/arxiv_math/2503.07925_pg2_pg1_repeat1.md @@ -0,0 +1,31 @@ +introduction applications, and highlights their relevance to the study of TDI systems and to a long standing conjecture of Paul Seymour on ideal clutters. + +### 1.1 Totally dual in \(\mathbb{L}\) systems + +Consider a prime \(p\). A rational number is \(p-\)adic if it is of the form \(\frac{r}{s} \text{where}r,s\in\mathbb{Z}\text{and}s\) is an integer power of \(p[2]\). A number is dyadic if it is 2-adic. Let \(\mathcal{S}\) be a (possibly infinite) set of primes. We + +denote by \(\mathbb{L}(\mathcal{S})\) the set of all rationals of the form \(\frac{r}{s} \text{where}r,s\in\mathbb{Z}\text{and}s\) is a product of integer powers of primes in \(\mathcal{S}\). Observe that \(\mathbb{L}(\{p\})\) denotes the \(p-\)adic rationals. To keep the notation light, we write \(\mathbb{L}_{p}\text{for}\mathbb{L}(\{p\})\). For any set of primes \(\mathcal{S},\mathbb{L}(\mathcal{S})\) is a heavy set (this follows from [3, Lemma 2.1]). + +For a system \(Mx\leq b\) to be TD in \(\mathbb{L}\) we consider the dual \((D:M,b,w)\) for all choices of \(w\in\mathbb{L}^{n}\). + +However, for the aforementioned heavy sets, it suffices to consider the choices \(w\in\mathbb{Z}^{n}\). Namely, + +Remark 1.1. Let \(\mathbb{L}:=\mathbb{L}(\mathcal{S})\text{where}\mathcal{S}\) is a set of primes. Then \(Mx\leq b\) is TD in \(\mathbb{L}\) if and only if for + +every admissible \(w\in\mathbb{Z}^{n},(D:M,b,w)\) has an optimal solution in \(\mathbb{L}^{m}\). + +Proof. Necessity is clear, let us prove sufficiency. Consider an admissible \(w\in\mathbb{L}^{n}.\text{For some}\mu\) that is + +a product of integer powers of primes in \(\mathcal{S}\) we have \(\mu w\in\mathbb{Z}^{n}\). There exists an optimal solution \(\bar{y}\in\mathbb{L}^{n}\) for \((D:M,b,\mu w)\). However, then \(\frac{1}{\mu} \bar{y}\) is a solution for \((D:M,b,w)\) that is in \(\mathbb{L}^{n}\). + +For any prime \(p\), we can find a system \(Mx\leq b\) that is TD in \(\mathbb{L}_{p}\) but not TD in \(\mathbb{L}_{q}\) for any prime + +\(q\neq p\). Namely, one can pick \(Mx\leq b\) to consist of a unique constraint, \(px\leq1\). However, if we require a system to be TD in \(\mathbb{L}_{p}\) and TD in \(\mathbb{L}_{p^{\prime}}\) for distinct primes \(p\text{and}p^{\prime}\), then it is totally dual in \(\mathbb{L}_{q}\) for every prime \(q[2\), Theorem 1.4]. The following stronger statement (the aforementioned case + +corresponds to \(\mathcal{S}_{1}=\{p\},\mathcal{S}_{2}=\{p^{\prime}\}\text{and}k=2)\) holds in the full dimensional case, + +Theorem 1.2. Let \(Mx\leq b\) be a system where \(\{x:Mx\leq b\}\) is a full-dimensional polyhedron. For + +\(i=1,\ldots,k\), let \(\mathcal{S}_{i}\) be a set of primes and suppose that \(Mx\leq b\) is TD in \(\mathbb{L}(\mathcal{S}_{i}).If\cap_{i\in[k]}\mathcal{S}_{i}=\emptyset\) then \(Mx\leq b\) is TD in \(\mathbb{L}_{q}\) for every prime \(q\). This will be an immediate consequence of Theorem 2.2. Consider an integral matrix \(M\) and an integral vector \(b\). A necessary condition for \(Mx\leq b\) to be + +TDI is that the polyhedron \(Q=\{x\in\mathbb{R}^{n}:Mx\leq b\}\) be integral, i.e., that every minimal proper face of \(Q\) contains an integral vector [10], see also [20, Corollary 22.1a]. However, this is not a sufficient condition [20, Equation (3) in Chapter 22]. Let us define a stronger necessary condition for a system to be TDI. We say that \(Mx\leq b\) is near-TDI if for every prime \(p,Mx\leq b\) is TD in \(\mathbb{L}_{p}\). Since \(\mathbb{Z}\subset\mathbb{L}_{p}\) for every prime \(p\), it then follows from Remark 1.1 that if a system is TDI, it is near-TDI. Furthermore, we have the following result [3, Theorem 1.5], diff --git a/outputs/arxiv_math/2503.07965_pg7_pg1_repeat1.md b/outputs/arxiv_math/2503.07965_pg7_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..782a323137227650df8984818d524d89206d9258 --- /dev/null +++ b/outputs/arxiv_math/2503.07965_pg7_pg1_repeat1.md @@ -0,0 +1,25 @@ +define the symplectic rotation matrix + +\[X(\theta_{i}): \begin{bmatrix} x_{i} \\ p_{i} \end{bmatrix} \mapsto \begin{bmatrix} \cos(\theta_{i}) & \sin(\theta_{i}) \\ -\sin(\theta_{i})\cos(\theta_{i}) \end{bmatrix} \begin{bmatrix} x_{i} \\ p_{i} \end{bmatrix} .\] (13) + +Replacing \(B\text{with}X(\theta_{i})B\) in Eq. (10) leaves the trace invariant so there is a \(SO(2)^{n}\) family of linear maps taking \(f_{0}\) to a minimal energy configuration. This family of matrices is enlarged when the symplectic eigenvalues of either \(V\text{or}H\) are degenerate. It is interesting to compare Eq. (8) and Equation Eq. (12). Notably, the AM-GM inequality shows that \(E_{Sp(2n)}\geq E_{SL(2n)}\) since + +\[2\sum_{i=1}^{n}\lambda_{i}^{H}\lambda_{n+1-i}^{V}\geq2n(\det(HV))^{1/2n},\] (14) + +with equality iff \(\lambda_{i}^{H}\lambda_{n+1-i}^{V}\equiv\) const. In particular, \(E_{Sp(2)}=E_{SL(2)}\) as expected. + +# C. Example 1 + +\(\epsilon^{2}y^{2}+p_{x}^{2}+p_{y}^{2}\). Suppose further that \(f_{0}(\mathbf{z},R)=\) + +\[\tfrac{6}{R^{2}|B(R)|} \chi_{B(R)}=\] + +We first compute an easy example in \(n=2.\text{For}\epsilon>0\), suppose that \(E(\mathbf{z},\epsilon)=x^{2}+ \frac{6}{R^{2}|B(R)|} \Theta(R^{2}-|\mathbf{z}|^{2})\) is a rescaled indicator function on the ball. We compute that \(N=1,\mathbf{c}=0,H(R)=\mathbb{I}_{4},V_{0}=0\), d = 0, and \(V(\epsilon)=\text{diag}(1,\epsilon^{2},1,1)\). The initial energy stored in \(f_{0}\) is \(E[f_{0}]=(3+\epsilon)\). For \(\epsilon\) small, we should expect that \(E_{SL(4)}\) is small since \(f_{0}\) can be squeezed onto the \(y\) axis via area-preserving maps. Indeed, Eq. (8) gives us that + +\[E_{SL(4)}=\epsilon^{1/2}.\] (15) + +Dividing this equation by \(E[f_{0}]\), we can alternatively compute the inaccessible energy fraction \(F_{SL(2n)}:= \frac{E_{SL(2n)}}{E[f_{0}]}\) to be + +\[F_{SL(4)}= \frac{4\epsilon^{1/2}}{3+\epsilon^{2}} .\] (16) + +The distribution function after an energy minimizing linear map is \(f_{0}\circ\phi^{-1}= \frac{1}{|B(R)|} \Theta(R^{2}- \epsilon^{-1/2}(x^{2}+p_{x}^{2}+p_{y}^{2})-\epsilon^{3/2}y^{2})\) which looks as expected. In contrast, the linear Gromov’s nonsqueezing theorem prohibits squeezing \(f_{0}\) onto the \(y-\)axis via linear symplectomorphisms. This implies we should find \(E_{Sp(2n)}\) to be finite in the limit \(\epsilon\rightarrow0^{+}.V(\epsilon)\) can be symplectically diagonalized by \(S_{V}=\text{diag}(1,\epsilon^{-1/2},1,\epsilon^{1/2})\) giving diff --git a/outputs/arxiv_math/2503.08011_pg19_pg1_repeat1.md b/outputs/arxiv_math/2503.08011_pg19_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..a261976afcabf595fbc29f946cf8e97c046b3243 --- /dev/null +++ b/outputs/arxiv_math/2503.08011_pg19_pg1_repeat1.md @@ -0,0 +1,51 @@ +[13] Y. S. Samoilenko, Spectral Theory of Families of Self-Adjoint Operators, vol. 57, Springer Science & Business Media, 1991. + +[14] K. Sato and R. Kawamura, Uniqueness Analysis of Controllability Scores and Their Application to Brain Networks, 2024, https://doi.org/10.48550/arXiv.2408.03023. + +[15] K. Sato and A. Takeda, Controllability Maximization of Large-Scale Systems Using Projected Gradient Method, IEEE Control Systems Letters, 4 (2020), pp. 821–826. + +[16] K. Sato and S. Terasaki, Controllability Scores for Selecting Control Nodes of Large-Scale Network Systems, IEEE Transactions on Automatic Control, 69 (2024), pp. 4673–4680. + +[17] T. H. Summers, F. L. Cortesi, and J. Lygeros, On Submodularity and Controllability in Complex Dynamical Networks, IEEE Transactions on Control of Network Systems, 3 (2016), pp. 91–101. + +[18] S. Terasaki and K. Sato, Minimal Controllability Problem on Linear Structural Descriptor Systems With Forbidden Nodes, IEEE Transactions on Automatic Control, 69 (2024), pp. 527–534. + +[19] N. K. Vishnoi, Algorithms for Convex Optimization, Cambridge University Press, 2021. + +[20] T. Yamamoto, On the Eigenvalues of Compact Operators in a Hilbert Space, Numerische Mathematik, 11 (1968), pp. 211–219. + +[21] K. Yosida, Functional Analysis, Springer Berlin Heidelberg, 2012. + +# Appendix A. Proof of Theorem 3.9. + +# In this section, we prove Theorem 3.9. Define \(S\) as + +\[(\mathrm{A}.1)S:=\left\{\{b_{i}\}_{i=1}^{\infty}\Bigg|0\leq b_{i}\leq a_{i},\sum_{i=1}^{\infty}b_{i}=1\right\},\] + +where \(\{a_{i}\}_{i=1}^{\infty}\) is a sequence defined in Assumption 3.6. Then, the following lemma holds. Lemma A.1. \(S\) is a compact subset of \(\ell^{1}(\mathbb{N})\). + +# Proof. Let \(\{\beta^{(n)}\}_{n=1}^{\infty}\) be a sequence in \(S,\text{where}\beta^{(n)}:=\{b_{i}^{(n)}\}_{i=1}^{\infty}\in S\). Since quence \(\{b_{1}^{(1_{n})}\}_{n=1}^{\infty}\). Similary, since \(0\leq b_{2}^{(1_{n})}\leq a_{2},\{b_{2}^{(1_{n})}\}_{n=1}^{\infty}\) also has a convergent + +## \(0\leq b_{1}^{(n)}\leq a_{1}\) holds for any \(n\in\mathbb{N}\), the sequence \(\{b_{1}^{(n)}\}_{n=1}^{\infty}\) has a convergent subsesubsequence \(\{b_{2}^{(2_{n})}\}_{n=1}^{\infty}\). Repeating this procedure for each coordinate and applying + +quence \(\{b_{i}^{(n_{n})}\}_{n=1}^{\infty}\) converges, and \(0\leq b_{i}^{(n_{n})}\leq a(i)\) holds. Define \(c_{i}:=\lim_{n\rightarrow\infty}b_{i}^{(n_{n})}\). + +# the diagonal argument, we obtain a subsequence \(\{\beta^{(n_{n})}\}_{n=1}^{\infty}.\text{For all}i\in\mathbb{N}\), the se-Since \(0\leq b_{i}^{(n_{n})}\leq a_{i}\), it follows that \(|b_{i}^{(n_{n})}-c_{i}|\leq a_{i}\). Moreover, since \(\sum_{i=1}^{\infty}a_{i}<\infty\), we + +can apply Lebesgue convergence theorem. This yields \(\lim_{n\rightarrow\infty}\sum_{i=1}^{\infty}|b_{i}^{(n_{n})}-c_{i}|=0\), + +which implies that the subsequence \(\{\beta^{(n_{n})}\}_{n=1}^{\infty}\) converges to \(c\) with respect to \(\ell^{1}\) norm. + +Furthermore, it is straightforward to verify that \(\{c_{i}\}_{i=1}^{\infty}\in S\). Thus, \(\{\beta^{(n)}\}_{n=1}^{\infty}\) has a subsequence that converges in \(S\). Therefore, \(S\) is compact. + +Lemma A.2. Under Assumption 3.6, the feasible region \(\mathcal{F}\) is defined as + +\[(\mathrm{A}.2)\mathcal{F}:=S\cap X,\] + +where + +\[(\mathrm{A}.3)X=\{\{p_{i}\}_{i=1}^{\infty}|\mu_{n}(W(p))>0\}.\] + +Then \(\mathcal{F}\) is a convex set. + +Proof. Since the convexity of \(S\) is straightforward, we show the convexity of \(X\). Then, it suffices to show that \(\mu_{n}(W((1-\theta)p+\theta q))>0\) for any \(\theta\in(0,1)\text{and}p,q\in X\). diff --git a/outputs/arxiv_math/2503.08031_pg36_pg1_repeat1.md b/outputs/arxiv_math/2503.08031_pg36_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..bf8c48d4134b9d3e52886d2bd760acee090d61e7 --- /dev/null +++ b/outputs/arxiv_math/2503.08031_pg36_pg1_repeat1.md @@ -0,0 +1,15 @@ +Lemma G.5 (Huang et al. (2023), Proposition 9). Let \(X_{1},\ldots,X_{n}\) be i.i.d. random vectors in \(\mathbb{R}^{p}\), and let + +\(h\) be a function \(h:\mathbb{R}^{p}\times\mathbb{R}^{p}\rightarrow\mathbb{R}\) satisfying \(\mathbf{E}(h(X_{1},X_{2})∣X_{1})=0\). Suppose there is a sequence of functions + +\(\phi_{1},\ldots,\phi_{K}:\mathbb{R}^{p}\rightarrow\mathbb{R}\text{such that}h\) can be represented as + +\[h(x,x^{\prime})=\sum_{k=1}^{K}\phi_{k}(x)\phi_{k}(x^{\prime})\] + +for all \(x,x^{\prime}\in\mathbb{R}^{p}\). Also, let \(\phi(X_{1})=(\phi_{1}(X_{1}),\ldots,\phi_{K}(X_{1}))\) and let \(\Sigma\in\mathbb{R}^{K\times K}\) denote the covariance matrix + +of \(\phi(X_{1})\). Lastly, let \(\tau^{2}=\text{var}(h(X_{1},X_{2}))\), and let \(Z_{1},\ldots,Z_{K}\) denote independent standard normal random + +variables. Then, + +\[\sup_{R_{t\in\mathbb{R}}R_{R_{RR}}}^{R_{R_{RR_{R}}}}R\mathbf{P}(\sum_{1\leq i\neq j\leq n} \frac{h(X_{i},X_{j})}{\sqrt{\tau^{2}n(n-1)}} \leq t)-\mathbf{P}(\sum_{k=1}^{K} \tfrac{1}{\tau} \lambda_{k}(\Sigma)(Z_{k}^{2}-1)\leq t)^{R_{R_{RR_{R}}}}R_{RR_{R_{RR}}}≲n^{- \frac{1}{5}}+n^{- \frac{1}{14}}( \tfrac{1}{\tau} \Vert h(X_{1},X_{2})\Vert_{L^{3}})^{\frac{3}{7}}.\] diff --git a/outputs/arxiv_math/2503.08059_pg6_pg1_repeat1.md b/outputs/arxiv_math/2503.08059_pg6_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..7328059e4d85b36494347133f0bb8c1f3c157296 --- /dev/null +++ b/outputs/arxiv_math/2503.08059_pg6_pg1_repeat1.md @@ -0,0 +1,99 @@ +![](figure) + +(a) Ground Truth Prediction Error \(1e^{-3}\) +\(u_1(t)\) + +(b) Ground Truth Prediction Error \(1e^{-2}\) +\(u_2(t)\) + +(c) Training Validation +Stage 1 loss +Epoch + +(d) Training Validation +Stage 2 loss +Steps +Epoch + +(e) Training Validation +Stage 3 loss +Euler \(\Rightarrow\) Dopri5 +Epoch[^inferred] + +![](figure) + +(a) Ground Truth Prediction Error \(1e^{-3}\) +\(u_1(t)\) + +(b) Ground Truth Prediction Error \(1e^{-2}\) +\(u_2(t)\) + +(c) Training Validation +Stage 1 loss +Epoch + +(d) Training Validation +Stage 2 loss +Steps +Epoch + +(e) Training Validation +Stage 3 loss +Euler \(\Rightarrow\) Dopri5 +Epoch[^inferred] + +Figure 4: Predicting DR and KS systems using SNODEs. (a) and (b) show the testing examples for the DR and KS systems, respectively. Here, the training set features a spatial resolution of \(N_{x}=32\), whereas the test set has \(N_{x}=128\). (c), (d), and (e) respectively illustrate the variations in training and validation losses across three training stages. + +We first consider the DR system. In practice, the source term may vary over time due to changes in environmental factors, yielding the following equation with a time-varying source term \(u(x,t)\): + +\[s_{t}=Ds_{xx}+Ks^{2}+u(x,t),x\in[0,1],t\in[0,1],(7)\] + +where \(D=0.01\) is the diffusion coefficient and \(K=0.01\) is the reaction rate. Here, we use \(u(x,t)=(\pi x)/5+u_{1}(t)\), where \(u_{1}(t)\) is a sampling function from the GRF. In addition, we consider the KS equation, which exhibits complex chaotic dynamics, of the following form + +\[s_{t}=-ssx^{x}\in^{-}[^{s}0^{x},^{x}32^{-}\pi^{u}],^{(x}t^{,}\in^{t)}[^{s}0^{x},^{x}2^{x}0^{x}],^{,}\] (8) + +where \(u(x,t)=x/16+u_{2}(t),\text{and}u_{2}(t)\) is the sampling function from the GRF. After training, our SNODEs framework exhibits excellent modeling capabilities on the aforementioned PDE systems. Moreover, our framework can be naturally applied to the super-resolution learning by using higher-resolution spatiotemporal data as the test set. In the temporal dimension, our approach is capable of estimating the system state for any given future moment through the ODESolve process. In terms of spatial dimension, our method allows for training with lower resolution data \((N_{x}=\) + +32) and testing in higher resolution data \((N_{x}=128)\). The corresponding outcomes are presented in Figures 4(a)-(b). Here, we take the DR system as an example and provide the training details for our SNODEs framework. In stage 1, the rapid capture of critical components of unknown dynamics is facilitated through flow matching pre-training, with the training error depicted in Figure 4(c), and we obtain \(\hat{F}_{1}=0.0099s_{xx}+0.9955u\). Herein, the regularization parameter \(\alpha\) is set to 0.01, resulting in the training loss exceeding the validation loss. In stage \(2,\hat{F}_{1}\) was fine-tuned through the ODESolve prediction, with the corresponding prediction error illustrated in Figure 4(d), culminating in \(\hat{F}_{1}=0.0098s_{xx}+u-0.0083s+0.0046\). Herein, we employ the ”Euler” method for the initial 120 epochs, incorporating + +![](figure) + +(a) \( u_3(t) \) + +Initial value \( t = 5 \) \( t = 5 \) + +Ground Truth y(Prediction) + +(b) Initial value + +Ground Truth \( t = 1 \) \( t = 2 \) \( t = 4 \) \( t = 20 \) + +\( u_3(t) \) + +y(Prediction)[^inferred] + +![](figure) + +(a) \( u_3(t) \) + +Initial value \( t = 5 \) \( t = 5 \) + +Ground Truth y(Prediction) + +(b) Initial value + +Ground Truth \( t = 1 \) \( t = 2 \) \( t = 4 \) \( t = 20 \) + +\( u_3(t) \) + +y(Prediction)[^inferred] + +Figure 5: Predicting NS systems using SNODEs. (a) The predicted result for training data with spatial resolution \(N_{x}=N_{y}=16\). (b) The predicted result for testing data with spatial resolution \(N_{x}=N_{y}=80\). + +progressively increasing prediction steps. Subsequently, for the remaining 80 epochs, the number of prediction steps is maintained at a constant 20, and we switch to the ”Dopri5” method for further training. In stage 3, we maintain \(\hat{F}_{1}\) fixed and employ a GeNN for residual learning, where the optimization objective is \(F-\hat{F}_{1}\) and the prediction error is presented in Figure 4(e). Herein, the training strategy remains consistent with that of stage 2. In fact, within the temporal-spatial domain of the DR experiment, the influence of \(Ks^{2}\) on the dynamics is minimal. Consequently, this term was not identified in stage 1. However, accurate predictions were achieved through the employment of simple substitute terms in stage 2. Then in the stage 3, the modeling accuracy was further enhanced through residual learning. Finally, we consider a 2-d NS system for a viscous, incompressible fluid in vorticity form, which reads + +\[s_{t}=\gamma_{x}s_{y}-\gamma_{y}s_{x}+\nu\Delta_{(x}s_{,}+_{y)}u_{\in}(x_{[},_{0}y_{,},_{2}t_{]})_{2},_{,t}\Delta_{\in}\gamma_{[}=_{0,2}-_{0}s_{]},_{,(9)}\] + +where \(\gamma\) represents the stream function, \(\Delta\) is the Laplacian operator, and \(\nu=0.001\). Additionally, \(u(x,y,t)= u_{3}(t)\times\{0.1\sin[2\pi(x+y)]+\cos[2\pi(x+y)]\}\) is the forcing function, where \(u_{3}(t)\) is a function obtained from a GRF. The system is defined over a square domain with dimensions \([0,2]^{2}\), and the time interval is \([0,20]\). Under this condition, we can express the stream function as the vorticity, i.e., \(\gamma=-\Delta^{-1}s\). In the discrete scenario within the Fourier domain, this corresponds to \(\tilde{\gamma}=-1/(\boldsymbol{k}^{2}+\boldsymbol{l}^{2})\tilde{s}\). To facilitate the training, we augment the first layer of SymNet with \(i\boldsymbol{k}/(\boldsymbol{k}^{2}+\boldsymbol{l}^{2})\tilde{s}\text{and}i\boldsymbol{l}/(\boldsymbol{k}^{2}+\boldsymbol{l}^{2})\tilde{s}\) in the Fourier domain. Then after training, Figure 5 demonstrates that our framework achieves the accurate operator learning in modeling the underlying dynamics, enabling the precise and stable prediction of system evolutions, even the initial values and parameter functions outside the training set distribution. Additional training details and experimental results for the above parametric PDE systems are provided in Appendix B.4. + +[^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/2503.08062_pg10_pg1_repeat1.md b/outputs/arxiv_math/2503.08062_pg10_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..23f491598c20a037aa8b08ac13f3a0986d133645 --- /dev/null +++ b/outputs/arxiv_math/2503.08062_pg10_pg1_repeat1.md @@ -0,0 +1,115 @@ +in Fig. 9. With the sensing procedures shown in Fig. 4, the sensing peak has a gain proportional to the number of subcarriers \(N\) and the number of OFDM symbols \(M\). With the sensing gain, the expected sensing power is increased by \(10\log_{10}(MN)\) dB, which brings the sensing power to \(-20.40\) dBm, as shown by the black dashed line in Fig. 9. + +-10 + +-20 + +-30 + +-40 + +-50 + +-60 + +-70 + +-80 + +-90 + +-100 + +0 + +Sensing processing gain + +Range Profile Rx Signal power Sensing Power + +50 100 150 200 250 300 350 Range (meters) + +![](figure) + +Fig. 9: Sensing profile for a target within the ISI-free range. + +Next, we consider the sensing of a single target beyond the ISI-free range, with \(d=304.96\mathrm{m}\), shown in Fig. 10. According to (37). The expected signal power is degraded by the factor \((1- \frac{N_{\tau}-N_{cp}}{N} )^{2}\), corresponding to 1.65 dB, and the expected ISI and ICI in the range profile is about \(-109.96\) dBm, which is negligible as compared with the noise level. Hence, the observed noise level is similar to the case in Fig. 9, which is \(P_{N}=-87.17\) dBm. On the other hand, the received signal \(y[n]\) has the average power -104.97 dBm at \(d=304.96\mathrm{m}\) and the sensing power is equivalent to -62.05 dBm after considering sensing processing gain and power degradation due to incomplete OFDM symbol. Fig. 10 shows that the simulation is consistent with the analysis. + +-50 + +-60 + +-70 + +-80 + +-90 + +-100 + +Range Profile Expected noise power Expected signal Power + +0 50 100 150 200 250 300 350 Range (meters) + +![](figure) + +Fig. 10: Sensing profile for a target beyond the ISI-free range. + +When multiple targets coexist, as shown in Fig. 11(a), the resultant ISI and ICI from targets beyond the ISI-free range adds up, which may lift the noise level up. However, according to (39), the total ISI/ICI resultant by all the 6 targets beyond the ISI-free distance is only \(-98\text{dBm}\), which slightly lifts the noise level from \(-87.17\)dBm to \(-86.83\text{dBm}\). In general, when the number of subcarriers is sufficiently large, the impact of ISI and ICI caused by CP limitation is negligible for sensing. According to the specifications in 5G NR, the bandwidth allocated for each user ranges from 34.56MHz to 396MHz when the subcarrier spacing is 120kHz, which corresponds to 288 to 3300 subcarriers. If the number of subcarriers in the simulation reduces from 2048 to 288 in Fig. 11(b), the interference power will increase to \(-97.86\text{dBm}\), which is still negligible as compared with the noise level. + +300 + +200 + +100 + +0 + +-100 + +Target 8 + +● BS ✗ Targets ISI-free distance + +Target 7 + +Target 4 + +Target 1 + +Target 3 Target 2 + +-200 + +Target 6 -400 -300 -200 -100 0 + +Target 5 100 + +200 + +(a) Target distribution + +-10 -20 -30 -40 -50 -60 -70 -80 -90 -100 + +Range Profile Noise+Interference power + +0 + +100 + +200 + +300 Range (meters) + +400 + +500 + +600 + +(b) Range profile with ISI/ICI + +![](figure) + +Fig. 11: The sensing of multiple targets. + +Consider a sensing detection threshold \(\rho\) in (42), the maximum sensing range of an OFDM-ISAC system can be obtained by solving \(\gamma(d)=\rho\), since the sensing SINR \(\gamma(d)\) is a monotonically decreasing function of \(d\). Let \(\rho=10\), the maximum sensing range is calculated to be 610m for the default CP length in Table I, which is much larger than the ISI free range 88.44m. Fig. 12 plots the change of the maximum sensing range with CP length. It is observed that if the CP is totally removed, i.e., \(T_{cp}=0\), the maximum sensing range can still reach about 590m for \(M=14\). Besides, the maximum sensing range increases from 610m to 800m when the CP length increases from \(0.59\mu s\) to \(5.3\mu s\), due to the reduced power degradation and ISI/ICI. For \(d>\) 800m, the received diff --git a/outputs/arxiv_math/2503.08077_pg42_pg1_repeat1.md b/outputs/arxiv_math/2503.08077_pg42_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..b6478250e778f4966fcb35b261dc5b50cfb5334f --- /dev/null +++ b/outputs/arxiv_math/2503.08077_pg42_pg1_repeat1.md @@ -0,0 +1,19 @@ +## 15. ANALYSING THE FLOW + +In section 13, we constructed a continuous vector field X on the closed surface S. Since in this setting \(\mathrm{F}\) cs is in ideal position, its vertical sublamination \(\Lambda^{cs}\) projects via \(\pi:M\rightarrow S\) to a geodesic lamination \(\Lambda\subset S\). Together, X and \(\Lambda\) have the following properties: + +1. the vector field X is tangent to the geodesic lamination \(\Lambda\); + +2. the vector field is zero at a point p if and only if \(\pi^{-1}(p)\) is a center circle and we refer to these as the critical points; + +3. each critical point lies on an isolated geodesic in \(\Lambda\) and we call these geodesics the critical geodesics; + +4. each critical geodesic contains exactly one critical point; + +5. at each critical point, the index of X at p is in \(\{-1,0,+1\}\); moreover, if we consider a small disk D centered at p and split D along the critical geodesic to produce two half-disks, then the index of X at p is in \(\{ \frac{-1}{2} ,0, \frac{+1}{2} \}\) on each of the half-disks. + +Item (3) is given by lemma 9.1 and item (5) by lemma 13.1. The plan now is to first excise the non-isolated leaves of \(\Lambda\) from S, producing a surface with boundary \(S_{0}\subset S\text{where}\Lambda\cap S_{0}\) consists only of isolated geodesic arcs, each of which is compact. We then split S0 into a number of “regions” Ri by cutting along the “critical arcs.” By applying the Poincaré–Hopf theorem to the vector field X restricted to a region Ri , we show that at least one of these regions has positive Euler characteristic and therefore must be a topological disk. Its pre-image \(\pi^{-1}(R_{i})\) must contain half of a vertical leaf of \(\mathrm{F}^{cu}\), and this causes a contradiction, completing the overall proof. For simplicity, we assume that the continuous vector field X integrates to a flow \(\varphi\) on S and we explain in the following remark how to adapt the proof when this is not the case. + +Remark. The definition of a dumbbell and the proof of proposition 14.1 do not actually rely on X integrating to a flow \(\varphi\) on all of S. We can instead define an outflow edge \(\sigma\) as having a small neighborhood U such that any integral curve starting at a point \(x\in U\) and tangent to X must remain in U until it hits \(\sigma\). The proof of proposition 14.1 only considers the flow \(\varphi\) on the lamination \(\Gamma\). In our setting, \(\Gamma\) will be a geodesic lamination and so the flow is well defined. After applying the proposition and restricting to a subset \(S_{0}\subset S\), we may replace X by a smooth approximation such that X is unchanged on the finitely many arcs of \(\Lambda\cap S_{0}\) which contain critical points. Thus, we may freely assume that X integrates to a flow \(\varphi\). + +Let \(\Gamma\) be the sublamination of \(\Lambda\) consisting of all of the non-isolated leaves in \(\Lambda\). Apply proposition 14.1 to \(\Gamma\) and let S0 denote the subset S \int(K ) given by the proposition. Each boundary component of S0 is therefore a dumbbell. diff --git a/outputs/arxiv_math/2503.08088_pg5_pg1_repeat1.md b/outputs/arxiv_math/2503.08088_pg5_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..1565f0056071944b0af68c044c14845b99b429e4 --- /dev/null +++ b/outputs/arxiv_math/2503.08088_pg5_pg1_repeat1.md @@ -0,0 +1,27 @@ +For a set \(X\subseteq V(G)\), the (open) neighborhood of \(X\), denoted by \(N_{G}(X)\), is the set \(\{v\in V(G)\setminus X:v\) is adjacent to a vertex in \(X\}\). The closed neighborhood of \(X\) is the set \(N_{G}[X]=N_{G}(X)\cup X\). We simply use \(N(v),N[v],N(X),\text{and}N[X]\) to denote \(N_{G}(v),N_{G}[v],N_{G}(X),\text{and}N_{G}[X]\), respectively when the context of the graph \(G\) is clear. We use \(G[X]\) to denote the subgraph of \(G\) induced by the vertices in \(X\). A vertex \(v\) dominates itself and all its neighbors. A set \(X\subseteq V(G)\) is called a clique of \(G\) if every pair of vertices of \(X\) are adjacent. For disjoint subsets \(X\text{and}Y\) of \(V(G),G[X,Y]\) denotes the set of edges \(\{e\in E(G):e\) has one endpoint in \(X\) and the other endpoint in \(Y\}\). If the graph \(G\) is clear from the context, we write \([X,Y]\) rather than \(G[X,Y]\). We say that \(G[X,Y]\) is complete if every vertex in \(X\) is adjacent to every vertex in \(Y\text{in}G\). For a given positive integer \(k\), we use the notation \([k]\) to denote the set \(\{1,\ldots,k\}\). For a set \(D\subseteq V(G)\) and a vertex \(u\in D\), the \(D-\)external private neighborhood of \(u\), denoted by \(\text{epn}(u,D)\), is the set \(\{v\in V(G)\setminus D:N(v)\cap D=\{u\}\}\). A vertex in \(\text{epn}(u,D)\) is a \(D-\)external private neighbor of \(u\). We say that a vertex \(v\in V(G)\setminus D\) is \(D-\)defended if there exists a vertex \(u\in N(v)\cap D\) such that \((D\setminus\{u\})\cup\{v\}\) is a dominating set of \(G\). Specifically, we say that \(v\in V(G)\setminus D\) is \(D-\) defended by a vertex \(u\text{if}u\in N(v)\cap D\text{and}(D\setminus\{u\})\cup\{v\}\) is a dominating set of \(G\). We note that \(D\) is a secure dominating set of \(G\) if and only if every vertex in \(V(G)\setminus D\) is \(D-\text{defended}\). We also note that if \(D\) is a dominating set of \(G\text{and epn}(u,D)=\emptyset\text{for some}u\in D\), then every neighbor of \(u\) in \(V(G)\setminus D\) is \(D-\)defended by \(u\). We state this observation formally as follows. + +Observation 2. If \(D\) is a dominating set of a graph \(G\text{and}\text{epn}(u,D)=\emptyset\) for some vertex \(u\in D\), then every neighbor of \(u\) that belongs to \(V(G)\setminus D\) is \(D-\)defended by \(u\). + +As a consequence of Observation 2, if \(D\) is a dominating set of \(G\text{and epn}(u,D)=\emptyset\) for every \(u\in D\), then every vertex in \(V(G)\setminus D\) is \(D-\text{defended}\), implying that \(D\) is a secure dominating set of \(G\). For a given dominating set \(D\) of \(G\), we partition \(D\text{as}A_{D}\cup B_{D}\), where the sets \(A_{D}\text{and}B_{D}\) are defined as follows. + +\((\mathcal{P}_{1})\) + +\(\bullet A_{D}=\{u\in D\) : there exists a neighbor of \(u\text{in}V(G)\setminus D\) that is not \(D-\text{defended}.\}\) + +\(\bullet B_{D}=D\setminus A_{D}.\) + +We now prove the following useful lemma. We remark that some of the facts that appear in the following lemma have appeared implicitly in several papers (see [4, 15]). + +Lemma 1. Let \(D\) be a dominating set of \(G\text{and}A_{D}\cup B_{D}\) be a partition of \(D\) as defined in \((\mathcal{P}_{1})\). + +(a) If \(A_{D}=\emptyset,\text{then}D\) is a secure dominating set of \(G\). + +(b) If a vertex \(vinV(G)\setminus D\) is not \(D-defended,\text{then}N(v)\cap D\subseteq A_{D}\). + +(c) If \(u\) is a vertex in \(A_{D},\text{then}\text{epn}(u,D)\neq \emptyset\). + +(d) If a vertex \(vinV(G)\setminus D\) is not \(D-defended\), then for every neighbor \(u\) of \(vinD\), the vertex \(v\) has a non-neighbor in \(\text{epn}(u,D)\), that is, \(\text{epn}(u,D)\setminus N(v)\neq \emptyset\). + +(e) If \(D^{\prime}\) is a superset of \(D,\text{then}A_{D^{\prime}}\subseteq A_{D}\text{and}B_{D}\subseteq B_{D^{\prime}},\text{where}A_{D^{\prime}}\cup B_{D^{\prime}}\) is a partition of \(D^{\prime}\) as defined in \((\mathcal{P}_{1})\). + +Proof. Parts (a) and (b) follow by the definition of the sets \(A_{D}\text{and}B_{D}\), while part (c) follows by Observation 2. diff --git a/outputs/arxiv_math/2503.08105_pg5_pg1_repeat1.md b/outputs/arxiv_math/2503.08105_pg5_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..e1cc8a86ee81bc092e1e8176a007033ee3633fe9 --- /dev/null +++ b/outputs/arxiv_math/2503.08105_pg5_pg1_repeat1.md @@ -0,0 +1,38 @@ +polynomial may not achieve the maximum allowable number of roots given by the Fundamental Theorem of Algebra, but this bound is sharp, in the sense that given a natural number \(n\) there exist a polynomial of degree \(n\) which has exactly \(n\) zeros counting multiplicities. + +Theorem 2.4. (Wilmshurst [14]) Let \(h(z) \underline{\text{and}} g(z)\) be analytic polynomials of degree \(n\text{and}m\) respectively with \(n>m.\text{Then}f(z)=h(z)+ \overline{g(z)} ,\) has at most \(n^{2}\) zeros counting multiplicities. The case \(n=m\) could have infinite number of zeros. + +It was shown by Bshouty et al. [3] that there exists a complex-valued harmonic polynomial \(f=h+ \overline{g} ,\) such that \(h\) is an analytic polynomial of degree \(n,g\) is an analytic polynomial of degree \(m\leq n\) and \(f\) has exactly \(n^{2}\) zeros counting with multiplicities in the field of complex numbers, \(\mathbb{C}.\) + +## 3 Main Results + +In this section we state and prove the main results of the paper. The first part of this section is results regarding products of complex-valued harmonic functions; the second part is results about the boundary values of the factors ans products; and the third part is an auxiliary result concerning the the maximum number of zeros of the product, provided the product and factors are complex-valued harmonic polynomials. + +### 3.1 Product of harmonic functions + +We know that the square of real harmonic function \(u\) cannot be harmonic, unless \(u\) is a constant. Of course the square of analytic function is analytic and hence harmonic as any analytic function is complex-valued harmonic function. Similarly the square of anti-analytic function is anti-analytic and hence complex-valued harmonic function. But for the complex-valued harmonic function which is neither analytic nor anti-analytic we have the following result + +Theorem 3.1. Suppose \(f(z)=h(z)+ \overline{g(z)} ,\) is harmonic on \(\Omega.\text{Then}f^{2},\) is harmonic on \(\Omega\) if and only if either \(horg\) is a constant on \(\Omega.\) + +Proof. If either \(h\text{or}g\) is a constant on \(\Omega,\) then it easy to see that the square of analytic function or the square of co-analytic function is harmonic on \(\Omega.\) Assume that the square of a complex-valued harmonic function is harmonic on \(\Omega.\) Then + +is harmonic on \(\Omega\) implies that, + +From which we obtain, + +\[f^{2}=(h+ \overline{g} )^{2}=h^{2}+ \overline{g}^{2} +2h \overline{g}\] + +\[\frac{\partial^{2}f^{2}}{\partial z \overline{z}} =0.\] + +\[\frac{\partial^{2}f^{2}}{\partial z \overline{z}} =2 \frac{\partial h}{\partial z} \frac{\partial \overline{g}}{\partial \overline{z}} =0,\] + +| ∂2f2 | ∂h ∂g | +| --- | --- | +| = 2 | = 0, | +| ∂zz | ∂z ∂z | + +This on turn implies that either of the following two is satisfied on \(\Omega,\) + +\[\frac{\partial h}{\partial z} =0, \frac{\partial \overline{g}}{\partial \overline{z}} =0.\] + +Hence either \(h\text{or} \overline{g}\) is a constant \(\Omega.\) Which is the desired result. diff --git a/outputs/arxiv_math/2503.08109_pg12_pg1_repeat1.md b/outputs/arxiv_math/2503.08109_pg12_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..055442eade7f5e740ca0085efb86db8c952b7003 --- /dev/null +++ b/outputs/arxiv_math/2503.08109_pg12_pg1_repeat1.md @@ -0,0 +1,19 @@ +These relations form the \(\mathcal{N}=2\) SUSY algebra. It describes SUSY quantum mechanics [38, 6]. We can also view it as generators of a \(\mathbb{Z}_{2}-\)grading of a Hilbert space. Using these generators it is easy to see that + +\[S=q\text{or}S=q^{\dagger}\] + +realizes nilpotent \(S’\mathrm{s}\). On the other hand when + +\[S=b\text{or}S=f,\] + +we obtain projector \(S’\mathrm{s}\). These operators help construct the constant \(Y_{ij}\) operator \(S_{i}S_{j}\). They can satisfy either of the three conditions set in (3.9). The corresponding Hamiltonians are either given by (3.4) or (3.8). The operator \(Y_{ij}=S_{i}T_{j}\) is Baxterized when either one of \(ST=0\text{or}TS=0\) is satisfied. This is easily realized using the SUSY generators4 as + +\[ST=0 \begin{cases} S=q,T=b \\ S=q^{\dagger},T=f \\ S=b,T=q^{\dagger} \\ S=f,T=q. \\ S=b,T=f, \end{cases}\] (3.15) + +The \(TS=0\) yields the same choices when \(S\text{and}T\) are interchanged in the above equation. The Hamiltonians in this case are of the form (3.8). All of the above SUSY realizations are algebraic or independent of the size of the representation of the SUSY algebra. For the \(\mathbb{C}^{2}\) representation the SUSY generators take the form + +\[\begin{aligned} q&= \begin{pmatrix} 01 \\ 00 \end{pmatrix} ;q^{\dagger}= \begin{pmatrix} 00 \\ 10 \end{pmatrix} , \\ b&=qq^{\dagger}= \begin{pmatrix} 10 \\ 00 \end{pmatrix} ;f=q^{\dagger}q= \begin{pmatrix} 00 \\ 01 \end{pmatrix} . \end{aligned}\] (3.16) + +It is clear that these matrices perform a \(\mathbb{Z}_{2}-\)grading of the Hilbert space \(\mathbb{C}^{2},\text{with}b\) and \(f\) projecting to the ‘bosonic’ and ‘fermionic’ parts of \(\mathbb{C}^{2}\). It is also important to note that the \(\mathbb{C}^{2}\) representation of SUSY (3.16) span the space of operators acting on \(\mathbb{C}^{2}\), + +4For solutions of the generalized Yang-Baxter equation using similar realizations see [31]. diff --git a/outputs/arxiv_math/2503.08118_pg5_pg1_repeat1.md b/outputs/arxiv_math/2503.08118_pg5_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..c716bb26ee18995895f4be32bd0106d914681546 --- /dev/null +++ b/outputs/arxiv_math/2503.08118_pg5_pg1_repeat1.md @@ -0,0 +1,61 @@ +## 4. Embedded Toric Resolution + +The rational double point singularities are given by one of the following equations: + +\[\begin{aligned} A_{n}:f(x,y,z)&=xy-z^{n+1},n\in\mathbb{N} \\ D_{n}:f(x,y,z)&=z^{2}-x(y^{2}+x^{n-2}),n\in\mathbb{N},n\geq4 \\ E_{6}:f(x,y,z)&=z^{2}+y^{3}+x^{4} \\ E_{7}:f(x,y,z)&=x^{2}+y^{3}+yz^{3} \\ E_{8}:f(x,y,z)&=z^{2}+y^{3}+x^{5} \end{aligned}\] + +Remark 4.1. Each member of \(ADE-\)singularities is Newton non-degenerate. + +Embedded toric resolutions of \(ADE-\)singularities were studied in [18, 19]. Here differently we construct regular refinements of Gr¨obner fans of each members of them. This leads us their embedded toric resolutions [1, 2]. Additionally, these resolutions will be minimal which means that: Definition 4.2. An embedded toric resolution is called minimal if + +\(i)\) The vectors in the regular refinement are irreducible. + +\(ii)\) The resolution graph obtaining from the refinement does not have any −1 curve means the self-intersection of each vertex is different from −1 (this gives us the resolution graph is minimal). + +Note that the self-intersection of a vertex is computed as: The number \(-s\in\mathbb{Z}_{\leq0}\) with \(s.v=\sum_{i=1}^{m}v_{i}\) is called the self-intersection of the vertex \(v\) in the graph where each \(v_{i}\) is adjacent to \(v\) in the regular refinement. + +### 4.1. \(A_{n}-\mathbf{singularities}\). Consider the hypersurface \(X=V(f)\subset\mathbb{C}^{3}\) where + +\[f(x,y,z)=xy-z^{n+1},n\in\mathbb{N}\] + +Given \(v_{1}\in(\mathbb{R}_{\geq0})^{3}\), let \(In_{v_{1}}(f)=f\). The polyhedral cone associated with \(v_{1}\) is + +\[C_{v_{1}}(f)=\langle(n+1,0,1),(0,n+1,1)\rangle\] + +Given \(v_{2}\in(\mathbb{R}_{\geq0})^{3}\), let \(In_{v_{2}}(f)=xy\). The polyhedral cone associated with \(v_{2}\) is + +\[C_{v_{2}}(f)=\langle(0,0,1),(n+1,0,1),(0,n+1,1)\rangle\] + +Given \(v_{3}\in(\mathbb{R}_{\geq0})^{3}\), let \(In_{v_{3}}(f)=-z^{n+1}\). The polyhedral cone associated with \(v_{3}\) is + +\[C_{v_{3}}(f)=\langle(1,0,0),(0,1,0),(n+1,0,1),(0,n+1,1)\rangle\] + +The Gr¨obner fan \(G(X)\) of \(X\) and its Gr¨obner cones can be seen as + +\[C_{v_{1}}(n+1,0,1)\] + +\[(0,n+1,1)\] + +\[(0,0,1)\] + +\[(0,0,1)\] + +\[C_{v_{2}}(n+1,0,1)\] + +\[(n+1,0,1)\] + +\[C_{v_{3}}\] + +\[(1,0,0)\] + +![](figure) + +Figure 1. singularities + +| (0, n + 1, 1) | (n + 1, 0, 1) | | (0, n + 1, 1) | +| --- | --- | --- | --- | +| (0, n + 1, 1) | (1, 0, 0) | | (0, 1, 0) | +| | | G(X) | | +| (0, 1, 0) | | | | +| | | | (0, n + 1, 1) | +| The Gr¨obner fan | and its | Gr¨obner cones | for An- | diff --git a/outputs/arxiv_math/2503.08119_pg4_pg1_repeat1.md b/outputs/arxiv_math/2503.08119_pg4_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..246f8032bc13431134e62752a3a405da779dae6a --- /dev/null +++ b/outputs/arxiv_math/2503.08119_pg4_pg1_repeat1.md @@ -0,0 +1,29 @@ +## Question: What is the ample cone of \(\mathcal{L}_{Y}(\lambda)\)? + +We begin by reviewing the history of this problem: + +1. In [2], the authors compute the ample cone for Hilbert modular surface, by constructing partial Hasse invariants and studying intersection theory on the surface directly. + +2. In [40], the authors conjectured the ampleness criterion for \(U(2)/\text{Hilbert}/\)quaternionic Shimura varieties. They studied the geometric description of Goren–Oort strata and proved the necessity part of the criterion. + +3. In [45], the author proved the ampleness criterion conjectured by [40], by computing the intersection numbers using the description of strata. + +4. In the unpublished paper [5], the authors proposed a sufficiency criterion for ampleness for Hodge type Shimura varieties. However, their bound is known to be not sharp. + +We know very little about the ampleness criterion beyond the above cases, especially when the group G is not type A1. The main difficulty is that the stratification of the flag space is usually very complicated, and computing intersection numbers \((\mathcal{L}_{Y}(\lambda)\cdot C)\) with a complete curve \(C\subseteq Y\) is non-trivial. In this paper, we give a sharp answer to the above question when the group G is of type A and \(G_{\mathcal{Q}_{p}}\) is split unitary. + +# Statement of the main result. + +Let F be a totally real field of degree N over \(\mathcal{Q}\) in which p is inert. We label the set of real embeddings of F by \(\{\tau_{1},\ldots,\tau_{N}\}\) as before. Let \(E/F\) be an imaginary quadratic extension in which \(p\mathcal{O}_{E}=qq^{c}\). Each \(\tau_{i}\) extends to complex embeddings \(\tau_{i}\) (over \(q)\text{and}\tau_{i}^{c}\) (over \(q^{c})\). For a tuple of integers \((m_{i},n_{i})_{1\leq i\leq N}\) such that \(m_{i}+n_{i}=n\), let G be the unitary group over \(\mathcal{Q}\), with signature at infinite places given by \((m_{i},n_{i})_{1\leq i\leq N}\) , i.e., + +\[G(\mathcal{R})=G\left(\prod_{i=1}^{N}U(m_{i},n_{i})\right).\] (7) + +In Section 3, we realize G as the group of symplectic similitudes of an n-dimensional E-vector space. Let X be the special fiber of the corresponding Shimura variety of signature \((m_{i},n_{i})\). Here we apologize for using the letter X again (but for different meaning) to maintain consistency with the rest of the paper. The signature condition implies that in the Hodge filtration, + +\[0\longrightarrow \omega_{\mathcal{A}^{\vee}/X,\tau_{i}}\longrightarrow H_{1}^{\text{dR}}(\mathcal{A}/X)_{\tau_{i}}\longrightarrow \text{Lie}_{\mathcal{A}/X,\tau_{i}}\longrightarrow 0,\] (8) + +\(\omega_{\mathcal{A}^{\vee}/X,\tau_{i}}\) (resp. \(\omega_{\mathcal{A}^{\vee}/X,\tau_{i}^{c}})\) is a subbundle of \(\omega_{\mathcal{A}^{\vee}/X}\) of rank mi (resp. ni). The polarization in the moduli problem induces a pairing + +\[〈\cdot,\cdot〉:H_{1}^{\text{dR}}(\mathcal{A}/X)_{\tau_{i}}\times H_{1}^{\text{dR}}(\mathcal{A}/X)_{\tau_{i}^{c}}\longrightarrow \mathcal{O}_{X}\] (9) + +such that \(\omega_{\mathcal{A}^{\vee}/X,\tau_{i}^{c}}=\omega_{\mathcal{A}^{\vee}/X,\tau_{i}}^{\perp}\) under this pairing. diff --git a/outputs/arxiv_math/2503.08127_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.08127_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..08ba74d9643c4451b6eee76bb6190db9ad45e186 --- /dev/null +++ b/outputs/arxiv_math/2503.08127_pg2_pg1_repeat1.md @@ -0,0 +1,5 @@ +conformation tensor, respectively. Here, the operator \(\otimes\) denotes \((\boldsymbol{u}\otimes\boldsymbol{v})_{ij}=u_{i}v_{j},(\boldsymbol{C}\otimes\boldsymbol{u})_{ijk}=C_{ij}u_{k}\). If \(\nabla\cdot\boldsymbol{u}=0\), we have \(\nabla\cdot(\boldsymbol{u}\otimes\boldsymbol{u})=(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+(\nabla\cdot\boldsymbol{u})\boldsymbol{u}=(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}\text{and}\nabla\cdot(\boldsymbol{C}\otimes\boldsymbol{u})=(\boldsymbol{u}\cdot\nabla)\boldsymbol{C}\). Therefore, \(\nabla\cdot(\boldsymbol{u}\otimes\boldsymbol{u})\text{and}\nabla\cdot(\boldsymbol{C}\otimes\boldsymbol{u})\) in (1.1a) and (1.1c) can be replaced by the convective terms \((\boldsymbol{u}\cdot\nabla)\boldsymbol{u}\text{and}(\boldsymbol{u}\cdot\nabla)\boldsymbol{C}\). The diffusive Peterlin model is discussed in [20], where the global existence of weak solutions for large data is proved. The existence of weak solutions in 3D is established in [4], as well as a conditional weak-strong uniqueness results. On the other hand, various discrete schemes have been proposed for the numerical simulation of the diffusion Peterlin model. The linear and nonlinear schemes of stabilized Lagrange-Galerkin FEM for the Oseen-type Peterlin viscoelastic model are proposed in [22, 23]. Three fully discrete schemes by using Newton’s iterative, Picard’s iterative and implicit-explicit time-stepping are presented in [16]. [36] introduces a linear decoupled scheme with stabilizing terms, where \((\boldsymbol{u},p)\) and each components of \(\boldsymbol{C}\) can be computed in parallel. Numerical studies of other viscoelastic models (such as the Oldroyd-B model) also exist [11, 25, 30, 39]. Moreover, some log-conformation formulations [1, 2, 12, 15, 35] are used to guarantee symmetric positive-definiteness of discrete conformation tensor. When \(\epsilon\) is very small, (1.1c) becomes convection-dominated. However, the numerical schemes used in most existing works [14, 16, 36, 38] are not specifically designed to address the convection-dominated issue. To tackle this problem, [22–24] propose the Lagrange-Galerkin method for the Oseen-type model. By carefully examining the existing results on error estimates, we find that the constant in the error bound often depends on the reciprocal of \(\epsilon.\text{As}\epsilon\) approaches 0, the parabolic equation (1.1c) transitions to a hyperbolic form, which brings difficulty to numerical approximation. To address the convection-dominated issue and to derive an error bound that does not depend on the reciprocal of \(\epsilon\), we propose a linear HDG scheme for the diffusion Peterlin model and study the error analysis. The HDG can reduce the computational cost of the DG method while preserving its attractive properties of conservation and stability [8, 18]. The method transfers the computation from the global domain to each element, resulting in a smaller stiffness matrix dimension that facilitates efficient inversion operation[10]. Currently the HDG method has been widely applied to elliptic problem [7, 28], convection-diffusion equation [27], linear elasticity system [29], Stokes equation [6, 9, 31] and Navier-Stokes equations [26, 32]. Especially, for the Navier-Stokes equations, the HDG scheme is mass and momentum conserving, energy stable, and pressure-robust. In addition, the discrete velocity is pointwise divergence-free, and the error estimates are optimal and independent of the viscosity \(\nu[17]\). The motivation of this work is to employ the HDG method to address the convection-dominated issue of (1.1). In particular, we aim to obtain error estimates that do not depend on \(\epsilon^{-1}\). In this work, we design a linear HDG scheme for the diffusion Peterlin problem. We prove the unique existences of the discrete problem. Furthermore, by utilizing the Ritz projections for Stokes and Poisson equations, we establish the error estimates. The numerical experiments are carried out to verify the theoretical convergence rates. The HDG scheme remains stable even for the cases with \(\epsilon=10^{-3}\text{and}\epsilon=0\). A comparison with the ordinary FEM in simulation shows that the HDG scheme performs better in preserving the positive definiteness of the conformation tensor. The remainder of this paper is organized as follows. Section 2 introduces the fully discrete semi-implicit linearizing HDG scheme which satisfies the mass conservation. Section 3 focuses on stability analysis. The theoretical convergence rate of the error estimates is rigorously proved under proper assumptions in Section 4. We provide several numerical examples in Section 5 to study the experimental convergence rates and illustrate the stability. + +## 2. The Linear HDG scheme + +Let \(\mathcal{T}:=\{K\}\) be a regular quasi-uniform triangulation of the domain \(\Omega\). The boundary of the element \(K\) is denoted by \(\partial K\) and the outward unit normal vector on \(\partial K\) by \(\boldsymbol{n}\). Two adjacent cells \(K^{+}\text{and}K^{-}\) share an interior facet \(F:=\partial K^{+}\bigcap\partial K^{-}\). A facet of \(\partial K\) that lies on the boundary \(\partial\Omega\) is called a boundary facet. The interior and boundary facet sets are denoted by \(\mathcal{F}_{I}\text{and}\mathcal{F}_{B}\), respectively. diff --git a/outputs/arxiv_math/2503.08151_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.08151_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..e9e3202a97d09b743964933202339f7184256ff3 --- /dev/null +++ b/outputs/arxiv_math/2503.08151_pg2_pg1_repeat1.md @@ -0,0 +1,16 @@ +around 2000 due to the studies of quantum computers. As well known, Grover’s algorithm can find targets much faster than the corresponding classical algorithm, and it can be considered as a quantum walk on a graph [4]. For database search in quantum computers, quantum walks have been focused on in quantum informations [5]. The major study in quantum walks is their finding probabilities and they have been investigated numerically or analytically. Analytic researches have resulted in long-time limit theorems in mathematics. The limit theorems are useful to understand how the quantum walkers distribute in space after they have repeated their unitary processes a lot of times. The first limit theorem was proved in 2002 and it stated a long-time limit distribution which was different from the limit distributions of classical random walks [6]. Since then, many limit theorems have been demonstrated and they told us interesting behavior of quantum walks [7]. In 2015, a remarkable behavior was discovered [8]. A quantum walk held a gap in distribution and the walker was never observed around the launching location. After the paper had been published, two types of quantum walk were reported and they also had a gap in distribution. One of them was a 3-state quantum walk in 2006 and its probability distribution showed a gap and localization [9]. The other was a 2-state walk in 2018 and its model was complicated [10]. Motivated by the quantum walk in 2018, we try to seek a simpler model which makes a probability distribution with a gap. We define a quantum walk in the next section and numerically observe its probability distribution. Since the numerical experiments let us expect that the quantum walk distributes with a gap, we will work on limit distributions in Sect. 3 so that the quantum walk is proved to hold a gap in distribution. + +## 2 Definition of a quantum walk + +The quantum walker with two coin states \(|0\rangle\text{and}|1\rangle\) is supposed to locate at points, whose set is represented by \(\mathbb{Z}=\{0,\pm1,\pm2,\ldots\}\), in superposition. Its system is described on a tensor Hilbert space \(\mathcal{H}_{p}\otimes\mathcal{H}_{c}\). The Hilbert space \(\mathcal{H}_{p}\) represents the locations and it is spanned by the orthogonal normalized basis \(\{|x\rangle:x\in\mathbb{Z}\}\). Also, the Hilbert space \(\mathcal{H}_{c}\) represents the coin states and it is spanned by the orthogonal normalized basis \(\{|0\rangle,|1\rangle\}\). Let us define + +\[\begin{vmatrix} \left[\right] \\ 1 \\ 0\rangle= & , \\ 0 \end{vmatrix} 1\rangle= \begin{bmatrix} 0 \\ 1 \end{bmatrix} ,\] (1) + +| 1 | 0 | | +| --- | --- | --- | +| \|0⟩ = , | \|1⟩ = , | (1) | +| 0 | 1 | | + +for the Hilbert space \(\mathcal{H}_{c}\) in this paper. Given two unitary operations, the coin states and the position states of the walker are operated with them. More precisely, the system of quantum walk at time \(t(=0,1,2,\ldots)\), represented by \(|\Psi_{t}\rangle\in\mathcal{H}_{p}\otimes\mathcal{H}_{c}\), gets a new system at time \(t+1\), represeted by \(|\Psi_{t+1}\rangle\in\mathcal{H}_{p}\otimes\mathcal{H}_{c}\), with unitary operations \(U_{1}\text{and}U_{2}\), + +\[|\Psi_{t+1}\rangle=U_{2}U_{1}|\Psi_{t}\rangle,\] (2) diff --git a/outputs/arxiv_math/2503.08169_pg16_pg1_repeat1.md b/outputs/arxiv_math/2503.08169_pg16_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..d4221413f59beb884a8c1127eb0819be9c53c824 --- /dev/null +++ b/outputs/arxiv_math/2503.08169_pg16_pg1_repeat1.md @@ -0,0 +1,31 @@ +so that the constants \(C(n_{0})>c(n_{0})\), given by + +\[\begin{aligned} C^{2}(n_{0}):&=_{x\in[-} \underline{\mathrm{m}_{1}\mathrm{a}} \mathrm{x}_{, \frac{1}{n_{0}+2} ]}(1-\eta x)^{2}+(\sigma x)^{2}, \\ c^{2}(n_{0}):&=_{x\in[-} \frac{\min_{1}^{n_{0}+2}}{n_{0}+2}_{,n_{0}^{1}+2^{]}} (1-\eta x)^{2}+(\sigma x)^{2}, \end{aligned}\] + +are as upper and lower bounds for \(S_{\boldsymbol{n}}(z)\). Clearly, + +\[C^{2}(n_{0})\leq1+ \frac{2|\eta|}{n_{0}+2} + \frac{|z|^{2}}{(n_{0}+2)^{2}} \leq\left(1+ \frac{|z|}{n_{0}+2} \right)^{2},\] + +which, in turn, provides an alternative proof of (4.16) for \(2-\text{norm}\). We now focus on proving (4.15). Without loss of generality we can assume \(\eta\geq0\). Observe that, with + +it follows that + +\[f(x):=1-2\eta x+|z|^{2}x^{2},\] + +\[\operatorname*{arg\,min}_{x\in\mathbb{R}}f(x)= \frac{\eta}{\sigma^{2}+\eta^{2}} = \frac{\eta}{|z|^{2}} =:x_{0},\min_{x\in\mathbb{R}}f(x)=f(x_{0})= \frac{\sigma^{2}}{\sigma^{2}+\eta^{2}} = \frac{\sigma^{2}}{|z|^{2}} .\] + +Thus, + +Furthermore, + +\[c^{2}(n_{0})\geq \begin{cases} \frac{\sigma^{2}}{2} , & \text{if}x_{0}\leq & \underline{1} & , \\ \overline{\Big(^{|z|}} & \Big) & \overline{n_{0}+2} \\ f & \frac{1}{n_{0}+2} & , & \text{if}x_{0}> \tfrac{1}{n_{0}+2} . \end{cases}\] + +\[f\left( \frac{1}{n_{0}+2} \right)=1- \frac{2\eta}{n_{0}+2} + \frac{\sigma^{2}+\eta^{2}}{(n_{0}+2)^{2}} =\left(1- \frac{\eta}{n_{0}+2} \right)^{2}+ \frac{\sigma^{2}}{(n_{0}+2)^{2}} ,\] + +which concludes the proof. Finally, (4.17) is a straightforward consequence of (4.15) and (4.16). + +- + +Remark 4.4 Theorem 4.3 implies the stability of the algorithm for any \(z\). Certainly, one might be concerned about the case where \(0<|\sigma|\ll|z|\) due to the factor \(|z|||\sigma^{-1}|\). Several key observations can be made in this situation. First, if \(\sigma\) is very small, we can incorporate this term into the function \(f\) and shift all calculations to the purely oscillatory integral case. On the other hand, if the second-phase algorithm is applied in its current form, we note that the estimate of the norm \(|(\mathrm{I}_{n}+ \frac{1}{2} z\mathrm{M}_{\boldsymbol{n}})^{-1}|2\) depends on the distance of \(z\) from the discrete set of eigenvalues of \(\mathrm{M}\boldsymbol{n}\). In practice, this distance is often larger than the pessimistic estimate derived in the proof. Therefore, the condition number of the matrix is, in practice, better than the bound established in the theorem for this case. Furthermore, a small increase or decrease in \(n_{0}\) by a few units can resolve any potential ill-conditioning issues. Finally, we emphasize that our computations never exhibited numerical instability, even in cases where the theorem suggests potential issues. + +- diff --git a/outputs/arxiv_math/2503.08172_pg25_pg1_repeat1.md b/outputs/arxiv_math/2503.08172_pg25_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..165d9712b76f50d9d6a18591fcc9088fae7367f4 --- /dev/null +++ b/outputs/arxiv_math/2503.08172_pg25_pg1_repeat1.md @@ -0,0 +1,39 @@ +Therefore, assuming that the first child of the root is \(j\in\{1,\ldots,d\}\) (equivalently, that the + +minimal label of \(\sigma^{(j)}⊔\bigsqcup_{l\neq j}\sigma^{(l)}\) belongs to the \(j\)th component), and using this counting principle, one get the recursive formula + +\[d(\emptyset,\sigma)=\left(\sum_{j=1}^{d} \frac{|\sigma^{(j)}}{|\sigma|-} \begin{vmatrix} \right) \\ \bigg( \\ w_{\emptyset,j} & (1) \\ 1 & \sigma \end{vmatrix} ||^{\sigma},\ldots^{|-},^{1}|\sigma^{(d)}|\bigg)\prod_{k=1}^{d}d_{k}(\emptyset,\sigma^{(k)}),\] (4.17) + +where \(d_{k}\) denotes the combinatorial dimension in which the weights \(w_{i,ij}\) are replaced by \(w_{ki,kij}\), and \(\left(_{k_{1},\cdots ,k_{d}}^{n}\right)\) denotes the usual multinomial coefficient. Thereafter, the proof of the lemma follows by induction. + +By using Lemma 4.1, we obtain results similar to those in Theorem 3.1. The proof follows from simple calculations and is omitted. + +Corollary 4.3. The normalized extremal NNHFs of this WBD take the form + +\[\varphi(\tau)=\prod_{i\in\tau}\alpha_{i}\prod_{i\in\tau^{\circ}} \frac{\alpha_{i}}{\sum_{j=1}^{d}\alpha_{ij}w_{i,ij}} ,\] (4.18) + +with \(\alpha_{\emptyset}=1\text{and}\alpha_{i}=\alpha_{i1}+\ldots+\alpha_{id}\in[0,1]\) for any \(i\in\mathcal{S}_{d}\). Again, the \(\alpha_{i}\) represents the asymptotic proportion of descendants of \(i\) in a regular path. Besides, the corresponding saturated ergodic central Markov kernel is then given by + +\[p(\tau,\tau⊔\{ij\})= \begin{cases} \underline{\alpha_{ij}w_{i,ij}} \\ \overline{\sum^{d}} & \alpha_{i}, & ifi\in\partial\tau, \\ \overline{\alpha_{k=1ik}w_{i,ik}} \\ \alpha_{ij}, & ifi\not\in \partial\tau\text{and}ij\not\in \tau, \end{cases}\] (4.19) + +for any \(i\in\tau\text{and}j\in\{1,\ldots,d\}\). As a consequence, we obtain the following generalization of Han’s hook length formula (4.3). Corollary 4.4. Let \(\mathcal{B}_{d}(n)\) be the set of all \(d-\)ary trees with \(n\) vertices. Then + +\[\begin{aligned} \sum_{\tau\in\mathcal{B}_{d}(n)} \frac{\prod_{\tau^{\circ}} \frac{\sum_{j=1}^{d}|\tau^{(vj)}|w_{j}}{|\tau^{(v)}|-1}}{|\tau^{(v)}|d^{|\tau^{(v)}|-1}} \left( \frac{w_{1}+\cdots +w_{d}}{d} \right)^{|\partial\tau|}&= \frac{1}{n!} \left( \frac{w_{1}+\cdots +w_{d}}{d} \right)^{n}. \\ \prod_{v\in\tau}^{v\in} \end{aligned}\] (4.20) + +Proof. Consider the uniform fragmentation measure case as in Section 4.1, that is \(\alpha_{i}=d^{-|i|}\) + +and \(w_{i,ij}\equiv w_{j}\text{for all}i\in\mathcal{S}_{d}\text{and}1\leq j\leq d\). By using (4.2) and by noting that + +\[\prod_{i\in\tau^{\circ}} \frac{\alpha_{i}}{\sum_{j=1}^{d}\alpha_{ij}w_{i,ij}} =\left( \frac{d}{\sum_{j=1}^{d}w_{j}} \right)^{|\tau|-|\partial\tau|},\] (4.21) + +| αi | | d | +| --- | --- | --- | +| Y Pd j=1 αijwi,ij | = | !\|τ\|−\|∂τ\| , Pd (4.21) j=1 wj | + +we deduce the result from (4.18) and the general combinatorial identity (7.5). + +As an example, when \(d=2\text{and}n=3\), we obtain + +\[\frac{1}{6} \left( \frac{w_{1}+w_{2}}{2} \right)^{3}= \frac{1}{12} \left( \frac{w_{1}+w_{2}}{2} \right)^{3}+ \frac{1}{48} \sum_{1\leq i,j\leq2}w_{i}w_{j}\left( \frac{w_{1}+w_{2}}{2} \right).\] + +It is much more difficult to describe the MERWs in full generality. However, drawing on the previous situation, one can construct them using appropriate IDLA-like random walks under some additional assumptions. diff --git a/outputs/arxiv_math/2503.08176_pg20_pg1_repeat1.md b/outputs/arxiv_math/2503.08176_pg20_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..828585be1b6fac58f26976752a79237b10136022 --- /dev/null +++ b/outputs/arxiv_math/2503.08176_pg20_pg1_repeat1.md @@ -0,0 +1,33 @@ +But the emptiness of the regions \(\mathcal{T}(3,2,k_{3})\text{for}k_{3}\geqslant3\) follows from lemma 3(3c). Therefore„ \(\mathcal{A}_{3}^{\circ}(3,1,2)\) consists of the triples (23). Further, if \((k_{1},k_{2},k_{3})\in\mathcal{A}_{3}^{*}(3,1,2)\), then we necessarily have \((k_{1},k_{2})\in\mathcal{A}_{2}^{*}(3,1,2)\). As it was stated before, we either have \(k_{1}\equiv0\pmod{3},k_{1}\geqslant3,k_{2}=1,\text{or}k_{1}=3,k_{2}=2\). For \(k_{2}=1\), the condition (19) corresponding to \(i=r=3\), reduces to + +\[k_{1}(k_{3}-1)\not\equiv 0\pmod{3}.