This monograph develops probability and stochastic-process ideas as a translation language for statistics: from designed observations and data objects to targets, stability statements, inference, and use. The chapters move from motivating examples and randomization through probability measures, kernels, likelihoods, data objects, weak convergence, empirical fields, functional data, M- and Z-estimation, testing, local approximations, event-time processes, and prediction. Historical and biomedical examples are used to keep abstract objects tied to records, mechanisms, and decisions. The aim is to give readers a common grammar for classical probability, modern data structures, and statistical practice.

In 1916, Hausdorff proved that the Baire hierarchy on $\mathbb{R}$, starting with the continuous functions, generates all Borel functions on $\mathbb{R}$. It remained open whether, starting with the a.e. continuous functions, the corresponding hierarchy generates all Lebesgue measurable functions on $\mathbb{R}$. We prove that, assuming the Axiom of Choice, the answer is negative.

The joint rate-distortion framework of Stavrou and Kountouris (IEEE Transactions on Communications 2023) characterises dual-fidelity tradeoffs for semantic communication on stochastic semantic sources. Many task-oriented communication systems instead use designed sources, where the semantic object is a deterministic oracle allocation $ϕ^(t)$ rather than a stochastic quantity given by nature. We isolate the subclass of designed sources under smooth concave utility with assumptions A1, A2 and Euclidean allocation codomain, and restrict the encoder class to deterministic common-category mappings. Within this subclass the SK exponential-tilting decoder and generalised Blahut--Arimoto iteration specialise to conditional-mean decoding and Lloyd--Max stationarity on $ϕ^(t)$. When the second fidelity is a monotone single-letter distortion, the joint problem stays inside the SK admissible class; the common-category SK rate is lower-bounded by the max of the corresponding Shannon rate-distortion functions, with equality only when the common-category reconstruction is compatible and RDF-optimal. When the second fidelity is aggregate verification, the joint problem leaves the SK single-letter class and admits a constrained-design feasibility band $R_{\min}(\varepsilon^) \leq R \leq R_{\max}(β^)$ of width $\log_2(K_{\max}/K_{\min})$ bits in partition cardinality. The reduction and the band are scope statements on the SK apparatus, not modifications to it. A smart-grid economic-dispatch example with a non-technical-loss-detection contrast illustrates the band.

Accurate evaluation of genome assemblies within highly repetitive regions, such as centromeres, remains a major open challenge in genomics. Conventional benchmarking relies on sequence alignment, which becomes problematic in regions of high homogeneity and divergence. Here, we framed centromere assembly evaluation as a comparative distribution problem in a compact centeny representation by computing genomic distances between functional motifs, rather than relying on nucleotide sequence. Our distribution-based metric assesses agreement between a query and a target chromosome by comparing their centromeric inter-motif distances rendered by KL divergence. When applied genome-wide to currently available human telomere-to-telomere (T2T) genomes, this approach yields an accuracy ranking for the entire assembly and for each individual chromosome. Altogether, we present a rapid and robust scoring system based on genomes numerical rendering of inter-motif distances, that provides a quantitative standard of assembly integrity in repetitive DNA regions and establishes a bona fide framework for chromosome-level genome-to-genome comparison.

We tackle two key difficulties in the simulation of the viscous hydrodynamics of a large dense collection of rigid particles: (i) the poor convergence rate of an iterative solution of the discretized linear system as particle gaps shrink, and (ii) the large number of unknowns needed to accurately discretize the resulting lubrication-driven flows. Our focus is the 2D Stokes resistance and mobility boundary value problems for nearly-touching disks. To address both challenges, we introduce a general two-body preconditioning strategy, and implement it with the method of fundamental solutions. For each close particle pair, the hard-to-resolve interaction is represented in a basis precomputed by solving a local boundary value problem on a fine grid. In an iterative solve, the resulting flow field corrects that obtained from a coarse representation of all particles. The local fine-grid correction can even be compressed so that all particles except the pair itself are affected by an equivalent set of coarse sources. Numerical experiments demonstrate rapid GMRES convergence in challenging multi-particle settings, with iteration counts remaining low even in densely packed suspensions. For example, the mobility problem is solved for a random close packing with area fraction $ϕ= 0.65$, $P = 10000$ monodisperse disks, and minimum separation $10^{-3}$, in just 47 GMRES iterations, achieving five digits of accuracy with 72 vector unknowns per body.

