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Modern networks are highly dynamic, and temporal graphs capture these changes through discrete edge appearances on a fixed vertex set, known in advance up to the graph's lifetime. The Vertex Cover problem extends to the temporal setting as Temporal Vertex Cover (TVC) and Sliding Window Temporal Vertex Cover (SW-TVC). In TVC, each edge is covered by one endpoint over the lifetime, while in SW-TVC, edges are covered within every $Δ$-step window. In always star temporal graphs, each snapshot is a star with a center that may change at each time step. TVC is NP-complete on always star temporal graphs, but an FPT algorithm parameterized by $Δ$ solves it optimally in $O(TΔ(n+m)\cdot 2^Δ)$. This paper presents two polynomial-time approximation algorithms for SW-TVC on always star temporal graphs, achieving $2Δ-1$ and $Δ-1$ approximation ratios with running times $O(T)$ and $O(TmΔ^2)$, respectively. These algorithms provide exact solutions for $Δ=1$ and $Δ\leq 2$. Additionally, we offer the first implementation and experimental evaluation of state-of-the-art approximation algorithms with $d$ and $d-1$ approximation ratios, where $d$ is the maximum degree of any snapshot. Our experiments on

We present a one-way shooting algorithm for transition path sampling that accepts every proposed trajectory, yet samples the correct transition path ensemble for systems with overdamped stochastic dynamics. The method is based on two key elements: a procedure to propose trajectories that are always reactive, and a reweighting scheme that corrects for the bias introduced by always accepting the proposed paths. This approach significantly improves the efficiency of transition path sampling by eliminating the cost associated with generating trajectories that are then rejected. We demonstrate the performances of the algorithm by investigating the formation of CO$_2$ clathrate hydrates along different reaction mechanisms, showing that the increased efficiency allows proper sampling of the formation of crystalline hydrates at temperatures and pressures that are difficult to access with conventional schemes.

We give a computer-assisted counterexample to the open question, posed by Rudin, Schapire, and Daubechies in COLT 2012, of whether exhaustive AdaBoost always converges to a finite cycle. The construction is based on a block-product gadget whose two factors share an exact period-2 orbit for their 5-step branch maps, but whose linearized return maps have dominant eigenvalues with an irrational logarithmic ratio. This irrationality forces the burst-winner sequence to have an irrational asymptotic frequency, precluding eventual periodicity. All assertions are certified by exact rational arithmetic. This work was developed in collaboration with GPT-5.4 Pro and Claude Opus 4.6.

We show that the claim in Ref. [PRL 131, 200202 (2023)], that the quantum time evolution always can be written as a product of a holonomy operator and a dynamic operator, is false, as it is based on a circular use of the time evolution operator.

The origin of cosmic structure is widely regarded as quantum, yet the Universe today appears classical. Standard lore attributes this to a "quantum-to-classical" transition on super-horizon scales during inflation. Gravity plays a central role: super-horizon dynamics squeeze quantum states, while the cosmological horizon enforces a system-environment split, leading to decoherence. But are these mechanisms always sufficient? We revisit this question, identifying assumptions and limitations in conventional arguments. We highlight recent work showing that beyond slow roll, non-linear dynamics of cosmological perturbations can generate non-classical features that may survive in observables. This raises the tantalizing possibility that quantum signatures may persist in cosmic structure. We propose a phase-space analysis based on the Wigner function as a concrete route to identifying and probing such signatures.

