TrueDoc run 97 on olmOCR-bench: 86.8, with all 1,403 page outputs and the scorer's log (part 2)

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+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
+
+\[C_{V}\left(u_{n}^{V}X_{n}^{\text{Test}}, \overline{\mathrm{x}}^{\text{Vacc}} \right).\]

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+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

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+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

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+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
+
+\[(\mathrm{P}3)|D|\leq\alpha N/(32d)\text{and}|N_{\Gamma}(D)\cap U|<d|D|/2.\]
+
+The first two properties imply that \(K_{k}\) is a minor of \(\Gamma\). By taking \(D=\emptyset,k=0\text{and}U=V(\Gamma)\), we have that \(P\) is not empty. Consider a partition
+
+\(V(\Gamma)=D\cup\left(\bigcup_{i=1}^{k}W_{i}\right)\cup U\text{in}P\) which maximises \(|D|\), tie-breaking by taking one which further maximises \(k\). We prove that then necessarily
+
+\[k\geq \frac{\alpha N}{64\ell} =:q.\]
+
+As \(q=\Theta( \sqrt{nt/\log n} )\), this establishes the theorem. Suppose, towards a contradiction, that this is note the case. That is, \(k<q\). Then
+
+\[|W|=\Bigg|\bigcup_{i=1}^{k}W_{i}\Bigg|<\alpha N/64,\]
+
+from which we conclude \(|U|\geq N/2\), with room to spare. The property (P3) in the proof of Theorem 1.4 is identical to the one used here, and the only property of \(W\) used in the proof of Claim 3.1 and Claim 3.2 is the upper bound on \(|W|\), which is identical to the one used here. Therefore, same as in the proof of Theorem 1.4, the following holds:
+
+- For every \(i\in[k]\) we have \(|N_{\Gamma}(W_{i})\cap U|\geq d|W_{i}|/2\), and
+
+- \(\Gamma[U]\) is a connected \((\beta/2,d/2)-\text{expander}\).
+
+Now we can finish the proof using Lemma 3.3. For each \(i\in[k]\), set \(U_{i}=N_{\Gamma}(W_{i})\cap U\). Then \(|U_{i}|\geq d\ell/2=:s.\text{For}k<i\leq q\), take \(U_{i}\subseteq U\) to be an arbitrary set of size \(s\). Apply Lemma 3.3 follows from \(d\geq t_{0}(\alpha)/4\) and the latter follows from \(d\leq \overline{n}\). We obtain a subset \(T\subseteq U\) of size
+
+with sets \(U_{1},\ldots,U_{q}\), which we indeed can do as \(qs>2|U|\text{and}|U|/s>\log|U|\), where the former
+
+\[|T|\leq K \frac{|U|}{s} \log\left( \frac{qs}{|U|} \right)_{\cdot} \frac{\log|U|}{\log d} \leq\ell,\]

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+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

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+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

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+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.

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+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).

outputs/arxiv_math/2503.06880_pg36_pg1_repeat1.md

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+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:
+
+\[1<r^{(}\sum_{i\leq n}\lambda_{i}f_{i}^{)}\leq\|r\|^{\|_{\|}}\|\sum_{\|_{i\leq n}}^{\|}\lambda_{i}f_{i}^{\|_{\|_{\|}}}\|_{\|}\leq1\]
+
+which is absurd. Hence for any \(\{f_{i}\}_{i\leq n}\subseteq X^{*}\text{and}\sigma\in(0,1)\):
+
+\[(r(f_{1}),\cdots ,r(f_{n}))\in\{(f_{1}(x),\cdots ,f_{n}(x))|x\in X_{\leq1}^{\}}\subseteq(1+\sigma)^{\{}(f_{1}(x),\cdots ,f_{n}(x))|x\in X_{\leq1}^{\}}\]
+
+or
+
+\[(r(f_{1}),\cdots ,r(f_{n}))\in^{\{}(f_{1}(x),\cdots ,f_{n}(x))|x\in X_{\leq1+\sigma}^{\}}\]
+
+- 
+
+Proposition 5.3. Any non-reflexive Banach space \(X\) will have a bounded bi-orthogonal system \((e_{i},f_{j})_{i,j\in\mathbb{N}}\) such that \(\sup_{n}\|\sum_{i\leq n}e_{i}\|<\infty or\sup_{n}\|\sum_{i\leq n}f_{i}\|<\infty\) Proof. Let \(r\in X^{**}\setminus Q(X)\) and suppose \(H\) is an arbitrary finite dimensional subspace of \(X\). Define:
+
+\[\rho=\sup^{\{}|r(y)|:y\in X_{=1}^{*}\cap H^{\}_{\perp}}.\]
+
+Suppose \(H=\text{Span}\{h_{1},h_{2},\cdots ,h_{n}\}\text{and}H^{*}=\text{Span}\{f_{1},f_{2},\cdots ,f_{n}\}\text{where}f_{i}(h_{j})=1\text{iff}i=j.\text{If}\rho=0\), then \(r\in(H^{\perp})^{\perp}\). Observe that \(Q(H)^{\perp}=Q^{\prime}(H^{\perp})\text{and}(H^{\perp})^{\perp}\subseteq Q^{\prime}(H^{\perp})_{\perp}.\text{Then}r\in Q(H)^{\perp_{\perp}}=Q(H)\subseteq Q(X)\), which is absurd. Hence \(\rho>0\). 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:
+
+\[\begin{aligned} \rho_{n}&=\sup^{\{}r(y)|y\in X_{=1}^{*}\cap E_{n}^{\}_{\perp}} \\ \text{such that}\beta_{1}&=r(y_{1})> \frac{1}{2} ,\beta_{n+1}=r(y_{n+1})\in^{(} \frac{1}{2} \rho_{n},\rho_{n}\text{and for each}i,j\in\{1,2,\cdots ,n\}: \\ y_{i}(b_{j})&= \begin{cases} r(y_{i}), & 1\leq i\leq j\leq n \\ 0, & 1\leq j<i\leq n \end{cases} \end{aligned}\]
+
+Since \(H_{n}^{\perp}\) is decreasing as \(n\) increases, we have \(\{\rho_{n}\}\) is non-increasing sequence in \((0,1]\) and hence convergent. Assume that \(\inf_{n}\rho_{n}=\lim_{n}\rho_{n}=0\). For each \(\epsilon\in(0,1)\) and then suppose \(\rho_{n}<\epsilon\text{for all}n\geq N\). Fix \(f\in X_{\leq1}^{*}\) and for each \(n\in\mathbb{N}\), define \(\tau_{n}(f)=\max_{i\leq n}|f(b_{i})|\) and:
+
+Since for each \(n\in\mathbb{N}\):
+
+\[z_{n}(f)= \frac{1}{\beta_{1}} f(b_{1})y_{1}+\sum_{1<i\leq n} \frac{1}{\beta_{i}} f(b_{i})-f(b_{i-1})y_{i}\]
+
+\[0< \frac{1}{2} \rho_{n+1}<\beta_{n}\leq\rho_{n}\]

outputs/arxiv_math/2503.06904_pg10_pg1_repeat1.md

@@ -0,0 +1,59 @@
+![](figure)
+
+1.5
+
+1.0
+
+0.5
+
+0.0
+
+\[-0.5\]
+
+\[-1.0\]
+
+\[-1.5\]
+
+\[-1.5-1.0-0.5\]
+
+0.0
+
+0.5
+
+1.0
+
+1.5
+
+![](figure)
+
+FIGURE 2. The left one is√ the nodal sets of \(U_{*}\) inside the cube \([-5/2,5/2]^{3} \csc_{j=1} \frac{j\pi}{m}\) . The right one is the intersection of nodal sets of \(q_{m}\text{with}z_{1}z_{2}-\text{plane}\).
+
+when \(m=16\text{and}d_{m}=\)
+
+and
+
+\(\left(\sum^{m-1}\right)^{-1}\)
+
+\(\overline{2} m\log m\)
+
+\[\zeta_{j}(z)= \begin{cases} \zeta(m\eta^{-1}|z|^{-2}|(z-\xi_{j}|z|)|), & \text{if} & |z|>1, \\ \zeta(m\eta^{-1}|z-\xi_{j}|), & \text{if} & |z|\leq1. \end{cases}\]
+
+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
+
+\[\Bigg\|\left(1-\sum_{j=1}^{m}\zeta_{j}\right)N\left(\sum_{j=1}^{m}\tilde{\phi}_{j}+\psi\right)\Bigg\|_{L_{w}^{q}}+\Bigg\||U_{*}|^{4}\sum_{j}(1-\zeta_{j})\tilde{\phi}_{j}\Bigg\|_{L_{w}^{q}}+\Vert V(z)\psi\Vert_{L_{w}^{q}}\leq \frac{C}{(\log m)^{2}} .\]

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
+
+\[\begin{aligned} \hat{}\hat{}\hat{} \\ f(y)&=\sum_{\chi\in\text{supp}(f)}f(\chi)\chi(y)=\sum_{\text{supp}}f(\chi)\chi(x)\chi(z)=\sum_{\text{supp}}f(\chi)\chi(x)=f(x). \\ \hat{}\hat{}\hat{} \\ \chi&\in(f)\chi\in(f) \end{aligned}\]

outputs/arxiv_math/2503.06939_pg20_pg1_repeat1.md

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+\[\sigma=\left[ \frac{1}{N} \sum_{k=1}^{N}(z_{k}-\nu)^{2}\right]^{\frac{1}{2}},\nu= \frac{1}{N} \sum_{k=1}^{N}z_{k}.(79)\]
+
+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

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
+
+=

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+# 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.

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+(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

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+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 \(\Delta_{n+1}\circ f^{\prime}\text{and}t_{n}\circ\Phi\).