The $K$-th nearest neighbor random walk $(X_n)_{n \geq 0}$ on a homogeneous Poisson point process $χ$ on $\R^d$ ($d\geq 1$), starts at the origin and at each step picks its next Poisson point among its closest neighbors according to i.i.d. labels having the same distribution as $K$. Our main result (Theorem 1) states that the number of Poisson points visited by $(X_n)_{n \geq 0}$ admits an exponential decay whenever the random variable $K$ has a bounded support (BS). In particular, the $K$-th nearest neighbor random walk visits finitely many Poisson points if and only if $K$ satisfies Assumption (BS). To prove it, we introduce the key notion of pioneer point which allows us to deal with the region of $\R^d$ already explored by $(X_n)_{n \geq 0}$. Still under Assumption (BS), we also prove an exponential decay for the Euclidean length of the trajectory performed by $(X_n)_{n \geq 0}$ (Theorem 2). Finally, and quite surprisingly, we exhibit an example of label distribution with bounded support for which the $K$-th nearest neighbor random walk discovers new Poisson points after a number of steps whose tail distribution is at least polynomial (Theorem 3).

This work investigates slowly rotating traversable wormholes supported by string fluids whose properties vary with distance from the throat. This radial variation allows the matter to transition smoothly from a de Sitter-like core near the center to a string-dominated environment further out, producing a regular, horizon-free, and asymptotically flat spacetime. By letting the transverse pressure depend on radius, the fluid naturally adapts to the surrounding geometry, resulting in a well-behaved energy density and shape function. Even modest rotation introduces frame-dragging effects that gently twist photon paths, creating subtle differences between co-rotating and counter-rotating trajectories. These effects are strongest near the throat, while at larger distances the spacetime is largely governed by the static gravitational potentials. Circular photon orbits reveal that the interplay of the redshift function, wormhole shape, and rotation shapes the photon-sphere structure. Different radial profiles of the string fluid generate distinctive photon-ring patterns, offering potential observational signatures of both the rotation and the internal matter distribution. Overall, radially varying string fluids provide a flexible and physically consistent source for traversable wormholes, bridging smoothly between vacuum-like and string-dominated regions while maintaining regularity and supporting slow rotation. This study highlights how anisotropic matter can influence both curvature and light propagation, providing a realistic framework for horizonless exotic spacetimes and suggesting new avenues to explore subtle observational effects around traversable wormholes.

For groups without strict hierarchy, collective decisions often emerge through compromise. We develop a second-order network model of collective decision-making using a dissipative Hamiltonian formulation, in which informed agents introduce preferred directions while uninformed participants contribute only direction-free dissipation. We show that under low conflict, the model admits a locally unique, exponentially stable compromise state. Using a structured modular network we further show that as conflict increases the local compromise branch terminates through a saddle-node fold rather than through a smooth mean-field symmetry-breaking transition. Modular polarized states persist on branches that are locally separated from the compromise branch. Direction-free dissipation does not shift the static structural threshold, but it delays escape from the saddle-node ghost and pushes the observable onset of polarization to larger conflicts. Our work identifies a dissipation-mediated mechanism, complementary to connectivity-based accounts, through which uninformed participants stabilize collective behavior in biological and engineered swarms.

The Schramm-Loewner Evolution (SLE) describes a family of fractal curves that arise in the study of the scaling limits of many planar Statistical Physics models. These curves are modeled using the Loewner Differential Equation for the conformal maps $g_t(z)$ with a Brownian motion driver. Using Euler's Method, in the current work we performed numerical experiments to study at a fixed time the quantities $|g_t(z) - \overline{g_t(z)}|$ and $Re(g_t(z)) - Re(\overline{g_t(z)})$, where $Re$ denotes the real part and $\overline{g_t(z)}$ refers to the sample average. These random variables measure the 'spread' of the dynamics from the average behavior at fixed time. One of the scopes of this work is to give numerical predictions for future theoretical investigations on these quantities. When investigating these quantities in the SLE case our experiments predict that the distribution is bimodal when the dynamics started close to the origin, and it can become bell-shaped if the dynamics is started further from the origin. In the second part, we performed experiments for a Multiple SLE model whose driver is Dyson Brownian Motion. Due to singularity in the dynamics of the drivers and the many data points needed, this part is challenging from a computational perspective. In the multiple SLE case, our experiments predict that the distribution is bell-shaped in all cases. In addition, we check the changes in the distributions as we vary the parameter $κ$ in the SLE case and $β$ in the Multiple SLE case.

