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We use the Banerjee--Mandal--Sahoo dipole-current Ward identity for the one-loop logarithmic soft-photon theorem as input and determine its finite-energy action on Mellin-difference hard currents. The commutator with such hard currents has a scheme-independent hard-hard residue that survives every one-particle redefinition. With the meromorphic continuation stated explicitly below, a two-particle Plancherel transform identifies this residue with an analytic two-particle primary module, and the coefficient map is a hard-current one-cocycle. The cocycle defines a minimal filtered abelian extension. It has a canonical two-particle primitive and integrates to an affine action. For scalar hard legs, the fixed-leg operator agrees coefficient by coefficient with the symmetry-governed long-range logarithmic tower of Choi, Kadhe, and Puhm. Applied to a tree-level scalar-QED photon-exchange block, the finite-energy analysis determines the logarithmic two-particle coefficient functional from the ordinary hard amplitude and the Banerjee--Mandal--Sahoo ordered-pair soft kernel. This gives a finite-energy relation between the Banerjee--Mandal--Sahoo dipole-current Ward identity and the exponenti

A hard real-time system cannot miss any deadline. A weakly-hard real-time system, on the contrary, is designed to tolerate a specific number of deadline misses. For instance, the AnyMiss(2, 300) weakly-hard constraint stipulates that in every window of 300 consecutive jobs, at most 2 deadlines are missed. The weakly-hard model is the state-of-the-art for industrial dependability-by-design of control systems that tolerate deterministic failures. Weakly-hard constraints correspond to regular languages. The size of the minimal finite state machine that recognizes whether a string satisfies the constraint (about 45k states for AnyMiss(2, 300)) is a notorious impediment for the verification of control system properties. This paper discusses an over-approximation of the language that allows us to provide sound safety guarantees for control systems under deadline misses that would be out of reach using the minimal finite state machine. We present a compressed language acceptor and prove that it simulates the original finite state machine. We study language cardinality properties, and report on empirical results that show how the new acceptor can be embedded in the control design workflow,

Composing independently trained LoRA adapters into a single large language model is useful for multi-domain adaptation, especially when the original training data cannot be shared. A common approach is to use MoE-style routing over LoRA experts, but for frozen pretrained adapters, soft weighted combinations can change the unit-scale additive update under which each LoRA module was originally trained. We propose \textbf{Hard-Routed MoR-LoRA}, a two-stage framework for composing frozen reasoning LoRA experts through unit-scale hard selection. First, domain-specific LoRA adapters are trained independently using reinforcement learning from verifiable feedback to obtain reasoning experts. Then, all experts are frozen, reasoning traces are distilled from them, and only a lightweight shared router together with a small attention LoRA is trained for integration. The router selects exactly one expert per token using hard top-1 routing, while a straight-through estimator enables gradient-based training. Experiments across five benchmarks, multiple model scales, and additional model families show that Hard-Routed MoR-LoRA preserves expert behavior while requiring substantially fewer trainable

Let $G$ be a complete edge-weighted graph on $n$ vertices. To each subset of vertices of $G$ assign the cost of the minimum spanning tree of the subset as its weight. Suppose that $n$ is a multiple of some fixed positive integer $k$. The $k$-matching problem is the problem of finding a partition of the vertices of $G$ into $k$-sets, that minimizes the sum of the weights of the $k$-sets. The case $k=3$ has been shown to be NP-hard [Johnsson et al.,1998]. In the Euclidean version, the vertices of $G$ are points in the plane and the weight of an edge is the Euclidean distance between its endpoints. We call this problem the Euclidean $k$-matching problem. We show that, for every fixed $k \ge 3$, the Euclidean $k$-matching is NP-hard. This resolves an open problem in the literature and provides the first theoretical justification for the use of known heuristic methods in the case $k=3$. We also show that the problem remains NP-hard if the trees are required to be paths.

Hard-jet correlations probe parton energy loss and the microscopic structure of the quark-gluon plasma formed in ultra-relativistic heavy-ion collisions. The correlation of high-$p_\mathrm{T}$ jets with other jets, hadrons, or electroweak bosons, offers differential sensitivity to medium-induced effects such as momentum broadening, color decoherence, and medium response in different types of nuclear reactions. Such correlations can also be used to study cold nuclear matter effects arising in $p$+A collisions. This proceeding summarizes recent advances achieved by studying hard-jet correlations in large and small systems discussed at Hard Probes 2024, complementing the experimental jet overview.

Overview of recent theoretical advances presented at Hard Probes 2024 in the study of hard-soft correlation in heavy ion collisions and small systems.