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+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 \(K\text{and}P_{i}\text{on}U,\text{where}1\leq i\leq3.\text{Then}\tilde{K}=KV/V,\tilde{P}_{i}=P_{i}V/V\text{and}KV<N\). According to Lemma 3.4(ii) and Lemma 2.8, we have
+
+\[(\widetilde{P}_{1}\times\widetilde{P}_{2})/(\tilde{K}\cap\widetilde{P}_{1}\times\widetilde{P}_{2})\cong \widetilde{P}_{3}/(\tilde{K}\cap\widetilde{P}_{3}).\]
+
+This implies that \(\tilde{K}\cap(\widetilde{P}_{1}\times\widetilde{P}_{2})\neq 1\). Suppose \(\tilde{K}\cap\widetilde{P}_{1}\neq 1\). Since \(\tilde{K}\) projects surjectively onto \(\widetilde{P}_{2}\times\widetilde{P}_{3}\), it follows that \(\tilde{K}\cap\widetilde{P}_{1}\) is a nontrivial normal subgroup of \(\widetilde{P}_{1}\). Thus, there exists a nontrivial normal subgroup \(D_{1}\) of \(P_{1}\text{such that}D_{1}\leq KV\) . Since \(G\) transitively permutes \(\{P_{1},\ldots,P_{5}\}\) under conjugation, we can find nontrivial normal subgroups \(D_{i}✂P_{i}\) for \(1\leq i\leq5\text{such that}G\) is transitive on \(\{D_{1},\ldots,D_{5}\}\). Hence \(\prod_{i=1}^{5}D_{i}✂X\). Considering that the stabilizer of \(G\text{on}\{P_{4},P_{5}\}\) is transitive on \(\{P_{1},P_{2},P_{3}\}\), we deduce that \(D_{1}\times D_{2}\times D_{3}\leq KV\) . Then \((\prod_{i=1}^{5}D_{i})\leq KV\text{as}D_{4}\times D_{5}\leq V\) . Recall that \(N=(\prod_{i=1}^{5}D_{i})K\). This implies that \((\prod_{i=1}^{5}D_{i})K\leq KV\) , which contradicts \(KV<N\). Therefore, we must have \(\tilde{K}\cap\widetilde{P}_{1}=1\), and similarly, \(\tilde{K}\cap\widetilde{P}_{2}=1\). Since
+
+\[\begin{aligned} \tilde{}\tilde{}\tilde{}\tilde{}\tilde{}\tilde{} \\ (K\cap(P_{1}\times P_{2}))\cap P_{1}&=K\cap P_{1}=1, \end{aligned}\]
+
+and \(\tilde{K}\cap(\widetilde{P}_{1}\times\widetilde{P}_{2})✁\widetilde{P}_{1}\times\widetilde{P}_{2}\), it follows that \(\tilde{K}\cap(\widetilde{P}_{1}\times\widetilde{P}_{2})\) centralizes \(\widetilde{P}_{1}\). Recalling that \(Z(\widetilde{P}_{1})=1\), the centralizer of \(\widetilde{P}_{1}\text{in}\widetilde{P}_{1}\times\widetilde{P}_{2}\) is \(\widetilde{P}_{2}\). Thus \(\tilde{K}\cap(\widetilde{P}_{1}\times\widetilde{P}_{2})\leq\widetilde{P}_{2}\). By

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+Moreover, the indicator \(1_{\{X_{(k-1)/n}<\rho_{0}\leq\rho_{0}+\theta<X_{k/n}\}}\) is larger, the smaller \(\theta\). Thus,
+
+\[\log\left( \frac{P_{3}^{\rho_{0}+\theta}(X_{(k-1)/n},X_{k/n};1/n)}{P_{3}^{\rho_{0}}(X_{(k-1)/n},X_{k/n};1/n)} \right)1_{I_{3,k}^{\theta}}\leq\exp\left(- \frac{(X_{(k-1)/n}-\rho_{0})^{2}}{2\alpha^{2}/n} \right)1_{I_{3,k}^{K \sqrt{n}}}\]
+
+and
+
+\[\begin{aligned} \sup_{\theta\in\Theta_{1}\cup\Theta_{2}}\sum_{k=1}^{n}\log\left( \frac{P_{3}^{\rho_{0}+\theta}(X_{(k-1)/n},X_{k/n};1/n)}{P_{3}^{\rho_{0}}(X_{(k-1)/n},X_{k/n};1/n)} \right)1_{I_{3,k}^{\theta}} \\ &\leq\sum_{k=1}^{n}\exp\left(- \frac{(X_{(k-1)/n}-\rho_{0})^{2}}{2\alpha^{2}/n} \right)1_{I_{3,k}^{K \sqrt{n}}}=:\Xi_{n}^{(3)}(K) \end{aligned}\]
+
+Again, with Corollary B.2 we find \(\sup_{K\geq0}\mathbb{E}_{\rho_{0}}[|\Xi_{n}^{(3)}(K)|/ \sqrt{n} ]\leq C_{\alpha,\beta}\).
+
+\(\mathbf{I}_{\mathbf{4}}(\boldsymbol{\theta})\): Follows by the same reasoning as \(\mathcal{I}_{\sqrt{2}} ( \theta)\) by interchanging the roles of \(\alpha\text{and}\beta\) and gives an upper bound \(\Xi_{n}^{(4)}\text{with}\mathbb{E}_{\rho_{0}}[|\Xi_{n}^{(4)}|/ \overline{n} ]\leq C_{\alpha,\beta}\).
+
+\(\mathbf{I}_{\mathbf{5}}(\boldsymbol{\theta})\): First, we split
+
+\[\begin{aligned} \frac{P_{1}^{\rho_{0}+\theta}(X_{(k-1)/n},X_{k/n};1/n)}{P_{2}^{\rho_{0}}(X_{(k-1)/n},X_{k/n};1/n)} \\ &= \frac{\beta}{\alpha} \frac{\exp- \tfrac{(X_{k/n}-X_{(k-1)/n})^{2}}{2\alpha^{2}/n}}{\exp- \tfrac{(X_{k/n}-X_{(k-1)/n})^{2}}{2\beta^{2}/n}} \left(\right) \frac{1- \tfrac{\alpha-\beta}{\alpha+\beta} \exp\left(- \tfrac{2}{\alpha^{2}/n} (X_{k/n}-\rho_{0}-\theta)(X_{(k-1)/n}-\rho_{0}-\theta)\right)}{1+ \tfrac{\alpha-\beta}{\alpha+\beta} \exp\left(- \tfrac{2}{\beta^{2}/n} (X_{k/n}-\rho_{0})(X_{(k-1)/n}-\rho_{0})\right)} . \end{aligned}\]
+
+Taking the logarithm on both sides then yields
+
+\[\begin{aligned} \log\left( \frac{P_{1}^{\rho_{0}+\theta}(X_{(k-1)/n},X_{k/n};1/n)}{P_{2}^{\rho_{0}}(X_{(k-1)/n},X_{k/n};1/n)} \right)&=\log\left( \frac{\beta}{\alpha} \right)- \frac{(X_{k/n}-X_{(k-1)/n})^{2}}{2/n} \left( \frac{1}{\alpha^{2}} - \frac{1}{\beta^{2}} \right)+R_{k,\theta} \\ \text{with} \end{aligned}\]
+
+\[R_{k,\theta}=\log\left( \frac{1- \tfrac{\alpha-\beta}{\alpha+\beta} \exp\left(- \tfrac{2}{\alpha^{2}/n} (X_{k/n}-\rho_{0}-\theta)(X_{(k-1)/n}-\rho_{0}-\theta)\right)}{1+ \tfrac{\alpha-\beta}{\alpha+\beta} \exp\left(- \tfrac{2}{\beta^{2}/n} (X_{k/n}-\rho_{0})(X_{(k-1)/n}-\rho_{0})\right)} \right).\]
+
+Denoting
+
+\[g_{k}:=\log\left( \frac{\beta}{\alpha} \right)- \frac{n}{2} \left( \frac{1}{\alpha^{2}} - \frac{1}{\beta^{2}} \right)(X_{k/n}-X_{(k-1)/n})^{2}\]
+
+(which is independent of \(\theta)\) and decomposing
+
+\[\begin{aligned} 1_{I_{5,k}^{\theta}}&=1_{\{\rho_{0}\leq X_{(k-1)/n}<\rho_{0}+\theta\}}1_{I_{5,k}^{\theta}} \\ &=1_{\{\rho_{0}+L/ \overline{n} \leq X_{(k-1)/n}<\rho_{0}+\theta\}}-1_{\{\rho_{0}+L/ \overline{n} \leq X_{(k-1)/n}<\rho_{0}+\theta\}}\left(1-1_{\{\rho_{0}<X_{k/n}\leq\rho_{0}+\theta\}}\right) \\ +1_{\{\rho_{0}\leq X_{(k-1)/n}<\rho_{0}+L/ \overline{n} \}}1_{\{\rho_{0}<X_{k/n}\leq\rho_{0}+\theta\}} \end{aligned}\]
+
+then gives the decomposition
+
+\[(\mathrm{F}.1)\text{log}\left( \frac{P_{1}^{\rho_{0}+\theta}(X_{(k-1)/n},X_{k/n};1/n)}{P_{2}^{\rho_{0}}(X_{(k-1)/n},X_{k/n};1/n)} \right)1_{I_{5,k}^{\theta}}=S_{1,\theta}(k)+S_{2,\theta}(k)+S_{3,\theta}(k)+S_{4,\theta}(k),\]

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+[4] CAI X, GUO W, QIU J-M. Comparison of Semi-Lagrangian Discontinuous Galerkin Schemes for Linear and Nonlinear Transport Simulations [J]. Communications on Applied Mathematics and Computation, 2019, 4: 3 - 33. 1
+
+[5] DING M, CAI X, GUO W, et al. A semi-Lagrangian discontinuous Galerkin (DG) - local DG method for solving convection-diffusion equations [J]. Journal of Computational Physics, 2020, 409: 109295.
+
+[6] CAI X, BOSCARINO S, QIU J. High order semi-Lagrangian discontinuous Galerkin method coupled with Runge-Kutta exponential integrators for nonlinear Vlasov dynamics [J]. Journal of Computational Physics, 2020, 427: 110036.
+
+[7] CAI X, GUO W, QIU J. A High Order Semi-Lagrangian Discontinuous Galerkin Method for the Two-Dimensional Incompressible Euler Equations and the Guiding Center Vlasov Model Without Operator Splitting [J]. Journal of Scientific Computing, 2019, 79.
+
+[8] YOUNES A, ACKERER P. Solving the advection-diffusion equation with the Eulerian-Lagrangian localized adjoint method on unstructured meshes and non uniform time stepping [J]. Journal of Computational Physics, 2005, 208(1): 384-402. 1
+
+[9] RUSSELL T F, CELIA M A. An overview of research on Eulerian-Lagrangian localized adjoint methods (ELLAM) [J]. Advances in Water Resources, 2002, 25(8): 1215-31. 1
+
+[10] QIU J-M, SHU C-W. Positivity preserving semi-Lagrangian discontinuous Galerkin formulation: Theoretical analysis and application to the Vlasov-Poisson system [J]. Journal of Computational Physics, 2011, 230(23): 8386-409. 1
+
+[11] RESTELLI M, BONAVENTURA L, SACCO R. A semi-Lagrangian discontinuous Galerkin method for scalar advection by incompressible flows [J]. Journal of Computational Physics, 2006, 216(1): 195-215. 1
+
+[12] YANG Y, CAI X, QIU J-M. Optimal convergence and superconvergence of semi-Lagrangian discontinuous Galerkin methods for linear convection equations in one space dimension [J]. Math Comput, 2020, 89: 2113-39. 1
+
+[13] FALCONE M, FERRETTI R. Convergence Analysis for a Class of High-Order Semi-Lagrangian Advection Schemes [J]. SIAM Journal on Numerical Analysis, 1998, 35(3): 909-40. 1
+
+[14] WEN G, WANG L. On Mixed Boundary Value Problems for Parabolic Equations [J]. Journal of Yantal University (Natural Science and Engineering), 1991, (01): 1-12+17. DOI: 10.13951/j.cnki.37-1213/n.1991.01.001. 1
+
+[15] SUN G, MIN C. The first boundary value problem for quasi-linear parabolic equations [J]. Journal of Qiqihar University (Natural Science Edition), 1982, (02): 7-11. 1
+
+[16] WEI H, HOU Y. Dimension Splitting Algorithms for Three Dimensional Steady/Unsteady Stokes Equations [J]. Chinese Journal of Engineering Mathematics, 2015, 32(3): 445-461. DOI: 10.3969/j.issn.1005-3085.2015.03.013.
+
+[17] GU C-H, LI D-Q, CHEN S-X, ZHENG S-M, TAN Y-J. Mathematical Physics Equations [M]. 3rd ed. HIGHER EDUCATION PRESS BEIJING. 2012: Chapter 5, pp. 138-142. 3.2
+
+# Appendix A. Differentiation formula for variable-limit integrals(Leibniz Rule)
+
+\(\Omega(t)\) denotes a time-dependent bounded domain, and at time \(t^{*},\Omega(t)\) is fixed as \(\Omega_{t}^{\star}\), i.e., \(\Omega_{t}^{\star}=\Omega(t^{\star})\). Let \((_{\overline{dt}}\cdot)\) d represent the material derivative (substantial derivative), and denote \(\mathbf{u}= \frac{d\mathbf{x}}{dt}\) . The differentiation formula for integrals with variable limits (also known as the Leibniz rule) is given by:

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+values for the sensitivity and the mismatch, this is a conservative estimate, as also indicated in Fig. 4c where the actual decrease is 6.1%.
+
+## V. DISCUSSION
+
+The paper provides formulas for analysis of sensitivity of OFO. The results are validated in a synthetic case study, showing that the formulas can be used instead of finite differences to compute the sensitivities in both unconstrained and constrained optimization problems. The synthetic case study also indicated that the sensitivities of OFO depend on how long the OFO controller has been running. If the time was insufficient, the sensitivities of OFO increase as the optimum is not reached. Conversely, if OFO reaches the optimum, then it becomes less sensitive to changes in parameters. These results suggest that future work could focus on analysis of sensitivity with respect to time. The second numerical case study presents an application of sensitivity analysis with respect to the model mismatch in a gas lift optimization problem. The case study further demonstrated that the sensitivity of OFO depends on how long the controller was running. In particular, the sensitivities to the model mismatch in a single time step decrease with time, which suggests that iterative algorithms with accuracy improving with iterations may be used for gradient estimation while preserving the optimum. The results for the case with coupling constraints emphasize the dependence on time in OFO. The formulas in Section III were derived by setting a constant final time \(T_{F}\) and treating OFO as a system of equations parametrised by p:
+
+\[\begin{aligned} \Phi^{T_{F}}&=\Phi(u^{T_{F}},y^{T_{F}},\mathbf{p}) \\ y^{0}&=h(u^{0}) \\ u^{1}&=u^{0}+\alpha\widehat{\sigma}_{\alpha}(u^{0},y^{0},\mathbf{p}) \\ \ddots \\ y^{k}&=h(u^{k}) \\ u^{k+1}&=u^{k}+\alpha\widehat{\sigma}_{\alpha}(u^{k},y^{k},\mathbf{p}) \\ \ddots \\ y^{T_{F}-1}&=h(u^{T_{F}-1}) \\ u^{T_{F}}&=u^{T_{F}-1}+\alpha\widehat{\sigma}_{\alpha}(u^{T_{F}-1},y^{T_{F}-1},\mathbf{p}) \end{aligned}\] (32)
+
+where \(\widehat{\sigma}_{\alpha}(u^{k},y^{k},\mathbf{p})\) is the solution to (9) at iteration \(k\). This assumption is equivalent to discretizing a continuous gradient flow algorithm with a constant time step [6]. However, (32) can be equivalently summarised as a nonlinear equation:
+
+\[F(\Phi^{T_{F}},\mathbf{u},\mathbf{y},\mathbf{p})=0\] (33)
+
+treating all arguments as independent variables, with no explicit dependence on time. A natural extension to the sensitivity analysis will now be to consider the impact of time, both as time step and the final time.
+
+## VI. CONCLUSIONS AND FUTURE WORKS
+
+## The objective of the paper is to facilitate the analysis of Online Feedback Optimization controllers with respect to
+
+their parameters. The importance of analysing the impact of the parameters of an optimization-based controller has been shown and used for traditional controllers, such as Model Predictive Control, but the analysis for Online Feedback Optimization remains under-explored. This paper addresses this gap by providing closed-form expressions for the sensitivity of Online Feedback Optimization to its parameters. In the future, the sensitivities can be used for tuning of the parameters of the controller, as well as checking robustness to model mismatch. Future work will also include the analysis of sensitivity with respect to time.
+
+### REFERENCES
+
+[1] B. Amos, I. Jimenez, J. Sacks, B. Boots, and J. Z. Kolter. Differentiable MPC for end-to-end planning and control. Advances in neural information processing systems, 31, 2018.
+
+[2] B. Amos and J. Zico Kolter. OptNet: Differentiable optimization as a layer in neural networks. In Doina Precup and Yee Whye Teh, editors, Proceedings of the 34th International Conference on Machine Learning, volume 70 of Proceedings of Machine Learning Research, pages 136–145. PMLR, 06–11 Aug 2017.
+
+[3] J. R. Andersen, L. Imsland, and A. Pavlov. Data-driven derivative-free trust-region model-based method for resource allocation problems. Computers & Chemical Engineering, 176:108282, August 2023.
+
+[4] J. C. G. Boot. On sensitivity analysis in convex quadratic programming problems. Operations Research, 11(5):771–786, October 1963.
+
+[5] M. Colombino, J. W. Simpson-Porco, and A. Bernstein. Towards robustness guarantees for feedback-based optimization. In 2019 IEEE 58th Conference on Decision and Control (CDC) Palais des Congres et
+
+des Expositions Nice Acropolis Nice, France, December 11-13, 2019. ` IEEE, 2019.
+
+[6] V. Häberle, A. Hauswirth, L. Ortmann, S. Bolognani, and F. Dörfler. Non-convex feedback optimization with input and output constraints. IEEE Control Systems Letters, 5(1):343–348, 2020.
+
+[7] A. Hauswirth, S. Bolognani, G. Hug, and F. Dörfler. Optimization algorithms as robust feedback controllers. Annual Reviews in Control, 57:100941, 2024.
+
+[8] Z. He, S. Bolognani, J. He, F. Dörfler, and X. Guan. Model-free nonlinear feedback optimization. IEEE Transactions on Automatic Control, pages 1–16, 2023.
+
+Matrix Differential Calculus with Applications in
+
+[9] J. R. Magnus. Statistics and Econometrics. Wiley, February 2019.
+
+[10] P. Mestres, A. Allibhoy, and J. Cortés. Regularity properties of optimization-based controllers. November 2023.
+
+[11] J. Nocedal and S. J. Wright. Numerical Optimization. Springer-Verlag New York, first edition, 1999.
+
+[12] L. Ortmann, F. Böhm, F. Klein-Helmkamp, A. Ulbig, S. Bolognani, and F. Dörfler. Tuning and testing an online feedback optimization controller to provide curative distribution grid flexibility. Electric Power Systems Research, 234:110660, 2024.
+
+[13] L. Ortmann, A. Hauswirth, I. Caduff, F. Dörfler, and S. Bolognani. Experimental validation of feedback optimization in power distribution grids. Electric Power Systems Research, 189:106782, 2020.
+
+automeris.io/
+
+[14] A. Rohatgi. Webplotdigitizer, v. 4.1, 2018. WebPlotDigitizer, accessed: 8 Nov 2024.
+
+[15] S. Surjanovic and D. Bingham. Virtual library of simulation experiments: Test functions and datasets. Retrieved November 8, 2024, from https://www.sfu.ca/˜ssurjano/stybtang.html.
+
+[16] M. Thombre, Z. J. Yu, J. Jäschke, and L. T. Biegler. Sensitivity-assisted multistage nonlinear model predictive control: Robustness, stability and computational efficiency. Computers & Chemical Engineering, 148:107269, 2021.
+
+[17] P.J. Werbos. Backpropagation through time: what it does and how to do it. Proceedings of the IEEE, 78(10):1550–1560, 1990.
+
+[18] M. Zagorowska, L. Ortmann, A. Rupenyan, M. Mercangöz, and L. Imsland. Tuning of online feedback optimization for setpoint tracking in centrifugal compressors. IFAC-PapersOnLine, 58(14):881– 886, 2024.

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+# The maximum number of cliques in disjoint copies of graphs
+
+Zhipeng Gaoa, Ping Lib, Changhong Luc, Rui Sun\(\mathrm{c},*\), Long-Tu Yuanc
+
+aSchool of Mathematics and Statistics, Xidian University, Xi’an, 710071, China.
+
+bSchool of Mathematics and Statistics, Shaanxi Normal University, Xi’an, Shaanxi, China. cSchool of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China.
+
+#### Abstract
+
+The problem of determining the maximum number of copies of \(T\) in an \(H-\)free graph, for any graphs \(T\text{and}H\), was considered by Alon and Shikhelman. This is a variant of Turán’s classical extremal problem. We show lower and upper bounds for the maximum number of \(s-\)cliques in a graph with no disjoint copies of arbitrary graph. We also determine the maximum number of \(s-\)cliques in an \(n-\)vertex graph that does not contain a disjoint union of \(k\) paths of length two when \(k=2,3,\text{or}s\geqslant k+2\), or \(n\) is sufficiently large, this partly confirms a conjecture posed by Chen, Yang, Yuan, and Zhang [1].
+
+## 1. Introduction
+
+A graph \(G\) is said to be \(H-\)free if it does not contain any subgraph isomorphic to \(H\). The classical extremal function ex \((n,H)\) denoted as \(\text{ex}(n,H)\), is formally defined as the maximum number of edges on an \(n-\text{vertex}H-\)free graph. This function extends naturally to a scenario where the objective is not merely to maximize the number of edges, but rather to maximize the number of copies of a specified graph \(T\) within an \(n-\)vertex graph that is \(H-\text{free}\). In accordance with the notation introduced by Alon and Shikhelman [2], the more general function is denoted as \(\text{ex}(n,T,H)\), which is commonly referred to the generalized Turán number of \(H\). Let \(P_{k}\) be the path with \(k\) vertices. Obviously, \(\text{ex}(n,P_{2},H)=\text{ex}(n,H)\). We denote the cycle with \(\ell\) vertices by \(C_{\ell}\) and the complete graph with \(s\) vertices by \(K_{s}\). The exploration of such problems dates back to the foundational work of Erdős [3], who established the values of \(\text{ex}(n,K_{s},K_{t})\) for any two cliques. After these seminal discoveries, a series of related findings were made, with one of the most notable being the resolution of \(\text{ex}(n,C_{5},C_{3})\) by Hatami et al. [4], and Grzesik [5], each working independently. A wealth of additional research has further expanded our understanding of generalized extremal numbers, as evidenced by various studies referenced in the literature [6, 7, 8, 9]. Scholars have studied the generalized Turán numbers for the vertex-disjoint union of graphs. Let \(kH\) represent the collection of \(k\) disjoint copies of the graph \(H\). Gerbner, Methuku, and Vizer [10] delved into the function \(\text{ex}(n,T,kH),\text{where}H\)
+
+∗Corresponding author. Email addresses: gaozhipeng@xidian.edu.cn (Zhipeng Gao), lp-math@snnu.edu.cn (Ping Li), chlu@math.ecnu.edu.cn (Changhong Lu), sunruicaicaicai@163.com (Rui Sun), ltyuan@math.ecnu.edu.cn (Long-Tu Yuan)

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+# Fluctuations of blowup time in a simple model of a super-Malthusian catastrophe
+
+Baruch Meerson1
+
+1Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
+
+Motivated by the paradigm of a super-Malthusian population catastrophe, we study a simple stochastic population model which exhibits a finite-time blowup of the population size and is strongly affected by intrinsic noise. We focus on the fluctuations of the blowup time \(T\) in the asexual binary reproduction model \(2A\rightarrow3A\), where two identical individuals give birth to a third one. We determine exactly the average blowup time as well as the probability distribution \(\mathcal{P}(T)\) of the blowup time and its moments. In particular, we show that the long-time tail \(\mathcal{P}(T\rightarrow\infty)\) is purely exponential. The short-time tail \(\mathcal{P}(T\rightarrow0)\) exhibits an essential singularity at \(T=0\), and it is dominated by a single (the most likely) population trajectory which we determine analytically.
+
+### I. INTRODUCTION
+
+It is argued that the human population of the Earth exhibits a super-Malthusian (that is a faster-than-exponential) growth that should ultimately lead to a finite-time blowup [1]. In general, a finite-time blowup in a population model can occur due to a positive feedback when the population growth rate increases faster than linearly with the population size. A simple example is provided by the following nonlinear ordinary differential equation (ODE) for the population size \(n=n(t)\):
+
+\[\dot{n}(t)=\beta n^{2}(t),\beta>0,\] (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 \(m\).