The advent of Extremely Large Telescopes (ELTs) ground-based optical and infrared observatories with primary mirrors exceeding 20m marks a transformative era in observational astronomy. This article explores the dawn of this new age, focusing on two of the next generation of optical/ infrared facilities: the Giant Magellan Telescope (GMT) and the Thirty Meter Telescope (TMT)\footnote{An accompanying paper by the author is on the European Extremely Large Telescope (ELT).} We describe the key technological innovations enabling the construction of these ambitious observatories, notably, segmented mirror design, advanced adaptive optics (AO), and laser guide star systems. Collectively, these breakthroughs yield more than an order of magnitude increase in both light-gathering area and spatial resolution, delivering image sharpness that, for widefield observations, surpassing space-based telescopes. We shall also discuss the comparison of space and ground-based telescopes and highlight their individual strengths and why space telescopes are not the solution to the satellite interference to ground-based telescopes. Ground based telescopes will continue to always be leaders in astronomical observations.

We consider the enumeration of $N$-ary convex structures on finite sets. Our main result shows that the total number of convexities is expressed as the binomial transform of the sequence of numbers of grounded convexities. We present the exact numbers of all $N$-ary and grounded $N$-ary convexities for $|X|\leqslant 5$. The obtained integer sequences have been cross-referenced with the OEIS. As a result, we identified both known sequences (such as A000798) and new ones not previously listed.

Using a sequence of high-school level mathematics questions that were not available on the Internet, we compare the performance of the popular AI software Claude with that of my friends and fellow human beings Pierre and Catherine. Pierre had solid scientific training as a young man, while Catherine studied literature. All three were subjected to a simulated pre-calculus oral exam with main questions and follow-up questions. Their performances are compared and the ones with the best and worst performances are identified. The outcome is that the current version of Claude, even though it is an extremely useful tool that has probably recorded the solution to nearly all calculus questions that are available on the Internet, {\em exhibits only a very limited understanding of the subject} and {\em does not exhibit the ability to make intelligent connections} between different features of a pre-calculus mathematics problem that it has never seen before.

We study degree-one persistent homology of the planar Wiener-sausage filtration generated by standard Brownian motion without drift. In the drifted case, regeneration along the drift direction leads to linear-in-time laws for persistent-homological observables. In the recurrent zero-drift case, this renewal structure disappears. The organizing mechanism is instead Brownian self-similarity: the persistence diagram at time $T$ is equal in law to the image of the unit-time diagram under spatial dilation by $\sqrt T$. Consequently, large-time questions on fixed radius windows are transformed into small-radius questions for the unit-time Brownian trace. Let $B$ be standard planar Brownian motion, let $K_T=B\left(\left[0,T\right]\right)$, and let $K_T^{\left(r\right)}$ be the radius-$r$ Wiener sausage. Since $K_T^{\left(r\right)}$ is connected, its first Betti number $β_1^T\left(r\right)$ is the number of bounded complementary components of $K_T^{\left(r\right)}$. For a bounded nonnegative Borel function $ψ$ supported in a compact interval $\left[a,b\right]\subset\left(0,\infty\right)$, we consider the smoothed Betti-curve observable $\left[r_0,r_1\right] \mathrmΦ_ψ\left(T\right) = \int_{r_0}^{r_1} β_1^T \left( r \right) ψ\left( r \right) dr$. We prove that there exist absolute constants 0