According to the "hard-steps" model, the origin of humanity required "successful passage through a number of intermediate steps" (so-called "hard" or "critical" steps) that were intrinsically improbable with respect to the total time available for biological evolution on Earth. This model similarly predicts that technological life analogous to human life on Earth is "exceedingly rare" in the universe. Here, we critically reevaluate the core assumptions of the hard-steps model in light of recent advances in the Earth and life sciences. Specifically, we advance a potential alternative model where there are no hard steps, and evolutionary novelties (or singularities) required for human origins can be explained via mechanisms outside of intrinsic improbability. Furthermore, if Earth's surface environment was initially inhospitable not only to human life, but also to certain key intermediate steps in human evolution (e.g., the origin of eukaryotic cells, multicellular animals), then the "delay" in the appearance of humans can be best explained through the sequential opening of new global environmental windows of habitability over Earth history, with humanity arising relatively quickly o

In order for quantum computations to be done as efficiently as possible it is important to optimise the number of gates used in the underlying quantum circuits. In this paper we find that many gate optimisation problems for approximately universal quantum circuits are NP-hard. In particular, we show that optimising the T-count or T-depth in Clifford+T circuits, which are important metrics for the computational cost of executing fault-tolerant quantum computations, is NP-hard by reducing the problem to Boolean satisfiability. With a similar argument we show that optimising the number of CNOT gates or Hadamard gates in a Clifford+T circuit is also NP-hard. Again varying the same argument we also establish the hardness of optimising the number of Toffoli gates in a reversible classical circuit. We find an upper bound to the problems of T-count and Toffoli-count of $\text{NP}^{\text{NQP}}$. Finally, we also show that for any non-Clifford gate $G$ it is NP-hard to optimise the $G$-count over the Clifford+$G$ gate set, where we only have to match the target unitary within some small distance in the operator norm.

A mini-review on the status of global analyses of nuclear parton distribution functions is given, focusing on the most relevant constraints for the hard-probes phenomenology in ultra-relativistic heavy-ion collisions.

The hot and dense QCD matter, known as the Quark-Gluon Plasma (QGP), is explored through heavy-ion collision experiments at the LHC and RHIC. Jets and heavy flavors, produced from the initial hard scattering, are used as hard probes to study the properties of the QGP. Recent experimental observations on jet quenching and heavy-flavor suppression have strengthened our understanding, allowing for fine-tuning of theoretical models in hard probes. The second conference, HOT QCD Matter 2024, was organized to bring the community together for discussions on key topics in the field. This article comprises 15 sections, each addressing various aspects of hard probes in relativistic heavy-ion collisions, offering a snapshot of current experimental observations and theoretical advancements. The article begins with a discussion on memory effects in the quantum evolution of quarkonia in the quark-gluon plasma, followed by an experimental review, new insights on jet quenching at RHIC and LHC, and concludes with a machine learning approach to heavy flavor production at the Large Hadron Collider.

The strength of modern generative models lies in their ability to be controlled through text-based prompts. Typical "hard" prompts are made from interpretable words and tokens, and must be hand-crafted by humans. There are also "soft" prompts, which consist of continuous feature vectors. These can be discovered using powerful optimization methods, but they cannot be easily interpreted, re-used across models, or plugged into a text-based interface. We describe an approach to robustly optimize hard text prompts through efficient gradient-based optimization. Our approach automatically generates hard text-based prompts for both text-to-image and text-to-text applications. In the text-to-image setting, the method creates hard prompts for diffusion models, allowing API users to easily generate, discover, and mix and match image concepts without prior knowledge on how to prompt the model. In the text-to-text setting, we show that hard prompts can be automatically discovered that are effective in tuning LMs for classification.

We highlight the STAR experiment's hard probes results, including jets and heavy flavor production in heavy-ion collisions to study the properties of the quark-gluon plasma. Various jet-substructure observables in proton-proton collisions are presented to explore both perturbative and non-perturbative regimes of quantum chromodynamics. Finally, we discuss the STAR experiment's hard probes physics program for ongoing data taking.

The in-medium dispersion of hard partons, encoded in their so-called asymptotic mass, receives large non-perturbative contributions from classical gluons, i.e. soft gluons with large occupation numbers. Here, we discuss how the analytical properties of thermal amplitudes allow for a non-perturbative determination of the infrared classical contribution through lattice determinations in the dimensionally-reduced effective theory of hot QCD, EQCD. We show how these lattice determinations need to be complemented by perturbative two-loop matching calculations between EQCD and QCD, so that the unphysical (classical) ultraviolet behavior of EQCD is replaced by its proper quantum QCD counterpart. We show how lattice and perturbative EQCD are in good agreement in the UV and present an outlook on the two-loop quantum QCD contribution.