Data mixing decides how to combine different sources or types of data and is a consequential problem throughout language model training. In pretraining, data composition is a key determinant of model quality; in continual learning and adaptation, it governs what is retained and acquired. Yet existing data mixing methods address only one phase of this lifecycle at a time: some require smaller proxy models tied to a single training phase, others assume a fixed domain set, and continual learning lacks principled guidance altogether. We argue that data mixing is fundamentally an online decision making problem -- one that recurs throughout training and demands a single, unified solution. We introduce OP-Mix (On-Policy Mix), a data mixing algorithm that operates across the entire language model training lifecycle. Our main insight is that candidate data mixtures can be cheaply simulated by interpolating between low-rank adapters trained directly on the current model, eliminating separate proxy models and ensuring the search is always grounded in the model's actual learning dynamics. Across pretraining, continual midtraining, and continual instruction tuning, OP-Mix consistently finds nea

We find that to the dynamics of a given dissipative system a $p=1$ differential form can be associated with a general decomposition into a potential term and a non-potential residual part. If the residual part is absent the form is closed and the system is gradient system or gradient like. If it is non-closed, in the differential form approach, it remains non-closed under a variable change of coordinates, i.e., the system is not a gradient one or a gradient like in any coordinate system. On the other hand, there are claims that a potential should always exists, i.e., the class of dissipative systems and the the class of gradient systems should coincide. We fix this conundrum by introducing a generalized change of coordinates that aims a transformation to a gradient system or a gradient like system. The condition of being closed in the new coordinates of a certain, through the generalized change of coordinates defined differential form, results in a nonlinear differential equation together with a consistency condition. We give examples of physical systems where an analytical solution for the transformation can be found, and hitherto, the potential, but even when the potential is not

This lecture note addresses the common misconception that the Gaussian distribution always yields the largest Cramér-Rao Bound (CRB). We show that this property only holds under restrictive conditions: specifically, when the mean and covariance parameters are decoupled in the Fisher Information Matrix (FIM), when the parameter of interest lies in the mean vector and when there are no additive nuisance parameters. Beyond this framework, we provide counterexamples demonstrating that non-Gaussian distributions can produce larger CRB.

In this paper, we provide a finite random iterated function system satisfying the open set condition, for which the random version of Bowen's formula fails to hold. This counterexample shows that analogous results established for random recursive constructions are not always obtained for random iterated function systems.

In this paper we study the interaction between logic and probability. In particular, we show that the convex hull of evaluations of a broad class of logics is always effectively axiomatizable. We define a Birkhoff-style calculus for probability axioms for which compactness, and finite completeness is proved. We give example for a logic for which probabilities are not finitely axiomatizable.

We discuss winning possibilities of players in various variants of cops and robber game played on large random graphs, a testbed for various kinds of network queries, search problems in particular. We explore the use of logic frameworks to investigate such results; in particular, we show that whenever a winning condition for either player can be expressed as a certain kind of formula in first-order logic, that player almost always wins. In the process, we obtain more insight into the logic-game connection from the zero-one law perspective.

Given a locale $L$, the collection $\mathsf{S}_c(L)$ of joins of closed sublocales forms a frame--somewhat unexpectedly, as it is naturally embedded in the coframe of all sublocales of $L$, where by coframe we mean the order-theoretic dual of a frame. This construction has attracted attention in point-free topology: as a maximal essential extension in the category of frames, for its (non-)functorial properties, its relation to canonical extensions and exact filters of frames, etc. A central open question of the theory, posed by Picado, Pultr, and Tozzi in 2019, asked whether $\mathsf{S}_c(L)$ is always a coframe, or whether there exists a locale for which this fails. In this paper, we resolve this question in the negative by constructing a locale $L$ such that $\mathsf{S}_c(L)$ is not a coframe. The main challenge in such questions lies in the difficulty of understanding exact infima in $\mathsf{S}_c(L)$; we circumvent this by analysing a certain separation property satisfied by $\mathsf{S}_c(L)$.

This note presents a proof that two principal ideals in a plactic monoid always intersect. Namely, this means that the plactic monoids are both left and right reversible. To the author's knowledge, this result has not yet appeared in the literature studying this monoid. This result holds for both finite rank plactic monoids and the infinite rank plactic monoid.

