Efron et al. (2004) introduced least angle regression (LAR) as an algorithm for linear predictions, intended as an alternative to forward selection with connections to penalized regression. However, LAR has remained somewhat of a "black box," where some basic behavioral properties of LAR output are not well understood, including an appropriate termination point for the algorithm. We provide a novel framework for inference with LAR, which also allows LAR to be understood from new perspectives with several newly developed mathematical properties. The LAR algorithm at a data level can viewed as estimating a population counterpart "path" that organizes a response mean along regressor variables which are ordered according to a decreasing series of population "correlation" parameters; such parameters are shown to have meaningful interpretations for explaining variable contributions whereby zero correlations denote unimportant variables. In the output of LAR, estimates of all non-zero population correlations turn out to have independent normal distributions for use in inference, while estimates of zero-valued population correlations have a certain non-normal joint distribution. These properties help to provide a formal rule for stopping the LAR algorithm. While the standard bootstrap for regression can fail for LAR, a modified bootstrap provides a practical and formally justified tool for interpreting the entrance of variables and quantifying uncertainty in estimation. The LAR inference method is studied through simulation and illustrated with data examples.

In this paper, we give explicit expressions about $q$-harmonic sums on $1-\cdots-1,A,1-\cdots-1$ indices. When $A=1$, many previous authors have studied and showed the identities, expressions, and properties. There are many results for explicit expressions about $q$-multiple zeta values or $q$-harmonic sums on $A-\cdots-A$ indices. Though there is the way to treat $q$-multiple zeta values unless the indices are the same, it has been successful to get the explicit expression of $q$-harmonic sums on $1-\cdots-1,A,1-\cdots-1$ indices when $A=2$. In this paper, we shall consider more general results when $A\ge 3$.

Representation theory of Lie (super)algebras has attracted significant research interest for many years, especially due to its applications in theoretical physics; in this regard, the representation theory of affine Lie (super)algebras is of central importance. To characterize simple modules over affine Lie (super)algebras, it is necessary to study the cases of nonzero and critical levels separately. Although a vast amount of research has been done on the representation theory of affine Lie (super)algebras $\mathcal{L}$, investigations concerning general modules at the critical level remain limited. In all existing studies, the characterization of the modules under consideration is reduced to the characterization of modules over some subalgebras of $\mathcal{L}$. Depending on the structure of the original modules, these subalgebras -- and the corresponding modules -- have different natures some of which are already known, while others need to be studied separately. In this paper, we give a complete characterization of the modules over specific subalgebras $\mathcal{G}$ of a twisted affine Lie superalgebra $\mathcal{L}$ that arise in the study of general zero level simple finite weight $\mathcal{L}$-modules. In particular, in the special case that $\\mathcal{G} = \mathcal{L}$, we obtain a complete characterization of quasi-integrable $\mathcal{L}$-modules of level zero.

In statistical physics, the multivariate hard-core model describes a system of particles, each of which receives its own fugacity. In graph-theoretic language, the partition function of the model translates to the multivariate independence polynomial, i.e., the multiaffine generalisation of the independence polynomial, defined by $Z_G(λ_1,\dots,λ_n) := \sum_{I\in\mathcal{I}(G)} \prod_{v\in I}λ_v$, where $\mathcal{I}(G)$ denotes the set of all independent sets in a graph $G$ on $[n]:=\{1,2,\dots,n\}$. We prove that for every simple graph $G$ on $[n]$ and $λ_1,\dots,λ_n\geq 0$, \[ Z_G(λ_1,\dots,λ_n) \geq \prod_{i=1}^n (1+(d_i+1)λ_i)^{1/(d_i+1)}, \] where $d_1,\dots,d_n$ is the degree sequence of $G$. This generalises a result of Sah, Sawhney, Stoner, and Zhao, who proved the univariate case $λ_1=\dots=λ_n=λ$. We further conjecture that our inequality should generalise to other antiferromagnetic models and give some evidence in support of it. In particular, for $λ_i,μ_i\geq 0$, $1\leq i\leq n$, we obtain a stronger inequality \[ \sum_{\substack{I,J\in \mathcal{I}(G) \\ I\cap J=\emptyset}} \prod_{v\in I}λ_v\prod_{u\in J}μ_u \geq \prod_{i=1}^n \left(1+(d_i+1)(λ_i+μ_i)+d_i(d_i+1)λ_iμ_i\right)^{1/(d_i+1)}, \] which proves our conjecture for a multiaffine generalisation of the semiproper colouring partition function with two proper colours. Our key technical steps for both theorems are obtained by using a custom mathematical research agent built on top of Gemini Deep Think, which can be seen as a benchmark demonstrating that the current state-of-the-art language models can, in part, assist with mathematical research.