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+## 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 \(0<q<1\). The \(q-\text{analog}[n]_{q}\) of any positive integer \(n\in\mathbb{N}^{*}\) and \(q-\)analog factorial \([n]_{q}!\) are defined as
+
+\[[n]_{q}= \frac{1-q^{n}}{1-q} \text{and}[n]_{q}!=[n]_{q}\times[n-1]_{q}\times\cdots \times[1]_{q}.\]
+
+The \(q-\)analog of \((x-a)^{n}\) is
+
+\[(x-a)_{q}^{n}= \begin{cases} 1 & \text{if}n=0 \\ (x-a)\times(x-qa)\times\cdots \times(x-q^{n-1}a) & \text{if}n\geqslant1, \end{cases}\]
+
+and we have
+
+\[(a-x)_{q}^{n}=(-1)^{n}q^{\frac{n(n-1)}{2}}\left(x-q^{1-n}a\right)_{q}^{n}.\]
+
+The \(q-\)analog of the exponential function \(e^{x}\) given by
+
+\[e_{q}^{x}=\sum_{k=0}^{+\infty} \frac{x^{k}}{[k]_{q}!} .\]
+
+The \(q-\)analog of identity \(e^{x}e^{-x}=1\) is defined by \(e_{q}^{x}E_{q}^{-x}=1\), where
+
+The \(q-\)analog of \(e^{- \frac{x2}{2}}\) is given by
+
+\[E_{q}^{x}=e_{\underline{x_{1}}}=\sum_{k^{\overline{q}}=0}^{+\infty}q^{\frac{k(k-1)}{2}} \frac{x^{k}}{[k]_{q}!} .\]
+
+\[E_{q^{2}}^{- \frac{q^{2}x}{[2]q}}=\sum_{k=0}^{+^{\underline{2}}\infty} \frac{q^{k(k+1)}(q-1)^{k}}{(1-q^{2})_{q^{2}}^{k}} x^{2k}.\]
+
+The \(q-\)analog of an improper integral is proper integral with limits \(-\nu\text{and}\nu\) where
+
+\[\nu=\nu(q)= \frac{1}{\sqrt{1-q}} .\]
+
+For \(a,b\in\mathbb{R}\), the Jackson integral or \(q-\)integral of arbitrary function \(f:\mathbb{R}\rightarrow\mathbb{R}\text{on}[a,b]\) is defined by
+
+\[\int_{a}^{b}f(x)d_{q}x=(1-q)\sum_{k=0}^{+\infty}q^{k}\left[bf(q^{k}b)-af(q^{k}a)\right],\] (2.1)
+
+When \(f\) is continue on \([a,b]\), we get that
+
+\[\lim_{q\rightarrow1}\int_{a}^{b}f(x)d_{q}x=\int_{a}^{b}f(x)dx.\]

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+# z
+
+\(\theta_{t}\)
+
+![](figure)
+
+## x BS
+
+y
+
+![](figure)
+
+Fig. 1. The massive MIMO system.
+
+systems [24], which can also be viewed as a kind of symplectic optimization, to solve the constrained optimization problem and provide an algorithm. The proposed algorithm achieves a faster rate than gradient descent. Simulation results indicate that the proposed precoder achieves a satisfying WSR performance while maintaining the smoothness of the effective channel. Meanwhile, the proposed precoder design exhibits lower computational complexity in comparison with traditional precoder designs. The rest of this paper is orgnized as follows: Section II presents the problem formulation of CSPD. Section III proposes the precoder design with symplectic optimization. Section IV presents simulation results illustrating the benefits of the CSPD. The paper concludes in Section V. Notations: The conjugate, transpose and hermitian of matrix A are denoted as \(\mathbf{A}^{*},\mathbf{A}^{T}\text{and}\mathbf{A}^{H}\), respectively. The trace of A is represented as the operator tr(A). The identity matrix of size \(M\times M\) is denoted as \(\mathbf{I}_{M}\). The operation diag(a) constructs a square diagonal matrix with the elements of vector a on its main diagonal. Additionally, \(\mathbf{A}=\text{Bdiag}\{\mathbf{A}_{1},\cdots ,\mathbf{A}_{\mathrm{N}}\}\) creates a block diagonal matrix with blocks \(\mathbf{A}_{1},\cdots ,\mathbf{A}_{N}\) along its main diagonal.
+
+## II. PROBLEM FORMULATION
+
+In this section, we formulate the CSPD as an optimization problem which balances the WSR performance and the smoothness of the effective channels.
+
+## A. System Model
+
+We consider a single-cell massive multiple-input multiple-output (MIMO) system consists of a BS equipped with a uniform planar array (UPA) comprising \(M=M_{z}M_{x}\) antennas, where \(M_{x}\text{and}M_{z}\) denote the number of horizontal and vertical antennas, respectively. The BS serves \(K\) single-antenna users. Orthogonal frequency division multiplexing (OFDM) modulation is employed. The number of subcarriers is \(N_{c},\text{and}N_{v}\) subcarriers are allocated for data transmission. Each time slot contains \(N_{b}\) OFDM symbols, and the channel parameters remain constant within each OFDM symbol but vary across different symbols. The diagram of the massive MIMO system is depicted in Fig.1
+
+## B. Per-subcarrier Precoder Design
+
+Let \(\mathbf{p}_{k}^{c}\in\mathbb{C}^{M\times1}\text{and}x_{k,c}\) denote the precoding vector and the transmitted signal of the \(k-\)th user at the \(c-\)th subcarrier, respectively. The transmitted signal after precoding can be presented as
+
+\[\mathbf{x}_{k,c}=\sum_{k=1}^{K}\mathbf{p}_{k}^{c}x_{k,c}.\] (1)
+
+The channel vector from the BS to user \(k\) at the \(c-\)th subcarrier is defined as \(\mathbf{h}_{k,c}\). The received signal \(y_{k,c}\) of user \(k\) at the \(c-\)th subcarrier can be presented as
+
+\[\begin{aligned} y_{k,c}&=\mathbf{h}_{k,c}\mathbf{x}_{k,c}+z_{k,c} \\ &=\mathbf{h}_{k,c}\mathbf{p}_{k}^{c}x_{k,c}+\mathbf{h}_{k,c}\sum_{l\neq k}^{K}\mathbf{p}_{l}^{c}x_{l,c}+z_{k,c} \end{aligned}\] (2)
+
+\(K\)
+
+where the noise \(z_{k,c}\) is distributed as \(\mathcal{C}\mathcal{N}(0,\sigma_{z}^{2})\). We treat the aggregate interference-plus-noise \(z_{k^{\prime},c}=\mathbf{h}_{k,c}\sum\mathbf{p}_{l}^{c}x_{l,c}+z_{k,c}\) as Gaussian noise. Let \(\Gamma_{k,c}\) be denoted as the variance of \(z^{\prime}\) we have
+
+\[l\neq k\]
+
+\[k,c^{,}\]
+
+\[\Gamma_{k,c}=\sigma_{z}^{2}+\sum_{l\neq k}^{K}\mathbf{h}_{k,c}\mathbf{p}_{l}^{c}(\mathbf{p}_{l}^{c})^{H}\mathbf{h}_{k,c}^{H}.\] (3)
+
+Assuming that user \(k\) can obtain \(\Gamma_{k,c}\). The rate of user \(k\) at the \(c-\)th subcarrier can be computed as
+
+\[\mathcal{R}_{k,c}=\log(1+\Gamma_{k,c}^{-1}\mathbf{h}_{k,c}\mathbf{p}_{k}^{c}(\mathbf{p}_{k}^{c})^{H}\mathbf{h}_{k,c}^{H}).\] (4)
+
+To maximize the WSR for each subcarrier, the problem formulation of the conventional per-subcarrier precoder design with total power constraint is given as
+
+\[\begin{aligned} \mathbf{p}_{1}^{c},\cdots ,\mathbf{p}&=\operatorname*{arg\,max}_{K}^{c}\sum_{k=1}^{K}w_{k}\mathcal{R}_{k,c} \\ \mathrm{s}.\mathrm{t}.\sum_{k=1}^{K}(\mathbf{p}_{k}^{c})^{H}\mathbf{p}_{k}^{c}&\leq P_{c} \end{aligned}\] (5)
+
+where \(w_{k}\) is used to ensure the fairness of different users and \(P_{c}\) is the power budget at \(c-\)th subcarrier. Since the precoders in (5) are designed independently for each subcarrier, the smoothness of the resulting frequency domain effective channels is not satisfied. This implies a weak correlation among effective channels, thereby complicating effective channel estimation and signal detection at the receiver.
+
+## C. CSPD with Channel Smoothing
+
+To significantly improve the performance of the channel estimation and signal detection, we optimize precoders across subcarriers, and the smoothness of the frequency domain effective channel is considered as a objective function in the optimization problem. The delay spread can be adjusted by setting related parameters. Let \(\mathbf{p}_{k}\) be a vector stacked by \(\mathbf{p}_{k}^{c}\text{as}\mathbf{p}_{k} [(\mathbf{p}_{k}^{1})^{T},\cdots ,(\mathbf{p}_{k}^{N_{v}})^{T}]^{T}\in\mathbb{C}^{MN_{v}\times1}\text{and}\mathbf{H}_{k}\) be a block diagonal matrix defined as \(\mathbf{H}_{k}=\text{Bdiag}\{\mathbf{h}_{k,1},\cdots ,\mathbf{h}_{k,N_{v}}\}\in=\)

