I re-examine a recent work by G. Landi and G. E. Landi. [arXiv:1808.06708 [physics.ins-det]], in which the authors claim that the resolution of a tracker ca vary linearly with the number of detection layers, $N$, that is, faster than the commonly known $\sqrt{N}$ variation, for a tracker of fixed length, in case the precision of the position measurement is allowed to vary from layer to layer, i.e. heteroscedasticity, and an appropriate analysis method, a weighted least squares fit, is used.

General reasoning represents a long-standing and formidable challenge in artificial intelligence (AI). Recent breakthroughs, exemplified by large language models (LLMs)1,2 and chain-of-thought (CoT) prompting3, have achieved considerable success on foundational reasoning tasks. However, this success is heavily contingent on extensive human-annotated demonstrations and the capabilities of models are still insufficient for more complex problems. Here we show that the reasoning abilities of LLMs can be incentivized through pure reinforcement learning (RL), obviating the need for human-labelled reasoning trajectories. The proposed RL framework facilitates the emergent development of advanced reasoning patterns, such as self-reflection, verification and dynamic strategy adaptation. Consequently, the trained model achieves superior performance on verifiable tasks such as mathematics, coding competitions and STEM fields, surpassing its counterparts trained through conventional supervised learning on human demonstrations. Moreover, the emergent reasoning patterns exhibited by these large-scale models can be systematically used to guide and enhance the reasoning capabilities of smaller models. A new artificial intelligence model, DeepSeek-R1, is introduced, demonstrating that the reasoning abilities of large language models can be incentivized through pure reinforcement learning, removing the need for human-annotated demonstrations.

In the quiet backwaters of cs.CV, cs.LG and stat.ML, a cornucopia of new learning systems is emerging from a primordial soup of mathematics-learning systems with no need for external supervision. To date, little thought has been given to how these self-supervised learners have sprung into being or the principles that govern their continuing diversification. After a period of deliberate study and dispassionate judgement during which each author set their Zoom virtual background to a separate Galapagos island, we now entertain no doubt that each of these learning machines are lineal descendants of some older and generally extinct species. We make five contributions: (1) We gather and catalogue row-major arrays of machine learning specimens, each exhibiting heritable discriminative features; (2) We document a mutation mechanism by which almost imperceptible changes are introduced to the genotype of new systems, but their phenotype (birdsong in the form of tweets and vestigial plumage such as press releases) communicates dramatic changes; (3) We propose a unifying theory of self-supervised machine evolution and compare to other unifying theories on standard unifying theory benchmarks, where we establish a new (and unifying) state of the art; (4) We discuss the importance of digital biodiversity, in light of the endearingly optimistic Paris Agreement.

See math-ph/0205036 for an expanded version.

This is a survey of the existing digital collections of French mathematical literature, run by non-profit organizations. This includes research monographs, serials, proceedings, Ph. D. theses, collected works, books and personal websites.

The unitary representation theory of locally compact contraction groups and their semi-direct products with $\mathbb{Z}$ is studied. We put forward the problem of completely characterising such groups which are type I or CCR and this article provides a stepping stone towards a solution to this problem. In particular, we determine new examples of type I and non-type-I groups in this class, and we completely classify the irreducible unitary representations of the torsion-free groups, which are shown to be type I. When these groups are totally disconnected, they admit a faithful action by automorphisms on an infinite locally-finite regular tree; this work thus provides new examples of automorphism groups of regular trees with interesting representation theory, adding to recent work on this topic.

In this paper it is argued that Barad's Agential Realism, an approach to quantum mechanics originating in the philosophy of Niels Bohr, can be the basis of a 'theory of everything' consistent with a proposal of Wheeler that observer-participancy is the foundation of everything. On the one hand, agential realism can be grounded in models of self-organisation such as the hypercycles of Eigen, while on the other agential realism, by virtue of the 'discursive practices' that constitute one aspect of the theory, implies the possibility of the generation of physical phenomena through acts of specification originating at a more fundamental level. Included in phenomena that may be generated by such a mechanism are the origin and evolution of life, and human capacities such as mathematical and musical intuition.

We analyze certain subgroups of real and complex forms of the Lie group E8, and deduce that any "Theory of Everything" obtained by embedding the gauge groups of gravity and the Standard Model into a real or complex form of E8 lacks certain representation-theoretic properties required by physical reality. The arguments themselves amount to representation theory of Lie algebras in the spirit of Dynkin's classic papers and are written for mathematicians.

In light of Gödel's undecidability results (incomplete theorems) for math, quantum indeterminism indicates that physics and the Universe may be indeterministic, incomplete, and open in nature, and therefore demand no single unification theory of everything. The Universe is dynamic and so are the underlying physical models and spacetime. As the 4-d spacetime evolves dimension by dimension in the early universe, consistent yet different models emerge one by one with different sets of particles and interactions. A new set of first principles are proposed for building such models with new understanding of supersymmetry, mirror symmetry, and the dynamic phase transition mechanism - spontaneous symmetry breaking. Under this framework, we demonstrate that different models with no theory of everything operate in a hierarchical yet consistent way at different phases or scenarios of the Universe. In particular, the arrow of time is naturally explained and the Standard Model of physics is elegantly extended to time zero of the Universe.

In this paper, we introduce an approach to the protein folding problem from the point of view of statistical physics. Protein folding is a stochastic process by which a polypeptide folds into its characteristic and functional 3D structure from random coil. The process involves an intricate interplay between global geometry and local structure, and each protein seems to present special problems. We introduce CSAW (conditioned self-avoiding walk), a model of protein folding that combines the features of self-avoiding walk (SAW) and the Monte Carlo method. In this model, the unfolded protein chain is treated as a random coil described by SAW. Folding is induced by hydrophobic forces and other interactions, such as hydrogen bonding, which can be taken into account by imposing conditions on SAW. Conceptually, the mathematical basis is a generalized Langevin equation. To illustrate the flexibility and capabilities of the model, we consider several examples, including helix formation, elastic properties, and the transition in the folding of myoglobin. From the CSAW simulation and physical arguments, we find a universal elastic energy for proteins, which depends only on the radius of gyration $R_{g}$ and the residue number $N$. The elastic energy gives rise to scaling laws $R_{g}\sim N^ν$ in different regions with exponents $ν=3/5,3/7,2/5$, consistent with the observed unfolded stage, pre-globule, and molten globule, respectively. These results indicate that CSAW can serve as a theoretical laboratory to study universal principles in protein folding.