Quantum computing has recently been emerging in theoretical chemistry as a realistic avenue meant to offer computational speedup to challenging eigenproblems in the context of strongly-correlated molecular systems or extended materials. Most studies so far have been devoted to the quantum treatment of electronic structure and only a few were directed to the quantum treatment of vibrational structure, which at the moment remains not devoid of unknowns. In particular, we address here a formal problem that arises when simulating a vibrational model under harmonic second quantization, whereby the disruption of the closure relation (resolution of the identity) -- which occurs when truncating the infinite bosonic basis set -- may have some serious effects as regards the correct evaluation of Hamiltonian matrix elements. This relates intimately to the normal ordering of products of ladder operators. In addition, we discuss the relevance of choosing an adequate primitive basis set within the present context with respect to its variational convergence properties. Such fundamental, yet consequential, aspects are illustrated numerically in the present work on a one-dimensional anharmonic Hamiltonian model corresponding to a double-well potential showing strong tunneling, of interest both for vibrational spectroscopy and chemical reactivity.

For $ 0 < r < 1 $, let $ \mathbb{A}_r = \{ z \in \mathbb{C} : r < |z| < 1 \} $ be the annulus with boundary $ \partial \overline{\mathbb{A}}_r = \mathbb{T} \cup r\mathbb{T} $, where $ \mathbb{T} $ is the unit circle in the complex plane $\mathbb C$. We study the class of operators \[ C_{1,r} = \{ T : T \text{ is invertible and } \|T\|, \|rT^{-1}\| \leq 1 \}, \] introduced by Bello and Yakubovich. Any operator $T$ for which the closed annulus $\overline{\mathbb{A}}_r$ is a spectral set is in $C_{1,r}$. The class $C_{1, r}$ is closely related to the \textit{quantum annulus} which is given by \[ QA_r = \{ T : T \text{ is invertible and } \|rT\|, \|rT^{-1}\| \leq 1 \}. \] McCullough and Pascoe proved that an operator in $ QA_r $ admits a dilation to an operator $ S $ satisfying $(r^{-2} + r^2)I - S^*S - S^{-1}S^{-*} = 0$. An analogous dilation result holds for operators in $ C_{1,r}$ class. We extend these dilation results to doubly commuting tuples of operators in quantum annulus as well as in $C_{1,r}$ class. We also provide characterizations and decomposition results for such tuples.

Despite their apparent simplicity, coupled oscillators exhibit surprisingly complex phenomena. Two notable examples are Berry phase (a geometric or topological aspect of the oscillators' memory) and non-Hermiticity (the often counterintuitive impact of dissipation), both of which possess rich mathematical structures. Here, we demonstrate that combining Berry phase and non-Hermiticity leads to a fundamentally new form of amplification. Specifically, we show that this combination allows a lossy oscillator system to be converted into one with gain via slow modulation of its parameters. This is distinct from other amplification mechanisms, as it results specifically from the complex-valued Berry phase that is unique to non-Hermitian systems. We show that this mechanism produces continuous, useful gain in an optomechanical system, and that similar results can be realized in a very wide range of settings.

Households play a crucial role in the propagation of infectious diseases due to the frequent and prolonged interactions that typically occur between their members. Recent studies have emphasized the need to include socioeconomic variables in epidemic models to account for the heterogeneity induced by human behavior. While sub-Saharan Africa suffers the highest burden of infectious disease diffusion, few studies have investigated the mixing patterns in the countries and their relation with social indicators. This work analyzes household contact matrices measured with wearable proximity sensors in a rural and an urban village in South Africa. Leveraging a rich data collection describing additional individual and household attributes, we investigate how the household contact matrix varies according to the household type (whether it is composed only of a familiar nucleus or by a larger group), the gender of its head (the primary decision-maker), the rural or urban context, and the season in which it was measured. We show the household type and the gender of its head induce differences in the interaction patterns between household members, particularly regarding child caregiving, suggesting they are relevant attributes to include in epidemic modeling.