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+define the set \(S\subseteq\{1,2,\ldots,n\}\times\{1,2,\ldots,f+g\}\), and assume the entry \(\left[A_{\text{true}}B_{\text{true}}\right]_{ij}\) is given for all \((i,j)\in S\). The set of systems compatible with the prior knowledge is given by
+
+\[\Sigma_{pk}(S):=\left\{\left[A(A_{B},B\right])_{ij}|_{=}\left[A_{\text{true}}B_{\text{true}}\right]_{ij}\forall(i,j)\in S\right\}.^{(6)}\]
+
+Subsequently, we define the set of systems compatible with both the data and the prior knowledge as
+
+\[\Sigma:=\Sigma_{d}\cap\Sigma_{pk}(S).\] (7)
+
+Note that the case that all entries of \(A_{\text{true}}\text{and}B_{\text{true}}\) are unknown can be captured by setting \(S=\emptyset\), which implies \(\Sigma=\Sigma_{d}\). It is clear from (2) and (6) that the system \((A_{\text{true}},B_{\text{true}})\) belongs to \(\Sigma\). However, in general, \(\Sigma\) contains other systems because the data may not uniquely determine \(A_{\text{true}}\text{and}B_{\text{true}}\), even if some entries of \(A_{\text{true}}\) and \(B_{\text{true}}\) are known.
+
+The goal of this paper is to find a controller that stabilizes the origin of the system \((A_{\text{true}},B_{\text{true}})\). Since on the basis of the data and the prior knowledge we cannot distinguish between \((A_{\text{true}},B_{\text{true}})\) and any other system in \(\Sigma\), we need to find a single controller that stabilizes the origin of all systems in \(\Sigma\). This motivates the following definition of informative data for stabilization of polynomial systems. In the rest of the paper, we assume that
+
+\[F(0)=0.\]
+
+Definition 1 The data \((\dot{\mathcal{X}},\mathcal{X},\mathcal{U})\) are called informative for stabilization if there exist a radially unbounded function \(V\in\mathcal{V}\) and a continuous controller \(K:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}\) such that \(K(0)=0\) and
+
+\[\frac{\partial V(x)}{\partial x} (AF(x)+BG(x)K(x))<0\forall x\in\mathbb{R}^{n}\setminus\{0\},(8)\]
+
+for all \((A,B)\in\Sigma\).
+
+Note that for a controller \(K\) satisfying \(K(0)=0\), the origin of the closed-loop system
+
+\[\dot{x}=AF(x)+BG(x)K(x),\] (9)
+
+is an equilibrium point, as \(F(0)=0\). If (8) holds then the origin is globally asymptotically stable for all closed-loop systems obtained by interconnecting any system \((A,B)\in\Sigma\) with the controller \(u=K(x)\).
+
+In this paper, we study the following two problems.
+
+# Problem 1 (Informativity) Find conditions under
+
+which the data \((\dot{\mathcal{X}},\mathcal{X},\mathcal{U})\) are informative for stabilization.
+
+# Problem 2 (Controller design) Suppose the data
+
+\((\dot{\mathcal{X}},\mathcal{X},\mathcal{U})\) are informative for stabilization. Find a controller \(u=K(x)\) satisfying \(K(0)=0\) and (8).
+
+## 3 Connection to previous work
+
+Current approaches for data-driven control of polynomial systems [8,9] build on the model-based method proposed in [15]. These methods do not incorporate prior knowledge and instead focus on designing a common stabilizing controller for all systems compatible with the data. In these works, the controller is considered to be of the form
+
+\[K(x)=Y(x)PZ(x),\]
+
+where \(Y\in\mathbb{R}^{m\times p}[x],P\in\mathbb{S}^{p}\) is positive definite, and \(Z\in\mathbb{R}^{p}[x]\) is radially unbounded satisfying
+
+\[F(x)=H(x)Z(x),\] (10)
+
+for some \(H\in\mathbb{R}^{f\times p}[x]\). The choice of candidate Lyapunov function
+
+\[V(x)=Z^{\top}(x)PZ(x),\] (11)
+
+then leads to
+
+\[\frac{\partial V}{\partial x} (x)(AF(x)+BG(x)K(x))=2Z^{\top}(x)P\Theta(x)PZ(x),\]
+
+where
+
+\[\Theta(x):= \frac{\partial Z}{\partial x} (x)\left[AB\right] \begin{bmatrix} H(x)P^{-1} \\ G(x)Y(x) \end{bmatrix} .\]
+
+The main idea in this line of work is to find \(P\text{and}Y(x)\) such that
+
+\[-\Theta(x)-\Theta^{\top}(x)>0\forall x\in\mathbb{R}^{n}\setminus\{0\},(12)\]
+
+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

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+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 \(L>0\).

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+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
+
+\[\varphi_{\Theta}(s):= \begin{cases} 0 & ifs=0, \\ s|\log s|\Theta(|\log s|) & if0<s<e^{-d-1}, \\ e^{-d-1}(d+1)\Theta(d+1) & ifs\geq e^{-d-1}. \end{cases}\]
+
+Indeed, \(\varphi_{\Theta}\) is the modulus of continuity of the force field given that the macroscopic density is Yudovich, which is enforced by Assumption 1.4 (see Crippa, Inversi, Saffirio and Stefani [4, Lemma 1.1 and Assumption 1.3]); as explained in [4], the value \(e^{-d-1}\) in the definition of \(\varphi_{\Theta}\) is essentially irrelevant and included solely to make \(\varphi_{\Theta}\) more appealing. Under Assumption 1.4 and Assumption 1.5, one can define weak solutions \(f\) to (VP) through
+
+\[\int_{0}\int_{\mathcal{X}\times\mathbb{R}^{d}}^{T}[(\partial_{t}\phi+v\cdot\nabla_{x}\phi-\nabla U_{f}\cdot\nabla_{v}\phi)f](t;x,v)dxdvdt=-\int_{\mathcal{X}\times\mathbb{R}^{d}}\phi(0,x)f(0;x,v)dxdv\]
+
+for all test functions \(\phi\in C_{c}^{\infty}([0,T)\times(\mathcal{X}\times\mathbb{R}^{d}))\), since then the product of the solution with the force field is integrable; i.e., \(\Vert f(t)\nabla U_{f}(t)\Vert_{L^{1}(\mathcal{X}\times\mathbb{R}^{d})}\in L^{1}([0,T])\). While [4, Theorem 1.6] assumed \(\varphi_{\Theta}\) to be nondecreasing concave in some regime for the 1-Wasserstein stability of (VP), in the \(p-\)Wasserstein setting, we instead assume a \(p-\)modified version of \(\varphi_{\Theta}\) to be nondecreasing concave in some regime as follows:
+
+Assumption 1.6. The growth function \(\Theta\) is such that \(\varphi_{p,\Theta}:[0,+\infty)\rightarrow[0,+\infty)\) is nondecreasing concave on \([0,c_{p,\Theta;d})\) for some positive constant \(c_{p,\Theta;d}<1/e\) that depends only on \(p,\Theta,\text{and}d,\text{where}\varphi_{p;\Theta}\) is given by
+
+\[\varphi_{p,\Theta}(s):= \begin{cases} 0 & ifs=0, \\ s|\log s|^{p}\Theta^{p}(|\log s|) & if0<s\leq c_{p,\Theta;d}, \\ \varphi_{p,\Theta}(c_{p,\Theta;d}) & ifs\geq c_{p,\Theta;d}. \end{cases}\]
+
+This encompasses for instance the bounded case with \(\Theta(r)=1(c_{p,\Theta;d}=e^{-\max\{p,d+1\}})\), the exponential Orlicz space with \(\Theta(r)=r^{1/\alpha}\text{and}1\leq\alpha<+\infty(c_{p,\Theta;d}=e^{-\max\{p\beta,d+1\}},\beta:=1+1/\alpha)\), and also a countable family of iterated logarithms due to Yudovich [18, Section 3] for two-dimensional Euler’s equations in vorticity form; \(\Theta_{n}:[0,+\infty)\rightarrow[0,+\infty)(c_{p,\Theta;d}=\min\{\exp^{-2p}\)
+
+\[(_{n+1}^{-2p}1),e^{-d-1}\})\text{given by}\]
+
+\[\Theta_{n}(r):= \begin{cases} r|\log_{1}(r)|^{2}|\log_{2}(r)|^{2}\cdots |\log_{n}(r)|^{2} & \text{if}r\geq\exp_{n}(1), \\ \Theta_{n}(\exp_{n}(1)) & \text{else}, \end{cases}\]
+
+where \(\exp_{0}(1):=1,\exp_{n+1}(1):=e^{\exp_{n}(1)}\), and
+
+\[\log_{n}(r):= \begin{cases} r & \text{if}n=0, \\ \mathrm{l} \underline{\text{og}\circ\log} \circ \underline{\cdots \circ\text{lo}} \mathrm{g}\circ|\log r| & \text{otherwise}. \\ & \overline{(n-} \text{imes} & \end{cases}\]
+
+Moreover, each of these cases satisfies the following two assumptions:

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+Here \(B_{n}\) denotes the ball of radius \(n\) centred at the origin. The following property is readily deduced from parabolic estimates:
+
+\[\lim_{n\rightarrow+\infty}u_{n}(t,x)= \overline{p} (x)\text{locally uniformly in}(t,x)\in[0,+\infty)\times\mathbb{R}^{n}.\] (4.10)
+
+We claim that the function \(u_{n}\) fulfils (4.9) for \(n\) sufficiently large, depending on \(\varepsilon\), hence by the previous step it satisfies (4.8). Assume by contradiction that this is not the case. Then, for any \(n\in\mathbb{N}\), it holds that
+
+\[\begin{aligned} \tilde{} \\ t_{n}:&=\inf\{t\geq0:\exists x\in tW,u_{n}(t,x)\leq \underline{u} (x)\}<+\infty. \end{aligned}\]
+
+We know from (4.10) that \(t_{n}\rightarrow+\infty\text{as}n\rightarrow+\infty\). In particular, \(t_{n}>0\text{for}n\) 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

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+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.

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+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
+
+\[|X|\geq|N_{G,r}(U-B)|\geq \frac{\varepsilon|U-B|}{(\log n)^{c}} \geq \frac{\varepsilon|U|}{2(\log n)^{c}} ,\]

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+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.