\] + +Since \(k_{1}\equiv0\) (mod 3) in the first case, this condition can not be satisfied for any \(k_{3}\). In the second case, the same condition takes the form + +\[7k_{3}\not\equiv 0\pmod{3}.\] + +For \(k_{3}\geqslant3\), the emptiness of \(\mathcal{T}(3,2,k_{3})\) follows from lemma 3 (3c). Therefore, non-empty regions correspond to the values \(k_{3}=1,k_{3}=2\) only. Hence, the set \(\mathcal{A}_{3}^{*}(3,1,2)\) consists of the triples + +\[(3,2,1),(3,2,2).\] (24) + +It turns out that for any \(r\geqslant4\) the set \(\mathcal{A}_{r}^{*}(3,1,2)\) consists of two sets: \((3,2^{r-2},1)\) and \((3,2^{r-1})\). Let us prove it by induction with respect to \(r\). For \(r=3\), it was proved earlier. Suppose now that this assertion holds for any \(r\), \(3\leqslant r\leqslant m-1\), and check it for \(r=m\). Indeed, let \((k_{1},\ldots,k_{m})\in\mathcal{A}_{m}^{*}(3,1,2).\text{Then}(k_{1},\ldots,k_{m-1})\in\mathcal{A}_{m-1}^{*}(3,1,2)\), so, by the induction, \((k_{1},\ldots,k_{m-1})\) is either \((3,2^{m-3},1)\text{or}(3,2^{m-2})\). Hence, \((k_{1},\ldots,k_{m})\) has the form \((3,2^{m-3},1,k)\text{or}(3,2^{m-2},k),\text{where}k\geqslant1\) is some integer. In the first case, the condition (19) corresponding to \(i=m\), has the form + +\[\mathbb{K}_{m}(3,2^{m-3},1,k)+\mathbb{K}_{m-1}(2^{m-3},1,k)\not\equiv 2\pmod{3}.\] (25) + +By explicit formulas for the continuants, we get + +\[\mathbb{K}_{m}(3,2^{m-3},1,k)=2(k-m)+3,\mathbb{K}_{m-1}(2^{m-3},1,k)=k-m+2,\] + +so (25) has the form \(3(k-m)+5\not\equiv 2\) (mod 3). Therefore, it is not satisfied for any \(k\). In the second case, we have the condition + +Using the formulas + +\[\mathbb{K}_{m}(3,2^{m-2},k)+\mathbb{K}_{m-1}(2^{m-2},k)\not\equiv 2\pmod{3}.\] + +\[\mathbb{K}_{m}(3,2^{m-2},k)=2km-k-2m+3,\mathbb{K}_{m}(2^{m-2},k)=km-k-m+2,\] + +we rewrite it as follows + +\[3m(k-1)-2k+5\not\equiv 2\pmod{3},\text{that is},k\not\equiv 0\pmod{3}.\] + +Now let us find all the tuples that contain the set \(\mathcal{A}_{r}^{\circ}(3,1,2)\text{for}r\geqslant4\). diff --git a/outputs/arxiv_math/2503.08178_pg15_pg1_repeat1.md b/outputs/arxiv_math/2503.08178_pg15_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..622418e9b138d9aded1bde61d4ce9eef9184c989 --- /dev/null +++ b/outputs/arxiv_math/2503.08178_pg15_pg1_repeat1.md @@ -0,0 +1,9 @@ +\(v_{c_{r}}\) be two nodes in \(G(A(H))\) connected by a path \(P=(v_{c_{1}},\ldots,v_{c_{r}})\text{with}r\geq2\). We add a new hyperplane \(h\) to \(A(H)\). Let \(A_{i}\text{and}A_{i+1}\) be the corresponding arrangement before and after \(h\) is added, respectively. For each cell \(c_{j}\text{in}A_{i}\), we denote the corresponding cell in \(A_{i+1}\) by \(\bar{c}_{j}.\text{If}c_{j}\neq \bar{c}_{j}\), we denote the two connected cells that bisect \(\bar{c}_{j}\) by \(c_{j}^{\prime}\text{and}c_{j}^{\prime\prime}\) , respectively. We show that there exists a \((v_{\bar{c}_{1}},v_{\bar{c}_{2}})-\text{path}P^{\prime}\text{in}G(A_{i+1})\). First, let \(c_{1}=\bar{c}_{1}\). If \(c_{2}=\bar{c}_{2}\), we can choose \(P^{\prime}=(v_{c_{1}},v_{c_{2}}).\text{If}c_{2}\neq \bar{c}_{2}\), the connectivity of \(c_{2}^{\prime}\text{and}c_{2}^{\prime\prime}\) ensures that we can choose \(P^{\prime}\in\left\{(v_{c_{1}},v_{c_{2}^{\prime}}),(v_{c_{1}},v_{c_{2}^{\prime\prime}})\right\}\). Second, let \(c_{1}\neq \bar{c}_{1}.\text{As}c_{1}\text{and}c_{2}\) are connected in \(G(A_{i})\), it follows that \(c_{1}^{\prime}\text{or}c^{\prime\prime} P^{\prime}\in\left\{(v_{c_{1}^{\prime}},v_{c_{2}}),(v_{c_{1}^{\prime}},v_{c_{2}^{\prime}}),(v_{c_{1}^{\prime}},v_{c_{2}^{\prime\prime}})\right\}\) depending on whether \(c_{2}\) is divided by \(h\) or not. In the second case, we can replace \(c_{1}^{\prime}\) by \(c_{1}^{\prime\prime}\) and obtain the sought path. Inductively, there exists a \((v_{\bar{c}_{1}},v_{\bar{c}_{r}})-\text{path}P^{\prime}\text{in}G(A_{i+1})\) and the claim follows. + +1 is connected to \(\bar{c}_{2}\text{in}G(A_{i+1})\). In the first case, we can choose + +The graph connectivity ensures that, starting from any node \(v_{c^{\prime}}\), all nodes \(v_{c}\) in \(G(A(H^{=}))\) can be reached using a breadth-first search. The choice of the incidence graph \(G(A(H^{=}))\) prevents us from reaching nodes that do not belong to a cell of \(A(H^{=})\). The incidence graph \(G(A(H^{=}))\) can be easily derived from the incidence lattice \(G^{\prime}(A(H^{=}))\) constructed in the algorithm of Edelsbrunner et al. (1986) to compute the arrangement \(A(H^{=})\). If two cells \(c_{1}\text{and}c_{2}\) share a common face \(f\) of dimension \(p-1\), there exists a node \(v_{f}^{\prime}\text{in}G^{\prime}(A(H^{=}))\) and an edge from \(v_{f}^{\prime}\) to each of the nodes \(v_{c_{1}}^{\prime}\text{and}v_{c_{2}}^{\prime}\) belonging to the cells \(c_{1}\text{and}c_{2}.\text{In}G(A(H^{=}))\), there is an edge between \(v_{c_{1}}\text{and}v_{c_{2}}\) and we store the information to which separating hyperplane \(h(e,f)\) this edge belongs. This requires constant time per face of dimension \(p-1\) and, thus, the graph \(G(A(H^{=}))\) can be obtained in time linear in the number of faces of dimension \(p-1\) of \(A(H^{=})\) from \(G^{\prime}(A(H^{=}))\). Of course, we store the representative point from the + +relative interior of each cell. We are now ready to state our improved algorithm. + +Theorem 3.6. Let \(p\geq2\). Algorithm 2 solves the \(p-\)parametric matroid problem in time \(O(m^{2p}f(m))\). diff --git a/outputs/arxiv_math/2503.08185_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.08185_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..c912d85d1d6684dba316095a2f21311348b71bdf --- /dev/null +++ b/outputs/arxiv_math/2503.08185_pg2_pg1_repeat1.md @@ -0,0 +1,31 @@ +where Ent\((_{\pi}f^{2})=\mathbb{E}_{\pi}[f^{2}\log f^{2}]-\mathbb{E}_{\pi}[f^{2}]\log\mathbb{E}_{\pi}[f^{2}]\) is the entropy of \(f^{2},\text{and where}\mathcal{E}_{P}(f,f)\) is the Dirichlet form defined by + +\[\mathcal{E}_{P}(f,f)=\sum_{x,y\in\Omega}\pi(x)P(x,y)(f(x)-f(y))f(x).\] + +A classical result is that + +\[t_{\text{mix}}(P,\varepsilon)\leq\left\lceil\mathrm{C}_{\text{LS}}(P)\left(\log\log\left( \frac{1}{\pi_{\star}} \right)+2\log\left( \frac{1}{\varepsilon} \right)\right)\right\rceil,\] + +\[\frac{1}{2^{n2}} ,\] + +where \(\pi_{\star}=\min_{x\in\Omega}\pi(x)\) (see for instance [5] and [11]). Since in our case we have \(\pi_{\star}≍^{1}\) we see that Theorem 1 will follow from the following proposition. + +## Proposition 2. The logarithmic Sobolev constant of \(P\) satisfies + +\[\mathrm{C}_{\text{LS}}(P)≲n^{2}.\] + +In the above proposition and in the rest of the paper, the notation \(u_{n}≲v_{n}\) means that there exists an absolute constant \(c>0\) such that for all \(n\geq1\), we have \(u_{n}\leq cv_{n}\). Interestingly, a crucial ingredient in the proof of Proposition 2 is the bound of \(O(n)\) on the Poincaré constant (inverse of the spectral gap), obtained by Kassabov [9]. Remark 1. Proposition 2 even yields an upper bound of \(O(n^{2}\log n)\) for the \(\ell^{2}-\)mixing time, at least for the continuous-time and lazy versions of the chain. + +We also study another chain on \(\Omega\), whose kernel is denoted by \(Q\), and whose dynamics is as follows: if the current state is a matrix \(x\in\Omega\) with rows \(\ell_{1},\ldots,\ell_{n}\), then an index \(i\) between 1 and \(n\) is chosen uniformly at random, and \(\text{row}\ell_{i}\) is replaced by \(\ell_{i}+\sum_{j\neq i}a_{j}\ell_{j}\), wit \((a_{j})_{j\neq i}\) independent Bernoulli random variables with parameter \(1/2\). In other words, the chain \(Q\) randomizes a uniformly chosen row, conditionally on the event that the resulting matrix is invertible (it is the Gibbs sampler for the joint distributions of rows). The following proposition shows that the mixing time of \(Q\) lies between \(n\log n\text{and}4n\log n\) whatever the value of the threshold \(\varepsilon\), establishing a pre-cutoff. + +## Proposition 3. For all \(\varepsilon\in(0,1)\), there exists a constant \(c_{\varepsilon}>0\) such that + +\[n\log n-c_{\varepsilon}n\leq t_{\text{mix}}(Q,\varepsilon)\leq4n\log n-2n\log\log n+c_{\varepsilon}n.\] + +In particular, the chain \(Q\) has a pre-cutoff. + +## 2. Related work + +To our knowledge, the chain \(P\) was first studied by Diaconis and Saloff-Coste [6], who used a comparison with another chain on \(\Omega\) studied by Hildebrand [8] to get an upper bound of \(O(n^{2})\) on the relaxation time (the inverse of the spectral gap), implying an upper bound of \(O(n^{4})\) on the \(\ell^{2}-\)mixing time. Then the results of Kassabov [9] on the Kazhdan constant yield an upper bound of \(O(n^{3})\) on the \(\ell^{2}-\)mixing time, which was the best known upper bound for total variation mixing as well. As for the lower bound, a simple counting argument shows that the total-variation mixing time can be lower bounded by \(\Omega\) (which is actually an estimate of the diameter of the underlying graph, see Andrén et al. + +\[\left( \frac{n^{2}}{\log n} \right)\] diff --git a/outputs/arxiv_math/2503.08186_pg7_pg1_repeat1.md b/outputs/arxiv_math/2503.08186_pg7_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..0acfb34639b6911e37452afa46e4500ad208b8cd --- /dev/null +++ b/outputs/arxiv_math/2503.08186_pg7_pg1_repeat1.md @@ -0,0 +1,43 @@ +with \(C^{\prime\prime}:=C_{\mathrm{N}\text{ash},d}^{-2}|||E|||^{-2- \frac{4}{d}}\) , and finally + +\[\frac{1}{2} \frac{d}{dt} \|\Gamma_{\Omega}(t,\cdot,y)\|_{\mathrm{L}^{2}(\Omega)}^{2}\leq-C^{\prime\prime}\|\Gamma_{\Omega}(t,\cdot,y)\|_{\mathrm{L}^{2}\Omega)}^{2+ \frac{4}{(^{d}}}+\|\Gamma_{\Omega}(t,\cdot,y)\|_{\mathrm{L}^{2}(\Omega)}^{2}.\] + +\[\left(\|\Gamma_{\Omega}(t,\cdot,y)\|_{\mathrm{L}^{2}(\Omega)}^{2}\right)^{- \frac{2}{d}}\] + +\[q^{\prime}(t)\geq \frac{4}{d} (C^{\prime\prime}-q(t)),\] + +which implies + +so that in the end + +\[q(t)\geq C^{\prime\prime} \frac{4}{d} t\mathrm{e}^{- \frac{4}{d} t},\] + +\[\|\Gamma_{\Omega}(t,\cdot,y)\|_{\mathrm{L}^{2}(\Omega)}^{2}\leq\left(C^{\prime\prime} \frac{4}{d} \right)^{- \frac{d}{2}}\mathrm{e}^{2T}t^{- \frac{d}{2}}.\] + +This yields the claimed L2 bound. + +For the \(\mathrm{L}^{\infty}\) bound, note first that the Laplace operator is self-adjoint, so that \(\Gamma_{\Omega}(t,x,y)= \Gamma_{\Omega}(t,y,x)\). By the semigroup property, we therefore find that + +\[\begin{aligned} \Gamma_{\Omega}(t,x,y)&=\int_{\Omega}\Gamma_{\Omega}(t/2,x,z)\Gamma_{\Omega}(t/2,z,y)dz \\ &\leq\|\Gamma_{\Omega}(t/2,x,\cdot)\|_{\mathrm{L}^{2}(\Omega)}\|\Gamma_{\Omega}(t/2,\cdot,y)\|_{\mathrm{L}^{2}(\Omega)} \\ &\leq\|\Gamma_{\Omega}(t/2,\cdot,x)\|_{\mathrm{L}^{2}(\Omega)}\|\Gamma_{\Omega}(t/2,\cdot,y)\|_{\mathrm{L}^{2}(\Omega)}, \end{aligned}\] + +and we use the already established L2 bound to conclude. A direct interpolation between the L1 and \(\mathrm{L}^{\infty}\) estimates yields the \(\mathrm{L}^{p}\) estimate stated in the Lemma. Finally, the self-adjointness argument leads to the second group of estimates. + + + +In the context of the domains appearing in the proof of Proposition 5 (but keeping the notations of Lemma 11), we write down for the related geometry the following statement: Lemma 12. Fix \(T>0\). Then there exists a constant \(C_{T,d}^{*}\) such that for any domain + +\[\Omega:=B_{d}(0,R)\cap\{(x^{\prime},x_{d})\in\mathbb{R}^{d}:x_{d}>\phi(x^{\prime})\}\] + +for \(R>0\text{and}\phi:\mathbb{R}^{d-1}\rightarrow\mathbb{R}\text{with}\|\nabla\phi\|_{\infty}\leq \frac{1}{11d} \text{with}\partial\Omega_{n}=B_{d}(0,R)\cap\{x^{\prime},x_{d})\in\mathbb{R}^{d}:x_{d}=\phi(x^{\prime})\}\) and + +\(\partial\Omega_{d}=\partial\Omega\setminus\partial\Omega_{n}\) the constant \(C_{\Omega,T,d}\) appearing in Lemma 11 is bounded by \(C_{T,d}^{*}\). Proof. For the domain considered in the statement, we need an extension operator \(E\) for functions + +We can first extend \(f\) to \(\Omega_{+}:=\{(x^{\prime},x_{d})\in\mathbb{R}^{d}:x_{d}>\phi(x^{\prime})\}\) by setting \(E_{1}f=0\text{on}\Omega_{+}\setminus\Omega\), and + +\(f\in\mathrm{H}^{1}(\Omega)\text{such that}f|_{\partial\Omega_{d}}=0\), for the norms H1 and L1. \(E_{1}f=f\text{on}\Omega\). Recalling that \(f|_{\partial\Omega_{d}}=0\), we see that \(||E_{1}||_{H^{1}\rightarrow H^{1}}=1\text{and}||E_{1}||_{L^{1}\rightarrow L^{1}}=1\). We then use Lemma 10 (with \(\Omega\) in this lemma corresponding to \(\Omega_{+}\) here) to build the operator \(E_{2}\) from + +\(H^{1}(\Omega_{+})\) to \(H^{1}(\mathbb{R}^{d})\), so that still thanks to Lemma \(10,|||E_{2}E_{1}|||=|||E_{2}|||\leq2 \sqrt{1+ \frac{1}{121d^{2}}}\) , and we conclude the Lemma by setting \(E:=E_{2}E_{1}\). + + + +We now write down an estimate which is a direct consequence of Lemma 11, and which will be used several times in the sequel. diff --git a/outputs/arxiv_math/2503.08187_pg4_pg1_repeat1.md b/outputs/arxiv_math/2503.08187_pg4_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..c40142efb4adccde0846f887ab141de035ce8bb7 --- /dev/null +++ b/outputs/arxiv_math/2503.08187_pg4_pg1_repeat1.md @@ -0,0 +1,17 @@ +where \(\alpha>0\). The FWI problem in equation 7 ensures that the estimated model, while satisfying the data (and the wave equation), is smooth along the direction \(\theta\). Figure 1, top row, shows the anisotropic regularization balls for only one term of the regularizer equation 4 and for different values of \([\sigma]_{i}=\sigma\text{and}[\theta]_{i}=\theta\). We observe that for \(\sigma=0.5\), the anisotropic regularization becomes equivalent to the standard isotropic regularization. When the value of \(\sigma\) increases, reaching the maximum considered value of 0.9, the degree of regularization applied in the direction \(\theta\) and its normal is maximally different, resulting in a needle-shaped ellipse. This shape favors models with elongated features aligned with \(\theta\), while allowing variations in the normal direction to it, as illustrated in the bottom row. + +## 3 Algorithm + +The augmented Lagrangian function of equation 7 is + +\[\begin{aligned} \mathcal{L}(\theta,\sigma,u_{s},m,\lambda,\nu)&= \frac{1}{2} \sum_{s=1}^{n_{s}}\Vert Pu_{s}-d_{s}\Vert_{2}^{2}+\alpha\mathcal{R}(m,\theta,\sigma) \\ + \frac{\mu}{2} \sum_{s=1}^{n_{s}}\Vert A(m)u_{s}-b_{s}\Vert_{2}^{2}-\lambda_{s}^{T}(A(m)u_{s}-b_{s}) \\ + \frac{\tau}{2} \Vert\theta-z\Vert_{2}^{2}-\nu^{T}(\theta-z), \end{aligned}\] + +where \(z(\theta)=\min(\max(\theta,- \frac{\pi}{2} ), \frac{\pi}{2} ),\lambda_{s}\text{and}\nu\) are Lagrange multipliers associated to the constraints, and \(\mu>0\) and \(\tau>0\) are the penalty parameters. The regularization parameter \(\alpha>0\) is given, being typically determined by the discrepancy principle. This problem can be efficiently solved using the alternating direction method of multipliers (ADMM, [15]), resulting in the following iterative procedure starting from initial guess \(m\) (typically encoding some prior information), (spatially invariant) \(\sigma=0.5\) and initial multipliers \(\lambda_{s}=\nu=0\): + +\[\begin{aligned} \theta&=\operatorname*{arg\,min}_{\theta}^{+}\mathcal{L}(\theta,\sigma,u_{s},m,\lambda,\nu) \\ \sigma&=\operatorname*{arg\,min}_{\sigma}^{+}\mathcal{L}(\theta^{+},\sigma,u_{s},m,\lambda,\nu) \\ u_{s}&=\operatorname*{arg\,min}_{u}^{+}\mathcal{L}(\theta^{+},\sigma^{+},u,m,\lambda,\nu) \\ m&=\operatorname*{arg\,min}_{m}^{+}\mathcal{L}(\theta^{+},\sigma^{+},u_{s}^{+},m,\lambda,\nu) \\ \lambda_{s}^{+}&=\lambda_{s}-\mu(A(m^{+})u_{s}^{+}-b_{s}) \\ \nu^{+}&=\nu-\tau(\theta^{+}-z^{+}). \end{aligned}\] (8a) (8b) (8c) (8d) (8e) (8f) + +Here, the updated variables at each iteration are denoted by a superscript “+”. Each step of the algorithm addresses a specific subproblem, as described more in details below. Subproblem equation 8a involves minimizing the regularization function \(\mathcal{R}\), as defined in equation 4, with respect to \(\theta\), given the current values of \(m\text{and}\sigma\). This can can be efficiently achieved using a single iteration of the Gauss-Newton method. To ensure the stability of the \(\theta\) update, it is essential to incorporate a smoothing term, as suggested by [11]. Subproblem equation 8b involves minimizing \(\mathcal{R}\) with respect to \(\sigma\), resulting in: + +\[[\sigma]_{i}= \frac{[g_{z^{\prime}}]_{i}^{2}}{[g_{x^{\prime}}]_{i}^{2}+[g_{z^{\prime}}]_{i}^{2}} ,\] (9) + +where \(([g_{x^{\prime}}]_{i},[g_{z^{\prime}}]_{i})^{T}=R([\theta]_{i})[\nabla m]_{i}\) represent the rotated gradient of the current model at the \(i\)th pixel. This expression for equation 9 is justified by considering that, when \([\theta]_{i}\) corresponds to the correct orientation angle, we typically expect \(|[g_{x^{\prime}}]_{i}|\ll|[g_{z^{\prime}}]_{i}|\). In this case, directly minimizing \([g_{x^{\prime}}]_{i}^{2}+[g_{z^{\prime}}]_{i}^{2}\) would heavily penalize \([g_{z^{\prime}}]_{i}\), which is undesirable because we aim to apply more smoothing along the direction defined by \([\theta]_{i}\). The weights \([\sigma]_{i}\) in equation 9 balance the contributions of gradient components in the weighted sum \([\sigma]_{i}^{2}[g_{x^{\prime}}]_{i}^{2}+(1-[\sigma]_{i})^{2}[g_{z^{\prime}}]_{i}^{2}\) , ensuring that the magnitudes of both terms are approximately equal before penalization. This adaptive weighting mechanism effectively enhances smoothing along the structural direction \([\theta]_{i}\) while allowing variations across it, eventually enabling the recovery of dominating structures and details in the model \(m\). To maintain stability during the \(\sigma\) update, the computed weights are smoothed using a convolution with a \(5\times5\) averaging filter; alternatively, one could incorporate a smoothing term. A comprehensive analysis of the methods for solving subproblems equation 8c and equation 8d can be found in [14]. diff --git a/outputs/arxiv_math/2503.08206_pg4_pg1_repeat1.md b/outputs/arxiv_math/2503.08206_pg4_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..adddcae7c9d55d60c18af0066cd777802baa1c89 --- /dev/null +++ b/outputs/arxiv_math/2503.08206_pg4_pg1_repeat1.md @@ -0,0 +1,25 @@ +such that + +\[|B(x)-2B_{0}^{+}(x)|\leq C^{+}\text{and}|W(x)-2B_{0}^{-}(x)|\leq C^{-}\] + +uniformly for all irrational numbers \(x\). + +The theorem can be deduced from the work of Wilton, by a modification of his application of Voronoi’s summation formula, but the proof we present here will be elementary in the sense that it relies on no complex analysis, and in fact only on elementary properties of \(\alpha-\)continued fractions as established, e.g., in [LMNN10]. Since the odd part \(B^{-}\) and the even part \(W^{+}\) of the Brjuno respectively Wilton functions are uniformly bounded functions — see Proposition 2.1 for closed form expressions — the functions of interest following Theorem 1.1 are the differences + +\[\Delta^{+}(x):=B^{+}(x)-2B_{0}^{+}(x)\text{and}\Delta^{-}(x):=W^{-}(x)-2B_{0}^{-}(x).\] + +Numerical computations show that \(\Delta^{\pm}\) behave widely differently. + +![](figure) + +![](figure) + +![](figure) + +Figure 3. Plots of \(\Delta^{+}\text{and}\Delta^{-}\) at 10000 randomly chosen points + +In [LM21, Theorem 1], the second and third authors established that \(\Delta^{+}(x)\) extends to a 12-Hölder continuous function on the real line. This is in stark contrast to what one observes in the graphical plots of \(\Delta^{-}(x)\), which seems to indicate that \(\Delta^{-}(x)\) is discontinuous at each rational with a jump of order \(2/q\text{for}x=p/q\). In that respect, it is reminiscent of the ‘popcorn function’ + +\[f(x)= \begin{cases} 1/q & \text{if}x=p/q\text{with}(p,q)=1; \\ 0 & \text{if}x\text{is irrational}; \end{cases}\] + +that staple of calculus illustrating the existence of a Riemann integrable function with infinitely many discontinuities. We confirm these numerical observations with the following theorem. Theorem 1.2. The function \(\Delta^{-}(x)\) is continuous on \((0, \frac{1}{2} )\setminus\mathbf{Q}\). At each rational \(x=p/q\), it has an increasing jump of \(2/q\). In particular \(\Delta^{-}(x)\) is Riemann integrable and nowhere differentiable. diff --git a/outputs/arxiv_math/2503.08227_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.08227_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..f70576a078047e7bbef3b98ccb751b5911602415 --- /dev/null +++ b/outputs/arxiv_math/2503.08227_pg2_pg1_repeat1.md @@ -0,0 +1,15 @@ +![](figure) + +![](figure) + +Figure 1: Example of a domain reduction using known internal symmetries. + +on the border \(\partial\Omega[1]\), having for instance the form + +\[a_{ij} \frac{\partial^{2}f}{\partial x_{i}\partial x_{j}} =g,\] (1) + +where \(a_{ij}\text{and}g\) are known functions of \(\vec{r}(x_{1},x_{2},x_{3})\text{and}f(\vec{r})\) is the function searched on \(\Omega\). Usual solving methods are the finite element (FEM) [2] and the finite difference (FDM) [3]. They both rely on the discretization of space forming a mesh of \(N\) points \(M_{n},n=1,...,N\) covering \(\Omega\) and the linearization of the PDE to solve. The methods lead to the build up of a linear problem + +\[AX=b,\] (2) + +where \(X\) is the unknown vector of component \(f(M_{n}),n=1,...,N,b\) is a constant vector defined by specific values given on \(\Omega\text{and}\partial\Omega\text{and}A\) is a square matrix of \(\text{rank}N\). The boundary conditions (BC) on \(\partial\Omega\) are usually of Dirichlet type \((\mathcal{D}),\text{when}f\) is known, or Neumann \((\mathcal{N}),\text{when}\partial f/\partial\vec{n}\) is known \((\vec{n}\) normal to \(\partial\Omega)\), or a combination of the two. When \(\Omega\) is large or the need for accuracy is high, \(N\) becomes rapidly very large \((N\gg10^{6})\) and the computer solving and the numerical storing of the inverse matrix \(A^{-1}\), solution of Eq. 2, can become problematic. It is generally of interest to reduce the problem size by considering all the symmetries applying both on \(\Omega,\partial\Omega\text{and}g\). An example of such a domain reduction from \(\Omega\) down to diff --git a/outputs/arxiv_math/2503.08235_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.08235_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..a1cf351f8bf00cf4f181643923cb7d3e740527c6 --- /dev/null +++ b/outputs/arxiv_math/2503.08235_pg3_pg1_repeat1.md @@ -0,0 +1,25 @@ +## 2. Multiparameter quantum estimation: precision, sloppiness and incompatibility + +In this section, we provide the theoretical framework, definitions and metrics used throughout the paper. We consider finite dimensional systems and a family of quantum states \(\rho_{\boldsymbol{\lambda}}\) encoding the values of \(d\) real parameters, denoted as a vector \(\boldsymbol{\lambda}=(\lambda_{0},\lambda_{1},\ldots,\lambda_{d})^{T}\) . If we perform a positive operator-valued measurement (POVM) \(\boldsymbol{\Pi}\) with elements \(\{\Pi_{k}\}\) satisfying \(\Pi_{k}\geq0\text{and}\sum_{k}\Pi_{k}=\mathbb{I}\), the measurement outcome \(k\) is obtained with probability \(p_{\boldsymbol{\lambda}}(k)=\text{Tr}[\rho_{\boldsymbol{\lambda}}\Pi_{k}]\). The estimator function based on the result is denoted as \(\hat{\boldsymbol{\lambda}}(k)\). The performance of the estimator is assessed by the covariance matrix \(\mathbf{V}(\hat{\boldsymbol{\lambda}})\) with elements + +\[V_{\mu\nu}=\sum_{k}p_{\boldsymbol{\lambda}}(k)[\hat{\lambda}_{\mu}(k)-E_{k}(\hat{\lambda}_{\nu})][\hat{\lambda}_{\nu}(k)-E_{k}(\hat{\lambda}_{\nu})].\] + +where \(E_{k}(\hat{\lambda}_{\mu})\) is the expectation value of \(\hat{\lambda}_{\mu}\) over the probability distribution \(p_{\boldsymbol{\lambda}}(k)\). In classical multiparameter estimation, when the estimators satisfying the locally unbiased conditions: + +\[\boldsymbol{E}_{\nu}(\hat{\boldsymbol{\lambda}})=\hat{\boldsymbol{\lambda}}\partial_{\mu}E_{k}(\hat{\lambda}_{\nu})=\delta_{\mu}^{\nu},\] + +where \(\partial_{\mu}= \frac{\partial}{\partial\lambda_{\mu}}\) , then the CRB [51] holds + +\[\mathbf{V}(\hat{\boldsymbol{\lambda}})\geq \frac{1}{M\mathbf{F}} ,\] + +where \(M\) is the number of repeated measurements and F is the FI matrix with elements defined by + +\[F_{\mu\nu}=\sum_{k}p_{\boldsymbol{\lambda}}(k)\partial_{\mu}\log p_{\boldsymbol{\lambda}}(k)\partial_{\nu}\log p_{\boldsymbol{\lambda}}(k)=\sum_{k} \frac{\partial_{\mu}p_{\boldsymbol{\lambda}}(k)\partial_{\nu}p_{\boldsymbol{\lambda}}(k)}{p_{\boldsymbol{\lambda}}(k)} .\] + +The CRB can be saturated in the asymptotic limit of an infinite number of repeated experiments using Bayesian or maximum likelihood estimators [52]. Due to the non-commutativity of the operators on \(\mathcal{H}\), the quantum analogue of the FI cannot be uniquely introduced. In fact, there exist several different definitions of quantum Fisher information. The most celebrated and useful approaches are based on the so-called symmetric logarithmic derivative (SLD) operators \(L_{\mu}^{S}[53]\) and right logarithmic derivative (RLD) operators \(L_{\mu}^{R}[54,55]\), defined as follows + +\[\begin{aligned} \partial_{\mu}\rho_{\boldsymbol{\lambda}}&= \frac{L_{\mu}^{S}\rho_{\boldsymbol{\lambda}}+\rho_{\boldsymbol{\lambda}}L_{\mu}^{S}}{2} , \\ \partial_{\mu}\rho_{\boldsymbol{\lambda}}&=\rho_{\boldsymbol{\lambda}}L_{\mu}^{R}. \end{aligned}\] + +We denote the corresponding SLD and RLD quantum Fisher information matrices (QFIM) as Q and J, respectively, with elements + +\[\begin{aligned} Q_{\mu\nu}&= \frac{1}{2} \text{Tr}\left[\rho_{\boldsymbol{\lambda}}\{L_{\mu}^{S},L_{\nu}^{S}\}\right], \\ J_{\mu\nu}&=\text{Tr}\left[\rho_{\boldsymbol{\lambda}}L_{\mu}^{R}L_{\nu}^{R\dagger}\right]. \end{aligned}\] diff --git a/outputs/arxiv_math/2503.08240_pg9_pg1_repeat1.md b/outputs/arxiv_math/2503.08240_pg9_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..1d3ce7bebc765f5620a6fca42377d4304401a636 --- /dev/null +++ b/outputs/arxiv_math/2503.08240_pg9_pg1_repeat1.md @@ -0,0 +1,23 @@ +in Figure 1a indicate the expectation a random vector will lie in the tangent space. The expectation is approximately \(\sqrt{n/d} ,\text{where}n\text{and}d\) are the dimensions of the tangent space approximation and manifold, respectively [9]. An explanation is therefore sufficiently aligned with the tangent space when that fraction in the tangent space is greater than \(\sqrt{n/d}\). We see in Figure 1a that standard base-point choices on CIFAR10 are significantly below the vertical line. It is the goal of future work to determine if the dimension of the tangent space of CIFAR10 of \(n=144\) or the parameter of the Gaussian base-point impacts the tangential alignment of IG on CIFAR10. We provide in Figure 2, example integrated gradient explanations for a point on MNIST32, FER2013, and Fashion-MNIST with differing base-point choice. We see that our method provides tangentially aligned explanations with \(\mu_{x}>0.91\) for all datasets. The tangentially aligned integrated gradient attributions are clear and perceptually aligned with the object to classify in the image. We see in Figure 2 that uniform, maximum \(\ell_{2}\) distance, and Gaussian are consistently random noise. + +### 4.5. Comparison of Gradient Explainability Models with Tangential Integrated + +# Gradients + +In this section we compare tangentially aligned integrated gradients with three common gradient explainability models: Gradient, Smooth Grad and Input *Gradient. The aforementioned gradient explainability models do not require a base-point choice. We demonstrate that tangentially aligned integrated gradients significantly improves upon integrated gradients. The gradient explainability models for a given model are defined as follows: + +## 1. Gradient The gradient of a model \(f\) at \(x\in\mathbb{R}^{d}\) for class \(i\) is defined as: + +\[\text{grad}(\mathrm{x})_{\mathrm{i}}:= \frac{\partial\mathrm{f}(\mathrm{x})_{\mathrm{i}}}{\partial\mathrm{x}} .\] (33) + +## 2. Smooth Grad We define Smooth Grad with \(n\) samples and standard deviation \(\sigma\) as: + +\[\text{SmoothGrad}(x)= \frac{1}{n} \sum_{i=1}^{n}\nabla f(x+a),\] (34) + +where, \(a\sim\mathcal{\mathcal{N}}(0,\sigma^{2})\). Following [9] we take \(\sigma=0.02\text{and}n=25\). + +## 3. Input*Gradient Input*Gradient is defined as: + +\[\text{Input}*\text{Gradient}:=x\odot \frac{\partial f(x)_{i}}{\partial x} .\] (35) + +In Figure 1b we provide density plots of the fraction an attribution is in the tangent space for: Gradient, Smooth Grad, Input*Gradient and tangentially aligned integrated gradients. We see in Figure 1b that tangentially aligned integrated gradients provides attributions consistently in the tangent space, out-performing the aforementioned gradient explainability models. In Figures 1a and 1b Gradient, Smooth Grad and Input*Gradient provide better tangential alignment than the common base-point choices provided in Section 2.2 on MNIST and CIFAR10. On Fashion-MNIST we see that the zero base-point choice provides comparable performance with Gradient, Smooth Grad and Input*Gradient. In Figures 1a and 1b we see that on FER2013, Gradient, Smooth Grad and Input*Gradient perform similarly to Gaussian, maximum \(\ell_{2}\) distance and zero base-point choices for integrated gradients. All gradient models on FER2013 outperform the uniform base-point choice for integrated gradients. We see in Figures 1a and 1b, Gradient, Smooth Grad, and Input*Gradient tend to out-perform Integrated gradients with standard the standard base-point choices. Tangential integrated gradients out-performs the aforementioned gradient explainability models and standard base-point choices. diff --git a/outputs/arxiv_math/2503.08244_pg9_pg1_repeat1.md b/outputs/arxiv_math/2503.08244_pg9_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..eefe5e9ffee1a44a46ff321d6b4e993b8d13ae3a --- /dev/null +++ b/outputs/arxiv_math/2503.08244_pg9_pg1_repeat1.md @@ -0,0 +1,37 @@ +single symbol \(\omega\in\Omega_{\vartheta}\), we write \(\mathbb{E}[X]=\int_{\Omega_{\vartheta}}X(\omega)d\mathbb{P}(\omega)\). Recall that a measure \(\mu\text{on}\mathbb{T}\) is called a stationary measure if + +\[\int_{\Sigma_{\vartheta}}\mu\left((T_{\omega})^{-1}(A)\right)d\mathbb{P}(\omega)=\mu(A),\] + +for any (Borel) measurable set \(A\subset\mathbb{T}\). The skew product map \(\Theta:\Sigma_{\vartheta}\times\mathbb{T}\rightarrow\Sigma_{\vartheta}\times\mathbb{T}\) is defined by + +\[\Theta(\omega,x):=(\sigma\omega,T_{\omega_{0}}(x)).\] + +Here \(\sigma\) is the left shift operator \(\sigma\omega:=(\omega_{i+1})_{i\in\mathbb{N}}\). With a slight abuse of notation we write + +\[T_{\omega}^{n}(x):=T_{\omega_{n-1}}\circ\cdots \circ T_{\omega_{0}}(x)\] + +for iterates. We compares two different trajectories by studying the random dynamical system. For \(\omega\in\Sigma_{\vartheta}\), the two-point map \((x,y)\mapsto T_{\omega}^{(2)}(x,y)\text{on}\mathbb{T}^{2}\) is the product + +\[(x,y)\mapsto(T_{\omega}(x),T_{\omega}(y)).\] + +This yields the random dynamical system + +\[(x_{n+1},y_{n+1})=T_{\omega_{n}}^{(2)}(x_{n},y_{n}).\] (2.3) + +The two-point skew product map \(\Theta^{(2)}:\Sigma_{\vartheta}\times\mathbb{T}^{2}\rightarrow\Sigma_{\vartheta}\times\mathbb{T}^{2}\) is denoted by + +\[\Theta^{(2)}(\omega,x,y)=(\sigma\omega,T_{\omega}^{(2)}(x,y)).\] + +A measure \(\mu^{(2)}\text{on}\mathbb{T}^{2}\) is a stationary measure of the random dynamical system \(T_{\omega}^{(2)}\text{on}\mathbb{T}^{2}\) if + +\[\int_{\Sigma_{\vartheta}}\mu^{(2)}\left(\left(T_{\omega}^{(2)}\right)^{-1}(A)\right)d\mathbb{P}(\omega)=\mu^{(2)}(A),\] + +for any (Borel) measurable set \(A\subset\mathbb{T}^{2}\). + +2. 1. Hypotheses. We focus on random circle endomorphisms whose trajectories are not confined to subintervals of the circle but spread over the entire circle. + +\[\text{There is}k>0\text{so that for any}x,y\in\mathbb{T},\text{there is}\omega\in\Sigma_{\vartheta}\text{so that}T_{\omega}^{k}(x)=y.(\mathbf{H}\mathbf{2})\] + +This hypothesis guarantees the existance of a unique absolutely continuous stationary measure of full support, but also has further applications that are used throughout the paper. + +Proposition 2.1. Suppose the random dynamical system described by (2.2) with \(\omega_{n}\) i.i.d. picked from a uniform distribution for \([-\vartheta,\vartheta]\), adheres to Hypotheses (H1), (H2). Then the random dynamical system admits an absolutely continuous stationary measure \(\mu\) with full support and smooth density. diff --git a/outputs/arxiv_math/2503.08261_pg14_pg1_repeat1.md b/outputs/arxiv_math/2503.08261_pg14_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..ac6eb4c86491c379af7b2704fec2b21f61ecb7ab --- /dev/null +++ b/outputs/arxiv_math/2503.08261_pg14_pg1_repeat1.md @@ -0,0 +1,31 @@ +Proof. We split the proof into two cases. Case-I: We assume \(0<\lambda<\Lambda_{\epsilon}.\) In view of Perron’s method, we consider the solutions \(\underline{u}\) and \(\overline{u}\) to the equations + +\[\begin{cases} \mathcal{M}_{\epsilon}u= \tfrac{\lambda}{u^{\delta(x)}} \text{in}\Omega, \\ u>0\text{in}\Omega\text{and}u=0\text{in}\mathbb{R}^{2}\setminus\Omega, \end{cases}\] (4.10) + +and + +\[\begin{cases} \mathcal{M}_{\epsilon}u= \tfrac{\overline{\lambda}}{u^{\delta(x)}} +h(x,u)\text{in}\Omega, \\ u>0\text{in}\Omega\text{and}u=0\text{in}\mathbb{R}^{2}\setminus\Omega, \end{cases}\] (4.11) + +respectively, where \(\overline{\lambda} \in(\lambda,\Lambda_{\epsilon}).\) For the existence of \(\underline{u}\), we refer to [25]. It is immediate to show that \(\underline{u} \text{and} \overline{u}\) are sub and super solutions of the equation \((P_{\lambda}^{\epsilon}),\) respectively. Claim: \(\underline{u} \leq \overline{u} \text{in}\Omega.\) To this concern, define a non-decreasing smooth function \(\psi:\mathbb{R}\rightarrow\mathbb{R}\) such that + +\[\psi(t)= \begin{cases} 1 & \text{if}t\geq1, \\ 0 & \text{if}t\leq0. \end{cases}\] (4.12) + +Let \(\psi_{\gamma}(t)=\psi( \frac{t}{\gamma} ),\text{for}\gamma>0.\) Incorporating \(\phi=\psi_{\gamma}( \underline{u} - \overline{u} )\) in the weak formulations of (4.10) and (4.11), then subtracting one from the other, we obtain + +\[B_{\epsilon}( \overline{u} - \underline{u} ,\phi)=\int_{\geq\lambda}\int_{\Omega}^{\Omega}\left( \frac{\overline{\lambda}}{\overline{\bigg(u}^{\delta} \frac{1^{(x)}}{\overline{u}^{\delta(}}}_{x} \frac{-}{)} \frac{\lambda}{\frac{u}{-}^{\delta} \frac{1^{(x)}}{\underline{u}^{\delta(}}}_{x} \frac{\bigg)}{)} \right)\phi_{\phi}dx+\int_{dx.^{\Omega}}h(x, \overline{u} )\phi dx\] (4.13) + +By using the monotone property of \(\psi,(4.13)\) yields + +\[\begin{aligned} \lambda\int_{\Omega}\left( \frac{1}{\overline{u}^{\delta(x)}} - \frac{1}{\underline{u}^{\delta(x)}} \right)\phi dx&\leq\int_{\mathbb{R}}-\int_{+_{2}}\int_{\mathbb{R}}^{\Omega}|\nabla_{2} \frac{(( \overline{(u} \overline{u} -_{-} \underline{u} \underline{u)} )^{|^{2}}(^{\psi}x)^{(_{\gamma}^{\prime}} \frac{u}{-} ( \frac{-}{u} \overline{u_{-}} ) \underline{ud} )^{x}(y))(\phi(x)-\phi(y))}{|x-y|^{2+2s}} dxdy \\ &\leq0, \end{aligned}\] + +which reveals + +\[|\{x\in\Omega: \underline{u} > \overline{u} \}|=0.\] + +Consequently, \(\underline{u} \leq \overline{u} \text{in}\Omega.\) Now, we define + +\[\mathcal{M}:=\{u\in X: \underline{u} \leq u\leq \overline{u} \text{in}\Omega\},\] + +which is a closed and convex subset of \(X\). Define \(m:=\inf_{\mathcal{M}}E_{\lambda,\epsilon}(u).\) We claim that this + +infimum is achieved by a solution of the equation \((P_{\lambda}^{\epsilon}).\) To this end, we suppose \(\{u_{n}\}_{n\in IN}\) be diff --git a/outputs/arxiv_math/2503.08272_pg14_pg1_repeat1.md b/outputs/arxiv_math/2503.08272_pg14_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..7acc585db5e4d1a407137d14441266734c0d0276 --- /dev/null +++ b/outputs/arxiv_math/2503.08272_pg14_pg1_repeat1.md @@ -0,0 +1,27 @@ +Finally, we briefly summarise the link between local and global utility in the monotone case, which too is completely analogous to the previous subsection. The expression for maximal expected monotone quadratic utility from Proposition 2.3 and Theorem 2.16 translates to + +\[1-2u_{\text{mmv}}(0)=\mathscr{E}(-\tilde{K}^{\text{mmv}})_{T},\] + +where we may interpret \(\tilde{K}^{\text{mmv}}=B^{2g_{\text{mmv}}\circ(\hat{\lambda}\cdot R)}\) as the cumulative maximal local squared MHR. Using the conversion between the (monotone) Hansen and the (monotone) Sharpe ratio, we also obtain + +\[1+2v_{\text{mmv}}(0)=\mathscr{E}(\widehat{K}^{\text{mmv}})_{T},\] + +where \(\widehat{K}^{\text{mmv}}=_{1-\Delta\tilde{K}}^{1}\) mmv \(\cdot\tilde{K}^{\text{mmv}}\) is the cumulative local squared monotone Sharpe ratio. + +## 4. Examples + +Example 4.1 illustrates that the locally optimal strategy \(\hat{\lambda}\) may not be unique even if there is no redundancy among the traded assets. Example 4.2 contrasts classical mean–variance and monotone mean–variance optimal portfolios in a continuous-time setting. Example 4.3 shows that the equivalences in Theorem 2.25 fail when \(R\) is not square integrable. Example 4.4 illustrates that Qˆ can be equivalent to P without being a \(\sigma-\)martingale measure for \(R\), again highlighting the importance of the square-integrability assumption in Theorem 2.25. Example 4.5 constructs a model with square-integrable yield \(R\), where the maximal Sharpe ratio available by trading is finite but the maximal monotone Sharpe ratio is infinite. Example 4.6 exhibits a complete market model with an equivalent martingale measure, where both the MV utility and the MMV utility are infinite. + +Example 4.1. Consider a one-period model with four atoms and two assets, whose yield distribution is given by + +| | ω1 | ω2 | ω3 | ω4 | +| --- | --- | --- | --- | --- | +| P[{ω}] | 0.2 | 0.6 | 0.1 | 0.1 | +| ∆R1(ω) | -0.5 | 0.5 | 1 | 1.2 | +| ∆R2(ω) | -0.5 | 0.5 | 1.2 | 1 | + +Choosing \(q=(0.62,0.18,0.1,0.1)\) yields an equivalent martingale measure Q for \(R\), hence the model is arbitrage-free. Observe that in this model, all separating measures are martingale measures for \(R\). One easily verifies that \(\hat{q}=(0.5,0.5,0,0)\) defines a martingale measure Qˆ whose density satisfies + +\[\text{Var}\left( \frac{\text{dQ}}{\text{dP}} \right)= \frac{2}{3} .\] + +The duality constraint in Proposition 2.20 and (2.7) yield that twice the maximal utility cannot exceed \(1-(1+ \frac{2}{3} )^{-1}=0.4\). On the other hand, one has \(\mathrm{E}[2g_{\text{mmv}}(R^{1})]=\mathrm{E}[2g_{\text{mmv}}(R^{2})]=0.4\) hence we have two distinct optimal strategies, each investing one dollar in one of the assets, respectively, and borrowing one dollar at the zero risk-free rate. Any affine combination of these two strategies is again optimal. The wealth of all optimal strategies coincides below the bliss point of the utility function. In this example, 2/3 is the maximal squared monotone Sharpe ratio among zero-cost strategies and 0.4 (twice the maximal expected utility) is the maximal squared monotone Hansen ratio among zero-cost strategies. diff --git a/outputs/arxiv_math/2503.08279_pg19_pg1_repeat1.md b/outputs/arxiv_math/2503.08279_pg19_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..056fad61ad271ebcf942c8d46f28b1cc9d23cc20 --- /dev/null +++ b/outputs/arxiv_math/2503.08279_pg19_pg1_repeat1.md @@ -0,0 +1,15 @@ +where we used the coarea formula in the first inequality [Fed59], and the fact that the map from \(\hat{\mathcal{D}}_{r}(\Omega)\cap S_{t}\) to \((\hat{V}_{\hat{j}},\angle_{Tits})\) defined by \(\sigma(t)\mapsto[\sigma]\) is locally \(\pi/(2C_{k}t)-\)Lip due to (23); see for instance [BBI01, Proposition 1.7.8]. To conclude, we note that \(\mathcal{H}^{n-1}(\hat{V}_{\hat{j}})\) has a positive lower bound independent of \(\hat{j}\), and thus of \(\Omega\). Indeed, because of the volume growth assumption on each end of \(M\), all the connected components of \(M_{\infty}\) are Alexandrov \(CBB(0)\) spaces of dimension \(n-1\); see Lemma 3.1,(4). Accordingly, all the open sets \(\hat{V}_{j},j=1,\ldots,J\), have strictly positive \((n-\) 1)-dimensional Hausdorff measure, so that \(\min_{j}\mathcal{H}^{n-1}(\hat{V}_{j})>0\). □ + +Remark 4.6. In the stronger assumptions of Corollary 1.4, that is when the sectional curvature of \(M\) is non-negative outside some compact set \(K\) of \(M\), there is no need to conformally deform the metric of \(M\). In particular, one can replace \(\varphi\equiv1\text{and}f\equiv0\) in the above proof. An even simpler case is when \(M\) has nonnegative sectional curvatures outside a compact set \(K\subset M,M\) has only one end, and diam \(M_{\infty}<\pi\). Indeed in this case, we claim that there exists \(R_{0}\) large enough so that any minimizing geodesic connecting two points of \(M\setminus B_{R_{0}}(o)\) does not intersect \(K\). One can thus implement Brendle’s approach to prove the isoperimetric inequality on \(M\setminus B_{R_{0}}(o)\). Namely, it is enough to consider \(\hat{A}_{r}:=\{y\in A_{r}:\Phi_{r}(y)\not\in B_{R_{0}}\}\), so that the minimizing geodesics from \(y\in\hat{A}_{r}\) to \(\Phi_{r}(y)\) is totally contained in \(M\setminus K\), and the curvature is non-negative along that geodesic. Since removing \(B_{R}(o)\) from the image of \(\Phi_{r}\) does not affect the asymptotic volume growth, one can conclude the proof. To show the above claim, suppose there exist two sequences of points \(\{x_{j}\},\{y_{j}\}\) such that \(\text{dist}(o,x_{j})>j,\text{dist}(o,y_{j})>j\) and a unit-speed minimizing geodesic \(\gamma_{j}\) from \(x_{j}\) to \(y_{j}\) which intersects \(K\). On the one hand, \(\text{dist}(x_{j},y_{j})\geq2j-2\) diam \(K\). On the other hand \(\lim_{j\rightarrow\infty}\text{dist}(x_{j},y_{j})/j<2\) by Lemma 3.1 (2) and the assumption on \(M_{\infty}\), thus giving a contradiction. A similar argument permits also to deal the case in which there are several ends, but each connected component of \(M_{\infty}\) has diameter \(<\pi\). Remark 4.7. According to the previous remark, when all the connected components of \(M_{\infty}\) have diameter \(<\pi\) the isoperimetric constant in (9) restricted to \(C_{c}^{\infty}(E\setminus B_{R_{0}}(o)),\text{where}E\) is an end of \(M\), is the sharp Euclidean one, i.e. \(n\omega_{n}^{1/n}\theta^{1/n}\text{with}\theta=\limsup_{r\rightarrow\infty}\omega_{n}^{-1}r^{-n}\text{vol}(B_{r}(0)\cap E)\) the AVR of \(E\). Clearly, the constant in (9) for general functions in \(C_{c}^{\infty}(M)\) necessarily depends on the whole manifold and can not be estimated only in term of the quantities appearing in our assumptions. For instance, adding a tubular handle \([0,L]\times\epsilon\mathbb{S}^{n-1}\) to \(M\) near \(o\) makes the isoperimetric constant explode when \(L\rightarrow\infty\) without adding negative curvature outside a compact neighborhood of \(o\). + +## 5. Submanifolds in ambient manifolds with nonnegative sectional curvature outside a compact set + +Throughout this section we will assume that \((M^{n+m},g)\) is a complete smooth \((n+m)-\) dimensional Riemannian manifold satisfying + +\[\text{Sect}^{M}\geq0\] + +outside some compact set \(K\subset M\). Moreover, we will assume that, given a reference point \(o\in M,\text{AVR}>0\) on each \(\text{end}E_{j}\) of \(M\), i.e. + +\[\theta_{j}=\limsup_{r\rightarrow\infty} \frac{\text{vol}(B_{r}^{M}(o)\cap E_{j})}{r^{n+m}} >0.\] + +Let \(\Sigma\) be a compact submanifold of dimension \(n\). We will denote by \(\overline{\nabla}\) the Levi-Civita connection on \(M\) and by \(\nabla^{\Sigma}\) the induced connection on \(\Sigma\). The second fundamental form \(II\) of \(\Sigma\) is given diff --git a/outputs/arxiv_math/2503.08283_pg7_pg1_repeat1.md b/outputs/arxiv_math/2503.08283_pg7_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..fba239dcce24b19ae9a3345389d819cdf64075d5 --- /dev/null +++ b/outputs/arxiv_math/2503.08283_pg7_pg1_repeat1.md @@ -0,0 +1,5 @@ +them initially, longer time horizons and higher initial energies are usually necessary. Once an initial condition that gives rise to a nonlinear optimal is found, this resulting nonlinear optimal can in turn also be used as an initial conditions for calculations with different parameters. Most of the parameter combinations – including all that we focus the following discussion on – were also rerun from different initial conditions to ensure that the same optimal was reached, although, since this is a nonlinear optimisation problem, there is no rigorous way to ensure that a global optimum has indeed been found. Nevertheless, the conclusions made in the following are quite robust to changes in the computational setup, including the domain size, as is further discussed in Appendix A.2. Since the code written for this study supports automatic differentiation, it would also be possible to obtain the necessary gradient information without solving the adjoint system. However, it has been found that using the classical adjoint-based approach performs better than automatic differentiation, both in terms of memory and runtime. In particular, the ability to more easily control the trade-off between memory usage and runtime through checkpointing makes the adjoint approach preferable since we are currently already close to the limits of available GPU-memory on an NVIDIA A100 card with 40 GB memory. Moreover, the gradients obtained from the adjoint method are fairly smooth, whereas in gradients obtained from automatic differentiation, the discretisation error manifests itself in the form of noise, so that some processing of these gradients would likely be necessary. Automatic differentiation does have the advantage of requiring less development time and thus being more readily applied to new problems, so that this feature will likely be valuable in future studies. + +## 4. Results + +As discussed in Section 2, the main parameters are the time horizon \(T\) and the initial disturbance energy \(e_{0}\). Unlike in the transition problem, it is not useful to look at very large time horizons, as the timescale on which disturbances can grow is constrained by large-scale disruptive turbulent events. This argument has been used by Butler & Farrell (1993) to determine the timescale for linear optimal growth, and the general idea also applies to the nonlinear case. From previous investigations (Lozano-Durán et al. 2021; Markeviciute & Kerswell 2024), we can estimate that the primary and the secondary linear process both operate on a timescale of around \(t_{p}=t_{s}=0.35ℎ/u_{\tau}\), and that bursting events are roughly \(t_{\text{burst}}=4ℎ/u_{\tau}\) apart. This motivates an initial focus on the time horizon \(T=0.7ℎ/u_{\tau}=2t_{p}\), as this would be enough to see both the primary and the secondary linear process. This is addressed in Section 4.1. To get a deeper understanding of the role of this parameter, we additionally study times \(t_{p},1.5t_{p},3t_{p},4t_{p},6t_{p}\text{and}8t_{p}\) in Section 4.2. For any \(T,\text{as}e_{0}\rightarrow0\), the linear optimisation problem is recovered. Consequently, for small \(e_{0}\), a quasilinear optimal, which closely resembles the linear optimal, is obtained. As \(e_{0}\) increases, there comes a point where a structurally completely different nonlinear optimal arises. For large \(e_{0}\), the final state of the disturbances may be well in the turbulent regime, which, in the current formulation, usually leads the optimisation algorithm to fail, as the final state dependency on the initial condition is too sensitive and chaotic for the optimisation problem to remain well-conditioned. The precise determination of this boundary is difficult and not very important to the present study, but values up to \(e_{0}/E_{0}=10^{-4}\) were found to converge without issues even for the longest time horizons considered here, where \(E_{0}\) denotes the energy of the base flow. Naturally, we are most interested in nonlinear optimals whose initial energy is below this value. diff --git a/outputs/arxiv_math/2503.08285_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.08285_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..5465c3ce9b978ff7f344748bbd49ea310bbcd9c5 --- /dev/null +++ b/outputs/arxiv_math/2503.08285_pg2_pg1_repeat1.md @@ -0,0 +1,5 @@ +a bypass operation, the resulting device would be able to sort only the identity permutations, whereas adding a bypass gives rise to a much more interesting device. In the latter case, an optimal sorting algorithm, called Queuesort, is described in [M] and further studied in [CF1], and sortable permutations can be characterized as those avoiding the pattern 321. When we move to the “pop” versions, again in the case of a pop queue a bypass operation is necessary in order to have nontrivial algorithms [CF2]. On the other hand, in the case of a pop stack, having or not having a bypass operation gives rise to two different (and nontrivial) sorting devices (unlike what happens for a classical stack). Curiously, there is no evidence in the literature of papers investigating the sorting properties of a pop stack with a bypass. This paper aims to fill this gap. Clearly, adding a bypass to a pop stack slightly increases the sorting power of the device and poses interesting combinatorial questions on its properties, which we analyze in some detail. More specifically, the content of our paper is the following. In Section 3 we characterize the set of sortable permutations in terms of two forbidden patterns; the enumeration of the resulting class of pattern avoiding permutations was already known (odd-indexed Fibonacci numbers, sequence A001519 in [Sl]); however, in Section 4 we give an independent proof of this enumerative result by describing a bijective link with a restricted class of Motzkin paths. In Section 5 we describe an algorithm to compute the preimage of a given permutation and use it to characterize and enumerate permutations having 0,1, and 2 preimages. Section 6 contains a complete description of the preimages of principal classes of permutations, by determining in which cases we obtain classes (and in such cases by determining the basis of the resulting classes). Section 7 is devoted to the characterization and (partly) enumeration of the sets of sortable permutations for the compositions of our sorting algorithm with other classical sorting algorithms (the characterization being given in terms of forbidden patterns). Finally, in Section 8 we consider the device consisting of two pop stacks in parallel with a bypass, and we determine the basis of the associated set of sortable permutations, and in Section 9 we give some hints for further work. + +## 2 Preliminaries on permutations and sorting algorithms + +Let \([1,n]=\{1,\ldots,n\}\). A permutation \(\pi\) of size \(n\) is a bijection from \([1,n]\) to \([1,n]\). It can be represented in linear form as \(\pi=\pi_{1}\cdots \pi_{n},\text{where}\pi_{i}=\pi(i),\text{for all}1\leq i\leq n\). The identity permutation of size \(n\) is \(12\cdots n\), and is denoted \(id_{n}\). We denote by \(S_{n}\) the set of all permutations of size \(n\) and by \(S=\bigcup_{n\in\mathbb{N}}S_{n}\) the set of all permutations. For a given permutation \(\pi\), we say that two elements are adjacent when their positions are consecutive integers, whereas we say that they are consecutive when their values are consecutive integers. For instance, if \(\pi=3645712\), the elements 1 and 7 are adjacent, the elements 5 and 6 are consecutive, and the elements 4 and 5 are both adjacent and consecutive. We warn the reader that we will strictly adhere to this terminology regarding the terms “consecutive” and “adjacent” throughout the paper. Given \(\pi=\pi_{1}\cdots \pi_{n}\in S_{n}\), an element \(\pi_{i}\) is called a left-to-right maximum when it is larger than all the elements to its left (that is, \(\pi_{i}=\max\{\pi_{j}|j\leq i\})\). We denote by \(LTR(\pi)\) the set of left-to-right maxima of \(\pi\). Therefore, the set of left-to-right maxima of the above permutation is \(LTR(3645712)=\{3,6,7\}\). Moreover, the reverse of \(\pi\) is the permutation \(\pi^{r}=\pi_{n}\cdots \pi_{1}\). Given \(\pi\in S_{n}\text{and}\tau=\tau_{1}\cdots \tau_{m}\in S_{m}\), the direct sum of \(\pi\text{and}\tau\) is the permutation \(\pi\oplus\tau\) obtained by concatenating \(\pi\) with the sequence \((\tau_{1}+n)\cdots (\tau_{m}+n)\) (obtained from \(\tau\) by adding \(n\) to each of its elements). For instance, the direct sum of the permutations 3142 and 42315 is 314286759. We note that any sequence of \(n\) distinct integers can be regarded as a permutation by appropriately rescaling its elements to \([1,n]\). Indeed, two sequences are said to be order-isomorphic if diff --git a/outputs/arxiv_math/2503.08326_pg9_pg1_repeat1.md b/outputs/arxiv_math/2503.08326_pg9_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..447f51606404eea463224a6437e8a24b3d143cae --- /dev/null +++ b/outputs/arxiv_math/2503.08326_pg9_pg1_repeat1.md @@ -0,0 +1,19 @@ +7. and (9) gives + +\[\begin{aligned} u_{n}^{(1)}&=u_{n-3}^{(1)}+u_{n-7}^{(2)}+u_{n-11}^{(3)}+u_{n-8}^{(2)}+u_{n-12}^{(3)} \\ &=u_{n-3}^{(1)}+u_{n-7}^{(2)}+u_{n-8}^{(2)}+u_{n-5}^{(1)}-u_{n-8}^{(1)} \\ &=u_{n-3}^{(1)}+u_{n-5}^{(1)}-u_{n-8}^{(1)}+u_{n-7}^{(2)}+u_{n-8}^{(2)} \end{aligned}\] (10) + +and (6) and (7) gives + +\[\begin{aligned} u_{n}^{(2)}&=u_{n-1}^{(1)}+u_{n-6}^{(2)}+u_{n-10}^{(3)} \\ &=u_{n-1}^{(1)}+u_{n-6}^{(2)}+u_{n-5}^{(2)}-u_{n-6}^{(1)} \\ &=u_{n-1}^{(1)}-u_{n-6}^{(1)}+u_{n-5}^{(2)}+u_{n-6}^{(2)}. \end{aligned}\] (11) + +Finally, (10) and (11) gives + +\[\begin{aligned} u_{n}^{(1)}&=u_{n-3}^{(1)}+u_{n-5}^{(1)}-u_{n-8}^{(1)}+u_{n-8}^{(1)}-u_{n-13}^{(1)}+u_{n-12}^{(2)} \\ +u_{n-13}^{(2)}+u_{n-9}^{(2)}-u_{n-14}^{(1)}+u_{n-13}^{(1)}+u_{n-14}^{(2)} \\ &=u_{n-3}^{(1)}+u_{n-5}^{(1)}+u_{n-9}^{(1)}-u_{n-13}^{(1)}-u_{n-1}^{14}+u_{n-5}^{(1)}-u_{n-8}^{(1)} \\ -u_{n-10}^{(1)}+u_{n-13}^{(1)}+u_{n-6}^{(1)}-u_{n-9}^{(1)}-u_{n-11}^{(1)}+u_{n-14}^{(1)} \\ &=u_{n-3}^{(1)}+2u_{n-5}^{(1)}+u_{n-6}^{(1)}-u_{n-8}^{(1)}-u_{n-10}^{(1)}-u_{n-11}^{(1)} \end{aligned}\] + +which gives us a universal characteristic polynomial \(x^{11}-x^{8}-2x^{6}-x^{5}+x^{3}+x+1\). However, it factorizes as + +\[x^{11}-x^{8}-2x^{6}-x^{5}+x^{3}+x+1=(x-1)(x^{4}+x^{3}+x^{2}+x+1)(x^{6}-x^{3}-x-1),\] + +and we can check whether some proper divisor is also a universal characteristic polynomial. Specifically, we can let \(j\in\{1,\ldots,13\}\) and set \(u_{3}^{(j)}=1\text{and}u_{3}^{(i)}=0\) for \(i\neq j\). We set the initial values for \(n=k\), as (1) is only defined for \(n\geq k\). It turns out that \(u_{13}^{(i)}-u_{10}^{(i)}-u_{8}^{(i)}-u_{7}^{(i)}=0\text{for all}i\in\{1,\ldots,13\}\) regardless of + +\(j\). By Lemma 3.2, we then have \(u_{n}^{(i)}-u_{n-3}^{(i)}-u_{n-5}^{(i)}-u_{n-6}^{(i)}=0\text{for all}n\geq13\) for all \(i\), and since any set of initial values is a linear combination of the ones we have tried out, \(x^{6}-x^{3}-x-1\) is a universal characteristic polynomial on \((\xi,I_{3})\). Since it is irreducible and \(\xi\) is nontrivial, it must be minimal. As an example of an execution of the flowchart in Algorithm 1, let \(v= (4,16,\{(L_{2},L_{3}),(R_{1},R_{2})\})\). The subgraph \(\rho(v)\subset\sigma_{3}\) consists of the SCCs diff --git a/outputs/arxiv_math/2503.08341_pg5_pg1_repeat1.md b/outputs/arxiv_math/2503.08341_pg5_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..8388e25e9fcd5dcb354a3a2935a327fed699ce9c --- /dev/null +++ b/outputs/arxiv_math/2503.08341_pg5_pg1_repeat1.md @@ -0,0 +1,29 @@ +Proposition 2.2. There exists an involution \(i:\mathcal{H}(W)\rightarrow\mathcal{H}(W)\), defined as \(i(T_{\omega})=T_{\omega^{-1}}\), which is an anti-homomorphism, i.e. + +\[i(T_{r}T_{r-1}\ldots T_{2}T_{1})=i(T_{1})i(T_{2})\ldots i(T_{r-1})i(T_{r}).\] + +In particular, if all the \(T_{i}\) are indexed by the generators of the Hecke algebra, we have + +\[i(T_{s_{r}}\ldots T_{s_{1}})=T_{s_{1}}\ldots T_{s_{r}}.\] + +Many particle systems can be characterized using the Hecke Algebra or random walk on the Hecke Algebra, see [8] for more examples. In this paper, we focus on the oriented swap process. Let \(W=S_{n}\) be the permutation group, \(S\) the set of \(n-1\) nearest neighbor transpositions as the generators, and \(l\) the length function, defined as the inversion number of a permutation. We set \(a_{s}=1\text{and}b_{s}=0\) and define a map \(\mathcal{P}:S_{n}\rightarrow\mathcal{H}(S_{n})\) such that a permutation \(\omega\) is mapped to an element \(\mathcal{P}(\omega)=T_{\omega}\) in the basis of the Hecke algebra. When a swap attempt occurs at position \(i,\omega\) transitions to \(\tau_{i}\omega\) if and only if \(\omega(i)<\omega(i+1)\). On the other hand, \(T_{\tau_{i}}T_{\omega}=T_{\tau_{i}\omega}\) if and only if \(\omega(i)<\omega(i+1)\). Now, consider a continuous-time random walk on Hecke algebra, i.e. \(W_{t}=\prod_{i=1}^{r}T_{i}\), where each \(T_{i}\) is chosen uniformly from all the nearest neighbor transposition, and \(r=Poi(t)\). Writing this random walk as a linear combination of the Hecke algebra basis defines a random measure on the basis. The probability that a finite swap process is in configuration \(\omega\) at time \(t\) corresponds to the expectation of coefficient of \(T_{\omega}\) in the decomposition of \(W_{t}\). + +## Corollary 2.3. [8, Cor. 2.2] + +1. For a finite oriented swap process at a fixed time \(t,\pi_{t}^{n}\text{and}(\pi_{t}^{n})^{-1}\) have the same law, the inverse is in the sense of the inversion of a permutation. + +2. For a infinite swap process at a fixed time \(t\), and its first \(k\) particles, + +\[(\pi_{t}(1),\pi_{t}(2),\ldots\pi_{t}(k))^{(}=^{d)}(\pi_{t}^{-1}(1),\pi_{t}^{-1}(2),\ldots\pi_{t}^{-1}(k)).\] + +Remark 2.4. The symmetry property for the finite case is a direct corollary of the property of the involution. For the infinite case, we need again the Harris’ construction and apply the symmetry property to each segment, see also the proof of [2, Thm 1.4]. + +### 2.3 Properties of TASEP and the oriented swap process + +Our analysis of the oriented swap process relies on its connection with TASEP. In this section, we present some propositions about TASEP and how they relate to the oriented swap process. Firstly, we define certain operations on configurations of TASEP or the oriented swap process. A configuration \(\rho\in\mathbb{Z}^{I}\) specifies the type of particle at each position and the set of all such configurations is denoted as \(\mathcal{S}\). + +Definition 2.5. For any \(k\in\mathbb{Z}\), and any configuration \(\rho,\text{where}I\) is a finite or infinite interval, we define the projection operator \(T_{k}:\mathcal{S}\rightarrow\mathcal{S}\) as + +\[(T_{k}\rho)(x)=\mathbf{1}_{\{\rho(x)\leq k\}}.