We study the energy-momentum tensor of a baryon in a top-down holographic QCD. In holographic QCD, the baryons are represented as solitons in a 5-dimensional gauge theory. We obtain the soliton solution by solving the equations of motion numerically. Using this result, the energy-momentum tensor and related quantities such as the mass, mean square radii, and the D-term (druck term) are computed. The evaluated D-term is about -2.05, whose absolute value is significantly larger than that in the previous work arXiv:2206.06578.

(119951) 2002 KX14 is a large classical TNO with limited previous observations and unresolved questions regarding its physical properties. Five stellar occultations by 2002 KX14 were observed from 2020 to 2023, involving multiple telescopes across different locations in Europe and the Americas. The five occultations resulted in 15 positive chords, accurately measuring the 2002 KX14's shape and size. The projected ellipse has semi-major and semi-minor axes of $241.0 \pm 7.2$ km and $157.1 \pm 5.2$ km, respectively, corresponding to an average area-equivalent diameter of $389.2 \pm 8.7$ km. The geometric albedo was estimated at $11.9 \pm 0.7\%$.

This article is concerned with a new filtered two-step variational integrator for solving the charged-particle dynamics in a mildly non-uniform moderate or strong magnetic field with a dimensionless parameter $\varepsilon$ inversely proportional to the strength of the magnetic field. In the case of a moderate magnetic field ($\varepsilon=1$), second-order error bounds and long-time near-conservation of energy and momentum are obtained. Moreover, the proof of the long-term analysis is accomplished by the backward error analysis. For $0<\varepsilon \ll 1$, the proposed integrator achieves uniform second-order accuracy in the position and the parallel velocity for large step sizes, while attaining first-order accuracy with respect to the small parameter $\varepsilon$ for smaller step sizes. The error bounds are derived from a comparison of the modulated Fourier expansions of the exact and numerical solutions. Moreover, long-time near-conservation of the energy and the magnetic moment is established using modulated Fourier expansion and backward error analysis. All the theoretical results of the error behavior and long-time near-conservation are numerically demonstrated by four numerical experiments.

Physics-informed deep learning approaches have been developed to solve forward and inverse stochastic differential equation (SDE) problems with high-dimensional stochastic space. However, the existing deep learning models have difficulties solving SDEs with high-dimensional spatial space. In the present study, we propose a scalable physics-informed deep generative model (sPI-GeM), which is capable of solving SDE problems with both high-dimensional stochastic and spatial space. The sPI-GeM consists of two deep learning models, i.e., (1) physics-informed basis networks (PI-BasisNet), which are used to learn the basis functions as well as the coefficients given data on a certain stochastic process or random field, and (2) physics-informed deep generative model (PI-GeM), which learns the distribution over the coefficients obtained from the PI-BasisNet. The new samples for the learned stochastic process can then be obtained using the inner product between the output of the generator and the basis functions from the trained PI-BasisNet. The sPI-GeM addresses the scalability in the spatial space in a similar way as in the widely used dimensionality reduction technique, i.e., principal component analysis (PCA). A series of numerical experiments, including approximation of Gaussian and non-Gaussian stochastic processes, forward and inverse SDE problems, are performed to demonstrate the accuracy of the proposed model. Furthermore, we also show the scalability of the sPI-GeM in both the stochastic and spatial space using an example of a forward SDE problem with 38- and 20-dimension stochastic and spatial space, respectively.

We investigate the use of low-precision first-order methods (FOMs) within a fix-and-propagate (FP) framework for solving mixed-integer programming problems (MIPs). We employ GPU-accelerated PDLP, a variant of the Primal-Dual Hybrid Gradient (PDHG) method specialized to LP problems, to solve the \LP-relaxation of our MIPs to low accuracy. This solution is used to motivate fixings within our FP framework. We evaluate the performance of our heuristic on MIPLIB 2017, demonstrating that low-accuracy LP solutions do not lead to a loss in the quality of the FP heuristic solutions. Further, we use our FP framework to produce high-accuracy solutions for large-scale (up to 243 million nonzeros and 8 million decision variables) unit commitment-based dispatch and expansion planning problems created with the modeling framework REMix. For the largest problems, we can generate solutions with a primal-dual gap of under 2% in less than 4 hours, whereas state-of-the-art commercial solvers cannot produce feasible solutions within 2 days of runtime.