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+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
+
+\[\Phi(U)(t)=e^{tA}U^{\text{in}}+\int_{0}^{t}e^{(t-s)A}F^{M}(U(s))ds,\forall t\geq0.\]

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+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, \(e^{-}<e<e^{+}\). Noting that \(v=v_{-}+v_{+}\), a double counting argument similar to the proof of Lemma 2.2 together with Lemma 2.1 gives that the number of common symbols summed over all pairs of a row and a column is
+
+\[rc\lambda_{rc}=v_{-}(e^{-})^{2}+v_{+}(e^{+})^{2}=v(e^{+}-e)(e^{-})^{2}+v(e-e^{-})(e^{+})^{2}.\]
+
+Dividing by \(v\) and using \(e=rc/v\), we get
+
+\[e\lambda_{rc}=(e^{+}-e)(e^{-})^{2}+(e-e^{-})(e^{+})^{2}=e((e^{+})^{2}-(e^{-})^{2})+e^{-}e^{+}(e^{-}-e^{+}).\] (1)
+
+Note that \(e^{+}-e^{-}=1\), so \((e^{+})^{2}-(e^{-})^{2}=e^{+}+e^{-}\), thus
+
+\[e\lambda_{rc}=e(e^{+}+e^{-})-e^{-}e^{+},\]
+
+from which (a) follows. In order to prove (b) it suffices to show that \(e^{-}<\lambda_{rc}<e^{+}\), which follows from (a):
+
+\[\begin{aligned} e^{-}&<e<e+ \frac{(e^{+}-e)(e-e^{-})}{e} =\lambda_{rc}, \\ \lambda_{rc}&=e+ \frac{(e^{+}-e)(e-e^{-})}{e} =e+(e^{+}-e)\cdot \frac{e-e^{-}}{e} <e+(e^{+}-e)=e^{+}. \end{aligned}\]
+
+Again using double counting and Lemma 2.1, the number of common symbols summed over all pairs of rows is
+
+\[\begin{pmatrix} r \\ 2 \end{pmatrix} \lambda_{rr}=v_{-} \begin{pmatrix} e^{-} \\ 2 \end{pmatrix} +v_{+} \begin{pmatrix} e^{+} \\ 2 \end{pmatrix} =v(e^{+}-e) \begin{pmatrix} e^{-} \\ 2 \end{pmatrix} +v(e-e^{-}) \begin{pmatrix} e^{+} \\ 2 \end{pmatrix} .\]
+
+Dividing by \(v/2\), we get
+
+\[\begin{aligned} \frac{r(r-1)\lambda_{rr}}{v} &=(e^{+}-e)e^{-}(e^{-}-1)+(e-e^{-})e^{+}(e^{+}-1) \\ &=e((e^{+})^{2}-(e^{-})^{2})+e^{+}e^{-}(e^{-}-e^{+})+e(e^{-}-e^{+}) \\ &=e\lambda_{rc}-e, \end{aligned}\]
+
+where the last equality follows from (1). This implies (c), and a similar count gives (d).
+
+Having done this ground work, we now formally define our main object of study.
+
+Definition 2.5. An \((r\times c,v)-\)near triple array is a binary (near) equireplicate \(r\times c\) row-column design on \(v\) symbols in which
+
+1. any row and column have either \(\lambda_{rc}^{-}or\lambda_{rc}^{+}\) common symbols,
+
+2. any two rows have either \(\lambda_{rr}^{-}or\lambda_{rr}^{+}\) common symbols,
+
+3. any two columns have either \(\lambda_{cc}^{-}or\lambda_{cc}^{+}\) common symbols.
+
+Note that the definition allows a near triple array to be equireplicate. To avoid having to calculate the quantities involved in the definition explicitly, it will often be convenient to instead use the following alternative characterization of near triple arrays.

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+of \(f\), denoted by \(c(f)\), as any directed \(S-\)path from \(u\) to \(v\) that utilizes only forward arcs from \(A_{B}(T_{\sigma},P_{\sigma})\).
+
+▶ Definition 13 (certification of an arc set). Consider an ordered partition \(P_{\sigma}=(V_{1},\ldots,V_{\ell})\) of an ordered tournament \(T_{\sigma}\) and let \(F\) be a set of some backward arcs of \(T_{\sigma}\). We say that we can certify \(F\) whenever it is possible to find a set \(F=\{c(f):f\in F\}\) of arc-disjoint certificates for the arcs in \(F\) .
+
+▶ Definition 14 (safe partition). Consider an ordered partition \(P_{\sigma}=(V_{1},\ldots,V_{\ell})\) of an ordered tournament \(T_{\sigma}\). Let \(B_{E}\) denote the set of all backward arcs in \(A_{B}(T_{\sigma},P_{\sigma})\). For a vertex subset \(S\subseteq V(T)\), we say \(P_{\sigma}\) is a \(S-\)safe partition if we can certify \(B_{E}\).
+
+Let \(D_{\sigma}=(V_{\sigma},A)\) be an ordered directed graph. We use \(\text{Sfas}(D_{\sigma},S)\) to denote the size of a minimum subset feedback arc set in the directed graph \(D_{\sigma}\). Specifically, given a directed graph \(D_{\sigma}\), and a vertex subset \(S\subseteq V_{\sigma}\), it refers to the cardinality of the smallest set \(F\subseteq A\) of arcs whose removal (or reversal) renders \(D_{\sigma}S-\text{acyclic}\). Given a tournament, for an arc subset \(A^{\prime}\subseteq A[T]\), the directed graph \(T[A^{\prime}]\) is defined by the subgraph \(T\) containing the arc set \(A^{\prime}\). More specifically, \(T[A^{\prime}]=(V(T),A^{\prime})\). For a ease of notation, when the subset is clear to the context, we use \(\text{Sfas}(D_{\sigma})\) instead of \(\text{Sfas}(D_{\sigma})\).
+
+▶ Lemma 15. Consider an ordered partition \(P_{\sigma}=(V_{1},\ldots,V_{\ell})\) of an ordered tournament \(T_{\sigma}\) and a terminal set \(S\subseteq V(T).IfP_{\sigma}\) is \(aS-\)safe partition then,
+
+\[Sfas(T_{\sigma})=Sfas(T_{\sigma}[A_{I}(T_{\sigma},P_{\sigma})])+Sfas(T_{\sigma}[A_{B}(T_{\sigma},P_{\sigma})])\]
+
+Moreover, there exists a minimum sized \(S-\)feedback arc set of \(T_{\sigma}\) containing \(B_{E},\text{where}B_{E}\) denote the set of all backward arcs in \(A_{B}(T_{\sigma},P_{\sigma})\).
+
+Proof. Given any bipartition of the arc set \(A\) into \(A_{1}\text{and}A_{2}\), we have
+
+\[\text{Sfas}(T_{\sigma})\geq\text{Sfas}(T_{\sigma}[A_{1}])+\text{Sfas}(T_{\sigma}[A_{2}]).\]
+
+Specifically, for the partition of \(A\) into \(A_{I}(T_{\sigma},P_{\sigma})\text{and}A_{B}(T_{\sigma},P_{\sigma})\), it follows that
+
+\[\text{Sfas}(T_{\sigma})\geq\text{Sfas}\left(T_{\sigma}[A_{I}(T_{\sigma},P_{\sigma})]\right)+\text{Sfas}\left(T_{\sigma}[A_{B}(T_{\sigma},P_{\sigma})]\right).\]
+
+Next, we need to show that
+
+\[\text{Sfas}(T_{\sigma})\leq\text{Sfas}\left(T_{\sigma}[A_{I}(T_{\sigma},P_{\sigma})]\right)+\text{Sfas}\left(T_{\sigma}[A_{B}(T_{\sigma},P_{\sigma})]\right).\]
+
+This assertion holds because, after reversing all arcs in \(B_{E}\), each remaining directed \(S-\)cycle is contained within \(T_{\sigma}[V_{i}]\text{for some}i\in[\ell]\). In other words, once all arcs in \(B_{E}\) are reversed, every \(S-\)cycle is entirely contained within \(T_{\sigma}[A_{I}(T_{\sigma},P_{\sigma})]\). Observe that as \(P_{\sigma}\) is \(S-\text{safe}\), the set of all backward arcs of \(A_{B}(T_{\sigma},P_{\sigma})\), i.e., \(B_{E}\) can be certified using only arcs from \(A_{B}(T_{\sigma},P_{\sigma})\). This concludes the proof of the first part of the lemma. Essentially, we have shown the existence of a minimum-sized \(S-\)feedback arc set for \(T_{\sigma}\) that includes \(B_{E}\). This completes the proof of the lemma.
+
+◀
+
+# Construction of \(a\mathbf{S}-\)safe partition.
+
+Recall that an \(S-\)triangle is a directed cycle of length three that includes at least one vertex from \(S\). It is clear that the number of arc-disjoint \(S-\) triangles provides a lower bound for the size of the smallest \(S-\)feedback arc set in a tournament. We illustrate how this set can be utilized to identify a safe partition in polynomial time.

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+By compactness we can now cover the interval \([a,b]\) with a finite number of such neighborhoods \(U_{\lambda_{1}},\ldots,U_{\lambda_{n}}\subseteq\mathbb{R}\). If
+
+\[V:=N(T(\lambda_{1})^{*})+N(T(\lambda_{2})^{*})+\ldots+N(T(\lambda_{n})^{*})\subset L^{\infty}(\mathbb{R}),\] (3.1)
+
+then \(\dim V<\infty\text{and}R(T(\lambda))+V=L^{\infty}(\mathbb{R})\text{for all}\lambda\in[a,b]\). consider the family of operators
+
+(IV) This step is inspired by Step 3 in the proof of [41, Thm. 5.3]. Keeping \(\tau>0\) 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)
+
+<table>
+<tr>
+<th colspan="3">S−(λ)⊕S+(λ)</th>
+</tr>
+<tr>
+<td rowspan="4">(3.3)</td>
+<td>X− ⊕ X+</td>
+<td>Y− ⊕ Y+</td>
+</tr>
+<tr>
+<td>Jλ</td>
+<td>J</td>
+</tr>
+<tr>
+<td>S(λ)</td>
+<td></td>
+</tr>
+<tr>
+<td>D(S(λ))</td>
+<td>L∞[−τ, τ ],</td>
+</tr>
+</table>
+
+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

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+## 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 \(0<t_{j}+t_{p}<t<t_{j+1}\), for any \(j=0,\cdots ,N-1\). This case will describe the homogeneous problem, which is governed without source \((f=0)\), at one pulse-to-pulse interval \(\Delta t\). For the sake of simplicity, we denote the initial instant of time \(t_{j}+t_{p}\) by \(t_{0}\) throughout this subsection.