\] + +The operator projects a configuration of multi species particle system(e.g. the infinite oriented process) to a TASEP configuration on the same segment with only particles and holes. diff --git a/outputs/arxiv_math/2503.08345_pg8_pg1_repeat1.md b/outputs/arxiv_math/2503.08345_pg8_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..4958f55d3f9a7a8c484a4d2370c4800e7413d8da --- /dev/null +++ b/outputs/arxiv_math/2503.08345_pg8_pg1_repeat1.md @@ -0,0 +1,63 @@ +### 4.3 Spectral analysis + +We can now derive specific spectral properties of the Koopman operator that we will leverage for state estimation. Those properties require the following additional assumptions on the nonlinear system (6). + +# Assumption 3 (stable hyperbolic equilibrium). + +The dynamics admits a stable equilibrium at the origin, whose basin of attraction contains the polydisc \(\overline{\mathbb{D}^{n}}\). Moreover, the eigenvalues \(\lambda_{j}\) of the Jacobian matrix \(\mathcal{J}_{F}(\mathbf{0})\) are simple and satisfy \(\text{Re}(\lambda_{j})<0\) for all \(j=1,\ldots,n\). + +# Assumption 4 (non-resonant eigenvalues). The + +eigenvalues \(\lambda_{j}\) of the Jacobian matrix \(\mathcal{J}_{F}(\mathbf{0})\) are nonresonant, i.e. for all \((m_{1},\ldots,m_{n})\in\mathbb{N}^{n}\) satisfying \(\sum_{l=1}^{n}m_{l}\geqslant2\), + +\[\lambda_{j}\neq \sum_{l=1}^{n}m_{l}\lambda_{l},\forall j=1,\ldots,n.\] + +# Assumption 5 (output map). The components of + +the output map \(h\) belong to \(\mathbb{H}^{2}(\mathbb{D}^{n})\). + +Remark 14 Nonresonant eigenvalues are required to rely on the Poincaré-Dulac linearization theorem. However, other linearization theorems exist, with different assumptions, see e.g. the Siegle-Bruno theorem in [Bernard, 2023]. In [Krener and MingQing, 2001], similar assumptions were considered to obtain a necessary and sufficient condition for the existence of a change of variable that linearizes the dynamics up to a nonlinear injection term. + +The first remarkable property resulting from those assumptions is the fact that the operators \(A_{F}\text{and}A_{F}^{*}\) admit a series expansion that allows to represent both operators as infinite matrices. + +## Lemma 15 For all \(f\in D(A_{F})\), + +\[A_{F}f=\sum_{\alpha\in\mathbb{N}^{n}}\left(\sum_{\beta\in\mathbb{N}^{n}}A_{F\alpha,\beta}f_{\beta}\right)e_{\alpha}\] + +where \(f_{\beta}=\langle f,e_{\beta}\rangle\text{and}A_{F\alpha,\beta}=\langle A_{F}e_{\beta},e_{\alpha}\rangle\). + +PROOF. Since \(\{e_{\alpha}\}_{\alpha\in\mathbb{N}^{n}}\) is an orthonormal basis of \(\mathbb{H}^{2}(\mathbb{D}^{n})\), any \(f\in D(A_{F})\) can be expanded as + +\[A_{F}\left(\sum_{\beta\in\mathbb{N}^{n}}f_{\beta}e_{\beta}\right)=\sum_{\beta\in\mathbb{N}^{n}}f_{\beta}A_{F}e_{\beta}.\] (15) + +Indeed, in [Mugisho and Mauroy, 2024], it is shown that the right-hand side is given by + +\[\begin{aligned} \sum_{\beta\in\mathbb{N}^{n}}f_{\beta}A_{F}e_{\beta} \\ &=\sum_{l=1}^{n}F_{l}\sum_{\beta\in\mathbb{N}^{n}}(\beta_{l}+1)f_{(\beta_{1},\ldots,\beta_{l-1},\beta_{l}+1,\beta_{l+1},\ldots,\beta_{n})}e_{\beta}. \end{aligned}\] + +It remains to show that the left-hand side of (15) is also equal to the right-hand side of (16). To do so, consider the operators \(A_{1}\text{and}A_{2}\) given by + +\[A_{1}f=\nabla f\text{and}A_{2}\omega=F\cdot\omega,\] + +for all \(f\in D(A_{F})\text{and}\omega\in(\text{Hol}(\mathbb{D}^{n}))^{n}\), respectively. Observe that \(A_{F}=A_{2}A_{1}\text{on}D(A_{F})\). Moreover, for all \(f\in D(A_{F})\text{and for all}l=1,\ldots,n\), + +\[(A_{1}f)_{l}=\sum_{\beta\in\mathbb{N}^{n}}(\beta_{l}+1)f_{(\beta_{1},\ldots,\beta_{l-1},\beta_{l}+1,\beta_{l+1},\ldots,\beta_{n})}e_{\beta}.\] + +Hence, for all \(f\in D(A_{F})\), + +\[\begin{aligned} A_{F}f&=A_{2}A_{1}f \\ &=\sum_{l=1}^{n}F_{l}\sum_{\beta\in\mathbb{N}^{n}}(\beta_{l}+1)f_{(\beta_{1},\ldots,\beta_{l-1},\beta_{l}+1,\beta_{l+1},\ldots,\beta_{n})}e_{\beta}. \end{aligned}\] + +In view of (16), it follows that identity (15) holds, which implies that + +\[\begin{aligned} A_{F}f&=\langle A_{F}f,e_{\alpha}\rangle e_{\alpha} \\ &=_{\alpha}^{\alpha}\sum_{\alpha\in^{\in}\mathbb{N}^{\mathbb{N}_{n}^{n}}}^{\in\mathbb{N}^{n}}\left\langle A_{F}\left(\sum_{\beta\in\mathbb{N}^{n}}f_{\beta}e_{\beta}\right),e_{\alpha}\right\rangle e_{\alpha} \\ &=\sum_{\beta\in\mathbb{N}^{n}}\langle A_{F}e_{\beta},e_{\alpha}\rangle f_{\beta}e_{\alpha}. \\ ■ \end{aligned}\] + +It is proved in [Mugisho and Mauroy, 2024] that the infinite matrix representation \(\mathcal{A}_{F}\) of the operator \(A_{F}\) is lower block triangular of the form + +\[\mathcal{A}_{F}f= \begin{pmatrix} [0] & \cdots \\ [0][\mathcal{A}_{11}] & [0] & \cdots \\ [0][\mathcal{A}_{21}][\mathcal{A}_{22}][0]\cdots \\ \ddots & \ddots & \ddots & \ddots & \ddots \end{pmatrix} \begin{pmatrix} [f_{0}] \\ [f_{1}] \\ [f_{2}] \\ \ddots \end{pmatrix} ,(17)\] + +| f = Pβ∈Nn fβeβ. Hence, we can prove that | [0] · · · | | [f0] | +| --- | --- | --- | --- | +| | [0] [A11] [0] | · · · | [f1] | +| | AFf = [0] [A21] [A22] | [0] | · · · [f2] , (17) | +| AF X fβeβ = X fβAF eβ. (15) | | | | +| | ... ... ... | ... | ... ... | +| β∈Nn β∈Nn | | | | diff --git a/outputs/arxiv_math/2503.08351_pg5_pg1_repeat1.md b/outputs/arxiv_math/2503.08351_pg5_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..48e137d440634f83e73a74530405a0d0258724e7 --- /dev/null +++ b/outputs/arxiv_math/2503.08351_pg5_pg1_repeat1.md @@ -0,0 +1,35 @@ +The Michael Selection Theorem guarantees the existence of a continuous selection for a lower semicontinuous multimap with convex values. However, this Theorem doesn’t hold for multimap with nonconvex values. In such cases, the author of [27] introduces the concept of decomposable sets to establish the existence of a continuous selection. Let us define decomposable sets now. Formally, the concept of decomposability resembles that of convexity, and as we will see in this section, decomposable sets behave like convex sets. For this reason, decomposable sets play a central role in many applications. Definition 2.6. A set \(K\subset L^{1}(I,X)\) is said to be decomposable if for every triple \((D,f_{1},f_{2})\in\Sigma\times K\times K\) we have + +\[\chi_{D}f_{1}+\chi_{I\setminus D}f_{2}\in K.\] (2.12) + +We now consider the nonconvex-decomposable version of Michael’s selection theorem. + +Theorem 2.7. [28, Theorem 4.5.32] If \(Z\) is a separable metric space, \(X\) is a separable Banach space and \(F:Z⊸L^{1}(I,X)\) is lower semicontinuous and has closed decomposable values, then \(F\) admits a continuous selection. + +## 3. Navier Stokes Equations + +In this section, we study the existence of local strong solutions for the nonstationary multivalued version of Navier-Stokes equations given by (1.1)-(1.4). Throughout this section, we assume that 3 < \(p<\infty\text{and}10\). Also let \(u_{0}\in D_{p}^{q}\text{with}30\) and + +\[\begin{aligned} u&\in L^{q}([0,b],D(A_{p}))\text{with}\partial_{t}u\in L^{q}([0,b],H_{p}), \\ p&\in L^{q}([0,b],W^{1,p}(D)^{3}) \end{aligned}\] (3.2) (3.3) + +such that \((u,p)\) is a solution to the problem (1.1)-(1.4) with \(f\in L^{q}([0,b],L^{p}(D)^{3})\text{with}f(t)\in F(t,u(t))\) for a.a. \(t\in[0,b]\). + +Proof. For \(b>0\), we introduce the space + +\[U(b)=\{u\in L^{q}([0,b],D(A_{p})):\partial_{t}u\in L^{q}([0,b],H_{p})\}.\] (3.4) + +Consider the selection multimap \(S_{F}:U(b)\subset L^{q}([0,b],H_{p})⊸L^{q}([0,b],L^{p}(D)^{3})\) as follows: + +\[S_{F}(u)=\{f\in L^{q}([0,b],L^{p}(D)^{3}):f(t)\in F(t,u(t)),\text{a.a.}t\in[0,b]\}.\] (3.5) + +By virtue of assumption (F1), the multimap \(F\) is product measurable. Hence, the multimap \(t⊸F(t,u(t))\) is closed valued and measurable for every \(u\in L^{q}([0,b],H_{p})\). In accordance with the Ryll-Kurtowski diff --git a/outputs/arxiv_math/2503.08355_pg1_pg1_repeat1.md b/outputs/arxiv_math/2503.08355_pg1_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..b03d5048967d06c3df119a32369fe932fee56375 --- /dev/null +++ b/outputs/arxiv_math/2503.08355_pg1_pg1_repeat1.md @@ -0,0 +1,17 @@ +# Pointwise Minimax Vector Field Reconstruction from Noisy ODE + +##### Hugo Henneuse 1,2 hugo.henneuse@universite-paris-saclay.fr March 12, 2025 + +###### Abstract + +This work addresses the problem of estimating a vector field from a noisy Ordinary Differential Equation (ODE) in a non-parametric regression setting with a random design for initial values. More specifically, given a vector field \(f:\mathbb{R}^{D}\rightarrow\mathbb{R}^{D}\) governing a dynamical system defined by the autonomous ODE: \(y^{\prime}=f(y)\), we assume that the observations are \(\tilde{y}_{X_{i}}(t_{j})=y_{X_{i}}(t_{j})+\varepsilon_{i,j}\text{where}y_{X_{i}}(t_{j})\) is the solution of the ODE at time \(t_{j}\) with initial condition \(y(0)=X_{i},X_{i}\) is sampled from a probability distribution \(\mu,\text{and}\varepsilon_{i,j}\) some noise. In this context, we investigate, from a minimax perspective, the pointwise reconstruction of \(f\) within the envelope of trajectories originating from the support of \(\mu\). We propose an estimation strategy based on preliminary flow reconstruction and techniques from derivative estimation in non-parametric regression. Under mild assumptions on \(f\), we establish convergence rates that depend on the temporal resolution, the number of sampled initial values and the mass concentration of \(\mu\). Importantly, we show that these rates are minimax optimal. Furthermore, we discuss the implications of our results in a manifold learning setting, providing insights into how our approach can mitigate the curse of dimensionality. + +## 1 Introduction + +Longitudinal data commonly arise in the study of dynamical systems across various disciplines, from physics to biology and economics (see e.g. Hirsch et al., 2013). By observing multiple trajectories in a given space, we aim to extract key information about the underlying dynamics, such as governing laws, stability properties, or long-term trends. A common approach to modeling such phenomena is to assume that the observed trajectories follow an underlying differential system, which provides a structured framework for inference, prediction, and control. However, real-world data are often subject to noise, missing observations, and irregular sampling, making their analysis particularly challenging and raising interesting statistical questions. In particular, given noisy observations of trajectories governed by an autonomous differential equation: + +\[y^{\prime}=f(y),\] (1) + +where \(f:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}\) is the vector field governing the system, can we reconstruct \(f\) from the observed data ? This question has recently garnered significant interest among the machine learning and statistical communities. Most existing works focus on the case where \(f\) is assumed to belong to a parametric class of functions (e.g. see the surveys McGoff et al., 2015; Ramsay and Hooker, 2017; Dattner, 2021). In contrast, the non-parametric literature is scarcer and primarily numerical (Chen et al., 2018; Heinonen et al., 2018; Bhat et al., 2020; Gottwald and Reich, 2021; Lahouel et al., 2024). More recently, some studies have provided theoretical analyses of the statistical performance of their proposed estimation strategies. Notably, Schötz and Siebel (2024); Schötz (2025) introduce two statistical models: the snake model and the stubble model. In the snake model, a few long trajectories are observed, covering the space of interest. In the stubble model, many short trajectories are observed, with initial values forming a deterministic and uniform cover of + +1Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay, Orsay, France 2DataShape, Inria Saclay, Palaiseau, France diff --git a/outputs/arxiv_math/2503.08360_pg10_pg1_repeat1.md b/outputs/arxiv_math/2503.08360_pg10_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..ac484d2aa8351edebbef83a035d42954687966d3 --- /dev/null +++ b/outputs/arxiv_math/2503.08360_pg10_pg1_repeat1.md @@ -0,0 +1,29 @@ +in \(P_{\ell}(D)\). We introduce the space of piecewise-polynomial functions + +\[P_{\ell}(T_{h}):=\{v\in L^{2}(T_{h}):v|_{K}\in P_{\ell}(K),\forall K\in T_{h}\}\] + +with respect to the partition \(T_{h}\) and the space of piecewise-polynomial functions + +\[P_{\ell}(F_{h}):=\{\hat{v}\in L^{2}(F_{h}):\hat{v}|_{F}\in P_{\ell}(F),\forall F\in F_{h}\}\] + +with respect to the skeleton \(F_{h}\) of the mesh \(T_{h}\). The subspace of \(L^{2}(T_{h},\mathbb{R}^{m\times n})\) with components in \(P_{\ell}(T_{h})\) is denoted \(P_{\ell}(T_{h},\mathbb{R}^{m\times n})\). Likewise, \(P_{\ell}(F_{h},\mathbb{R}^{m\times n})\) stands for the subspace of \(L^{2}(F_{h},\mathbb{R}^{m\times n})\) with components in \(P_{\ell}(F_{h})\). We finally consider + +\[P_{\ell}(\partial T_{h},\mathbb{R}^{m\times n}):=\left\{\phi\in L^{2}(\partial T_{h},E);\phi|_{\partial K}\in P_{\ell}(\partial K,\mathbb{R}^{m\times n}),\forall K\in T_{h}\right\},\] + +where \(P_{\ell}(\partial K,\mathbb{R}^{m\times n}):=\prod_{F\in F(K)}P_{\ell}(F,\mathbb{R}^{m\times n})\). Remark 2. It is important to keep in mind that, by definition, the functions in \(L^{2}(\partial T_{h},\mathbb{R}^{m\times n})\) and \(P_{\ell}(\partial T_{h},\mathbb{R}^{m\times n})\), are multi-valued on every interior face \(F\) , whereas the functions in \(L^{2}(F_{h},\mathbb{R}^{m\times n})\) and \(P_{\ell}(F_{h},\mathbb{R}^{m\times n})\) are single-valued on each face \(F\) . We consider \(\mathbf{n}\in P_{0}(\partial T_{h},\mathbb{R}^{d}),\text{where}\mathbf{n}|_{\partial K}=\mathbf{n}_{K}\) is the unit normal vector of \(\partial K\) oriented toward the exterior of \(K\). Obviously, if \(F=K\cap K^{\prime}\) is an interior edge/face of \(F_{h},\text{then}\mathbf{n}_{K}=-\mathbf{n}_{K^{\prime}}\) on \(F\) . If \(\varphi\in H^{s}(T_{h},\mathbb{R}^{m\times n}),\text{with}s>^{1}/2\), the function \(\varphi|_{\partial T_{h}}\in L^{2}(\partial T_{h},\mathbb{R}^{m\times n})\) is meaningful by virtue of the trace theorem. For the same reason, if \(\varphi\in H^{1}(\Omega,\mathbb{R}^{m\times n})\text{then}\hat{\varphi}:=\varphi|_{F_{h}}\) is well defined in \(L^{2}(F_{h},\mathbb{R}^{m\times n})\). For \(k\geq0\), we introduce the finite-dimensional subspaces of \(H_{1}\text{and}H_{2}\) given by + +\[H_{1,h}:=P_{k+1}(T_{h},\mathbb{R}^{d\times2})\text{and}H_{2,h}:=P_{k}(T_{h},\mathbb{R}_{\text{sy}\mathrm{m}}^{d\times d})\times P_{k}(T_{h}),\] + +respectively. We consider the following discrete trace inequality. + +Lemma 2. There exists a constant \(C>0\) independent of \(h\text{and}k\) such that + +\[\Vert \tfrac{h_{F}^{/^{1}2}}{k+1} q\Vert_{0,\partial T_{h}}\leq C\Vert q\Vert_{0,T_{h}}\forall q\in P_{k}(T_{h}).\] (31) + +Proof. See [22, Lemma 3.2]. + +For any integer \(\ell\geq0\text{and}K\in T_{h}\), we denote by \(\Pi_{K}^{\ell}\) the \(L^{2}(K)-\)orthogonal projection onto \(P_{\ell}(K)\). The global projection \(\Pi_{T}^{\ell}\text{in}L^{2}(T_{h})\) onto \(P_{\ell}(T_{h})\) is then given by \((\Pi_{T}^{\ell}v)|_{K}=\Pi_{K}^{\ell}(v|_{K})\) for all \(K\in T_{h}\). Similarly, the global projection \(\Pi_{F}^{\ell}\text{in}L^{2}(F_{h})\) onto \(P_{\ell}(F_{h})\) is given, separately for all \(F\in F_{h}\), by \((\Pi_{F}^{\ell}\hat{v})|_{F}=\Pi_{F}^{\ell}(\hat{v}|_{F}),\text{where}\Pi_{F}^{\ell}\) is the \(L^{2}(F)\) orthogonal projection onto \(P_{\ell}(F)\). In the following, we maintain the notation \(\Pi_{T}^{\ell}\) to refer to the \(L^{2}-\)orthogonal projection onto \(P_{\ell}(T_{h},\mathbb{R}^{m\times n})\). It should be noted that the tensorial version of \(\Pi_{T}^{\ell}\) inherently preserves the symmetry of the matrices, as it is derived by applying the scalar operator component-wise. Similarly, we will also use \(\Pi_{F}^{\ell}\) to denote the \(L^{2}\) orthogonal projection onto \(P_{\ell}(F_{h},\mathbb{R}^{m\times n})\). In the remainder of this section, we provide approximation properties for the projectors defined above. A detailed proof of these results can be found in [22, Section 3] and the references therein. + +Lemma 3. There exists a constant \(C>0\) independent of \(h\text{and}k\) such that + +\[\Vert q-\Pi_{T}^{k}q\Vert_{0,T_{h}}+\Vert \tfrac{h_{F}^{/^{1}2}}{k+1} (q-\Pi_{T}^{k}q)\Vert_{0,\partial T_{h}}\leq C \tfrac{h_{K}^{\min\{r,k\}+1}}{(k+1)^{r+1}} \Vert q\Vert_{1+r,\Omega},\] (32) diff --git a/outputs/arxiv_math/2503.08365_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.08365_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..53fc7f1f8e44bfefc555d6e9c209cf341f6766a5 --- /dev/null +++ b/outputs/arxiv_math/2503.08365_pg2_pg1_repeat1.md @@ -0,0 +1,21 @@ +that transforms one copy of an edge to another passes over a vertex. Interestingly, the upper bound for 3-planar graphs is tight in this more general setting only [4, 6]. The crossing number of a drawing \(\Gamma\) is the number of edge crossings in \(\Gamma\). The crossing number \(\text{cr}(G)\) of a graph \(G\) is the minimum crossing number over all drawings of \(G\). By definition every \(k-\)planar graph \(G\) admits a \(k-\)plane drawing and thus + +\[\text{cr}(G)\leq \frac{km}{2} ,\] (S) + +where \(m\) denotes the number of edges in \(G\). For a \(k-\)planar graph, this simple inequality connects upper bounds on the number of edges with lower bounds on the crossing number. Both of these come together in the well-known Crossing Lemma [2, Chapter 45], as the best constants in the Crossing Lemma are obtained by analyzing \(k-\)plane drawings [1, 6, 10, 11]. Conversely, combining the lower bound on \(\text{cr}(G)\) from the Crossing Lemma with an upper bound on \(\text{cr}(G)\) we obtain an upper bound on the number of edges in \(G\). While (S) would work here, it is probably not an ideal choice because the graphs for which (S) is tight might be very different from those graphs that have a maximum number of edges, for any fixed \(n\). For instance, for a 1-planar graph \(G\) we have \(\text{cr}(G)\leq n-2[17\), Proposition 4.4], which beats the bound we get by plugging \(m\leq4n-8\) into (S) by a factor of two. Can we obtain similar improvements by bounding \(\text{cr}(G)\) in terms of \(n\), rather than \(m,\text{for}k\geq\) 2? Indeed, very recently it has been shown that \(\text{cr}(G)\leq3. \overline{3} n\text{if}G\) is 2-planar and \(\text{cr}(G)\leq 6.6n\text{if}G\) is 3-planar [3]. There is some indication that the bound for 2-planar graphs could be tight up to an additive constant, as it is achieved by the standard drawings of optimal 2-planar graphs (Figure 1). But the crossing number of these graphs is not known. + +![](figure) + +![](figure) + +● Figure 1 Construction by Pach and Tóth [15, Figure 3]. Left: A planar drawing with pentagonal faces. Right: To each pentagonal face all diagonals are added. + +In contrast, there exists a family of simple 3-planar graphs with \(5.5n-15\) edges whose standard drawings have \(5.5n-21\) crossings (Figure 2). Thus, there is a gap of \(1.1n\) between the lower and the upper bound for the crossing number of 3-plane drawings. + +![](figure) + +![](figure) + +![](figure) + +● Figure 2 Construction from [11, Figure 8]. Left: A cylinder with two layers, each consisting of three hexagonal faces. Right: To each face of a layer all but one diagonal is added. To the top and bottom face six diagonals are added. Missing diagonals are represented by dashed lines. diff --git a/outputs/arxiv_math/2503.08374_pg13_pg1_repeat1.md b/outputs/arxiv_math/2503.08374_pg13_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..d2ec9bd14636b810acab5f26854913afd4903405 --- /dev/null +++ b/outputs/arxiv_math/2503.08374_pg13_pg1_repeat1.md @@ -0,0 +1,27 @@ +Lemma 4. The function \(\det A_{n;3}\) has the same sign as \(\delta_{n}\) in some neighborhood of the triangular boundary of the region \(00\) be an integer such that \(N:=\omega_{X}^{\otimes k}|_{Z}\) is very ample and let \(i:Z֒\rightarrow X\) be the inclusion map (here \(Z\) is equipped with the reduced induced subscheme structure). Then the Hilbert polynomial of \(P_{i^{*}\mathcal{H}^{j}}(m)=\chi(i^{*}\mathcal{H}^{j}\otimes N^{\otimes m})\) has degree equal to + +\[s_{j}:=\text{dimSupp}(i^{*}\mathcal{H}^{j})=\text{dimSupp}(\mathcal{H}^{j})\] + +([Ser55, p. 276, Proposition 6]). Moreover, by tensoring (3) with positive powers of \(\omega_{X}\) and by restricting the isomorphisms to \(Z\), we find + +\[i^{*}\mathcal{H}^{j}\otimes N\simeq i^{*}\mathcal{H}^{j}\text{for all}j.\] + +Therefore \(P_{i^{*}\mathcal{H}^{j}}(m)=P_{i^{*}\mathcal{H}^{j}\otimes N}(m)=P_{i^{*}\mathcal{H}^{j}}(m+1)\text{for all}m\in\mathbb{Z}\) which is impossible if \(\deg P_{i^{*}\mathcal{H}^{j}}>0\). Hence \(s_{j}=0\text{for all}j\text{and}\iota_{Z}P\simeq\mathbb{C}(p)[r]\) for some closed point \(p\in Z\text{and}r\in\mathbb{Z}\) by Lemma 3.4. It follows that \(P\simeq\iota_{Z}^{!}\mathbb{C}(p)[r]\). □ + +Example 3.9. We construct further instances of equivalences between derived categories with support extending the equivalences (1). Denote by + +\[R:\text{Aut}^{0}(X)\times\text{Pic}^{0}(X)\rightarrow\text{Aut}^{0}(Y)\times\text{Pic}^{0}(Y)\] + +the Rouquier isomorphism induced by an equivalence \(F:\mathrm{D}^{b}(\text{Coh}(X))\rightarrow\mathrm{D}^{b}(\text{Coh}(Y))\) (cf. [Rou11, Théor`eme 4.18] or [Huy06, Proposition 9.45], and [PS11, p.531, footnote (1)]). By following [Lom14, Proposition 3.1], if \(\alpha\in\text{Pic}^{0}(X)\) is a topologically trivial line bundle such that \(H^{0}(X,\omega_{X}^{\otimes k_{0}}\otimes\alpha)\neq0\) for some \(k_{0}\in\mathbb{Z},\text{then}R(\text{id}_{X},\alpha)=(\text{id}_{Y},\beta)\text{for some}\beta\in\text{Pic}^{0}(Y)\) and moreover there are isomorphisms \(R_{k}:H^{0}(X,\omega_{X}^{\otimes k}\otimes\alpha)\rightarrow H^{0}(Y,\omega_{Y}^{\otimes k}\otimes\beta)\text{for all}k\in\mathbb{Z}\). As in [Tod06], one can prove + +that if \(E\in\big|\omega_{X}^{\otimes k}\otimes\alpha\big|\text{and}E^{\prime}=R_{k}(E)\in\big|\omega_{Y}^{\otimes k}\otimes\beta\big|\) is the corresponding divisor, then \(F\) restricts to an equivalence of triangulated categories \(\mathrm{D}_{\text{Supp}(E)}^{b}(\text{Coh}(X))\simeq\mathrm{D}_{\text{Supp}(E^{\prime})}^{b}(\text{Coh}(Y))\). diff --git a/outputs/arxiv_math/2503.08443_pg27_pg1_repeat1.md b/outputs/arxiv_math/2503.08443_pg27_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..4737415dabaf5251e263aaf21a6ca36104d0382b --- /dev/null +++ b/outputs/arxiv_math/2503.08443_pg27_pg1_repeat1.md @@ -0,0 +1,39 @@ +Proof. If \(\overline{\lambda} \not\in \sigma(D|\mathcal{J})\) the resolvent \(R=(D- \overline{\lambda} I|\mathcal{J})^{-1}\) has the form + +\[Rx=\ell(x)\phi_{\overline{\lambda}}+V(I- \overline{\lambda} V)^{-1}x,\] + +with \(\ell\) a continuous linear functional on \(C^{\infty}(D)\). Since \(\mathcal{W}\varphi(\lambda)=0\) it follows that + +\[\varphi_{\lambda}(x)=\varphi(V(I- \overline{\lambda} V)^{-1}x)=\varphi(Rx)=0,\] + +for all \(x\in\mathcal{J}\) . By the form of \(\varphi\) it follows easily that \(\varphi_{\lambda}\) has order \(N-1\text{if}N>0\). Moreover, + +\[\mathcal{W}\varphi_{\lambda}(z)=\varphi(V(I- \overline{\lambda} V)^{-1}\phi_{z})^{\#}=\varphi((z- \overline{\lambda} )^{-1}(\phi_{z}-\phi_{\overline{\lambda}}))^{\#}= \frac{\mathcal{W}\varphi(z)}{z-\lambda} .\] + +(ii) follows by a repeated application of (i). + +### 6.3 Residual subspaces and their annihilators + +As pointed out before, we are especially interested in \(D-\)invariant subspaces \(\mathcal{J}\text{with}\sigma(D|\mathcal{J})=\emptyset\). Such subspaces will be called residual. They possess an alternative characterization based on the following simple observation. Lemma 3. If \(\mathcal{J}\in\mathscr{J}\text{and}p\) is a polynomial then \(p(D)\mathcal{J}\) is closed. Proof. It suffices to show that \((D-\lambda I)\mathcal{J},\lambda\in\mathbb{C}\) is closed. Let \((x_{k})\) be a sequence in \(\mathcal{J}\) such that \((D-\lambda I)x_{k}\rightarrow y\in H\). Then there is a sequence \((c_{k})\text{in}\mathbb{C}\) such that + +\[V(I-\lambda V)^{-1}(D-\lambda I)x_{k}=x_{k}-c_{k}\phi_{\lambda}\rightarrow V(I-\lambda V)^{-1}y,\] (30) + +when \(k\rightarrow\infty.\text{If}\phi_{\lambda}\in\mathcal{J}\) , it follows that \((x_{k})\) converges to \(x\in\mathcal{J}\text{and}y=(D-\lambda I)x\in(D-\lambda I)\mathcal{J}\) . + +If \(\phi_{\lambda}\not\in \mathcal{J}\) , let \(\varphi\) be a continuous linear functional in \(\mathcal{J}^{\perp}\text{with}\varphi(\phi_{\lambda})=1\). Then applying \(\varphi\) to both sides of (30) we obtain that \((c_{k})\) converges and as above, we obtain that \((x_{k})\) converges to \(x\in \underline{\mathcal{J}}\) and \(y=(D-\lambda I)x\in(D-\lambda I)\mathcal{J}\) . With the lemma in hand, the characterization of residual subspaces is as follows. Proposition 12. A subspace \(\mathcal{J}\in\mathscr{J}\) is residual if and only if + +\[\mathcal{J}=\bigcap\{p(D)\mathcal{J}:ppolynomial\}.\] + +Proof. If \(\sigma(D|\mathcal{J})=\emptyset\) we have \((D-\lambda)\mathcal{J}=\mathcal{J}\text{for all}\lambda\) and the equality in the statement holds. + +Conversely, if the equality holds, then \((D-\lambda I)\mathcal{J}=\mathcal{J}\) for each \(\lambda\). Suppose for a contradiction that + +\(D-\lambda_{0}I\) is not injective for some \(\lambda_{0}\). Then for each \(j\geq0\) there exists a nonzero \(x_{j}\in\mathcal{J}\) , such + +that \((D-\lambda_{0})^{j}x_{j}=\phi_{\lambda_{0}}\). From this it follows that \(V_{\lambda_{0}}^{j}\phi_{\lambda_{0}}\in\mathcal{J}\text{for all}j\geq0,\text{and}\mathcal{J}=C^{\infty}(D)\) by Proposition 8 (ii). Note that for every subspace \(\mathcal{J}\in\mathscr{J}\) we can consider its residual part + +\[\mathcal{J}_{res}=\bigcap\{p(D)\mathcal{J}:p\text{polynomial}\}.\] + +By the above proposition this is a closed residual subspace of \(\mathcal{J}\) Let us now turn to annihilators of residual subspaces. It turns out that these spaces can be described using the generalized Fourier transform \(\mathcal{W}\). \(\alpha\in\mathbb{R}\). + +To state our result we use the same notation as in Theorem 8, that is \(\mathcal{W}H=e^{-i\alpha z}\mathcal{H}(E)\), with diff --git a/outputs/arxiv_math/2503.08451_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.08451_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..b220b3d66f9933f42e2bc7314c35e74f2eba745e --- /dev/null +++ b/outputs/arxiv_math/2503.08451_pg2_pg1_repeat1.md @@ -0,0 +1,41 @@ +10x slower to train [10]. Therefore, subsequent works focused on convolutional neural network based blocks, which are faster to train. However, recently there is renewed interest in efficient RNNs and state space models both of which aim to address the inefficiencies of traditional RNNs. These newer developments aim to combine the sequential processing capabilities of RNNs with improved computational and memory efficiency, making them viable for longer sequences and higher throughput requirements. Key advancements include: 1.) State Space Models (SSMs): Recent models like S4 (Structured State Space for Sequence Modeling) [16] and its variants propose replacing traditional recurrent operations with structured linear systems that can process sequences in parallel while retaining the ability to model long-range dependencies. By leveraging the mathematical properties of state space representations, these models reduce the need for sequential backpropagation, offering significant speed and memory improvements. These efforts culminated in the recently introduced S6 model [17], which uses a specific Mamba block around the SSM core, gaining considerable attention due to its strong performance on sequential tasks. 2.) Work on parallelization of RNNs: where [18] linearized and diagonalized the recurrence; whereas [19] built on improving the LSTM model; and [20] introduced a new minimal version of GRUs and LSTMs, called minLSTM and minGRU. In this work, we analyze the minGRU model for its use in the Turbo autoencoder structure. For that we combine it with the Mamba block for SSMs and integrate it into the parallel Turbo structure of the encoder. We will show that it matches and at times outperforms the convolutional network performance, while being as fast to train as the convolutional networks for smaller sequence length, and having significant advantages at long block lengths. + +a) Notation: In this work, we distinguish between random vectors in the communication model and deterministic vectors in the neural network: an \(n-\)dimensional random vector is denoted by an uppercase letter with a superscript, e.g. \(X^{n}\), whose components are \(X_{1},X_{2},\ldots\) , and whose realized value is \(x^{n}\). In contrast, neural-network vectors are written in bold lowercase with a time subscript, e.g. \(\mathbf{x}_{t}\), to represent the input at time \(t\) in a recurrent model. For example, \(X^{n}\) corresponds to a random codeword of length \(n\) transmitted over the channel (Section III), whereas \(\mathbf{x}_{t}\) is the deterministic input signal at time \(t\) in the MinGRU-based block (Section II). + +## II. MINGRU-BASED BLOCK + +## A. Introduction to Recurrent Neural Networks and Gated Recurrent Units + +Recurrent Neural Networks (RNNs) are a class of neural networks designed to process sequential data by maintaining a hidden state that captures information about previous inputs. Unlike feedforward neural networks, RNNs share parameters across time steps, enabling them to model temporal dependencies effectively. Formally, given an input sequence \(\{\mathbf{x}_{t}\}_{t=1}^{T}\), the hidden state \(\mathbf{h}_{t}\) at time \(t\) is computed as: + +\[\mathbf{h}_{t}=f(\mathbf{h}_{t-1},\mathbf{x}_{t};\theta),\] (1) + +where \(f\) is a non-linear function (e.g., a combination of matrix multiplications and activation functions), \(\mathbf{h}_{t-1}\) is the hidden state from the previous time step, \(\mathbf{x}_{t}\) is the input at time \(t\), and \(\theta\) are the network parameters. One of the key challenges in training RNNs is the vanishing/exploding gradient problem, which limits their ability to capture long-term dependencies. Gated architectures like the Gated Recurrent Unit (GRU) were introduced to address this limitation. + +## B. Gated Recurrent Unit (GRU) + +The GRU is a simplified yet effective variant of the Long Short-Term Memory (LSTM) network. It uses gating mechanisms to control the flow of information, which helps in preserving long-term dependencies and mitigating the vanishing gradient issue. The GRU maintains a single hidden state \(\mathbf{h}_{t}\), updated as follows: + +1. Update Gate: The update gate \(\mathbf{z}_{t}\) determines how much of the past hidden state \(\mathbf{h}_{t-1}\) should be retained: + +\[\mathbf{z}_{t}=\sigma(\mathbf{W}_{z}\mathbf{x}_{t}+\mathbf{U}_{z}\mathbf{h}_{t-1}+\mathbf{b}_{z}),\] + +where \(\sigma\) is the sigmoid activation function, and \(\mathbf{W}_{z},\mathbf{U}_{z},\mathbf{b}_{z}\) are learnable parameters. + +2. Reset Gate: The reset gate \(\mathbf{r}_{t}\) controls how much of the past hidden state should influence the current computation: + +\[\mathbf{r}_{t}=\sigma(\mathbf{W}_{r}\mathbf{x}_{t}+\mathbf{U}_{r}\mathbf{h}_{t-1}+\mathbf{b}_{r}).\] + +3. Candidate Hidden State: A candidate hidden state \(\tilde{\mathbf{h}}_{t}\) is computed using the reset gate: + +\[\tilde{\mathbf{h}}_{t}=\tanh(\mathbf{W}_{h}\mathbf{x}_{t}+\mathbf{U}_{h}(\mathbf{r}_{t}\odot\mathbf{h}_{t-1})+\mathbf{b}_{h}),\] + +where ⊙ denotes the element-wise product. + +4. Final Hidden State: The final hidden state \(\mathbf{h}_{t}\) is a convex combination of the previous hidden state and the candidate hidden state, weighted by the update gate: + +\[\mathbf{h}_{t}=(1-\mathbf{z}_{t})\odot\mathbf{h}_{t-1}+\mathbf{z}_{t}\odot\tilde{\mathbf{h}}_{t}.\] + +Neglecting the bias terms, and streamlining the matrices as linear operations Linear\(d_{h}\) with a certain matching hidden dimension \(d_{h}\), a GRU cell has the following simplified form: + +\[\begin{aligned} \boldsymbol{z}_{t}&=\sigma(\text{Linear}_{d_{h}}([\boldsymbol{x}_{t},\boldsymbol{h}_{t-1}])) \\ \boldsymbol{r}_{t}&=\sigma(\text{Linear}_{d_{h}}([\boldsymbol{x}_{t},\boldsymbol{h}_{t-1}])) \\ \tilde{\boldsymbol{h}}_{t}&=\tanh(\text{Linear}_{d_{h}}([\boldsymbol{x}_{t},\boldsymbol{r}_{t}\odot\boldsymbol{h}_{t-1}])) \\ \boldsymbol{h}_{t}&=(\mathbf{1}-\boldsymbol{z}_{t})\odot\boldsymbol{h}_{t-1}+\boldsymbol{z}_{t}\odot\tilde{\boldsymbol{h}}_{t}. \end{aligned}\] diff --git a/outputs/arxiv_math/2503.08453_pg23_pg1_repeat1.md b/outputs/arxiv_math/2503.08453_pg23_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..33b8087c43d8c06107b0152e6a2f090a41a9b739 --- /dev/null +++ b/outputs/arxiv_math/2503.08453_pg23_pg1_repeat1.md @@ -0,0 +1,31 @@ +Proof. One has to impose \(d_{1,1}=0\text{and}d_{p,j}=0\) in eq. (5.7). Then, coefficients \(\omega_{i,j}\) vanish, because all the successive commutators \([K(h),\ldots,[K(h),\tilde{K}(h)]\ldots]\) appearing in the BCH formula applied to \(\mathrm{e}^{K(h)}\mathrm{e}^{\tilde{K}(h)}\) contain only pure imaginary terms up to order \(\mathcal{O}(h^{2p+1})\) . + +In consequence, if the problem (5.1) is defined in a Lie group \(\mathcal{G}\) , then the scheme \(S_{h}\) of order \(p\) is conjugate to a method that preserves the Lie group structure up to order \(2p+1\) .This is so up to order \(2p+2\) if \(\Psi_{h}\) is symmetric-conjugate and the order \(p\) is even, due to the particular structure of \(K_{h}\) in that case [5]. On the other hand, if \(\Psi_{h}\) is symmetric-conjugate and its order \(p\) is odd, then the resulting AC method is time-symmetric and of order \(p+1\) . + +### 5.2 New alternating-conjugate methods + +The analysis of the previous subsection shows that, in addition to concatenating a given method \(\Psi_{h}\) of order \(p\) (say palindromic or symmetric-conjugate) with the same scheme with complex conjugate coefficients, one can also get an alternating-conjugate method of order \(p\geq2\) by considering \(\Psi_{h}\) as in Proposition 5.1, namely by requiring the following order conditions: + +\[\begin{aligned} \Re(k_{1,1})&= \frac{1}{2} ,\Re(k_{p,j})=0;j=1,\ldots,c(p) \\ k_{\ell,j}&=0,\ell=2,\ldots,p-1;j=1,\ldots,c(\ell). \end{aligned}\] (5.12) + +The simplest AC method of order \(p=2\) corresponds of course to the composition \(S_{h}=\Phi_{\alpha h}\Phi_{\overline{\alpha} h}\) with + +\[\Phi_{\alpha h}=\mathrm{e}^{K(h)}\text{and}K(h)=\alpha hM+\alpha^{2}h^{2}Y_{2}+\alpha^{3}h^{3}Y_{3}+\cdots .\] + +By imposing \(\Re(\alpha)= \frac{1}{2} ,\Re(\alpha^{2})=0\) we get \(\alpha= \frac{1}{2} (1\pm i)\) , i.e., we recover method (5.2). Analogously, for an AC method of order 3 within this family one has to take \(\Psi_{h}=\Phi_{\alpha_{1}h}\Phi_{\alpha_{2}h}\Phi_{\alpha_{3}h}\) to satisfy the 5 required conditions (5.12). Although there are solutions with \(\alpha_{3}\in\mathbb{R}\) , it is more efficient to consider \(\Phi_{h}\) in the composition (5.4) as a 2nd-order time-symmetric method, namely + +\[\Psi_{h}=\Psi_{\alpha_{1}h}^{[2]}\Psi_{\alpha_{2}h}^{[2]}\cdots \Psi_{\alpha_{r}h}^{[2]}.\] (5.13) + +Now the number of order conditions (5.12) to achieve a method of order 3, 4, 5, 6 is, respectively, 2, 4, 7 and + +11. This is the strategy we follow next to construct higher order schemes with the minimum number of basic methods (or stages). We denote, for brevity, the whole AC method by its sequence of coefficients: + +\[S_{h}=(\alpha_{1},\alpha_{2},\ldots,\alpha_{r}, \overline{\alpha}_{1} , \overline{\alpha}_{2} ,\ldots, \overline{\alpha}_{r} ).\] + +Order 3. The two order conditions, \(\Re(k_{1,1})= \frac{1}{2} ,\Re(k_{3,1})=0\) can be satisfied with just one basic scheme, + +\(\Psi_{\alpha_{1}h}^{[2]}\) , if + +\[\alpha_{1}= \frac{1}{2} \pm i \frac{1}{2 \sqrt{3}} .\] + +In this way we recover the scheme (2.5), which is both symmetric-conjugate and AC. Notice that if \(\Psi_{h}^{[2]}\) is taken as the Strang splitting (2.3) for two operators, then the number of exponentials is 5 instead of 12 with a composition of the Lie–Trotter scheme. diff --git a/outputs/arxiv_math/2503.08457_pg18_pg1_repeat1.md b/outputs/arxiv_math/2503.08457_pg18_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..896947056eacbcff680d4f3a7f66b362789233aa --- /dev/null +++ b/outputs/arxiv_math/2503.08457_pg18_pg1_repeat1.md @@ -0,0 +1,33 @@ +## (1) The global section functor + +\[\Gamma:\text{Mod}_{\underline{\mathcal{S}}_{X}}\rightarrow\text{Mod}_{\underline{\mathcal{S}}_{X} (X)}\] + +is exact and establishes an equivalence of categories between the category of sheaves of right \(\underline{\mathcal{S}}_{X} -\)modules and the category of right modules over the global sections \(\underline{\mathcal{S}}_{X} (X)\) of \(\underline{\mathcal{S}}_{X}\). + +2. If \(\underline{F} \in\text{Mod}_{\underline{\mathcal{S}}_{X}}\) locally has finite resolutions by finitely generated free \(\underline{\mathcal{S}}_{X} -\text{modules}\), then \(\Gamma(X, \underline{F} )\) has a finite resolution by finitely generated projective modules. + +3. The derived category of perfect complexes of sheaves \(\mathrm{D}_{\text{Perf}}(\text{Mod}_{\underline{\mathcal{S}}_{X}})\) is equivalent to the derived category of perfect complexes of modules \(\mathrm{D}_{\text{Perf}}(\text{Mod}_{\underline{\mathcal{S}}_{X} (X)} \overline{)}\). + +By this theorem, there is a (strict) perfect complex of \(A^{0}-\text{modules}(E,E^{0})\) and a quasi-isomorphism e0 : \((E^{\bullet},E^{0})\rightarrow(F^{\bullet},F^{0})=(\Gamma(M, \underline{C}_{F}^{\infty} ),D)\). We shall follow the argument of Theorem 3.2.7 of [Blo05] to construct the higher components \(E^{i}\) of \(Z-\)connection along with the higher components of a morphism ei. On \(F^{\bullet}\), we have a \(Z-\)connection + +\[F=D\otimes1+1\otimes d:F^{\bullet}\rightarrow F^{\bullet}\otimes_{A^{0}}A^{\bullet}\] + +. The idea is to transfer this \(Z-\)connection to \(E^{\bullet}\) which is compatible with the quasi-isomorphism on H0’s. Note that we have an induced connection + +\[H^{k}:H^{k}(F^{\bullet},F^{0})\rightarrow H^{k}(F^{\bullet},F^{0})\otimes_{A^{0}}A^{1}\] + +for each k. First we will transfer this connection to a connection on \(H^{k}(E^{\bullet},E^{0})\), and we have the following commutative diagram + +\[\begin{aligned} H^{k}(E^{\bullet},E^{0})^{\underline{H^{k}}}H^{k}(E^{\bullet},E^{0})\otimes_{A^{0}}A^{1} \\ e^{0}e^{0}\otimes1 \\ H^{k}(F^{\bullet},F^{0})^{\underline{H^{k}}}H^{k}(F^{\bullet},F^{0})\otimes_{A^{0}}A^{1} \end{aligned}\] + +Note that \(e^{0}\otimes1\) is a quasi-isomorphism since \(A^{\bullet}\) is flat over A0. We need the following lemma. + +Lemma 2.10. Given a bounded complex offinitely generated projective \(A^{0}-\text{modules}(E^{\bullet},E^{0})\) with connections \(H^{k}:H^{k}(F^{\bullet},F^{0})\rightarrow H^{k}(F^{\bullet},F^{0})\otimes_{A^{0}}A^{1}\) for each k, there exists connections + +lifting \(H^{k}\), i.e. + +\[\tilde{H}^{k}:E^{k}\rightarrow E^{k}\otimes_{A^{0}}A^{1}\] + +\[\tilde{H}^{k}E^{0}=(E^{0}\otimes1)\tilde{H}^{k}\] + +for each k and the connection induced on the cohomology is \(H^{k}\). Proof. This is Lemma 3.2.8 in [Blo05] and Lemma 4.6 in [BS14]. Since \(E^{\bullet}\) is bounded, let [N, M] be its magnitude. Pick some arbitrary connection \(\nabla\) on EM. Consider the following diagram whose rows are exact diff --git a/outputs/arxiv_math/2503.08465_pg8_pg1_repeat1.md b/outputs/arxiv_math/2503.08465_pg8_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..8bad95e8a93383bf1ab8478ffc567b7db8303892 --- /dev/null +++ b/outputs/arxiv_math/2503.08465_pg8_pg1_repeat1.md @@ -0,0 +1,49 @@ +\[w_{\perp}^{T} \overline{A} w_{\perp}\geq\rho\Lambda w_{\perp}^{T}Mw_{\perp}\text{for all}w_{\perp}\in \overline{E}_{\leq\rho\Lambda}^{\perp}\] (4.1) + +Let \(W_{\perp}\) be the basis matrix of \(\overline{E}_{\leq\rho\Lambda}^{\perp} .\) Note that \(W\) can be obtained by solving few lowest eigenmodes of the pencil \(( \overline{A} ,M)\), whereas \(W_{\perp}\) is large, dense, and never available in a practical computation. In the following, we use the notation + +\[B_{\perp}:=W_{\perp}^{T}BW_{\perp},\forall B\in\mathbb{R}^{n\times n}.\] (4.2) + +We proceed to split eigenvectors of (1.1) as the sum of a component in \(\overline{E}_{\leq\rho\Lambda}\) and a correction term in \(\overline{E}_{\leq\rho\Lambda}^{\perp} .\) A suitable splitting is given in Lemma 4.2 after a technical result. + +Lemma 4.1. Assume that \(A:S\rightarrow\mathbb{S}_{++}^{n\times n}\) is spectrally equivalent to \(\overline{A}\) as in (3.1) with constants positive definite for all \((\sigma,t)\in S\times(0;\Lambda)\). + +\(\alpha\text{and}\beta\). Let \(\Lambda>0\text{and}\rho\) be such that \(\alpha\rho>1\text{and}\rho>1\). Then the matrix \(A_{\perp}(\sigma)-tM_{\perp}\) is + +Proof. For all \((\sigma,t)\in S\times(0;\Lambda)\text{and}w_{\perp}\in \overline{E}_{\leq\rho\Lambda}^{\perp} \setminus\{0\},\) from the spectral equivalence of \(A(\sigma)\) with \(\overline{A}\), we get + +\[w_{\perp}^{T}A(\sigma)w_{\perp}-tw_{\perp}Mw_{\perp}\geq\alpha w_{\perp}^{T} \overline{A} w_{\perp}-tw_{\perp}Mw_{\perp}\geq(\alpha\rho\Lambda-t)\|w_{\perp}\|_{M}^{2}>0\] (4.3) + +\[\perp\leq\rho\Lambda^{.}\] + +as \(w_{\perp}\in \overline{E}\) Lemma 4.2. Make the same assumptions as in Lemma 4.1. Let \(\sigma\in S\text{and}(\lambda \underline{(} \sigma),x(\sigma))\in (0;\Lambda)\times\mathbb{R}^{n}\setminus\{0\}\) be a solution to (1.1). In addition, let \(W_{\perp}\) be a basis matrix of \(\overline{E}_{\leq\rho\Lambda}^{\perp} .\) Then there holds that + +\[x(\sigma)= \overline{x} (\sigma)+Z(\sigma,\lambda(\sigma)) \overline{x} (\sigma),\] (4.4) + +for \(\overline{x} (\sigma)\in \overline{E}_{\leq\rho\Lambda}\) and correction operator \(Z(\sigma,t):S\times(0,\Lambda)\mapsto\mathbb{R}^{n\times n}\) defined as + +\[Z(\sigma,t)=W_{\perp}(W_{\perp}^{T}(A(\sigma)-tM)W_{\perp})^{-1}W_{\perp}^{T}\delta A(\sigma)\] (4.5) + +with \(\delta A(\sigma)=A(\sigma)- \overline{A}\). Proof. Let \(\sigma\in S\) and split \(x(\sigma)= \overline{x} (\sigma)+r(\sigma)\text{with} \overline{x} (\sigma)\in \overline{E}_{\leq\rho\Lambda} \text{and}r(\sigma)=W_{\perp}\gamma(\sigma)\in \overline{E}\) + +\[\perp\leq\rho\Lambda^{.}\] + +Then as \((\lambda(\sigma),x(\sigma))\in(0;\Lambda)\times\mathbb{R}^{n}\) is a solution of (1.1), we get + +\[(A(\sigma)-\lambda(\sigma)M) \overline{x} (\sigma)=-(A(\sigma)-\lambda(\sigma)M)W_{\perp}\gamma(\sigma).\] (4.6) + +By orthogonality, we first observe that \(W_{\perp}^{T}M \overline{x} (\sigma)=0\text{and}W_{\perp}^{T} \overline{A} \overline{x} (\sigma)=0\). By multiplication with basis matrix \(W_{\perp}^{T}\) and using these identities gives + +\[W_{\perp}^{T}(A(\sigma)- \overline{A} ) \overline{x} (\sigma)=-W_{\perp}^{T}(A(\sigma)-\lambda(\sigma)M)W_{\perp}\gamma(\sigma).\] (4.7) + +Then from Lemma 4.1, matrix in the right-hand side of equation (4.7) is invertible and we obtain + +\[-W_{\perp}(W_{\perp}^{T}(A(\sigma)-\lambda(\sigma)M)W_{\perp})^{-1}W_{\perp}^{T}\delta A(\sigma) \overline{x} (\sigma)=W_{\perp}\gamma(\sigma).\] + +It is straightforward to estimate the \(\overline{A} -\)norm of \(\overline{x} (\sigma)\) appearing in (4.4). Recall that \(\overline{x} (\sigma)^{T} \overline{A} W_{\perp}=0\) and using spectral equivalence (3.1) gives + +\[\| \overline{x} (\sigma)\|_{\overline{A}}\leq\|x(\sigma)\|_{\overline{A}}\leq \frac{1}{\alpha} \|x(\sigma)\|_{A(\sigma)}.\] (4.8) + +As \(x(\sigma)\) is an eigenvector of the pencil \((A(\sigma),M)\) corresponding to an eigenvalue on \((0,\Lambda)\) and \(\|x(\sigma)\|_{M}=1\), it follows that + +\[\|x(\sigma)\|_{A(\sigma)}\leq\Lambda^{1/2}\text{and thus}\| \overline{x} (\sigma)\|_{\overline{A}}\leq \frac{1}{\alpha} \Lambda^{1/2}.\] (4.9) diff --git a/outputs/arxiv_math/2503.08475_pg18_pg1_repeat1.md b/outputs/arxiv_math/2503.08475_pg18_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..1439ab2a8c1da960c5304ba2810fb6b59c0a8706 --- /dev/null +++ b/outputs/arxiv_math/2503.08475_pg18_pg1_repeat1.md @@ -0,0 +1,35 @@ +We start with the inclusion \(\supseteq\) and hence it is enough to show for \(n\preceq m\) aperiodic that \(S_{gen}(n)\subseteq W(\rho\times S_{gen}(m^{\prime}))\). We first note that since \(m^{\prime}\text{and}m\) are not aperiodic, we decompose \(m=m_{p}+m_{ap}\), where \(m_{p}=[a,b]_{\rho}+\ldots+[a+o(\rho)-1,b+o(\rho)-1]_{\rho}\text{and}[a,b]_{\rho}\) is the minimal \([a,b]_{\rho}\) found in \(m\) with \([a,b]_{\rho}+\ldots+[a+o(\rho)-1,b+o(\rho)-1]_{\rho}\) contained in \(m\). Note that \(m\neq m_{p}\), since otherwise there exists no \(m^{\prime}\text{such that}\rho*m^{\prime}=m\). Let \(n\preceq m\) be an aperiodic multisegment. There exists \(n_{p}\prec m_{p}\) obtained via one elementary operation \([c,d]_{\rho}+[c+1,d+1]_{\rho}\mapsto[c,d+1]_{\rho}+[c+1,d]_{\rho}\text{such that}m\preceq n_{p}+m_{ap}\) by Lemma 2.2.3. Let \([1,b^{\prime}]_{\rho}\) be the segment in \(m^{\prime}\text{such that}m=m^{\prime}-[1,b^{\prime}]_{\rho}+[0,b]_{\rho}\). Let us first assume that either \(c\neq1\mod o(\rho)\text{or}c=1\text{mod}o(\rho)\text{and}l([1,b^{\prime}]_{\rho})\geq l([c,d+1]_{\rho})\). Then it is easy to see that there exists \(n^{\prime}\text{such that}\rho*n^{\prime}=n_{p}+m_{p}\text{and}n^{\prime}\preceq m^{\prime}\). In this case we have by induction + +\[S_{gen}(n)\subseteq S_{gen}(\rho*n^{\prime})=W(\rho\times S_{gen}(n^{\prime}))\subseteq W(\rho\times S_{gen}(m^{\prime})).\] + +If \(c=1\mod o(\rho)\text{and}l([1,b^{\prime}]_{\rho})\leq l([c,d+1]_{\rho})\) we obtain that \([1,b^{\prime}]=[c,d]_{\rho}\). In this case we differentiate again \(o(\rho)>2\text{and}o(\rho)=2\). In the former, we let \(m^{\prime\prime}=m-[c,d+1]_{\rho}-[c-1,d]_{\rho}+[c+ 1,d+1]_{\rho}-[c,d]_{\rho}\) and note that \(\rho∣-∣*\rho*n^{\prime\prime}=m\rho*\rho∣-∣*n^{\prime\prime}=n\). From here the claim follows as in the aperiodic case. If \(o(\rho)=2\) we let \(m^{\prime\prime}=m-[c,d+1]_{\rho}-[c-1,d]_{\rho}+[c+2,d+1]_{\rho}-[c,d]_{\rho}\) and note that \(\rho∣-∣*\rho*\rho*n^{\prime\prime}=m\rho*\rho∣-∣*\rho*n^{\prime\prime}=n\) and we again finish the argument as in the aperiodic case. It remains to show that \(W(\rho\times S_{gen}(m^{\prime}))\subseteq S_{gen}(m)\). Since \(m^{\prime}\) cannot be aperiodic, it is enough to show for aperiodic \(n\preceq m^{\prime},\rho*m^{\prime}\preceq m\), which follows from Lemma 2.4.1. + +\[\mathbf{Claim}\mathbf{4}:m^{\prime}*\rho=m\Rightarrow W(\rho\times S_{gen}(m^{\prime}))=S_{gen}(m)\] + +\[\text{This can be proven as Claim}1\text{and Claim}3.\] + +Finally for \(m\in Mult_{R,◻}\), let \(m=m_{1}+\ldots+m_{k}\in Mult_{R}\) be a multisegment with \(m_{i}\in Mult_{\overline{\mathbb{F}}_{\ell}}(\rho_{i}),\rho_{i}\in C^{◻}\) and in pairwise different cuspidal lines. Then we set + +\[S_{gen}(m):=W(S(m_{1})\times\ldots\times S(m_{k}))\] + +and if \(m^{\prime}=m_{1}^{\prime}+\ldots+m_{k}^{\prime}\in Mult_{R,◻},m_{i}\in Mult_{\overline{\mathbb{F}}_{\ell}}(\rho_{i})\) is a second product, we define we + +\[m*m^{\prime}:=m_{1}*m_{1}^{\prime}+\ldots+m_{k}*m_{k}^{\prime},\] + +extending the product ∗ to \(*Mult_{R,◻}\times Mult_{R,◻}\rightarrow Mult_{R,◻}.\) Lemma 3.4.2. Let \(m=m_{1}+\ldots+m_{k}\in Mult_{R,◻}\) as above. Then + +\[W(S_{gen}(m))\cong S_{gen}(m_{1})\times\ldots\times S_{gen}(m_{k})\] + +and hence for \(m^{\prime}\in Mult_{R,◻}a\) second multisegment + +\[W(S_{gen}(m)\times S_{gen}(m^{\prime}))=S_{gen}(m*m^{\prime}).\] + +Moreover, if \(n\preceq m\text{then}S_{gen}(n)\subseteq S_{gen}(m)\). + +Proof. Note that there exists a non-zero map + +\[S_{gen}(m_{1})\times\ldots\times S_{gen}(m_{k})\rightarrow W(S_{gen}(m)).\] + +Let \(\pi\) be the unique irreducible subrepresentation of the latter representation. It is then enough to show that \(\pi\) is the unique subrepresentation of the former representation. To see this note that by Theorem 2.2.1 and Lemma 2.2.1, the representation \(\pi_{1}\times\ldots\times\pi_{k}\) is irreducible, where \(\pi_{i}\) is the unique irreducible subrepresentation of \(S_{gen}(m_{i})\). Since \(\pi_{1}\times\ldots\times\pi_{k}\) is generic, it has to be isomorphic to \(\pi\). Thus \(\pi\) is a subrepresentation of + +\[S_{gen}(m_{1})\times\ldots\times S_{gen}(m_{k}).\] diff --git a/outputs/arxiv_math/2503.08488_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.08488_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..16e78c1dc39a0b30ecc9ded2e5cd66386cd6bb49 --- /dev/null +++ b/outputs/arxiv_math/2503.08488_pg2_pg1_repeat1.md @@ -0,0 +1,31 @@ +| 5 Proof of Theorem 2.1 | 15 | +| --- | --- | +| 6 Proof of Proposition 4.1 | 17 | +| 6.1 Edge pairing: a preliminary version | 20 | +| 6.2 Implication of Assumption 2.2: Resolution of Difficulties [A1] and [B1] . . . | 23 | +| 6.3 Edge pairing and weight assignments | 25 | +| 6.4 Refined pairing and weight assignments | 27 | +| 6.4.1 Proof of Proposition 4.1 by assuming Proposition 6.11 | 29 | +| 6.4.2 Switching between graphs of equal weights: Proof of Proposition 6.11 | 31 | + +## 1 Introduction + +In the present paper, we are interested in the phase transition at the critical inverse temperature of the XY model on the three-dimensional square lattice. To formulate the problem we start by considering a finite subset of a three-dimensional square lattice + +\[\mathcal{L}=[-L,L]^{3}\bigcap\mathbb{Z}^{3}.\] (1.1) + +On this lattice the Hamiltonian is defined as + +\[H_{L,\nu}:=-\sum_{k,l\in\mathcal{L}}J_{k,l}S_{k}\cdot S_{l}\] (1.2) + +and \(\nu=+\text{or}0.\) Here, depending on the location of the site \(k,S_{k}\in\mathbb{S}^{1}\) satisfies different conditions: the lattice is decomposed into two parts, + +\[\mathcal{L}=\mathcal{L}^{o}\cup\partial\mathcal{L},\] (1.3) + +with \(\mathcal{L}^{o}\) being the interior, and \(\partial\mathcal{L}=\{(z_{1},z_{2},z_{3})|\max\{|z_{1}|,|z_{2}|,|z_{3}|\}=L\}\) the boundary; for \(k\in\mathcal{L}^{o}\), the only requirement for \(S_{k}\) is that \(|S_{k}|=1;\text{when}k\in\partial\mathcal{L}\) i.e. it is on the boundary, we consider two boundary conditions: when \(\nu=0\), we use the free boundary condition, specifically + +\[S_{k}=0,\text{if}k\in\partial\mathcal{L};\] (1.4) + +when \(\nu=+\), the plus boundary condition, + +\[S_{k}=(1,0)^{T},\text{if}k\in\partial\mathcal{L}.\] (1.5) diff --git a/outputs/arxiv_math/2503.08494_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.08494_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..349ebf790e7e3c05bf0c171115cd71de67289e87 --- /dev/null +++ b/outputs/arxiv_math/2503.08494_pg2_pg1_repeat1.md @@ -0,0 +1,63 @@ +vacy protection across computer science, control engineering, and communication technology. Multiple types of research have been illustrated for distributed NE seeking with DP method [17]–[20]. A privacy-preserving distributed algorithm for seeking the NE was designed in [17] for aggregative games and [18] focused on quadratic network games. [19] extended privacy-preserving mechanisms for NE seeking to directed graph. Considering coupling constraints, [20] proposed a GNE seeking strategy for multi-cluster games. Though [21] investigated GNE seeking with privacy preservation, information exchange efficiency was not considered. Privacy and communication efficiency are seldom explored together in distributed GNE seeking for aggregative games though it is of great interest. The motivation of this paper is to study aggregative games with coupling constraints and to design an efficient distributed algorithm to seek GNE of the games. The principal contributions of this paper can be outlined as follows: + +1. We provide a novel communication-efficient distributed GNE seeking algorithm (Algorithm 1), combining an event-triggered mechanism to decrease transmission rounds and a stochastic compressor to minimize transmitted bits. + +2. Besides achieving efficient communication, our algorithm ensures convergence to exact GNE by developing precise step size conditions (Theorem 1). In contrast to existing GNE seeking works ensuring accuracy [21], our proposed algorithm further reduces the communication cost. + +3. We show that \((0,\delta)-\)differential privacy is achieved under a stochastic compressor (Theorem 2). Particularly, the algorithm guarantees asymptotic convergence to the GNE and privacy protection simultaneously. + +The organization of this paper is as follows: Section II presents the fundamental concepts and establishes the formulation for the problem. Section III provides an event-triggered and compressed distributed GNE seeking algorithm with an analysis of convergence and privacy. Section IV provides simulation examples to confirm the obtained results, and Section V concludes the paper. Notations: Let \(\mathbb{R}(\mathbb{R}^{+})\) denote the set of real (positive real) numbers, \(\mathbb{Z}\) the integers, and \(\mathbb{N}\) the natural numbers. For \(n\in\) matrices). Let \(P\) T and \([P]_{ij}\) denote the transpose and \((i,j)-\) th entry of matrix \(P\) , respectively. The gradient of \(f\) at x is denoted by \(\nabla f(\mathbf{x}),\text{with}\nabla_{i}f(\mathbf{x})\) as its \(i-\)th component. Define \(\mathbf{1}_{N}(\mathbf{0}_{N})\) as the \(N-\)dimensional all-ones (all-zeros) vector, and \(I_{d}\) as the \(d-\)dimensional identity matrix. The inner product and Euclidean (induced-2) norm are denoted by \(\langle\cdot,\cdot\rangle\text{and}|\cdot|\), respectively. Let \(\Pi_{\mathcal{D}}(\cdot)\) be the Euclidean projection onto a closed convex set \(\mathcal{D}\). For probability and expectation, we use + +\(\mathbb{N},\mathbb{R}^{n}(\mathbb{R}^{n\times d})\) represents \(n-\)dimensional real vectors \((n\times d\) real + +\(\mathbb{P}(\cdot)\text{and}\mathbb{E}[x]\). Finally, \(\text{col}(x_{1},...,x_{m})=[x_{1}^{\mathrm{T}},...,x_{m}^{\mathrm{T}}]^{\mathrm{T}}\) stacks vectors \(x_{1},...,x_{m}\). + +## A. Problem Formulation + +Let us examine an aggregative game comprising \(N\) players, where each participant \(i\in\mathcal{V}=\{1,2,...,N\}\) is associated + +with a decision vector \(x_{i}\in\Omega_{i},\text{with}\Omega_{i}\subseteq\mathbb{R}^{d}\) representing a compact convex set, and they share the constraint + +\[g(\bar{x})\leq0,\bar{x}= \frac{1}{N} \sum_{i=1}^{N}x_{i}\in\Upsilon\subset\mathbb{R}^{d}\] (1) + +where \(g:\mathbb{R}^{d}\rightarrow\mathbb{R}\) is the constrained function. We assume that each player aims at minimizing its local cost function \(J_{i}(x_{i},h(\mathbf{x})):\Omega\rightarrow\mathbb{R},\text{where}\mathbf{x}=\text{col}(x_{1},x_{2},...,x_{N})\in\Omega= \Omega_{1}\times\Omega_{2}\times...\times\Omega_{N}\text{and}h(\mathbf{x})=\bar{x}\) is the aggregation function. The function \(J_{i}(x_{i},h(\mathbf{x}))\) is continuously differentiable and convex in \(x_{i}\) for every fixed \(h(\mathbf{x})\). In particular, the problem is formulated as follows: Problem 1: Each player \(i\) intends to + +\[\begin{aligned} \mathrm{m}_{x}\mathrm{i}_{i}\mathrm{n}_{\in}\text{im}_{\Omega}\text{iz}_{i}\mathrm{e}J_{i}(x_{i},\bar{x}) \\ \text{subject to}g(\bar{x})&\leq0. \end{aligned}\] + +The resolution of Problem 1 yields a GNE, characterized as follows. [22] A strategy profile \(\mathbf{x}^{*}\) + += + +1. + +Definition + +\(\text{col}(x_{1}^{*},x_{2}^{*},...,x_{N}^{*})\in\Omega\cap\Upsilon\) is called a GNE of the constrained game if + +\[J_{i}(x_{i}^{*},h(\mathbf{x}^{*}))\leq J_{i}(x_{i}, \frac{1}{N} x_{i}+\sum_{j\neq i} \frac{1}{N} x_{j}^{*}),\] + +for all \(x_{i}:(x_{i},x_{-i}^{*})\in\Omega\cap\Upsilon,\text{where}x_{-i}^{*}= \text{col}(x_{1}^{*},\ldots,x_{i-1}^{*},x_{i+1}^{*},\ldots,x_{N}^{*})\). If \(J_{i}(x_{i},h(x))\) is regarded as a function of \(x\), then \(J_{i}(x_{i},h(x))\) can be written as \(J_{i}(\mathbf{u}),u\in\Omega\) for simplism and the following assumptions about the gradient are associated with the GNE. \(1)J_{i}(\cdot)\) is Lipschitz continuous, i.e., there exists a constant \(l_{J}\) such that for all \(\mathbf{u},\mathbf{v}\in\Omega\text{and}i\in\mathcal{V}\), + +Assumption 1. + +\[\Vert J_{i}(\mathbf{u})-J_{i}(\mathbf{v})\Vert\leq l_{J}\Vert\mathbf{u}-\mathbf{v}\Vert.\] + +\(2)g(\cdot)\) is Lipschitz continuous, i.e., there exists a constant + +\(l_{g}\) such that for all \(u,v\in\mathbb{R}^{d}\), + +\[\Vert g(u)-g(v)\Vert\leq l_{g}\Vert u-v\Vert.\] + +on the compact set \(\mathbb{R}^{d}\). In other words, there exists a constant + +The Lipschitz continuous condition implies \(g(\cdot)\) is bounded \(C_{g}>0\text{such that}\Vert g(\cdot)\Vert\leq C_{g}. 1)\nabla J_{i}(\cdot)\) is Lipschitz continuous, i.e., there exists a constant \(G_{J}\) such that for all \(\mathbf{u},\mathbf{v}\in\Omega\) and + +Assumption 2. + +\[i\in\mathcal{V},\] + +\[\Vert\nabla_{i}J_{i}(\mathbf{u})-\nabla_{i}J_{i}(\mathbf{v})\Vert\leq G_{J}\Vert\mathbf{u}-\mathbf{v}\Vert.\] + +\(2)\nabla_{i}g(\cdot)\) is Lipschitz continuous, i.e., there exists a constant \(G_{g}\) such that for all \(u,v\in\mathbb{R}^{d}\), + +\[\Vert\nabla_{i}g(u)-\nabla_{i}g(v)\Vert\leq G_{g}\Vert u-v\Vert.\] diff --git a/outputs/arxiv_math/2503.08498_pg10_pg1_repeat1.md b/outputs/arxiv_math/2503.08498_pg10_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..8d67897776956fdb52d9b882473c78f3f8963990 --- /dev/null +++ b/outputs/arxiv_math/2503.08498_pg10_pg1_repeat1.md @@ -0,0 +1,32 @@ +| n | p(z) | g(z) | a, b | N z4 p(z) | +| --- | --- | --- | --- | --- | +| 4 | z4 − 1 | - | - | z(z4+3) | +| | | | | 4 | +| 3 | (z − 1)2(z2 + az + b) | (a − 2)z2+ (−3a + 2b)z − 4b | a = 2 b = 3 | z(z3+z2+z+9) 12 | +| 2 | (i) (z − 1)2(z − a) 2 | 2(2a − (a + 1)z) | a = −1 | z(z2+3) 4 | +| 2 | (ii) (z − 1)3(z − a) | −(a + 3)z + 4a | a = −3 | z(z2+2z+9) | +| | | | | 12 | +| 1 | (z − 1)4 | 4 | - | z(z+3) | +| | | | | 4 | + +Table 1: Newton maps \(N_{\frac{z4}{p(z)}}\) with an exceptional point + +1. Let \(p\) be generic. If \(p\) is linear, then it follows from the Scaling property that \(N_{R}(z)\) is conjugate to \(N^{z}\) + +\(\overline{z-1}\) . + +Let \(p\) be non-linear. As \(p\) is generic and \(N_{\frac{zd}{p(z)}}\) is a polynomial, we have \(zp^{\prime}(z)- dp(z)=\alpha\) for some non-zero \(\alpha\). Letting \(y=p(z)\), we have the first-order linear differential equation \(y^{\prime}- \frac{d}{z} y= \frac{\alpha}{z}\) . The solution is + +\[y\cdot \frac{1}{z^{d}} =\int \frac{\alpha}{z} \cdot \frac{1}{z^{d}} dz+\beta=- \frac{\alpha}{dz^{d}} +\beta,\] + +for an arbitrary constant \(\beta\). Therefore, \(p(z)=\beta z^{d}- \frac{\alpha}{d} \text{and}R(z)= \frac{z^{d}}{\beta z^{d}- \frac{\alpha}{d}}\) . Consider \(c\text{such that}c^{d}= \frac{\alpha}{d\beta}\) and use the Scaling property to see that \(N_{R}\) is conjugate to \(N_{\beta R(cz)}\). We are done since \(\beta R(cz)= \frac{z^{d}}{z^{d}-1}\) and the resulting Newton map is . + +\[\frac{z^{d+1}+(d-1)z}{d}\] + +2. Let \(n\) denote the number of distinct roots of \(p\). Using the Scaling property, we assume without loss of generality that 1 is a multiple root of \(p\text{whenever}p\) is not generic. Along with this, what is going to be repeatedly used in all the following cases is that \(g(z)\) is a non-zero constant (see Equation (3.1)). + + (a) Let \(\deg(p)=3\). Then there are three cases depending on the values of \(n\). If \(n=3,\text{then}p\) is generic, and from the first part of this theorem, it follows that \(p(z)=z^{3}-1\), and hence \(N_{R}(z)= \frac{1}{3} z(z^{3}+2).