Altermagnets, which exhibit the advantages of both antiferromagnets and ferromagnets, have attracted significant attention recently. Among them, ruthenium dioxide (RuO2), a prototypical altermagnet candidate, is under intensive debate on its magnetic order and altermagnetic characters. In this work, we provide a comprehensive study of the spin-to-charge conversion in epitaxial RuO2 thin films with various orientations and fabrication methods. By utilizing thermal spin injections from a ferrimagnetic insulator, we unambiguously reveal a negative spin Hall angle for RuO2, which is opposite to all the previous reports using ferromagnetic metals. Most importantly, we observe robust anisotropic spin-to-charge conversion in RuO2, with voltage ratios of 30% for the (100)- and (110)-orientations and 40% for the (101)-orientations. The ratio remains consistent across RuO2 films fabricated by sputtering, pulsed laser deposition, and molecular-beam epitaxy. These results conclusively show a robust and anisotropic spin Hall effect in RuO2 with the absence of altermagnetic spin-splitting contributions. Our study provides crucial insights and advances the understanding of spin-to-charge conversions in emerging materials with low crystal symmetries.

We reinterpret publicly available $K^+ \to π^+ ν\barν$ data collected by NA62 from 2016 to 2024 to constrain the fundamental vectorial coupling of the QCD axion to down and strange quarks. Using a fully reproducible likelihood analysis and a complete renormalization-group evolution of the axion couplings from the Peccei-Quinn (PQ) scale to the kaon scale, we translate the experimental limit into bounds on both the low-energy flavour-violating coupling and the fundamental UV parameters. In the generic regime where strong contributions dominate the decay amplitude, we obtain $|(F_V)_{sd}(μ_K)| > 1.6 \times 10^{12}\,\text{GeV}$. The coexistence of parametrically suppressed weak contributions implies a second, conceptually distinct constraint: the fact that weak-amplitude dominance arises from highly tuned UV coupling configurations, translates into a conservative general lower limit on the PQ scale, $f_a > 4.9 \times 10^4\,\text{GeV}$. These results provide the strongest accelerator-based constraints on axion-induced $d \leftrightarrow s$ transitions and establish a robust lower bound on $f_a$, complementary to astrophysical limits.

Momentum conservation in the nucleon is examined in terms of continuous flow of the momentum current density (or in short, momentum flow), which receives contributions from both kinetic motion and interacting forces involving quarks and gluons. While quarks conduct momentum flow through their kinetic motion and the gluon scalar (anomaly) contributes via pure interactions, the gluon stress tensor has both effects. The quarks momentum flow encodes the information of the color-Lorentz force density on them, and the momentum conservation allows to trace its origin to the gluon tensor and anomaly (a ``negative pressure'' potential). From the state-of-the-art lattice calculations and experimental fits on the form factors of the QCD energy-momentum tensor, we exhibit pictures of the momentum flow and the color-Lorentz forces on the quarks in the nucleon. In particular, the anomaly contributes a critical attractive force with a strength similar to that of a heavy-quark confinement potential.

We propose an experimental scheme to load ultracold Fermi gases from the ground orbital band of a one-dimensional optical lattice into the first excited orbital band. Unlike the narrow momentum distribution of a Bose-Einstein Condensate, Fermi gases exhibit a broad momentum distribution. To address this, we define the average loading efficiency across all quasi-momentum states and theoretically perform the loading operation simultaneously for each Bloch state. Using a multiparameter global optimization method, we determine the loading efficiency at various lattice depths. We can enhance the loading efficiency by adjusting the phase of the lattice, which leverages the different symmetries of Bloch wavefunctions in various optical lattice orbitals. We also identified that the primary factor hindering higher loading efficiency in the Fermi gas is the multiple occupancy of the quasi-momentum states. Our simulations of various occupancies revealed a decreasing trend in mean loading efficiency as the number of occupied quasi-momentum states increases. Finally, we compare our method with other loading techniques and assess its experimental feasibility.