This paper bridges combinatorial geometry and music theory to solve the fundamental challenge of embedding classical Western harmony into the microtonal 19-tone equal temperament (19-TET). Inspired by Roger Penrose's observations on the mathematical elegance of 19-TET, we provide the theoretical foundation for a physical 19-TET acoustic piano currently under construction. However, playing classical 12-TET music on such an instrument poses a topological problem: emvedding the classical Euler-Riemann Tonnetz into the 19-TET universe inevitably distorts structural chords, creating dissonant ``wolves.'' By formalizing these harmonic spaces as incidence configurations (the 12_3 and 19_3 graphs) and utilizing integer cuts in our optimization model, we exhaustively prove that exactly 32 out of 36 Neo-Riemannian harmonic connections can be preserved. We demonstrate a strict 5-fold degeneracy of this optimum: there exist exactly 5 mathematically equivalent local packings for the wolf chords. Among these, we identify a unique canonical realization in which the 14 excised vertices form a perfectly contiguous geometric void along the primary Hamiltonian cycle. We reveal that the 4 inevitably broken edges represent the exact topological scars of the historical enharmonic diesis, and we formulate the Vicentino Hypothesis regarding 16th-century microtonal composition. Finally, to make this theoretical geometry physically playable, we design a novel 19-TET split-key keyboard, formalized through a biomechanical cost function that optimizes the performer's hand span. This work provides the complete theoretical, historical, and ergonomic blueprint for the next generation of microtonal acoustic instruments.

Conscious access in the human brain is often described as the outcome of a competition among candidate representations, but this competition is usually left at the level of mechanism or metaphor rather than analyzed as a strategic allocation problem. We introduce an access contest in which internal modules compete for a scarce broadcast slot by choosing a costly amplification effort. Access is allocated by a smooth probabilistic rule, allowing the model to interpolate between diffuse selection and winner-take-all competition. We establish pure-strategy equilibrium existence under standard convexity and bounded-benefit assumptions, and give sufficient conditions for uniqueness using diagonal strict concavity. We then analyze capture in the two-module case, and for quadratic costs, we derive a sharp threshold in the competition intensity above which capture occurs. For strongly convex costs, we prove an if-and-only-if capture criterion in terms of the cost-adjusted amplification advantage of the lower-value module. Under the same curvature-dominance condition that guarantees uniqueness, we show that the unique pure Nash equilibrium of the general \(M\)-module access contest can be approximated efficiently by projected pseudo-gradient dynamics, with logarithmic dependence on the desired accuracy. Finally, we prove an impossibility theorem for single-slot access mechanisms. Exact winner-take-all efficiency is incompatible with robustness to small score perturbations. Thus, smooth probabilistic access rules are not merely analytically convenient, but structurally motivated. These results provide a game-theoretic foundation for studying competition for conscious access, connecting equilibrium analysis, capture, computation, and mechanism-level limitations under a common formal model.

Atom probe tomography (APT) literature is broad, rapidly growing, and dispersed across a wide range of journals, which can make it difficult to identify prior work on a given material system, instrument, or analytical approach. Conventional search engines (e.g., Google Scholar) excel at general retrieval but do not preserve the domain-specific metadata that often determines the relevance of an APT publication (e.g., analysis mode, laser wavelength, or instrument configuration). Herein, APTLAS is presented, which is an indexed repository of published APT literature. At present, the database contains ~2,300 records, each accompanied by metadata extracted from the source publication. The accompanying web tool, available at https://aptlas.bavleyguerguis.com/, allows users to browse and filter by material system, instrument, application, publication type, or keyword search.

Anisotropic surface phonon polaritons (SPhPs) offer extremely strong light confinement and unique light propagation characteristics, particularly on anisotropic polar crystals. Despite the classical importance of α-quartz as a prototypical uniaxial bulk crystal, real-space observation of anisotropic SPhP propagation on α-quartz has remained elusive. In this study, we report the first direct observation of SPhP propagating on an α-quartz surface using scattering-type near-field optical microscopy (s-SNOM). We demonstrate that the dispersion relation and propagation length of SPhPs exhibit remarkable anisotropy depending on the propagation direction relative to the optic axis of α-quartz. Furthermore, we verify that these experimental behaviors agree with theoretical calculations based on the dielectric permittivity tensors. Our results establish α-quartz as a robust, highly promising platform for light-controlling nanodevices and mid-infrared on-chip sensing.