\text{If}n=2,\text{then}p\) has a root with multiplicity 2 and therefore \(p(z)=(z-1)^{2}(z-a),\text{where}a\neq 0,1\). In this case, \(g(z)=-(a+2)z+3a\) and therefore \(a=-2\). Thus, \(N_{R}(z)= \frac{z}{6} (z^{2}+z+4)\). If \(n=1,\text{then}p(z)=(z-1)^{3}\), and we get \(N_{R}(z)= \frac{1}{3} z(z+2)\). + + (b) Let \(\deg(p)=4\). All possible cases of \(p\) and the resulting Newton maps are given in Table 1. + + (c) For \(\deg(p)=5\), all the possible forms of \(p\) and resulting Newton maps are given in Table 2. diff --git a/outputs/arxiv_math/2503.08500_pg11_pg1_repeat1.md b/outputs/arxiv_math/2503.08500_pg11_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..95ce9db769662c9acfe7f3ba58f991631e284b3b --- /dev/null +++ b/outputs/arxiv_math/2503.08500_pg11_pg1_repeat1.md @@ -0,0 +1,49 @@ +# \(\rightarrow\) + +![](figure) + +Figure 12. Two ♯-moves convert \(T(9,2)\) (left) to the unknot (right). + +the standard projection. It follows that \(C(4_{1})=2\). Clark also showed that \(C(K)\leq2\cdot g(K)+1\) for any knot \(K\) and asked if this upper bound was ever achieved. Clark’s question was finally answered in 1995 by Murakami and Yasuhara [30] using the Gordon-Litherland pairing. To do this, they first proved the following necessary condition for a knot \(K\) to + +bound a \(\beta_{1}=2\) nonorientable spanning surface \(F\) . + +Theorem 4.1 (Murakami-Yasuhara [30]). There is a pair of curves generating \(H_{1}(F;\mathbb{Z})\) such that the Goeritz matrix of \(F\) takes the form: + +\[G_{F}(K)= \begin{bmatrix} l & m \\ m & n \end{bmatrix}\] + +and satisfies \(\sigma(K)=\text{sign}(G_{F}(K))-(l+2m+n),\text{where}l,n\) are odd and \(m\) is even. + +Consider, as in [30], the knot 74. This knot has 3-genus 1 and hence \(C(7_{4})\leq3\). Since 74 is + +hyperbolic, we have \(C(7_{4})\geq2\). It therefore suffices to show that \(C(K)\neq2\). Now, suppose that + +\(C(K)=2\). Since 74 has determinant 15, one can write out integral binary quadratic forms with determinant 15 (see e.g. Conway-Sloane [8]). Using a case analysis, it is then possible to show that + +no such binary quadratic form satisfies Theorem 4.1. Hence, it follows that \(C(7_{4})=3=2\cdot g(7_{4})+1\). Crosscap numbers are in general difficult to compute. Yet progress is currently being made. For example, Kalfagianni-Lee [19] determined upper and lower bounds on the crosscap numbers for several families of knots in terms of the coefficients of the Jones polynomial. + +4. 3. Non-orientable slice genus. The (smooth) 4-genus of a knot \(K\) is the smallest genus among + +all surfaces properly embedded in \(\mathbb{B}^{4}\text{such that}\partial F=K\). A well-known lower bound on the 4-genus knot \(K\text{in}\mathbb{S}^{3}\) is defined as the smallest first Betti number of all compact connected non-orientable + +is \(∣\sigma(K)∣/2\). In analogy with crosscap number, the (smooth) non-orientable slice genus \(\gamma^{*}(K)\) of a + +surfaces \(F\) smoothly embedded in \(\mathbb{B}^{4}\text{such that}\partial F=K.\text{If}K\) is slice, then \(\gamma^{*}(K)\) is defined to be + +0. In 1975, Viro [39] showed that \(\gamma^{*}(4_{1})=2\), and hence 41 does not bound a Möbius band in \(\mathbb{B}^{4}\). + +Another proof that \(\gamma^{*}(4_{1})=2\) follows from a result of Yasuhara, which is proved using the + +Gordon-Litherland pairing. Recall that the quadratic form \(q:H_{1}(F;\mathbb{Z}/2\mathbb{Z})\rightarrow\mathbb{Z}/2\mathbb{Z}\) of a knot \(K\) + +is defined by \(q(x)=x^{⊺}Ax,\text{where}F\) is a Seifert surface of \(K\text{and}A\) is a Seifert matrix for \(F\) . The + +Arf invariant of \(K\) is the Arf invariant of this quadratic form. Hence, \(\text{Arf}(K)=0\text{if}q\) maps a + +majority of elements of \(H_{1}(F;\mathbb{Z}/2\mathbb{Z})\) to 0 and otherwise \(\text{Arf}(K)=1\). The Arf invariant is also the + +\(\pmod{2}\) coefficient of \(z^{2}\) in the Conway polynomial of \(K\). Note that there is a well-defined injection + +\(\mathbb{Z}/2\mathbb{Z}\rightarrow\mathbb{Z}/8\mathbb{Z}\) given by \(x\rightarrow4\cdot x\). Theorem 4.2 (Yasuhara [40]). If a knot \(Kin\mathbb{S}^{3}\) bounds a Möbius band in \(\mathbb{B}^{4}\), then + +\[\begin{aligned} \sigma(K)+4\cdot Arf(K)&\equiv0or\pm2\pmod{8}. \\ 11 \end{aligned}\] diff --git a/outputs/arxiv_math/2503.08504_pg4_pg1_repeat1.md b/outputs/arxiv_math/2503.08504_pg4_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..c0d3d62151b79c808c450b5b4d8d843b6b382610 --- /dev/null +++ b/outputs/arxiv_math/2503.08504_pg4_pg1_repeat1.md @@ -0,0 +1,37 @@ +which gives another necessary condition for (7) on the sphere \(S^{d}\) + +\[\frac{1}{\beta} \geq d- \frac{2d}{q} -\sigma.\] (9) + +If the sharp Schr¨odinger admissible condition \(\frac{1}{p} = \frac{d}{2} ( \frac{1}{2} - \frac{1}{q} )\) holds, then (9) is equivalent to + +\[\frac{1}{\beta} \geq \frac{4}{p} -\sigma.\] (10) + +If the sharp wave admissible condition \(\frac{1}{p} = \frac{d-1}{2} ( \frac{1}{2} - \frac{1}{q} )\) holds, then (9) is equivalent to + +\[\frac{1}{\beta} \geq \frac{d}{d-1} \frac{4}{p} -\sigma.\] (11) + +| | 1 | d | 4 | +| --- | --- | --- | --- | +| (11) | ě | | ´ σ. | + +Now, we introduce our main new results on the fractional Schr¨odinger, the wave and the Klein-Gordon equations on compact manifolds. + +supercritical regime + +subcritical regime + +0 + +![](figure) + +Figure 1. Sharp Schr¨odinger admissible pairs + +1. 2. Fractional Schr¨odinger equation. Suppose \(p\geq2,q<\infty\text{and} \frac{1}{p} = \frac{d}{2} ( \frac{1}{2} - \frac{1}{q} )\). We divide these sharp Schr¨odinger admissible pairs \((p,q)\) into four groups. See Figure 1. + +\[\frac{2(d+1)}{d-1}\] + +\[\frac{2(d+1)}{d-1}\] + +(i) Subcritical regime: \(d\geq1,2\leq q<\) (ii) Critical point: \(d\geq2,q=\) (iii) Supercritical regime: \(d=2, \frac{2(d+1)}{d-1} 0\) is the scale up to which the isoperimetric inequality holds in X, CI is as in (9.1), and kI is as in (3.1). Let \(00:E_{\mathbf{I}}(u|_{B(p,r)})\geq5^{-1}\varepsilon_{\mathbf{I}}\right\}\] + +and let ¯r > 0 be the infimum of the r(p) over all \(p\in S^{2}\). We clearly have + +\[E_{\mathbf{I}}(u|_{B(p,\bar{r})})\leq \frac{\varepsilon_{\mathbf{I}}}{5}\] + +for every \(p\in S^{2}\) and there exists p¯ \(\in S^{2}\) such that equality holds for p = p¯. If r¯ \(\geq r_{0}\) then the proposition holds with \(\eta\) being the identity mapping, so we may assume that ¯r \(E_{\mathbf{I}}(u)- \frac{\varepsilon_{\mathbf{I}}}{5} .\] (9.2) + +The proposition easily follows from this together with Lemma 9.3. Indeed, let \(\eta:S^{2}\rightarrow S^{2}\) be as in the lemma applied with p0 = p¯ and r = r¯. Then for every \(p\in S^{2}\) we have + +\[\eta(B(p,r_{0}))\subset B(\eta(p),\bar{r})\text{or}\eta(B(p,r_{0}))\cap B(\bar{p},L\bar{r})=\emptyset.\] + +In the first case we obtain + +\[E_{\mathbf{I}}(u\circ\eta|_{B(p,r_{0})})\leq E_{\mathbf{I}}(u|_{B(\eta(p),\bar{r})})\leq \frac{\varepsilon_{\mathbf{I}}}{5}\] + +and in the second case + +\[E_{\mathbf{I}}(u\circ\eta|_{B(p,r_{0})})\leq E_{\mathbf{I}}(u|_{S^{2}\setminus B(\bar{p},L\bar{r})})\leq \frac{\varepsilon_{\mathbf{I}}}{5} .\] + +This establishes the proposition assuming (9.2). diff --git a/outputs/arxiv_math/2503.08554_pg1_pg1_repeat1.md b/outputs/arxiv_math/2503.08554_pg1_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..eb89b7dfe216f8f52caea78f7d0ac41060684579 --- /dev/null +++ b/outputs/arxiv_math/2503.08554_pg1_pg1_repeat1.md @@ -0,0 +1,37 @@ +# LoS Blockage in Pinching-Antenna Systems: Curse or Blessing? + +Zhiguo Ding, Fellow, IEEE, and H. Vincent Poor, Life Fellow, IEEE + +Abstract—This letter investigates the impact of line-of-sight (LoS) blockage on pinching-antenna systems. Analytical results are developed for both single-user and multi-user cases to reveal that the presence of LoS blockage is beneficial for increasing the performance gain of pinching antennas over conventional antennas. This letter also reveals that LoS blockage is particularly useful in multi-user cases, where co-channel interference can be effectively suppressed by LoS blockage. Index Terms—Pinching antennas, line-of-sight blockage, large-scale path loss, multiple-input multiple-input systems. + +## I. INTRODUCTION + +Pinching-antenna systems have recently been recognized as a promising transmission technique for next-generation mobile networks due to the following three features [1], [2]. The first feature is their low costs since pinching antennas are simple dielectric particles, e.g., clothes pinches, applied on waveguides. The second feature is the capability of pinching antennas to create strong line-of-sight (LoS) connections between a base station and its user, e.g., it is possible to activate a pinching antenna right next to the user, and hence the path loss experienced by the user can be very small. The third feature is that the multi-input multi-output (MIMO) systems created by pinching antennas can be flexibly reconfigured, e.g., adding (or removing) antennas becomes straightforward. The fundamental limits of pinching-antenna systems with different configurations, e.g., different numbers of pinching antennas, users, and waveguides, have been identified in [3]. These obtained analytical results reveal that pinching antennas achieve a significant performance gain over conventional antennas. In [4], a low-complexity implementation to activate pinching antennas, instead of moving them, was studied, and the array gain achieved by pinching-antenna systems has been identified in [5]. In addition, sophisticated resource allocation algorithms have been developed for uplink and downlink transmissions in pinching-antenna systems [6] and [7], respectively. Recall that one of the key features of pinching antennas is to reduce the transceiver distance, which means that in pinching-antenna systems, a user experiences less large-scale path loss and LoS blockage, compared to conventional antenna systems. However, in the literature, there is no study that formally investigated the impact of pinching antennas on LoS blockage, which motivates this letter. The contribution of this letter is two-fold. One is to focus on the single-user special case, where analytical results are developed to analyze the outage probability achieved by pinching antennas. The presented analytical and simulation results confirm the intuition that the presence of LoS blockage is beneficial for increasing the performance gain of pinching antennas over conventional antennas, compared to the case without + +Z. Ding is with the University of Manchester, Manchester, M1 9BB, UK, and Khalifa University, Abu Dhabi, UAE. H. V. Poor is with the Department of Electrical and Computer Engineering, Princeton University, Princeton, NJ 08544, USA. + +blockage. The second contribution focuses on a general multiuser scenario. For conventional antenna systems, the existence of strong co-channel interference severely degrades the system performance. The presence of LoS blockage makes it more difficult to combat co-channel interference since the LoS blockage can make a user’s channel matrix no longer full rank, and hence, many interference cancellation methods, such as zero-forcing approaches, become not applicable. However, in pinching-antenna systems, LoS blockage becomes a blessing since a user’s interference link is likely to be subject to blockage since the corresponding interfering pinching antenna could be far away from the user. This intuition is confirmed by the presented analytical results, which show that the ergodic data rate gain of pinching antennas over conventional antennas is unbounded at a high signal-to-noise ratio (SNR). + +## II. SYSTEM MODEL + +Consider a downlink pinching-antenna system with \(M\) single-antenna users, denoted by \(\mathrm{U}_{m}\), in a rectangular-shaped service area, denoted by \(\mathcal{A}\), whose two sides are denoted by \(D_{\mathrm{W}}\text{and}D_{\mathrm{L}}\). Assume that the service area is divided by \(M\) parallelly installed waveguides into \(M\) identical rectangles with sides being \(\frac{D_{\mathrm{W}}}{M} \text{and}D_{\mathrm{L}}\). To facilitate the performance analysis, assume that \(\mathrm{U}_{m}\) is uniformly distributed in the rectangle centered by the \(m-\)th waveguide, and \(\mathrm{U}_{m}’\mathrm{s}\) location is denoted by \(\boldsymbol{\psi}_{m}=(x_{m},y_{m},0)\). It is further assumed that a single pinching antenna is activated on the \(m-\)th waveguide at the location closest to \(\mathrm{U}_{m}\), and hence its location can be denoted by \(\boldsymbol{\psi}_{m}^{\text{Pin}}=(x_{m},\beta_{m},d),\text{where}d\) denotes the height + +of the waveguide and \(\beta_{m}=- \frac{D_{\mathrm{W}}}{2} +(m-1) \frac{D_{\mathrm{W}}}{M} + \frac{D_{\mathrm{W}}}{2M}\) . + +Similar to \([3],\mathrm{U}_{m}’\mathrm{s}\) observation is given by + +\[y_{m}=\sum_{k=1}^{M}\tilde{h}_{mk}p_{mk} \sqrt{P} s_{m}+\sum_{i\neq m}\sum_{k=1}^{M}\tilde{h}_{mk}p_{ik} \sqrt{P} s_{i}+w_{m},(1)\] + +\[-2\pi j\bigg( \frac{1}{\lambda} |\boldsymbol{\psi}m-\boldsymbol{\psi}_{k}^{\text{Pin}}|+ \frac{1}{\lambda g} |\boldsymbol{\psi}_{0}^{\text{Pin}}-\boldsymbol{\psi}_{k}^{\text{Pin}}\] + +\(|\boldsymbol{\psi}_{m}-\boldsymbol{\psi}^{\text{Pin}}\) + +\(|_{k}\) + +where \(\tilde{h}_{mk}=\alpha_{mk}h_{mk},h_{mk}\) denotes the channel gain between the \(k-\)th antenna and \(\mathrm{U}_{m}\), i.e., \(h_{mk}= \sqrt{\eta} e\) , the overall transmit power budget is denoted by \(P,p_{mk}\) is the precoding coefficient, \(\lambda\text{and}\lambda_{g}\) denote the carrier and waveguide wavelengths, respectively, \(\eta= \frac{c^{2}}{16\pi^{2}f_{c}^{2}} ,c\) is the speed of light, the carrier frequency is denoted by \(f_{c},w_{m}\) denotes the additive noise with power \(\sigma^{2},s_{m}\) denotes \(\mathrm{U}_{m}’\mathrm{s}\) signal, and \(\alpha_{mk}\) is an indicator function for the LoS blockage. In particular, if there is LoS blockage between the pinching antenna on the \(k-\)th waveguide and \(\mathrm{U}_{m},\alpha_{mk}=0.\text{Otherwise},\alpha_{mk}=1\). In general, LoS blockage can be modeled as follows: [8] + +\[\mathbb{P}(\alpha_{mk}=1)=e^{-\phi|\boldsymbol{\psi}_{i}^{\text{Pin}}-\boldsymbol{\psi}_{m}|},\] (2) + +and for ultra-dense indoor environments, the following LoS blockage model can also be used: [9] + +\[\mathbb{P}(\alpha_{mk}=1)=e^{-\phi|\boldsymbol{\psi}_{i}^{\text{Pin}}-\boldsymbol{\psi}_{m}|^{2}},\] (3) diff --git a/outputs/arxiv_math/2503.08572_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.08572_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..360efb5ab0e4598a1bc9dc7b85e7177230bac29f --- /dev/null +++ b/outputs/arxiv_math/2503.08572_pg3_pg1_repeat1.md @@ -0,0 +1,43 @@ +values; ii) in the strongly underdamped case, the second-order phase transition turns into a first-order phase transition (note the jump in the curve for \(m=10)\), which is in line with what has been reported before [26, 27]. In any case, for all inspected values of the mass, the asynchronous state \((r\ll1)\) always exists over a finite range of mean coupling strengths. In this parameter region the temporal statistics of the dynamics is independent of \(K\). In the remainder of the study, we will set \(K=0\). + +# B. Power Spectrum Calculation + +To further analyze the fluctuations in the incoherent regime, we compute the power spectrum of individual oscillators, denoted as \(S_{x_{\ell}}(\omega),\text{where}x_{\ell}=e^{i\theta_{\ell}}\). The power spectrum is defined as: + +\[S_{x_{\ell}}(\omega)=\lim_{T\rightarrow\infty} \frac{1}{T} \left\langle|\tilde{x}_{\ell}(\omega)|^{2}\right\rangle,\tilde{x}_{\ell}(\omega)=\int_{0}^{T}e^{i\omega t}x_{\ell}(t)d_{(5}t,_{)}\] + +where \(\tilde{x}_{\ell}(\omega)\) is the Fourier transform of \(x_{\ell}(t)\) computed over a finite time interval \(T\) . Common temporal statistics are characterized by the averaged power spectrum, + +\[S_{z}(\omega)= \frac{1}{N} \sum_{\ell=1}^{N}S_{x_{\ell}}(\omega).\] (6) + +We obtain these spectra by integrating the full set of second-order differential equations (Eq. (1)) using a fourth-order Runge-Kutta method with a relative tolerance of \(10^{-8}\). We refer to this direct numerical integration as the network dynamics (ND) method and use it as a benchmark against the iterative mean-field (IMF) method introduced in the following section. In Fig. 2 and Fig. 3, the ND results are shown as black dashed lines for comparison with the IMF method. The simulation is run for a total time period \(T\) (with \(T=10^{5}\) for Fig. 2 and \(T=10^{4}\) otherwise), discarding the initial time \(t_{d}=1000\) to eliminate transients. To allow oscillator-resolved analysis, the natural frequencies \(\omega_{\ell}\) are drawn from the prescribed distribution and held fixed, as is the Gaussian-distributed connectivity matrix \(K_{\ell m}\). The system size is chosen as \(N=10^{4}\) to minimize dependence on the specific realization of the connectivity matrix \(K_{\ell m}\), which is also drawn from a Gaussian distribution. The power spectrum is computed using the fast Fourier transform (FFT). The single-oscillator power spectrum \(S_{x_{\ell}}(\omega)\) is evaluated for a selected oscillator \((\ell=3\) in our case), while the mean power spectrum \(S_{z}(\omega)\) is obtained by averaging over all oscillators: + +\[S_{z}(\omega)= \frac{1}{N} \sum_{\ell=1}^{N}|\tilde{x}_{\ell}(\omega)|^{2}.\] (7) + +The results are smoothed by binning neighboring frequency points to improve spectral accuracy. We divided the frequency axis into \(M=2000\) uniform bins, with \(T/(Mdt)\) frequency points averaged per bin using an accumulation method to reduce statistical noise. + +# III. ITERATIVE MEAN FIELD METHOD + +In this study, we employ the iterative mean field (IMF) method, often also referred to as dynamic mean field method [28], to analyze the asynchronous states of the Kuramoto model with inertia and disorder in coupling and natural frequencies. This method provides a self-consistent approach to compute the power spectra of single oscillators, approximating the network noise experienced by each oscillator as a Gaussian process, iteratively refined until convergence. The method has been used for spin systems in the early 1990’s [29], for networks of spiking neurons [19–21], and specifically for the Kuramoto model with disorder in the coupling coefficients (but without inertia) in [4–6, 30]. For our system with inertia, we start from the equations of motion for the oscillators Eq. (1) and express the coupling term in the dynamics as a multiplicative network noise: + +\[m\ddot{\theta}_{\ell}(t)+\dot{\theta}_{\ell}(t)=\omega_{\ell}+\text{Im}\left(e^{-i\theta_{\ell}(t)}\zeta_{\ell}(t)\right),(8)\] + +where the network noise \(\zeta_{\ell}(t)\) is defined as: + +\[\zeta_{\ell}(t)=\sum_{m=1}^{N}K_{\ell m}e^{i\theta_{m}(t)}.\] (9) + +For large networks in the asynchronous regime (excluding the Volcano transition case observed with symmetric coupling coefficients, see [30, 31]), this noise behaves as a superposition of independent Gaussian processes due to the central limit theorem. The autocorrelation of the network noise can be written as: + +\[\begin{aligned} \langle\zeta_{\ell}(t)\zeta_{\ell}(t^{\prime})\rangle&=\left\langle K_{\ell m}K_{\ell n}e^{i(\theta_{m}(t^{\prime})-\theta_{n}(t))}\right\rangle \\ &=\sum_{m,n}\left( \frac{k^{2}}{N} \delta_{mn}+ \frac{K^{2}}{N^{2}} \right)\left\langle e^{i(\theta_{m}(t^{\prime})-\theta_{n}(t))}\right\rangle. \end{aligned}\] + +Assuming uncorrelated oscillators yields + +\[\left\langle e^{i(\theta_{m}(t^{\prime})-\theta_{n}(t))}\right\rangle=\delta_{mn}\left\langle e^{i(\theta_{m}(t^{\prime})-\theta_{m}(t))}\right\rangle,(11)\] + +leading to the simplified form + +\[\langle\zeta(t)\zeta(t^{\prime})\rangle=\left(k^{2}+K^{2}/N\right)\left\langle e^{i(\theta(t^{\prime})-\theta(t))}\right\rangle.(12)\] + +The right-hand side thus represents the autocorrelation of a single oscillator’s phase pointer. In the Fourier domain, the network noise spectrum \(S_{\zeta}(\omega)\) is proportional to the power spectrum of a single oscillator averaged over the network: + +\[S_{\zeta}(\omega)=(k^{2}+K^{2}/N)S_{z}(\omega),\] (13) diff --git a/outputs/arxiv_math/2503.08577_pg6_pg1_repeat1.md b/outputs/arxiv_math/2503.08577_pg6_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..09225139ae9964b59ae8519dcb7963f4dba68456 --- /dev/null +++ b/outputs/arxiv_math/2503.08577_pg6_pg1_repeat1.md @@ -0,0 +1,35 @@ +However, in our work, we are interested in heat kernels on Lie groups. To make sense of the heat equation on a Lie group, the proper Riemannian structure needs to be chosen. For compact semi-simple Lie group \(G\), the Riemannian structure \((G,g)\) can be defined via Ad-invariant positive definite inner product \((\cdot,\cdot)\) stemming from the Killing form + +2. + +Notice that although the group \(\mathrm{U}(d)\) is compact, it is not semi-simple. Hence, the metric tensor stemming from the Killing form is only positive semi-definite. Indeed, one can check that such a metric tensor for \(\mathrm{U}(1)\cong S^{1}\) is identically zero, so it does not equip \(S^{1}\) with the Riemannian structure. This is in contrast with the construction from Example 1. Of course, general Lie groups are not commutative. Hence, in order to study the heat equation on a compact Lie group \(G\), non-commutative Fourier/harmonic analysis is needed. Fourier coefficients on a compact Lie group are calculated with respect to the irreducible representations (irreps) of the group. Generally, the object being transformed is the regular Borel measure on \(G\). However, we focus on the related case of integrable functions \(f\). In this case (see e.g. [34]), the Fourier coefficient \(\hat{f}(\lambda)\) is the operator in \(\text{End}(V_{\pi_{\lambda}})\) defined via + +\[\hat{f}(\lambda)=\int_{G}\pi(g^{-1})f(g)d\mu(g),\] (22) + +where by \(V_{\pi_{\lambda}}\) we denote the representation space of irrep \(\pi_{\lambda}\) with highest weight \(\lambda\). Equipping the space \(\text{End}(V_{\pi_{\lambda}})\) + +with the norm \(\sqrt{d_{\lambda}} ||\cdot||_{HS},\text{where}d_{\lambda}:=\dim(V_{\pi_{\lambda}})\) and the Hilbert-Schmidt norm \(||u||_{HS}^{2}=\text{Tr}(uu^{*})\), one can show that such the Fourier transform is an isomorphism of Hilbert spaces + +\[L^{2}(G)\cong \bigoplus_{\pi\in\hat{G}}\text{End}(V_{\pi_{\lambda}}),\] (23) + +where \(\hat{G}\) is the set of equivalence classes of irreps of \(G\). Namely, we obtain a generalisation of the Plancherel’s theorem + +\[||f||_{2}^{2}=\int_{G}|f(g)|^{2}d\mu(g)=\sum_{\lambda\in\hat{G}}d_{\lambda}||\hat{f}(\lambda)||_{HS}^{2}.\] (24) + +This is a consequence of the Peter-Weyl theorem. + +Remark 1. The transform (22) is a generalization of the Fourier series. Indeed, suppose a compact group \(G\) is additionally abelian and connected (so is a torus). Take one-dimensional torus \(\mathrm{U}(1)\cong S^{1}\) for example. The unitary irreps \(\pi_{\lambda}\) of U(1) are the homomorphisms \(\mathrm{U}(1)\rightarrow\mathrm{U}(1)\) so they are of the form \(e^{i\phi}\mapsto e^{i\lambda\phi}\) for some integer \(\lambda\). All irreps are one-dimensional and \(\hat{S}^{\sim^{1}}=\mathbb{Z}\). The Fourier coefficients of a function \(f:\mathrm{U}(1)\rightarrow\mathbb{C}\) are + +\[\hat{f}(\lambda)= \frac{1}{2\pi} \int_{-\pi}^{\pi}e^{-i\lambda\phi}f(e^{i\phi})d\phi,\] (25) + +which coincides with the Fourier coefficients of the corresponding \(2\pi-\)periodic complex-valued function \(\tilde{f}:\mathbb{R}\rightarrow\mathbb{C}\), \(\tilde{f}(x)=f(e^{ix})\). Similarly, other results such as the completeness, orthogonality relations and Plancherel’s theorem generalise to the non-abelian case via the Peter-Weyl theorems and representation theory. + +Heat kernels on simply-connected compact semi-simple Lie groups were studied in Ref. [32], together with a useful Poisson form. In Section IV we show how to apply those results for \(\text{PU}(d)\), which is not simply-connected. + +# III. MAIN RESULTS AND APPLICATIONS + +Below we summarise the main results of this paper and outline some of their applications. + +Result 1. The main technical result of the paper is the construction of the polynomial approximation to the Dirac delta function \(H_{P}^{(t)}(\cdot,\sigma)on\text{PU}(d)\), together with some of its properties. This allows us to summarise the key properties of the family of polynomial approximations of Dirac delta based on the trimmed heat kernels (see Theorem 3 for a precise statement. + +2 Taking the negative of the negative-definite Killing form leads to the positive-definite scalar product. diff --git a/outputs/arxiv_math/2503.08578_pg6_pg1_repeat1.md b/outputs/arxiv_math/2503.08578_pg6_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..0185cba5bf27a9da0a55240256f0e5e9d3dbf2cc --- /dev/null +++ b/outputs/arxiv_math/2503.08578_pg6_pg1_repeat1.md @@ -0,0 +1,19 @@ +[11, Assumption 4.1] imposes additional conditions, such as \(f\in\mathcal{C}^{2}(\mathbb{R}^{d})\) and the boundedness of \(\Vert\nabla^{2}f\Vert_{\infty}\). In the framework of [22, Definition 8], some additional local growth conditions on \(f\) around the global minimizer are required. For our rescaled CBO we do not require these additional assumptions for the convergence proof. + +- Multiple–minimizer case: Although our rescaled CBO is not specifically designed to address the multiple minimizer problem, we can still establish certain asymptotic properties in cases where the objective function \(f(\cdot)\) has multiple minimizers, whether they are discrete points or compact subsets (see Theorem 3.9). Let us mention here that specially designed CBO models aimed at addressing multiple-minimizer problems have been introduced in [9, 24]. + +Our results rely on two main ingredients: uniform estimates on the moments of the measure, and on a Harnack inequality on the invariant elliptic CBO equation. Indeed we need Harnack’s inequality in order to ensure that the measure has a non–null mass around minimizers, which will then allow us to prove Laplace’s principle. A summary of our findings is in section 4 at the end of the manuscript. + +## 2. Useful estimates + +We start by recalling some notation. When a measure \(\mu\) has a density \(\varrho\) with respect to Lebesgue measure that we denote by \(dx,\text{then}\mu\) is absolutely continuous with respect to \(dx\), we write \(\mu\ll dx\text{and}\varrho= \frac{d\mu}{dx}\) is Radon–Nikodym derivative of \(\mu\) with respect to \(dx\). Unless a confusion arises, we shall use the same notation for a measure and for its density (when it exists). We shall also denote by \(W^{k,p}(\Omega),p\geq1,k\geq0\) the standard Sobolev space of functions whose generalized derivatives up to order \(k\) are in \(L^{p}(\Omega)\), and we denote by \(W_{loc}^{k,p}(\Omega)\) the class of functions \(f\text{such that}\chi f\in W^{k,p}(\Omega)\) for each \(\chi\in\mathcal{C}_{c}^{\infty}(\Omega)\) the class of infinitely differentiable functions with compact support in \(\Omega\). In what follows, \(\Vert\cdot\Vert\) denotes the Frobenius norm of a matrix and \(|\cdot|\) is the standard Euclidean norm in \(\mathbb{R}^{d};\mathscr{P}(\mathbb{R}^{d})\) denotes the space of probability measures on \(\mathbb{R}^{d}\), and \(\mathscr{P}_{p}(\mathbb{R}^{d})\text{with}p\geq1\) contains all \(\mu\in\mathscr{P}(\mathbb{R}^{d})\text{such that}\mu(|\cdot|^{p}):=\int_{\mathbb{R}^{d}}|x|^{p}\mu(dx)<\infty\); it is equipped with \(p-\)Wasserstein distance \(W_{p}(\cdot,\cdot)\). Our main assumption concerns the function \(f(\cdot)\) to be minimized, and can be stated as follows. + +Assumption 2.1. We assume the following properties for the objective function. + +\((1)f:\mathbb{R}^{d}\rightarrow\mathbb{R}_{+}\) is bounded from below by \(\underline{f} :=\min f>0\), and there exists some constant \(L_{f}>0\) such that + +\[|f(x)-f(y)|\leq L_{f}(1+|x|+|y|)|x-y|,\forall x,y\in\mathbb{R}^{d}.\] + +2. There exist constants \(c_{\ell},c_{u}>0\text{and}M>0\) such that + +\[f- \underline{f} \leq c_{u}(|x|^{2}+1),\forall x\in\mathbb{R}^{d}\text{and}c_{\ell}|x|^{2}\leq f- \underline{f} \text{for}|x|\geq M.\] diff --git a/outputs/arxiv_math/2503.08595_pg4_pg1_repeat1.md b/outputs/arxiv_math/2503.08595_pg4_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..e4677491556b1feb593cdcb0bb7f94d66bf9899d --- /dev/null +++ b/outputs/arxiv_math/2503.08595_pg4_pg1_repeat1.md @@ -0,0 +1,23 @@ +dynamical delocalization result. However, assumption (1.3) is stronger than simply assuming the spectrum is AC, and this stronger assumption seems to be essentially necessary for ergodicity, see [7, Prp. 1.5] for a related result. Some basic examples were given in [2, §3.2]: if \(\nu=1\), i.e. the fundamental cell is simply one vertex, then the limiting measure is uniform. This covers the continuous quantum walk \(\mathrm{e}^{\mathrm{i}t\mathcal{A}}\text{on}\mathbb{Z}^{d}\) and the triangular lattice for example. The same property is true for the hexagonal lattice and the infinite ladder, both of which have \(\nu=2\). It was also shown that in the cases where \(\Gamma\) is a \(1d\) strip of width 3, or a cylinder \(\mathbb{Z}□C_{4}\), then the limiting distribution is not uniform. + +1. 2. Main results. In this note we analyze further families of graphs which satisfy our assumptions, and compute the limiting average \(\langle a\rangle_{p}\) explicitly. + +We first give the following result, extracted from [7]. Proposition 1.2 (Case of Cartesian and Tensor Products). Suppose \(\Gamma_{0}\) is a periodic graph with \(\nu=1\) (for example \(\Gamma_{0}=\mathbb{Z}^{d}\) or the triangular lattice), and let \(G_{F}\) be any finite graph with \(\nu_{F}=|G_{F}|\) vertices. Let \(\Gamma_{1}=\Gamma_{0}□G_{F}\) be the Cartesian product , \(\Gamma_{2}=\Gamma_{0}\times G_{F}\) be the tensor product and \(\Gamma_{3}=\Gamma_{0}⊠G_{F}\) be the strong product of \(\Gamma_{0}\text{and}G_{F}\) . Let \(E_{\Gamma_{0}}(\theta)\) be the band function of \(\Gamma_{0},(w_{j})\) an orthonormal eigenbasis of \(A_{G_{F}}\text{and}(\mu_{j})\) the corresponding eigenvalues, \(j\leq\nu_{F}\) . Then + +1. The band functions of \(\mathcal{A}_{\Gamma_{1}}\) are given by \(E_{j}(\theta)=E_{\Gamma_{0}}(\theta)+\mu_{j}\). + +2. The band functions of \(\mathcal{A}_{\Gamma_{2}}\) are given by \(E_{j}(\theta)=\mu_{j}E_{\Gamma_{0}}(\theta)\). + +3. The band functions of \(\mathcal{A}_{\Gamma_{3}}\) are given by \(E_{j}(\theta)=(1+\mu_{j})E_{\Gamma_{0}}(\theta)+\mu_{j}\). + +4. Assumption (1.3) is satisfied for \(\mathcal{A}_{\Gamma_{1}}\) but not necessarily for \(\mathcal{A}_{\Gamma_{2}}or\mathcal{A}_{\Gamma_{3}}\). + +5. For each of \(\mathcal{A}_{\Gamma_{1}},\mathcal{A}_{\Gamma_{2}}\text{and}\mathcal{A}_{\Gamma_{3}}\), we have + +\[\langle a\rangle_{p}=\sum_{q=1}^{\nu_{F}}\langle a(\cdot+v_{q})\rangle\sum_{s=1}^{\nu_{F}^{\prime}}|P_{\mu_{s}}(v_{p},v_{q})|^{2},\] + +where \(P_{\mu_{s}}(v_{p},v_{q})=\sum_{j\mu_{j}=\mu_{s}}w_{j}(v_{p}) \overline{w_{j}(v_{q})}\) is the (kernel) of the orthogonal projection for the distinct eigenvalues of the finite graph. For example, if \(\Gamma_{0}=\mathbb{Z}^{d},\text{then}E_{\Gamma_{0}}(\theta)=2\sum_{i=1}^{d}\cos2\pi\theta_{i}\text{and if}\Gamma_{0}\) is the triangular lattice, then \(E_{\Gamma_{0}}(\theta)=2\cos2\pi\theta_{1}+2\cos2\pi\theta_{2}+2\cos2\pi(\theta_{1}+\theta_{2}),\text{for}\theta_{i}\in[0,1)\). Here \(\Gamma_{1,2,3}\) are viewed as periodic graphs with fundamental domain containing \(\nu_{F}\) vertices, cf. [7, Lemma 3.1, §3.4], with \((v_{i})\) the vertices of \(G_{F}\) . Arguing as before, we get for these more special graphs that + +\[\mu_{T,v_{p}+\mathbf{n}_{a}}^{N}(\mathbf{k}_{a}+v_{q})\approx \frac{1}{N^{d}} \sum_{s=1}^{\nu^{\prime}}|P_{\mu_{s}}(v_{p},v_{q})|^{2}=: \frac{1}{N^{d}} d(p,q).\] (1.6) + +whenever (1.3) is satisfied. This gives a more satisfactory concept of a quantum limiting distribution than in [7] where quantum ergodicity was assessed by the behaviour of eigenvector bases, and it was shown in [7, §4.5] that such a limiting distribution depends on the eigenvector basis. In contrast, here the RHS of (1.6) depends only on the graph. Our main target now is to compute the weights \(d(p,q)\) for specific finite graphs \(G_{F}\) . Because of point (4) above, the theorems are illustrated only for the Cartesian product, but some hold more generally. For definiteness, the reader can assume \(\Gamma_{0}=\mathbb{Z}\) in all these results, which is already interesting. However, nothing changes for any \(\Gamma_{0}\) having a single vertex in its fundamental domain, such as \(\mathbb{Z}^{d}\) and the triangular lattice. A nice simplification in the family of Cartesian products is that the limiting weight \(d(p,q)\) in (1.6) depends only on the finite graph \(G_{F}\) , compared to the general case (1.5), where the weight depends on the full Floquet matrix and computations become more daunting. Still, as we will see, Cartesian products already offer interesting contrasting diff --git a/outputs/arxiv_math/2503.08597_pg10_pg1_repeat1.md b/outputs/arxiv_math/2503.08597_pg10_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..4916f812389ae2c0e8acf07371af5ab58e08619e --- /dev/null +++ b/outputs/arxiv_math/2503.08597_pg10_pg1_repeat1.md @@ -0,0 +1,27 @@ +DoF tuples achievable by classical coding schemes. The NS-assisted DoF region \(\mathcal{D}^{\text{NS}}\) is defined as the closure of all DoF tuples achievable by NS-assisted coding schemes. In particular, the classical sum-DoF \(d_{\Sigma}\), the NS-assisted sum-DoF \(d_{\Sigma}^{\mathrm{N}\mathrm{S}}\) are defined as \(d_{\Sigma}≜\max_{(d_{1},\cdots ,d_{K})\in\mathcal{D}}(d_{1}+\cdots +d_{K})\), \(d_{\Sigma}^{\mathrm{N}\mathrm{S}}≜\max_{(d_{1},\cdots ,d_{K})\in\mathcal{D}^{\text{NS}}}(d_{1}+\cdots +d_{K})\), respectively. + +## 3 Results + +### 3.1 Fully Connected CoMP BC + +Our first result, stated in the following theorem, shows that NS-assistance does not improve the DoF region, or even the capacity region, in the fully-connected (Definition 3) CoMP BC. + +Theorem 1 (Fully Connected CoMP BC Capacity and DoF Regions). For a fully-connected CoMP BC network, the capacity regions with and without NS-assistance are characterized, under the \(\mathbb{F}_{q}\) model as, + +\[\mathcal{C}^{\text{NS}}(q)=\mathcal{C}(q)=\left\{(R_{1},,\cdots ,R_{K})\in\mathbb{R}_{+}^{K}|R_{1}+R_{2}+\cdots +R_{K}\leq C_{1}(q)\right\},\] (17) + +and under the Gaussian model as, + +\[\mathcal{C}^{\text{NS}}(P)=\mathcal{C}(P)=\left\{(R_{1},\cdots ,R_{K})\in\mathbb{R}_{+}^{K}|R_{1}+R_{2}+\cdots +R_{K}\leq C_{1}(P)\right\},\] (18) + +where \(C_{1}(q),C_{1}(P)\) represent the single-user capacity of Rx-1 under the two models. Note that \(C_{1}(q)= \log_{2}(q)\). The corresponding DoF regions, under both the \(\mathbb{F}_{q}\) model and the Gaussian model, are characterized as, + +\[\mathcal{D}^{\text{NS}}=\mathcal{D}=\left\{(d_{1},d_{2},\cdots ,d_{K})\in\mathbb{R}_{+}^{K}|d_{1}+d_{2}+\cdots +d_{K}\leq1\right\}.\] (19) + +The key to Theorem 1 is the statistical-equivalence, or the same-marginals property of the receivers, which makes them indistinguishable from the transmitter’s perspective. The same-marginals argument [25,26] is a standard line of reasoning in classical literature that makes use of the fact that the probabilities of error experienced by the receivers for an arbitrary (classical) coding scheme depend only on the marginal channel distribution of each receiver. In our fully connected CoMP BC since the marginal distributions are identical across receivers, the same-marginals property ensures that the capacity and DoF regions remain unchanged if every receiver has exactly the same channel realizations as Rx-1, in every channel-use. Once all receivers observe the same channel output, even allowing full cooperation among the receivers cannot change the capacity or DoF regions. Therefore the sum-capacity cannot exceed the single-user capacity, and any allocation of rates across messages that does not exceed the single-user capacity is trivially achievable, implying immediately the classical capacity and DoF regions in Theorem 1. Beyond the classical case, in order to show that NS-assistance cannot improve the capacity and DoF regions, two additional facts are needed. + +Fact 1: For a point to point (single user) channel, NS assistance cannot improve the capacity. Fortunately, this non-trivial fact is already well-established, as noted in Lemma 1 [8–10]. + +Fact 2: The same-marginals property still holds under NS-assistance, even when NS-assistance is available to all parties.5 + +5The same-marginals property has been shown in [16] for a BC with NS-assistance available to only the decoders, i.e., receivers, when the metric of interest is the sum of probabilities of error of the receivers. diff --git a/outputs/arxiv_math/2503.08602_pg38_pg1_repeat1.md b/outputs/arxiv_math/2503.08602_pg38_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..30177bb9738c6c763f8914d5d82cb4c164d0cd5d --- /dev/null +++ b/outputs/arxiv_math/2503.08602_pg38_pg1_repeat1.md @@ -0,0 +1,21 @@ +We start by defining so-called ‘off-shell’ Bethe vectors: fix some \(0\leq k\leq n\) and let \(x_{1},\ldots,x_{k}\) be pairwise commuting indeterminates. Then we set + +\[\mathbf{b}(x_{1},\ldots,x_{k})=t_{10}(-x_{1})\cdots t_{10}(-x_{k})v_{o},\] (8.31) + +where \(v_{o}\) is the highest weight vector (8.29). (For \(k=0\) we shall simply take \(v_{o}\) instead.) The following is a restatement of [GK17, Prop 4.3] which gives the expansion of the off-shell Bethe vectors in the spin basis \(v_{\lambda}=v_{i_{n}}\otimes\cdots \otimes v_{i_{1}}\text{where}I=i_{1}\ldots i_{n}\) is the unique 01-word corresponding to \(\lambda\): + +## Proposition 8.12. We have the expansion + +\[(8.32)\mathbf{b}(x_{1},\ldots,x_{k})=x_{1}\cdots x_{k}\sum_{\lambda}\varepsilon_{\lambda}^{-1}G_{\lambda^{\vee}}(1-x_{1},\ldots,1-x_{k}|1-\varepsilon_{w(n)}^{-1},\ldots,1-\varepsilon_{w(1)}^{-1})v_{\lambda},\] + +\[\varepsilon_{j\in J_{\lambda}w(j)}\] + +where the sum runs over all \(\lambda\subset(n-k)^{k},\lambda^{\vee}\) denotes the partition whose Young diagram is the complement of the one of \(\lambda\) in the \(k\times(n-k)\) bounding box and \(\varepsilon_{\lambda}=\prod\) with \(J_{\lambda}\subset\{1,\ldots,n\}\) being the positions of 0-letters in the 01-word corresponding to \(\lambda\). Using the commutation relations from Lemma 8.2 one now shows by a standard computation in the literature on quantum integrable systems (see e.g. [Fad90]) that the Bethe vectors are eigenvectors of the images of the elements \(t(z)\) in the evaluation module \(\mathbf{V}_{w}\) provided the indeterminates \(x_{i}\) satisfy the so-called Bethe ansatz equations, + +\[\prod_{j=1}^{n}(1-x_{i}/\varepsilon_{j})\prod_{j\neq i}^{k}(x_{j}/x_{i})+(-1)^{k}q=0,i=1,\ldots,k.\] (8.33) + +The solutions of these equations are called ‘Bethe roots’ and the Bethe vectors evaluated on the Bethe roots are called ‘on-shell’. (Physically, the Bethe roots are the momenta of quasi-particles in the associated quantum spin chain.) The statement which in is general difficult to prove is that all eigenvectors are obtained this way and that they from an eigenbasis in each subspace \(V_{k,n}=\text{span}\{v_{\lambda}:\lambda\subset(n-k)^{k}\}\subset\mathbf{V}_{w}\). Expanding the Bethe roots as power series in \(q\) both statements have been proven in [GK17, Lemma 4.6 and Theorem 4.8]. In particular, each solution \(x^{\lambda}=(x_{1}^{\lambda},\ldots,x_{k}^{\lambda})\text{with}\lambda\subset(n-k)^{k}\) is uniquely identified by its formal limit \(q\rightarrow0\) where it coincides with \(\varepsilon_{\lambda}\). Theorem 8.1. The ‘on-shell’ Bethe vectors \(\{\mathbf{b}_{\lambda}=\mathbf{b}(x_{1}^{\lambda},\ldots,x_{k}^{\lambda}):\lambda\subset(n-k)^{k}\}\) provide an eigenbasis in each subspace \(V_{k,n}\) and we have the eigenvalue equations + +\[t(z)\mathbf{b}_{\lambda}=\left( \frac{(_{j=1}^{n}1+z/\varepsilon_{j})}{(_{i=1}^{k}1+z/x_{i}^{\lambda})} \prod + \frac{q}{\prod_{i=1}^{k}(1+x_{i}^{\lambda}/z)} \right)\mathbf{b}_{\lambda}.\] (8.34) + +N.B. the eigenvalues are polynomial in \(z\) because of the Bethe ansatz equations. In fact, one verifies that the condition that the residues at \(z=-x_{i}\) vanish for \(i=1,\ldots,k\) is equivalent to the Bethe ansatz equations (8.33). The polynomial form of the eigenvalues can be made explicit using the so-called level-rank duality which we discuss next. diff --git a/outputs/arxiv_math/2503.08614_pg8_pg1_repeat1.md b/outputs/arxiv_math/2503.08614_pg8_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..2ae60eb81d60342ed7ce1726a9af7b8a63d57a80 --- /dev/null +++ b/outputs/arxiv_math/2503.08614_pg8_pg1_repeat1.md @@ -0,0 +1,33 @@ +Proof. Let \(g\in\Lambda\) have a non-trivial projection on \(\mathbb{R}_{\mathbf{H}}\), and let \(z\in\Lambda\cap Z\). We have \(g^{k}zg^{-k}= a^{k}(z),\text{where}a=r(g)\). Since the actions of \(\mathbb{R}_{\mathbf{L}}\text{and}K\) on the center of Heis are trivial, and the action of \(\mathbb{R}_{\mathbf{H}}\) is contracting (up to forward or backward iteration), it follows that the sequence + +\((a^{k}(z))_{k\in\mathbb{Z}}\) has an accumulation point at 1. As \(\Lambda\) is discrete, the set \(\{a^{k}(z),k\in\mathbb{N}\}\subset\Lambda\) must be finite, which forces \(z=1\). + +- + +Lemma 3.7. ([17, Lemma 5.3]) Let \(Q\) be the semi-direct product \(Q=\mathbb{R}\ltimes C\text{where}C\) is a + +compact connected Lie group. Then \(Q\) is isomorphic to the product \(\mathbb{R}\times C\). The last lemma we will need in this section is Lemma 3.8 below. It is similar to [16, Proposition 5.4], but we present it here in a form that allows for systematic application. The proof however is the same, so to avoid unnecessary lengthening of the paper, we will not rewrite the proof. Let \(Q\ltimes C\subset\text{Aut}(\text{Heis})\) preserving the decomposition \(heis=a\oplus z,\text{and such that}C\) is compact. Define + +\[G:=(Q\ltimes C)\ltimes N.\] + +Let \(\Gamma\) be a discrete subgroup of \(G\). Under some conditions, the proposition below ensures the existence of a nilpotent syndetic hull of \(\Gamma\), i.e. a connected nilpotent subgroup of \(G\) containing \(\Gamma\) as a lattice. Lemma 3.8. Let \(\Gamma\) be a discrete subgroup of \(G\), whose projection on \(Q\) is dense. Then, up to finite index, \(\Gamma\) is a lattice in a connected closed nilpotent subgroup of \(G\). 3.2. Proof of Theorem 3.1. We will examine all possible closures of the projection of \(\Gamma\) + +to \(\mathbb{R}_{\mathbf{H}}\times\mathbb{R}_{\mathbf{L}}\simeq\mathbb{R}^{2}\). + +Proposition 3.9. If \(\overline{p(\Gamma)} =\mathbb{R}_{\mathbf{H}}\times\mathbb{R}_{\mathbf{L}},\text{then}\Gamma\) does not act properly discontinuously and cocompactly on \(X\). Proof. By Lemma 3.8, \(\Gamma\) is a cocompact lattice in a connected nilpotent subgroup \(N\) of \(G\). We will show that \(N\cap\) Heis \(\neq \{1\}\). Our claim is that \(\dim N\geq n+2\). Indeed, \(\Gamma\) acts + +properly and cocompactly on the \(K(\pi,1)\) space \(N/C\simeq\mathbb{R}^{k},\text{where}C\) is the maximal compact subgroup of \(N\). So its cohomological dimension is equal to \(\dim(N/C)\). On the other hand, since \(\Gamma\) also acts properly and cocompactly on \(X=\mathbb{R}^{n+2}\), its cohomological dimension must be \(n+2\). This implies that \(\dim N/C=n+2\). Now, decompose \(N\text{as}N=\mathbb{R}\ltimes N_{1}\), where the \(\mathbb{R}-\)factor is generated by a one-parameter subgroup of \(G\) with a nontrivial projection on \(\mathbb{R}_{\mathbf{L}}\), and where + +\[N_{1}:=N\cap(\mathbb{R}_{\mathbf{H}}\times K)\ltimes\text{Heis},\] + +satisfies \(\dim N_{1}\geq n+1\). We have \(p_{\mathbf{H}}(N_{1})=\mathbb{R}\), so we further decompose \(N_{1}\text{as}N_{1}=\mathbb{R}\ltimes N_{0}\), where the \(\mathbb{R}-\)factor is a one-parameter subgroup \(s(t)\) of \((\mathbb{R}_{\mathbf{H}}\times K)\ltimes\)Heis that has a nontrivial projection on \(\mathbb{R}_{\mathbf{H}}\), and + +\[N_{0}:=N_{1}\cap(K\ltimes\text{Heis}),\] + +\(\mathrm{O}(n)\), whose maximal possible dimension is \(\lfloor \frac{n}{2} \rfloor\). Consequently, \(N_{0}\) must intersect Heis nonwith \(\dim N_{0}\geq n\). Since \(N_{0}\) is nilpotent, its projection to \(K\) is an abelian subgroup of trivially. Moreover, \(N_{1}\) contains the subgroup \(s(t)\ltimes(N_{0}\cap\text{Heis})\), which is therefore nilpotent. However, since the action of \(s(t)\text{on}N_{0}\cap\) Heis is semisimple (and unipotent), it must be trivial. This is impossible since \(s(t)\) projects non-trivially on \(\mathbb{R}_{\mathbf{H}}\), so its real eigenvalues are all different from 1 (the real eigenvalues of its projection to \(K\) are all equal to 1). + +- + +Proposition 3.10. If \(\overline{p(\Gamma)} =\mathbb{R}\), the action of \(\Gamma\) is properly discontinuous and cocompact if and only if \(\Gamma\) is contained in the isometry group. Proof. We have a short exact sequence \(1\rightarrow K\ltimes\) Heis \(\rightarrow G\rightarrow \underline{\mathbb{R}_{\mathbf{L}}\times} \mathbb{R}_{\mathbf{H}}\rightarrow1\). Let \(\mathbb{R}_{\mathbf{T}}= \{e^{tT},t\in\mathbb{R}\}\) be a one-parameter subgroup of \(G\text{suchthat}p(\mathbb{R}_{\mathbf{T}})= \overline{p(\Gamma)} .\text{Then},\Gamma\) is contained in \(G_{1}:=\mathbb{R}_{\mathbf{T}}\ltimes(K\ltimes\) Heis). We will show that this group is, in fact, isomorphic to a group of the form \((\mathbb{R}\times K)\ltimes\text{Heis}\). Write \(\mathbf{T}=\lambda+\Phi+\omega,\text{with}\lambda\in\mathbb{R}H\oplus\mathbb{R}L,\Phi\in k,\text{and}\omega\in heis\). For any \(\Psi\in k\), we have + +\[[\mathbf{T},\Psi]=[\Phi,\Psi]+[\omega,\Psi]\in k\ltimes heis.\] (2) diff --git a/outputs/arxiv_math/2503.08618_pg9_pg1_repeat1.md b/outputs/arxiv_math/2503.08618_pg9_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..2b76234e95c5ccbe7cd62275e8e11ba4667fe41e --- /dev/null +++ b/outputs/arxiv_math/2503.08618_pg9_pg1_repeat1.md @@ -0,0 +1,23 @@ +On removing the restriction from \(\beta_{n_{j}},j=0,1,2,\ldots,n\) in Corollary 4.1 i.e. \(\alpha_{n_{0}}\leq \alpha_{n_{1}}\leq\cdots \leq\alpha_{n_{k}}\text{and for all}\beta_{n_{j}}\text{in}p(q)=\sum_{\nu=0}^{k}q^{n_{\nu}}a_{n_{\nu}}\) , we have from \((4.18)\) + +\[\begin{aligned} |\xi_{1}(q)|&\leq MR^{n}\{\sum_{j=1}^{k}(|\alpha_{n_{j}}-\alpha_{n_{j-1}}|+|\beta_{n_{j}}-\beta_{n_{j-1}}|)+|\alpha_{n_{0}}|+|\beta_{n_{0}}|\} \\ &\leq MR^{n}\{(\alpha_{n_{k}}-\alpha_{n_{0}}+|\alpha_{n_{0}}|)+2\sum_{j=0}^{k-1}|\beta_{n_{j}}|\} \end{aligned}\] + +and on combining it with (4.20), we have the following; + +Corollary 4.2. Let \(p\in\mathbb{P}_{n}\text{with}a_{n_{l}}=\alpha_{n_{l}}+\beta_{n_{l}}i\) such that no two adjacent + +coefficients \(a_{n_{j}}^{\prime}s\) are equal, i.e. \(a_{n_{j}}\neq a_{n_{j+1}},j=0,1,2,\cdots ,k\text{and}\alpha_{n_{0}}\leq\alpha_{n_{1}}\leq \cdots \leq\alpha_{n_{k}}\text{and for all}\beta_{n_{j}},0\leq j\leq n.\) Further let \(M_{n_{j}},1\leq j\leq k\text{with}M\) as defined in Theorem 2.1. Then all the zeros lie in + +\[|q|\leq \frac{M}{|a_{n}|} \{(\alpha_{n}-\alpha_{0}+|\alpha_{0}|)+2\sum_{j=0}^{k-1}|\beta_{n_{j}}|+|\beta_{n}|\}.\] (4.24) + +Setting \(\beta_{n_{j}}=0,j=0,1,\ldots,n\) and taking \(\alpha_{n_{j}}=a_{n_{j}},j=0,1,\ldots,n\), we have the following result from Corollary 4.2. Corollary 4.3. Let \(p(q)=\sum_{\nu=0}^{n}q^{n_{j}}a_{n_{j}}\) such that no two adjacent coefficients + +\(a_{n_{j}}^{\prime}s\) are equal, i.e. \(a_{n_{j}}\neq a_{n_{j+1}},j=0,1,2,\cdots ,k\text{and}\alpha_{n_{0}}\leq\alpha_{n_{1}}\leq\cdots \leq\alpha_{n_{k}}\). + +Further let \(M_{n_{j}},1\leq j\leq k\text{with}M\) as defined in Theorem 2.1. Then all the zeros of \(p(q)\) lie in + +\[|q|\leq \frac{M}{|a_{n}|} (\alpha_{n}-\alpha_{0}+|\alpha_{0}|).\] (4.25) + +## 5. Conclusion + +Over the past ten years, the regular functions of the quaternionic variable have been introduced and thoroughly examined. Their diverse applications across a wide range of scientific disciplines have contributed significantly to their rapid development and have shown themselves to be a rich topic for investigation. We note that it became interesting to determine the regions containing some or all of the zeros of a regular polynomial of quaternionic variable following the study of the structure of zero sets and the Fundamental Theorem of Algebra for regular polynomials. The distribution of zeros for polynomials with quaternionic variables and quaternionic coefficients was not well covered in the literature.Here, we derive regions that contain all of the zeros of a regular polynomial of a quaternionic variable when the structure of the zero sets established in the recently developed theory of regular functions and polynomials of a quaternionic variable, as well as a maximum modulus theorem, restrict the real and imaginary parts of its coefficients. diff --git a/outputs/arxiv_math/2503.08630_pg15_pg1_repeat1.md b/outputs/arxiv_math/2503.08630_pg15_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..b01e3b5a84883fd1ca1358f0f5328788dc665204 --- /dev/null +++ b/outputs/arxiv_math/2503.08630_pg15_pg1_repeat1.md @@ -0,0 +1,25 @@ +\[\begin{aligned} n-1 \\ \sum_{v\in\Lambda i=0} \end{aligned}\] + +\(i=0Sv,w_{i}\). Consequently, + +therefore implies that \(1=\sum\) + +\[\begin{aligned} U_{j}U_{j}^{*}&=\left(\sum_{v\in\Lambda^{0}}S_{(v,f_{i,j})}^{*}\right)\sum_{u\in\Lambda^{0}}\sum_{m=0}^{n-1}S_{(u,f_{m,j})}\Bigg) \\ &=\sum_{v\in\Lambda}\sum_{i=_{v}0}^{n-1}\sum_{\in\Lambda(_{0}v,f_{i,j})}^{n_{i=}-_{0}1_{*}}SS_{(v,f_{i,j})}=\sum_{v\in\Lambda}\sum_{i=0}^{n-1}S_{v,w_{i}} \\ &=1=\left(\sum_{i=0}^{n-1}S_{(v,f_{i,j})}\right)\sum_{u\in\Lambda^{0}}\sum_{m=0}^{n-1}S_{(u,f_{m,j})}^{*}\Bigg) \\ &=U_{j}^{*}U_{j}. \end{aligned}\] + +We conclude that \(\{U_{j}\}\) is a unitary representation of \(\mathbb{Z}^{k_{2}}\) contained in \(C^{*}(\Omega)\). To see that conjugation by \(U_{j}\) will induce the action of \(\rho_{j}\), fix \(\lambda\in\Omega_{1}^{w_{m}}(\cong \Lambda).\) By Lemma 2.9, + +\[\begin{aligned} \hat{} \\ (\lambda,w_{m})(s(\lambda),f_{m-1,j})&=(r(\lambda),f_{m-1,j})((f_{m-1,j})_{\rhd}(\lambda),w_{m-1}) \end{aligned}\] + +since \(C_{n,k_{2}}\) is a stable quasi-factor. (CK2) now implies that + +\[\begin{aligned} U_{j}S_{(\lambda,w_{m})}U_{j}^{*}&=\left(\sum_{i=0v}^{n-1}\sum_{\in\Lambda^{0}}S_{(v,f_{i,j})}^{*}\right)S_{(\lambda,w_{m})}\left(\sum_{\ell=0u}^{n-1}\sum_{\in\Lambda^{0}}S_{(u,f_{\ell,j})}\right) \\ &=S_{(r(\lambda),f_{m-1,j})}^{*}S_{(\lambda,w_{m})}\left(\sum_{\ell=0u}^{n-1}\sum_{\in\Lambda^{0}}S_{(u,f_{\ell,j})}\right) \\ &=S_{(r(\lambda),f_{m-1,j})}^{*}S_{(\lambda,w_{m})}S_{(s(\lambda),f_{m-1,j})} \\ &=S_{(r(\lambda),f_{m-1,j})}^{*}S_{(r(\lambda),f_{m-1,j})}S_{((\widehat{f}_{m-1,j})_{\rhd}(\lambda),w_{m})} \\ &=S_{((\widehat{f}_{m-1,j})_{\rhd}(\lambda),w_{m})}=\rho_{m-1,j}(S_{(\lambda,w_{m})}). \end{aligned}\] + +As each \(\rho_{i,j}\) is a \(*-\text{homomorphism},\text{and}C^{*}(\Omega_{1}^{w_{m}})\) is generated by \(\{S_{(\lambda,w_{m})}\}_{\lambda}\), we conclude that the automorphism \(\rho_{j}\) of \(\bigoplus_{i=0}^{n-1}C^{*}(\Omega_{1}^{w_{i}})\) is indeed given by conjugation by \(U_{j}\). This demonstrates that \(\{U_{j}\}_{j=1}^{k_{2}}\) yields a unitary representation of \(\mathbb{Z}^{k_{2}}\) which induces \(\rho\text{on}C^{*}(\Omega_{1})\cong\) the crossed product now gives a ∗-homomorphism + +\[\begin{aligned} n-1 \\ \bigoplus_{i=0} \end{aligned}\] + +\(C_{i=0}^{*}(\Omega_{1}^{w_{i}})\) via conjugation. The universal property of + +\[\Phi:\left(\bigoplus_{i=0}^{n-1}C^{*}(\Omega_{1}^{w_{i}})\right)\rtimes_{\rho}\mathbb{Z}^{k_{2}}\rightarrow C^{*}(\Omega)\] + +with image \(\langle C^{*}(\Omega_{1}),U_{j}\rangle\). It remains to show that \(\Phi\) is bijective. We demonstrate surjectivity by observing first that each vertex projection \(S_{(v,w_{i})}\) lies in \(C^{*}(\Omega_{1})\) and therefore \(S_{(v,w_{i+1})}U_{j}^{*}=S_{(v,f_{i,j})}\in\text{Im}\Phi\text{for all}v,i,j\). In particular, every generator of \(C^{*}(\Omega_{2})\) lies in \(\text{Im}\Phi\), making the image all of \(C^{*}(\Omega)\). To demonstrate injectivity, we construct a Cuntz–Krieger \(\Omega-\)family inside of \((\bigoplus_{i}C^{*}(\Omega_{1}^{w_{i}}))\rtimes_{\rho}\mathbb{Z}^{k_{2}}\), such that the associated \(*-\text{homomorphism}\Psi:C^{*}(\Omega)\rightarrow (\bigoplus_{i}C^{*}(\Omega_{1}^{w_{i}}))\rtimes_{\rho}\mathbb{Z}^{k_{2}}\) satisfies \(\Psi\Phi=\) id. To that end, write \(V_{j}\) for the canonical generators of \(\mathbb{Z}^{k_{2}}\) in the crossed product, and for each \(e\in G(\Lambda)^{1},f_{i,j}\in C_{n,k_{2}}^{1}\) and each vertex \(v\in\Lambda^{0},w_{i}\in C_{n,k_{2}}^{0},\) define \(T_{(e,w_{i})},T_{(v,w_{i})},T_{(v,f_{i,j})}\in(\bigoplus_{i}C^{*}(\Omega_{1}^{w_{i}}))\rtimes_{\rho}\mathbb{Z}^{k_{2}}\) diff --git a/outputs/arxiv_math/2503.08634_pg6_pg1_repeat1.md b/outputs/arxiv_math/2503.08634_pg6_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..e2e6414e7a1e0a5fd65140eab7aff21d17f437b5 --- /dev/null +++ b/outputs/arxiv_math/2503.08634_pg6_pg1_repeat1.md @@ -0,0 +1,55 @@ +[Case i] The following error bounds hold true for some \(M>0\). + +(i-1) \(0\leq\mathbb{E}[h(\hat{x}_{\eta})]-h^{*}\leq Err_{\eta}+\eta M\). + +\[\begin{aligned} (i-2) \tfrac{\mu_{f}}{2} \mathbb{E}[\Vert\hat{x}_{\eta}-x^{*}\Vert^{2}]-\Vert\nabla f(x^{*})\Vert\mathbb{E}[\text{dist}(\hat{x}_{\eta},X_{h}^{*})]&\leq\mathbb{E}[f(\hat{x}_{\eta})]-f^{*}\leq \tfrac{1}{\eta} Err_{\eta}. \\ \Big( \end{aligned}\] + +\()_{h}]+ \frac{1}{\eta} Err_{\eta}\Big)\) + +(i-3) If \(\mu_{f}>0,\text{then}\mathbb{E}[\Vert\hat{x}_{\eta}-x^{*}\Vert^{2}]\leq \frac{2}{\mu_{f}} \Big(\Vert\nabla f(x^{*})\Vert\mathbb{E}[\text{dist}(\hat{x}_{\eta},X^{*}\) . + +[Case ii] Suppose \(X_{h}^{*}\) is \(\alpha-\)weak sharp with order \(\kappa\geq1\). Then, the following holds for some \(M>0\). + +(ii-1) \(0\leq\mathbb{E}[h(\hat{x}_{\eta})]-h^{*}\leq Err_{\eta}+\eta M\). + +\[((_{i}i_{-}i-_{3}2_{)})_{I} \frac{\mu_{f}}{f^{2}\mu} \mathbb{E}_{f}[\Vert_{>}\hat{x}_{0^{\eta},}-_{th}x_{e^{*}n}\Vert_{\mathbb{E}}^{2}]_{[}-_{\Vert\hat{x}}\Vert_{\eta}\nabla_{-}f(_{x}x_{\Vert^{*}}^{*})_{2}\Vert_{]\leq^{\kappa}} \sqrt{\frac{\tfrac{1}{\alpha}_{2} (}{\mu_{f}} E\Big(r_{\Vert}r_{\nabla^{\eta}f}+_{(x}\eta_{*}M_{)\Vert})} \sqrt{\kappa}\leq \frac{\mathbb{E}[f(\hat{x}_{\eta})]-f^{*}}{\tfrac{1}{\alpha} (Err_{\eta}+\eta M)} \leq_{+} \tfrac{1}{\frac{\eta_{1}}{\eta}} E_{E}r_{r}r_{r_{\eta}^{\eta}}.\Big).\] + +[Case iii] If \(X_{h}^{*}\) is \(\alpha-\)weak sharp with order \(\kappa=1\text{and}\eta\leq \frac{\alpha}{2\Vert\nabla f(x^{*})\Vert} ,\) then the following holds. + +(iii-1) \(0\leq\mathbb{E}[h(\hat{x}_{\eta})]-h^{*}\leq2Err_{\eta}.\) + +\[(\text{iii}-2) \tfrac{\mu_{f}}{2} \mathbb{E}[\Vert\hat{x}_{\eta}-x^{*}\Vert^{2}]- \tfrac{2\Vert\nabla f(x^{*})\Vert}{\alpha} Err_{\eta}\leq\mathbb{E}[f(\hat{x}_{\eta})]-f^{*}\leq \tfrac{1}{\eta} Err_{\eta}.\] + +(iii-3) If \(\mu_{f}>0,\text{then}\mathbb{E}[\Vert\hat{x}_{\eta}-x^{*}\Vert^{2}]\leq \frac{2}{\eta\mu_{f}} Err_{\eta}.\) + +Proof. (i-1) In view of the definition of Err\(\text{and}_{\eta}f_{\eta}\), we have \(\mathbb{E}[h(\hat{x}_{\eta})+\eta f(\hat{x}_{\eta})]-f_{\eta}^{*}\leq\text{Err}_{\eta}.\) Also, from the definition of \(f_{\eta}^{*}\) , we have \(f_{\eta}^{*}\leq f_{\eta}(x^{*})=h^{*}+\eta f^{*}\). From the preceding two relations, we obtain + +\[\mathbb{E}[h(\hat{x}_{\eta})]-h^{*}+\eta(\mathbb{E}[f(\hat{x}_{\eta})]-f^{*})\leq\text{Err}_{\eta}.\] (2) + +By invoking Assumption 1, we have \(\mathbb{E}[f(\hat{x}_{\eta})]>-\infty\). Therefore, there exists some \(M>0\) such that \(f^{*}-\mathbb{E}[f(\hat{x}_{\eta})]0\) such that + +\[\begin{aligned} d&\in C^{2}(\Gamma_{\sigma}^{\circ})\cap C^{0}( \overline{\Gamma}_{\sigma} ), \\ |\nabla d(x)|&=1\text{for all}x\in\Gamma_{\sigma}, \end{aligned}\] (2.4) (2.5) + +Moreover, for every measurable nonnegative function \(g:(0,\sigma)\rightarrow\mathbb{R}\) + +\[g\circ d\in L^{1}(\Gamma_{\sigma})\Longleftrightarrow g\in L^{1}(0,\sigma).\] (2.6) + +Proof. For (2.4) and (2.5) see e.g. [6] Appendix 14.6. Here’s a brief outline of how to prove (2.6): from the coarea formula and (2.4) and (2.5), we have, since \(|\nabla d(x)|=1\) + +\[\int_{\Gamma_{\sigma}}g(d(x))dx=\int_{0}^{\sigma}g(t)\mathcal{H}^{N-1}(\Gamma_{\sigma}\cap\{d=t\})dt,\] + +where \(\mathcal{H}^{N-1}\) is the Hausdorff measure of \(\Gamma_{\sigma}\cap\{d=t\}\). Since the \(\partial\Omega\) is \(C^{2}\), there exist two positive constants \(c_{1}\text{and}c_{2}\) such that + +\[c_{1}H^{N-1}(\partial\Omega)\leq H^{N-1}(\Gamma_{\sigma}\cap\{d=t\})\leq c_{1}H^{N-1}(\partial\Omega)\] + +(see e.g. [9] , Appendix 2.12.3). This ends the proof. + +- diff --git a/outputs/arxiv_math/2503.08675_pg30_pg1_repeat1.md b/outputs/arxiv_math/2503.08675_pg30_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..b674af67dfb58e88de81feaf0c5824f8ac0d547e --- /dev/null +++ b/outputs/arxiv_math/2503.08675_pg30_pg1_repeat1.md @@ -0,0 +1,39 @@ +for some almost surely finite random variable \(O^{\prime}\). Remark 6.8. Under the assumption that \(\liminf_{i\rightarrow\infty}d(i)\geq R\) we can adapt the proof of (6.23) (and by using (6.1) rather than (6.2)) to show that for any \(\varepsilon>0\) there exists \(p>0\) such that + +\[\lim_{t\rightarrow\infty}\mathbb{P}_{\mathcal{S}}\left(\forall s\in[(1-p)t,(1+p)t]:\Big| \frac{O_{s}^{\text{cont}}}{t} - \frac{R}{\lambda^{*}+R} \Big|<\varepsilon\right)=1.\] + +◀ + +It yields a weaker bound on \(O_{s}^{\text{cont}}\), but with a weaker condition on \(d\) and a larger range of \(s\). Remark 6.9. When setting \(s=t\) in (6.23) and (6.24), the quantity in the absolute values converges to zero in probability. Though in some cases where \(\mathcal{K}\) grows sufficiently fast it may be possible to strengthen the convergence in probability to almost sure convergence, we are unable to derive the stronger bounds due to the use of (6.18) and (6.19) in Corollary 6.6. Stronger versions of these results would be required to improve to almost sure convergence. Remark 6.10. Proposition 6.7 provides the main step in proving the asymptotic behaviour of \(O_{n}\), as presented in Theorems 2.3 through 2.10. It remains to translate the asymptotic behaviour of \(O_{t}^{\text{cont}}\) into the asymptotic behaviour of \(O_{n}\), which is carried out in Section 9. + +◀ + +◀ + +Intuitively, an individual born at time \(Tt-T)\), independently of all other individuals. At the same time, the number of individuals born around time \(T\) is roughly \(\mathrm{e}^{\lambda^{*}T}\) by Proposition 6.5. The proof uses first and second moment bounds to establish the optimal choice of \(T\) such that no individuals born before \(T\) are alive at time \(t\), whereas many individuals born after \(T\) are alive at time \(t\). + +Proof. We first prove (6.23). Fix \(K,p>0\) and define + +\[\ell_{t}:= \frac{R}{\lambda^{*}+R} t-K\log t,u_{t}:= \frac{R}{\lambda^{*}+R} t+K\log t,\tilde{u}_{t}:= \frac{R}{\lambda^{*}+R} t,\] (6.26) + +and + +\[\underline{r} (t):=t-p\log t, \overline{r} (t):=t+p\log t.\] + +We then observe that \(O_{t}^{\text{cont}}\) is increasing in \(t\), so that + +\[\mathbb{P}_{\mathcal{S}}\left(\Big|O_{s}^{\text{cont}}- \frac{R}{\lambda^{*}+R} t\Big|0\), recalling that \(\mathcal{A}_{t}^{\mathrm{c}\text{ont}}\) denotes the set of alive individuals at time \(t\), + +\[\begin{aligned} \mathbb{P}_{\mathcal{S}}\left(O_{\underline{r} (t)}^{\text{co}\mathrm{n}\mathrm{t}}\leq\ell_{t}\right)_{\leq}&= \frac{\mathbb{P}_{1^{\mathcal{S}}}\big(\mathcal{B}}{\mathbb{P}(\mathcal{S})} \Big[(_{\mathbb{P}}0\left(,\ell_{\mathcal{B}^{t}})_{(0}\cap_{,\ell}\mathcal{A}_{t} \frac{r}{)}^{(} \cap^{t)}\mathcal{A}^{\neq _{\underline{r} (}\emptyset_{t)}}\right)\neq \emptyset,|\mathcal{B}(0,\ell_{t})|\leq\mathrm{e}^{\lambda^{*}\ell_{t}+\delta\log t}\Big) \\ +\mathbb{P}\left(|\mathcal{B}(0,\ell_{t})|\geq\mathrm{e}^{\lambda^{*}\ell_{t}+\delta\log t}\right)\Big]. \end{aligned}\] (6.28) + +The second probability in the brackets converges to zero by Corollary 6.6 (with \(r=0,s=\ell_{t}\), and \(u=\delta\log t)\). Furthermore, lifetimes among individuals are i.i.d. and each individual born before time \(\ell_{t}\) needs to live for at least \(\underline{r} (t)-\ell_{t}\) time to be alive at time \(\underline{r} (t)\). Hence, by a union bound and using the upper bound in (6.2) of Lemma 6.1, we arrive for some large constant \(C>0\) at the upper bound + +\[\mathbb{P}(\mathcal{S})^{-1}\mathrm{e}^{\lambda^{*}\ell_{t}+\delta\log t}\mathbb{P}(L\geq \underline{r} (t)-\ell_{t})+o(1)\leq\mathrm{e}^{\lambda^{*}\ell_{t}+\delta\log t-R( \underline{r} (t)-\ell_{t})+C\log( \underline{r} (t)-\ell_{t})}+o(1).(6.29)\] + +By the definition of \(\ell_{t}\text{and} \underline{r} (t)\), this equals + +\[\exp\left(\left[-(\lambda^{*}+R)K+\delta+Rp+C\right]\log t+\mathcal{O}(1)\right)+o(1).\] + +Hence, for any \(K\) larger than \(K_{0}:=C(\lambda^{*}+R)^{-1}\), we can choose \(\delta\text{and}p\) sufficiently small so that the terms in the square brackets are negative. As a result, the upper bound in (6.29) tends to zero with \(t\) for these choices of \(K,p,\text{and}\delta\). diff --git a/outputs/arxiv_math/2503.08705_pg1_pg1_repeat1.md b/outputs/arxiv_math/2503.08705_pg1_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..26b276a9520df6c675dfaf958992d39269c2f360 --- /dev/null +++ b/outputs/arxiv_math/2503.08705_pg1_pg1_repeat1.md @@ -0,0 +1,41 @@ +# A Block-Based Heuristic Algorithm for the Three-Dimensional Nuclear Waste Packing Problem + +#### Yajie Wena,1, Defu Zhangb,2 + +Xiamen University, Siming south street 443, Xiamen, 361000, Fujian, China + +## ARTICLE INFO + +Keywords: Three-dimensional packing Nuclear waste Heuristic + +##### ABSTRACT + +In this study, we present a block-based heuristic search algorithm to address the nuclear waste container packing problem in the context of real-world nuclear power plants. Additionally, we provide a dataset comprising 1600 problem instances for future researchers to use. Experimental results on this dataset demonstrate that the proposed algorithm effectively enhances the disposal pool’s space utilization while minimizing the radiation dose within the pool. The code and data employed in this study are publicly available to facilitate reproducibility and further investigation. + +## 1. Introduction + +This paper addresses the challenge of nuclear waste treatment in the context of a real-life nuclear power plant. During regular operation, nuclear power plants generate radioactive waste, which must be collected, classified into distinct boxes, and stored in pre-dug disposal pools. The waste packages are sealed with cement to ensure the safety of the environment and human health. The waste packages within the pools must be strategically arranged to maximize disposal pool usage and reduce radiation exposure per unit time (measured in Sieverts per second) in each pool. This problem can be formulated as a three-dimensional knapsack problem with an additional constraint on the minimum radiation limit. To tackle this, the paper proposes a block-based heuristic algorithm, BSNA, to identify optimal placement strategies for the waste boxes. Furthermore, we generate 1600 synthetic nuclear waste packing problems based on real-world nuclear waste data to evaluate the proposed algorithm’s performance. These generated problems are intentionally more complex than typical real-world scenarios, incorporating larger disposal pools and a greater variety and volume of nuclear waste boxes. This provides a comprehensive test bed for assessing the algorithm’s effectiveness in more challenging settings. + +## 2. Problem Description + +The dose rate of a nuclear waste box at a given position, measured in Sieverts per second, is determined by the number of radioactive decay occurring per second, known as the activity, multiplied by the dose rate constant of the radioactive element. This value is then divided by the square of the distance between the nuclear element and the observation point. Since the dose rate constant for a specific nuclear element is a fixed value, the radiation dose of the nuclear waste box is directly proportional to the activity and inversely proportional to the square of the distance. The + +↑ wynne.jei@gmail.com (Y. Wen); dfzhang@xmu.edu.cn (D. Zhang) ORCID(s): 0009-0004-2053-0583 (Y. Wen) + +relationship is mathematically expressed as (1). + +\[\dot{H}= \frac{\Gamma\cdot A}{r^{2}}\] (1) + +- Dose Rate \(\dot{H}[Sv/t]\): The amount of radiation absorbed per unit time, adjusted for biological effects. + +- Dose Rate Constant \(\Gamma[Sv\cdot m^{2}/(Bq\cdot t)]\): A factor that describes how much radiation dose is received per unit of radioactive source activity at a given distance. + +- Activity \(A[Bq]\): The number of radioactive decay occurring per second. + +- Distance \(r[m]\): The distance from the radiation source to the point of measurement. + +For simplicity, let the vertical distance from the center point of the nuclear waste box to the top of the disposal pool be \(r\). Given a disposal pool with length \(L\), width \(W\), and height \(H\), and a set of \(n\) nuclear waste boxes, each with length \(l_{i}\), width \(w_{i}\), height \(ℎ_{i}\), activity \(A_{i}\), and dose rate constant \(\Gamma_{i}\), the objective of the problem is to place as many nuclear waste boxes as possible into the disposal pool to maximize the utilization rate of the disposal pool, while minimizing the sum of the dose rates of all the nuclear waste boxes in the disposal pool. This problem is similar to a variant of the three-dimensional bin packing problem, as it has an additional optimization objective. + +## 3. Literature Review + +The three-dimensional bin packing problem (3D-BPP or 3D-CLP) is a well-known NP-hard optimization problem. Over the years, various approaches have been developed to tackle this problem, including exact algorithms, heuristic algorithms, metaheuristic algorithms, and more recently, deep learning-based methods. Among these, heuristic and metaheuristic algorithms are the most widely adopted due to their practical efficiency in solving large-scale problems. diff --git a/outputs/arxiv_math/2503.08711_pg7_pg1_repeat1.md b/outputs/arxiv_math/2503.08711_pg7_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..b17bdcd7b34d94428435af0253e872baf5400840 --- /dev/null +++ b/outputs/arxiv_math/2503.08711_pg7_pg1_repeat1.md @@ -0,0 +1,180 @@ +Table 2 + +Running Time of BSPA for Different \(p\) Values. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Datasets