This paper explores the critical behavior of the semilinear heat equation $u_t+\mathcal{L}_{a, b}u=|u|^p+f(x)$, considering both the presence and absence of a forcing term $f(x).$ The mixed local-nonlocal operator $\mathcal{L}_{a, b}=-aΔ+b(-Δ)^s,\,a,\,b \in \mathbb{R}_+,$ incorporates both local and nonlocal Laplacians. We determine the Fujita-type critical exponents by considering the existence or nonexistence of global solutions. Interestingly, the critical exponent is determined by the nonlocal component of the operator and, as a result, coincides with that of the fractional Laplacian. In the case without a forcing term, our results improve upon recent findings by Biagi et al. [Bull. London Math. Soc. 57 (2025), 265-284] and Del Pezzo et al. [Nonlinear Analysis 255 (2025), 113761]. When a forcing term is included, our results refine those of Wang et al. [J. Math. Anal. Appl., 488 (1) (2020), 124067] and complement the work of Majdoub [La Matematica, 2 (2023), 340-361].

AI models have demonstrated strong predictive capabilities for various tokamak instabilities--including tearing modes (TM), ELMs, and disruptive event--but their opaque nature raises concerns about safety and trustworthiness when applied to fusion power plants. Here, we present a physics-based interpretation framework using a TM prediction model as a first demonstration that is validated through a dedicated DIII-D TM avoidance experiment. By applying Shapley analysis, we identify how profiles such as rotation, temperature, and density contribute to the model's prediction of TM stability. Our analysis shows that in our experimental scenario, a large density profile is lightly destabilizing, but core electron temperature and rotation peaking play the primary role in TM stability. This work offers a generalizable ML-based event prediction methodology, from training to physics-driven interpretability, bridging the gap between physics understanding and opaque ML models.

In this paper, we give a geometric interpretation of virtual knotoids as arcs in thickened surfaces. Then we show that virtual knotoid theory is a generalization of classical knotoid theory. This gives a proof of a conjecture of Kauffman and the first author.

Retrieval-Augmented Generation (RAG) improves large language models (LLMs) by retrieving relevant information from external sources and has been widely adopted for text-based tasks. For structured data, such as knowledge graphs, Graph Retrieval-Augmented Generation (GraphRAG) retrieves and aggregates information along graph structures. More recently, GraphRAG has been extended to general text settings by organizing unstructured text into graph representations, showing promise for reasoning and grounding. Despite these advances, existing GraphRAG systems for text data are often tailored to specific tasks, datasets, and system designs, resulting in heterogeneous evaluation protocols. Consequently, a systematic understanding of the relative strengths, limitations, and trade-offs between RAG and GraphRAG on widely used text benchmarks remains limited. In this paper, we present a comprehensive benchmark study comparing RAG and GraphRAG on established text-based tasks, including question answering and query-based summarization. We introduce a unified evaluation protocol that standardizes data preprocessing, retrieval configurations, and generation settings, enabling fair and reproducible comparisons. Our results highlight the distinct strengths of RAG and GraphRAG across different tasks and evaluation perspectives. Building on these findings, we explore selection and integration strategies that combine the strengths of both paradigms, leading to consistent performance improvements. We further analyze failure modes, efficiency trade-offs, and evaluation biases, and highlight key considerations for designing and evaluating retrieval-augmented generation systems.

We introduce the Morse parametric qualification condition for bilevel programming. Generic semi-algebraic functions are Morse parametric in a piecewise sense. Thus, bilevel programs with a Morse parametric lower level constitute a relevant intermediate class between strongly convex and fully generic lower levels. In this framework, we study bilevel gradient algorithms with two strategies: the single-step multi-step strategy, which involves a sequence of steps on the lower-level problems followed by one step on the upper-level problem, and a differentiable programming strategy that optimizes a smooth approximation of the bilevel problem. While the first is shown to be a biased gradient method on the problem with rich properties, the second, inspired by meta-learning applications, is less stable but offers simplicity and ease of implementation.