The Castelnuovo-Mumford polynomials are the maximal degree components of Grothendieck polynomials. The support of each Castelnuovo-Mumford polynomial is conjectured to be M-convex, i.e. the set of integer points of a generalized permutahedron (Mészáros and St. Dizier, 2020). This conjecture is known to hold in certain special cases but remains open in general. We define new families of permutations whose Castelnuovo-Mumford polynomials we show to have M-convex support. Specifically, we investigate which permutations have Castelnuovo-Mumford polynomials whose supports are the set of integer points in a schubitope.

We consider a run-and-tumble particle (RTP) with stochastic resetting confined to the half line $[0,\infty)$ with a sticky boundary at $x=0$. In the bulk the RTP tumbles at a constant rate $α>0$ between velocity states $\pm v$ with $v>0$ and randomly resets to its initial position and orientation $(x_0,k_0)\in(\mathbb{R}^+,\pm)$. When the RTP reaches the target at $x=0$ it attaches to the boundary for a random waiting time before either detaching and continuing to navigate the bulk domain or (permanently) entering the target. These events are the analogs of adsorption, desorption, and absorption of a particle by a partially reactive surface in physical chemistry. We use renewal theory to characterize the particle trajectory in terms of successive binding events under two distinct desorption protocols: via resetting to $(x_0,k_0)$ and via continuous movement from $x=0$ with velocity $+v$. First we derive the nonequilibrum stationary state (NESS) in the case of no absorption and characterize the accumulation at the boundary. Second, we compute the mean first passage time (MFPT) statistics. In addition to observing the usual unimodal dependence of the MFPT on bulk resetting, both the NESS and MFPT strongly depend on the initial orientation $k_0$ and the desorption protocol. For instance, if the initial orientation is toward the boundary, we find that the desorption-induced resetting protocol can reduce the MFPT more effectively than the non-resetting desorption protocol. We also show how matching the desorption kinetics with the bulk resetting or tumbling rate introduces a trade-off between minimizing the adsorption and absorption times. In this setting we find that the desorption protocol which minimizes the absorption MFPT for a given set of parameters is almost always the opposite of that favored when desorption and bulk kinetics are not the same.

This paper derives non-asymptotic error bounds for nonlinear stochastic approximation algorithms in the Wasserstein-$p$ distance. To obtain explicit finite-sample guarantees for the last iterate, we develop a coupling argument that compares the discrete-time process to a limiting Ornstein-Uhlenbeck process. Our analysis applies to algorithms driven by general noise conditions, including martingale differences and functions of ergodic Markov chains. Complementing this result, we handle the convergence rate of the Polyak-Ruppert average through a direct analysis that applies under the same general setting. Assuming the driving noise satisfies a non-asymptotic central limit theorem, we show that the normalized last iterates converge to a Gaussian distribution in the $p$-Wasserstein distance at a rate of order $γ_n^{1/6}$, where $γ_n$ is the step size. Similarly, the Polyak-Ruppert average is shown to converge in the Wasserstein distance at a rate of order $n^{-1/6}$. These distributional guarantees imply high-probability concentration inequalities that improve upon those derived from moment bounds and Markov's inequality. We demonstrate the utility of this approach by considering two applications: (1) linear stochastic approximation, where we explicitly quantify the transition from heavy-tailed to Gaussian behavior of the iterates, thereby bridging the gap between recent finite-sample analyses and asymptotic theory and (2) stochastic gradient descent, where we establish rate of convergence to the central limit theorem.

We prove a priori subelliptic estimates, near a non-characteristic boundary point, for the heat operators associated to a wide class of maximally subelliptic quadratic forms. This is the third paper in a series devoted to studying general maximally subelliptic boundary value problems.