10 t(s)30 t(s)60 t(s)90 t(s)120 t(s)
gapgapgapgapgap
C60.001.4051.431.3250.001.3251.431.4051.431.40
N60.000.6853.080.6853.080.6853.080.6853.080.68
NT-N132.862.6675.432.6660.862.6660.002.6660.002.66
NT-T135.432.6674.572.6660.862.6960.002.6660.002.66
Avg109.902.1667.502.1457.692.1557.402.1657.402.16
KR117.502.7070.002.7060.002.7060.002.7060.002.70
BWMV435.847.62177.127.65113.287.6691.447.6681.127.66
Avg428.387.51174.617.53112.037.5590.707.5480.637.54
+ +Table 3 + +BSPA Results Under the OF and RF Scenario. + +| Datasets | gap | BSPA(0.1,30,30,90) t(s) | OF gap | BSPA(0.1,60,60,90) t(s) | gap | BSPA(0.1,30,30,90) t(s) | RF gap | BSPA(0.1,60,60,90) t(s) | +| --- | --- | --- | --- | --- | --- | --- | --- | --- | +| C | 1.40 | 51.43 | 1.32 | 100.00 | 0.79 | 45.7 | 0.75 | 91.43 | +| N | 0.68 | 53.08 | 0.67 | 106.15 | 0.72 | 53.1 | 0.60 | 101.54 | +| NT-N | 2.66 | 60.00 | 2.59 | 120.00 | 2.14 | 60.0 | 2.04 | 120.00 | +| NT-T | 2.66 | 60.00 | 2.63 | 120.00 | 2.13 | 60.0 | 2.06 | 120.00 | +| Avg. | 2.16 | 57.40 | 2.10 | 114.23 | 1.69 | 56.3 | 1.61 | 111.92 | +| KR | 2.70 | 60.00 | 2.67 | 120.00 | 1.60 | 60.0 | 1.60 | 120.00 | +| BWMV | 7.62 | 91.44 | 7.56 | 181.68 | 3.03 | 72.7 | 2.79 | 141.12 | +| Avg. | 7.51 | 90.70 | 7.44 | 180.23 | 3.00 | 72.4 | 2.76 | 140.63 | + +### 3.4. Experiment and analysis + +Table 3 presents the computational results of the algorithm under both OF and RF scenarios. In this table, + +\(p\) + +proposed in this paper is limited to such high-end configurations, as this setup was chosen solely to meet the specific requirements of the experiments. Our algorithm is + +the notation \(BSPA_{0.1,30,30,90}\) denotes the execution of the capable of running on personal computers used in daily algorithm with parameters \(b=0.1,T_{1}=30\) s, \(T_{3}=30\) s, life and can even outperform the results obtained in the and \(p=90\). This notation is consistently applied across other aforementioned experiments because personal computers + +tables. As shown in Table 3, under both OF and RF scenarios, the proposed algorithm demonstrates a consistent reduction in the gap value across all datasets as search time increases. However, despite a twofold increase in search time, the reduction in the gap value remains marginal. Nevertheless, the gap values obtained by our algorithm across all datasets are within an acceptable range. Specifically, for the zero-waste datasets C, N, NT-N, and NT-T, the average fill length exceeds the minimum length by only 2.1%. For the non-zero-waste dataset KR, this deviation is merely 2.67%, while for the large-scale dataset BWMV, it remains limited to 7.57%. These results substantially outperform those of manual filling, indicating that the proposed method can be effectively applied to industrial material cutting to assist enterprises in reducing material waste during the cutting process. In the previous experiments, we used a platform with a 256-core processor and 120 GB of RAM to run our algorithm. However, this does not imply that the algorithm usually have higher CPU frequency. Furthermore, if the algorithm is implemented using more efficient programming languages, such as C or C++, the performance of loading results within the same time limit will be further enhanced. Table 4 presents the algorithm’s performance on a widely used PC with an Intel i7-13700K CPU, 16GB of RAM, and the Windows 11 operating system, using standard datasets. + +## 4. Conclusions + +This paper presents a beam search-based parallel method for solving the two-dimensional strip packing problem. Experimental results demonstrate that the proposed algorithm can find an efficient placement strategy to minimize the used length of the strip as much as possible. To facilitate future research, the code and datasets are available at https: //github.com/Yzhjdj/BSPA diff --git a/outputs/arxiv_math/2503.08736_pg1_pg1_repeat1.md b/outputs/arxiv_math/2503.08736_pg1_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..a01691bf12bb17690ad52241ae200078e6ab8997 --- /dev/null +++ b/outputs/arxiv_math/2503.08736_pg1_pg1_repeat1.md @@ -0,0 +1,39 @@ +# A Note on Clifford Stabilizer Codes for Ising Anyons + +Sanchayan Dutta∗ + +University of California, Davis + +We provide a streamlined elaboration on existing ideas that link Ising anyon (or equivalently, Majorana) stabilizer codes to certain classes of binary classical codes. The groundwork for such Majorana-based quantum codes can be found in earlier works (including, for example, Bravyi [BTL10] and Vijay et al. [VF17]), where it was observed that commuting families of fermionic (Clifford) operators can often be systematically lifted from weakly self-dual or self-orthogonal binary codes. Here, we recast and unify these ideas into a classification theorem that explicitly shows how q-isotropic subspaces in \(\mathbb{F}_{2}^{2n}\) yield commuting Clifford operators relevant to Ising anyons, and how these subspaces naturally correspond to punctured self-orthogonal codes in \(\mathbb{F}_{2}^{2n+1}\). + +## 1. INTRODUCTION + +Ising anyons are non-Abelian excitations in certain two-dimensional topological phases of matter. They can encode and process quantum information by virtue of their fusion and braiding properties. A particularly relevant physical instantiation comes from Majorana zero modes at the boundaries of topological superconductors [Kit01, Iva01]. Adapting quantum error-correcting strategies to the Isinganyon (or Majorana) context entails generalizing the familiar notion of Pauli stabilizer codes to accommodate fermionic degrees of freedom and parity constraints. Early progress in this direction was made by Bravyi, who introduced Majorana fermion codes [BTL10]. Building on this, Vijay et al. [VF17] highlighted how these fermionic stabilizers map naturally onto weakly self-dual or self-orthogonal binary codes. Related developments have investigated the broader class of Clifford stabilizer codes [Oka23], emphasizing that commuting, even-parity operators underlie valid stabilizer constraints. In this note, we revisit and unify these ideas by focusing on the concept of q-isotropic subspaces of \(\mathbb{F}_{2}^{n}\) [Oka23]. We show how such subspaces serve as the building blocks for defining Clifford stabilizer codes in Ising-anyon systems. Furthermore, by enlarging the ambient space to \(\mathbb{F}_{2}^{n+1}\) via a convenient mapping, these q-isotropic subspaces turn out to correspond exactly to punctured images of classical self-orthogonal codes, providing a direct route to analyzing quantum code properties (such as distance) in purely classical terms. One advantage of this viewpoint is that it allows us to utilize standard techniques from classical coding theory: in particular, distance bounds such as the Varshamov bound imply that one can construct families of codes with growing length, good distance, and nonvanishing rate, guaranteeing robust error protection in a broad sense. While these results have appeared in various forms in the literature, here we aim to offer a transparent exposition of why these classical arguments continue to apply in the Majorana (or Ising-anyon) context. + +∗ dutta@ucdavis.edu + +## 2. QUANTUM METRICS AND CLIFFORD CODES + +Here we give a concise overview of the foundational concepts from Chapter 2 of [Oka23], focusing on quantum metrics as a unifying framework for assigning distances to quantum errors. This is analogous to the Hamming distance for classical coding but is adapted to genuinely quantum settings. In finite dimensions, one can specify a generating set of errors, from which all higher-distance errors are built. We also introduce the Kuperberg-Weaver notion of a quantum metric [KW12], specialized here to finite dimensions, which assigns distances to operators by specifying how they are generated at each level of the metric. Examples include classical and quantum Hamming spaces, as well as spinorial, semispinorial, and full Clifford metrics. + +### 2.1. Kuperberg–Weaver Metrics in Finite Dimension + +We work in a finite-dimensional Hilbert space \(\mathcal{H}\) with associated operator space \(\mathcal{B}(\mathcal{H})\). A Kuperberg–Weaver quantum metric on \(\mathcal{B}(\mathcal{H})\) is a nested family of subspaces + +\[\{\mathcal{E}_{t}\}_{t\geq0}\subseteq\mathcal{B}(\mathcal{H}),\] + +satisfying the following axioms: + +\(\mathbf{1}.\mathcal{E}_{0}=\mathbb{C}I_{\mathcal{H}}\). Only scalar multiples of the identity have distance zero. + +\(\mathbf{2}.\mathcal{E}_{t}^{*}=\mathcal{E}_{t}\) for each t. Each level is closed under adjoints, so these subspaces consistently contain all relevant “errors” of a given size. + +\(\mathbf{3}.\mathcal{E}_{s}\mathcal{E}_{t}\subseteq\mathcal{E}_{s+t}\). This is a “triangle inequality” condition on errors, ensuring that composing an error of degree s with another of degree t yields an error of degree at most s +t. + +Graph Metrics. A convenient way to build such a quantum metric in finite dimensions is to specify a generating subspace + +\[\mathcal{E}\subseteq\mathcal{B}(\mathcal{H}),\] + +containing the identity \(I_{\mathcal{H}}\) and closed under adjoints. We interpret \(\mathcal{E}\) as the space of “lowest-degree” or single-step nontrivial errors [Oka23]. Define + +\[\mathcal{E}_{0}=\mathbb{C}I_{\mathcal{H}},\text{andforeachinteger}t\geq1,\] diff --git a/outputs/arxiv_math/2503.08770_pg37_pg1_repeat1.md b/outputs/arxiv_math/2503.08770_pg37_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..42080061bcb1d9fc9b1147573a45a91a766d05f6 --- /dev/null +++ b/outputs/arxiv_math/2503.08770_pg37_pg1_repeat1.md @@ -0,0 +1,39 @@ +We therefore only need to show that \(\sigma\) intertwines the differential, or equivalently, + +\[\sigma(\mathbf{r}(t_{1}+z-t_{2}))=\mathbf{r}(t_{1}-z+t_{2}).\] (4.46) + +This follows from applying \(\tau_{z}\) to equation (4.20), as well as the fact that r is difference-dependent. + +Remark 4.6. The commutativity condition of Proposition 4.5 will be called weak commutativity. + +From Proposition 4.5, we can define a “meromorphic” tensor product \(M_{z}\otimes N_{0}\) for a pair of FSF modules of \(U_{\hbar}(d(\mathcal{O}))\). However, the resulting module is not FSF anymore. Nevertheless, there is still a natural generalization of associativity condition as follows. Consider now a triple of FSF modules \(M=U[[\hbar]],N=V[[\hbar]]\text{and}P=W[[\hbar]]\), the tensor producr \(M_{z+w}\otimes(N_{w}\otimes P_{0})\) is naturally identified with the vector space + +\[U\otimes V\otimes W[[\hbar]]((w))((z+w))\] (4.47) + +where as \((M_{z}\otimes N)_{w}\otimes P_{0}\) is identified with + +\[U\otimes V\otimes W[[\hbar]]((z))((w)).\] (4.48) + +These two are not immediately comparable, but they both receive a map from the vector space + +\[U\otimes V\otimes W[[\hbar]][[z,w]][z^{-1},w^{-1},(z+w)^{-1}]\] (4.49) + +via formal expansion, just as in [FBZ04, Chapter 3]. We claim the following result, which is mostly clear from the definition. + +Proposition 4.7. For a triple of FSF modules \(M=U[[\hbar]],N=V[[\hbar]]\text{and}P=W[[\hbar]]\), the following statements are true. + +## • The vector space + +\[U\otimes V\otimes W[[\hbar]][[z,w]][z^{-1},w^{-1},(z+w)^{-1}]\] (4.50) + +has a well-defined DG module structure of \(U_{\hbar}(d(\mathcal{O}))\), whose action is induced by \((1\otimes \Delta_{w})\Delta_{z+w}=(\Delta_{z}\otimes1)\Delta_{w}\), and whose differential is given by + +\[d_{\mathbf{r}}(z,w):=d_{M}+d_{N}+d_{P}-2\hbar^{(}\mathbf{r}^{12}(t_{1}+z-t_{2})+\mathbf{r}^{13}(t_{1}+z+w-t_{2})+\mathbf{r}^{23}(t_{1}+w-t_{2})^{)}.\] + +We denote this module by \(M_{z+w}\otimes N_{w}\otimes P_{0}\), signifying its independence of order. + +- Formal expansion as discussed above gives the following maps of DG modules of \(U_{\hbar}(d(\mathcal{O}))\), both of which are equivalences after appropriate ring extension: + +\[M_{z+w}\otimes N_{w}\otimes P_{0}\] + +\[(M_{z}\otimes N)_{w}\otimes P_{0}M_{z+w}\otimes(N_{w}\otimes P_{0})\] diff --git a/outputs/arxiv_math/2503.08773_pg19_pg1_repeat1.md b/outputs/arxiv_math/2503.08773_pg19_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..ff5fbc4b74a86ed7fc91e4df23fdfc194e3f7c4e --- /dev/null +++ b/outputs/arxiv_math/2503.08773_pg19_pg1_repeat1.md @@ -0,0 +1,35 @@ +### 4.1 Entanglement temperatures + +Let a general multi-interval region in \(d=2\) (at the line \(t=0)\) be \(A=\cup_{i=1}^{N}(l_{i},r_{i})\). In a CFT the modular Hamiltonian \(K\) of the vacuum is non local except for the case of a single interval. However, it was conjectured in [9, 10] that for multi-intervals there is a local term in \(K\) having the same universal structure for any CFT. This writes + +\[K_{\text{loc}}=\int dx\beta(x)T_{00}(x),\] (4.1) + +where the “local inverse temperatures” are + +\[\beta(x)= \frac{2\pi}{\sum_{i=1}^{n}\left( \tfrac{1}{x-l_{i}} - \tfrac{1}{x-r_{i}} \right)} .\] (4.2) + +The modular Hamiltonian for multi-interval regions in \(d=2\) is known explicitly for free massless scalar and fermions [24, 25], and the local term agrees with (4.1). A clear operational definition of these entanglement temperatures is that they measure the ratio between the expectation value of the modular Hamiltonian and the energy for very localized high energy excitations: + +\[\beta(x)=\lim_{E\rightarrow\infty,\Delta x\rightarrow0} \frac{\langle\psi|K|\psi\rangle}{\langle\psi|H|\psi\rangle} .\] (4.3) + +Here \(|\psi\rangle\) is an excitation above the vacuum localized in \(\Delta x\) around the point \(x\) and has energy \(E\). This limit can be argued to exist based on monotonicity of the relative entropy [9]. We will now show the universality of these entanglement temperatures in a more formal way. To test the entanglement temperatures we will be using a chiral field operator acting on the vacuum as the excited state probe. We would like to compute (for a real chiral field \(\phi)\) + +\[\beta(x)=\lim_{y\rightarrow x} \frac{\langle0|\phi(x)K\phi(y)|0\rangle}{\langle0|\phi(x)H\phi(y)|0\rangle} .\] (4.4) + +This measures the ratio between the modular energy \(K\) and the ordinary energy \(H\). As a + +regularization we set \(x\neq y\) and take the limit \(x\rightarrow y.^{7}\) The modular operator \(K\) can be + +obtained from the modular flow \(\Delta^{i\tau}=e^{-iK\tau}\text{with}K=-\log\Delta\). We get the entanglement temperature as a limit of a modular evolved correlator + +\[\beta(x)=\lim_{y\rightarrow x}\lim_{\tau\rightarrow0}- \frac{\partial_{\tau}\langle0|\phi(x)\Delta^{i\tau}\phi(y)|0\rangle}{\partial_{y^{0}}\langle0|\phi(x)\phi(y)|0\rangle} .\] (4.5) + +For a general chiral field \(\phi\) in a CFT, S. Hollands obtained a general structure of the modular evolved correlator [26]. From a clever use of the KMS condition for the modular flow and analyticity in the Euclidean plane Hollands maps the problem into a Riemann-Hilbert problem in the plane with a cut at \(A\). His expression will be enough to get the entanglement temperatures by the limit (4.5). Take a chiral field \(\phi\) of dimension \(h\). By locality \(h\) is a half integer number. The field is normalized such that + +\[\langle0|\phi(x)\phi(y)|0\rangle= \frac{e^{-i\pi h}}{(x-y-i0^{+})^{-2h}} .\] (4.6) + +Defining + +\[\Pi_{l}(x)=\prod_{j=1}^{N}(x-l_{j}),\Pi_{r}(x)\equiv\prod_{i=1}^{N}(x-r_{i}),Z(x)= \frac{1}{2\pi} \ln(-\Pi_{l}(x)/\Pi_{r}(x)),(4.7)\] + +7The structure of the singularity turns out to be the same in numerator and denominator, so we can equivalently take fields smeared in a short interval around a point. diff --git a/outputs/arxiv_math/2503.08800_pg9_pg1_repeat1.md b/outputs/arxiv_math/2503.08800_pg9_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..beca2994f28f410e664ab1d54a0c640566d1803f --- /dev/null +++ b/outputs/arxiv_math/2503.08800_pg9_pg1_repeat1.md @@ -0,0 +1,57 @@ +Remark 2.10. In Proposition 2.8 and all of our computation examples, there is a symmetry between the \(x_{i}\text{and}y_{i}\) coordinates of \(X_{\Delta_{n}}(\mathbb{N}^{2n})\text{for all}i\). Antoine de Saint-Germain has kindly pointed out to the second author that the \(x_{i}\mapsto y_{i}\) direction of this symmetry has the following names in different contexts: + +- the Auslander–Reiten translate in finite-dimensional representation theory; + +- the Fomin–Zelevinsky twist in Lie theory (cf. [dSG23, §5.6]; + +- the Donaldson–Thomas transformation in Calabi–Yau theory; + +- the maximal green sequence in combinatorics. + +2. 4. Effective bounds. As before, let \(A_{\Delta_{n}}=(a_{i,j})\) be a Cartan matrix for Dynkin type \(\Delta_{n}\). Denote its inverse by \(A_{\Delta_{n}}^{-1}=(a_{i,j}^{-1})\). Define the following associated values: + +\[\begin{aligned} b_{i,\Delta_{n}}:&=\prod_{j=1}^{n}2^{a_{i,j}^{-1}} \\ c_{i,\Delta_{n}}:&=\prod_{j=1}^{n}\left(1+2\sum_{k\neq i}a_{j,k}\right)^{a_{i,j}^{-1}}. \end{aligned}\] + +Muller [Mul23, Proposition 2.3 and Example 3.1] gives bounds on frieze entries in terms of these values. + +Lemma 2.11. [Mul23, Proposition 2.3 and Example 3.1] Let \(F\) be a positive integral \(\Delta_{n}-frieze\). Then there is the following upper bound on the product of entries in its + +\(i-\)th row: \(\prod_{j=1}^{P_{\Delta n}}F_{i,j}\leq b_{i,\Delta_{n}}^{P_{\Delta n}}\). Furthermore, if all entries in \(F\) are at least 2, then + +\(P_{\Delta n}\) + +\(i,\Delta_{n}\). + +\(\prod_{j=1}^{P_{\Delta n}}F_{i,j}\leq c\) + +\()_{16384}^{16}\approx\) + +Remark 2.12. There is a misprint in [Mul23, Example 3.1]: the bound \(( \underline{151875} =_{8,E_{8}}( \underline{177347025604248046875}\) + +\(2^{51}\) on the eighth row of an \(E_{8}-\)frieze should be \(c^{16}\) + +\()_{144115188075855872}^{16}\approx2^{164}\). + +An immediate consequence of Lemma 2.11 and Proposition 2.3 is that if the frieze + +\[P_{\Delta n}\] + +\[P_{\Delta n}\] + +Furthermore, if all entries in \(F\) are at least \(2,\text{then}x_{i}\leq c\) + +\(F\) corresponds to the point \((x_{1},\ldots,x_{n};y_{1},\ldots,y_{n})\in X_{\Delta_{n}}(\mathbb{N}^{2n}),\text{then}x_{i}\leq b_{i,\Delta_{n}}\). \(i,\Delta_{n}\). The Diophantine model of friezes allows us to find a minimal element in each \(\mathbb{Z}/P_{\Delta_{n}}\mathbb{Z}-\)orbit on which we can reduce existing bounds by a power of \(\frac{1}{P_{\Delta n}}\) . + +Proposition 2.13. Let \(F\) be a positive integral \(\Delta_{n}-\)frieze and fix an \(i\in\{1,\ldots,n\}\). + +There is a \(\Delta_{n}-friezeF^{\prime}\) corresponding to \((x_{1},\ldots,x_{n};y_{1},\ldots,y_{n})\in X_{\Delta_{n}}(\mathbb{N}^{2n})\) such + +that \(F\) is a horizontal translation of \(F^{\prime}\text{and}x_{i}\leq b_{i,\Delta_{n}}\). Furthermore, if all entries in + +\(F\) are at least \(2,\text{then}x_{i}\leq c_{i,\Delta_{n}}\). Proof. Each \(\Delta_{n}-\text{frieze}F\) is a horizontal translation of a frieze \(F^{\prime}\) in its \(\mathbb{Z}/P_{\Delta_{n}}\mathbb{Z}-\)orbit + +such that \(F_{i,1}^{\prime}\leq F_{i,j}^{\prime}\text{for all}1\leq j\leq P_{\Delta_{n}}\). By Lemma 2.11, there is a bound on + +\(\prod_{j=1}^{P_{\Delta n}}F_{i,j}^{\prime}\leq b_{i,\Delta_{n}}^{P_{\Delta n}}\). But \(x_{i}=F_{i,1}^{\prime}\) is the smallest factor in the product of \(P_{\Delta_{n}}-\)many translation \(F^{\prime}\) are also all at least 2. Hence the same argument gives the bound in terms of \(c_{i,\Delta_{n}}\) instead of \(b_{i,\Delta_{n}}\). + +terms, hence \(x_{i}\leq b_{i,\Delta_{n}}\). If the entries of \(F\) are all at least 2, then the entries of the □ diff --git a/outputs/arxiv_math/2503.08808_pg5_pg1_repeat1.md b/outputs/arxiv_math/2503.08808_pg5_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..09754f557fff618cfb773d8f993841ab45d65947 --- /dev/null +++ b/outputs/arxiv_math/2503.08808_pg5_pg1_repeat1.md @@ -0,0 +1,29 @@ +The above substitution emphasizes the fact that we are now dealing directly with the exponential distribution of the squared magnitudes \(X_{1}\text{and}X_{2}\), rather than with the complex fields \(Z_{1}\text{and}Z_{2}\) because \(\sigma^{2}\text{and}\rho\) are, respectively, the variance and the correlation coefficient of \(X_{1}\text{and}X_{2}\), as shown in the previous section. We now seek to extend this result to the sum of \(k\) independent correlated exponential variables, leading to correlated Gamma-distributed variables. Following an approach analogous to that used in deriving the Nakagami distribution [1], we make use of the following two-dimensional Laplace transform formula: + +\[\begin{aligned} \frac{1}{\Gamma(k)b^{k-1}} \int_{0}\int_{0}^{\infty\infty}(x_{1}x_{2})^{\frac{k-1}{2}}e^{-\alpha(x_{1}+x_{2})}I_{k-1}\left(2b \sqrt{x_{1}x_{2}} \right)e^{-z_{1}x_{1}-z_{2}x_{2}}dx_{1}dx_{2} \\ &= \frac{1}{\left[(z_{1}+a)(z_{2}+a)-b^{2}\right]^{k}} .(20) \end{aligned}\] + +This formula, reported as equation (79) in [12], holds for \(\Re(k)>0.\) By setting the parameters to match our exponential distribution, \(k=1,\alpha=\) characteristic function of the exponential distribution: + +\[\tfrac{1}{\sigma(1-\rho)} ,b=\] + +\(\overline{\sigma(1-\rho)}\) , we obtain the joint + +\[\phi_{\text{exp}}(z_{1},z_{2})= \frac{1}{\sigma^{2}(1-\rho)\left[(z_{1}+a)(z_{2}+a)-b^{2}\right]} .\] (21) + +Since the sum of \(k\) independent exponential variables follows a Gamma distribution, the characteristic function of two correlated Gamma variables can be directly deduced: + +\[\phi_{\text{Gamma}}(z_{1},z_{2})=\left[\phi_{\text{exp}}(z_{1},z_{2})\right]^{k}= \frac{1}{\sigma^{2k}(1-\rho)^{k}\left[(z_{1}+a)(z_{2}+a)-b^{2}\right]^{k}} .(22)\] + +To invert this characteristic function and recover the probability density function, we use the two-dimensional Mellin inversion formula [13]: + +\[\begin{aligned} \left( \frac{1}{2\pi i} \right)2\int_{c-i\infty}c+i\infty\int_{c-i\infty}c+i\infty \frac{1}{\left[(z_{1}+a)(z_{2}+a)-b^{2}\right]^{k}} e^{z_{1}x_{1}+z_{2}x_{2}}dz_{1}dz_{2} \\ &= \frac{1}{\Gamma(k)b^{k-1}} (x_{1}x_{2})^{\frac{k-1}{2}}e^{-\alpha(x_{1}+x_{2})}I_{k-1}\left(2b \sqrt{x_{1}x_{2}} \right). \end{aligned}\] (23) + +Applying this inversion formula to the characteristic function in (22), we finally obtain the joint PDF of two correlated Gamma-distributed variables: + +\[f_{\text{Gamma}}(x_{1},x_{2})= \frac{(x_{1}x_{2})^{\frac{k-1}{2}}}{\Gamma(k)\sigma^{k+1}(1-\rho)\rho^{\frac{k-1}{2}}} \exp\left(- \frac{x_{1}+x_{2}}{\sigma(1-\rho)} \right)I_{k-1}\left( \frac{2 \sqrt{\rho x_{1}x_{2}}}{\sigma(1-\rho)} \right).(24)\] + +This expression generalizes the previous result for the exponential PDF with \(k=1\) and agrees with results reported in the literature (see [14, 15]). + +## 4. Probability Density Function of the Ratio of two correlated Gamma-distributed random variables + +In this section, we derive the PDF of the ratio \(Z=X_{1}/X_{2},\text{where}X_{1}\text{and}X_{2}\) are two correlated Gamma-distributed random variables with the same shape and scale parameters. By definition, the PDF of the ratio \(Z\) for two non-negative random variables \(X_{1}\text{and}X_{2}\) is given by: diff --git a/outputs/arxiv_math/2503.08813_pg19_pg1_repeat1.md b/outputs/arxiv_math/2503.08813_pg19_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..8ec895f10b88353be9ea93c5c15593e58fc9898d --- /dev/null +++ b/outputs/arxiv_math/2503.08813_pg19_pg1_repeat1.md @@ -0,0 +1,23 @@ +\[\begin{aligned} w_{1}^{(6)}(u_{12}\otimes f_{h})&=0\text{for}h=1,2,6,w_{1}^{(6)}(u_{12}\otimes f_{h})=b_{12h}\text{for}h=3,4,5, \\ w_{1}^{(6)}(u_{1234}\otimes f_{h})&=0\text{for}h\neq 5,w_{1}^{(6)}(u_{1234}\otimes f_{5})=b_{12345}. \end{aligned}\] + +To compute the next map we use the fact that for \(1\leq h\leq5,w_{1}^{(6)}(u_{H}\otimes f_{h})\) is nonzero if and only if \(h\not\in H\). This gives + +\[q_{2}^{(6)}(f_{1})=b_{1}b_{12345}f_{6}+b_{12345}(f_{1}+b_{1}f_{6})+P_{1}f_{6}+w_{2}^{(1)}(\hat{e}_{1})=(b_{1}b_{12345}+2P_{1})f_{6},\] + +and similarily for other indices \(\neq 6\). Thus we obtain + +\[w_{2}^{(6)}(f_{h})=b_{1}b_{12345}+2P_{h}\text{for}1\leq h\leq5\text{and}w_{2}^{(6)}(f_{6})=-b_{12345}.\] + +The ideal generated by the entries of \(w_{2}^{(6)}\) is \((P_{1},\ldots,P_{5},b_{12345})\). It is a hyperplane section of the ideal of submaximal pfaffians of the skew-symmetric matrix defined by the variables \(b_{ijk}\). The matrix of \(w_{2}^{(1)}\) is a presentation matrix of this ideal. On the other hand, the top component of the critical representation \(W_{0}\) contains information about the spinor structure of this related ideal. The computation of \(w_{1,1}^{(0)}\) is straightforward. Using the formulas for \(w_{0}^{(0)}\) we simply get \(w_{1,1}^{(0)}(1)=b_{12345}\). Also for \(w_{1,2}^{(0)}\) we easily obtain \(w_{1,2}^{(0)}(\theta_{ij})=w_{1,2}^{(0)}(\theta_{\widehat{i}j})=w_{1,2}^{(0)}(\theta_{i\widehat{j}})=0\) and \(w_{1,2}^{(0)}(\theta_{\widehat{i}\widehat{j}})=b_{H}\text{where}H=\{1,2,3,4,5\}\setminus\{i,j\}.\text{For}w_{2}^{(0)}\) , if \(|H|=3,5\) we get \(q_{2}^{(0)}(u_{h})= b_{12345}\cdot w_{1}^{(1)}(u_{I})\text{and}w_{2}^{(0)}(u_{H})=b_{H}b_{12345}.\text{For}H=\{i\}\subset\{1,\cdots ,5\}\), we get + +\[q_{2}^{(0)}(u_{i})=w_{2}^{(1)}(\hat{e}_{i})+(f_{i}+b_{i}f_{6})b_{12345}=P_{i}f_{6}\] + +and consequently \(w_{2}^{(0)}(u_{i})=P_{i}\). We will shortly see that the entries of \(w_{2}^{(0)}\) give a choice of spinor coordinates, i.e. a choice of the structure map \(w_{0}^{(0)}\) , for the minimal free resolution of the ideal \((P_{1},\ldots,P_{5},b_{12345})\). + +## 2.4 The minimal free resolution of a hyperplane section of pfaffians + +Let us now analyze higher structure maps in the case of the complex \(\mathbb{F}\) which is the minimal free resolution of a hyperplane section of the ideal generated by submaximal pfaffians of a \(5\times5\) skew-symmetric matrix. Define the skew-symmetric matrix on variables \(x_{ij}=-x_{ji}\) and denote its pfaffians using the notation + +\[P_{\widehat{1}}=x_{23}x_{45}-x_{24}x_{35}+x_{25}x_{34},\] + +with the other pfaffians obtained from this one by permutation of indices. Let \(y\) be an element of \(R\) regular modulo the ideal \((P_{\widehat{1}},\ldots,P_{\widehat{5}})\). The complex \(\mathbb{F}\) resolving the Gorenstein ideal \((P_{\widehat{1}},\ldots,P_{\widehat{5}},y)\) has format \((1,6,10,6,1)\). We consider bases on \(F\text{and}G\) such that the diff --git a/outputs/arxiv_math/2503.08820_pg1_pg1_repeat1.md b/outputs/arxiv_math/2503.08820_pg1_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..b2562a186b05e405e868140daa8420bf093bdd5f --- /dev/null +++ b/outputs/arxiv_math/2503.08820_pg1_pg1_repeat1.md @@ -0,0 +1,13 @@ +# Ghost Kohnert posets + +##### Kelsey Hanser* and Nicholas Mayers* + +##### *Department of Mathematics, North Carolina State University, Raleigh, NC, 27605 March 13, 2025 + +###### Abstract + +Recently, Pan and Yu showed that Lascoux polynomials can be defined in terms of certain collections of diagrams consisting of unit cells arranged in the first quadrant. Starting from certain initial diagrams, one forms a finite set of diagrams by applying two types of moves: Kohnert and ghost moves. Both moves cause at most one cell to move to a lower row with ghost moves leaving a new “ghost cell” in its place. Each diagram formed in this way defines a monomial in the associated Lascoux polynomial. Restricting attention to diagrams formed by applying sequences of only Kohnert moves in the definition of Lascoux polynomials, one obtains the family of key polynomials. Recent articles have considered a poset structure on the collections of diagrams formed when one uses only Kohnert moves. In general, these posets are not “well-behaved,” not usually having desirable poset properties. Here, as an intermediate step to studying the analogous posets associated with Lascoux polynomials, we consider the posets formed by restricting attention to those diagrams formed by using only ghost moves. Unlike in the case of Kohnert posets, we show that such “ghost Kohnert posets” are always ranked join semi-lattices. In addition, we establish a necessary condition for when ghost Kohnert posets are bounded and, consequently, lattices. + +## 1 Introduction + +In recent work [2, 10, 13, 18, 19], it has been found that many families of polynomials arising in algebraic combinatorics can be viewed as generating polynomials for certain collections of “diagrams.” Such families include Schubert [12], key [6, 7], and Lascoux polynomials [11]. Within this framework, diagrams are collections of unit cells (some of which may be decorated) arranged in the first quadrant (see Figures 1 and 2) which are encoded by (signed) monomials where the power of \(x_{i}\) is the number of cells in \(\text{row}i\). For the polynomials known to have such definitions currently, the encoded diagrams are formed by starting from an initial “seed” diagram and applying sequences of appropriately defined moves, i.e., Kohnert or K-Kohnert moves (see below as well as Section 2.1). Such moves effect the position of a single cell in the diagram and can additionally add decorated cells to the diagram as well. Here, we study a family of posets on decorated diagrams which arise within this framework. Definitions for families of polynomials in terms of diagrams and moves, as described above, originated in the thesis of Kohnert [10] where he showed that key polynomials admit such a definition and conjectured the same for Schubert polynomials; this conjecture was eventually proven by multiple authors [2, 18, 19]. The families of key and Schubert polynomials both form bases of the polynomial ring \(\mathbb{Z}[x_{1},x_{2},\ldots]\) which consist of homogeneous polynomials that are indexed by weak compositions in the case of keys and permutations in the case of Schuberts. Given a weak composition \(\alpha\in\mathbb{Z}_{\geq0}^{n}\) and a permutation \(w\in S_{n}\), we denote the associated key and Schubert polynomials by \(\mathcal{K}_{\alpha}\text{and}S_{w}\), respectively. Moreover, we associate with \(\mathcal{K}_{\alpha}\) a corresponding “key diagram,” denoted \(\mathbb{D}(\alpha),\text{and with}S_{w}\) a corresponding “Rothe diagram,” denoted \(\mathbb{D}(w)\). See Figure 1 (a) for \(\mathbb{D}(\alpha)\) with the weak composition \(\alpha=(1,1,3,2)\) and Figure 1 (d) for \(\mathbb{D}(w)\) with the permutation \(w=[4,2,5,3,1]\). For both key and Schubert polynomials, one starts from the associated diff --git a/outputs/arxiv_math/2503.08848_pg1_pg1_repeat1.md b/outputs/arxiv_math/2503.08848_pg1_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..7ff30614f76a1ed76331eb7d29d751d41b594c42 --- /dev/null +++ b/outputs/arxiv_math/2503.08848_pg1_pg1_repeat1.md @@ -0,0 +1,429 @@ +# Uniform convergence of Pfaffian point process to the Airy line ensemble + +##### Zhengye Zhou + +###### Abstract + +We consider a family of Pfaffian Schur processes whose first coordinate marginal relates to the half– space geometric last passage percolation. We show that the line ensembles corresponding to the Pfaffian Schur processes with geometric weights converge uniformly over compact sets to the Airy line ensemble. By detailed asymptotic analysis of the kernels, we can verify the conditions for the finite dimensional weak convergence introduced in [7]. By utilizing the tightness criteria of the line ensembles established in [8], we can further improve the finite dimensional convergence to the uniform convergence over compact sets. Moreover, using the same methodology we also show that sequences of spiked Pfaffian Schur processes converge uniformly over compact sets to the Airy wanderer line ensembles constructed in [7]. + +#### Contents + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
1Introduction and main results1
2Definitions3
2.1 Airy line ensemble.........................3
2.2 Pfaffian Schur process.........................4
2.3 Correlation kernel and deformation of contours ..........................6
3Kernel Convergence8
3.1 Asymptotic analysis.........................8
3.2 Limits of the kernel.........................12
4Upper tail estimates18
4.1 Kernel estimate.........................18
4.2 Probability estimate.........................20
5Weak convergence of line ensembles20
5.1 Finite dimensional convergence.........................21
5.2 Tightness of the line ensemble.........................23
+ +## 1 Introduction and main results + +In recent years, models related to KPZ growth in a half-space were extensively studied. For example, [1] studied the fluctuations of last passage percolation with exponential weights in a half-quadrant within the framework of Pfaffian Schur processes, [17] shows the fluctuations of a one-dimensional polynuclear growth model in a half-space, the log-gamma polymer in a half space was also considered [13]. For a large amount of half-space models, the fluctuations follow the GUE Tracy Widom distribution in the bulk. It is interesting to further investigate if these models converges to the Airy line ensemble. + +The Airy line ensemble \(\mathcal{L}^{\text{Airy}}\) is a random sequence of continuous functions that arises as a scaling limit in random matrix theory and various models within the KPZ universality class [5, 6, 15, 18]. In particular, diff --git a/outputs/arxiv_math/2503.08868_pg80_pg1_repeat1.md b/outputs/arxiv_math/2503.08868_pg80_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..a5974e4bf48c7fe7b770ba8d842902edd8631e42 --- /dev/null +++ b/outputs/arxiv_math/2503.08868_pg80_pg1_repeat1.md @@ -0,0 +1,29 @@ +cyclic order around the right hand root point. (Compare Remark 2.30.) In fact they always seem to land as indicated, with + +\[\alpha^{\prime}=\widehat{\beta},\beta^{\prime}=\widehat{\gamma},\text{and}\delta^{\prime}=\widehat{\alpha}.\] + +Recall that \(\alpha\text{and}\gamma\) always belong to the same grand orbit, which also contains \(\widehat{\gamma}=\beta^{\prime}\) . It follows that \(\alpha\text{and}\beta^{\prime}\) belong to the same grand orbit: \(((\alpha))=((\beta^{\prime})).\) Similarly, since \(\beta\) and \(\delta\) belong to the same orbit, and \(\alpha^{\prime}=\widehat{\beta}\), consequently \(((\alpha^{\prime}))=((\beta))\) . Assuming as in Equation (12) that \(((\alpha))\neq ((\beta))\) it follows that + +\[((\alpha))=((\gamma))=((\beta^{\prime}))=((\delta^{\prime}))\neq ((\alpha^{\prime}))=((\gamma^{\prime}))=((\beta))=((\delta)).\] + +The key to understanding this situation is to look at the Julia set for the parabolic root points. Since the two root points of an A component are dual to each other, they have Julia sets which are identical, except for the choice of which component to label as \(a\) and which + +to label as \(-a\) . + +Nearly all of the examples we know are for rabbit regions. As one specific example, let us concentrate on the \(1/3\) -rabbit region as shown in Figure 54, and on the A component in the upper left of this Figure. Figure 60 illustrates the Julia set for either of its two root points. (See Figure 21 for an analogous Julia set for the \(1/2\) rabbit, and Figure 23 for the \(1/5\) rabbit.) + +corresponding periodic angle \(3\theta\) with denominator \(3^{p}-1\) . Note that two co-periodic angles + +Given any co-periodic angle \(\theta\) with denominator \(3(3^{p}-1)\) , it is useful to consider the \(\theta\text{and}\theta^{\prime}\) satisfy \(3\theta=3\theta^{\prime}\) if and only if they are twins. In our example, the root point of each \(\pm2a\) component in the Julia set is the landing point of six dynamic rays. However, only four of the six dynamic angles correspond to parameter angles for rays landing at the specified root point of A. Below is a table describing the angles for the six dynamic rays landing at the left hand root point in Figure 54 (or 60), and specifying which four are the angles of parameter rays landing on a root of A. + +\[\begin{aligned} \gamma\delta\alpha\beta \\ \text{angles}:161922314041/78 \\ \times3:16192251415/26 \end{aligned}\] + +Here is the corresponding table for the right hand root points. + +\[\begin{aligned} \alpha^{\prime}\beta^{\prime}\gamma^{\prime}\delta^{\prime} \\ \text{angles}:67687174514/78 \\ \times3:15161922514/26 \end{aligned}\] + +Evidently the last row of the second table is just the last row of the first table permuted cyclically. It follows that the angles in the second table are just the twins of the angles in the first table, again permuted cyclically. In particular + +\[\alpha^{\prime}=\widehat{\beta},\beta^{\prime}=\widehat{\gamma},\text{and}\delta^{\prime}=\widehat{\alpha}.\] + +However the twin \(\widehat{\delta}\) is an angle which does not correspond to any parameter ray landing at a root point of this A component, and the same is true for \(\widehat{\gamma}^{\prime}\) . diff --git a/outputs/arxiv_math/2503.08898_pg56_pg1_repeat1.md b/outputs/arxiv_math/2503.08898_pg56_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..9ca58071d652bb072386e044a41e25f372bd6639 --- /dev/null +++ b/outputs/arxiv_math/2503.08898_pg56_pg1_repeat1.md @@ -0,0 +1,33 @@ +Remark 9.2. We point out that one can also define analogues of these operators in the quasi-parabolic descendent algebras \(\mathbb{D}_{\alpha_{\bullet}}^{\mathrm{q}\mathrm{p}\text{ar}}\) ; recall that this is a supercommutative algebras generated by symbols ch\((_{k}\gamma)\text{for}\gamma\in H^{*}(C)\text{and ch}_{k}(e_{j})\text{for}0\leq j\leq l+1\), modulo certain simple relations; alternatively, it is generated by ch\((_{k}^{H}\gamma),\text{ch}_{k}^{H}(e_{j})\). One defines \(\mathrm{L}_{n}^{\mathrm{q}\text{par}}:\mathbb{D}_{\alpha_{\bullet}}^{\mathrm{q}\mathrm{p}\text{ar}}\rightarrow\mathbb{D}_{\alpha_{\bullet}}^{\mathrm{q}\mathrm{p}\text{ar}}\) by \(\mathrm{L}_{n}^{\mathrm{q}\text{par}}=\mathrm{R}_{n}^{\mathrm{q}\text{par}}+\mathrm{T}_{n}^{\mathrm{q}\text{par}}\text{where R}_{n}^{\mathrm{q}\text{par}}\) is defined exactly in the same way as in the parabolic case, and \(\mathrm{T}_{n}\) is + +\[\mathrm{T}_{n}^{\mathrm{q}\text{par}}=\sum_{a+b=n}a!b!^{(}(1-g)\text{ch}_{a}(\text{pt})\text{ch}_{b}(\text{pt})+\sum_{j=0}^{l}\text{ch}_{a}(e_{j})\text{ch}_{b}(e_{j+1}-e_{j})^{)}.\] + +In particular, the Virasoro constraints do not depend on a choice of weights within a fixed chamber. + +The main result of this section are the Virasoro constraints for moduli spaces of parabolic bundles: + +\[\begin{aligned} \mathbf{Theorem}\mathbf{9}.\mathbf{3}.\text{Let}\alpha&=(r,d,f_{\bullet},c)\in C(I).\text{For every}D\in\mathbb{D}^{\text{par}}\text{we have} \\ \int_{[M_{\alpha}]}\mathrm{L}_{\text{wt}_{0}}(D)&=0. \end{aligned}\] + +These generalize the Virasoro constraints for moduli spaces of stable bundles shown in [BLM]. For the proof, we will need compatibility statements between the Virasoro constraints and the structures that we have discussed earlier in the paper: wall-crossing, flag bundles, and Hecke operators. We develop these compatibilities first, and then give the proof in Section 9.5. + +9. 1. Primary states and wall-crossing compatibility. Compatibility with + +wall-crossing is already the key ingredient in [BLM]. To state it, let us introduce the Lie subalgebra of primary states. + +Definition 9.4. The space of primary states of weight \(i\) on Vpartr is + +\[P_{i}=^{\{}v\in\mathbf{V}_{\text{tr}}^{\text{par}}:L_{0}(v)=iv\text{and}L_{n}(v)=0\text{for}n>0^{\}}.\] + +The space of primary states on the Lie algebra \(\check{\mathbf{V}}_{\text{tr}}^{\text{par}}\) is + +\[\check{P}_{0}=P_{1}/T(P_{0})\subseteq\check{\mathbf{V}}_{\text{tr}}^{\text{par}}.\] + +By [LM, Corollary 5.7], a class \(u\in\check{\mathbf{V}}_{\text{tr}}^{\text{par}}\) (of non-trivial topological type) is a primary state if and only if \(L_{\text{wt}_{0}}(u)=0\), where + +\[L_{\text{wt}_{0}}=\sum_{n\geq-1} \frac{(-1)^{n}}{(n+1)!} L_{-1}^{n+1}\circ L_{n}:\check{\mathbf{V}}_{\text{tr}}^{\text{par}}\rightarrow\mathbf{V}_{\text{tr}}^{\text{par}}\] + +is the dual of Lwt0.17 In particular, the moduli space \(M_{\alpha}\) satisfies the Virasoro constraints (i.e. Theorem 9.3 holds) if and only if \([M_{\alpha}]\in\check{P}_{0}\) is a primary state. We recall the reader that Vpar When the symmetrized Euler pairing \(\chi_{\mathcal{P}}^{\mathrm{s}\text{ym}}\) is non-degenerate, this vertex algebra + +tr is a lattice vertex algebra (cf. Proposition 3.1). + +17When Vpartr admits a conformal element \(\omega,L_{\text{wt}_{0}}\) is a canonical lift of the operator \([-,\omega]\), see [BLM, Proposition 3.13, Lemma 3.15]. diff --git a/outputs/arxiv_math/2503.08910_pg26_pg1_repeat1.md b/outputs/arxiv_math/2503.08910_pg26_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..907b3f65215fb55f5df48852a61bedecdc183e20 --- /dev/null +++ b/outputs/arxiv_math/2503.08910_pg26_pg1_repeat1.md @@ -0,0 +1,13 @@ +\(\sigma(i^{\prime})=1\text{for some}i^{\prime}\in J,\text{then}\Xi_{1}(b_{i^{\prime}}^{\sigma(i^{\prime})})=\Xi_{1}(\sim b_{i^{\prime}})=0\), so \(\Xi\left(g(\sigma)\wedge\bigwedge_{i\in J}b_{i}^{\sigma(i)}\right)=0\). In conclusion, \(\Xi(a)=\Xi_{0}(g(\sigma^{0})),\text{where}\sigma^{0}\) is the constant map onto 0. + +\(□3.15\) We wonder whether the results of this section are valid when restricting to fams with the uap. These are not in general. For example, let \(a\subseteq\omega\) be the set of even numbers, \(\mathscr{B}:=\mathcal{P}(\omega)\), let \(\mathscr{B}_{0}\) be the field of sets over \(\omega\) generated by \(a\) and let \(F\subseteq\mathcal{P}(\omega)\) be the filter of subsets of \(\omega\) containing \(\{0,1\}\). Let \(\Xi_{0}\) be the (unique) probability measure on \(\mathscr{B}\text{such that}\Xi(a)= \frac{2}{3}\) (which is clearly uniformly supported). By Theorem 3.15, there is a unique probability fam \(\Xi\text{on}\langle\mathscr{B}_{0}\cup F\rangle\) extending \(\Xi_{0}\text{such that}\Xi(b)=1\) for all \(b\in F\) , in particular, \(\Xi(\{0,1\})=1.\text{Then},\Xi(\{0\})=\Xi(a\cap\{0,1\})=\Xi_{0}(a)= \frac{2}{3}\) and, naturally, \(\Xi(\{1\})= \frac{1}{3}\). However, \(\Xi\) cannot have the uap because no singleton can have measure 23 under a probability fam with the uap (by Theorem 2.18). This also shows that Theorem 3.8 is not valid in general when restricting to the uap: both \(\Xi_{0}\text{and}\Xi_{F}\) are uniformly supported, but any common extension extends \(\Xi\) and thus cannot have the uap. We can validate Theorem 3.15 to fams with the uap under further conditions. Theorem 3.16. In Theorem 3.15, assume that \(\mathscr{B}=\mathcal{P}(X)\text{and that}\mathscr{B}_{0}\) contains all the finite subsets of \(X.If\Xi_{0}\) has the uap, then so does \(\Xi\). Likewise for “uniformly supported”. + +Proof. If all finite sets have \(\Xi_{0}-\)measure zero, then the same is true for \(\Xi\). So assume that \(\Xi_{0}\) is uniformly supported with support \(P\) . Since \(\mathscr{B}_{0}\) contains all finite sets, all finite sets in \(P\) with positive measure must be singletons (by minimality). It is not hard to show that \(P\) is a support of \(\Xi\), so \(\Xi\) is uniformly supported. + +\(□3.16\) Similarly, we can validate the extension results when adding one set to the subalgebra. Lemma 3.17. In the context of Lemma 3.11, assume that \(\mathscr{B}=\mathcal{P}(X)\text{and that}\mathscr{C}\) contains all the singletons. Then, if \(\Xi\) has the uap, then \(\Xi^{\prime}\) can be found with the uap. As a consequence, \(\Xi\) can be extended to a fam into \(K\) with the uap on the whole \(\mathcal{P}(X)\). Here, “uap” can be replaced by “uniformly supported”, even more, extensions of \(\Xi\) can be found with the same support. + +Proof. The result is trivial when all singletons have Ξ-measure zero. So assume that \(\Xi\) is uniformly supported. The argument in the proof of Lemma 3.11, when replacing \(A\) by a support of \(\Xi\), allows to find a uniformly supported \(\Xi^{\prime}\) with the same support as \(\Xi\). The rest follows by Zorn’s Lemma. + +\(□3.17\) We ask whether (iii) \(\Rightarrow\) (i) of Theorem 3.8 is valid when \(\Xi_{0}\text{and}\Xi_{1}\) have the uap for finding \(\Xi\) with the uap, under the condition that both \(\mathscr{B}_{0}\text{and}\mathscr{B}_{1}\) contain all singletons. If, say, \(\Xi_{0}\) is free then \(\Xi_{1}\text{and}\Xi\) must be free, but we do not know how to obtain an uniformly supported \(\Xi\text{when}\Xi_{0}\text{and}\Xi_{1}\) are uniformly supported. Question 3.18. Assume that \(\mathscr{B}_{0}\text{and}\mathscr{B}_{1}\) are fields of sets over \(X\), containing all the singletons, and \(\Xi_{0}\text{and}\Xi_{1}\) are uniformly supported fams with the uap on \(\mathscr{B}_{0}\text{and}\mathscr{B}_{1}\), respectively. Assume that \(\Xi_{0}\text{and}\Xi_{1}\) are compatible (in the sense of Theorem 3.8 (iii)). Is there a uniformly supported fam that extends both \(\Xi_{0}\text{and}\Xi_{1}\)? + +A strategy to solve this problem may suggest a short proof of Theorem 3.8. diff --git a/outputs/arxiv_math/2503.08911_pg10_pg1_repeat1.md b/outputs/arxiv_math/2503.08911_pg10_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..b7fb00c1cc579f59e10d3258fe8f41aa96bb2e64 --- /dev/null +++ b/outputs/arxiv_math/2503.08911_pg10_pg1_repeat1.md @@ -0,0 +1,15 @@ +The decaf order has many nice properties. It is the product of Bruhat order for Sk and the poset determined by pushbacks on the subset \(\{w\in\mathcal{W}_{n,k}|\pi(w)=id\}\). The decaf order is a ranked poset on \(\mathcal{W}_{n,k}\), and its rank generating function is the same as the Poincaré polynomial in (1.4). The medium roast and espresso orders are not ranked posets in general. For \(n\geq5\) and most values of k, there are covering relations in the medium roast Fubini-Bruhat order \((\mathcal{W}_{n,k},\leq)\) with a dimension difference of 2 or more, causing the medium roast Fubini-Bruhat order to be unranked in general. For example, in \(\mathcal{W}_{5,4},44312\) covers 41321, but 44312 has dimension 1, and 41321 has dimension 3. + +Theorem 4.2. The Superpushback Rule. Suppose \(w\in\mathcal{W}_{n,k},i\in\) [k \(-1],andj\in\) [n] such that \(w_{j}=\pi_{i}\) is a redundant letter in w. If \(i+p\leq k\) and v is obtained from w by replacing wj by \(\pi_{i+p}(w),thenv⇀w\) and this is a covering relation in both espresso and medium roast orders. + +Theorem 4.3. The Lifting Property. Suppose \(v,w\in\mathcal{W}_{n,k},i\in\) [k \(-1],\alpha_{i+1}(v)<\alpha_{i}(v)\), and \(\alpha_{i+1}(w)<\alpha_{i}(w).Ifv\leq w\) in medium roast Fubini-Bruhat order, then \(s_{i}v\leq s_{i}w\). Furthermore, if \(v⇀w,thens_{i}v⇀s_{i}w\). + +## 5 Essential Sets + +We extend the notion of a Rothe diagram from Definition 2.1 to Fubini words. This allows us to define the essential set for a Fubini word. We then show the essential set determines a minimal set of rank equations on the corresponding PR variety, generalizing Fulton’s essential set for permutations and Schubert varieties [7]. This leads to an essential set characterization of \(v\leq w\) in medium roast order. + +Definition 5.1. [16] A Fubini word \(w\in\mathcal{W}_{n,k}\) is called convex if \(h0\) consists of clones of an initial resistant subpopulation that is assumed to be present from the very beginning. This hypothesis is consistent with the heterogeneity and polyclonal nature of solid cancers, which in the clinical practice is particularly realistic in the case of advanced or metastatic disease (see e.g. [37]). We assume that the initial + +fraction \(\hat{f}_{0}\) of resistant cells ranges in \([0.002,0.1]\ni\hat{f}_{0}\), meaning that the ratio \(\hat{f}_{0}:= r(0)/(r(0)+s(0))\) varies between \(0.2\%\) (most favourable scenario) and 10% (worst-case scenario). In future work, we plan to incorporate a feature that accounts for the secondary resistance mechanism into the model. + +Space of parameters and ensemble measure. We are in a position to provide a formal definition for the space of parameters \(\Theta\text{where}\theta:=(\hat{d}_{D},\hat{d}_{T},\hat{r}_{R},\hat{f}_{0})\) takes values. According diff --git a/outputs/arxiv_math/2503.08947_pg7_pg1_repeat1.md b/outputs/arxiv_math/2503.08947_pg7_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..28d8111938348728f7ef1fe18bf453349820de6b --- /dev/null +++ b/outputs/arxiv_math/2503.08947_pg7_pg1_repeat1.md @@ -0,0 +1,33 @@ +with the surface affine connection given by + +\[\gamma_{ab}^{c}=-\hat{\upsilon}^{c}\partial_{a}\tau_{b}+ \frac{1}{2} h^{cd}\left(\partial_{a}\bar{h}_{bd}+\partial_{b}\bar{h}_{ad}-\partial_{d}\bar{h}_{ab}\right)\] (45) + +\[=\Gamma_{ab}^{c}+u_{\mu}^{c}\partial_{a}u_{b}^{\mu}=u_{\mu}^{c}u_{a}^{\nu}\nabla_{\nu}u_{b}^{\mu},\] (46) + +and associated torsion tensor \(F_{ab}\) defined through + +\[2\gamma_{[ab]}^{c}=-\hat{\upsilon}^{c}F_{ab}.\] (47) + +A submanifold Riemann tensor \(\mathcal{R}_{bcd}^{a}\) can be constructed from \(\gamma_{ab}^{a}\) through the standard form + +\[\mathcal{R}_{bcd}^{a}=\partial_{c}\gamma_{db}^{a}-\partial_{d}\gamma_{cb}^{a}+\gamma_{cf}^{a}\gamma_{db}^{f}-\gamma_{df}^{a}\gamma_{cb}^{f}.\] (48) + +The boost invariant extrinsic curvature of the submanifold \(K_{ab}^{I}\) can be defined from the covariant derivative of the tangential vector through + +\[K_{ab}^{I}=-u_{a}^{\mu}u_{b}^{\nu}\nabla_{(\mu}n_{\nu)}^{I}=n_{\mu}^{I}D_{a}u_{b}^{\mu}+ \frac{1}{2} \hat{\upsilon}^{I}\tau_{ab},\] (49) + +and transforming under mass U(1) transformations as + +\[\delta K_{ab}^{I}= \frac{1}{2} F_{Ia}\partial_{b}\sigma+ \frac{1}{2} F_{Ib}\partial_{a}\sigma.\] (50) + +Notice that the covariant derivative \(D\) does not act on normal indices \(I\). It is also possible to define a covariant derivative \(\mathcal{D}\) that acts on all types of indices. In particular we will find that + +\[\mathcal{D}_{a}T^{I}=D_{a}T^{I}-\omega_{aJ}^{I}T^{J}.\] (51) + +The tensor \(\omega_{aJ}^{I}\) is called the external rotation tensor and can be interpreted as a SO(d-p) connection, it is defined from the covariant derivative of the normal 1-form + +\[\omega_{aJ}^{I}=n_{J}^{\mu}D_{a}n_{\mu}^{I},\] (52) + +it is antisymmetric in \(I,J\) indices, and transforms under U(1) mass transformations as + +\[\delta\omega_{aJ}^{I}=- \frac{1}{2} \left(F_{aJ}\partial^{J}\sigma+F_{J}^{I}\partial_{a}\sigma+\tau_{a}^{I}\partial_{J}\sigma\right).\] (53) diff --git a/outputs/arxiv_math/2503.08949_pg8_pg1_repeat1.md b/outputs/arxiv_math/2503.08949_pg8_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..06fe8c2f3924bb9f2ae1645421975a3dfd6651c2 --- /dev/null +++ b/outputs/arxiv_math/2503.08949_pg8_pg1_repeat1.md @@ -0,0 +1,27 @@ +2. Shape of the Eigenfunction \(v_{s,E}\): To precisely understand how \(\lambda_{s,E}\) varies in \(E\), we must not only understand the operator \(F_{s,E}\) but also the associated eigenfunction \(v_{s,E}\). This differs from prior work, such as [Bap14], which circumvented the analysis of \(v_{s,E}\) by instead estimating the action of \(F_{s,E}\) on certain test vectors (which loses careful control on \(\lambda_{s,E}\text{for}E\approx E_{0})\). We approximate the eigenfunction \(v_{s,E}\) in Section 8, where we begin by defining an explicit function \(\tilde{u}\) by + +\[\tilde{u}(x)=(t^{2}\ln K)^{-1}\cdot1\{|x|\leq Ct^{2}\}+|x|^{-(2-s)}\cdot1\{Ct^{2}\leq|x|\}.