This article introduces Levy-driven graph supOU processes, a parsimonious parametrisation for high-dimensional time series in which dependence between components is governed by a graph structure. Specifically, the model bridges short- and long-range dependence within a single parametric family while accommodating a wide range of marginal distributions. We further develop a generalised method of moments estimator, establish its consistency and asymptotic normality, and assess its finite-sample performance through a simulation study. Finally, we illustrate the practical relevance of our model and estimation method in an empirical study of wind capacity factors in a European electricity network context.

In this paper, we aim to study the quantum transition between a Schwarzschild black hole and a string black hole in the large $D$ limit. Classically, such a transition between these two distinct black hole geometries is forbidden. The only feasible discussion is centered on how a black hole evaporates, loses mass, and transitions into highly excited fundamental strings. Building upon our previous work on T-duality between the Schwarzschild and string black holes, we reduce the problem to two dimensions, where the corresponding Wheeler-De Witt equation can be derived. Using this equation, we identify the two black hole geometries as distinct wave function states. This allows us to easily compute the transition probability between these two geometries, driven by the string coupling.

Whenever an anomaly in the flavour sector appears, analyses are performed examining whether it can be explained by adding a small number of carefully-chosen flavour non-universal four-fermion SMEFT operators. These analyses are typically carried out in the down or the up basis, i.e., it is assumed that the weak and mass eigenstates are the same for the left-handed down-type or up-type quarks. In these bases, there is no dependence on the matrices that transform from the weak to the mass basis, and which are unmeasurable in the Standard Model. In this paper, we argue that it is better to use a generic weak basis, in which no assumptions about the alignment of weak and mass eigenstates are made. The analysis now directly includes elements of the transformation matrices. By doing a fit to the data, it is possible to both determine if the flavour anomaly can be explained and extract the transformation matrices. In principle, this can be extended to a complete reconstruction of the Yukawa matrices.

We generalize the eta-pairing theory to very general non-Hermitian Hubbard models and find many novel phenomena without Hermitian analogs. For instance, the Hermitian conjugate of an eta-pairing eigenoperator may not be an eigenoperator, eta-pairing eigenoperators can be spatially modulated, and the $SU(2)$ pseudospin symmetry may not be possible even if $H$ commutes with the eta-pairing operators. Remarkably, these novel non-Hermitian phenomena are closely related to each other by several theorems we establish and can lead to, for example, new types of eta-pairing operators (e.g., the notion of non-Hermitian angular-momentum operators) and the anomalous localization of eta-pairing eigenstates. Some issues on the $SO(4)$ and particle-hole symmetries are clarified. Our general eta-pairing theory also reveals a previously unnoticed unification of these symmetries of the Hubbard model. To exemplify these findings, we first propose the one-dimensional Hatano-Nelson-Hubbard model (with or without the bulk translation invariance) and show that the right and left two-particle eta-pairing eigenstates are exponentially localized at opposite boundaries of the chain. Then, we generalize this model to two dimensions and find that the eta-pairing eigenstates can exhibit the first- or second-order skin effect. Finally, to realize all of the non-Hermitian eta-pairing phenomena, we construct a general two-sublattice model that is defined on an arbitrary lattice; this model can also reveal the eta-pairing structure in systems with Hermitian hoppings, including the original eta-pairing theory for square lattice, the extension to triangular lattice, and some topological systems. Our results establish a new and rigorous theoretical framework for studying novel quantum phenomena in interacting non-Hermitian many-body systems, even in arbitrary spatial dimensions and without the bulk translation symmetry.

