We point out that T. Tanaka's recent criticism [quant-ph/0603075] of the results of J. Math. Phys. 43, 3944 (2002) [math-ph/0203005] is based on an assumption which was never made in the latter paper, namely that the diagonalizability of an operator implies that it is normal. Therefore, Tanaka's objections regarding this paper are not valid.

We study cash-flow forecasting for derivatives used in liquidity management and clarify its relation to risk-neutral valuation and replication. While it is well known that expectations under different measures (e.g., $\mathbb{P}$ vs. $\mathbb{Q}$) can yield different undiscounted cash-flows, further inconsistencies arise when payment times are stochastic. We show that using discounting sensitivities (funding-curve hedge ratios) instead of "expected cash-flows" aligns forecasting with the self-financing replication strategy and avoids measure-mixing/aggregation issues. We then illustrate how a standard valuation model delivers pathwise funding requirements and propose a simple liquidity valuation adjustment to capture settlement lags and related timing frictions. The note provides implementation hints (American Monte Carlo with adjoint differentiation) and clarifies when "expected cash-flows" are informative and when sensitivities should be used instead.

Mathematical models, calibrated to data, have become ubiquitous to make key decision processes in modern quantitative finance. In this work, we propose a novel framework for data-driven model selection by integrating a classical quantitative setup with a generative modelling approach. Leveraging the properties of the signature, a well-known path-transform from stochastic analysis that recently emerged as leading machine learning technology for learning time-series data, we develop the Sig-SDE model. Sig-SDE provides a new perspective on neural SDEs and can be calibrated to exotic financial products that depend, in a non-linear way, on the whole trajectory of asset prices. Furthermore, we our approach enables to consistently calibrate under the pricing measure $\mathbb Q$ and real-world measure $\mathbb P$. Finally, we demonstrate the ability of Sig-SDE to simulate future possible market scenarios needed for computing risk profiles or hedging strategies. Importantly, this new model is underpinned by rigorous mathematical analysis, that under appropriate conditions provides theoretical guarantees for convergence of the presented algorithms.

Electrospray ion-beam deposition (ES-IBD) is a versatile tool to study structure and reactivity of molecules from small metal clusters to large protein assemblies. It brings molecules gently into the gas phase where they can be accurately manipulated and purified, followed by controlled deposition onto various substrates. In combination with imaging techniques, direct structural information of well-defined molecules can be obtained, which is essential to test and interpret results from indirect mass spectrometry techniques. To date, ion-beam deposition experiments are limited to a small number of custom instruments worldwide, and there are no commercial alternatives. Here we present a module that adds ion-beam deposition capabilities to a popular commercial MS platform (Thermo Scientific$^{\mathrm{TM}}$ Q Exactive$^{\mathrm{TM}}$ UHMR). This combination significantly reduces the overhead associated with custom instruments, while benefiting from established high performance and reliability. We present current performance characteristics including beam intensity, landing-energy control, and deposition spot size for a broad range of molecules. In combination with atomic force microscopy (AFM) and transmission electron microscopy (TEM), we distinguish near-native from unfolded proteins and show retention of native shape of protein assemblies after dehydration and deposition. Further, we use an enzymatic assay to quantify activity of an non-covalent protein complex after deposition an a dry surface. Together, these results indicate a great potential of ES-IBD for applications in structural biology, but also outline the challenges that need to be solved for it to reach its full potential.

Questions of noise stability play an important role in hardness of approximation in computer science as well as in the theory of voting. In many applications, the goal is to find an optimizer of noise stability among all possible partitions of $\mathbb{R}^n$ for $n \geq 1$ to $k$ parts with given Gaussian measures $μ_1,\ldots,μ_k$. We call a partition $ε$-optimal, if its noise stability is optimal up to an additive $ε$. In this paper, we give an explicit, computable function $n(ε)$ such that an $ε$-optimal partition exists in $\mathbb{R}^{n(ε)}$. This result has implications for the computability of certain problems in non-interactive simulation, which are addressed in a subsequent work.

This essay recounts my personal journey towards a deeper understanding of the mathematical foundations of algorithmic music composition. I do not spend much time on specific mathematical algorithms used by composers; rather, I focus on general issues such as fundamental limits and possibilities, by analogy with metalogic, metamathematics, and computability theory. I discuss implications from these foundations for the future of algorithmic composition.

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.