We consider a time-slotted job-assignment system consisting of a central server, $N$ task-specific networks of machines, and multiple users. Each network specializes in executing a distinct type of task. Users stochastically generate jobs of various types and forward them to the central server, which routes each job to the appropriate network of machines. Due to resource constraints, the server cannot serve all users' jobs simultaneously, which motivates the design of scheduling policies with possible preemption. To evaluate scheduling performance, we introduce a novel timeliness metric, the age of job, inspired by the well-known metric, the age of information. We study the problem of minimizing the long-term weighted average age of job. We first propose a max-weight policy by minimizing the one-step Lyapunov drift and then derive the Whittle index (WI) policy when the job completion times of the networks of machines follow geometric distributions. For general job completion time distributions, we introduce a Whittle index with max-weight fallback (WIMWF) policy. We also investigate the Net-gain maximization (NGM) policy. Numerically, we show that the proposed WIMWF policy achieves the best performance in the general job completion time setting. We also observe a scaling trend: two different max-weight policies can outperform the NGM policy in small systems, whereas the NGM policy improves as we scale the system size and becomes asymptotically better than max-weight policies. For geometric service times, the WI policy yields the lowest age across all considered system sizes.

Bayesian optimization (BO) is a popular, sample-efficient technique for expensive, black-box optimization. One such problem arising in manufacturing is that of maximizing the reliability, or equivalently minimizing the probability of a failure, of a design which is subject to random perturbations - a problem that can involve extremely rare failures ($P_\mathrm{fail} = 10^{-6}-10^{-8}$). In this work, we propose two BO methods based on Thompson sampling and knowledge gradient, the latter approximating the one-step Bayes-optimal policy for minimizing the logarithm of the failure probability. Both methods incorporate importance sampling to target extremely small failure probabilities. Empirical results show the proposed methods outperform existing methods in both extreme and non-extreme regimes.

We develop the theory of translating solitons for the Mean Curvature Flow (MCF) in hyperbolic space of dimension $n+1\ge 3$. More specifically, we establish that horospheres are dynamically stable as radial graphical solutions to MCF. To that end, we construct rotationally invariant translators analogous to the winglike solitons introduced by Clutterbuck, Schnürer and Schulze, which serve as barriers in an argument based on White's avoidance principle and the strong maximum principle for parabolic PDEs.

Topological Hochschild homology (THH) is an invariant of ring spectra developed by Bökstedt. In recent years many equivariant analogues to THH have emerged. One example is twisted THH which is an invariant of $C_n$-equivariant ring spectra developed by Angeltveit, Blumberg, Gerhardt, Hill, Lawson, and Mandell. In this paper, we study the algebraic structure of twisted THH, and perform some computations. Specifically, we compute $C_2$-twisted THH of the Real bordism spectrum and show that the $C_p$-twisted THH of geometric ring $C_p$-spectra reduces to a computation of classical THH. We extend the algebraic structure of twisted THH to the twisted Bökstedt spectral sequence of Adamyk, Gerhardt, Hess, Klang, and Kong. We show that, under appropriate flatness conditions and for $R$ a commutative ring $C_p$-spectrum, the $C_p$-twisted Bökstedt spectral sequence is a spectral sequence of commutative $\underline{E}_\star(R)$-algebras.

Kato developed an exotic Deligne-Langlands correspondence using a geometric model for the multiparameter affine Hecke algebra of type C, based on his exotic nilpotent cone. Achar-Henderson and Springer showed that this exotic nilpotent is intimately related to another, apparently simpler variety called the enhanced nilpotent cone. Each of these is defined as the Hilbert nullcone of a polar module, the exotic Sp(2n)-module and the enhanced GL(n)-module, respectively. In this paper we conduct a detailed study of the geometry of these two modules, by introducing the Jordan stratification, simultaneously generalising classical results on the adjoint representation as well as the symmetric space associated to (gl(2n), sp(2n)). One of the key tools we develop is the theory of induced orbits in the enhanced and exotic nilpotent cones, following the work of Lusztig-Spaltenstein. Our main application is a classification of sheets in these modules, inspired by a theorem of Borho.

We study the asymptotic behaviour of sequences of multivariate random variables representing the number of occurrences of a given set of symbols in a word of length $n$ generated at random according to a rational stochastic model. Assuming primitive the matrix of the total weights of transitions of the model, we first determine asymptotic expressions for the mean values and the covariances of such statistics. Then we establish two asymptotic results that generalize known univariate cases to different regimes: a large deviation principle with speed $n$, implying almost sure convergence, and a multivariate Gaussian limit. Additionally, we introduce a novel moderate deviation result as a bridge between these regimes. Central to our proofs is a quasi-power property for the moment generating function of the statistics, allowing us to employ the Gärtner-Ellis Theorem for both large and moderate deviations.