Following a leading vehicle is a daily but challenging task because it requires adapting to various traffic conditions and the leading vehicle's behaviors. However, the question `Does the following vehicle always actively react to the leading vehicle?' remains open. To seek the answer, we propose a novel metric to quantify the interaction intensity within the car-following pairs. The quantified interaction intensity enables us to recognize interactive and non-interactive car-following scenarios and derive corresponding policies for each scenario. Then, we develop an interaction-aware switching control framework with interactive and non-interactive policies, achieving a human-level car-following performance. The extensive simulations demonstrate that our interaction-aware switching control framework achieves improved control performance and data efficiency compared to the unified control strategies. Moreover, the experimental results reveal that human drivers would not always keep reacting to their leading vehicle but occasionally take safety-critical or intentional actions -- interaction matters but not always.

We provide the first counterexample showing that the ground state energy of electrons in an external Coulomb potential is not always a convex function of the number of electrons. This convexity has been conjectured for decades and plays an important role in quantum chemistry. Our counterexample involves six nuclei with small fractional charges placed far apart. The ground state energy of 3 electrons is shown to be higher than the average of the energies for 2 and 4 electrons. We also show that the nuclei can bind 2 or 4 electrons, but not 3. This article raises the question of whether the energy convexity really holds for all possible molecules (with nuclei of integer charge).

Recurrence problems are fundamental in dynamics, and for example, sizes of the set of points recurring infinitely often to a target have been studied extensively in many contexts. For example, the problem of finding the dimension for shrinking target set in an iterated function system is an active research area. In the current work, we consider a set with a finer recurrence quality, the eventually always hitting set. In a sense, the points in the intersection of an eventually always hitting set and a shrinking target set not only return infinitely often but also at a bounded rate. We study this set in the context of self-conformal iterated function systems, and compute upper and lower bounds for its Hausdorff dimension. Additionally, as an intermediate theorem, we obtain a Hausdorff dimension result for the intersection of eventually always hitting and shrinking target sets.

Consider an open quantum system which interacts with its environment. Assuming that the experimenter has access only to the system, an interesting question is whether it is possible to detect initial correlations between the system and the environment by performing measurements only on the system. Various methods have been proposed to detect correlations by local measurements on the system. After reviewing these methods, we will show that initial correlations between the system and the environment are always detectable. In particular, we will show that one can always find a unitary evolution, for the whole system-environment, such that the trace distance method, proposed to witness correlations locally, succeeds. We also find the condition for existence of the optimal unitary evolution, for which the entire correlation is locally detectable. Next, we address the case where the system and the environment interact through a time-independent Hamiltonian. For this case we will see that if the initial correlation can be detected locally at some time t , then it can be detected for almost all the other times too. On the other hand, we see that one can find cases for which initial correla

Ambrosio, Honda, and Tewodrose proved that the regular Weyl's law is equivalent to a mild condition related to the infinitesimal behavior of the measure of balls in compact finite dimensional RCD spaces. Though that condition is seemed to always hold for any such spaces, however, Dai, Honda, Pan, and Wei recently show that for any integer n at least 2, there exists a compact RCD space of n dimension fails to satisfy the regular Weyl's law. In this short article we prove that one dimensional RCD spaces always satisfy the regular Weyl's law.

Recent works for time-series forecasting more and more leverage the high predictive power of Deep Learning models. With this increase in model complexity, however, comes a lack in understanding of the underlying model decision process, which is problematic for high-stakes application scenarios. At the same time, simple, interpretable forecasting methods such as ARIMA still perform very well, sometimes on-par, with Deep Learning approaches. We argue that simple models are good enough most of the time, and that forecasting performance could be improved by choosing a Deep Learning method only for few, important predictions, increasing the overall interpretability of the forecasting process. In this context, we propose a novel online model selection framework which learns to identify these predictions. An extensive empirical study on various real-world datasets shows that our selection methodology performs comparable to state-of-the-art online model selections methods in most cases while being significantly more interpretable. We find that almost always choosing a simple autoregressive linear model for forecasting results in competitive performance, suggesting that the need for opaque