\] (1.10) + +Observe that, if \(s\) is very close to 1, the function \(\tilde{u}(x)\) behaves as \((t^{2}\ln K)^{-1}\text{for}x\leq Ct^{2}\text{and as}x^{-1}\) for \(x\geq Ct^{2}\). This is the general behavior we expect for the eigenfunction \(v_{s,E}\). However, the precise location and nature of the crossover around \(x=O(t^{2})\), at which \(v_{s,E}(x)\) changes from behaving as \(x^{-1}\sim t^{-2}\) to as \((t^{2}\ln K)^{-1}\), is not transparent to us. So, instead of comparing \(v_{s,E}\) directly to \(\tilde{u}\), we will compare \(F_{s,E}v_{s,E}\) to \(F_{s,E}\tilde{u}\). Specifically, as Lemma 8.5, we show that + +\[c(F_{E}\tilde{u})(x)\leq(F_{s,E}v_{s,E})(x)\leq C(F_{E}\tilde{u})(x),\] (1.11) + +where the constants \(c\text{and}C\) are uniform in \(K\). While useful, this does not explain how \(v_{s,E}\) varies with \(E\) and, indeed, we do not know how to do this. We bypass this by making use of the identity + +\[\ln\lambda_{s,E}=\lim_{n\rightarrow\infty} \frac{\ln\Vert F_{E}^{n}\tilde{u}\Vert_{1}}{n} .\] (1.12) + +The benefit of this representation is that it replaces \(v_{s,E}\) with the function \(\tilde{u}\) that is independent of \(E\). However, it comes at the expense of having to compute high powers of \(F_{s,E}\) when applied to \(\tilde{u}\). + +3. Estimating Iterates of \(F_{s,E}\): To deduce the monotonicity of \(\lambda_{s,E}\), we analyze the difference \(\ln\lambda_{s,E_{2}}-\ln\lambda_{s,E_{1}}\) using (1.12). We have + +\[\Vert F_{s,E_{2}}^{n}\tilde{u}\Vert_{1}-\Vert F_{s,E_{1}}^{n}\tilde{u}\Vert_{1}=\int_{-\infty}^{\infty}\left((F_{s,E_{2}}^{n}-F_{s,E_{1}}^{n})\tilde{u}\right)(x)dx.\] (1.13) + +We note the identity + +\[F_{s,E_{2}}^{n}-F_{s,E_{1}}^{n}=\sum_{j=0}^{n-1}F_{s,E_{2}}^{n-j-1}(F^{\circ})F_{s,E_{1}}^{j},\] (1.14) + +where we define + +\[(F^{\circ}u)(x)=(F_{s,E_{2}}-F_{s,E_{1}})u(x)= \frac{t^{2-s}}{|x|^{2-s}} \int_{-\infty}^{\infty}\left(\rho_{s,E_{2}}(-y-t^{2}x^{-1})-\rho_{s,E_{1}}(-y-t^{2}x^{-1})\right)u(y)dy.\] + +When \(|x|\geq Ct^{2}\text{and}y\) is close to 0 (which is where the above integral will mainly be supported), (1.9) implies that \(\rho_{s,E_{2}}(-y-t^{2}/x)-\rho_{s,E_{1}}(-y-t^{2}/x)0\). For any \(M_{0}\in\mathbf{N},\delta>0\), there exists \(p>0\) such that the following is true. Suppose that \(\Gamma\) is a multichordal \(CLE_{\kappa}in(D; \underline{x} ;\beta)\in D^{\text{ext}}\) procedure described just above to \(\Gamma\). Let \(E_{\delta}\) be the event that for every \(w\in D_{W,r}\) we have that \(\Gamma_{\text{ou}\mathrm{t}}^{*,B(w,2r),B(w,r)}\) is \(\delta-\text{separated}\). Then for any choice of target pivotals \(F_{W,r}=\{(\ell_{i}^{1},\ell_{i}^{2},y_{i})\}_{i=1,\ldots,M^{*}}\) with \(M^{*}\leq M_{0}\) we have + +\[\mathbf{P}[G_{W,r}^{\mathrm{r}\text{es}}|\Gamma]\mathbf{1}_{E_{\delta}}\geq p\mathbf{1}_{E_{\delta}}\] + +\(2N\). Apply the resampling + +where \(G_{W,r}^{\mathrm{r}\text{es}}\) is the event that the resampling is \(F_{W,r}-\text{successful}\). + +Proof. Suppose we are given any target pivotals \(F_{W,r}=\{(\ell_{i}^{1},\ell_{i}^{2},y_{i})\}_{i=1,\ldots,M^{*}}\text{with}M^{*}\leq M_{0}\). If we perform the resampling operation, then with uniformly positive probability (depending on \(M_{0},r)\) we sample \(\tilde{M}=M^{*}\) and the points \(w_{i}\in D_{W,r/2}\text{such that}w_{i}\in B(y_{i},r/2)\text{for every}i=1,\ldots,M^{*}\). In particular, the strands \(\ell_{i}^{1},\ell_{i}^{2}\) cross the annulus \(A(w_{i},r,2r)\). On the event \(E_{\delta}\) the strands of \(\Gamma_{\text{ou}\mathrm{t}}^{*,B(w_{i},2r),B(w_{i},r)}\) are \(\delta-\text{separated}\). Therefore, by Corollary 3.16, the probability that \(\ell_{i}^{1},\ell_{i}^{2}\) switch their linking pattern after performing the resampling is bounded from below uniformly in \(\delta\). Moreover, by continuity (Theorem 1.7) we see that with positive probability bounded from below the new strands \(\ell^{\text{res},1},\ell^{\text{res},2}\) to keep track of the intersection points between the chords, the proof of Theorem 1.7 shows that continuity also holds when we require the intersection sets to converge.) Repeating this argument \(M^{*}\) times for each target pivotal, we conclude. + +intersect again inside \(B(w_{i},r/2)\). (Although we have not phrased the topology (1.2) + +\(i\) + +\(i\) + +- + +Recall the setup of Section 4.3. Suppose that \(D\subseteq\mathbf{C}\) is a simply connected domain and let \(\Gamma\) be a nested \(\text{CLE}_{\kappa}\text{in}D\). Let \(u\in(0,1/8)\) be fixed. Let \(j_{0}\in\mathbf{Z}\text{and}z\in D\) be such that \(B(z,2^{-j_{0}})\subseteq D\). For \(j>j_{0}\), let \(U_{j}=B(z,2^{-j}),V_{j}=B(z,(1+u)2^{-j})\), and let \(G_{j}\) be the filtration generated by the partially explored \(\text{CLE}_{\kappa}\Gamma_{\text{ou}\mathrm{t}}^{*,V_{j},U_{j}}\) . Let \(\varphi_{j}\) be the unique conformal map from \(V_{j}^{*,U_{j}}\) to D with \(\varphi_{j}(z)=0,\varphi_{j}^{\prime}(z)>0\). Let \((\mathbf{D}; \underline{s}^{j} ;\beta_{j})\) be the exterior link pattern decorated marked domain which arises under \(\varphi_{j}\). Let \(\Gamma_{\underline{s}^{j} ,\beta_{j}}\) be the image of \(\Gamma\) under \(\varphi_{j}\). Conditionally on \(G_{j}\), we have that \(\Gamma_{\underline{s}^{j} ,\beta_{j}}\) is a multichordal \(\text{CLE}_{\kappa}\mathrm{i} \overline{\mathrm{n}} (\mathbf{D}; \underline{s}^{j} ;\beta_{j})\). Fix \(5/80,r>0,\text{and}W=A(0,r_{1},r_{2})\). Suppose that \((\mathbf{D}; \underline{x} ;\beta)\in D_{2N}^{\mathrm{e}\mathrm{x}\mathrm{t}}\) is \(\delta_{0}-\)separated as in Definition 4.3 and let \(\Gamma\) be a multichordal \(CLE_{\kappa}in(\mathbf{D}; \underline{x} ;\beta)\). Let \(D_{W,r}\) be as in (5.1). Let \(E_{\delta}\) be the event that for every \(w\in D_{W,r}\) we have that \(\Gamma_{\text{ou}\mathrm{t}}^{*,B(w,2r),B(w,r)}\) is \(\delta-\text{separated}\). Then + +\[W,r/100\] + +\[\mathbf{P}_{(D; \underline{x} ;\beta)}^{\mathrm{C}\mathrm{L}\mathrm{E}_{\kappa}}[E_{\delta}]\rightarrow1as\delta\rightarrow0\] + +at a rate that depends only on \(\delta_{0}\text{and}r\). diff --git a/outputs/arxiv_math/2503.08964_pg1_pg1_repeat1.md b/outputs/arxiv_math/2503.08964_pg1_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..ddf132847a5ffb787470657eb4c1f3a9ceaff9a9 --- /dev/null +++ b/outputs/arxiv_math/2503.08964_pg1_pg1_repeat1.md @@ -0,0 +1,27 @@ +# List rainbow connection number of graphs + +##### Rongxia Tang† + +##### Henry Liu∗† + +##### Yueping Shi† + +## 12 March 2025 + +###### Abstract + +##### Chenming Wang† + +An edge-coloured path is rainbow if all of its edges have distinct colours. Let \(G\) be a connected graph. The rainbow connection number of \(G\), denoted by \(rc(G)\), is the minimum number of colours in an edge-colouring of \(G\) such that, any two vertices are connected by a rainbow path. The strong rainbow connection number of \(G\), denoted by \(src(G)\), is the minimum number of colours in an edge-colouring of \(G\) such that, any two vertices are connected by a rainbow geodesic (i.e., a path of shortest length). These two notions of connectivity of graphs were introduced by Chartrand, Johns, McKeon and Zhang in 2008. In this paper, we introduce the list rainbow connection number \(rc^{\ell}(G)\), and the list strong rainbow connection number \(src^{\ell}(G)\). These two parameters are the versions of \(rc(G)\text{and}src(G)\) that involve list edge-colourings. Among our results, we will determine the list rainbow connection number and list strong rainbow connection number of some specific graphs. We will also characterise all pairs of positive integers \(a\text{and}b\) such that, there exists a connected graph \(G\text{with}src(G)=a\text{and}src^{\ell}(G)=b\), and similarly for the pair \(rc^{\ell}\text{and}src^{\ell}\). Finally, we propose the question of whether or not we have \(rc(G)=rc^{\ell}(G)\), for all connected graphs \(G\). + +###### AMS Subject Classification (2020): 05C15 + +Keywords: List colouring; (strong) rainbow connection number + +## 1 Introduction + +In this paper, all graphs are finite, simple, and undirected. Let \(K_{n}\text{and}C_{n}\) denote the complete graph and cycle on \(n\) vertices. Let \(K_{n_{1},\ldots,n_{t}}\) denote the complete \(t-\)partite graph with class sizes \(n_{1},\ldots,n_{t}\). In particular, \(K_{m,n}\) is the complete bipartite graph with class sizes \(m\text{and}n.\text{For}t_{1},t_{2},\ldots\geq0\), let \(K_{t_{1}\times1,t_{2}\times2,\ldots}\) denote the complete multipartite graph with \(t_{i}\) classes of size \(i,\text{for all}i\geq1\). Any subscript with \(t_{i}=0\) is usually omitted. In particular, \(K_{t\times n}\) denotes the complete \(t-\)partite graph where every class has size \(n\). For a graph \(G\), + +E-mail addresses: + +∗Corresponding author †School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China. tangrx3@mail2.sysu.edu.cn (R. Tang), liaozhx5@mail.sysu.edu.cn (H. Liu), shiyp9@mail2.sysu.edu.cn (Y. Shi), wangchm33@mail2.sysu.edu.cn (C. Wang) diff --git a/outputs/arxiv_math/2503.08975_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.08975_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..3848d8e2d45b137752946b78af9c80d07b39d409 --- /dev/null +++ b/outputs/arxiv_math/2503.08975_pg3_pg1_repeat1.md @@ -0,0 +1,29 @@ +Theorem 1.6 (Bars [5]). The modular curve \(X_{0}(N)\) has infinitely many points of degree 2 over \(\mathbb{Q}\) if and only if + +\[N\in\{1-33,35-37,39-41,43,46-50,53,59,61,65,71,79,83,89,101,131\}.\] + +Theorem 1.7 (Jeon [18]). The modular curve \(X_{0}(N)\) has infinitely many points of degree 3 over \(\mathbb{Q}\) if and only if + +\[N\in\{1-29,31,32,34,36,37,43,45,49,50,54,64,81\}.\] + +Theorem 1.8 (Hwang, Jeon; Derickx, Orlić [17, 11]). The modular curve \(X_{0}(N)\) has infinitely many points of degree 4 over \(\mathbb{Q}\) if and only if + +\[\begin{aligned} N&\in\{1-75,77-83,85-89,91,92,94-96,98-101,103,104,107,111, \\ 118,119,121,123,125,128,131,141-143,145,155,159,167,191\}. \end{aligned}\] + +From these results, it is easy to determine the modular curves of the form \(X_{0}(N)\) with density degree 1, 2, 3 and 4. For example, \(X_{0}(N)\) has density degree 4 if and only if \(N\) occurs in the set of Theorem 1.8 but not in the sets of Theorems 1.6 and 1.7. Motivated by the recent work of Kadets and Vogt [23] on density degrees, we determine all curves \(X_{0}(N)\) with density degree equal to 5. Our main result is the following theorem: Theorem 1.9. The modular curve \(X_{0}(N)\) has density degree \(\min(\delta(X_{0}(N)/\mathbb{Q}))\) equal to 5 if and only if \(N=109\). This main theorem is an important step in determining all curves \(X_{0}(N)\) with infinitely many quintic points. In fact, if a curve has infinitely many quintic points, then it has to have density degree \(\leq5\). And our result completes the classification of all modular curves of density degree \(\leq5\). So, in order to determine all curves \(X_{0}(N)\) with infinitely many quintic points, it suffices to determine for the explicit list \(X_{0}(N)\) of density degree \(\leq4\) whether they have infinitely many quintic points. With some extra work, we are able to prove the following: Theorem 1.10. The modular curve \(X_{0}(N)\) has infinitely many points of degree 5 for + +\[N\in\{1-45,47-58,61,63,64,65,67,68,72,73,75,80,81,91,109,121,125\}.\] + +The modular curve \(X_{0}(N)\) has only finitely many points of degree 5 for \(N\geq192\) and + +\[\begin{aligned} N&\in\{46,59,60,62,66,69,70,71,76,78,84,87,90,93,94,95,97,102,104,105, \\ 106,108,110,112-117,119,120,122,124,126,127,129,130,132-140, \\ 144,146-154,156,157,158,160-166,168-190\}. \end{aligned}\] + +Remark 1.11. With the above theorem, there are 30 levels \(N\) for which we do not know yet if there are infinitely many quintic points. These levels are + +\[\{74,77,79,82,83,85,86,88,89,92,96,98,99,100,101,103,107,\] + +\(111,118,123,128,131,141,142,143,145,155,159,167,191\}.\) + +The outline of the paper is as follows: + +- In Section 2 we present the previously known results that will be used in later sections of the paper. diff --git a/outputs/arxiv_math/2503.08982_pg4_pg1_repeat1.md b/outputs/arxiv_math/2503.08982_pg4_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..72d1c16896bda1732521aea77e58080ae5b18d6f --- /dev/null +++ b/outputs/arxiv_math/2503.08982_pg4_pg1_repeat1.md @@ -0,0 +1,21 @@ +agent seeks to maximize its cumulative reward. Let vector \(\boldsymbol{b}=(b(1),b(2),\ldots,b(|\mathcal{S}|))\) denote the belief state where \(b(s)\) is the probability that the true system state is \(s\in\mathcal{S}\). Starting from belief \(\boldsymbol{b}\) at time \(t\), the agent updates belief to \(\boldsymbol{b}^{\prime}\) at time \(t+1\) after executing action \(a\) and observing \(o\) as + +\[b^{\prime}(s^{\prime})= \frac{P(o|a,s^{\prime})}{P(o|b,a)} \sum_{s\in\mathcal{S}}P(s^{\prime}|s,a)b(s)= \frac{P(o|a,s^{\prime})\sum_{s\in\mathcal{S}}P(s^{\prime}|s,a)b(s)}{\sum_{s^{\prime}\in\mathcal{S}}P(o|a,s^{\prime})\sum_{s\in\mathcal{S}}P(s^{\prime}|s,a)b(s)} ,\forall s\in\mathcal{S}.(2.1)\] + +Naturally, \(\sum_{s\in\mathcal{S}}b(s)=1\). The initial belief, \(\boldsymbol{b}_{\mathbf{0}}\), provides the probability distribution of the state at the beginning of the planning horizon. The beliefs derived from an initial belief, via a feasible sequence of actions and observations are called reachable belief points. Figure 1 shows the belief states that are reachable from the initial belief state, \(\boldsymbol{b}_{\mathbf{0}}=(0.5,0.5)\) in one stage for the well-known tiger problem presented by Kaelbling et al. (1998), considering three actions, Listen \((a_{1})\), Open Left Door \((a_{2})\), Open Right Door \((a_{3})\), and two observations resulting from each action, under the assumption that opening a (left or right) door restarts the problem and resets the belief to \((0.5,0.5)\). Appendix B contains a detailed description of the tiger problem. + +![](figure) + +![](figure) + +Figure 1: Tree structure of a two-stage tiger problem (Kaelbling et al. 1998). Circles represent belief states observed right before taking an action in each stage. + +A policy \(\pi\) provides the sequence of actions to be taken over the planning horizon as a function of the belief state. The value function represents the expected cumulative reward obtained under the optimal policy, \(\pi^{*}\). For a finite horizon problem, the following backward recursion computes the value function, \(V_{t}(\boldsymbol{b})\), as, + +\[V_{t}(\boldsymbol{b})=\max_{a\in\mathcal{A}}\left[\sum_{s\in\mathcal{S}}b(s)R(s,a)+\sum_{o\in\mathcal{O}}P(o|\boldsymbol{b},a)V_{t+1}^{\pi^{*}}(\boldsymbol{b}^{\prime})\right],\forall\boldsymbol{b}\in\mathcal{B}_{⊔},⊔\in\{\prime,\infty,\ldots,\mathcal{T}-\infty\},(2.2)\] + +where \(\mathcal{B}_{⊔}\) denotes the \(|\mathcal{S}|-\)dimensional belief space. The optimal policy for belief state \(\boldsymbol{b}\) at time period \(t\) is defined as + +\[\pi_{t}^{*}(\boldsymbol{b})=\operatorname*{arg\,max}_{a\in\mathcal{A}}\left[\sum_{s\in\mathcal{S}}b(s)R(s,a)+\sum_{o\in\mathcal{O}}P(o|\boldsymbol{b},a)V_{t+1}(\boldsymbol{b}^{\prime})\right].\] (2.3) + +Sondik (1971) showed that the value function is represented exactly by a piecewise-linear and convex function such that the value function for a specific period is represented by a set of \(|\mathcal{S}|-\) diff --git a/outputs/arxiv_math/2503.08984_pg35_pg1_repeat1.md b/outputs/arxiv_math/2503.08984_pg35_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..cb9385373edf0c001b4c28843393e45139784882 --- /dev/null +++ b/outputs/arxiv_math/2503.08984_pg35_pg1_repeat1.md @@ -0,0 +1,17 @@ +\[\text{where}\psi(x)=e^{|x|}(1+|x|)-1.\] + +We construct the coupling recursively. For the base case with \(t=0\), clearly \(\mathbb{P}\{G_{e}^{0}=T_{e}^{0}\}=1.\) Condition on \(T_{e}^{2t}=G_{e}^{2t}\) (with an appropriate vertex mapping) and event \(C^{2t}\). We aim to construct a coupling so that \(T_{e}^{2t+1}=G_{e}^{2t+1}\text{and}T_{e}^{2t+2}=G_{e}^{2t+2}\) with probability at least \(1-n^{-\Omega(1)}.\) Each vertex \(u\text{in}\partial G_{e}^{2t}\) has \(B_{u}\) number of incident blue edges connecting to vertices in \([n]\setminus V(G_{e}^{2t})\), where the \(B_{u}’\mathrm{s}\) are i.i.d. \(\text{Binom}(n-|V(G_{e}^{2t})|,\lambda/n)\). Similarly, each vertex \(u\text{in}\partial T_{e}^{2t}\) has \(\tilde{B}_{u}\) number of incident blue edges, where the \(\tilde{B}_{u}’\mathrm{s}\) are i.i.d. \(\text{Pois}(\lambda).\) Thus, we can couple \(B_{u}’\mathrm{s}\) to \(\tilde{B}_{u}^{\prime}s\) using (26) and take a union bound over \(u\in\partial G_{e}^{2t}\equiv\partial T_{e}^{2t}\) . In particular, + +\[\begin{aligned} \mathbb{P}\left\{B_{u}=\tilde{B}_{u},\forall u\in\partial G_{e}^{2t}|G_{e}^{2t}=T_{e}^{2t},C^{2t}\right\} \\ &\geq\geq^{1}1^{-}-\big|(^{\partial}2k^{G}\lambda^{2_{e}t}+\big|\left(2^{\lambda})^{/_{t}^{2}}\log^{n+}n^{\psi}\left(\lambda\left(_{2}\lambda/n^{-}+^{(n}O^{-}\left(^{|}(^{V}2k^{(}\lambda^{G}+^{)_{e}^{2t}}1^{|)})^{\lambda_{t+}/_{1}n}\lambda\right)/\right)n\right)\right) \\ &\geq1-n^{-1+o(1)}, \end{aligned}\] + +where the second inequality holds because conditional on \(C^{2t},|\partial G_{e}^{2t}|\leq(2k\lambda+2)^{t}\log n\) and \(|V(G_{e}^{2t})|\leq(2k\lambda+1)^{t+1}\log n\). Thus, we have constructed a coupling such that \(B_{u}=\tilde{B}_{u}\) for all \(u\in\partial G_{e}^{2t}\) with probability at least \(1-n^{-1+o(1)}\). Recall that if event \(E^{2t}\) occurs, the set of blue edges added to \(G_{e}^{2t+1}\) connect to distinct vertices in \([n]\setminus V(G_{e}^{2t})\). Thus, on event \(E^{2t}\cap\{B_{u}=\tilde{B}_{u},\forall u\in\partial G_{e}^{2t}\}\), there exists a one-to-one mapping from the vertices in \(\partial G_{e}^{2t+1}\) to vertices in \(\partial T_{e}^{2t+1}\text{such that}G_{e}^{2t+1}=T_{e}^{2t+1}.\) Further, recall that on event \(E^{2t+1}\), each vertex \(u\text{in}\partial G_{e}^{2t+1}\) has exactly \(k\) incident red edges, and these red edges connect to distinct vertices in \([n]\setminus V(G_{e}^{2t+1})\). Thus, on the event \(E^{2t+1}\cap E^{2t}\cap\{B_{u}=\tilde{B}_{u},\forall u\in\partial G_{e}^{2t}\}\), there exists a one-to-one mapping from the vertices in \(\partial G_{e}^{2t+2}\) to the vertices in \(\partial T_{e}^{2t+2}\), so that \(=T_{e}^{2t+2}.\) In conclusion, we get that + +\[G_{e}^{2t+2}\] + +\[\begin{aligned} \mathbb{P}\left\{G_{e}^{2t+2}=T^{2t+2}|G_{e}^{2t}=T^{2t},C^{2t}\right\} \\ &\geq\mathbb{P}\left\{E^{2t+1}\cap E^{2t}\cap\{B_{u}=\tilde{B}_{u},\forall u\in\partial G_{e}^{2t}\}|G_{e}^{2t}=T^{2t},C^{2t}\right\} \\ &\geq\mathbb{P}\left\{B_{u}=\tilde{B}_{u},\forall u\in\partial G_{e}^{2t}|G_{e}^{2t}=T^{2t},C^{2t}\right\}-\mathbb{P}\left\{\left(E^{2t+1}\cap E^{2t}\right)^{c}|G_{e}^{2t}=T^{2t},C^{2t}\right\} \\ &\geq1-n^{-1+o(1)}, \end{aligned}\] + +where the last inequality holds by Lemma F.3, since we are assuming \((2k\lambda+2)^{t}\log n=n^{o(1)}\). Moreover, + +\[\begin{aligned} \mathbb{P}\left\{G_{e}^{2t+2}=T^{2t+2},C^{2t+2}|G_{e}^{2t}=T^{2t},C^{2t}\right\} \\ &\geq\mathbb{P}\left\{G_{e}^{2t+2}=T^{2t+2}|G_{e}^{2t}=T^{2t},C^{2t}\right\}-(1-\mathbb{P}\left\{C^{2t+2}|G_{e}^{2t}=T^{2t},C^{2t}\right\}) \\ &\geq1-n^{-\Omega(1)}, \end{aligned}\] + +where the last inequality holds by combining the last displayed equation with Lemma F.2. It diff --git a/outputs/arxiv_math/2503.08986_pg11_pg1_repeat1.md b/outputs/arxiv_math/2503.08986_pg11_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..5cea52aef38967a78bcb9b70880e281fc1d975b7 --- /dev/null +++ b/outputs/arxiv_math/2503.08986_pg11_pg1_repeat1.md @@ -0,0 +1,15 @@ +![](figure) + +Fig. 8. The sum AC results of FAS-assisted STAR-RIS RSMA versus the average SNR \(\overline{\gamma}\) with phase errors and ideal phases for \(K=30\). + +especially in channels with rich scattering. The results in Fig. 8 study the impact of phase errors on the sum AC, \(\overline{\mathcal{C}}_{\text{sum},u} = \overline{\mathcal{C}}_{\mathrm{c},u} + \overline{\mathcal{C}}_{\mathrm{p},u}\). First of all, as expected, a small gap between the simulation results and the theoretical predictions is observed. This discrepancy arises from the use of a heuristic technique to approximate the expectation of the maximum of \(N_{u}\) correlated Gamma RVs, as discussed in Remark 2. The heuristic method, based on simplifying assumptions, provides a computationally efficient approximation that captures the essential behavior of the system, especially in cases involving moderate correlations. Despite the gap, the results indicate that this approach performs well across a range of scenarios, making it a useful approximation for evaluating the AC performance under correlated conditions. For both FAS and TAS, we see that as \(\overline{\gamma}\) increases, the sum AC initially improves but eventually saturates. This occurs because in the RSMA scheme, rate splitting effectively manages interference and optimizes power allocation at moderate SNR levels. As \(\overline{\gamma}\) grows, the SINR expressions in (8) and (11) converge to deterministic values, depending on the power allocation factors for the common and private streams. Beyond this threshold, further increases in SNR do not significantly improve the sum AC, resulting in saturation. Additionally, the case with ideal phase results in a higher level of sum AC compared to the scenario with phase errors. This is because with ideal phase, STAR-RIS can perfectly align the transmitted signals with the desired reflection coefficients, thereby maximizing the signal power at the receiver and minimizing interference. In contrast, phase errors introduce misalignment between the transmitter and STAR-RIS, leading to suboptimal signal reflection and increased interference. As a result, the effective channel gains are reduced in the presence of phase errors, causing a degradation in the sum AC. Noteworthy, the asymptotic behavior for the FAS and TAS schemes is independent of the phase errors, so does the performance degradation because the phase + +![](figure) + +Fig. 9. The sum AC results of FAS-assisted STAR-RIS RSMA versus the average SNR \(\overline{\gamma}\) with phase errors for different numbers of STAR-RIS elements \(K\). + +mismatch is reduced as \(\overline{\gamma}\) grows. Finally, Fig. 9 studies how the number of STAR-RIS elements, \(K\), affects the sum AC. As we can see, a larger \(K\) provides a higher sum AC for both FAS and TAS. Increasing \(K\) also results in a higher sum AC for both FAS and TAS. This gain occurs because a larger number of STAR-RIS elements increases the surface’s ability to adaptively reconfigure the channel, allowing for better alignment of the reflected signals. With more elements, STAR-RIS can have more precise control over the spatial distribution of the reflected signals, effectively enhancing the SNR and reducing interference. Furthermore, a larger \(K\) enables better exploitation of spatial diversity, thereby improving the capacity for both RSMA users. As such, the increase in the number of STAR-RIS elements contributes to a more efficient channel utilization, leading to a higher sum AC. We can also see that FAS provides a higher sum AC than the TAS counterpart. This is because FAS can dynamically adjust its configuration, improving channel alignment and spatial diversity, while TAS, with its fixed setup, lacks this flexibility, limiting its performance in varying conditions. As in the previous figure, the sum AC saturates to a value that only depends on the RSMA power allocation factors, as the dependence on \(K\) is cancelled out for sufficiently large \(\overline{\gamma}\). + +## V. CONCLUSION + +This paper presented a comprehensive investigation of FAS-assisted STAR-RIS communication systems employing RSMA signaling, with a particular focus on the impact of phase errors under the ES protocol. By modeling the phase errors with a generic distribution and deriving the equivalent channel gain characterized by the multivariate \(t-\text{distribution}\), we provided a realistic and robust analysis of the system performance. Compact analytical expressions for key performance metrics, such as outage OP and AC were derived, with the latter being diff --git a/outputs/arxiv_math/2503.08991_pg14_pg1_repeat1.md b/outputs/arxiv_math/2503.08991_pg14_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..82727c05286873634e4977f0c9256c5ed433c54b --- /dev/null +++ b/outputs/arxiv_math/2503.08991_pg14_pg1_repeat1.md @@ -0,0 +1,27 @@ +Proof of Theorem 3.7. First, note that \(f_{A}\) has a unique measure of maximal entropy \(\eta\) and its push forward measure \(\mu=\pi_{*}\eta\) is a measure of maximal entropy of \(g_{A}\), because \(\pi\) is bounded to one which guarantees \(h_{\mu}(g_{A})=h_{\eta}(f_{A})\) (e.g. see [39]) and, hence, \(h(g_{A})=h(f_{A})=h_{\eta}(f_{A})=h_{\mu}(g_{A})\) (see [25] for the definition of metric entropy). Next let us prove the uniqueness of the measure of maximal entropy of \(g_{A}\). To do so, let us fix any measure of maximal entropy \(\hat{\mu}\text{for}g_{A}\). Theorem 2.12 implies that \(g_{A}\) has the periodic specification property, which ensures the existence of a sequence \(\hat{\mu}_{n}\) of measures supported on periodic points \(p_{n}\text{such that}\hat{\mu}\) is the weak* limit of \(\hat{\mu}_{n}\) (see [25, Proposition 21.8]). Let \(q_{n}\) be any periodic point \(q_{n}\in\pi^{-1}(p_{n})\) and let \(\hat{\eta}_{n}\) be the measure supported on \(q_{n}.\text{Then}\hat{\mu}_{n}=\pi_{*}(\hat{\eta}_{n})\). Without loss of generality we may assume that \((\hat{\eta}_{n})_{n\in\mathbb{N}}\) converge in the weak* topology to a measure \(\hat{\eta}\). But then, by continuity of the push-forward operator \(\pi_{*}\) we have \(\pi_{*}\hat{\eta}=\hat{\mu}\) and as a consequence + +\[h(g_{A})=h_{\hat{\mu}}(g_{A})\leq h_{\hat{\eta}}(f_{A})\leq h(f_{A})=h(g_{A}).\] + +This proves that \(\hat{\eta}=\eta\) and consequently that \(\hat{\mu}=\mu\), that is, \(\mu\) is the unique measure of maximal entropy of \(g_{A}\). It remains to prove that \(\mu\) is the weak* limit of \(\mu_{n}\). A difficulty that arise is how to lift precisely the sequence \((\mu_{n})_{n\in\mathbb{N}}\) to the Torus since the sequence \((\eta_{n})_{n\in\mathbb{N}}\) defined by + +\[\eta_{n}= \frac{1}{\text{Per}_{n}(f_{A})} \sum_{p\in P_{n}(f_{A})}\delta_{p},\] + +which converge to \(\eta\) by Bowen’s proof, does not project to \((\mu_{n})_{n\in\mathbb{N}}\). Indeed, the antipodal periodic points are also in the pre-image of periodic points of \(g_{A}\). An attempt would be to include the antipodal periodic points in the definition of these measures and consider the sequence \((\hat{\eta}_{n})_{n\in\mathbb{N}}\) defined by + +\[\hat{\eta}_{n}= \frac{1}{\text{Per}_{n}(f_{A})+\text{Per}_{n}^{-}(f_{A})}_{p\in} \sum_{P_{n}(f_{A})\cup P_{n}^{-}(f_{A})}\delta_{p}.\] + +But \(\hat{\eta}_{n}\) still does not project to \(\mu_{n}\) and the problem relies on the existence of the spines (points with a single pre-image). Thus, we rule out these points as follows: for each \(n\in\mathbb{N}\), let + +\[\begin{aligned} P_{n}^{*}(g_{A})&=\{p\in P_{n}(g_{A}):\#\pi^{-1}(p)=2\},\text{Per}_{n}^{*}(g_{A})=\#P_{n}^{*}(g_{A}), \\ \text{and}\hat{\mu}_{n}&= \frac{1}{\text{Per}_{n}^{*}(g_{A})} \sum_{p\in P_{n}^{*}(g_{A})}\delta_{p}. \end{aligned}\] + +Since there are only four points with a single pre-image, we have + +\[|\text{Per}_{n}(g_{A})-\text{Per}_{n}^{*}(g_{A})|\leq4.\] + +Thus, \((\mu_{n})_{n\in\mathbb{N}}\text{and}(\hat{\mu}_{n})_{n\in\mathbb{N}}\) converge weakly* to exactly the same measure, provided the limit exists. For each \(n\in\mathbb{N}\), let + +\[P_{n}^{*}(f_{A})=\pi^{-1}(P_{n}^{*}(g_{A}))\text{and Per}_{n}^{*}(f_{A})=\#P_{n}^{*}(f_{A})=2\text{Per}_{n}^{*}(g_{A}),\] + +and note that for \(n\) sufficiently large we have + +\[\frac{\text{Per}_{n}(f_{A})}{2} \leq\text{Per}_{n}^{*}(f_{A})\leq3\text{Per}_{n}(f_{A}).\] diff --git a/outputs/arxiv_math/2503.08996_pg21_pg1_repeat1.md b/outputs/arxiv_math/2503.08996_pg21_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..a5f89c1d6acdad93465fae89490e0f336771586f --- /dev/null +++ b/outputs/arxiv_math/2503.08996_pg21_pg1_repeat1.md @@ -0,0 +1,23 @@ +![](figure) + +![](figure) + +Figure 5.1. Frequency domain for \(\mathbb{C}_{\mathbf{k}\approx\mathbf{K}_{\star}}(t;v,\theta)\) + +For the proof, we recall that changing the variable, the sectorial domain of integration for \(\mathbb{C}_{\mathbf{k}\approx\mathbf{K}_{\star}}(t;v,\theta)\) is transformed into the rectangular one (see Figure 5.1). To refine the bound in the previous section, we find the curve \(k_{2}=\gamma(k_{1})\) where the modified phase function \(\tilde{\varphi}(\mathbf{k};v,\theta)\) is stationary in the \(k_{2}-\)direction at the frequency \(\mathbf{k}=(k_{1},\gamma(k_{1}))\). Then, near the curve (the red region), we only consider the oscillation in the \(k_{1}-\text{direction}\), but the additional decay is obtained from the small measure. On the other hand, away from the curve (the blue region), we take the additional decay by integration by parts in the \(k_{2}-\text{direction}\). The \(O(|t|^{-c})-\)width of the red region will be chosen later to optimize the sum of the two bounds. For this, using Lemma 5.11, we differentiate the phase in the \(k_{2}-\)direction and write + +? + +\[\partial_{k_{2}}\tilde{\varphi}(\mathbf{k};v,\theta)= \frac{\overline{3} k_{1}}{2} G(\mathbf{k}),\] (5.12) + +where + +\[G(\mathbf{k}):=_{+}k_{(^{2}}+ \frac{3^{(}a_{\theta}^{-}b_{\theta}}{4} \frac{3b_{\theta}}{+2} \frac{3^{-}(4 \frac{7a_{\theta}}{4a^{2_{2_{\theta}}}} -^{k_{2}}3^{+})}{32} k \frac{15b_{\theta}}{4_{-_{2}}} \frac{8^{k}7^{22}}{8} )a_{\theta}^{k}b_{\theta}^{1}k_{2}^{)_{2}}k_{1}^{2}+k_{1}^{3}\mathcal{O}_{0}(\mathbf{k})+k_{2}^{3}\tilde{\mathcal{O}}_{0}(\mathbf{k}).^{(5.13)}\] + +By the implicit function theorem, we determine the frequencies where \(G(\mathbf{k})\) vanishes, which is equivalent to \(\partial_{k_{2}}\tilde{\varphi}(\mathbf{k};v,\theta)=0\text{when}k_{1}>0\). + +Lemma 5.14. There exist small \(\delta_{0}>0\) and an analytic curve \(\gamma:(-\delta_{0},\delta_{0})\rightarrow\mathbb{R}\) such that if \(|k_{1}|<\delta_{0},\text{then}G(k_{1},\gamma(k_{1}))=0\) and + +\[\gamma(k_{1})= \frac{3b_{\theta}}{2} k_{1}+ \frac{9a_{\theta}b_{\theta}}{2} k_{1}^{2}+\mathcal{O}_{3}(k_{1}),\] + +where \(\mathcal{O}_{3}(k_{1})\) is an analytic function of \(k_{1}\) near the origin such that \(|\mathcal{O}_{3}(k_{1})|\lesssim|k_{1}|^{3}\) (see (2.16)). diff --git a/outputs/arxiv_math/2503.09012_pg8_pg1_repeat1.md b/outputs/arxiv_math/2503.09012_pg8_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..58c86e46157fb34ee847925c0628e9cee2a22642 --- /dev/null +++ b/outputs/arxiv_math/2503.09012_pg8_pg1_repeat1.md @@ -0,0 +1,15 @@ +![](figure) + +![](figure) + +FIG. 3. The one-shot work cost \(W^{\epsilon}(\rho_{AB};\sigma_{A^{\prime}B^{\prime}})\) of converting Alice’s system in the presence of Bob’s side information is calculated as follows. Initially, Alice’s and Bob’s systems are in the state \(\rho_{AB}\), and Alice has access to a battery system \(A_{0}\) in a pure state representing her initial work storage. Supposedly without doing work, they perform a conditionally uniformity-covariant operation \(N_{A_{0}AB\rightarrow A_{1}A^{\prime}B^{\prime}}\) such that the state \(\tau_{A_{1}A^{\prime}B^{\prime}}\) after the operation is \(\epsilon-\)close to the target state \(\sigma_{A^{\prime}B^{\prime}}\) in tensor product with a pure state of a battery system \(A_{1}\) representing Alice’s final work storage. According to Landauer’s principle, Alice has to invest \((1/\beta_{\mathrm{b}})(\log_{2}|A_{0}|-\log_{2}|A_{1}|)\) or extract \((1/\beta_{\mathrm{b}})(\log_{2}|A_{1}|-\log_{2}|A_{0}|)\) work to “recharge” or “discharge” her battery afterwards, returning her work storage to the initial level. This completes the the desired conversion from \(A\) to \(A^{\prime}\) without causing significant change to the environment (specifically, the battery), and the work cost of the whole process boils down to the amount of work involved in Alice’s recharging or discharging her battery. + +that the desired conversion can always be completed. In the end, the decrease or increase in the number of pure qubits in the battery gives a precise account for the overall amount of work invested to or extracted from the conversion. Third, as argued in Sec. II A, every legitimate operation that can be performed without work supply must fall within the scope of conditionally uniformity-covariant operations. Here, we assume that the converse is also true, namely that all such operations can be performed freely, so that the work cost defined under such operations is equal to the actual work cost in its literal sense. Note that even in a less permissive setting without this assumption, the work cost defined under conditionally uniformity-covariant operations still provides a no-go limit on the actual work cost. Fourth, we accommodate the practical need for approximate conversion, where a bounded error is tolerated. Formally, for a pair of states \(\rho_{AB},\sigma_{A^{\prime}B^{\prime}}\), we define the one-shot work cost of converting \(A\) to \(A^{\prime}\) with error \(\epsilon\in[0,1]\) as + +\[W^{\epsilon}(\rho_{AB};\sigma_{A^{\prime}B^{\prime}})≔ \frac{1}{\beta_{\mathrm{b}}}_{N_{A_{0}AB\rightarrow}A_{A}} \inf_{,_{0_{1A}}A_{\prime_{B}1},_{\in^{\prime}\text{CUCO}}}\left\{\log_{2}|A_{0}|-\log_{2}|A_{1}|:\Delta\left(|0\rangle\langle0|_{A_{1}}\otimes\sigma_{A^{\prime}B^{\prime}},N_{A_{0}AB\rightarrow A_{1}A^{\prime}B^{\prime}}\left[|0\rangle\langle0|_{A_{0}}\otimes\rho_{AB}\right]\right)\leq\epsilon\right\},\] + +where CUCO denotes the set of conditionally uniformity-covariant operations. In Eq. \((40),A_{0}\text{and}A_{1}\) are the battery systems representing Alice’s initial and final work storage, respectively. A positive work cost means that Alice has to release this amount of work from her storage and invest it into the conversion; a negative work cost means that Alice can extract an amount of work, equal to the absolute value of the work cost, from the conversion and keep it in her storage. See Fig. 3 for an illustration. We now focus on two particular conversion tasks, namely preparation and erasure with side information. For a state \(\rho_{AB}\), the one-shot work costs of preparing and erasing the system \(A\) with error \(\epsilon\in[0,1]\) are defined, respectively, as + +\[\begin{aligned} W_{\mathrm{p}\text{rep}}^{\epsilon}(\rho_{AB})≔W^{\epsilon}(|0\rangle\langle0|_{A};\rho_{AB}), \\ W_{\mathrm{e}\text{ras}}^{\epsilon}(\rho_{AB})≔W^{\epsilon}(\rho_{AB};|0\rangle\langle0|_{A}). \end{aligned}\] (41) (42) + +In other words, the preparation task refers to putting Alice’s system in a given state, coupled with Bob’s system, when it is initialized to a default pure state; the erasure task refers to the reverse process of overwriting Alice’s system with the default pure state when it is initialized to the given state. Our first diff --git a/outputs/arxiv_math/2503.09021_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.09021_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..4e954024fde5256d6b6545daef88be3e366d471b --- /dev/null +++ b/outputs/arxiv_math/2503.09021_pg3_pg1_repeat1.md @@ -0,0 +1,13 @@ +with an unknown regularization functional of the inverse problem as an equivalent constrained minimization problem with an unknown feasible region depending on the undetermined regularization functional. The CNN in [24] is then designed to learn the a priori information of the shape of the unknown contrast by using a normalization technique in the training process and trained to act like a projector which is helpful for projecting the solution into the feasible region of the constrained optimization problem associated with the inverse problem; see [24] for details. In this paper, we propose two iterative regularization algorithms that incorporate the a priori information of the shape and location (i.e., the support) of the unknown contrast, which is learned by a deep neural network from a direct imaging method, as regularization strategies for recovering the contrast of the inhomogeneous medium from the far-field data. Precisely, we first train a deep neural network to retrieve the a priori information of the support of the unknown contrast from a direct imaging method (e.g., the one in [36]). Then, the learned a priori information is incorporated into the projected Landweber method in our first algorithm, whilst the learned a priori information is used to construct the regularization functional for the variational regularization formulation of the inverse problem which is finally solved by an iteration algorithm in our second algorithm. It is worth noting that the trained deep neural network in this paper is used to provide a good approximation of the support of the unknown contrast (which is indeed confirmed in the numerical examples), while the trained deep neural network in our previous work [24] focused on learning the a priori information of the shape of the unknown contrast. Extensive numerical experiments demonstrate that our algorithms have a satisfactory reconstruction performance, strong robustness to noise and good generalization ability. The rest of this paper is organized as follows. Section 2 presents the direct and inverse medium scattering problems considered in this paper. In Section 3, we propose a deep neural network that can retrieve the support of the unknown contrast based on the direct imaging method. In Section 4, we present two iterative reconstruction algorithms that incorporate the a priori information of the support of the unknown contrast for solving the inverse medium scattering problem. Numerical experiments are carried out in Section 5 to illustrate the effectiveness of our algorithms. Some conclusions and remarks are given in Section 6. + +## 2 Problem formulation + +In this section, we introduce the direct and inverse medium scattering problems considered in this paper. Consider an inhomogeneous medium in \(\mathbb{R}^{2}\) characterized by the piecewise smooth refractive index \(n(x)>0.\) Define \(m(x):=n(x)-1\) which is the contrast of the inhomogeneous medium and assumed to be compactly supported in a disk with radius \(\rho\), i.e., \(\text{supp}(m)\subset B_{\rho}:= \{x\in\mathbb{R}^{2}:|x|<\rho\}.\) Let \(u^{i}=u^{i}(x,d):=e^{ikx\cdot d}\) be an incident plane wave with the incident \(u=u^{i}+u^{s}\), which is the sum of the incident field \(u^{i}\) and the scattered field \(u^{s}\), satisfies the reduced wave equation + +direction \(d\in\mathbb{S}^{1}:=\{x\in\mathbb{R}^{2}:|x|=1\}\) and the wave number \(k>0\). Then the total field + +\[△u(x)+k^{2}n(x)u(x)=0\text{in}\mathbb{R}^{2},\] (2.1) + +and the scattered field \(u^{s}\) is assumed to satisfy the Sommerfeld radiation condition + +\[\lim_{r\rightarrow\infty}r^{\frac{1}{2}}\left( \frac{\partial u^{s}}{\partial r} -iku^{s}\right)=0,r=|x|.\] (2.2) diff --git a/outputs/arxiv_math/2503.09034_pg7_pg1_repeat1.md b/outputs/arxiv_math/2503.09034_pg7_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..ba9c7963f0706297dd40b469c778652232e941e2 --- /dev/null +++ b/outputs/arxiv_math/2503.09034_pg7_pg1_repeat1.md @@ -0,0 +1,29 @@ +Proposition 3.6 ( [15, Corollary 1.4]). Consider the assumption in Theorem 3.5. Then, Sel\((_{p}^{\pm_{\infty}}E/K_{\infty})\) has no proper Λ-submodules with finite index. By the above proposition and a property of the Fitting ideal (cf. [10, Lemma A.7]), we have + +\[\text{Fitt}_{\Lambda}\left(\text{Sel}_{p^{\infty}}^{\pm}(E/K_{\infty})^{\vee}\right)=\text{char}_{\Lambda}\left(\text{Sel}_{p^{\infty}}^{\pm}(E/K_{\infty})^{\vee}\right).\] (3.1) + +Last, we prove the control theorem for Sel\((_{p}^{\pm_{\infty}}E/K_{\infty})\) in our situation by the similar arguments to [1]. We prepare the following lemma for the proof. Let \(w\) be a prime of \(K_{\infty}\) above \(p\). Lemma 3.7. The natural map \(f_{n}^{\pm}:H_{\pm}^{1}(K_{n,p},E[p^{\infty}])\longrightarrow H_{\pm}^{1}(K_{\infty,w},E[p^{\infty}])[\omega_{n}^{\pm}]\) is injective, and the cokernel of \(f_{n}^{\pm}\) is a finite group for any positive integer \(n\). Here, we define \(H^{1}\) Proof. Note that we have \(E[p^{\infty}]^{G_{K\infty,p}}=0\) by [15, Proposition 3.2]. By Inflation-Restriction exact sequence, we see that the canonical map + +\[(_{\pm}^{1}K_{\infty,w},E[p^{\infty}]):=\lim_{\longrightarrow }H_{\pm}^{1}(K_{n,p},E[p^{\infty}]).\] + +\[f_{n}:H^{1}(K_{n,p},E[p^{\infty}])\longrightarrow H^{1}(K_{\infty,p},E[p^{\infty}])^{\text{Gal}(K_{\infty,p}/F_{n,p})}=H^{1}(K_{\infty,p},E[p^{\infty}])[\omega_{n}]\] + +is injective. Since the map \(f_{n}^{\pm}\) is the restriction of \(f_{n}\) to \(H_{\pm}^{1}(K_{n,p},E[p^{\infty}]),f_{n}^{\pm}\) is also injective. For the claim for Coker \(f_{n}^{\pm}\) , it suffices to show that both the \(\mathbb{Z}_{p}-\)coranks of \(H_{\pm}^{1}(K_{n,p},E[p^{\infty}])\) and \(H_{\pm}^{1}(K_{\infty,w},E[p^{\infty}])[\omega_{n}^{\pm}]\) are same. Since we have \(H_{\pm}^{1}(K_{\infty,w},E[p^{\infty}])^{\vee}\simeq\Lambda^{2}\) as Λ-module by the Rubin conjecture, we see that + +\[\begin{aligned} H_{\pm}^{1}(K_{\infty,w},E[p^{\infty}])[\omega_{n}^{\pm}]^{\vee}&\simeq(H_{\pm}^{1}(K_{\infty,w},E[p^{\infty}]))^{\vee}/\omega_{n}^{\pm}(H_{\pm}^{1}(K_{\infty,w},E[p^{\infty}]))^{\vee} \\ &\simeq\Lambda^{2}/\omega_{n}^{\pm}\Lambda^{2}. \end{aligned}\] + +On the other hand, we have \(H_{\pm}^{1}(K_{n,p},E[p^{\infty}])=\widehat{E}(K_{n,p})^{\pm}\otimes(\mathbb{Q}_{p}/\mathbb{Z}_{p})\) by the definition. Thus, we see that \(H_{\pm}^{1}(K_{n,p},E[p^{\infty}])^{\vee}\simeq\Lambda_{n}^{2}/\omega_{n}^{\pm}\Lambda_{n}^{2}\) by Proposition 3.4. Therefore, the \(\mathbb{Z}_{p}-\text{rank}\) of \(H_{\pm}^{1}(K_{n,p},E[p^{\infty}])\text{and}H_{\pm}^{1}(K_{\infty,w},E[p^{\infty}])[\omega_{n}^{\pm}]\) are the same. Proposition 3.8. Consider the assumption in Theorem 3.5. Then, the canonical homomorphism + +- + +\[\text{Sel}_{p^{\infty}}^{\pm}(E/K_{\infty})[\omega_{n}^{\pm}]\longrightarrow \text{Sel}_{p^{\infty}}^{\pm}(E/K_{n})[\omega_{n}^{\pm}]\] + +is injective, and the order of the cokernel is finite for any \(n\). Proof. We take the finite subset \(\Sigma=\{p\}\cup\{\)bad primes of \(E\}\). Then, we have the following commutative diagram: + +\[\begin{aligned} 0\text{Sel}_{p}^{\pm}(E/K_{n})[\omega_{n}^{\pm}]H^{1}(K_{\Sigma}/K_{n},E[p^{\infty}])[\omega_{n}^{\pm}]^{\underline{a}}\prod_{v_{n}|v} \frac{H^{1}(K_{n,v_{n}},E[p^{\infty}])[\omega_{n}^{\pm}]}{H_{\pm}^{1}(K_{n,v_{n}},E[p^{\infty}])} , \\ v&\in\Sigma \\ s_{n}^{\pm}h_{n}^{\pm}g_{n}&=\prod^{\pm}g_{n,vn}^{\pm} \\ 0\text{Sel}_{p}^{\pm}(E/K_{\infty})[\omega_{n}^{\pm}]H^{1}(K_{\Sigma}/K_{\infty},E[p^{\infty}])[\omega_{n}^{\pm}]\prod_{v_{\infty}|v} \frac{H^{1}(K_{\infty,v_{\infty}},E[p^{\infty}])[\omega_{n}^{\pm}]}{H_{\pm}^{1}(K_{\infty,v_{\infty}},E[p^{\infty}])[\omega_{n}^{\pm}]} . \\ v&\in\Sigma \end{aligned}\] + +Since we have \(E[p^{\infty}]^{G_{K\infty}}=0\), the map \(H^{1}(K_{\Sigma}/K_{n},E[p^{\infty}])\longrightarrow H^{1}(K_{\Sigma}/K_{\infty},E[p^{\infty}])[\omega_{n}]\) induced by the restriction map is isomorphism by the Inflation-Restriction exact sequence. Therefore, \(h_{n}^{\pm}\) is also isomorphism. By the snake lemma, \(s_{n}^{\pm}\) is injective, and it suffices to calculate the kernel of \(g_{n}^{\pm}\) . We first consider the case \(v\in\Sigma\setminus\{p\}.\text{Then},K_{\infty,v_{\infty}}/K_{n,v_{n}}\) is the trivial extension or the unramified \(\mathbb{Z}_{p}-\text{extension}\). If the extension \(K_{\infty,v_{\infty}}/K_{n,v_{n}}\) is trivial, then it is clear that \(\text{Ker}g_{n,v_{n}}^{\pm}=0\). Assume that \(K_{\infty,v_{\infty}}/K_{n,v_{n}}\) is the unramified \(\mathbb{Z}_{p}-\text{extension}\). Write \(B_{v_{\infty}}:=E[p^{\infty}]^{G_{K\infty,v\infty}}\) . We + +consider the exact sequence + +\[0H^{1}(K_{\infty,v_{\infty}}/K_{n,v_{n}},B_{v_{\infty}})H^{1}(K_{n,v_{n}},E[p^{\infty}])H^{1}(K_{\infty,v_{\infty}},E[p^{\infty}])\] diff --git a/outputs/arxiv_math/2503.09042_pg5_pg1_repeat1.md b/outputs/arxiv_math/2503.09042_pg5_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..a991864063638ab32e9aabb6c69143b7d4c8db8f --- /dev/null +++ b/outputs/arxiv_math/2503.09042_pg5_pg1_repeat1.md @@ -0,0 +1,35 @@ +\[\mathbb{P}[g(X)\geq k]\leq\epsilon.\] (3.1) + +Then + +\[\mathbb{E}\left[X_{f(Y)}Y_{g(X)}\right]\leq \frac{1}{2} -(1-4\epsilon)2^{-k+1}.\] + +Proof. Recall our notation \(W_{f,g}=\mathbb{E}\left[X_{f(Y)}Y_{g(X)}\right]\). Since \(X\) is independent of \(Y\) , since \(X\text{and}-X\) have the same distribution, and since \(Y\text{and}-Y\) have the same distribution, \((X,Y)\) has the same joint distribution as \((X,-Y)\), \((-X,Y)\text{or}(-X,-Y)\). Hence + +\[W_{f,g}=\mathbb{E}\left[-X_{f(-Y)}Y_{g(X)}\right]=\mathbb{E}\left[-X_{f(Y)}Y_{g(-X)}\right]=\mathbb{E}\left[X_{f(-Y)}Y_{g(-X)}\right].\] + +Summing these and rearranging yields + +\[4W_{f,g}=\mathbb{E}\left[(X_{f(Y)}-X_{f(-Y)})(Y_{g(X)}-Y_{g(-X)})\right].\] + +Let \(B_{Y}\) be the event that \(Y_{1}=Y_{2}=\cdots =Y_{k-1}\), and note that \(\mathbb{P}[B_{Y}]= 2^{-(k-2)}\). Write + +\[\begin{aligned} 4W_{f,g}&=\mathbb{E}\left[(X_{f(Y)}-X_{f(-Y)})(Y_{g(X)}-Y_{g(-X)})1_{\{B_{Y}\}}\right] \\ +\mathbb{E}\left[(X_{f(Y)}-X_{f(-Y)})(Y_{g(X)}-Y_{g(-X)})1_{\{B_{Y}^{c}\}}\right]. \end{aligned}\] (3.2) + +Consider the first term on the right hand side. Since \(\mathbb{P}[g(-X)\geq k]= \mathbb{P}[g(X)\geq k]\leq\epsilon\), we have that \(\mathbb{P}[g(X)\geq k\text{or}g(-X)\geq k]\leq2\epsilon\), by the union bound. Now, if \(g(X)0\). The weighted Laplacian \(L_{\alpha}(G)\) of \(G\) with respect to \(\alpha\) is defined as: \((L_{\alpha}(G))_{ij}= -\sum_{\mathbf{e}\in\mathbf{E}_{o(\alpha)}(G),\mathbf{e}(0)=v_{i},\mathbf{e}(1)=v_{j}}c_{\mathbf{e}}\text{forall}i\neq j,\text{and}(L_{\alpha}(G))_{ii}=\) for all diagonal entries. Removing the row and column of \(L_{\alpha}\) with respect to \(w\), we get the reduced weighted Laplacian \(L_{\alpha}^{\mathrm{r}\text{ed}(w)}(G)\). + +\[\sum_{\mathbf{e}\in\mathbf{E}_{o(\alpha)}(G),\mathbf{e}(0)=v_{i},\mathbf{e}(1)\neq v_{i}}c_{\mathbf{e}}\] diff --git a/outputs/arxiv_math/2503.09088_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.09088_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..1bc7212d3972ee48bed9035b224ef04814559f7b --- /dev/null +++ b/outputs/arxiv_math/2503.09088_pg2_pg1_repeat1.md @@ -0,0 +1,33 @@ +Bona and Tzvetkov [1] proved that the BBM equation is globally well - posed in + +\(H^{s}(\mathbb{R})\text{for}s\geq0\). The same result also holds for the periodic case [2, 3]. This well posedness is optimal, since if \(s<0\), the equation (1) is ill - posed in \(H^{s}(\mathbb{R})\). In fact, Bona and Tzvetkov [1] also proved that for \(s<0\), the solution map \(u_{0}\mapsto u(t)\) of the Later, Panthee [4] improved this result by showing that the solution map is not + +equation (1) is not a \(C^{2}\) map from \(H^{s}(\mathbb{R})\) to \(C([0,T];H^{s}(\mathbb{R}))\). + +continuous from \(H^{s}(\mathbb{R})\) to \(\mathcal{D}^{\prime}(\mathbb{R})\) for any fixed \(t>0\) small enough. Carvajal and Panthee [5] considered the following generalized BBM equation + +\[\begin{cases} u_{t}+u_{x}+Lu_{t}+(u^{k+1})_{x}=0, \\ u(0,x)=u_{0}(x), \end{cases}\] (2) + +where \(L\) is defined via the Fourier transform as \(\widehat{Lu}(\xi)=|\xi|^{\alpha+1}\widehat{\omega}(\xi),\text{with}\alpha>0\). + +Carvajal and Panthee [5] proved that the equation (2) is locally well - posed in \(H^{s}(\mathbb{R})\) + +for some \(s\) depending on \(\alpha\text{and}k\). Moreover, if \(s<\max\{0, \frac{1}{2} - \frac{\alpha}{k} \}\), the equation (2) is ill + +- posed in \(H^{s}(\mathbb{R})\), which means that the solution map is not \(C^{k+1}-\) differentiable from \(H^{s}(\mathbb{R})\) to \(C([0,T];H^{s}(\mathbb{R}))\) for any \(T>0.\text{When}\alpha=1\), some global well - posedness results for (2) with periodic boundary conditions were obtained [3]. + +## 2 Model Derivation + +From 2002 to 2004, Bona, Chen and Saut [6, 7] started from the free - surface Euler equations. Following the derivation context of the Boussinesq original equations, they considered the equivalent relationships among the physical quantities in the shallow - water region and combined the following expanded physical perturbation parameters + +\[\alpha= \frac{A}{h_{0}} \ll1,\beta= \frac{h_{0}}{l^{2}} \ll1,S= \frac{\alpha}{\beta} = \frac{Al^{2}}{h_{0}^{3}} \approx1,\] + +where \(A\text{and}l\) represent the wave amplitude and wavelength in the classical sense, \(h_{0}\) represents the undisturbed water depth, and \(S\) represents the Stokes number. The first + +- order form of the \(abcd-\)system is + +\[\begin{cases} \partial_{t}\eta+\partial_{x}w+\partial_{x}(w\eta)+a\partial_{x}^{3}w-b\partial_{x}^{2}\partial_{t}\eta=0, \\ \partial_{t}w+\partial_{x}\eta+w\partial_{x}w+c\partial_{x}^{3}\eta-d\partial_{x}^{2}\partial_{t}w=0. \end{cases}\] (3) + +Here, \(\eta\) is proportional to the deviation of the free - surface position at \(x;w=w(x,t)\) is proportional to the horizontal velocity at the point \((x,z_{0},t)\) at the depth \(z_{0}\) in the flow domain. The constants \(a,b,c,d\) are not arbitrary, and they satisfy the relationships + +\[\begin{cases} \Big( & \Big) & \Big( & \Big) \\ a= \tfrac{1}{2} & \theta^{2}- \tfrac{1}{3} & \lambda, & b= \tfrac{1}{2} & \theta^{2}- \tfrac{1}{3} & (1-\lambda), \\ c= \tfrac{1}{2} (1-\theta^{2})\mu, & d= \tfrac{1}{2} (1-\theta^{2})(1-\mu), \end{cases}\] (4) diff --git a/outputs/arxiv_math/2503.09107_pg9_pg1_repeat1.md b/outputs/arxiv_math/2503.09107_pg9_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..fd702d2554ef673dc80d737624713d624d9e4e10 --- /dev/null +++ b/outputs/arxiv_math/2503.09107_pg9_pg1_repeat1.md @@ -0,0 +1,7 @@ +Similarly, the second inequality uses the boundedness of entries of \(Q-\text{matrix};\text{and}\cdots\) represents the first two terms in a fold. Thus by Grönwall’s inequality, it follows + +\[\int_{I}|\hat{u}^{n,x}(s,\cdot)-\hat{u}^{x}(s,\cdot)|dx\leq C\left(\int\int_{+_{I}}\int^{I}\int_{s^{T}s}^{T}\Big|^{-}\Big|q_{f^{e,_{x}e^{\prime}}}\left(\left(_{t,}\hat{\phi}_{e^{x},}(_{\hat{Z}}t,_{tn}e_{,\boldsymbol{K}^{,}})_{x},_{,}\hat{Z}_{\hat{Z}_{t,\boldsymbol{I}}^{t^{x},_{n}\boldsymbol{K}_{,x}}},_{,}\hat{Z}_{\hat{\phi}^{t_{n}^{x},\boldsymbol{I}_{,x}}}\right)_{(t}-_{,e}q_{)^{e}}\right)^{,e}+^{\prime}\left(_{f}\hat{\phi}_{x}^{n}\left(^{,x}t(_{,}t_{e},_{,}e_{\hat{Z}}),_{tx}\hat{Z}_{,_{,\boldsymbol{K}}^{t,\boldsymbol{K}^{,x}}\hat{Z}}^{n},_{tx}\hat{Z}_{,_{,\boldsymbol{I}}^{t,}\hat{\phi}^{\boldsymbol{I},x_{x}}}^{n}\right)_{(}\Big|_{t^{2},e}d_{)}t\right)dx\Big|^{d^{2}tdx}\right)^{(14)}\] + +Again the continuity of \((\hat{Z}_{\boldsymbol{K}},\hat{Z}_{\boldsymbol{I}})\) induces the convergence of the two \(q\) functions. This completes the proof. Lemma 1.1: \(\mathcal{Z}\) is closed in \(C\left([0,T];L^{2}(I;\mathbb{R}\otimes\mathbb{R})\right)\) equipped with norm \(\Vert\cdot\Vert_{\mathcal{Z}}\). Proof: First note \(\mathcal{Z}\) is closed if and only if any convergent sequence \((f_{n})_{n=1}^{\infty}\text{in}\mathcal{Z}\) converges to some \(f\in\mathcal{Z}\). For the sake of contradiction, suppose there exists \((f_{n})_{n=1}^{\infty}\in\mathcal{Z}\) converges to \(f\not\in \mathcal{Z}\). Then there exists \(t_{0}\in[0,T]\) such that there exists a measurable set \(\tilde{I}\subset I\text{with}m(\tilde{I})>0,\text{that}f(t_{0})(x)>\bar{A}+\epsilon_{0}\text{or}f(t_{0})(x)<-\bar{A}-\epsilon_{0}\text{for some}\epsilon_{0}>0,x\in\tilde{I}\). Here \(m\) is a measure equipped on \(I.\text{Then}\forall n\in\mathbb{N}\), + +\[\begin{aligned} \int_{I}|f_{n}(t_{0})(x)-f(t_{0})(x)|dx&\geq\int_{\tilde{I}}|f_{n}(t_{0})(x)-f(t_{0})(x)|dx>\epsilon_{0}\cdot m(\tilde{I}) \\ \text{nd thus completes the proof}. \end{aligned}\] diff --git a/outputs/arxiv_math/2503.09115_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.09115_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..0bec3a9f77855edb839ac12b488572d1c8a28dff --- /dev/null +++ b/outputs/arxiv_math/2503.09115_pg3_pg1_repeat1.md @@ -0,0 +1,19 @@ +## Related work. + +Visibility graphs of polygons (and other geometric creatures) have been studied extensively both from a combinatorial and a computational point of view, see, e.g., [8, 10, 9, 13, 25]. It is worth mentioning a recent result of Davies, Krawczyk, McCarty and Walczak [3] who proved that curve pseudo-visibility graphs are \(\chi-bounded\), that is, their chromatic number can be bounded from above in terms of their clique number. If the maximum size of any \(K_{t,t}-\text{free}n-\)vertex graph in a certain family of graphs is at most \(c_{t}n,\text{where}c_{t}\) depends only on \(t\), then this family of graphs is called degree-bounded.4 By a remarkable result of Gir˜ao and Hunter [11], if a hereditary family of graphs is degree-bounded, then \(c_{t}\) can be bounded by a polynomial function. For a further discussion on degree-boundedness see the recent survey of Du and McCarty [4]. + +## Organization. + +The bounds for curve pseudo-visibility graphs are proved in Section 3. In Section 4 we prove the bounds for polygon visibility graphs including the special cases of star-shaped and \(x-\)monotone polygons. + +## 2 Terminology and tools + +An ordered graph is a graph whose vertices are linearly ordered. In a cyclically-ordered graph \(G=(V,E,<)\) the vertices are ordered cyclically. For any vertex \(v\text{in}G\) we denote by \(G_{v}=(V,E,<_{v})\) the ordered graph such that \(<_{v}\) is the linear order induced by < when setting \(v\) as the smallest element. Let \(G=(V,E,<)\) be an ordered graph and let \(A,B\subseteq V\) be two vertex subsets. We write \(AN_{G^{\prime}}(v)\) for every vertex \(v\in R\text{and}|E^{\prime}|\geq|E|/4\). + +Proof. For every vertex \(v\) we toss a fair coin. If the result is ‘heads’, then we place \(v\text{in}L\) and delete all the edges \((u,v)\text{such that}uv\). Clearly the resulting subgraph \(G^{\prime}\) is bipartite, \(vN_{G^{\prime}}(v)\) for every vertex \(v\in R\). Since every edge of \(G\) survives with probability \(1/4\), the expected number of edges in \(G^{\prime}\) is \(|E|/4\). Therefore, there is a series of coin tosses that yields at least that many edges. + +For points lying on a Jordan curve \(K\) we denote by \(<_{K}\) their clockwise cyclic order along \(K\). The clockwise sub-curve from a point \(p\in K\) to a point \(q\in K\) is denoted by \(p❀_{K}q\). Any curve pseudo-visibility graph \(G_{\mathcal{L}}(K,V)\) can naturally be considered as a cyclically-ordered graph where \(<_{K}\) is the cyclic order of its vertices. Furthermore, edges cross in the sense of cyclically-ordered graphs if and only if they cross in the geometric sense in these graphs. This fact is trivial for polygon and curve visibility graphs, and is not hard to prove for curve pseudo-visibility graphs, see [3, Lemma 2.3]. + +Let \(u\text{and}v\) be two vertices of an ordered graph \(G\text{such that}u0\), there exists \(\delta>0\) such that for any sufficiently large prime number \(p\), there exist \(A,B\subseteq\mathbb{F}_{p}\text{with}|A|+|B|>(1+\delta)p-O(1),|B|\leqslant\varepsilon p\), and a relation \(\mathcal{R}\subseteq A\times B\) with maximum degree 1, such that \(|A+_{\mathcal{R}}B|=p-3\). Plugging \(\ell=\lfloor \frac{p-k-1}{2k-1} \rfloor\text{and}\ell=1\) into Theorem 1.11(i), we have the following. + +## Corollary 1.12. Suppose \(p\) is a prime number. + +(i) For any integer \(k\geqslant1\), there exist subsets \(A,B\subseteq\mathbb{F}_{p}\) and a function \(R:B\rightarrow A\) such that \(|A|=p-(k-1)\lfloor \frac{p-k-1}{2k-1} \rfloor-k+1,|B|=k\lfloor \frac{p-k-1}{2k-1} \rfloor+2\text{and}|A+_{\mathcal{R}}B|=p-k\). + +(ii) For any \(\varepsilon>0\), there exist \(A,B\subseteq\mathbb{F}_{p}\) and a function \(R:B\rightarrow A\text{such that}|A|=(1-2\varepsilon)p+O(1)\), + +\(|B|=\varepsilon p+O(1)\text{and}|A+_{\mathcal{R}}B|=|A|+|B|-4\), where the value of the \(O(1)\) term within the expressions of \(|A|\text{and}|B|\) are smaller than 3. + +Thus, for potential extensions of Corollary 1.8(i), in the regime where \(|A|+|B|\geqslant p\), it is necessary to assume at least \(|A|+|B|\geqslant p-k+\lfloor \frac{p-k-1}{2k-1} \rfloor+4> \frac{2k}{2k-1} p-k+2\) in order to establish \(|A+_{\mathcal{R}}B|\geqslant p-k+1\). In the regime \(|A|+|B|\leqslant(1-\varepsilon)p\), an additional assumption \(|B|\leqslant\varepsilon p\) is required. Combining Corollary 1.8(i) and the examples above, we propose the following conjecture. The main obstacle in proving this conjecture lies in the fact that, in Theorem 1.6, the parameter \(c_{\varepsilon}\) is not linear in \(\varepsilon\). So it remains unclear how to prove \(|A+_{\mathcal{R}}B|\geqslant|A|+|B|-3\) under assumptions such as \(|A|+c|B|0,\) and \(c\in\mathbb{C}^{*}\). Make change of variables in \(A(t)\) giving \(\tilde{A}(t)\) by replacing each \(t^{\beta}\) by \(\exp(-\beta\gamma^{-1}\sigma)t^{\alpha\beta\gamma^{-1}},\text{where}\sigma=\log(c)\). This comes from the necessity to obtain the image of the term \(ct^{\gamma}\) in the desired form as \(t^{\alpha}\), and it is an automorphism of \(\mathbb{K}\) fixing \(\mathbb{C}.\text{Then}\tilde{A}(t)=t^{\alpha}B+o(t^{\alpha})\in V(\mathbb{K})\cap PSL_{2}(\mathbb{K})\) and \(\text{VAL}(\tilde{A}(t))=(\alpha,[B]_{\mathbb{R}^{*}})\). + +- + +## 3. Examples + +Example 1. We describe the image under VAL of a particular line \(L\) tangent to the quadric \(Q(\mathbb{K}).\) Consider \(L\subset\mathbb{K}P^{3}\) parametrized by + +\[\mathbb{K}P^{1}\ni[z:w]\longmapsto Z(t)= \begin{bmatrix} tw & z \\ 0 & t^{-1}w \end{bmatrix} \in \overline{\text{PSL}_{2}\mathbb{K}} =\mathbb{K}P^{3},\] + +\[\text{is}B_{\infty}= \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \text{corresponding to}w=0.\] + +For all other values of \([z:1]\in\mathbb{K}P^{1}\text{and}z=ct^{\gamma}+o(t^{\gamma}),c\in\mathbb{C}^{*},\) we have three cases: \(\gamma>1,\gamma=1\text{and}\gamma<1\). Look for the limit at \(\infty\text{for}z\) represented by decreasing exponents. In the first case, the dominant term of \(Z(t)\) is on the cone picture of \(\overline{PSL}_{2} \mathbb{C}=\mathbb{C}P^{3}\), it is a ray over \(B_{\infty}\), each point with an \(S^{1}\) in the circle bundle \(\mathcal{S}\). + +\[\begin{bmatrix} 0 & ct^{\gamma} \\ 0 & 0 \end{bmatrix}\] + +. Projected by VAL + +\[\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \text{nt term}_{\text{an}}\text{is}_{d} \begin{bmatrix} t & ct \\ 0 & 0 \end{bmatrix} \bigg[0_{0}1_{0}\] + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
1001
bundle over Q(C) with pointsandexcluded.