We discuss how to write down three specific natural numbers $A$, $B$, $C$ such that for any real number $r$ you've probably ever thought of, it is consistent with $\mathsf{ZFC}$ set theory that $$\def\Rb{\mathbb{R}}\def\Nb{\mathbb{N}}r = \log\left(\sup_{x_0,x_1 \in \Rb} \inf_{x_2 \in \Rb} \sup_{x_3 \in \Rb}\inf_{x_4 \in \Rb}\sup_{m \in \Nb}\inf_{n_0,\dots,n_{A} \in \Nb} x^2_0 \begin{bmatrix} \phantom{+}(n_0 - 2)^2 + (n_1-m)^2 \\ + n_2 + (n_B - n_C)^2 \\ + n_3 \sum_{k=0}^4 ( x_k - \frac{n_{k+5}}{1+n_4} +n_4)^2 \\ + \sum_{i,j = 0}^B (n_{9+2^i3^j} - n_i^{n_j})^2 \end{bmatrix} \right).$$ We also discuss why it's possible, assuming the existence of certain large cardinals, for there to be a real number $s$ which cannot be the value of this formula for our particular $A$, $B$, $C$. This involves set-theoretic mice.

Both natural and synthetic chemical systems not only exhibit a range of non-trivial dynamics, but also transition between qualitatively different dynamical behaviours as environmental parameters change. Such transitions are called bifurcations. Here, we show that recurrent neural chemical reaction networks (RNCRNs), a class of chemical reaction networks based on recurrent artificial neural networks that can be trained to reproduce a given dynamical behaviour, can also be trained to exhibit bifurcations. First, we show that RNCRNs can inherit some bifurcations defined by smooth ordinary differential equations (ODEs). Second, we demonstrate that the RNCRN can be trained to infer bifurcations that allow it to approximate different target behaviours within different regions of parameter space, without explicitly providing the bifurcation itself in the training. These behaviours can be specified using target ODEs that are discontinuous with respect to the parameters, or even simply by specifying certain desired dynamical features in certain regions of the parameter space. To achieve the latter, we introduce an ODE-free algorithm for training the RNCRN to display designer oscillations, such as a heart-shaped limit cycle or two coexisting limit cycles.

We investigate the geometric and topological properties of the group of locally conformally symplectic (LCS) diffeomorphisms, utilizing the LCS flux homomorphism defined by S. Haller. By analyzing the flux map from the universal cover of the identity component $(\ker Φ)_0$ to the first Lichnerowicz cohomology group $H_ω^1(M)$, we establish a short exact sequence characterizing the Hamiltonian subgroup $\Ham_Ω(M)$ and provide conditions for its topological splitting as a semidirect product. We develop LCS analogues of fundamental symplectic results, including a Weinstein neighborhood theorem, a flux rigidity theorem for homotopies, and a characterization of LCS structures on mapping tori. A central theme of this work is the influence of the Hodge decomposition of the Lee form $ω= dh + l$. In the exact case ($l=0$), we utilize the global conformal equivalence to symplectic structures to establish energy-capacity inequalities, an LCS Hofer metric, and non-displaceability results. We explicitly analyze the relationship between the LCS Calabi invariant and its symplectic counterpart, showing they are controlled by a multiplicative factor depending on the conformal weight. For the general non-exact case ($l \neq 0$), we introduce a Twisted Calabi invariant that captures the interaction between Hamiltonian dynamics and the harmonic component of the Lee form.

Generalising the two-dimensional case, we provide estimates for the mean-values of the lengths of the edges of an integral box with given volume and minimal surface.

We study the evolution of the population size distribution of a critical Galton-Watson process with infinite variance of the offspring size of particles assuming that the population size is unusually small at the distant moment $n$ of observation.

A major barrier to wide adoption of Electric Vehicles (EVs) is the absence of reliable and equitable charging infrastructure. Poorly located charging stations create coverage gaps and slow down EV adoption, especially in underserved communities. This paper proposes a two-stage stochastic mixed-integer programming model for the optimal deployment of Fixed and Mobile Charging Stations (FCSs and MCSs) across multiple zones and periods. Initially, a finite dominating set of candidate locations is identified using the Edge Scanning Algorithm for a Single Refueling Station (ESS), an exact continuous-location method. We modify the ESS algorithm to incorporate existing public charging stations, thereby avoiding redundant coverage. In the first stage of our model, FCSs are allocated based on long-term traffic patterns, budgetary constraints, and socioeconomic factors to ensure stable baseline coverage. The second stage dynamically assigns MCSs in response to short-term demand fluctuations and uncertainties, aiming to minimize relocation costs while maximizing coverage. We use a scenario-based framework to capture demand variability. Numerical experiments on realistic networks demonstrate the model's capacity to enhance system resilience and reduce unmet demand. These findings offer practical insights for planners and policymakers seeking to develop accessible and demand-responsive EV charging infrastructure.