We study Bermudan option pricing under the Gatheral Double Mean-Reverting (GDMR) stochastic volatility model. The model features a variance process together with a stochastic long-run mean variance process and allows Constant Elasticity of Variance (CEV)-type exponents in the diffusion coefficients. This model is attractive since it provides a flexible specification for volatility dynamics. However, the pricing of early-exercise derivatives under the GDMR model remains largely unexplored in the literature. To address this challenge, we adapt a Hybrid Least-Squares Monte Carlo-Partial Differential Equation (LSMC-PDE) framework to the GDMR model and provide a detailed model-specific implementation. Conditioning on simulated variance paths, the pricing problem reduces to a one-dimensional problem in the asset price, which is solved by a Fourier-based approach, while the remaining dependence on the variance variables is approximated by least-squares regression. Our numerical experiments demonstrate that the Hybrid LSMC-PDE approach yields accurate pricing estimates and often lower pricing errors than plain LSMC, particularly for low and moderate numbers of simulation paths, showing the benefit of using the model structure in early-exercise option pricing.

In recent years, the Large Area Telescope (LAT) onboard the \emph{Fermi} satellite has detected more than 300 pulsars in the high energy range. The population studies of high energy pulsars show that the gamma ray luminosity of a pulsar ($L_γ$) can be expressed in terms of the spin down luminosity ($\dot{E}$) as $L_γ\propto \dot {E}^s$ having exponent $s\sim 0.68$. This high energy emission, assumed to originate far from the stellar surface and near the light cylinder, is usually studied in either purely curvature or purely synchrotron regime. In this work, we adopt a synchro-curvature radiation framework to understand the origin of gamma ray emission from the pulsar and its implications at the population-level. By comparing the observed cutoff energies of the differential gamma-ray spectra with the theoretical synchro-curvature predictions and enforcing radiation reaction approximation, we determine the equilibrium Lorentz factor and pitch angle of the emitting charged particles. This approach allows to quantify the relative roles of curvature and synchrotron radiation to the radiative losses, thereby providing a physically grounded interpretation of the luminosity trend across the pulsar population.

The detection of astrophysical neutrino point sources remains challenging due to atmospheric backgrounds obscuring signal and statistical penalties from the look-elsewhere effect. The Tau Air-shower Mountain-Based Observatory (TAMBO) is a neutrino telescope that achieves unprecedented signal-to-background discrimination in the 1-1000 PeV energy range. Leveraging its unique deep valley geometry, TAMBO will generate an exceptionally pure neutrino sample, enabling precise investigations of neutrino sources. Preliminary sensitivity studies demonstrate TAMBO's potential to map diffuse and point-source neutrino emissions, representing a significant advancement in high-energy neutrino astronomy.

Thermomagnetic generators (TMGs) are devices that convert waste heat to electricity through a change in magnetization of a solid material. This causes a changing flux through a coil, which induces an electromotive force per Faraday's law. However, the influence of the coil on the performance of the TMG has not been investigated and existing TMG prototypes merely utilize some coil, not the optimal coil for a given device. In this work we present an analytical and numerical model of a TMG that calculates power by explicitly coupling the TMGs magnetic and electric circuits and use this to analyze the influence of the coil on the TMG performance. We show that analytically TMG power has a linear dependence on coil volume, independent of the specific combination of wire radius and coil turns. The model is validated with experimental data, and finally used to study prototype TMGs presented in literature, where we show that the power of these literature TMGs can be increased by a factor of 10-400 times, had larger coils been used in the prototypes.

Second Harmonic Generation (SHG) in nonlinear crystals has been extensively investigated, but most existing models still rely on simplifying assumptions. In realistic settings, thermal effects introduce complications that are difficult to capture analytically because the governing equations are highly coupled and nonlinear. Direct experimental characterization is also limited, since studying thermal effects would require spatiotemporal temperature data at every point in the crystal, which is not experimentally accessible. To address these limitations, we have developed a SHG Computational Toolkit Suite, a coordinated collection of independent modeling toolkits that cover different SHG scenarios under various physical conditions. Each toolkit focuses on a particular configuration or coupling mechanism, while the suite as a whole provides well-documented numerical implementations, reproducible workflows, and illustrative examples. Together, this article and the Toolkit Suite provide a coherent infrastructure for computational studies of SHG. It enables researchers to replicate, adapt, and extend our methods without duplicating foundational development efforts, thereby accelerating SHG research and promoting reproducibility.