Let $T_1,\ldots, T_m$ be a family of $d\times d$ invertible real matrices with $\|T_i\|<1/2$ for $1\leq i\leq m$. For ${\bf a}=(a_1,\ldots, a_m)\in {\Bbb R}^{md}$, let $π^{\bf a}\colon Σ=\{1,\ldots, m\}^{\Bbb N}\to {\Bbb R}^d$ denote the coding map associated with the affine IFS $\{T_ix+a_i\}_{i=1}^m$, and let $K^{\bf a}$ denote the attractor of this IFS. Let $W$ be a linear subspace of ${\Bbb R}^d$ and $P_W$ the orthogonal projection onto $W$. We show that for $\mathcal L^{md}$-a.e.~${\bf a}\in {\Bbb R}^{md}$, the Hausdorff and box-counting dimensions of $P_W(K^{\bf a})$ coincide and are determined by the zero point of a certain pressure function associated with $T_1,\ldots, T_m$ and $W$. Moreover, for every ergodic $σ$-invariant measure $μ$ on $Σ$ and for $\mathcal L^{md}$-a.e.~${\bf a}\in {\Bbb R}^{md}$, the local dimensions of $(P_Wπ^{\bf a})_*μ$ exist almost everywhere, here $(P_Wπ^{\bf a})_*μ$ stands for the push-forward of $μ$ by $P_Wπ^{\bf a}$. However, as illustrated by examples, $(P_Wπ^{\bf a})_*μ$ may not be exact dimensional for $\mathcal L^{md}$-a.e.~${\bf a}\in {\Bbb R}^{md}$. Nevertheless, when $μ$ is a Bernoulli product measure, or more generally, a supermultiplicative ergodic $σ$-invariant measure, $(P_Wπ^{\bf a})_*μ$ is exact dimensional for $\mathcal L^{md}$-a.e.~${\bf a}\in {\Bbb R}^{md}$.

This is the first article of a series of our recent works, addressing an open question of Bonk-Heinonen-Koskela [5], to study the relationship between (inner) uniformality and Gromov hyperbolicity in infinite dimensional spaces. Our main focus of this paper is to establish a dimension-free Gehring-Hayman inequality for quasigeodesics. A well-known theorem of J. Heinonen and S. Rohde in 1993 states that if $D\subset \mathbb{R}^n$ is quasiconformally equivalently to an uniform domain, then the Gehring-Hayman inequality holds in $D$: quasihyperbolic geodesics in $D$ minimizes the Euclidean length among all curves in $D$ with the same end points, up to a universal dimension-dependent multiplicative constant. In this paper, we develop a new approach to strengthen the above result in the following three aspects: 1) obtain a dimension-free multiplicative constant in the Gehring-Hayman inequality; 2) relax the class of quasihyperbolic geodesics to more general quasigeodesics; 3) relax the quasiconformal equivalence to more general coarsely quasihyperbolic equivalence. As a byproduct of our general approach, we are able to prove that the above improved Gehring-Hayman inequality indeed holds in Banach spaces. This answers affirmatively an open problem raised by J. Heinonen and S. Rohde in 1993 and reformulated by J. Väisälä in 2005.

Data assimilation (DA) integrates observations with model forecasts to produce optimized atmospheric states, whose physical consistency is critical for stable weather forecasting and reliable climate research. Traditional Bayesian DA methods enforce these nonlinear, flow-dependent physical constraints through empirical and tunable covariance structures, but with limited accuracy and robustness. Here, we introduce Latent Data Assimilation (LDA), a framework that performs Bayesian DA in a latent space learned from multivariate global atmospheric data via an autoencoder. We demonstrate that the autoencoder can largely capture nonlinear physical relationships, enabling LDA to produce balanced analyses without explicitly modeling physical constraints. Assimilation in latent space also improves both analysis quality and forecast skill compared to traditional model-space DA, under both idealized and real observational settings. Furthermore, LDA exhibits strong robustness across latent dimensions and remains effective even when the autoencoder is trained on inaccurate but physically realistic forecasts, highlighting its flexibility for real-world applications.