Data assimilation plays a crucial role in modern weather prediction, providing a systematic way to incorporate observational data into complex dynamical models. The paper addresses continuous data assimilation for a model arising as a singular limit of the three-dimensional compressible Navier-Stokes-Fourier system with rotation driven by temperature gradient. The limit system preserves the essential physical mechanisms of the original model, while exhibiting a reduced, effectively two-and-a-half-dimensional structure. This simplified framework allows for a rigorous analytical study of the data assimilation process while maintaining a direct physical connection to the full compressible model. We establish well posedness of global-in-time solutions and a compact trajectory attractor, followed by the stability and convergence results for the nudging scheme applied to the limiting system. Finally, we demonstrate how these results can be combined with a relative entropy argument to extend the assimilation framework to the full three-dimensional compressible setting, thereby establishing a rigorous connection between the reduced and physically complete models.

We study the Holomorphic and Random Dynamics of some rank 2 free groups generated by two Hénon type maps. For these simply constructed examples we prove that the Fatou set is non-empty and that the stationary measures are supported on a compact set. With some further care this allows us to construct examples having no stationary measures. These examples illustrate the types of phenomena that may arise when studying holomorphic group actions on non-compact manifolds.

Testing whether two multivariate samples exhibit the same extremal behavior is an important problem in various fields including environmental and climate sciences. While several ad-hoc approaches exist in the literature, they often lack theoretical justification and statistical guarantees. On the other hand, extreme value theory provides the theoretical foundation for constructing asymptotically justified tests. We combine this theory with Kullback-Leibler divergence, a fundamental concept in information theory and statistics, to propose a test for equality of extremal dependence structures in practically relevant directions. Under suitable assumptions, we derive the limiting distributions of the proposed statistic under null and alternative hypotheses. Importantly, our test is fast to compute and easy to interpret by practitioners, making it attractive in applications. Simulations provide evidence of the power of our test. In a case study, we apply our method to show the strong impact of seasons on the strength of dependence between different aggregation periods (daily versus hourly) of heavy rainfall in France.

On a compact Lie group $G$, we consider the reproducing kernel Hilbert space $\mathcal{H}_K$ associated with the integral kernel $K$ of a left-invariant, positive, symmetric, trace class integral operator on $L^2(G)$. We present lower and upper asymptotic estimates for the entropy Kolmogorov numbers (also called covering numbers) for the embedding of $\mathcal{H}_K$ into the space $C(G)$ of continuous functions on $G$.

The Bodirsky-Pinsker conjecture asserts a P vs. NP-complete dichotomy for the computational complexity of Constraint Satisfaction Problems (CSPs) of first-order reducts of finitely bounded homogeneous structures. Prominently, two structures in the scope of the conjecture have log-space equivalent CSPs if they are pp-bi-interpretable, or equivalently, if their polymorphism clones are topologically isomorphic. The latter gives rise to the algebraic approach which regards structures with topologically isomorphic polymorphism clones as equivalent and seeks to identify structural reasons for hardness or tractability in topological clones. We establish that the equivalence relation of pp-bi-interpretability underlying this approach is reasonable: On the one hand, we show that it is decidable under mild conditions on the templates; this improves a theorem of Bodirsky, Pinsker and Tsankov (LICS'11) on decidability of equality of polymorphism clones. On the other hand, we show that within the much larger class of transitive $ω$-categorical structures without algebraicity, the equivalence relation is of lowest possible complexity in terms of descriptive set theory: namely, it is smooth, i.e., Borel-reduces to equality on the real numbers. On our way to showing the first result, we establish that the model-complete core of a structure that has a finitely bounded Ramsey expansion (which might include all structures of the Bodirsky-Pinsker conjecture) is computable, thereby providing a constructive alternative to previous non-constructive proofs of its existence.