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+A dyadic rational is a rational number whose denominator is a power of 2. In the next theorem, we construct isostatic grids with points whose coordinates are dyadic rationals. Theorem 6.4. For each \(N\geq0\) there exists an independent, infinitesimally rigid configuration \(S=(P,L,I)\) of points and lines such that every geometric point/line incidence between \(\mathbf{p}_{i}\) and \(\mathbf{l}_{j}\) in the plane is recorded as an incidence \((i,j)\in I\), and such that all points with coordinates \((n/2^{N}:m/2^{N}\) : 1), for integers \(n,m\) satisfying \(0\leq n,m\leq2^{N}\), is either a point of the configuration or is the point of intersection of at least two lines.
+
+## Proof. We construct an example for each \(N\) recursively.
+
+At the 0-th level, first take the four points of the unit square \(a=(0\) : 0 : \(1),b=(0\) : 1 : 1), \(c=(1\) : 1 : 1), and \(d=(1\) : 0 : 1). Then form the complete quadrilateral by constructing the lines \(ab,ac,bc\text{and}cd\) and the intersection \(p_{1}=(1\) : 0 : 0) of the lines \(ad\text{and}bc\), and the intersection \(p_{2}=(0\) : 1 : 0) of the lines \(ab\text{and}cd\). Note that the complete quadrangle is independent, and infinitesimally rigid. Then add the diagonal lines, \(ac\text{and}bd\), and their intersections \(p_{3}=(1\) : 1 : 0) and \(p_{4}=(1:-1\) : 0) with the line through the points \(p_{1}\text{and}p_{2}\), which is the line at infinity. Then construct the lines \(ap_{4},bp_{3},cp_{4}\text{and}dp_{3}\). Then construct the points \(e\text{and}f\) as the intersections of the lines \(ap_{4}\) and \(bp_{3}\) and the lines \(bp_{3}\text{and}cp_{4}\) respectively. Finally, construct the horizontal line through \(e\), which is the line \(ep_{1}\), and the vertical line through \(f\), which is the line \(fp_{2}\). See Figure 11a. This completes the 0-th level, and yields an independent
+
+| b | f ● |  | c |
+| --- | --- | --- | --- |
+| ● e ● |  | ● |  |
+| ● |  | ● |  |
+| a |  |  | d |
+
+| b | ● | f ● |  | ● | c |
+| --- | --- | --- | --- | --- | --- |
+| ● |  |  |  | ● |  |
+| ● e ● ● |  |  |  | ● |  |
+|  |  |  |  |  |  |
+| ● ● |  |  |  | ● |  |
+| a |  |  |  |  | d |
+
+\[adad\]
+
+a)
+
+b)
+
+Figure 11: Isostatic dyadic grids for \(N=0\text{and}N=1\).
+
+and infinitesimally rigid configuration by Theorem 6.3. Notice for the purpose of the recursion that the segments \(ab\text{and}cd\) have points with dyadic coordinates at level \(N\) and crossings, by single lines, exactly at heights \((2k+1)/2^{N+1},\text{with}0\leq k\leq 2^{N}-1\). Notice also that segments \(ae,eb,bf\text{and}fc\) have points exactly those with dyadic coordinates at level \(N+1\), and are only crossed by lines at those points. For the recursive step start by the points of intersection of \(ab\text{and}cd\) with the horizontal lines of the configuration. Add the lines between each of the new points and \(p_{3}\text{and}p_{4}\). Each such new line bisects one of the segments along \(ae,eb,bf\text{and}fc\). Add points at each of these intersections. Finally, add horizontal lines through the new points along \(ae\text{and}eb\) and vertical lines through the new points along \(bf\text{and}fc\). These subdivide the grid, with the points along \(ab\text{and}cd\) again alternating between points and pairs of crossed lines. The dyadic grids at levels \(N=0\text{and}N=1\) are illustrated in Figures 11b and 11a. Each level is created from the previous by intersections of two lines and drawing lines through two points, and so each level is independent and infinitesimally rigid by Theorem 6.3.
+
+In the construction, the points at level \(N\), i.e. the points with coordinates \((n/2^{N}:m/2^{N}\) : 1), in the interior of \(abcd\) occur at the crossings of four lines. If a point is introduced there by the Cayley algebra, then that point will have two incidences recorded in the set \(I\), and two implied geometric incidences. Adding either of these geometric incidences to \(I\) results in a loss of constraint independence, in other words, the introduction of an equilibrium stress. This illustrates the connection between self-stresses, constraint independence and projective theorems: certain incidences are implied by projective theorems, and such implied incidences will be dependent of the remaining incidences. The points at the level \(N+1\) inside \(abcd\) which occur at the crossings of two lines are also the points of a square lattice, and all of these points may be added to form a rigid and independent configuration. See Figure 11b.

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+by the WINGS-FMSP program at the Graduate School of Mathematical Sciences, the University of Tokyo.
+
+## 2. isometries on hyperbolic plane
+
+Let \(\mathbb{H}\) be the upper half-plane model in hyperbolic geometry. The group of isometries of \(\mathbb{H}\) is given by
+
+\[\text{PSL}_{2}^{\pm}(\mathbb{R})= \begin{Bmatrix} \bigg( & \bigg)\bigg| \\ a & b \\ \pm & a,b,c,d\in\mathbb{R},ad-bc=\pm1 \\ c & d \end{Bmatrix} .\]
+
+Each isometry \(g\in\text{PSL}_{2}^{\pm}(\mathbb{R})\) is classified according to the absolute value of its trace (for example, see [5]):
+
+- Case 1: \(\det g=1\) (Orientation-preserving isometries)
+
+  - Elliptic if \(|\text{Tr}g|<2\). In this case, \(g\) has a unique fixed point in \(\mathbb{H}\).
+
+  - Parabolic if \(|\text{Tr}g|=2\). In this case, \(g\) has a unique fixed point on the real axis.
+
+  - Hyperbolic if \(|\text{Tr}g|>2\). 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.

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+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 \(d^{e}<d\) traditional states with random inflow or outflow, and G contains the LG new states with deterministic transitions. Let us consider the microscopic CTMC model given in Equation (2.1) but restrict this point of view to the time intervals \([0,T]\). To account for transitions between cascade states,

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+and
+
+\[\int Ne^{3a}Gdt= \frac{2}{N^{3}} e^{3a}\left(\dot{a}+\dot{\beta}_{+}\right)\left(2\dot{a}-\dot{\beta}_{+}-\dot{\beta}_{-}\right)\left(2\dot{a}-\dot{\beta}_{-}+\dot{\beta}_{+}\right).\] (11)
+
+### 2.2. Minisuperspace description
+
+By replacing (10) and (11) in the Action Integral (1) and integrating by parts we end with the following point-like Lagrangian function
+
+\[L\left(N,a,\dot{a},\beta_{\pm},\dot{\beta}_{\pm}\right)= \frac{e^{3a}}{N_{+}} \left( \frac{-_{e^{3}}3_{\alpha}}{N^{3}} \dot{a}_{\dot{\phi}^{2}}\left(+_{\dot{a}} \frac{3}{4_{+}} \dot{\beta}_{\dot{\beta}_{+}^{+2}}+\right) \frac{1}{\Big(4} 2^{\dot{\beta}}\dot{a}^{-2}-^{-}\dot{\beta}^{2_{+}\Lambda}-^{N}\bigg)\dot{\beta}_{-}\right)\left(2\dot{a}-\dot{\beta}_{-}+\dot{\beta}_{+}\right)\bigg).(12)\]
+
+We remark that when \(\dot{\phi}=0\), i.e. \(\phi=0\), then the limit of General Relativity is recovered. The field equations follow from the variation of the aforementioned point-like Lagrangian function with respect to the dynamical variables \(N,a,\beta_{\pm}\text{and}\phi\) . Without loss of generality we assume the lapse function \(N=1\). Thus, for the constant lapse function the gravitational field equations read
+
+\[0=3H^{2}- \frac{3}{4} \dot{\beta}_{+}^{2}- \frac{1}{4} \dot{\beta}_{-}^{2}-3\dot{\phi}\left(H+\dot{\beta}_{+}\right)\left(2H-\dot{\beta}_{+}-\dot{\beta}_{-}\right)\left(2H-\dot{\beta}_{-}+\dot{\beta}_{+}\right)-2\Lambda,(13)\]
+
+\[\begin{aligned} 0&=2\dot{H}+H^{2}\left(3-4\ddot{\phi}\right)-2\Lambda+\dot{\beta}_{+}^{3}\dot{\phi}-8H\dot{H}\dot{\phi}-\dot{\beta}_{+}\dot{\phi}\left(\dot{\beta}_{-}^{2}-2\ddot{\beta}_{+}\right) \\ +\dot{\beta}_{+}^{2}\left( \frac{3}{4} +\ddot{\phi}\right)+ \frac{1}{12} \dot{\beta}_{-}\left(\dot{\beta}_{-}\left(3+\ddot{\phi}\right)+8\dot{\phi}\ddot{\beta}_{-}\right), \end{aligned}\] (14)
+
+\[0=- \frac{3}{2} +\ddot{\beta} \frac{3^{+}}{2} +_{H}1\left(8_{2}H_{\dot{\phi}^{2}}\left(\dot{\beta}_{\dot{\beta}_{-}^{+}}\dot{\phi}_{+}+_{2}2_{\ddot{\beta}}\dot{\phi}_{+}\left(\right)3_{+}\dot{\beta}_{\dot{\beta}^{+}}\left(_{+}\dot{H}\left(4-_{\ddot{\phi}-}\ddot{\beta}_{3^{-}}\right)\right)+_{-}\dot{\beta}_{6^{-}\dot{\beta}}\ddot{\beta}_{\dot{\phi}_{+}^{2-}}\right)\right)+_{,}\ddot{\phi}\left(\dot{\beta}_{-}^{2}-3\dot{\beta}_{+}^{2}\right)\] (15)

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+right ideal (resp., finitely generated right ideal) of \(R\) is projective. Similarly, left hereditary and left semihereditary rings are defined.
+
+## 2. Decompositions into Projective and Stable Submodules
+
+In this section, we mention some rings over which the decomposition of some modules into projective and stable submodules occurs. A submodule \(K\) of \(M\) is called essential in \(M\), if for every submodule \(L\) of \(M,L\cap K=0\) implies \(L=0\), and it is denoted by \(K\leq_{e}M\). A nonzero module \(M\) is said to be a uniform module if every nonzero submodule of \(M\) is an essential submodule. An \(R-\text{module}M_{R}\) is said to have uniform dimension (=Goldie dimension) \(n\), denoted by \(\mathrm{u}.\dim(M)=n,\text{where}n\) is a positive integer, if there is an essential submodule \(N\) of \(M\text{such that}N\) is the direct sum of \(n\) (nonzero) uniform submodules. The zero module is defined to have a uniform dimension 0. If for a nonzero module \(M\), there exists no positive integer \(n\text{such that u}.\dim(M)=n\), then we write \(\mathrm{u}.\dim(M)=\infty\) (this will hold if and only if \(M\) contains an infinite direct sum of nonzero submodules); otherwise, we write \(\mathrm{u}.\dim(M)<\infty\). A submodule \(K\) of \(M\) is said to be small in \(M\) if for every submodule \(L\) of \(M,K+L=M\) implies \(L=M\). An \(R-\text{module}M\) is said to be hollow ( =couniform) if \(M\neq 0\) and every proper submodule \(N\) of \(M\) is small in \(M\). A finite
+
+set \(\{N_{i}|i\in I\}\) of proper submodules of \(M\) is said to be coindependent if \(N_{i}+\left(\bigcap_{j\neq i}N_{j}\right)=M\) for every \(i\in I\), or, equivalently, if the canonical injective mapping \(M/\bigcap_{i\in I}N_{i}\rightarrow\oplus_{i\in I}M/N_{i}\) is bijective. An arbitrary set \(A\) of proper submodules of \(M\) is said to be coindependent if its finite subsets are coindependent. A module \(M\) is said to have finite hollow dimension(=couniform dimension = dual Goldie dimension) \(n\), denoted by \(\mathrm{h}.\dim(M)=n,\text{where}n\) is a positive integer, if there exists a coindependent set \(\{N_{1},N_{2},\ldots,N_{n}\}\) of proper submodules of \(M\) with \(M/N_{i}\) hollow for all \(i\text{and}N_{1}\cap N_{2}\cap\cdots \cap N_{n}\) is small in \(M\). The zero module is defined to have a hollow dimension 0. If for a nonzero module \(M\), there exists no positive integer \(n\) such that
+
+\(\mathrm{h}.\dim(M)=n\), then we write \(\mathrm{h}.\dim(M)=\infty\) (this holds if and only if there exist an ifinite coindependent set of proper submodules of \(M)\); otherwise, we write \(\mathrm{h}.\dim(M)<\infty\). See [11, Sections 2.6, 2.7, 2.8] and [19, Section 6A].
+
+Theorem 2.1. If a module \(M\) cannot be decomposed as \(M=P\oplus N\text{where}P\) is a projective submodule and \(N\) is a stable submodule, then there exists a sequence \((P_{k})_{k=1}^{\infty}\) of nonzero proper projective submodules of \(M\) and a sequence \((N_{k})_{k=1}^{\infty}\) of nonzero proper submodules of \(M\) such that for every \(k\in\mathbb{Z}^{+}\),
+
+\[M=N_{k}\oplus P_{k}\oplus P_{k-1}\oplus\cdots \oplus P_{1}\text{with}N_{k}=N_{k+1}\oplus P_{k+1},\]
+
+and so \(\mathrm{u}.\dim(M)=\infty=\mathrm{h}.\dim(M)\). Proof. If \(M\) were a projective module or a stable module, then it would have a decomposition of the required form trivially. So \(M\) must be a module which is not projective, and since it is not stable, it can be decomposed as \(M=P\oplus N\) for some submodules \(P\text{and}N\text{where}P\) is a nonzero projective module. Then \(N\) is not projective since otherwise, \(M=P\oplus N\) would be a projective module. In particular, \(N\neq 0.\text{If}N\) were stable, then \(M=P\oplus N\) would be a decomposition into a direct sum of a projective submodule and a stable submodule. So \(N\) is not a stable module. Thus, \(N\) is neither projective nor stable. Now argue as for \(M\). The module \(N\) should have a nonzero projective direct summand, that is, \(N=P_{1}\oplus N_{1}\) for some nonzero projective submodule \(P_{1}\) and a submodule \(N_{1}\) which is not projective and not stable. Continuing in this way by induction, we obtain a sequence \((P_{k})_{k=1}^{\infty}\) of nonzero proper projective submodules of \(M\) and a sequence \((N_{k})_{k=1}^{\infty}\) of nonzero proper submodules of \(M\) such that
+
+\[M=N_{k}\oplus P_{k}\oplus P_{k-1}\oplus\cdots \oplus P_{1}\text{with}N_{k}=N_{k+1}\oplus P_{k+1}\text{for all}k\in\mathbb{Z}^{+}.\]
+
+Since \(P_{i}\neq 0\text{for all}i\in I,\mathrm{u}.\dim(P_{i})\geq1\) and so for every \(n\in\mathbb{Z}^{+},\mathrm{u}.\dim(M)=\mathrm{u}.\dim(N_{n}\oplus P_{n}\oplus P_{n-1}\oplus\cdots \oplus P_{1})\geq n\) by [19, 6.6]. Therefore \(\mathrm{u}.\dim(M)=\infty\). Similarly \(\mathrm{h}.\dim(M)=\infty\) should hold using the properties of hollow dimension, see [11, Section 2.8]
+
+-