We introduce a highly structured family of hard satisfiable 3-SAT formulas corresponding to an ordered spin-glass model from statistical physics. This model has provably "glassy" behavior; that is, it has many local optima with large energy barriers between them, so that local search algorithms get stuck and have difficulty finding the true ground state, i.e., the unique satisfying assignment. We test the hardness of our formulas with two Davis-Putnam solvers, Satz and zChaff, the recently introduced Survey Propagation (SP), and two local search algorithms, Walksat and Record-to-Record Travel (RRT). We compare our formulas to random 3-XOR-SAT formulas and to two other generators of hard satisfiable instances, the minimum disagreement parity formulas of Crawford et al., and Hirsch's hgen. For the complete solvers the running time of our formulas grows exponentially in sqrt(n), and exceeds that of random 3-XOR-SAT formulas for small problem sizes. SP is unable to solve our formulas with as few as 25 variables. For Walksat, our formulas appear to be harder than any other known generator of satisfiable instances. Finally, our formulas can be solved efficiently by RRT but only if the pa

Preference analysis is widely applied in various domains such as social choice and e-commerce. A recently proposed framework augments the relational database with a preference relation that represents uncertain preferences in the form of statistical ranking models, and provides methods to evaluate Conjunctive Queries (CQs) that express preferences among item attributes. In this paper, we explore the evaluation of queries that are more general and harder to compute. The main focus of this paper is on a class of CQs that cannot be evaluated by previous work. These queries are provably hard since relate variables that represent items being compared. To overcome this hardness, we instantiate these variables with their domain values, rewrite hard CQs as unions of such instantiated queries, and develop several exact and approximate solvers to evaluate these unions of queries. We demonstrate that exact solvers that target specific common kinds of queries are far more efficient than general solvers. Further, we demonstrate that sophisticated approximate solvers making use of importance sampling can be orders of magnitude more efficient than exact solvers, while showing good accuracy. In ad

In very recent work the mean field theory of the jamming transition in infinite dimensional hard spheres models was presented. Surprisingly, this theory predicts quantitatively numerically determined characteristics of jamming in two and three dimensions. This is a rare and unusual finding. Here we argue that this agreement in non-generic: only for hard sphere models it happens that sufficiently close to jamming the effective interactions are in agreement with mean-field theory, justifying the truncation of many body interactions (which is the exact protocol in infinite dimensions). Any softening of the bare hard sphere interactions results in effective interactions that are not mean-field all the way to jamming, making the discussed phenomenon non generic.

Non thermal emission from galaxy clusters demonstrates the existence of relativistic particles and magnetic fields in the Intra Cluster Medium (ICM). Present instruments do not allow to firmly establish the energy associated to these components. In a few years gamma ray observations will put important constraints on the energy content of non thermal hadrons in clusters, while the combination of radio and hard X-ray data will be crucial to measure the energy content in the form of relativistic electrons and magnetic field. SIMBOL-X is expected to drive an important breakthrough in the field also because it is expected to operate in combination with the forthcoming low frequency radio telescopes (LOFAR, LWA). In this contribution we report first estimates of statistical properties of the hard X-ray emission in the framework of the re-acceleration model. This model allows to reproduce present radio data for Radio Halos and to derive expectations for future low frequency radio observations, and thus our calculations provide hints for observational strategies for future radio and hard--X-ray combined observations.

Shadow removal from a single image is generally still an open problem. Most existing learning-based methods use supervised learning and require a large number of paired images (shadow and corresponding non-shadow images) for training. A recent unsupervised method, Mask-ShadowGAN~\cite{Hu19}, addresses this limitation. However, it requires a binary mask to represent shadow regions, making it inapplicable to soft shadows. To address the problem, in this paper, we propose an unsupervised domain-classifier guided shadow removal network, DC-ShadowNet. Specifically, we propose to integrate a shadow/shadow-free domain classifier into a generator and its discriminator, enabling them to focus on shadow regions. To train our network, we introduce novel losses based on physics-based shadow-free chromaticity, shadow-robust perceptual features, and boundary smoothness. Moreover, we show that our unsupervised network can be used for test-time training that further improves the results. Our experiments show that all these novel components allow our method to handle soft shadows, and also to perform better on hard shadows both quantitatively and qualitatively than the existing state-of-the-art sha

This manuscript is the outcome of the subgroup ``PDFs, shadowing and $pA$ collisions'' from the CERN workshop ``Hard Probes in Heavy Ion Collisions at the LHC''. In addition to the experimental parameters for $pA$ collisions at the LHC, the issues discussed are factorization in nuclear collisions, nuclear parton distributions (nPDFs), hard probes as the benchmark tests of factorization in $pA$ collisions at the LHC, and semi-hard probes as observables with potentially large nuclear effects. Also, novel QCD phenomena in $pA$ collisions at the LHC are considered. The importance of the $pA$ program at the LHC is emphasized.