0000
diff --git a/outputs/arxiv_math/2503.09135_pg6_pg1_repeat1.md b/outputs/arxiv_math/2503.09135_pg6_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..94b7be72d6bca9e78ee80d25d55830a066dffc97 --- /dev/null +++ b/outputs/arxiv_math/2503.09135_pg6_pg1_repeat1.md @@ -0,0 +1,43 @@ +\[\begin{aligned} [\rho(t_{1})]&=\rho(t_{1})+3\sigma_{\rho(t_{1})}\cdot\delta\rho(t_{1}) \\ [\bar{Az}(t_{1})]&=\bar{Az}(t_{1})+CI_{\bar{Az}(t_{1})}\cdot\delta\bar{Az}(t_{1}) \\ [[_{\rho}\bar{El}_{(t}(_{3}t_{)^{1}]})_{=}]&=_{\rho(}\bar{El}_{t_{3}}(_{)}t_{1})+C_{+}I_{3^{\bar{El}}\sigma_{\rho(t_{3}}^{(t_{1})}}\cdot_{)}\delta_{\cdot}\bar{El}_{\delta\rho(}(_{t}t_{)_{3}^{1}}) \\ [\bar{Az}(t_{3})]&=\bar{Az}(t_{3})+CI_{\bar{Az}(t_{3})}\cdot\delta\bar{Az}(t_{3}) \\ [\bar{El}(t_{3})]&=\bar{El}(t_{3})+CI_{\bar{El}(t_{3})}\cdot\delta\bar{El}(t_{3}) \end{aligned}\] (5) + +where \(CI\) indicates the confidence interval for the measurements or search space for the optimization variables. We then perform steps 1–7 of Algorithm 1 with ADS. This operation produces + +\[\Delta^{T}\Delta=\bigcup_{i=1}^{N_{s}}\mathcal{T}_{\Delta^{T}\Delta}^{i}(\boldsymbol{x})\] (6) + +with \(\boldsymbol{x}=(\rho(t_{1}),\bar{Az}(t_{1}),\bar{El}(t_{1}),\rho(t_{3}),\bar{Az}(t_{3}),\bar{El}(t_{3}))\text{and}N_{s}\) the number of domains generated by ADS. A minimization of the residual squares is then performed on each domain. Only those for which \(\Delta^{T}\Delta\) is zero or below a certain thresholds are retained. These domains describe the set of spacecraft states compatible with the available observations, which can propagated forward in time to predict future detections. + +### 3.2 Accurate angular measurements on one tracklet + +The second case considers the availability of accurate angular measurements on one of the two tracklets. For illustration, we consider these measurements together with the range and range-rate to be available on the first tracklet at \(t_{1}\). Range and range-rate are also available for the second trackelet, at \(t_{2}\). Concerning the angular data and their rates, we work with the regressed values and their rate. The procedure is summarized in Algorithm 2. + +## Algorithm 2: IOD with accurate angular measurements on a single tracklet + +\[\bar{Az}^{\mathrm{r}_{˙}\bar{Az}}(^{˙}t_{1}^{(})^{t},^{)_{1}}\bar{El}^{,_{˙}\bar{E}}(^{˙}t\] + +Input: Initial guesses for ¯˙ Output: The values of ¯˙ + +\(El(t_{1}) El(t_{1})\) that solve \(\Delta=\mathbf{0}\) + +1. Guess initial values for ¯˙ + +\(Az(t_{1})\), ¯˙ + +\(El(t_{1})\); + +2. Take observations \(\rho(t_{1}),\dot{\rho}(t_{1}),\bar{Az}(t_{1}),\bar{El}(t_{1})\); + +3. Compute the Cartesian state: \(Az(t_{1}),\) ¯˙ \(Az(t_{1}),\) ¯˙ + +\(\boldsymbol{r}(t_{1})(\rho(t_{1}),\rho(t_{1}),\bar{Az}(t_{1}),\bar{El}(t_{1}),\) ¯˙ \(\boldsymbol{v}(t_{1})(\rho(t_{1}),\dot{\rho}(t_{1}),\bar{Az}(t_{1}),\bar{El}(t_{1}),\) ¯˙ + +\(El(t_{1})) El(t_{1}))\); + +4. Propagate to the the state forward in time: + +\[\boldsymbol{r}(t_{2})(\rho(t_{1}),\dot{\rho}(t_{1}),\bar{Az}(t_{1}),\bar{El}(t_{1}),\bar{Az}^{˙}(t_{1}),\bar{El}^{˙}(t_{1}))\] + +\[\boldsymbol{v}(t_{2})(\rho(t_{1}),\dot{\rho}(t_{1}),\bar{Az}(t_{1}),\bar{El}(t_{1}),\bar{Az}^{˙}(t_{1}),\bar{El}^{˙}(t_{1}));\] + +\[\Delta(\rho(t_{1}),\dot{\rho}(t_{1}),\bar{Az}(t_{1}),\bar{El}(t_{1}),\bar{Az}^{˙}(t_{1},\bar{El}^{˙}(t_{1})))= \begin{pmatrix} \rho^{c}(t_{2})-\rho(t_{2}) \\ \dot{\rho}^{c}(t_{2})-\dot{\rho}(t_{2}) \end{pmatrix} ;\] + +\[6.\text{Compute}\bar{Az}^{˙}(t_{1}),\bar{El}^{˙}(t_{1})\text{such that}\Delta=\mathbf{0}\] diff --git a/outputs/arxiv_math/2503.09155_pg16_pg1_repeat1.md b/outputs/arxiv_math/2503.09155_pg16_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..ac2863ffb8943c4e5b3f8187947cc4c061f48b86 --- /dev/null +++ b/outputs/arxiv_math/2503.09155_pg16_pg1_repeat1.md @@ -0,0 +1,25 @@ +of system (29) we can see that \(e_{n}\) is a real and positive root of the polynomial + +\[Q(s):=\alpha s^{m+1}+\alpha s-1,\] (30) + +and, in view of Descartes’ rule of signs, there is a unique such \(e_{n}\). Then + +\[e_{j}=\left(_{k}\prod_{\geq j+1}\alpha_{k}\right)e_{n},\text{for all}j\in[n-1],\] (31) + +and hence \(e\in\text{int}(\mathcal{B}_{G})\) is unique. Several studies (see e.g. Sanchez (2009b) and the references therein) derived conditions guaranteeing that the equilibrium \(e\in\text{int}(\mathcal{B}_{G})\) is globally asymptotically stable. Tyson (1975) analyzed the special case of (29) with \(n=3\). He noted that if \(e\) is locally asymptotically stable, then one may expect that all solutions converge to \(e\), and proved that system (29) admits a periodic solution whenever \(e\) is unstable. For \(n=3\), the model can also be studied using the theory of competitive dynamical systems (Smith, 1995). The case \(n=3\) has also been analyzed using the theory of Hopf bifurcations (Woller et al., 2014). For a general \(n\), the analysis using Hopf bifurcations becomes highly non-trivial and results exist only for special cases, e. g. under the additional assumption that all the \(\alpha_{i}’\mathrm{s}\) are equal, see Invernizzi and Treu (1991). Hastings et al. (1977) studied the general \(n-\)dimensional case and proved that, if the Jacobian of the vector field at the equilibrium has no repeated eigenvalues and at least one eigenvalue with a positive real part, then the system admits a non-trivial periodic orbit; the proof relies on the Brouwer fixed point theorem. + +Our Theorem 2 allows us to prove the following result. + +Corollary 3 Consider the \(n-\)dimensional Goodwin model (29) with \(n\geq3\), and let \(e\) denote the + +unique equilibrium in \(\text{int}(\mathcal{B}_{G})\). Let \(J:\mathbb{R}_{\geq0}^{n}\rightarrow\mathbb{R}^{n\times n}\) denote the Jacobian of the vector field of the Goodwin model. Suppose that \(J(e)\) has at least one eigenvalue with a positive real part. Then, for + +any initial condition \(a\in\mathbb{R}_{\geq0}^{n}\setminus\{e\}\text{such that}s^{-}(a-e)\leq1\), the solution \(x(t,a)\) of (29) converges to a (non-trivial) periodic orbit as \(t\rightarrow\infty\). + +# PROOF. + +The Jacobian of (29) + +\[J(x)= \begin{bmatrix} -\alpha_{1} & 0 & 0 & \ldots & 0 & - \frac{mx_{n}^{m-1}}{(1+x_{n}^{m})^{2}} \\ 1 & -\alpha_{2} & 0 & \ldots & 0 & 0 \\ 0 & 1 & -\alpha_{3}\ldots & 0 & 0 \\ \ddots & \ddots & \ddots & \ddots & \ddots & \ddots \\ 0 & 0 & 0 & \ldots-\alpha_{n-1} & 0 \\ 0 & 0 & 0 & \ldots & 1 & -\alpha_{n} \end{bmatrix}\] + +has the sign pattern \(\bar{A}_{2}\) in (3) for all \(x\in\mathbb{R}_{\geq0}^{n}\), hence the system is 2-cooperative on \(\mathbb{R}_{\geq0}^{n}\). We diff --git a/outputs/arxiv_math/2503.09157_pg22_pg1_repeat1.md b/outputs/arxiv_math/2503.09157_pg22_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..2732b552f84d302c070c7511166f827de9f32ca7 --- /dev/null +++ b/outputs/arxiv_math/2503.09157_pg22_pg1_repeat1.md @@ -0,0 +1,45 @@ +So + +\[\Vert\partial_{I}\Phi(n,I)\cdot h\Vert_{\infty}\geq\Vert h\Vert_{\infty}.\] (43) + +Therefore, \(\partial_{I}\Phi(n,I)\) is a global diffeomorphism on \(C^{0}([0,T],\mathbb{R})\) and according to (43) + +\[\Big\|\left(\partial_{I}\Phi(n,I)\right)^{-1}\Big\|\leq1.\] + +Hence, according to the implicit function Theorem, we can find a function \(F\in C^{1}\left(X,C^{0}([0,T],\mathbb{R}^{+})\right)\) such that + +\[\forall n\in X,\Phi(n,F(n))=0.\] + +i.e. + +\[\forall t\in[0,T],F(n)(t)=\int_{0}^{\infty}r(F(n)(t),x)n(t,x)dx.(44)\text{eq}:\text{fonctio}\] + +Finally + +\[\Vert\partial_{n}F(n)\Vert\leq\Big\|\left(\partial_{I}\Phi(n,I)\right)^{-1}\Big\|\Big\|\partial_{n}\Phi(n,I)\Big\|\leq1\times r_{M}.\] + +Hence, \(F\) is \(r_{M}-\text{lipschitzian on}X\). Now, if \(\mu_{1},\mu_{2}\) are two solutions of (26), associated to the activity function \(I_{1},I_{2}\in C^{0}([0,T],\mathbb{R})\text{with}\mu_{1}(0)=\mu_{2}(0)=n_{\text{ini}}\), then, according to Lemma 6.2, + +\[\forall t\in[0,T],\Vert\mu_{1}(t)-\mu_{2}(t)\Vert_{1}\leq\Vert\mu_{1}(0)-\mu_{2}(0)\Vert_{1}+2\Vert\partial_{I}r\Vert_{\infty}\int_{0}^{t}\Vert I_{1}(\tau)-I_{2}(\tau)\Vert_{\infty}d\tau\] + +Then, since \(\mu_{1}(0)=\mu_{2}(0)=n_{\text{ini}}\), we obtain + +\[\forall t\in[0,T],\Vert\mu_{1}(t)-\mu_{2}(t)\Vert_{1}\leq2T\Vert\partial_{I}r\Vert_{\infty}\Vert I_{1}-I_{2}\Vert_{\infty}.\] + +And so + +\[\Vert\mu_{1}-\mu_{2}\Vert_{X}\leq2T\Vert\partial_{I}r\Vert_{\infty}\Vert I_{1}-I_{2}\Vert_{\infty}.\] + +Finally, if \(n_{1},n_{2}\in X,\text{and}\mu(n_{1}),\mu(n_{2})\in X\) are the two solutions associated to the activity functions \(I_{1}=F(n_{1})\text{and}I_{2}=F(n_{2})\text{and with}\mu_{1}(0)=\mu_{2}(0)=n_{\text{ini}}\), then, + +\[\begin{aligned} \Vert\mu(n_{1})-\mu(n_{2})\Vert_{X}&\leq2T\Vert\partial_{I}r\Vert_{\infty}\Vert I_{1}-I_{2}\Vert_{\infty}. \\ &\leq2T\Vert\partial_{I}r\Vert_{\infty}r_{M}\Vert n_{1}-n_{2}\Vert_{X} \end{aligned}\] + +Now, if we take \(T>0\text{such that}2T\Vert\partial_{I}r\Vert_{\infty}r_{M}<1\), then the function \(n\mapsto\mu(n)\) is a contraction on \(X\) and therefore, according to Picard Fixed point Theorem, there exist a unique function on \(X\text{suchthat}\mu(n)=n\). This function is the solution of (1) on \([0,T]\). + +Finally, we use the same argument on every interval \([kT,(k+1)T],\text{with}k\in\mathbb{N}\) to prove that the solution \(n\) is uniquely and well-defined on \(\mathbb{R}^{+}\). + +apport a r + +Lemma 6.2. Let \(\mu_{1}\text{and}\mu_{2}\) be the solution of (26) with respectively \(r_{1}(t,x):=r(x,I_{1}(t))\) and \(r_{2}(t,x)=r(x,I_{2}(t)),\mu_{1}(0)=\mu_{1}^{0}\text{and}\mu_{2}(0)=\mu_{2}^{0}\). Then, + +\[\forall t\geq0,\Vert\mu_{1}(t)-\mu_{2}(t)\Vert_{1}\leq\Vert\mu_{1}^{0}-\mu_{2}^{0}\Vert_{1}+2\Vert\partial_{I}r\Vert_{\infty}\int_{0}^{t}\Vert I_{1}(\tau)-I_{2}(\tau)\Vert_{\infty}d\tau.\] diff --git a/outputs/arxiv_math/2503.09174_pg2_pg1_repeat1.md b/outputs/arxiv_math/2503.09174_pg2_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..336fd04d68fdc2f1c95fefee3ab47cf3c4f9a05c --- /dev/null +++ b/outputs/arxiv_math/2503.09174_pg2_pg1_repeat1.md @@ -0,0 +1,3 @@ +under arbitrary scattering conditions. The main result is that the number of DoF per unit area is equal to the number of Nyquist samples per square meter to be able to reconstruct the field within a specified accuracy. The authors of [28] used the Fourier plane-wave series expansion for electromagnetic fields to evaluate the number of additional DoF in the reactive near-field, which is determined by the evanescent waves. The author of [29] introduced a general approach to evaluate the number of effective communication modes and their strength between two volumes. The solution is cast in terms of an eigenproblem obtained from the wave equation and Green’s function. An interesting result is that the number of DoF between two parallel communicating surfaces of area \(A_{T}\) and \(A_{R}\), located at a distance \(r\), is equal to \(A_{T}A_{R}/(\lambda^{2}r^{2})\), provided that the two surfaces are in the paraxial regime. Under the paraxial setting, to elaborate, \(r\) is such that the Fresnel approximation can be applied, that is, only the first two terms of the Taylor series expansion of \(r\) are retained, and the surface apertures are small compared to the propagation distance. Following a similar methodology, the authors of [30] derived a closed form expression for the DoF of LoS channels as a function of the orientations of the arrays, while still assuming the paraxial setting. Naturally, analysis of the DoF requires, especially in free-space channels, careful consideration of the transmission distance, the size of the surfaces, and the relative geometry of the surfaces. Thus, when the paraxial approximation, which is often assumed in the literature, is not fulfilled, that is, the transmission distance is comparable with the sizes of the surfaces, the calculation of the DoF is not straightforward, and alternative methods are required. More specifically, besides the numerical solution of the eigenproblem formulated in, e.g., [29], the DoF can be obtained by applying two general methods, which are known as (1) the spatial bandwidth [31]-[34] and (2) Landau’s methods [35]- [37], with the latter being highly coupled with the eigenproblem in [29]. Even though these methods are quite general and can be applied to general network scenarios, including non-paraxial settings, explicit analytical expressions for the DoF are difficult to obtain in general network deployments, e.g., in non-paraxial settings. Approximation approaches based on the spatial bandwidth and Landau’s eigenproblem, which can be applied to nonparaxial settings, can be found in [12] and [37], respectively. The impact of spatial blocking on the DoF has recently been analyzed in [38]. A short overview based on [4] and applied to continuous linear arrays can be found in [39]. To overcome these limitations, some ad hoc methods have recently been proposed. For example, the authors of [40], have provided closed-form solutions for the DoF between a linear LIS and a linear small intelligent surface (SIS), which is of practical interest since the considered setting models the link between a base station and a mobile terminal. While the approach in [40] is a key step towards understanding non-paraxial settings, a limited geometric setup is considered, and the solutions are restricted to specific conditions. In the present paper, we are mainly interested in linear continuous arrays. In this context, the authors of [41] have recently introduced an analytical approach to compute the DoF based on the spatial bandwidth method. A crucial assumption in [41] is that the considered linear arrays are composed of an infinite number of ideal point sources and that the transmitting linear array is characterized by an azimuth-independent radiation pattern. Under these assumptions, it is sufficient to parameterize the relative direction of the receiving linear array by using a single polar angle. This allows for a simplified analysis, since the transmitting and receiving linear arrays are always in the visibility region of one another. The same authors generalize the analysis in [42], by introducing analytical asymptotic expressions for spatial bandwidth, highlighting the importance of the network geometry on the DoF. Although DoF evaluation has been investigated in conventional geometric settings, especially in paraxial setup, a comprehensive performance analysis remains elusive, especially from a system-level (or statistical) point of view [43]. Using fundamental results from stochastic geometry theory, new spatial models and channel statistics can be conceptualized to capture the peculiarities of near-field communications. This will allow for the characterization of location-dependent DoF in the receiver’s association policy, with major applications in network performance analysis. In the present work, a new analytical framework for calculating the DoF of a communication link between a large continuous linear array, and a small intelligent line, still modeled as a continuous linear array, is introduced. The proposed approach can be applied to paraxial and non-paraxial settings. To this end, the arrays are modeled as electromagnetic lines in free space, obeying the Huygens-Fresnel principle. The two arrays are assumed to radiate only in one of the two half-planes, in order to consider realistic deployments in which the transmitting array radiates only towards the intended receiving array. The transmitting array uses a steering function to concentrate the energy towards the direction of a focal point on the receiving array in the far-field. The calculation of the DoF is based on the calculation of the number of orthogonal steering functions that fit within the length of the receiving array. This approach allows us to account for near-field channels and non-paraxial settings. Therefore, the effective lengths of the two mutually visible arrays play an important role in the calculation of the DoF, since the mutual visibility of the arrays also determines the steering functionality. Another factor that determines the DoF is the projected lengths of the two linear arrays onto the line that is perpendicular to the line that connects the centers of the two arrays. These projections are highly dependent on the positions and orientations of the arrays. To elaborate, the contributions of this paper are summarized as follows: + +- For a pair of linear arrays lying on the same plane, we provide a methodology to characterize the complete or partial visibility conditions between the arrays, without any restrictions on their relative positions and orientations. The considered unconstrained geometric setting, in which the lines can transmit only in a half hemisphere, can, in fact, lead to the lack of visibility or to a partial visibility condition. In the latter case, the effective length diff --git a/outputs/arxiv_math/2503.09178_pg16_pg1_repeat1.md b/outputs/arxiv_math/2503.09178_pg16_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..a67d48c70aa93e300d9621995baff76a49eb2923 --- /dev/null +++ b/outputs/arxiv_math/2503.09178_pg16_pg1_repeat1.md @@ -0,0 +1,67 @@ +polynomial degree \(N\) at \(M=11\), while Figure 3(b) displays the \(L^{2}-\)errors with respect to the angular discretization number \(M+1\) at \(N=30\). + +100 + +10-5 + +10-10 + +10-15 + +3 + +4 + +5 + +6 + +7 + +8 + +![](figure) + +(a) Numerical errors \(\Vert u-u_{N}^{M}\Vert_{L^{2}}\) vs. \(N(M=11)\). + +10-5 + +10-10 + +10-15 + +2 + +4 + +6 + +8 + +10 + +12 + +14 + +(b) Numerical errors \(\Vert u-u_{N}^{M}\Vert_{L^{2}}\) vs. \(M+1(N=30)\). + +Figure 3: The \(L^{2}-\)errors versus \(N\text{and}M+1\) of Example 1 using the spectral method. + +From Figure 3(a), it is evident that the proposed method exhibits spectral accuracy concerning the polynomial degree \(N\). As the degree \(N\) increases to 6, the \(L^{2}-\)errors approach the machine epsilon, indicating the convergence of the numerical solution to the exact solution. Similarly, in Figure 3(b), it can be observed that when the angular discretization number \(M+1\) is small (e.g., \(M=1)\), the \(L^{2}-\)errors also approach the machine epsilon. This occurs because the exact solution is a sixth-degree polynomial solely in the spatial variable \(x\) and is entirely independent of the angular variable \(\mu\). As a result, selecting \(N=6\text{and}M=1\) enables the \(L^{2}-\)error to reach machine epsilon. Furthermore, by comparing Figure 2(b) and Figure 3(a), it is evident that for problems with a higher regularity of the exact solution, the spectral method requires fewer degrees of freedom compared to the Hermite WENO fast sweeping method to achieve the same level of accuracy. + +### 5.2 Example 2 + +In this test, we solve the absorbing-scattering transfer problem described by the equation (1.1) with + +\[\Sigma_{t}=22000,\Sigma_{s}=1,\] + +\[s(x,\mu)=-4\pi\mu^{3}\cos^{3}\pi x\sin\pi x+\Sigma_{t}(\mu^{2}\cos^{4}\pi x+const)-\Sigma_{s}(const+ \frac{\cos^{4}\pi x}{3} ).\] + +Here \(const=10^{-14}\) is a small positive constant which is used to ensure the source term to be nonnegative. The computational domain is \(D=(0,1)\). The boundary condition is given as follows + +\[\begin{cases} \varphi(0,\mu)=\mu^{2}+const, & \text{if}\mu>0, \\ \varphi(1,\mu)=\mu^{2}+const, & \text{if}\mu<0. \end{cases}\] + +For this problem, we have the exact solution given as [25] + +\[\varphi(x,\mu)=\mu^{2}\cos^{4}\pi x+const.\] diff --git a/outputs/arxiv_math/2503.09181_pg5_pg1_repeat1.md b/outputs/arxiv_math/2503.09181_pg5_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..07adfa2decbffa437bc5949d334cac254bc0757e --- /dev/null +++ b/outputs/arxiv_math/2503.09181_pg5_pg1_repeat1.md @@ -0,0 +1,13 @@ +\(x_{s}^{(n)}=\{x_{\lambda}:\lambda\in s^{(n)}\}\subset x^{(n)}\) be the subset of features, which represents the selected and revealed features. Our task is to select features \(s^{(n)}\) and make a final prediction about \(y^{(n)}\) while \(s^{(n)}\) is different for each instance. Our purpose is to find two functions, implemented as neural networks; the policy to select features and the predictor to make predictions. The policy takes the revealed features \(x_{s}^{(n)}\) and the whole features of features \(z^{(n)}\) as input. Then, it returns one index and the corresponding feature will be selected and revealed. We denote the policy as \(\pi(x_{s}^{(n)},z^{(n)})\in\Lambda^{(n)}\). The predictor also takes \(x_{s}^{(n)}\) and \(z^{(n)}\) as input and it returns the prediction about \(y^{(n)}\). We denote the prediction and the predictor as \(\hat{y}^{(n)}=f(x_{s}^{(n)},z^{(n)})\) We innovate a way to select features one by one with the policy until the predetermined number of features is selected. Firstly, we determine the budget \(k\), which is the number of selected features in the end. In the initial state, we suppose \(s^{(n)}=\phi\text{and}x_{s}^{(n)}=\phi\), that is, any features has not revealed. Note that, we assume that features of features \(z\) are fully revealed in the initial state and they are given as prior information. This assumption is natural because we consider features of features as the description or the property of each feature on the natural language or numerical expression and they are usually given before the feature values are revealed. Then, we select a feature with the policy \(\pi(x_{s},z)\) and reveal the selected feature. (In other words, \(s\leftarrow s\cup\pi(x_{s},z)\) and the size of \(x_{s}\) increases by one.) We repeat this process for \(k\) times and make the final prediction with the predictor \(f(x_{s},z)\) from \(|x_{s}|=k\) features. This procedure is shown in Fig. 1b and 1c. Our ultimate goal is to minimize the loss of the prediction made by the predictor after selecting features by the policy. Given the loss function \(l(\hat{y},y)\), the objective function of our problem is + +\[\min_{\theta,\phi}\mathbb{E}_{\mathbf{y},\mathbf{x}}l(f_{\theta}(\mathbf{x}_{s},z),\mathbf{y}),\] (1) + +where \(\theta\text{and}\phi\) are parameters for parameterized predictor and policy respectively. In addition, \(s(|s|=k)\) is constructed by the iterative selection of the policy \(\pi_{\phi}(x_{s},z)\). + +## 4 Greedy Dynamic Feature Selection + +This section explains the existing DFS method based on conditional mutual information (CMI) [3]. Note that this method handles the special case of our problem definition that the feature set is fixed. Namely, \(\Lambda=[d]\text{and}\Lambda^{(n)}=\Lambda\) for all \(n=1,...,N\). Therefore, in this section, we note \(x=\{x_{1},...,x_{d}\}\) and \(x_{s}=\{x_{i}:i\in s\subset[d]\}\). Also, the policy \(\pi(x_{s})\) and the predictor \(f(x_{s})\) take only revealed features \(x_{s}\) as input. + +### 4.1 Greedy Dynamic Feature Selection + +This method aims to find a greedy policy based on CMI and the predictor based on the naive Bayesian, implemented as neural networks such as MLP. Concretely, diff --git a/outputs/arxiv_math/2503.09182_pg4_pg1_repeat1.md b/outputs/arxiv_math/2503.09182_pg4_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..25617f3b990f93cbb235def6ed662274a50e807e --- /dev/null +++ b/outputs/arxiv_math/2503.09182_pg4_pg1_repeat1.md @@ -0,0 +1,35 @@ +is of \(\mu-\)measure zero. (Here the key observation is that, since \(\Vert\lambda a(\omega)+\nu b(\omega)\Vert= \Vert\lambda a(\omega)\Vert+\Vert\nu b(\omega)\Vert\) for a.e. \(\omega\in\Omega\), one has \(\Vert\alpha a(\omega)+\beta b(\omega)\Vert=\alpha\Vert a(\omega)\Vert+\beta\Vert b(\omega)\Vert\) whenever \(\alpha,\beta\geq0\) for a.e. \(\omega\in\Omega.)\) Redefining the values of \(a,b,x,y,\text{and}z\text{on}H\) to become 0, the redefined \(a,b,x,y,\text{and}z\) are the same functions in \(L_{1}(\mu,X)\) as the original ones. The existence of the desired \(W\) now follows from \((♯)\). + +- + +The following two lemmata may be known; however, we do not know of any references for them. + +Lemma 2.2. Let \(\mu\) be a non-zero finite (countably additive non-negative) measure on a \(\sigma-\text{algebra}\Sigma\) of subsets of a set \(\Omega\), and let \(X\) be a Banach space. Let \(z\in S_{L_{1}(\mu,X)}\), let \(E\in\Sigma\), and let \(0<\varepsilon<1\). + +(a) If \(z(\omega)\neq 0\text{for every}\omega\in E\), then there are \(n\in\mathbb{N}\) and pairwise disjoint measurable subsets \(E_{0},E_{1},\ldots,E_{n}\) of \(E\text{with}\bigcup_{i=0}^{n}E_{i}=E\) such that + + \((1)\mu(E_{0})<\varepsilon\); + + \((2)\big\|\int_{E_{i}}zd\mu\big\|>(1-\varepsilon)\int_{E_{i}}\Vert z\Vert d\mu\text{for every}i\in\{1,\ldots,n\}\text{with}\mu(E_{i})>0\); + + 3. for every \(i\in\{1,\ldots,n\}\text{with}\mu(E_{i})>0\), there is a neighbourhood \(W_{i}\) of \(z\) in the relative weak topology of \(B_{L_{1}(\mu,X)}\) such that + +\[\bigg\|\int_{E_{i}}wd\mu\bigg\|>(1-\varepsilon)\int_{E_{i}}\Vert z\Vert d\mu\text{for every}w\in W_{i};\] (2.1) + +this neighbourhood can be chosen to be of the form \(W_{i}:=\big\{w\in\) + +\[B_{L_{1}(\mu,X)}:\] + +\(\big|\int_{E_{i}}w_{i}^{*}(w-z)d\mu\big|<\delta_{i}\big\}\text{for some}w_{i}^{*}\in S_{X^{*}}\text{and}\delta_{i}>0\). + +(b) There is a neighbourhood \(W\) of \(z\) in the relative weak topology of \(B_{L_{1}(\mu,X)}\) such that + +\[(2.2)\int_{E}\Vert z\Vert d\mu-\varepsilon<\int_{E}\Vert w\Vert d\mu<\int_{E}\Vert z\Vert d\mu+\varepsilon\text{for every}w\in W.\] + +Proof. (a). Assume that \(z(\omega)\neq 0\text{for every}\omega\in E\). Choose a real number \(\sigma>0\) so that \(\mu(E\setminus D)< \frac{\varepsilon}{2} \text{where}D:=\{\omega\in\Omega:\Vert z(\omega)\Vert>\sigma\}\). Since the function \(z\) is essentially separably-valued, there is a subset \(N\) of \(D\text{such that}\mu(N)=0\) and the set \(z(D\setminus N)\) is separable. It follows that there are pairwise disjoint measurable subsets \(E_{i}\) of \(E,i=1,2,\ldots\) , such that diam \(z(E_{i})< \frac{\varepsilon\sigma}{2} \text{for every}i\in\mathbb{N}\), and \(D\setminus N=\bigcup_{i=1}^{\infty}E_{i}\). Pick an \(n\in\mathbb{N}\) so that \(\mu(C)< \frac{\varepsilon}{2} \text{where}C:=\bigcup_{i=n+1}^{\infty}E_{i}\), and set \(E_{0}:=(E\setminus D)\cup N\cup C\). Fix an \(i\in\{1,\ldots,n\}\) and suppose that \(\mu(E_{i})>0\). Letting \(z_{i}\in z(E_{i})\) be arbitrary and picking a \(z_{i}^{*}\in S_{X^{*}}\) so that \(\text{Re}z_{i}^{*}(z_{i})=\Vert z_{i}\Vert\), one has + +\[\begin{aligned} \bigg\|\int_{E_{i}}zd\mu\bigg\|&\geq\text{Re}z_{i}^{*}\left(\int_{E_{i}}zd\mu\right)=\int_{E_{i}}\left(\text{Re}z_{i}^{*}(z_{i})+\text{Re}z_{i}^{*}(z-z_{i})\right)d\mu \\ &\geq(\Vert z_{i}\Vert-\Vert z-z_{i}\Vert)d\mu\geq\int_{E_{i}}(\Vert z\Vert-2\Vert z-z_{i}\Vert)d\mu \\ \int_{E_{i}}^{E_{i}} \\ &>(\Vert z\Vert-\varepsilon\sigma)d\mu\geq\int_{E_{i}}(\Vert z\Vert-\varepsilon\Vert z\Vert)d\mu=(1-\varepsilon)\int_{E_{i}}\Vert z\Vert d\mu. \end{aligned}\] + +Now pick a \(w_{i}^{*}\in S_{X^{*}}\) satisfying \(\text{Re}w_{i}^{*}\left(\int_{E_{i}}zd\mu\right)=\big\|\int_{E_{i}}zd\mu\big\|\) and set \(W_{i}:=\big\{w\in\) + +\[B_{L_{1}(\mu,X)}:\big|\int_{E_{i}}w_{i}^{*}(w-z)d\mu\big|<\delta_{i}\big\}\text{where}\delta_{i}:=\big\|\int_{E_{i}}zd\mu\big\|-(1-\varepsilon)\int_{E_{i}}\Vert z\Vert d\mu>0.\] diff --git a/outputs/arxiv_math/2503.09190_pg4_pg1_repeat1.md b/outputs/arxiv_math/2503.09190_pg4_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..50171554fe780a0aa3e9b8e6b6b3d2c2ff7ab4ac --- /dev/null +++ b/outputs/arxiv_math/2503.09190_pg4_pg1_repeat1.md @@ -0,0 +1,65 @@ +Lemma 3.1. Assume that \(T\in\mathcal{T}_{h}\text{and}S\in\mathcal{S}_{h}\) satisfy \(\emptyset\neq T\cap\Gamma_{h}\subset S\). Then we have + +\[\Vert\nabla^{m}(v_{h}-\mathring{I}_{h}v_{h})\Vert_{L^{p}(T)}\leq Ch_{T}^{1/p-m}\Vert v_{h}\Vert_{L^{p}(S)}\forall v_{h}\in V_{h},\] + +where \(m=0,1,\ldots\text{and}1\leq p\leq\infty\). We recall from [3, Theorem 5] standard interpolation error estimates for \(I_{h}\): + +Lemma 3.2. Let \(l,m\in\mathbb{N}\) satisfy \(0\leq l\leq m\leq k+1\text{and}p\in[1,\infty]\). Assume the embedding \(W^{m,p}֒\rightarrow C^{0}\) holds for + +subsets in \(\mathbb{R}^{N}\). Then we have + +\[\Vert v-I_{h}v\Vert_{W^{l,p}(T)}\leq Ch^{m-l}\Vert v\Vert_{W^{m,p}(T)}(T\in\mathcal{T}_{h},v\in W^{m,p}(T)).\] + +Estimates for \(\mathring{I}_{h}\) are, however, more involved because of domain perturbation \((u=0\text{on}\Gamma\) does not necessarily imply \(\tilde{u}=0\text{on}\Gamma_{h})\). We state it in the following form, whose proof is similar to that of Proposition 5.1 below (we only have to consider global \(\Omega_{h}\) and set \(v_{2}=0\) there) and thus omitted here. + +Proposition 3.1. Under the assumptions of Lemma 3.2, let \(m\geq2\text{and}v\in W^{m,p}(\tilde{\Omega})\) satisfy \(v=0on\Gamma\). Then we have + +\[\left(\sum_{T\in\mathcal{T}_{h}}\Vert\nabla^{l}(v-\mathring{I}_{h}v)\Vert_{L^{p}(T)}^{p}\right)^{1/p}\leq Ch^{m-l}\Vert v\Vert_{W^{m,p}(\Omega_{h})}+Ch^{k+1-l}\Vert\nabla^{2}v\Vert_{L^{p}(\Gamma(\delta))},\] + +with the obvious modification for \(p=\infty\). + +## 4. Reduction to \(W^{1,1}-\)analysis of a regularized Green function + +Fixing arbitrary \(K\in\mathcal{T}_{h}\text{and}z\in K\), we try to bound the pointwise error \(\tilde{u}(\boldsymbol{z})-u_{h}(\boldsymbol{z})\). We construct a regularized delta function; the proof is given in the appendix. + +Proposition 4.1. For \(K\in\mathcal{T}_{h}\text{and}\boldsymbol{z}\in K\), there exists \(\eta=\eta_{K,\boldsymbol{z}}\in C_{0}^{\infty}(K)\) such that dist(supp \(\eta,\partial K)\geq Ch_{K}\), + +\[\Vert\nabla^{m}\eta\Vert_{L^{\infty}(K)}\leq Ch_{K}^{-N-m}(m=0,1),\text{and}\] + +\[(v_{h},\eta)_{K}=v_{h}(\boldsymbol{z})\text{for}v_{h}=\hat{v}_{h}\circ\boldsymbol{F}_{K}^{-1}\text{with arbitrary}\hat{v}_{h}\in\mathbb{P}_{k}(\hat{T}),\] + +where the constant \(C\) is independent of \(K,\boldsymbol{z},\text{and}h_{K}\). Next we introduce a “dyadic decomposition” of \(\Omega_{h}\). We set a sequence of scales: + +\[d_{0}=Lh,d_{j}=2^{j}d_{0}\text{for}j=1,\ldots,J:=\left\lceil \frac{\log(\text{diam}\Omega_{h}/d_{0})}{\log2} \right\rceil,\] + +where \(L\) means the ratio of the “initial stride” \(d_{0}\) to the “minimum scale” \(h\). As we see later, \(L\) will be taken sufficiently large (but independently of \(h)\). Then we define a subset \(\Omega_{h,j}\) of \(\Omega_{h}-\)which has the scale \(d_{j}\) in terms of the distance from \(K-\)by + +\[\Omega_{h0}=\bigcup\{T\in\mathcal{T}_{h}|d(T,K)\leq d_{0}\},\] + +\[\Omega_{h,j}=\bigcup\{T\in\mathcal{T}_{h}|d_{j-1}0,\text{for}0\leq\ell_{1}\leq\ell_{2}\leq J\). + +Now let us start the first part of the proof of Theorem 1.1. For any \(v_{h}\in\mathring{V}_{h}\) we use the regularized delta function \(\eta\) constructed in Proposition 4.1 to get + +\[(\tilde{u}-u_{h})(\boldsymbol{z})=(\tilde{u}-v_{h})(\boldsymbol{z})+(v_{h}-\tilde{u},\eta)_{\Omega_{h}}+(\tilde{u}-u_{h},\eta)_{\Omega_{h}}.\] + +The first two terms on the right-hand side are bounded by \(C\Vert\tilde{u}-v_{h}\Vert_{L^{\infty}(K)}\). To address the last term we define a + +regularized Green function \(g\in W^{3,\infty}(\Omega)\) by solving + +\[-\Delta g=\eta\text{in}\Omega,g=0\text{on}\Gamma,\] + +where \(\eta\) is extended by zero outside \(\text{supp}\eta\subset\Omega\) (this inclusion holds if \(h\) is small). We also utilize its finite element + +approximation \(g_{h}\in\mathring{V}_{h}\) obtained by solving + +\[a_{h}(v_{h},g_{h})=(\nabla v_{h},\nabla g_{h})_{\Omega_{h}}=(v_{h},\eta)_{\Omega_{h}}\forall v_{h}\in\mathring{V}_{h}.\] diff --git a/outputs/arxiv_math/2503.09195_pg13_pg1_repeat1.md b/outputs/arxiv_math/2503.09195_pg13_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..379d81dee3c4f10e7da79f418f8083f3daf914c9 --- /dev/null +++ b/outputs/arxiv_math/2503.09195_pg13_pg1_repeat1.md @@ -0,0 +1,17 @@ +arbitrary pair \((j_{-},j_{+})\). We can see that each ellipsoid can be chosen one after another and disjointly, for each critical point of the function \(\pi2,1,1|S_{0,c_{G}}\) corresponding to each vertex \(v\) of degree at least 3. We can put an ellipsoid of the standard form centered at some point and containing the point \((\pi_{2,1,1}(v_{0})-\epsilon_{v_{0},+,j},a_{v_{0},+,j})\in F_{D_{\{S_{0,c_{G}}\}},1}((\pi_{2,1,} \underline{(_{1}v_{0}} )+\epsilon_{v_{0},-,j},a_{v_{0},-,j})\in F_{D_{\{S_{0,c_{G}}\}},1})\) in the interior in such a way that the boundary \(\overline{D_{j^{\prime}}} -D_{j^{\prime}}\) of its closure contains a point \(p_{j^{\prime}}\in S_{0, \underline{c_{G}}}\text{with}\pi_{2,1,1} \underline{(p_{j^{\prime}})=} \pi_{2,1,1}(v_{0})\). We also do in such a way that the intersection \(( \overline{D_{j^{\prime}}} -D_{j^{\prime}})\bigcap \overline{D_{\{S_{0,c_{G}}\}}}\) is mapped by \(\pi_{2,1,1}\) into \(\mathbb{R}\) as an injective function with the minimum (resp. maximum) \(\pi_{2,1,1}(v_{0})\) . Remember that the relations \(0<\epsilon_{v_{0},-,j},\epsilon_{v_{0},+,j}<\epsilon^{\prime}<\epsilon\) are satisfied and that the numbers \(\epsilon v_{0},-,j_{1}\) and \(\epsilon_{v_{0},-,j_{2}}(\epsilon_{v_{0},+,j_{1}}\text{and}\epsilon_{v_{0},+,j_{2}})\) are mutually disjoint for distinct numbers \(j_{1}\) and \(j_{2}\) and sufficiently small. Remember also that the relation \(a_{v_{0},-,j_{1}}0\), another sufficiently small one \(\epsilon>0\), and sufficiently small ones with the relations \(0<\epsilon_{v,-,j},\epsilon_{v,+,j},\epsilon_{v_{0},-,j},\epsilon_{v_{0},+,} \underline{<_{j}} \epsilon^{\prime}<\epsilon\). We see the additional part. The set \(\overline{D_{j^{\prime}}}\) contains a point of the boundary \(\overline{D_{j^{\prime}}} - D_{j^{\prime}}\) such that the value of the projection \(\pi_{2,1,1}\) there is \(\pi_{2,1,1}(v)\text{or}\pi_{2,1,1}(v_{0})\). The set \(\overline{D_{j^{\prime}}}\) contains no point from \(F_{D_{\mathcal{S}},2}\) by considering the local shape of the curve \(S_{0,c_{G}}\). From the additional assumption, there exists at least one point different from this such that the value of the projection \(\pi_{2,1,1}\) there is \(\pi_{2,1,1}(v)\text{or}\pi_{2,1,1}(v_{0})\) in \(F_{D_{\mathcal{S}},1}\). We can easily check that the pointed set \((D_{j},x_{j^{\prime}})\) is \((\mathcal{S},D_{\mathcal{S}},A_{D_{\mathcal{S}}}\) )-PLS and it is not \((\mathcal{S},D_{\mathcal{S}},A_{D_{\mathcal{S}}}\) )-PS. + +- + +We present another example as our new result. + +Theorem 5. We also abuse the notation from Definition 4 here. In our steps here, for each pair of ellipsoids of the standard form centered at \((\pi_{2,1,1}(v_{0})-\) diff --git a/outputs/arxiv_math/2503.09208_pg18_pg1_repeat1.md b/outputs/arxiv_math/2503.09208_pg18_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..283b142910fe6ae1626dc1861bb52d123dfc870b --- /dev/null +++ b/outputs/arxiv_math/2503.09208_pg18_pg1_repeat1.md @@ -0,0 +1,29 @@ +where \(R^{\varepsilon}=R_{2}^{\varepsilon}+\text{div}(R_{1}^{\varepsilon})+R_{3}^{\varepsilon}+R_{4}^{\varepsilon}\text{and}C_{1}\) is some positive constant independent of \(\varepsilon\). Therefore, + +\[\lim_{\varepsilon\rightarrow0}\bigg\| \frac{G(I+\varepsilon h)-G(I)}{\varepsilon} -(\xi,\eta)\bigg\|_{X}=\lim_{\varepsilon\rightarrow0}\Big\|(\tilde{\xi},\tilde{\eta})\Big\|_{X}=0.\] + +This establishes the Gâteaux differentiability of \(G\) at \(I\) in the direction \(h,\text{with}G^{\prime}(I)h=(\xi,\eta)\). Now, we establish a result on the differentiability of the cost functional: Theorem 4.1 (On the Gâteau-differentiability of the Cost Functional). Let Assumptions (A1)-(A10) hold. Then the cost functional \(J\) is Gâteaux differentiable at any \(I\in\mathcal{U}_{ad}\). Furthermore, for any direction \(h\in L^{\infty}(0,T)\), the Gâteaux derivative of \(J\) at \(I\) in the direction \(h\) is given by: + +\[\langle J^{\prime}(I),h\rangle=\int_{0}\left[\alpha\int_{\mathbb{R}^{N}}^{T}\xi(t,x)dx+2\beta I(t)h(t)\right]dt+\gamma\int_{\mathbb{R}^{N}}\xi(T,x)dx\] (45) + +where \((\xi,\eta)\) is the solution to the linearized system (35). Proof. Let \(I\in\mathcal{U}_{ad}\) and define the functional \(L_{I}:L^{\infty}(0,T)\rightarrow\mathbb{R}\) by: + +\[\langle L_{I},h\rangle=\int_{0}\left[\alpha\int_{\mathbb{R}^{N}}^{T}\xi(t,x)dx+2\beta I(t)h(t)\right]dt+\gamma\int_{\mathbb{R}^{N}}\xi(T,x)dx\] (46) + +where \((\xi,\eta)\) solves the linearized system (35) with the direction \(h\). Now let \(h_{1},h_{2}\in L^{\infty}(0,T)\) and \(\lambda\in\mathbb{R}\), and let us consider \(h=h_{1}+\lambda h_{2},\text{with}(\xi_{1},\eta_{1})\text{and}(\xi_{2},\eta_{2})\) are the solutions to the linearized system corresponding to directions \(h_{1}\text{and}h_{2}\) respectively. By the linearity of system (35), the solution \((\xi,\eta)\) corresponding to \(h\) satisfies \(\xi=\xi_{1}+\lambda\xi_{2}\text{and}\eta=\eta_{1}+\lambda\eta_{2}\). Consequently: + +\[\begin{aligned} \langle L_{I},h_{1}+\lambda h_{2}\rangle&=\int_{0}\left[\alpha\int_{\mathbb{R}^{N}}^{T}(\xi_{1}+\lambda\xi_{2})dx+2\beta I(t)(h_{1}(t)+\lambda h_{2}(t))\right]dt+\gamma\int_{\mathbb{R}^{N}}(\xi_{1}+\lambda\xi_{2})(T,x)dx \\ &=\langle L_{I},h_{1}\rangle+\lambda\langle L_{I},h_{2}\rangle. \end{aligned}\] + +Therefore \(L_{I}\) is linear. Furthermore, for any \(h\in L^{\infty}(0,T)\), by using the estimates from Lemma 4.1 we obtain: + +\[|\langle L_{I},h\rangle|\leq\alpha\int_{0}\int_{\mathbb{R}^{N}}^{T}|\xi(t,x)|dxdt+2\beta\Vert I\Vert_{L^{\infty}}\Vert h\Vert_{L^{\infty}}T+\gamma\int_{\mathbb{R}^{N}}|\xi(T,x)|dx\leq C_{1}\Vert h\Vert_{L^{\infty}(0,T)},\] + +where \(C_{1}\) is a positive constant depending on \(\alpha,\beta,\gamma,T\) , and the bounds from Lemma 4.1. Therefore \(L_{I}\) is continuous. To show that \(L_{I}\) represents the Gâteaux derivative of \(J\), let \(h\in L^{\infty}(0,T),\varepsilon>0\), and let define \(I^{\varepsilon}=I+\varepsilon h\) and let \((p^{\varepsilon},d^{\varepsilon})=G(I^{\varepsilon})\text{and}(p,d)=G(I)\). Let define \(\Phi_{I}(\varepsilon)= \underline{J(I^{\varepsilon})-J(I)}\) , we have: \(\varepsilon\) + +\[\begin{aligned} \Phi_{I}(\varepsilon)&=\int\left[\alpha\int_{T}^{T} \frac{p^{\varepsilon}-p}{\varepsilon} dx+\beta \frac{(I^{\varepsilon})^{2}-I^{2}}{\varepsilon} \right]dt+\gamma\int_{\mathbb{R}^{N}} \frac{p^{\varepsilon}(T,x)-p(T,x)}{\varepsilon} dx \\ &=\int_{0}^{0}\left[\alpha\int_{\mathbb{R}^{N}}^{\mathbb{R}^{N}}\xi^{\varepsilon}dx+\beta(2Ih+\varepsilon h^{2})\right]dt+\gamma\int_{\mathbb{R}^{N}}\xi^{\varepsilon}(T,x)dx \end{aligned}\] + +where \(\xi^{\varepsilon}=(p^{\varepsilon}-p)/\varepsilon\). By Lemma 4.1, we have \(\xi^{\varepsilon}\rightarrow\xi\text{in}X\text{as}\varepsilon\rightarrow0,\text{where}\xi\) is the solution to the linearized system (35). Therefore: + +\[\lim_{\varepsilon\rightarrow0}\Phi_{I}(\varepsilon)=\int_{0}\left[\alpha\int_{\mathbb{R}^{N}}^{T}\xi(t,x)dx+2\beta I(t)h(t)\right]dt+\gamma\int_{\mathbb{R}^{N}}\xi(T,x)dx=\langle L_{I},h\rangle.\] (47) + +This establishes that \(J\) is Gâteaux differentiable at \(I\) with derivative \(J^{\prime}(I)=L_{I}\), which complete the proof. diff --git a/outputs/arxiv_math/2503.09230_pg13_pg1_repeat1.md b/outputs/arxiv_math/2503.09230_pg13_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..f934be18c93e564e021eb0913929da26f624d757 --- /dev/null +++ b/outputs/arxiv_math/2503.09230_pg13_pg1_repeat1.md @@ -0,0 +1,11 @@ +agrees with \(G\) in the neighborhood of \(\Pi ifH\cap\Pi=G\cap\Pi\) and for each face \(f\) of \(H\), there exists a corresponding face \(\varphi(f)\) of \(G\) incident to exactly the same vertices of \(G\cap\Pi\). + +We point out that we can take \(\varphi(f)=f\) in case \(f\) is contained in \(\Pi\), since we assume that the embeddings of \(H\text{and}G\) coincide within \(\Pi\). The nontrivial case arises when \(f\) is not contained in \(\Pi\). + +Lemma 13. Let \(G\) be a 3-connected graph embedded in a fixed surface \(\Sigma\text{with}\text{fw}(G)\geqslant3\) and let \((\Pi,\Pi^{+})\) be a pair of nested surfaces with boundary contained in \(\Sigma\) and contoured by \(G\). Then there exists a 3-connected surface minor \(H\) of \(G\cap\Pi^{+}\) that agrees with \(G\) in the neighborhood of \(\Pi\). Proof. We define \(H\) as the surface minor of \(G\cap\Pi^{+}\) obtained by contracting each component of \((G\cap\Pi^{+})-V(G\cap\Pi)\) into a single vertex. We obtain an embedding of \(H\text{in}\widehat{\Pi}^{+}\) by modifying the canonical embedding of \(G\cap\Pi^{+}\) in this surface. Specifically, we continuously contract the components of \((G\cap\Pi^{+})-V(G\cap\Pi)\) and their enclosed faces into single points. This contraction step can be modeled by a continuous map \(\sigma:\widehat{\Pi}^{+}\rightarrow\widehat{\Pi}^{+}\), in such a way that the embedding of \(H\) is the image under \(\sigma\) of the embedding of \(G\cap\Pi^{+}\). We choose \(\sigma\) to be injective outside of the components of \((G\cap\Pi^{+})-V(G\cap\Pi)\), and the identity within \(\Pi\). This can be done without loss of generality. + +![](figure) + +Figure 7: A graph \(G\cap\Pi^{+}\) restricted to some disk \(\Delta=\Delta_{i}\) which is a component of \(\widehat{\Pi}^{+}-\text{int}(\Pi)\) (left), and the corresponding minor \(H\) restricted to the same disk (right). The two larger vertices of \(H\) correspond to component vertices. The green and the blue cycle are cuffs (that is, boundary components) of \(\Pi\text{and}\Pi^{+}\) respectively. The figure shows, for each \(i\in[3]\), a face \(f_{i}\) of \(H\) contained in \(\Delta\) along with and the corresponding face \(\varphi(f_{i})=\sigma^{-1}(f_{i})\) of \(G\). + +We first argue that \(H\) agrees with \(G\) in the neighborhood of \(\Pi\). Since \(\sigma\) is the identity within \(\Pi\), we see immediately that \(H\cap\Pi=G\cap\Pi\). Now consider some face \(f\) of \(H\). We let \(\varphi(f):=\sigma^{-1}(f)\) denote the corresponding face of \(G\). We proceed to check that this is indeed a face of \(G\), which is moreover incident to the same vertices of \(G\cap\Pi\text{as}f\). If \(f\) is contained in \(\Pi\), then we have \(\sigma^{-1}(f)=f\) and there is nothing to check. Therefore, we may assume that \(f\) is contained in \(\widehat{\Pi}^{+}-\Pi\). We observe that \(\sigma^{-1}(f)\) is a face of \(G\) since we obtained \(H\) from \(G\cap\Pi^{+}\) only by contracting edges. No edge or vertex was deleted. Because \(\sigma\) is the identity within \(\Pi\), hence in particular on the boundary of \(\Pi\), we see that \(\sigma^{-1}(f)\) is incident to exactly the same vertices of \(G\cap\Pi\text{as}f\). diff --git a/outputs/arxiv_math/2503.09231_pg4_pg1_repeat1.md b/outputs/arxiv_math/2503.09231_pg4_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..5ed9cd40b8096486d78d9182aaeb93d23a9469d6 --- /dev/null +++ b/outputs/arxiv_math/2503.09231_pg4_pg1_repeat1.md @@ -0,0 +1,39 @@ +Let \(U=(P,S,M)\), we rewrite the system of equations (9) into the following form: + +\[\begin{cases} \underline{\partial U} \\ (t,x)+\mathcal{A}(U)(t,x)=G(U)(t,x), & (t,x)\in[0,T]\times\Omega_{R} \\ \overline{\partial t} \\ U(t,x)=U_{R}, & (t,x)\in[0,T]\times\partial\Omega_{R} \\ U(0,x)=U_{0}(x). & x\in\Omega_{R} \end{cases}\] (10) + +with + +\[\begin{aligned} \mathcal{A}(U)(t,x)&=\left[\text{div}\left(\alpha_{p}\vec{(}t,x)\left(\int_{\Omega_{R}}\gamma_{p}(x-y)P(t,y)dy\right)P(t,x)\right),\text{div}\left(\alpha_{s}\vec{(}t,x)\left(\int_{\Omega_{R}}\gamma_{s}(x-y)S(t,y)dy\right)S(t,x)\right),-D\Delta M\right], \\ G(U)&= \begin{pmatrix} g_{1}(U) \\ g_{2}(U) \\ g_{3}(U) \end{pmatrix} = \begin{pmatrix} mP(H(M)-a_{1}\lambda M) \\ -ma_{2}\lambda MS \\ m_{s}S(1-M)-\eta MP \end{pmatrix} ,U_{R}=(0,S_{R},M_{R})\text{and}U_{0}=(P_{0},S_{0},M_{0}). \end{aligned}\] + +## 2. Existence and uniqueness of solution + +Our aim in this section is to prove the existence and uniqueness of solutions for the system outlined in equation (9). We begin by examining the local dynamics through the isolation of the convolution term and leveraging established principles from the theory of semilinear evolution equations. Following this, we integrate insights from the theory of nonlocal balance equations, as elaborated in [21] and [22], to affirm the existence and uniqueness of the solutions for the system mentioned in equation (2). Before proceeding with the existence and uniqueness proofs, we need to define the following spaces. Let \(L^{\infty}(\Omega_{R})\) be the space of essentially bounded measurable functions on \(\Omega_{R}\), equipped with the norm: + +\[\Vert f\Vert_{L^{\infty}(\Omega_{R})}=\text{esssup}_{x\in\Omega_{R}}|f(x)|.\] + +Let \(C(\Omega_{R})\) be he space of continuous functions on \(\Omega_{R}\), with the uniform norm: + +\[\Vert f\Vert_{C(\Omega_{R})}=\sup_{x\in\Omega_{R}}|f(x)|.\] + +Let \(C_{b}^{1}(\Omega_{R})\) be the space of continuously differentiable functions on \(\Omega_{R}\) with bounded derivatives, normed by: + +\[\Vert f\Vert_{C_{b}^{1}(\Omega_{R})}=\Vert f\Vert_{C(\Omega_{R})}+\Vert\nabla f\Vert_{C(\Omega_{R})}.\] + +For a Banach space \((X,\Vert.\Vert_{X})\), let \(C([0,T];X)\) denote the space of continuous functions from \([0,T]\) to \(X\), with the norm: + +\[\Vert f\Vert_{C([0,T];X)}=\sup_{t\in[0,T]}\Vert f(t)\Vert_{X}.\] + +Let \(L^{1}([0,T];X)\) be the space of Bochner integrable functions from \([0,T]\) to \(X\), with the norm: + +\[\Vert f\Vert_{L^{1}([0,T];X)}=\int_{0}^{T}\Vert f(t)\Vert_{X}dt.\] + +Let \(W^{2,1}(\Omega_{R})\) be the Sobolev space defined as: + +\[W^{2,1}(\Omega_{R})=\{u\in L^{1}(\Omega_{R}):D^{\alpha}u\in L^{1}(\Omega_{R})\text{for all}|\alpha|\leq2\}\] + +### 2.1. Existence and uniqueness of the solution of the local system + +Let \(w_{p}\text{and}w_{s}\) be two fixed functions belonging to \(C\left([0,T],C_{b}^{1}(\mathbb{R}^{d})\right)\), and let \(w:=(w_{p},w_{s})\). We consider the associated local system to (10) and written as follows: + +\[\begin{cases} \underline{\partial U_{w}} \\ (t,x)+\mathcal{A}_{w}(U_{w})(t,x)=G(U_{w})(t,x), & (t,x)\in[0,T]\times\Omega_{R} \\ \overline{\partial t} \\ U_{w}(t,x)=U_{R}, & (t,x)\in[0,T]\times\partial\Omega_{R} \\ U(0,x)=U_{0}(x). & x\in\Omega_{R} \end{cases}\] (11) diff --git a/outputs/arxiv_math/2503.09233_pg16_pg1_repeat1.md b/outputs/arxiv_math/2503.09233_pg16_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..616a05f9bc9f592093a7d73154adde9311870fd2 --- /dev/null +++ b/outputs/arxiv_math/2503.09233_pg16_pg1_repeat1.md @@ -0,0 +1,27 @@ +Since we assume \(T_{j}\) is \(\mathcal{I}-\text{model for}j=1,\ldots,m\), Lemma 2.13 induces + +\[\begin{aligned} \int_{X}\langle\wedge_{j=1}^{m}T_{j,k}^{D}\rangle\wedge\omega^{n-m}&=\int\langle\wedge_{j=1}^{m}(\theta_{j}+\epsilon_{k}\omega)_{u_{j,k}^{D}}\rangle\wedge\omega^{n-m} \\ ց\int_{X}^{X}\langle\wedge_{j=1}^{m}(\theta_{j})_{P[u_{j}]_{\mathcal{I}}}\rangle\wedge\omega^{n-m}&=\int_{X}\langle\wedge_{j=1}^{m}T_{j}\rangle\wedge\omega^{n-m},k\longrightarrow \infty. \end{aligned}\] + +Combining (3.8), (3.9), (3.10) and let \(k\longrightarrow \infty\). Then, we get (3.7) + +\[\int_{X}\langle\wedge_{j=1}^{m}T_{j}^{\prime}\rangle\wedge\omega^{n-m}\leq\int_{X}\langle\wedge_{j=1}^{m}T_{j}\rangle\wedge\omega^{n-m}-(\lambda_{m}^{\prime}-\lambda_{m})\prod_{j=1}^{m-1}\left( \frac{\delta_{j}\epsilon}{2c} \right)^{m-1}\int_{\widehat{X}}[\widehat{V}]\wedge(\sigma^{*}\omega)^{n-m}\wedge\widehat{\omega}^{m-1}\] + +for the general case. Finally, since we assume that \(T_{1}^{\prime},\ldots,T_{m}^{\prime}\) is of relative full mass intersection with respect to \(T_{1},\ldots,T_{m}\), the inequality above does not hold and this make a contradiction. Therefore, \(\lambda_{j}^{\prime}-\lambda_{j}=\nu(T_{j}^{\prime},V)-\nu(T_{j},V)=0\text{for some}j=1,\ldots,m\). + +Step 3. We get rid of the assumption that \(V\) is smooth. By using Hironaka’s desingularization method, we get \(\sigma^{\prime}:X^{\prime}\longrightarrow X\), which is a composition of finite many blow-ups along smooth centers, such that \(V^{\prime}=\sigma^{\prime-1}(V)\) is smooth. Let \(R_{j}=(\sigma^{\prime})^{*}T_{j}\text{and}R_{j}^{\prime}=(\sigma^{\prime})^{*}T_{j}^{\prime}.\) In general, \(R_{1}^{\prime},\ldots,R_{m}^{\prime}\) do not necessarily have full mass intersection with respect to \(R_{1},\ldots,R_{m}\). However, we have + +\[\begin{aligned} \int_{X^{\prime}}\langle\wedge_{j=1}^{m}R_{j}^{\prime}\rangle\wedge((\sigma^{\prime})^{*}\omega)^{n-m}&=\int\langle\wedge_{j=1}^{m}T_{j}^{\prime}\rangle\wedge\omega^{n-m} \\ &=\int^{X}\langle\wedge_{j=1}^{m}T_{j}\rangle\wedge\omega^{n-m} \\ &=\int_{X^{\prime}}^{X}\langle\wedge_{j=1}^{m}R_{j}\rangle\wedge((\sigma^{\prime})^{*}\omega)^{n-m}. \end{aligned}\] + +Since \([V]\wedge\omega^{n-m}\neq 0\), it follows that \([V^{\prime}]\wedge(\sigma^{\prime*}(\omega))^{n-m}\neq 0\). Hence, we can apply step 1. and step 2. in this setting, and get + +\[\nu(T_{j}^{\prime},V)=\nu(R_{j}^{\prime},V^{\prime})=\nu(R_{j},V^{\prime})=\nu(T_{j},V)\] + +for some \(j\). This completes the proof of Theorem 1.1. + +### 3.2 Proof of Theorem 1.2 + +Let \(\mathscr{B}\) be a closed cone in the cone of big classes, and let \(V=\{x_{0}\}\) be a point in \(X\). First, note that to prove inequality (1.1), it suffices to consider \(\{\theta_{j}\}\in\mathscr{S}\cap\mathscr{B}\text{for}j=1,\ldots,m\), where \(\mathscr{S}\) is the unit sphere in \(H^{1,1}(X,\mathbb{R})\). Recall that in (3.7), we choose + +\[\delta_{j}=(c\Vert\{\theta_{j}\}\Vert+ \frac{\epsilon}{2c_{V}} )^{-1} \frac{\lambda_{j}^{\prime}-\lambda_{j}}{2} ,\] + +for \(j=1,\ldots,m\). Here, \(\epsilon\) is derived from the Kähler currents \(P_{j}\in\{\theta_{j}\}\text{such that}P_{j}\geq\epsilon\omega\). Since \(\mathscr{S}\cap\mathscr{B}\) is compact, we can choose \(\epsilon\) to be independent of \(\{\theta_{j}\}\in\mathscr{S}\cap\mathscr{B}\). Also, by diff --git a/outputs/arxiv_math/2503.09234_pg7_pg1_repeat1.md b/outputs/arxiv_math/2503.09234_pg7_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..35499c2a3461043f0196560ca8780c5f28469b97 --- /dev/null +++ b/outputs/arxiv_math/2503.09234_pg7_pg1_repeat1.md @@ -0,0 +1,23 @@ +with \(k-1\) punctures. We assume that both metrics are nondegenerate, that both metrics have one end with a common asymptotic necksize of \(\varepsilon_{*},\text{and that}(M_{1},g_{1})\) admits a Jacobi field deforming the necksize \(\varepsilon_{*}\) to first order. We then glue the two metrics together end-to-end, along the end with the common asymptotic necksize of \(\varepsilon_{*}\). Within this construction we have the freedom to rotate \((M_{1},g_{1})\) relative to \((M_{2},g_{2})\), using any element of \(SO(n-1)\) fixing the common axis of the Delaunay asymptotes. We vary this rotation over the whole group, which then gives us a family of solutions in \(\mathcal{M}_{k}\). In Section 7 we will complete the proof of Theorem 1.5 by showing the resulting submanifold cannot be contractible. The key tool we use is the forgetful map, sending a metric in \(\mathcal{M}_{k}\) to its conformal class. + +Acknowledgments: Frank Pacard suggested this construction to us, and we thank him for the prompt. We also thank Jie Qing for helpful conversations about the space of conformal structures on a finitely punctured sphere and Pedro Gaspar for generously providing figures. ASA is supported by Centro de Modelamiento Matemático (CMM) BASAL fund FB210005 for center of excellence from ANID-Chile, RC is supported by Fondecyt grant number 11230872 and by Centro de Modelamiento Matem´atico (CMM) BASAL fund FB210005 for center of excellence from ANID-Chile and ASS is supported by CNPq grant number 408834/2023-4, 312027/2023-0, 444531/2024-6 and 403770/2024-6. Part of this research was completed while ASA visited the Center for Mathematical Modeling at the University of Chile, which we thank for their hospitality. + +## 2. Preliminaries + +In this section we recall some useful material from other sources and perform some preliminary computations. + +2. 1. Delaunay metrics. The Delaunay metrics are all the constant \(Q-\)curvature metrics on a twice-punctured sphere and, as we will see later, play an important role in understanding the behavior of singular constant \(Q-\)curvature metrics with isolated singularities. + +\[n-4\] + +Consider a metric \(g=U^{\underline{4}}g^{\circ}\text{on}\mathbf{S}^{n}\setminus\{p,q\}\text{where}p\text{and}q\) are distinct. After a rotation and a dilation, we can assume \(p=N\) is the north pole and \(q=S\) is the south pole. Transferring to the Euclidean gauge we write \(g=u^{\frac{4}{n-4}}g_{\text{euc}}\text{where}u:\mathbf{R}^{n}\setminus\{0\}\rightarrow(0,\infty)\). This metric has \(Q-\)curvature equal to \(\frac{n(n^{2}-4)}{8}\) if and only if \(u\) satisfies (1.6) with a single singular point at the origin. Frank and König [5] classified all the solutions of + +\[u:\mathbf{R}^{n}\setminus\{0\}\rightarrow(0,\infty),\Delta^{2}u= \frac{n(n-4)(n^{2}-4)}{16} u^{\frac{n+4}{n-4}},\] (2.1) + +and we describe them here. First we perform the Emden-Fowler change of coordinates, defining + +\[F:\mathcal{C}^{\infty}(\mathbf{B}_{r}(0)\setminus\{0\})\rightarrow\mathcal{C}^{\infty}((-\log r,\infty)\times\mathbf{S}^{n-1}),F(u)(t,\theta)=e^{\frac{4-n}{2} t}u(e^{-t}\theta).(2.2)\] + +We can of course invert \(F\), obtaining + +\[F^{-1}(v)(x)=|x|^{\frac{4-n}{2}}v(-\log|x|,\theta).\] diff --git a/outputs/arxiv_math/2503.09254_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.09254_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..dd050dceced34c23b33bf0b0a60860009c822293 --- /dev/null +++ b/outputs/arxiv_math/2503.09254_pg3_pg1_repeat1.md @@ -0,0 +1,33 @@ +Some key theoretical results behind the Gr¨obner walk are stated below. Detailed proofs and additional context may be found in Chapters 1 and 2 of [10]. + +Theorem 2. For an ideal \(I\lhd R\), the following sets are in one-to-one correspondence: + +\[\begin{Bmatrix} \text{in}_{<}(I), \\ <\text{is a term order} \end{Bmatrix} \leftrightarrow \begin{Bmatrix} \text{marked Gr}\ddot{ö}bnerbases \\ ofI \end{Bmatrix} \leftrightarrow \begin{Bmatrix} \text{full}-\text{dimensional} \\ conesof\mathbb{G}(I) \end{Bmatrix}\] + +In our setting, a marked Gr¨obner basis is a reduced Gr¨obner basis with the leading terms identified (formally, each \(g\) is encoded as a pair \((g,x^{\alpha}),\text{where}x^{\alpha}=\text{in}_{<}(g))\). The first correspondence in Theorem 2 immediate, whilst the second correspondence is a consequence of [13, Theorem 1.11]: marked Gr¨obner bases encode the defining integer vectors of an H-description of the corresponding cone. Lower-dimensional cones in \(\mathbb{G}(I)\) correspond to generalized initial ideals in\((_{\omega}I),\text{where}\omega\) is any weight vector in the relative interior of said cone. Generically, such ideals are “almost monomial”, and may be retrieved with the help of the following lemma: + +Lemma 3. Let \(G_{<}\) be a marked Gröbner basis of \(I\) with regards to \(<\text{and}\omega\in\mathbb{R}_{\geq0}^{n}\) be a weight vector on the boundary of the corresponding cone in \(\mathbb{G}(I)\). The set + +\[\text{in}_{\omega}(G_{<})=\{\text{in}_{\omega}(g),g\in G_{<}\}\] + +is a marked Gröbner basis of in\((_{\omega}I)\) with respect to <. + +At every step of the Gr¨obner walk, a basis of this form converted with Buchberger’s algorithm and then lifted to the basis of \(I\) corresponding to the adjacent full-dimensional cone, which corresponds to \((<_{t})_{\omega}\), i.e. the refinement of the target ordering \(<_{t}\) by \(\omega\). + +Lemma 4. Let \(M=\{m_{1},...,m_{r}\}\) be the marked Gröbner basis of in\((_{\omega}I)\) with respect to the refinement ordering \((<_{t})_{\omega}\). Then + +\[G:=\{m_{1}- \overline{m_{1}}^{G_{<}} ,...,m_{r}- \overline{m_{r}}^{G_{<}} \}\] + +is a Gröbner basis of \(I\) with respect to \((<_{t})_{\omega}\text{where} \overline{f}^{G_{<}}\) denotes the normal form of \(f\) with respect to the basis \(G_{<}\). + +This process of subsequent passing to the generalized initial ideal and lifting to the adjacent basis is repeated until the target basis is computed. + +## 3. Functionality + +Our implementation of the Gr¨obner walk ships with OSCAR since version 1. 2. 0, thus it suffices to load OSCAR. There is a straightforward interface through the function groebner_walk. + +Example 5. Continuing from example Example 1, we can calculate a Gr¨obner basis of the ideal + +\[I=\langle y^{4}+x^{3}-x^{2}+x,x^{4}\rangle\lhd\mathbb{Q}[x,y]\] + +with respect to \(<_{\text{lex}}\) by starting from a Gr¨obner basis for the graded reverse lexicographic ordering \(<\text{degrevlex}\). Since \(<\text{degrevlex}\) is the default internal ordering of any polynomial ring in OSCAR, it suffices to call the Gr¨obner walk in the following way. diff --git a/outputs/arxiv_math/2503.09284_pg29_pg1_repeat1.md b/outputs/arxiv_math/2503.09284_pg29_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..4568c7a52da4272cd66adafd7a0ecf2857e72dcc --- /dev/null +++ b/outputs/arxiv_math/2503.09284_pg29_pg1_repeat1.md @@ -0,0 +1,29 @@ +exist some \(t_{1}\text{with}00\), it is possible to construct \(\delta-\text{isometries}f_{n}:(Z_{n},\rho_{n})\rightarrow(Z,\rho_{0})\) for all sufficiently large \(n\). Thus, by Remark 3.5, this will be sufficient to conclude \((Z_{n},\rho_{n})\xrightarrow{AIconv.