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+## 4. Open questions
+
+Question 1. Clearly, the first question we are interested in is whether there are inner functions \(\Theta\), other than the meromorphic inner functions \(e^{i\tau(\cdot)}\), such that the associated Toeplitz
+
+operator \(T_{\overline{\Theta}}:H_{\Theta}^{1}\rightarrow H^{1}(\mathbb{C}^{+})\) is bounded, or equivalently, for which inner functions \(\Theta\) the equality
+
+\[H_{\Theta}^{1}=K_{\Theta}^{1}\oplus\Theta H^{1}\]
+
+holds. Moreover, from Theorem 1.3 we know that, if \(\Theta\) is an inner function, \(T_{\overline{\Theta}}:H_{\Theta}^{1}\rightarrow H^{1}(\mathbb{C}^{+})\) is unbounded if and only if
+
+\[K_{\Theta}^{1}\oplus\Theta H^{1}⊊H_{\Theta}^{1},\]
+
+that is, if and only if the quotient \(H_{\Theta}^{1}/(K_{\Theta}^{1}\oplus\Theta H^{1})\) is non-trivial. When non-trivial, it would be interesting to describe it. Question 2. The second question delves deeper into the subject and it is related to another interesting problem. In analogy with the holomorphic case, we introduce a subspace of the
+
+real variable Hardy space \(H^{1}(\mathbb{R}).\text{For}b\in\text{BMO}(\mathbb{R})\) we consider the pre-annihilator of \(\{b\}\), that is,
+
+\[H_{b}^{1}(\mathbb{R}):=\left\{\phi\in H^{1}(\mathbb{R})\text{such that}\langle\phi,b\rangle=0\right\}.\] (4.1)
+
+Observe that, if \(b=\Theta\in H^{\infty}(\mathbb{C}^{+})\)
+
+\[H_{\Theta}^{1}(\mathbb{R})= \overline{H^{1}(\mathbb{C}^{+})}_{|_{\mathbb{R}}} \oplus(H_{\Theta}^{1})_{|_{\mathbb{R}}}.\]
+
+Following the ideas in Theorem 1.3, we reformulate the boundedness of \(T_{\overline{\Theta}}\) in terms of \([ \overline{\Theta} ,H]\). Proposition 4.1. Let \(\Theta\) be an inner function. Then, the following are equivalent:
+
+(a) the Toeplitz operator \(T_{\overline{\Theta}}:H_{\Theta}^{1}\rightarrow H^{1}(\mathbb{C}^{+})\) is bounded;
+
+(b) the commutator \([ \overline{\Theta} ,H]:H_{\Theta}^{1}(\mathbb{R} ) \rightarrow L^{1}(\mathbb{R})\) is bounded.
+
+(c) the multiplication operator \(M_{\overline{\Theta}}:H_{\Theta}^{1}(\mathbb{R})\rightarrow H^{1}(\mathbb{R})\) is bounded.
+
+Proof. We note that for every \(H_{\Theta}^{1}(\mathbb{R})\ni f=f_{1}+f_{2},\text{with}f_{1}\in \overline{H^{1}(\mathbb{C}^{+})}_{|_{\mathbb{R}}} \text{and}f_{2}\in H_{\Theta}^{1}\), by using (1.2) we have that
+
+Moreover,
+
+\[[ \overline{\Theta} ,H](f_{1})= \overline{\Theta} H(f_{1})-H( \overline{\Theta} f_{1})= \overline{\Theta} (-if_{1})+i \overline{\Theta} f_{1}=0.\]
+
+\[[ \overline{\Theta} ,H](f_{2})= \overline{\Theta} H(f_{2})-H( \overline{\Theta} f_{2})=-i \overline{\Theta} f_{2}+iP_{+}( \overline{\Theta} f_{2})-iP_{-}( \overline{\Theta} f_{2})=-2iH_{\overline{\Theta}}(f_{2}).\]
+
+Consequently, \([ \overline{\Theta} ,H]\) is bounded if and only if the Hankel operator \(H_{\overline{\Theta}}\) is bounded from \(H_{\Theta}^{1}\)
+
+to \(\overline{H^{1}(\mathbb{C}^{+})|_{\mathbb{R}}}\), which is equivalent to the boundedness of \(T_{\overline{\Theta}}:H_{\Theta}^{1}\rightarrow H^{1}(\mathbb{C}^{+})\), by Theorem 1.3. Thus, (a) and (b) are equivalent.
+
+Next, suppose (a) holds. Then we have that if \(f\in H_{\Theta}^{1}, \underline{\text{th}} \text{en}( \overline{\Theta} f)_{|_{\mathbb{R}}}\in H^{1}(\mathbb{R})\). Indeed, we
+
+wish to show that \(( \overline{\Theta} f)_{|_{\mathbb{R}}} \text{and}H(( \overline{\Theta} f)_{|_{\mathbb{R}}})\in L^{1} (\mathbb{R})\). Clearly \(\overline{\Theta} f \underline{\in} L^{1}(\mathbb{R})\), and
+
+\[H( \overline{\Theta} f)=-i\left(2P_{+}-\text{Id}\right)( \overline{\Theta} f)=-2iT_{\Theta}f+i \overline{\Theta} f\in L^{1}(\mathbb{R}),\]
+
+since (a) holds. Therefore, \(( \overline{\Theta} f)_{|_{\mathbb{R}}}\in H^{1}(\mathbb{R})\), and
+
+\[\Vert \overline{\Theta} f\Vert_{H^{1}(\mathbb{R})}\leq2\Vert T_{\Theta}f\Vert_{H^{1}(\mathbb{C}^{+})}+\Vert f\Vert_{H^{1}(\mathbb{C}^{+})}\leq C\Vert f\Vert_{H^{1}(\mathbb{C}^{+})},\]

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+Definition 4. The process defined on the interval \([0,1]\) as
+
+\[X(t):=\sum_{j=0}^{+\infty}\sum_{k=0}^{2^{j}-1}\left(\int_{0}^{1}(t-s)_{+}^{H_{j}(k/2^{j})-1/2}h_{j,k}(s)ds\right)\varepsilon_{j,k},\] (6)
+
+will be called the Gaussian Haar-based multifractional process. The abbreviation GHBMP will be used to denote processes from this class.
+
+Note that the GHBMP processes defined by (6) are centered Gaussian, but do not have stationary increments.
+
+## 3. On GHBMP and its Hölder continuity
+
+This section studies GHBMP and its basic properties. It also provides several key characteristics of Haar-based transforms that will be used in subsequent analyses. The first result shows that the series in Definition 4 is \(L_{2}-\)convergent and GHBMP is correctly defined.
+
+Theorem 3.1. For all fixed \(t\in[0,1]\) it holds
+
+\[\mathbb{E}X^{2}(t)=\sum_{j=0}^{+\infty}\sum_{k=0}^{2^{j}-1}\bigg|\int_{0}^{1}(t-s)_{+}^{H_{j}(k/2^{j})-1/2}h_{j,k}(s)ds\bigg|^{2}<+\infty.\]
+
+Therefore, the GHBMP processes are properly defined in \(L_{2}-\text{sense}\).
+
+The proof of Theorem 3.1 relies on the following result.
+
+Lemma 1. For any fixed value \(\lambda\in[ \underline{H} , \overline{H} ],\) let \(h^{[\lambda]}(\cdot)\) be the function defined on \(\mathbb{R}\) as
+
+\[\begin{aligned} h^{[\lambda]}(x):&=\int_{\mathbb{R}}(x-s)_{+}^{\lambda- \frac{1}{2}}h(s)ds \\ &=\left(\lambda+ \frac{1}{2} \right)^{-1}\left((x)_{+}^{\lambda+ \frac{1}{2}}-2\left(x- \frac{1}{2} \right)_{+}^{\lambda+ \frac{1}{2}}+(x-1)_{+}^{\lambda+ \frac{1}{2}}\right).(7) \end{aligned}\]
+
+Then, there is a constant \(c>0\) 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)

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+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.

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+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.

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+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].

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+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.

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+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
+
+\[\tilde{\psi}_{\tilde{\psi}_{3}^{1}}(_{(}r_{r},_{,}z_{z})_{)}:_{:}=_{=} \begin{bmatrix} \tfrac{ik_{1}(z)^{2}}{z} \tilde{p}_{+}(k_{1}(z)r) \\ \tilde{p}_{+} (k_{1}(z)r) \end{bmatrix} \begin{bmatrix} \tfrac{ik_{1}(z)^{2}}{z} \tilde{q}_{+}(k_{1}(z)r)_{,} & \tilde{\psi}_{4}(r,z):= & \tfrac{ik_{2}(z)^{2}}{z} \tilde{q}_{+}(k_{2}(z)r) \\ \tilde{q}_{+} (k_{1}(z)r) & \tilde{q}_{+} (k_{2}(z)r) \end{bmatrix} ,\tilde{\psi}_{2}(r,z):=\bigg[ \frac{ik_{2}(z)^{2}}{\tilde{p}_{+}^{z}} (p_{\tilde{k}_{2}^{+}}(_{(}k_{z^{2})r}(z_{)})r)_{,,}\]

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+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)

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+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 \(\Gamma\).