The power spectrum of the Brownian motion of probe microparticles with mass m and radius R immersed in a viscoelastic material reveals valuable information about repetitive patterns and correlation structures that manifest in the frequency domain. In this paper, we employ a viscous viscoelastic correspondence principle for Brownian motion and we show that the power spectrum of Brownian motion in any linear, isotropic viscoelastic material is proportional to the real part of the complex dynamic fluidity of a linear rheological network that is a parallel connection of the linear viscoelastic material within which the Brownian particles are immersed and an inerter, with distributed intrance with mass mR. The synthesis of this rheological analogue simplifies appreciably the calculation of the power spectrum for Brownian motion within viscoelastic materials such as Maxwell fluids, Jeffreys fluids, subdiffusive materials, or in dense viscous fluids that give rise to hydrodynamic memory.

Classical game theory is a powerful framework to analyze the strategic interactions among rational players. However, in many real-life scenarios, players choose actions based on their inherent natural tendencies rather than deliberate reasoning. In this paper, we develop an analytical framework to study large population games with an alpha-fraction of rational and (1-alpha)-fraction of herding players. We introduce a new notion of equilibrium called alpha-Rational Nash Equilibrium (in short, alpha-RNE) and discuss its interpretations. Some classical equilibria may disappear, and some new ones may emerge, but only for smaller alpha >0. Interestingly, rational players benefit from the presence of herding and may even achieve utility exceeding the socially optimum. Even more strikingly, in some cases, the herding players also benefit, attaining utility close to the social optimum. We further study the effect of the herding fraction on system performance using measures such as the Price of Anarchy (PoA). In transportation networks, a well-known paradox first studied by Pigou and later by Braess typically arises from rational decision-making: adding an extra link can reduce overall efficiency. Our analysis leads to a different conclusion. When a substantial fraction of users exhibit herding behavior, introducing a new link can increase efficiency, provided herding choices can be suitably influenced. The gains are larger when the herding fraction is higher and/or congestion is lower. By contrast, when herding decisions cannot be influenced, the added link may become detrimental. We also study a bandwidth sharing game in which herding tendencies improve system efficiency. Finally, we discuss the mechanism or influence design in the presence of herding, highlighting both opportunities and risks.

In algebraic geometry, trigonal curves can always be embedded into Hirzebruch surfaces. In tropical geometry, the notion of trigonality does not have a unique translation. We focus on the characterization in terms of the existence of a degree 3 morphism to a line, and discuss relations to possible embeddings into $\mathbb R^2$ reflecting an embedding into a Hirzebruch surface. Our results can be divided into three parts: for tropical curves of low genus 3 and 4, we discuss the relation between a trigonal morphism and an embedding dual to the polygon of a Hirzebruch surface, building on works on embeddings of hyperelliptic tropical curves and curves of low genus. We compare obstructions for embeddings with obstructions for the existence of a degree 3 morphism to a line. Finally, we showcase examples where a non-smooth embedding can be unfolded to reflect certain features of a degree 3 morphism to a line.

In this work, we propose the orthogonal delay-Doppler (DD) division multiplexing (ODDM) modulation with frequency modulated continuous wave (FMCW) (ODDM-FMCW) waveform to enable integrated sensing and communication (ISAC) with a low peak-to-average power ratio (PAPR). We first propose a square-root-Nyquist-filtered FMCW (SRN-FMCW) waveform to address limitations of conventional linear FMCW waveforms in ISAC systems. To better integrate with ODDM, we generate SRN-FMCW by embedding symbols in the DD domain, referred to as a DD-SRN-FMCW frame. A DD chirp compression receiver is designed to obtain the channel response efficiently. Next, we construct the proposed ODDM-FMCW waveform for ISAC by superimposing a DD-SRN-FMCW frame onto an ODDM data frame. A comprehensive performance analysis of the ODDM-FMCW waveform is presented, covering peak-to-average power ratio, spectrum, ambiguity function, and Cramer-Rao bound for delay and Doppler estimation. Numerical results show that the proposed ODDM-FMCW waveform delivers excellent ISAC performance in terms of root mean square error for sensing and bit error rate for communications.

Shallow Water Moment Equations (SWME) are extensions to the well-known Shallow Water Equations (SWE) for the efficient modeling and numerical simulation of free-surface flows. While the SWE typically assume a depth-averaged vertical velocity profile, the SWME allow for vertical variations of the velocity profile. The SWME therefore assume a polynomial profile and then derive additional evolution equations for the polynomial coefficients via higher order depth integration. In this work, we perform a new systematic derivation of the energy equation for a specific variant of the SWME, called the Shallow Water Linearized Moment Equations (SWLME). The derivation is based on the standard SWE energy equation derivation and includes the skew-symmetric formulation of the model. The new systematic derivation is beneficial for the extension to other SWME variants and their numerical solution.