} (Z,\rho_{0})\), i.e. \((\partial_{P}X_{n},\rho_{x_{n}})\xrightarrow{AIconv.} (\partial_{P}X,\rho_{x})\). Since \((\mathcal{M}(Z_{n}),\rho_{n})\xrightarrow{GHconv.} (\mathcal{M}(Z),\rho_{0}),\text{for every}R>0\), there exists \(\epsilon_{n}-\)isometries + +\[\mathcal{F}_{n}:B_{\mathcal{M}(Z)}(\rho_{0},R)\rightarrow B_{\mathcal{M}(Z_{n})}(\rho_{n},R)\] + +such that \(\epsilon_{n}\rightarrow0+\text{and}\mathcal{F}_{n}(\rho_{0})=\rho_{n}\). The antipodal space \((Z,\rho_{0})\) is of finite cardinality, let us label \(Z=\{1,2,\cdots ,m\}\). By [BP24, Theorem 1.1], the maximal Gromov hyperbolic spaces \(\mathcal{M}(Z,\rho_{0})\) is isometric to a polyhedral complex such that diff --git a/outputs/arxiv_math/2503.09285_pg9_pg1_repeat1.md b/outputs/arxiv_math/2503.09285_pg9_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..2733fbd0c612e740c9d0faa80676997bb0339841 --- /dev/null +++ b/outputs/arxiv_math/2503.09285_pg9_pg1_repeat1.md @@ -0,0 +1,33 @@ +where n is the outward normal to \(\partial D\). Denote \(P_{L}\) as the orthogonal projection of \(L^{2}(D)^{2}\) onto \(H\). The space of vector fields whose gradients are integrable in \(L^{2}(D)^{2}\) is also relevant and we define + +\[V:=\{\mathbf{u}\in H^{1}(D)^{2}:\nabla\cdot\mathbf{u}=0,\mathbf{u}|_{\partial D}=0\}.\] + +We denote the norms associated to \(H\text{and}V\) respectively as \(|\cdot|\text{and}\Vert\cdot\Vert.\) The Stokes operator is defined as \(Au=-P_{L}\Delta u\), for any vector field \(\mathbf{u}\in V\cap H^{2}(D)^{2}.\) Since \(A\) is self-adjoint with a compact inverse, we infer that \(A\) admits an increasing sequence of eigenvalues \(\lambda_{k}\sim k\) diverging to infinity with the corresponding eigenvectors \(e_{k}\) forming a complete orthonormal basis for \(H\). We denote by \(P_{N}\text{and}Q_{N}\) the projection onto \(H_{N}= \text{span}\{e_{k}:k=1,\ldots,N\}\) and its orthogonal complement, respectively. Recall the generalized Poincaré inequalities + +\[\Vert P_{N}\mathbf{u}\Vert^{2}\leq\lambda_{N}|P_{N}\mathbf{u}|^{2},|Q_{N}\mathbf{u}|^{2}\leq\lambda_{N}^{-1}\Vert Q_{N}\mathbf{u}\Vert^{2}\] (3.6) + +hold for all sufficiently smooth u and any \(N\geq1.\) We make the following Hypotheses: + +\(\mathbf{H}_{\mathbf{1}}.\) The function \(\sigma=(\sigma_{1},\ldots,\sigma_{m})\) is bounded and Lipschitz, i.e., there exist constants \(B_{0},L>\) such that + +\[\begin{aligned} |\sigma(\mathbf{u})|^{2}&=\sum_{k=1}^{m}|\sigma_{k}(\mathbf{u})|^{2}\leq B_{0},\text{for all}\mathbf{u}\in H; \\ |\sigma(\mathbf{u})-\sigma(\mathbf{v})|^{2}&=\sum_{k=1}^{m}|\sigma_{k}(\mathbf{u})-\sigma_{k}(\mathbf{v})|^{2}\leq L|\mathbf{u}-\mathbf{v}|^{2},\text{for all}\mathbf{u},\mathbf{v}\in H. \end{aligned}\] + +## \(\mathbf{H}_{\mathbf{2}}.\) There exists \(N\in\mathbb{N}\) such that + +\[P_{N}H\subset\text{Range}(\sigma(\mathbf{u})),\text{for all}\mathbf{u}\in H.\] + +Moreover, the corresponding pseudo-inverse operators \(\sigma_{k}^{-1}:P_{N}H\rightarrow H\) are uniformly bounded, i.e., there exists a constant \(C_{0}\) such that + +\[|\sigma(\mathbf{u})^{-1}(P_{N}\mathbf{w})|\leq C_{0}|P_{N}\mathbf{w}|,\text{for all}\mathbf{u},\mathbf{w}\in H.\] + +\(\mathbf{H}_{\mathbf{3}}.\) Finally, we assume that \(N\) is sufficiently large such that + +\[\lambda_{N}> \frac{L}{\nu} + \frac{C_{D}^{2}}{\nu^{3}} B_{0}.\] (3.7) + +The existence of a unique strong solution to equation (3.5) is standard, see e.g. [28]. Furthermore, as the energy estimates and Feller property can be obtained directly, we prove these two parts first. + +# Energy estimates + +# Proposition 3.3. Assume H1. Then + +\[\mathbb{E}|\mathbf{u}|^{2}\leq e^{-2\nu t}|x|^{2}+ \frac{B_{0}}{2\nu} .\] (3.8) diff --git a/outputs/arxiv_math/2503.09298_pg3_pg1_repeat1.md b/outputs/arxiv_math/2503.09298_pg3_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..193f36d4b0d85386f99510b367413d9594df6e7c --- /dev/null +++ b/outputs/arxiv_math/2503.09298_pg3_pg1_repeat1.md @@ -0,0 +1,21 @@ +versions of Whitney’s flat cochains. The next paragraph provides an alternative description of \(F_{m}^{\alpha}(K)^{*}\). An \(\alpha-\)fractional charge \(\omega\) over \(K\) is a linear functional on \(N_{m}(K)\), the space of normal \(m-\)currents supported in \(K\), that satisfies the continuity condition + +\[|\omega(T)|\leqslant CN(T)^{1-\alpha}F(T)^{\alpha}\] (1) + +for all \(T\in N_{m}(K).\text{When}K\) is sufficiently nice, for instance if it is convex, then \(\omega\) extends by density to a continuous linear functional on \(F_{m}^{\alpha}(K)\). Conversely, any element of \(F_{m}^{\alpha}(K)^{*}\), restricted to \(N_{m}(K)\), satisfies (1). Thus, the space \(CH^{m,\alpha}(K)\) of \(\alpha-\text{fractional}m-\)charges is (isomorphic to) the dual of \(F_{m}^{\alpha}(K)\) in the case where \(K\) is convex. One important class of \(\alpha-\)fractional charges consists of \(\alpha-\)Hölder continuous differential forms. More precisely, to each such form \(\eta\in\text{Lip}^{\alpha}(K;\wedge^{m}\mathbb{R}^{d})\), one associates the functional \(\Lambda(\eta)\) that acts on normal currents \(T\in N_{m}(K)\) by + +\[\Lambda(\eta)(T)=\int_{K}\langle\eta(x),T®(x)\rangle d\Vert T\Vert(x).\] + +In the simplest case where \(m=0\), the map \(\Lambda\) is in fact a Banach space isomorphism between Lip\((^{\alpha}K)\text{and}CH^{0,\alpha}(K)\). As a result, we can identify \(\alpha-\)fractional 0-charges over \(K\text{with}\alpha-\)Hölder continuous functions on \(K\). In this case, it is known that Lip\((^{\alpha}K)\) is the double dual of the little Hölder space, described for example in [28, Chapter 4]. We treat the general case \(m\geqslant0\) by analogy. We define the space of little \(\alpha-\text{fractional}m-\) charges ch\((^{m,\alpha}K)\) as the closure in \(CH^{m,\alpha}(K)\) of \(\{\Lambda(\omega_{|K}):\omega\text{smooth}\}\), and we establish that \(F_{m}^{\alpha}(K)\supset ch^{m,\alpha}(K)^{*}\). Thus any bounded sequence in \(F_{m}^{\alpha}(K)\) possesses a weakly convergent subsequence, by the Banach-Alaoğlu theorem. Convergence in flat norm is later obtained by adapting the classical deformation theorem to the setting of fractional currents. + +1. 3. Generalized Stokes’ formula. The exterior derivative operator for fractional charges is well-defined as the adjoint of the boundary operator. Specifically, \(d\omega\) is the \(\alpha-\)fractional \((m+1)-\)charge over \(K\), given by + +\[\langle d\omega,T\rangle=\langle\omega,\partial T\rangle\] (2) + +for \(T\in N_{m}(K).\text{If}K\) is convex, this formula extends more generally to \(\alpha-\)fractional currents over \(K\). A particular case of this formula is worthy of attention, as it results in a generalized Gauss-Green formula for Hölder vector fields over fractal boundaries. It extends the work of Harrison and Norton [17] (it should be however stressed that formula (2) is purely definitional, contrary to the more constructive Gauss-Green formulas in [17, 16]). Indeed, we can take for \(U\) a bounded open set with finite \((1-\alpha)-\)perimeter (meaning that \(1_{U}\in W_{c}^{1-\alpha,1}(\mathbb{R}^{d})\), or equivalently, \(⟦U⟧\) is \(\alpha-\text{fractional})\). We shall prove that this condition is weaker than the \((d-1+\alpha)-\)summability (see [17, §3] or Subsection 3.2) of \(\partial U\) that is required in the Gauss-Green formula of Harrison-Norton. In this case, formula (2) applies for \(T=⟦U⟧\text{and}\omega\) that is associated to an \(\alpha-\)Hölder continuous \((d-1)-\text{form}\). + +1. 4. Pushforward by Hölder maps. The pushforward of currents by Hölder maps \(f\) has been explored in the literature, notably in [14] and [29]. The results in [29] complement ours, as they address the case where the components \(f_{i}\) of \(f\) have different Hölder exponents. + +Our contribution is to provide a functional-analytical setting to this construction by showing that \(f_{\#}\) is a bounded linear map between spaces of fractional currents. Indeed, in Section 8, we define the pushforward \(f_{\#}T\) of an \(\alpha-\text{fractional}m-\text{current}T\) by a \(\gamma-\)Hölder continuous map \(f\). The construction makes sense as long as \(m+\alpha<\gamma(m+1)\). Under this assumption, we show that \(f_{\#}T\) is \(\beta-\text{fractional},\text{where}\beta\) is given by the equation + +\[\frac{m+\alpha}{\gamma} =m+\beta.\] diff --git a/outputs/arxiv_math/2503.09299_pg16_pg1_repeat1.md b/outputs/arxiv_math/2503.09299_pg16_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..de521452690449d3df8e955eca0439cf1c279294 --- /dev/null +++ b/outputs/arxiv_math/2503.09299_pg16_pg1_repeat1.md @@ -0,0 +1,44 @@ +graphon Trunc. \(\text{SVD}(A^{\prime})\) + +10 + +8 + +20 + +1000 2000 3000 4000 5000 + +n + +3 + +2 + +1 + +1000 2000 3000 4000 5000 + +n + +| | | | | [icon] | 13 | | +| --- | --- | --- | --- | --- | --- | --- | +| 10 | | | | | | | +| | | | | [icon] | 14 | | +| 12 | | | | | | | +| 14 | | | | | 15 | | +| 16 | | | | | 16 | | +| 0 | 200 | 400 600 n | 800 | 1000 | | 250 500 750 1000 1250 1500 n | + +Figure 1: Upper Left: difference between target functions for optimal interventions of a network on \(n\) vertices sampled from \((1,1/2)-\mathrm{H}¨\)older graphon \(W_{1}(x,y)= \sqrt{|x-y|}\) and interventions based on hard-thresholding estimator. Lower Left: Rank of the hard-thresholding estimator for a network on \(n\) vertices sampled from \(W_{1}(x,y)\). Midlle: Difference between target functions for optimal interventions of a network on \(n-\)vertices sampled from SBM with 4 comminties and interventions computed using a) graphon b) hard-thersholding estimator. Right: The difference between target functions for optimal interventions of a network \(\boldsymbol{A}\) of size 10000 sampled from SBM with 4 communities and interventions based on a) graphon b) hard-thersholding estimator comupted for other network \(\boldsymbol{A}^{\prime}\) of size \(n\). + +## 6 Conclusion + +Driven by applications in graphon games, we developed an estimator for an unknown graphon based on a sampled network. Under standard regularity conditions for the graphon, our estimator possesses two key features: (i) it approximates the unknown graphon in the operator norm, and + +(ii) it is of low rank. We established upper bounds for the convergence rate and rank of the estimator. Furthermore, we demonstrated that the estimator yields near-optimal solutions for the social welfare problem in linear-quadratic graphon games, with efficient computation. We also quantified the convergence rates of the associated costs and the computational complexity relative to the network size. + +# References + +[Bandeira and van Handel, 2016] Bandeira, A. S. and van Handel, R. (2016). Sharp nonasymptotic bounds on the norm of random matrices with independent entries. The Annals of Probability, 44(4):2479–2506. + +[Bisgard, 2020] Bisgard, J. (2020). Analysis and linear algebra: the singular value decomposition and applications, volume 94. American Mathematical Soc. diff --git a/outputs/arxiv_math/2503.09301_pg11_pg1_repeat1.md b/outputs/arxiv_math/2503.09301_pg11_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..b5d6f6db613aabbfa5b520b07e8291b290998ff8 --- /dev/null +++ b/outputs/arxiv_math/2503.09301_pg11_pg1_repeat1.md @@ -0,0 +1,11 @@ +where we use the notation \(\mathcal{K}^{p}:=\mathcal{B}^{p}[-1]\text{for all}p\in P\) . Further, we call \(\mathcal{K}\) the chain complex consisting of preboundaries. + +Example 3.1. Consider the \(P\) -graded chain complex associated to the filtration in Example 2.8. The chain group \(\mathcal{C}_{1}^{2}\) admits a splitting \(\mathcal{C}_{1}^{2}\cong\mathcal{M}_{1}^{2}\bigoplus\mathcal{K}_{1}^{2},\text{where}\mathcal{M}_{1}^{2}\) is generated uniquely by \(uv+vw\); see Fig. 1. On the other hand, the preboundary \(\mathcal{K}_{1}^{2}\) can either be chosen to be generated by \(uv\) or \(vw.\text{As for}\mathcal{C}^{1}\), the single edge \(uw\in\mathcal{C}_{1}^{1}\) generates a preboundary for the single boundary vertex \(w\in\mathcal{C}_{0}^{1}\). Throughout this section, we use the notation \(\mathcal{A}:=\bigoplus_{p\in P}\mathcal{A}^{p},\text{for}\mathcal{A}\in\{\mathcal{M},\mathcal{B},\mathcal{K}\}\). In addition, we fix an isomorphism \(\zeta:\mathcal{C}\cong\mathcal{M}\bigoplus\mathcal{B}\bigoplus\mathcal{K}\) such that, for each \(p\in P\text{and}n\in\mathbb{Z}\), it restricts to a splitting isomorphism \(\zeta_{n}^{p}:\mathcal{C}_{n}^{p}\cong\mathcal{M}_{n}\bigoplus^{p}\mathcal{B}_{n}\bigoplus^{p}\mathcal{K}_{n}^{p}\). Also, we abuse notation and use ‘the submodules \(\mathcal{M},\mathcal{B}\text{and}\mathcal{K}\) of \(\mathcal{C}’\), to refer to the actual submodules \(\zeta^{-1}(\mathcal{M}),\zeta^{-1}(\mathcal{B})\text{and}\zeta^{-1}(\mathcal{K})\), respectively. In particular, given a chain \(c\in\mathcal{C}\), there are unique chains \(z\in\mathcal{M},b\in\mathcal{B}\text{and}w\in\mathcal{K}\) such that \(c=z+b+w\). We also consider the inclusions and projections \(ı_{\mathcal{A}}:\mathcal{A}\hookrightarrow\rightarrow\mathcal{C}\text{and}_{\mathcal{A}}:\mathcal{C}։\mathcal{A}\) for \(\mathcal{A}\in\{\mathcal{M},\mathcal{K},\mathcal{B}\}\). Before moving on, we need to add another condition on \(\zeta\). Namely, we assume that \(d_{n}(\mathcal{K}_{n}^{p})\subseteq \mathcal{B}_{n-1}^{p}\text{for all}n\in\mathbb{Z}\text{and}p\in P\) and say that \(\zeta\) is consistent. Further, in this case, notice that \(d_{n}\) sends \(\mathcal{K}_{n}^{p}\) isomorphically to \(\mathcal{B}_{n-1}^{p}\). Later, in Proposition 4.1, we show that such a splitting exists and that can be easily computed. One can formulate the multicomplex \(\mathcal{C}\) as a based complex where, for each degree \(n\), there is an isomorphism + +\[\mathcal{C}_{n}\cong^{(}\bigoplus_{p\in P}\mathcal{M}_{n}^{)_{p}}\bigoplus(\bigoplus_{p\in P}\mathcal{B}_{n}^{)_{p}}\bigoplus(\bigoplus_{p\in P}\mathcal{K}_{n}^{)_{p}}=\bigoplus_{\mathcal{A}_{n}^{p}\in I_{n}}\mathcal{A}_{n}^{p}\] + +where we have used \(I_{n}=\{\mathcal{A}_{n}^{p}|\mathcal{A}\in\{\mathcal{M},\mathcal{K},\mathcal{B}\},p\in P,\mathcal{A}_{n}^{p}\neq0\}\). Thus, given \(p\leq q\) from \(P\) , the twisted differential \(t^{pq}:\mathcal{C}^{q}\rightarrow\mathcal{C}^{p}\) can be written as \(t^{pq}=\sum_{\mathcal{A},\mathcal{D}\in\{\mathcal{M},\mathcal{B},\mathcal{K}\}}t_{\mathcal{A}\mathcal{D}}^{pq}\), where the summands are given by \(t_{\mathcal{A}\mathcal{D}}^{pq}=ı_{\mathcal{A}}\circ t^{pq}\circ_{\mathcal{D}}\). Further, using that \(\zeta\) is a consistent splitting, it follows that \(t^{p}=t_{\mathcal{B}\mathcal{K}}^{p}\text{for all}p\in P\) since all other components are zero. Next, we proceed to apply algebraic Morse theory as presented in Section 2.4. To start, we consider the associated digraph \(G_{\mathcal{C}}\) with vertex set \(\bigcup_{n\in\mathbb{Z}}I_{n}\), together with the arrows \(\mathcal{D}_{n}^{q}\rightarrow \mathcal{A}_{n-1}^{p}\text{for}(t_{\mathcal{A}\mathcal{D}}^{pq})_{n}\neq0\text{and}p\leq q\) from \(P,n\in\mathbb{Z}\text{and}\mathcal{A},\mathcal{D}\in\{\mathcal{M},\mathcal{B},\mathcal{K}\}\). We define a partial matching \(\mu\text{on}G_{\mathcal{C}}\) which contains all arrows of the form \(\mathcal{K}_{n}^{p}\rightarrow\mathcal{B}_{n-1}^{p}\) which are in \(G_{\mathcal{C}}\). In fact, notice that \(\mu\) is a Morse matching. To see this, first, since \(\zeta\) is consistent, we know that \(t_{\mathcal{B}\mathcal{K}}^{p}:\mathcal{K}_{n}^{p}\rightarrow\mathcal{B}_{n-1}^{p}\) is an isomorphism for all \(p\in P\text{and all}n\in\mathbb{Z}\). On the other hand, since we assume that \(P\) is finite, it follows that \(G_{\mathcal{C}}\) is also finite. Also, one can see that there are no directed cycles in \(G_{\mathcal{C}}^{\mu}\). This is because, by definition, all arrows from \(G_{\mathcal{C}}^{\mu}\) are of the form \(\mathcal{D}_{n}^{q}\rightarrow\mathcal{A}_{n-1}^{p}\text{and such that}p\leq q,\text{and},\text{if}p=q\), the only possible arrow is \(\mathcal{B}_{n}^{p}\rightarrow\mathcal{K}_{n-1}^{p}\). Altogether, using Lemma 2.12, it follows that \(\mu\) is a Morse matching. In particular, notice that \(\mu_{n}^{0}=\{\mathcal{M}_{n}^{p}|p\in P\text{and}\mathcal{M}_{n}^{p}\neq0\}\). Hence, we are equipped with the based complex \(\mathcal{C}\) together with a Morse matching \(\mu\text{on}G_{\mathcal{C}}\), and so we can follow the exposure from Section 2.4 to obtain a specialised result. During the rest of this section, we follow such strategy. To start, we define a based complex \(\tilde{\mathcal{C}}\) which is equal to \(\mathcal{C}\), except for the differential \(\tilde{d}\), which is \(\tilde{d}=ı_{\mathcal{B}}^{(}\sum_{p\in P}t_{\mathcal{B}\mathcal{K}}^{)_{p}}_{\mathcal{K}}\). In particular, notice that \(\tilde{d}\) is a \(P\) -filtered map. On the other hand, we define the based complex \(\tilde{\mathcal{M}}\) which is given by \(\tilde{\mathcal{M}}_{n}=\bigoplus_{p\in P}\mathcal{M}_{n}^{p}\) together with the trivial differential. As in Section 2.4, using [Skö18, Lemma 1], there is a \(P\) -filtered contraction + +\[\begin{aligned} \tilde{} \\ \tilde{\mathcal{M}} \tfrac{\underline{f}}{\tilde{g}} \tilde{\mathcal{C}}_{\tilde{h}} \end{aligned}\] + +where (one can check that) \(\tilde{f}:\tilde{\mathcal{C}}\rightarrow\tilde{\mathcal{M}}\text{and}\tilde{g}:\tilde{\mathcal{M}}\rightarrow\tilde{\mathcal{C}}\) are given by \(\tilde{f}=_{\mathcal{M}}\text{and}\tilde{g}=ı_{\mathcal{M}}\), respectively, while \(\tilde{h}:\tilde{\mathcal{C}}\rightarrow\tilde{\mathcal{C}[}1]\) is given by \(\tilde{h}=-ı_{\mathcal{K}}^{(}\sum_{p\in P}(t_{\mathcal{B}\mathcal{K}}^{p})^{)_{-1}}_{\mathcal{B}}\). In fact, it is not difficult diff --git a/outputs/arxiv_math/2503.09337_pg14_pg1_repeat1.md b/outputs/arxiv_math/2503.09337_pg14_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..8b9b51a37d191140982b42a06d3181e36d3006fc --- /dev/null +++ b/outputs/arxiv_math/2503.09337_pg14_pg1_repeat1.md @@ -0,0 +1,25 @@ +Then we define the completion problem as follows: + +\[\min_{\mathcal{X}} \frac{1}{2} \Vert\mathcal{W}\odot(\mathcal{D}*_{t}\mathcal{X}-\mathcal{Y})\Vert_{F}+\lambda\Vert\mathcal{X}\Vert_{1},\] (17) + +such that ⊙ is the pointwise product. The following section presents the method to solve this problem. + +### 5.1 Iterative Shrinkage Thresholding with Anderson acceleration + +The Iterative Shrinkage Thresholding Algorithm (ISTA) is a class of first-order methods that can be seen as an extension of classical gradient methods, used to solve linear inverse problems such as (9). This algorithm is known for its simplicity and efficiency in addressing this type of problem. However, ISTA is known to converge slowly and exhibits a sublinear global rate of convergence. In [17], an extrapolation method is used to accelerate the algorithm, resulting in the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA). In this work, we employ a different acceleration method, known as Anderson Acceleration (AA), to improve the performance of ISTA. The Anderson Acceleration (AA) method for fixed-point iterations, originally developed by D. G. Anderson, was primarily used in electronic structure computations. Despite its effectiveness, this method has not been as widely adopted as other acceleration techniques. In our specific problem, however, it demonstrates superior results in terms of precision compared to FISTA and other acceleration methods, as will be shown in the numerical results section. The new accelerated algorithm is presented in Algorithm 5 and is based on the work in [8], [18], and [7]. Let’s turn to the problem (9): + +\[\min_{\mathcal{X}}L(\mathcal{X},\mathcal{D}^{(t)};\mathcal{Y})= \frac{1}{2} \Vert\mathcal{D}^{(t)}*_{t}\mathcal{X}-\mathcal{Y}\Vert_{F}^{2}+\lambda\Vert\mathcal{X}\Vert_{1},\] (18) + +\[=G(\mathcal{X})+\lambda\Vert\mathcal{X}\Vert_{1}\] (19) + +The general solution to the problem (18) is given by + +\[\mathcal{X}_{k}=\mathcal{T}_{\lambda t}(\mathcal{X}_{k-1}-t\mathcal{D}^{T}*_{t}(\mathcal{D}*_{t}\mathcal{X}_{k-1}-\mathcal{Y}),\] (20) + +such that \(\mathcal{T}_{\alpha}:\mathbb{R}^{M_{1}\times M_{2}\times\cdots \times M_{N}}\rightarrow\mathbb{R}^{M_{1}\times M_{2}\times\cdots \times M_{N}}\) is the shrinkage operator defined by: + +\[T_{\alpha}(\mathcal{X})_{i_{1}i_{2}\ldots i_{N}}=(|\mathcal{X}_{i_{1}i_{2}\ldots i_{N}}|-\alpha)_{+}sgn(\mathcal{X}_{i_{1}i_{2}\ldots i_{N}}).\] (21) + +The parameter \(t= \dfrac{1}{L}\) , such that \(L\) is the Lipschitz continuous gradient of the function \(G\). Proposition 3. Consider the function \(G(\mathcal{X})= \frac{1}{2} \Vert\mathcal{D}*_{t}\mathcal{X}-\mathcal{B}\Vert_{F}^{2}.G\) is differentiable, and the gradient is given by: + +\[\nabla G(\mathcal{X})=\mathcal{D}^{T}*_{t}(\mathcal{D}*_{t}\mathcal{X}-\mathcal{B}).\] diff --git a/outputs/arxiv_math/2503.09352_pg5_pg1_repeat1.md b/outputs/arxiv_math/2503.09352_pg5_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..85efa54dbfb552714efdfe40f3d0a161447acbfb --- /dev/null +++ b/outputs/arxiv_math/2503.09352_pg5_pg1_repeat1.md @@ -0,0 +1,45 @@ +which converges in \(\mathcal{O}^{L-\text{la}}\) + +\((_{\mathcal{M}_{\text{LT},\infty}}U)\). Later we will show \(D_{i}(s)\) acturally lies in the smaller subspace \(\mathcal{O}_{\mathcal{M}_{\mathrm{L}\mathrm{T},\infty}}^{L-\mathrm{l}\mathrm{a},m^{0}=0}(U)\). Using \(E_{d,i}(x_{i}-x_{i,n})=1\), a direct computation shows \(E_{d,i}.D_{i}(s)=0\text{for all}s\in\mathcal{O}_{\mathcal{M}_{\mathrm{L}\mathrm{T},\infty}}^{L-\mathrm{l}\mathrm{a},m^{0}=0}(U)\), and + +\[s=\sum_{l=0}^{\infty}a_{i,l}(x_{i}-x_{i,n})^{l}\] + +with \(a_{i,l}=D_{i}( \frac{(E_{d,i})^{l}.s}{l!} )\). Here we note the coefficients \(a_{i,l}\) are determined by \(s\). From the construction we know the coefficients \(a_{i,l}\) are killed by \(E_{d,i}\text{for all}l.\text{For}j\neq i\), we furthure express the coefficients \(a_{i,l}\) along \(E_{d,j}\), namely + +\[a_{i,l}=\sum_{k=0}^{\infty}b_{i,l,j,l^{\prime}}(x_{j}-x_{j,n})^{l^{\prime}}\] + +with \(b_{i,l,j,l^{\prime}}=D_{j}( \frac{(E_{d,j})^{l\prime}.a_{i,l}}{l^{\prime}!} ).\text{As}[E_{d,i},E_{d,j}]=0\text{for}i,j=0,1,...,d-1\), we see \(bi,l,j,l^{\prime}\) is killed by \(E_{d,i}\) and \(E_{d,j}\). After we expand \(s\) along each derivation \(E_{d,i}\text{for}i=0,1,...,d-1\), we arrive at an expression + +\[\begin{aligned} s&=\sum_{i_{0}=0}^{\infty}\sum_{i_{1}=0}^{\infty}\cdots \sum_{=_{d-}}^{\infty_{1}}c_{i_{0}\ldots i_{d-1}}(x_{0}-x_{0,n})^{i_{0}}\cdots (x_{d-1}-x_{d-1,n})^{i_{d-1}} \\ i0 \end{aligned}\] + +such that \(E_{d,i}c_{i_{0}\ldots i_{d-1}}=0\text{for all}i=0,1,...,d-1.\text{As}m^{0}\) acts trivially on \(\mathcal{O}_{\mathscr{F}\ell}\), we know \(m^{0}\) acts trivially on \(x_{i}-x_{i,n}\) for any \(i=0,...,d-1\). Therefore, + +\[0=\sum_{i_{0},\ldots,i_{d-1}}(m^{0}c_{i_{0}\ldots i_{d-1}})(x_{0}-x_{0,n})^{i_{0}}\cdots (x_{d-1}-x_{d-1,n})^{i_{d-1}}\] + +\(\cdots :z_{d-1}:1]\}\subset\mathbb{P}^{d}\), the matrix \(Z=\) is a lifting of the point \(z=[z_{0}:z_{1}:...:z_{d-1}\) : 1] + +This implies \(m^{0}c_{i_{0}\ldots i_{d-1}}=0\). Indeed, the coefficients \(ci_{0}\ldots i_{d-1}\) are determined by \(s\). Combined this with the fact that \(\mathcal{O}_{\mathcal{M}_{\mathrm{L}\mathrm{T},\infty}}^{L-\mathrm{l}\mathrm{a}}\) is killed by \(\pi^{-1}n^{0}\), we see \(p^{0}\) kills \(ci_{0}\ldots i_{d-1}\). Hence there is an induced action of \(\overline{n}^{0} =g^{0}/p^{0}\) on \(c_{i_{0}\ldots i_{d-1}}.\text{On}U, \overline{n}^{0}\) is generated by \(E_{d,i}\text{for}i=0,1,...,d-1\). Indeed, on the open locus \(V_{o}=\{[z_{0}:z_{1}\) : 0 \((z_{0},z_{1},...,z_{d-1})1\) as \(oZ=z\text{where}o=[0\) : 0 : \(\cdots\) : 0 : 1]. Then \(Z^{-1}E_{d,i}Z=E_{d,i}\text{for}i=0,1,...,d\) generates \(\overline{n}^{0} \text{on}V_{o}\). From the construction we know \(ci_{0}\ldots i_{d-1}\) is killed by \(E_{d,i}\text{for all}i=0,1,...,d-1\), hence it is killed by \(\overline{n}^{0}\). This shows that \(g^{0}\) acts trvially on \(ci_{0}\ldots i_{d-1}\), which implies \(c_{i_{0}\ldots i_{d-1}}\in\mathcal{O}_{\mathcal{M}_{\text{LT},\infty}}^{\text{sm}}(U)\). Corollary 0.3.2. For any \(U\in B_{\text{LT}}\), the image of \(\mathcal{O}_{\mathcal{M}_{\text{LT},\infty}}^{\text{sm}}(U)\otimes_{C}\pi^{-1}\mathcal{O}_{\mathscr{F}\ell}(U)\) inside \(\mathcal{O}_{\mathcal{M}_{\mathrm{L}\mathrm{T},\infty}}^{L-\mathrm{l}\mathrm{a},m^{0}=0}(U)\) is dense. Proof. This follows directly from Theorem 0.3.1. To simplify the notation, we denote \(\mathcal{O}_{\text{LT}}\) by the sheaf \(\mathcal{O}_{\mathcal{M}_{\mathrm{L}\mathrm{T},\infty}}^{L-\mathrm{l}\mathrm{a},m^{0}=0}\text{on}\mathcal{M}_{\text{LT},\infty}\). Put \(\mathcal{O}_{\mathrm{L}\mathrm{T}}^{\text{sm}}:=\mathcal{O}^{G-\text{sm}} \Omega_{\mathcal{M}_{\text{LT},n}}^{k}\) be the sheaf of \(k-\)differential forms on \(\mathcal{M}_{\text{LT},n}\text{for}k=0,1,...,d\) and put \(\Omega_{\text{LT}}^{k,\text{sm}}:=\) lim \(=_{\text{LT}}\Omega^{0,\text{sm}}\Omega^{1,\text{sm}}\) Proposition 0.3.3. There exists a differential operator \(d:\mathcal{O}_{\text{LT}}\rightarrow\mathcal{O}_{\text{LT}}\otimes_{\mathcal{O}_{\mathrm{L}\mathrm{T}}^{\text{sm}}}\Omega_{\mathrm{L}\mathrm{T}}^{1,\text{sm}}\) , such that + +\(\pi_{n}:\mathcal{M}_{\text{LT},\infty}\rightarrow\mathcal{M}_{\text{LT},n}\). Clearly \(\mathcal{O}^{\text{sm}}\) + +LT and \(\Omega^{k,\text{sm}}\) + +\(=_{\text{LT}}\wedge_{\mathcal{O}_{\mathrm{L}\mathrm{T}}^{\text{sm}}}^{k}\) + +(i) \(d\) is given by the usual derivation on \(\mathcal{O}_{\mathrm{L}\mathrm{T}}^{\text{sm}}\), + +(ii) \(d\) is \(\pi^{-1}\mathcal{O}_{\mathscr{F}\ell}-\text{linear}\). + +Proof. Define \(d|_{\mathcal{O}_{\mathrm{L}\mathrm{T}}^{\text{sm}}}\) as the differential map on finite levels \(\mathcal{O}^{\text{sm}}\) + +\(\mathcal{M}_{\text{LT},\infty}\). Let + +\(\longrightarrow n\pi_{n}^{-1}\Omega_{\mathcal{M}_{\text{LT},n}}^{k}\) with + +LT . + +LT , and define \(d|_{\pi^{-1}\mathcal{O}_{\mathscr{F}\ell}}\) to be the zero + +Moreover, \(d\) is uniquely determined by these two properties up to constants. \(\rightarrow_{\text{LT}}\Omega^{1,\text{sm}}\) map. By Theorem 0.3.1, for any \(U\in B_{\text{LT}}\text{such that}z_{d}\neq 0\text{on}U\), we may write any section \(s\in\mathcal{O}_{\text{LT}}(U)\) of the form for some sufficiently large \(n\): + +\[\begin{aligned} s&=\sum_{i_{0}=0}^{\infty}\sum_{i_{1}=0}^{\infty}\cdots \sum_{=_{d-}}^{\infty_{1}}c_{i_{0}\ldots i_{d-1}}(x_{0}-x_{0,n})^{i_{0}}\cdots (x_{d-1}-x_{d-1,n})^{id-1} \\ i0 \end{aligned}\] diff --git a/outputs/arxiv_math/2503.09367_pg9_pg1_repeat1.md b/outputs/arxiv_math/2503.09367_pg9_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..504be2382f782db5489158dfbc2e5e6346168621 --- /dev/null +++ b/outputs/arxiv_math/2503.09367_pg9_pg1_repeat1.md @@ -0,0 +1,292 @@ +\(z_{1}\) + +\[z_{c}\] + +\(F^{\prime}\) + +\[b_{q}\] + +\[a_{p-1}\] + +\[b_{q-1}\] + +\(a_{1}\) + +\(w_{s}\) + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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+ +\[x_{B_{s}^{1}}x_{2}\] + +\[B_{s-1}\] + +\[B_{s+1}\] + +Figure 3: Some notations on \(B_{s}\). + +a contradiction. If \(|G_{2}|\geq t,\text{then}|B_{r}|=|G_{2}|\geq t\text{and}m(B_{r})36k^{\log_{2}3}.\] (2) + +Let \(G\) be a \(2C_{k}-\)free plane graph with \(e(G)=ex_{\mathcal{P}}(n,2C_{k})\). It is clear that \(G\) is connected; otherwise we can add an edge between two components of \(G\) to make it remain \(2C_{k}-\text{free}\), a contradiction. If \(G\) is \(C_{k}-\text{free}\), then \(\overline{36k^{1+\text{log}_{2}3}} +^{\underline{k^{2}}}\) + +\(n\) + +by Theorem 1.3, \(e(G)\leq3n-6- \frac{n}{4k^{\text{log}_{2}3}} \leq3n-6-\) contains a \(C_{k}\). + +## Case \(\mathbf{1}.G\) is 2-connected. + +2 , the result holds. So, we assume that \(G\) + +In that case, the boundary of each face in \(G\) is a cycle. Assume that \(c\) is the smallest number of vertices that need to be removed from \(G\) to make it \(C_{k}-\text{free}\), and let \(v_{1},v_{2},\ldots,v_{c}\) be one such set of vertices. Then \(c\leq k\), since we can remove vertices of a \(k-\)cycle to make it \(C_{k}-\text{free}\). Moreover, for any \(i,j\in[c]\), there exists a \(k-\text{cycle in}G\text{such that}v_{i}\) is contained in the \(k-\text{cycle}\), but \(v_{j}\) not. + +## Case 2.1. \(c=1\). + +Since \(G\) is 2-connected, it follows that \(G^{\prime}=G-v_{1}\) is connected and is \(C_{k}-\text{free}\). Assume that \(G^{\prime}\) has \(r\) blocks \(B_{1},B_{2},\ldots,B_{r},\text{where}|B_{i}|\geq k^{\log_{2}3}\text{for}i\in[q]\text{and}|B_{i}|0\text{and}0<\delta\leq1\). Here, we will extend this proposition 2.2 to optimal control of fractional PDE in our main result. The first proposition represents about the Poincaré inequality in fractional Sobolev space and the second refers to the minimizer of a function defined on a suitable space. + +Proposition 3.2 (Poincaré inequality, [4]). Let \(s\in(0,1),\Omega\subset\mathbb{R}^{N}\) be an open and bounded set then we have + +\[\Vert u\Vert_{2}^{2}\leq C(N,\Omega,s)[u]_{H^{s}(\Omega)}^{2}.\] + +where some constant \(C(N,\Omega,s)\) depending upon \(N,\Omega\text{and}s\). + +Proposition 3.3. Let the admissible control space \(U_{ad}\) be weakly closed, bounded subset of \(L^{2}(\Omega)\text{with}J_{s}:U_{ad}\rightarrow\mathbb{R}\) is weakly lower semi-continuous. Then \(J_{s}\) has minimizer in \(U_{ad}\). + +The existence and uniqueness of optimal control via strict convex and lower semicontinuous is discussed in the following proposition. diff --git a/outputs/arxiv_math/2503.09398_pg1_pg1_repeat1.md b/outputs/arxiv_math/2503.09398_pg1_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..abd1b92611954279752b0c37e6ef186cdb81963e --- /dev/null +++ b/outputs/arxiv_math/2503.09398_pg1_pg1_repeat1.md @@ -0,0 +1,31 @@ +# Precoder Learning by Leveraging Unitary Equivariance Property + +Yilun Ge, Shuyao Liao, Shengqian Han, Chenyang Yang School of Electronics and Information Engineering, Beihang University, Beijing 100191, China Email: \(\{\text{yilunge}\), shuyaoliao, sqhan, cyyang\(\}@\text{buaa.edu}\).cn + +Abstract—Incorporating mathematical properties of a wireless policy to be learned into the design of deep neural networks (DNNs) is effective for enhancing learning efficiency. Multi-user precoding policy in multi-antenna system, which is the mapping from channel matrix to precoding matrix, possesses a permutation equivariance property, which has been harnessed to design the parameter sharing structure of the weight matrix of DNNs. In this paper, we study a stronger property than permutation equivariance, namely unitary equivariance, for precoder learning. We first show that a DNN with unitary equivariance designed by further introducing parameter sharing into a permutation equivariant DNN is unable to learn the optimal precoder. We proceed to develop a novel non-linear weighting process satisfying unitary equivariance and then construct a joint unitary and permutation equivariant DNN. Simulation results demonstrate that the proposed DNN not only outperforms existing learning methods in learning performance and generalizability but also reduces training complexity. + +## Index Terms—Precoding, deep learning, unitary equivariant, permutation equivariant + +## I. INTRODUCTION + +The optimization of multi-user multi-input multi-output (MU-MIMO) precoder is a challenging problem. Various numerical algorithms have been developed, e.g., the weighted minimum mean square error (WMMSE) algorithm [1], which, however, are with high computational complexity. Deep learning offers a promising solution for optimization due to its low inference complexity. In [2], a fully-connected deep neural network (DNN) was trained to approximate the performance of WMMSE with significantly reduced inference complexity. However, such DNNs often involve a large number of learnable parameters, requiring extensive training time and massive training samples. Incorporating inductive biases into DNNs architecture with mathematical properties of the policies being learned is crucial for enhancing learning efficiency [3,4]. A well-known example is convolutional neural network (CNN), which leverages the property of translational invariance and has significantly promoted the field of image recognition [5]. Recent studies have demonstrated the importance of integrating permutation equivariance properties to enhance learning efficiency for wireless communications [6–12]. In [8], the mismatch between the inductive bias of CNNs and the permutation equivariance property of the precoding policy was identified, and an edge-updated graph neural network (GNN) was proposed to leverage the permutation equivariance property. In [9], the attention mechanism was introduced into GNN and a graph attention network (GAT) was used to learn the energy efficient precoder. In [10], a GNN was designed by introduce Taylor’s expansion of matrix pseudo-inverse into the model, which improves learning efficiency. In [11], a multidimensional GNN was developed for hybrid precoder learning, which is able to exploit complicated permutation equivariance properties. In this paper, we study the learning of MU-MIMO precoder by investigating a stronger property than permutation equivariance, known as unitary equivariance. This property means that if the channel matrix is multiplied by a unitary matrix, then the optimal precoder will be multiplied by the same unitary matrix. The only instance of a unitary equivariant DNN design was introduced in [13]. However, it lacked a systematic analysis of how the unitary equivariance property impacts DNN design, and its performance in precoder learning is unsatisfactory, as to be demonstrated later. In this paper, we first show that a DNN with a parameter sharing structure derived from the unitary and permutation equivariance property is unable to learn the optimal precoder. To solve the problem, we then develop a non-linear weighting process that satisfies the unitary equivariance, and finally construct a joint unitary and permutation equivariant neural network (UPNN) for precoder learning. Simulation results demonstrate the advantages of the proposed UPNN in performance improvement and training complexity reduction. + +## II. EQUIVARIANCE PROPERTY OF A PRECODING POLICY + +Consider a downlink MU-MIMO system where a base station (BS) equipped with \(N\) antennas serves \(K\) single-antenna users. The channel matrix is denoted as \(\mathbf{H}=[\mathbf{h}_{1},\ldots,\mathbf{h}_{K}]\in \mathbb{C}^{N\times K},\text{where}\mathbf{h}_{k}\in\mathbb{C}^{N\times1}\) is the channel vector of user \(k\). The precoding matrix is denoted as \(\mathbf{V}=[\mathbf{v}_{1},\ldots,\mathbf{v}_{K}]\in\mathbb{C}^{N\times K}\), where \(\mathbf{v}_{k}\in\mathbb{C}^{N\times1}\) is the precoding vector of user \(k\). The precoder optimization problem that maximizes the sum rate subject to the total power constraint is + +\[\max_{\mathbf{V}}\sum_{k=1}^{K}R_{k}\] (1a) + +\[\mathrm{s}.\mathrm{t}.\text{Tr}\left(\mathbf{V}^{\mathrm{H}}\mathbf{V}\right)\leq P_{\text{max}},\] (1b) + +2 + +\(|\mathbf{h}^{\mathrm{H}}\) + +\[\mathbf{v}_{kk}^{\mathrm{H}}|\] + + is the data rate of + +\[\sum_{m\neq k}|\mathbf{h}_{k}^{\mathrm{H}}\mathbf{v}_{m}|^{2}+\delta_{k}^{2}\] + +where \(R_{k}=\log_{2}\Big(1+\) user \(k,\delta_{k}^{2}\) is the noise power, and \(P_{\text{max}}\) is the maximal transmit diff --git a/outputs/arxiv_math/2503.09421_pg9_pg1_repeat1.md b/outputs/arxiv_math/2503.09421_pg9_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..2a15f740020ba61d1752c8c5b79aae6bddfe6221 --- /dev/null +++ b/outputs/arxiv_math/2503.09421_pg9_pg1_repeat1.md @@ -0,0 +1,31 @@ +where the constant \(c\) is independent of \(A\text{and}n\) (but does depend on the dimension of \(\text{Ran}(P))\). We use that \(\int_{1}^{\infty}t^{s-2}dt<\infty\text{for}s<1\) to bound the second term. Thus, there exists a constant \(C(s)\) such that + +\[\int_{n}^{n+1}\Vert(A+x)^{-1}\Vert^{s}dx\leq C(s)\] + +\[\sum_{n\in\mathbb{Z}}\] + +for all maximally dissipative \(A,\text{where}C(s)\) is independent of \(n\text{and Ran}(P)\). Using that \(\frac{1}{1+(|n|-1)^{2}} <\infty\), we can therefore bound the right hand side of (9) for all \(|z|>1\). If \(|z|<1\), we drop a minus sign in both norms in (9) and use that \(i\widehat{F}_{z}\text{and}i\widehat{F}_{z}^{-1}\) are dissipative. Repeating the arguments from above yields the bound for all \(|z|<1\). The case \(x^{(i)}=y^{(j)}\) is significantly easier: We can directly take \(\alpha=\omega_{x}\) and do not need \(\beta\). The operators \(F_{z}\text{and}\widehat{F}_{z}\) act on \(\mathbb{C}^{3}\), while all other estimates still hold. The integrals in (9) can be similarly bounded by Lemma 1. + +We state a simplified version of Lemma 3.1 from [3], which we use to bound the integrals in the proof of Theorem 2. Lemma 1. Let \(\mathcal{H}\) be a separable Hilbert space, \(A\) be a maximally dissipative operator with strictly positive imaginary part and \(M_{1},M_{2}:\mathcal{H}\rightarrow\mathcal{H}\) be Hilbert-Schmidt operators. Then there exists a constant \(c\) independent of \(A,M_{1}\text{and}M_{2}\) such that for any \(t>0\): + +\[|\left\{x\in\mathbb{R}s.t.\Vert M_{1}(A+x)^{-1}M_{2}\Vert>t\right\}|\leq c\Vert M_{1}\Vert_{HS}\Vert M_{2}\Vert_{HS} \frac{1}{t} ,\] + +where \(|\cdot|\) denotes the Lebesgue measure. + +### 4.2 The boundary of a box + +\((L)\Lambda_{L}^{C}\) + +We want to define a ”box” \(\Lambda_{L}\) for any size \(L\in\mathbb{N}^{2}\), whose sides have lengths \(L_{1}\text{and}L_{2}\), such that the Quantum Walker is unable to cross the boundary of \(\Lambda_{L}\). Restricting the Walker to some box \(\Lambda_{L}\) is achieved by changing the coin matrix at specific lattice sites on the boundary of \(\Lambda_{L}\). In other words, we want to obtain unitary operators \(U_{\omega}=U_{\omega}^{\Lambda_{L}}\oplus U_{\omega}\) and subspaces \(\mathcal{H}_{L}\oplus\mathcal{H}_{L}^{C}=\mathcal{H}\text{such that}\mathcal{H}_{L}\), respectively \(\mathcal{H}_{L}^{C}\), are invariant under \(U_{\omega}^{\Lambda_{L}}\) , respectively \(U_{\omega}^{\Lambda_{L}^{C}}\) . Note that we call a subspace \(\mathcal{H}^{\prime}\subset\mathcal{H}\) invariant under \(U\text{if}U\mathcal{H}^{\prime}\subset\mathcal{H}^{\prime}\). Recalling the definition of \(\mathcal{H}^{j,k}(4)\), we use that \(C_{0}\) induces a fully localized Quantum Walk, see section 2, and define the invariant subspaces: + +\[\mathcal{H}_{L}=\bigoplus_{-_{j}-_{+L}L_{k^{21}>^{\leq}}\leq_{-k}j\leq_{\leq_{L}}L_{L_{-_{1}^{1_{2}}}}-_{-_{L}}1_{1_{2}}}\mathcal{H}^{j,k},\mathcal{H}_{L}^{C}=\mathcal{H}\setminus\mathcal{H}_{L}.\] + +The choice \(j+k>-L_{1}-L_{2}\) is not necessary, but simplifies the structure of \(\Lambda_{L}\), see Figure + +3. We call the number of \(\Gamma_{A}-\text{vertices in}\Lambda_{L}\) the volume of \(\Lambda_{L}\), that is: + +\[\text{vol}(\Lambda_{L})=4L_{1}L_{2}-1\text{and}|L|= \sqrt{L_{1}^{2}+L_{2}^{2}} .\] (10) + +To obtain a Quantum Walk such that these two subspaces are invariant, we need to change the coin matrix at specific lattice sites from \(C\) to \(C_{0}\). In particular, we use the coin matrix \(C_{0}\) at all \(\Gamma_{B}\) sites in + +\[\begin{aligned} \Gamma_{C_{0}}^{(L)}&= \begin{cases} \left(\right) & \Big( & \Big) \\ 0 \\ |j,k\rangle\otimes & \text{s.t.} & -L_{1}\leq j\leq L_{1}-1,k=L_{2}-1 & \text{or} \\ 1 \end{cases} \\ \left(-L_{1}+1\leq j\leq L_{1},k=-L_{2}-1\right)\text{or}\left(j=L_{1},-L_{2}\leq k\leq L_{2}-2\right) \\ \text{or}\left(j=-L_{1},-L_{2}\leq k\leq L_{2}-2\right)\Big\}. \end{aligned}\] diff --git a/outputs/arxiv_math/2503.09424_pg5_pg1_repeat1.md b/outputs/arxiv_math/2503.09424_pg5_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..a67ca91aa34452f2505b657eae92b8e34cf47807 --- /dev/null +++ b/outputs/arxiv_math/2503.09424_pg5_pg1_repeat1.md @@ -0,0 +1,31 @@ +al. [14], enhance fuel efficiency but impose high computational costs due to system nonlinearities. An alternative approach, the Equivalent Consumption Minimization Strategy (ECMS) [25], reduces computational complexity while delivering near-optimal fuel savings by relying on instantaneous slope data. Both predictive and non-predictive strategies are viable for connected automated vehicles (CAVs), yet achieving a balance between fuel efficiency and computational feasibility remains a key challenge for practical deployment. + +### 1.2 Outline of the paper + +In Section 2 we introduce a convex relaxation of the problem. In Section 3 we briefly sketch the exactness result for this relaxation (under suitable assumptions) when we drop the boundary condition \(w_{n}=w_{\text{fni}}\) at step \(n\), already proved in our paper [2]. Moreover, we show that the relaxation is not exact any more as soon as we include also the boundary condition. In Section 4 we introduce a feasibility-based bound tightening technique, while in Section 5 we prove that the addition of the lower limit for the variables obtained through the bound tightening procedure makes the convex relaxation exact. Next, in Section 6 we introduce further feasibility-based bound tightening techniques. In Section 7 we prove that the feasible region of our problem is a lattice and from that we derive a necessary and sufficient condition to establish its non-emptiness, also discussing a procedure to verify such condition, based on the iterated application of the bound tightening techniques discussed in Sections 4 and 6. Finally, in Section 8 we draw some conclusions. + +## 2 Convex relaxation of the problem + +The constraints of our problem are the following: + +\[\frac{M}{P_{\text{max}} \sqrt{w_{i}}} \geq \frac{1}{h} (w_{i+1}-w_{i})+\gamma w_{i}+g(\sin\alpha_{i}+c)i\in\{1,\ldots,n-1\}\] (4) + +\[\begin{aligned} \frac{1}{h} (w_{i+1}-w_{i})+\gamma w_{i}+g(\sin\alpha_{i}+c)&\leq g\mu i\in\{1,\ldots,n-1\} \\ \frac{1}{h} (w_{i+1}-w_{i})+\gamma w_{i}+g(\sin\alpha_{i}+c)&\geq-g\mu i\in\{1,\ldots,n-1\} \end{aligned}\] (5) (6) + +\[\begin{aligned} \overline{h} (w_{i+1}-w_{i})+\gamma w_{i}+g(\sin\alpha_{i}+c)&\geq-g\mu i\in\{1,\ldots,n-1\} \\ w_{i}&\leq w_{i}^{\text{max}}i\in\{1,\ldots,n\} \\ w_{i}&\geq0i\in\{1,\ldots,n\} \end{aligned}\] (6) (7) (8) + +\(w_{1}=w_{\text{init}},w_{n}=w_{\text{fni}}.\) + +(9) + +We rewrite constraints (7)–(9) as follows: + +\[\begin{aligned} w_{i}&\leq w_{i}^{\text{max}}i\in\{1,\ldots,n\} \\ w_{i}&\geq w_{i}^{\text{min}}i\in\{1,\ldots,n\}, \end{aligned}\] (10) (11) + +where \(w_{i}^{\text{min}}=0,\text{for}i\in\{2,\ldots,n-1\}\), while \(w_{1}^{\text{min}}=w_{1}^{\text{max}}=w_{\text{init}}\text{and}w_{n}^{\text{min}}=w_{n}^{\text{max}}=w_{\text{fni}}\) (note that the two boundary conditions (9) are split into the constraints \(w_{\text{init}}\leq w_{1}\leq w_{\text{init}}\) and \(w_{\text{fni}}\leq w_{n}\leq w_{\text{fni}})\). Moreover, after introducing the variables \(t_{i},f_{i},i=1,\ldots,n-1\), we can replace the maximum power constraints (4) with the following constraints: + +\[t_{i}= \frac{1}{\sqrt{w_{i}}} i\in\{1,\ldots,n-1\}\] (12) + +\[t_{i}\geq \frac{Mf_{i}}{P_{\text{max}}} i\in\{1,\ldots,n-1\}\] (13) + +\[f_{i}= \frac{1}{h} (w_{i+1}-w_{i})+\gamma w_{i}+g(\sin\alpha_{i}+c)i\in\{1,\ldots,n-1\}.\] (14) diff --git a/outputs/arxiv_math/2503.09432_pg6_pg1_repeat1.md b/outputs/arxiv_math/2503.09432_pg6_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..7a165490bfd5ce30161c202102741ea940856496 --- /dev/null +++ b/outputs/arxiv_math/2503.09432_pg6_pg1_repeat1.md @@ -0,0 +1,3 @@ +the Semisimplicity conjecture. Moreover, all rational maps on \(Y\) satisfies the Dynamical degree comparison conjecture. + +An explicit example is the Hilbert scheme \(X^{[2]}\) of zero cycles of length 2 of appropriate varieties \(X\) (see Section 2 for more detail). For proving Theorem 1.1, we combine the ideas in [HT24] to that in [Xie24]. The first main idea in [Xie24] is, over a finite field \(\mathbb{F}_{q}\), to each finite correspondence \(f\) and each \(k= 0,1,\ldots,n\), consider a composition \(Fr_{q}^{s}\circ f^{t}\text{where}s\) is an integer (it is important to allow negative values for \(s)\text{and}t\) a positive integer. If the dynamical degrees \(\lambda_{k}(f)\) are a log-concave sequence, using that Frobenius morphisms commute with correspondences defined over the same finite field, with a suitable choice of \(s\text{and}t\) one can make sure that \(\lambda_{k}(Fr_{q}^{s}\circ f^{t})\) is not too far away compared to \(\max_{j=0,\ldots,n}\lambda_{j}(Fr_{q}^{s}\circ f^{t})\). Note that the choice of \(\mu_{i}\) in the proof of Lemma 2.13 in [Xie24] is the same as the choice of \(r\) in the proof of Lemma 4.9 in [HT24]. Note that for a finite correspondence \(f\), as in the case of endomorphisms, the cohomological dynamical degree \(\chi_{2k}(f)\) is the same as the spectral radius of \(f^{*}|_{H^{2k}(X)}\). Therefore, a known result on spectral radii comparison (Theorem 1.2 in [Tru24]), which is a generalisation of Entropy comparison conjecture, can be used to show that \(\chi_{2k}(f)=\lambda_{k}(f)\). The inequality \(\chi_{2k+1}(f)^{2}\leq\lambda_{k}(f)\lambda_{k+1}(f)\) is obtained from this by the trick of using the product correspondence \(f\times f:X\times X\rightarrow X\times X\). Thus, Dynamical degree comparison conjecture holds for finite correspondences whose numerical dynamical degrees are a log-concave sequence. In the general case, the numerical dynamical degrees \(\lambda_{k}(f)\) may not be log-concave. The second main idea in [Xie24] is (see Theorem 1.14 therein, and Proposition 1.4 below) to establish some interesting relations between certain polytopes of two sequences satisfying certain properties. Theorem 1.14 in [Xie24] can be viewed as a continuous version of Lemma 4.9 in [HT24] (see Proposition 1.4 below). To prove Theorem 1.1, here we replace the cohomological operator \(G_{r}\) in the approach in [HT24] by composing with iterations of \(F^{*}\). There is a complication arising in case \(f\) is not a finite correspondence: in this case the dynamical degree is not the spectral radius but involves norms of dynamical iterates of the correspondence \(f^{♦t}\). In case the dynamical degrees \(\lambda_{k}(f)\) are a log-concave sequence, then again Lemma 4.9 in [HT24] can be used to conclude Theorem 1.1. In the general case, we make use of special properties of the Frobenius norm on matrices and of asymptotic singular values of powers of a matrix (this property was used in [Hu19, Hu24] to solve Dynamical degree comparison conjecture for endomorphisms of Abelian varieties), as well as use essentially the second main idea in [Xie24] mentioned above, but in the following form which is similar to the form of Lemma 4.9 in [HT24]. (More precisely, in Lemma 4.9 in [HT24], Condition 2 below is required, but instead of Condition 1 below it is required that the sequence \(a_{j}\) is log-concave.) Proposition 1.4. Let \(a_{0},a_{1},\ldots,a_{n},\text{and}b_{0},b_{1},\ldots,b_{2n}\) be non-negative numbers. Assume the following two conditions are satisfied: Condition 1: \(b_{2j}\geq a_{j}forallj=0,\ldots,n\). diff --git a/outputs/arxiv_math/2503.09434_pg25_pg1_repeat1.md b/outputs/arxiv_math/2503.09434_pg25_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..b38ac310d847415f821cec1298145f29c4c3fc65 --- /dev/null +++ b/outputs/arxiv_math/2503.09434_pg25_pg1_repeat1.md @@ -0,0 +1,33 @@ +Proof. In the first case, we set \(v_{p}=(\nabla X|_{p})^{-1}w_{p}\) and apply the Cauchy-Schwarz inequality to obtain + +\[\begin{aligned} \langle\nabla_{v_{p}}X,v_{p}\rangle&=\langle w_{p},(\nabla X|_{p})^{-1}w_{p}\rangle\leq\Vert(\nabla X|_{p})^{-1}\Vert\Vert w_{p}\Vert^{2}=\Vert(\nabla X|_{p})^{-1}\Vert\Vert\nabla_{v_{p}}X|\Vert^{2} \\ \text{so}-\sup_{p\in\mathcal{U}}\Vert(\nabla X|_{p})^{-1}\Vert\text{can be taken as an upper bound for}\alpha_{p}. \end{aligned}\] + +In the second case, we choose \(v_{p}=r_{p}+\beta n_{p}\text{where}r_{p}\in\text{Range}(\nabla X|_{p}^{*}),\beta\in\mathbb{R}\) and \(n_{p}\in\ker(\nabla X|_{p})\neq 0\). Then, the condition (25) reads + +\[\langle\nabla_{r_{p}}X,r_{p}\rangle+\beta\langle\nabla_{r_{p}}X,n_{p}\rangle\leq-\alpha\Vert\nabla_{r_{p}}X\Vert^{2}\] + +where \(\beta\) is arbitrary. Therefore, it is necessary that \(\langle\nabla_{r_{p}}X,n_{p}\rangle=0\) which is true for all choices of \(r_{p}\text{and}n_{p}\) if and only if \(\nabla_{r_{p}}X\in\text{Range}(\nabla X|_{p}^{*})\). By definition, the set of all \(\nabla_{r_{p}}X|_{p}\) is precisely \(\text{Range}(\nabla X|_{p})\), so we conclude that one must have \(\text{Range}(\nabla X|_{p})=\text{Range}(\nabla X|_{p}^{*})\). The proof is completed in the same way as in the non-singular case, by restricting \(\nabla X|_{p}\) to \(\text{Range}(\nabla X|_{p}^{*})\). + +Note that this proposition does not ensure that the vector field \(X\) is cocoercive since it might happen (as in the bound of the proof) that \(\alpha<0\). + +# A coordinate formula for the cocoercivity constant. + +We give a concrete formula for the cocoercivity constant when it exists. Suppose we have introduced local coordinates \(x_{1},\ldots,x_{d}\). Now \(g\) is to be interpreted as a \(d\times d\) matrix with elements \(g_{ij}=\langle\partial x_{i},\partial x_{j}\rangle\). The matrix of the operator \(\nabla X|_{p}\) is denoted \(\mathcal{A}_{X,p}\). We assume that we can compute its reduced singular value decomposition \(\mathcal{A}_{X,p}=U\Sigma V^{T}\text{where}U,V\in\mathbb{R}^{d\times r}\) have orthonormal columns and \(\Sigma\in\mathbb{R}^{r\times r}\) is diagonal and positive definite and where \(r\) is the rank of \(\mathcal{A}_{X,p}\). Thanks to Proposition 12 we can write \(U=VQ\text{where}Q\) is orthogonal and \(r\times r\), i.e. \(Q=V^{T}U\). We are interested in finding the largest possible \(\alpha_{p}\) such that + +\[v_{p}^{T}g\mathcal{A}_{X,p}v_{p}\leq-\alpha_{p}v_{p}^{T}\mathcal{A}_{X,p}^{T}g\mathcal{A}_{X,p}v_{p},\forall v_{p}\in\mathbb{R}^{d}\] + +It is easy to show that \(-\alpha_{p}\) can be chosen as the largest eigenvalue + +\[-\alpha_{p}=\max_{1\leq i\leq d}\lambda_{i}\left( \frac{M+M^{T}}{2} \right)\] + +with + +\[M=\tilde{g}^{- \frac{1}{2}}\Sigma^{-1}V^{T}U\tilde{g}^{\frac{1}{2}},\tilde{g}=U^{T}gU.\] + +To find \(\alpha\) it suffices to maximize \(\alpha_{p}\) over the domain \(\mathcal{U}\subset M\). Formulas for \(\mu_{+}\text{and}\mu_{-}\) in Theorems 6 and 9 can be derived similarly. In the case of \(\nabla X\) non-singular, we obtain + +\[\mu_{+}=\lambda_{\text{max}}\left( \frac{M_{+}+M_{+}^{T}}{2} \right),\mu_{-}=\lambda_{\text{max}}\left( \frac{M_{-}+M_{-}^{T}}{2} \right),\] + +where \(\lambda_{\text{max}}(A)\) denotes the maximum eigenvalue \(A\) and with + +\[M_{+}=-G^{1/2}(I-P_{X})\nabla X^{-1}G^{-1/2},M_{-}=-G^{1/2}P_{X}\nabla X^{-1}G^{-1/2}.\] diff --git a/outputs/arxiv_math/2503.09440_pg14_pg1_repeat1.md b/outputs/arxiv_math/2503.09440_pg14_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..1d1aed1fb8d8caf052f6dfc62d8a969683963586 --- /dev/null +++ b/outputs/arxiv_math/2503.09440_pg14_pg1_repeat1.md @@ -0,0 +1,51 @@ +subgraph of \(G\) is the intersection graph of a compatible collection of subtrees of \(T\) . Thus, by Theorem 3.2 it suffices to show that \(G\) has a simple vertex. Let \(V(G)=\{v_{1},\ldots,v_{n}\}\) and let \(T_{1},\ldots,T_{n}\) be subtrees of \(T\text{in}\Lambda\) satisfying \(v_{i}\sim v_{j}\text{in}G\) if and only if \(V(T_{i})\cap V(T_{j})\neq \emptyset\). For each \(i\), let \(w_{i}\) be the unique vertex of \(T_{i}\) of minimum weighted distance to \(r\). We may assume, without loss of generality, that \(d_{T}(w_{1},r)\geq d_{T}(w_{j},r)\) for \(j=2,\ldots,n\). We claim that \(v_{1}\) is simple in \(G\). Suppose that \(v_{i},v_{j}\in N_{G}(v_{1})\text{and}T_{i}>T_{j}\). It suffices to show that \(N_{G}(v_{j})\subseteq N_{G}(v_{i})\). Since + +and + +it follows that + +Similarly + +Since \(T_{i}>T_{j}\) we have + +Let \(v_{k}\in N_{G}(v_{j})\). Then + +\[d_{T}(w_{1},r)\geq d_{T}(w_{j},r)\] + +\[V(T_{1})\cap V(T_{j})\neq \emptyset\] + +\[w_{1}\in V(T_{j}).\] + +\[w_{1}\in V(T_{i}).\] + +\[w_{j}\in V(T_{i}).\] + +\[V(T_{j})\cap V(T_{k})\neq \emptyset.\] + +If \(w_{j}\in V(T_{k})\text{then}V(T_{i})\cap V(T_{k})\neq \emptyset\). Otherwise + +and + +whence + +since \(T_{i}>T_{j}\). In either event + +and so + +\[w_{k}\in V(T_{j})\] + +\[d_{T}(r,w_{k})\leq d_{T}(r,w_{1})\] + +\[w_{1}\in V(T_{i})\cap V(T_{j})\] + +\[w_{k}\in V(T_{i})\] + +\[V(T_{i})\cap V(T_{k})\neq \emptyset\] + +\[v_{k}\in N_{G}(v_{i}).\] + +Since \(v_{k}\) was an arbitrary vertex of \(N_{G}(v_{j})\), it follows that + +\[N_{G}(v_{j})\subseteq N_{G}(v_{i})\] + +. diff --git a/outputs/arxiv_math/2503.09455_pg21_pg1_repeat1.md b/outputs/arxiv_math/2503.09455_pg21_pg1_repeat1.md new file mode 100644 index 0000000000000000000000000000000000000000..c71635d9d885ae2e290b58c4c43857cfbc2815cc --- /dev/null +++ b/outputs/arxiv_math/2503.09455_pg21_pg1_repeat1.md @@ -0,0 +1,48 @@ +Proposition 5.1. Let \((M,\omega)\) be a closed symplectic manifold. Assume \(^^{c}(M,\omega)\) is not extendable to \(^_{0}^{c}(M,\omega).\text{Then}R_{0}\in\mathrm{H}^{2}(\text{Symp}_{0}^{c}(M,\omega))\) is non-zero. In particular, the Reznikov class \(R\in\mathrm{H}^{2}(\text{Symp}^{c}(M,\omega))\) is non-zero. + +that Shelukhin’s quasimorphism \(S_{M}\) on Ham Symp + +\(^^{c}(M,\omega)\) and set + +Before proving the proposition, we discuss the vanishing of the Reznikov class in a more general setting. Let \(M\) be a closed symplectic manifold. Let \(\tilde{G}\) be a subgroup of Symp \(G=p(\tilde{G}),\text{where}p\): Symp map. Consider the following commutative diagram: + +\(^_{0}^{c}(M,\omega)\) which contains Ham \(^_{0}^{c}(M,\omega)\rightarrow\text{Symp}_{0}^{c}(M,\omega)\) is the universal covering + +\[\begin{aligned} \tilde{}\tilde{} \\ \text{Ham}^{c}(M,\omega)^{\underline{i_{0}}}\tilde{G}^{\underline{i_{1}}}\text{Symp}_{0}^{c}(M,\omega) \\ ppp \\ \text{Ham}^{c}(M,\omega)^{\underline{i_{0}}}G^{\underline{i_{1}}}\text{Symp}_{0}^{c}(M,\omega). \end{aligned}\] + +| Ham ^c(M, ω) | ie0 | Ge | ie1 | Symp ^c0(M, ω) | +| --- | --- | --- | --- | --- | +| p | | p | | p | +| Hamc(M, ω) | i0 | G | i1 | Sympc0(M, ω). | + +Here \(\widetilde{i}_{0},\widetilde{i}_{1},i_{0}\text{and}i_{1}\) are the inclusions. Lemma 5.2. The following are equivalent. + +\((1)i_{1}^{*}R_{0}\in\mathrm{H}^{2}(G)\) is zero. + +2. There exists \(\phi\in\mathrm{Q}(G)\text{such that}\widetilde{i}_{0}^{*}p^{*}\phi=S_{M}\). + +Proof. To prove that (1) implies (2), we assume that \(i_{1}^{*}R_{0}=0\). Recall that \(b_{J}\) is a cocycle representing the Reznikov class. Then there exists \(u\in\mathrm{C}^{1}(G)\) such that \(i_{1}^{*}b_{J}=\delta u\). Since \(b_{J}\) is a bounded cocycle, \(u\) is a quasimorphism. Recall from (2.4) that \(-\delta\nu_{J}=p^{*}i_{0}^{*}i_{1}^{*}b_{J}\). Hence we have + +cause Ham + +\[-\delta\nu_{J}=p^{*}i_{0}^{*}\delta u=\delta(p^{*}i_{0}^{*}u)=\delta(\widetilde{i}_{0}^{*}p^{*}u).\] + +\(p^{*}u\): Ham + +\(p^{*}u\). Let \(\phi\) be the + +This implies that \(\nu_{J}+\widetilde{i}_{0}^{*}^^{c}(M,\omega)\rightarrow\mathbb{R}\) is a homomorphism. Be-\(^^{c}(M,\omega)\) is perfect ([Ban78]), we have \(\nu_{J}=-\widetilde{i}_{0}^{*}\) homogenization of \(-u\). Then we have \(S_{M}=\widetilde{i}_{0}^{*}p^{*}\phi\). To prove that (2) implies (1), we assume (2) and take \(\phi\in\mathrm{Q}(G)\) satisfying \(\widetilde{i}_{0}^{*}p^{*}\phi=S_{M}\). Since \(S_{M}\) is the homogenization of \(\nu_{J}\), there exists a bounded \(^^{c}(M,\omega)\rightarrow\mathbb{R}\text{such that}S_{M}=\nu_{J}+v\). Then we have + +function \(v\) : Ham + +\[-\delta\nu_{J}=-\delta(S_{M}-v)=\delta v-\delta\widetilde{i}_{0}^{*}p^{*}\phi=\delta v-p^{*}i_{0}^{*}\delta\phi.\] + +Together with \(-\delta\nu_{J}=p^{*}i_{0}^{*}i_{1}^{*}b_{J}\), we have + +\[p^{*}i_{0}^{*}(i_{1}^{*}b_{J}+\delta\phi)=\delta v.\] + +\(^^{c}(M,\omega)\) is zero. + +Note that \(i_{1}^{*}b_{J}+\delta\phi\) is a bounded cocycle on \(G\). Since \(v\) is a bounded function, the second bounded cohomology class \(p^{*}i_{0}^{*}[i_{1}^{*}b_{J}+\delta\phi]\) of Ham Since \(G/\text{Ham}^{c}(M,\omega)\) is abelian, the bounded cohomology \(\mathrm{H}_{b}^{2}(G/\text{Ham}^{c}(M,\omega))\) is zero. In particular, the map \(i_{0}^{*}:\mathrm{H}_{b}^{2}(G)\rightarrow\mathrm{H}_{b}^{2}(\text{Ham}^{c}(M,\omega))\) is injective by \(^^{c}(M,\omega)\rightarrow\text{Ham}^{c}(M,\omega)\) is surjective, the map + +(2.2). Moreover, since \(p\): Ham