The Reynolds equation from lubrication theory and the Stokes equations for zero Reynolds number flows are distinct models for an incompressible fluid with negligible inertia. Here we investigate the sensitivity of the Reynolds equation to large surface gradients, and explore flow recirculation in corner geometries in comparison to the Stokes equation. We compare the solutions for the Reynolds and Stokes equations in the backward facing step (BFS), the regularized BFS, and the lid-driven triangular cavity. For the BFS variations listed above, we compute the error in terms of the average pressure drop through the channel and show how the error increases with increasing expansion ratio and with increasing magnitude of surface gradients. We further investigate the phenomenology of corner flow recirculation that arises in the Stokes solutions. In particular, we observe that occluding the corner separated region in the Stokes solution to the BFS does not disrupt the bulk flow characteristics.

We propose a two-step procedure to detect cointegration in high-dimensional settings, focusing on sparse relationships. First, we use the adaptive LASSO to identify the small subset of integrated covariates driving the equilibrium relationship with a target series, ensuring model-selection consistency. Second, we adopt an information-theoretic model choice criterion to distinguish between stationarity and nonstationarity in the resulting residuals, avoiding dependence on asymptotic distributional assumptions. Monte Carlo experiments confirm robust finite-sample performance, even under endogeneity and serial correlation.

We study the porous medium equation (PME) in one space dimension in presence of additive non-conservative white noise, and interpreted as a stochastic growth equation for the height field of an interface. We predict the values of the two growth exponents $α$ and $β$ using the functional RG. Extensive numerical simulations show agreement with the predicted values for these exponents, however they also show anomalous scaling with an additional "local" exponent $α_{\rm loc}$, as well as multiscaling originating from broad distributions of local height differences. The stationary measure of the stochastic PME is found to be well described by a random walk model, related to a Bessel process. This model allows for several predictions about the multiscaling properties.

Sparse optimization seeks an optimal solution with few nonzero entries. To achieve this, it is common to add to the criterion a penalty term proportional to the $\ell_1$-norm, which is recognized as the archetype of sparsity-inducing norms. In this approach, the number of nonzero entries is not controlled a priori. By contrast, in this paper, our motivation is to find an optimal solution with at most~$k$ nonzero coordinates (or for short, $k$-sparse vectors), where $k$ is a given sparsity threshold (or ``sparsity budget''). For this purpose, we study the class of generalized $k$-support dual~norms that arise from any given so-called source norm. When added as a penalty term, we provide conditions under which such generalized $k$-support dual~norms promote $k$-sparse solutions. The result follows from an analysis of the exposed faces of closed convex sets generated by $k$-sparse vectors, and of how primal support identification can be deduced from dual information. Finally, we study some of the geometric properties of the unit balls for the $k$-support dual~norms and their dual norms when the source norm belongs to the family of $\ell_p$-norms. In particular, we show a striking structural property: every proper face of the unit balls for the $k$-support dual~norms is a hypersimplex, i.e., the convex hull of $0/1$-valued points with the same $\ell_0$-norm.

In this paper we propose a framework to analyze iterative first-order optimization algorithms for time-varying convex optimization. We assume that the temporal variability is caused by a time-varying parameter entering the objective, which can be measured at the time of decision but whose future values are unknown. We consider the case of strongly convex objective functions with Lipschitz continuous gradients under a convex constraint set. We model the algorithms as discrete-time linear parameter varying (LPV) systems in feedback with monotone operators such as the time-varying gradient. We leverage the approach of analyzing algorithms as uncertain control interconnections with integral quadratic constraints (IQCs) and generalize that framework to the time-varying case. We propose novel IQCs that are capable of capturing the behavior of time-varying nonlinearities and leverage techniques from the LPV literature to establish novel bounds on the tracking error. Quantitative bounds can be computed by solving a semi-definite program and can be interpreted as an input-to-state stability result with respect to a disturbance signal which increases with the temporal variability of the problem. As a departure from results in this research area, our bounds introduce a dependence on different additional measures of temporal variations, such as the function value and gradient rate of change. We exemplify our main results with numerical experiments that showcase how our analysis framework is able to capture convergence rates of different first-order algorithms for time-varying optimization through the choice of IQC and rate bounds.