Collective emission in light-harvesting assemblies is governed by the local transition dipole and finite geometry of emitting units, a fact that point-dipole approximation obscures. To go beyond this picture, we develop a non-Hermitian Hamiltonian using the quantum electrodynamic dyadic Green's tensor for a purple bacteria. We construct it for the isolated 24-bacteriochlorophyll conical frustum and its P42$_1$2 crystallographic assembly. The P42$_1$2 unit-cell symmetry is found to invert the bright-dark ordering of the single ring, placing subradiant states at the low-energy end and revealing the entire crystal to be the energy-harvesting entity. Tilt-driven switching is activated only in crystal geometries where the finite dipole-carrier (LH2) lies perpendicular to the growth plane. Vacancy and orientational disorder work only in cooperation to renormalize the switching threshold from higher polar angles to lower values.

Efficient thermal management is critical for the reliability and performance of power electronics systems in automotive applications. This work presents a computationally efficient modeling approach for transient thermal simulation of power electronic systems, with a focus on inverter modules using multiple MOSFETs mounted on a printed circuit board assembly (PCBA). A case study of an inverter module comprising six MOSFETs arranged as high-side and low-side pairs for a three phases system mounted on a PCBA, attached to a heat sink is considered. Computational fluid dynamic (CFD) simulations in Ansys Icepak are performed considering different heat transfer mechanisms, including natural convection, forced convection at constant velocity, and forced convection with varying flow velocity. A transient thermal model is developed using the Lumped Parameter Linear Superposition (LPLSP) method, a hybrid approach that combines lumped parameter modeling with the principle of linear superposition to capture transient thermal behavior efficiently. Temperatures of the components from the simulations are compared with temperatures from the LPLSP model and temperatures from a Linear Time Invariant (LTI) based reduced order model (ROM) developed for this system. It is observed that the LPLSP model is able to model a wide range of use cases very accurately with error of less than 5 %. This method enables rapid thermal performance evaluation of power electronics systems that have very fast transients in component level power dissipation and variations in ambient conditions, making it particularly well-suited for early-stage design iterations and long-duration mission profile simulations. The approach offers a practical path to reducing development cycles for automotive power electronics design.

Non-Hermitian (NH) topology has been extensively explored in wave and matter systems, typically relying on the routing of complex, non-reciprocal couplings in physical space. This work demonstrates the experimental realization of programmable NH topological phases within decentralized multi-robot networks. By digitally programming non-reciprocal interaction rules and establishing real-time state exchange among active robots, we observe emergent topological zero modes (TZMs) and NH skin effects in synthetic lattices spanning one to three dimensions. Dynamically tailoring non-reciprocal parameters enables the precise morphing of TZMs between localized and delocalized states, establishing a versatile framework for topological mode engineering across dimensionalities. This platform establishes multi-robot networks as highly reconfigurable systems for exploring non-equilibrium topological physics, while paving the way for topologically protected, robust collective behaviors in active matter.

Droplet and particle grouping can be influenced by applying an acoustic field and have practical applications such as particle scavenging and aerosol filters of engine exhaust and air purifiers. The present work experimentally investigates the influence of a standing acoustic wave on a single stream of droplets. The experimental setup consists of an acoustic transducer and a reflector plate through which the droplet stream passes in the presence or absence of an external pressure field generated by a standing acoustic wave. A droplet stream is generated with the help of a nozzle connected to a pressurized working fluid supply and piezoelectric transducer to control the spacing between droplets. The effect of the acoustic pressure field on the droplet stream generated by the nozzle operated at different piezoelectric excitation frequencies and fluid pressures is investigated. Droplet stream characteristics at every nozzle excitation frequency are observed with a high-speed camera when the acoustic field is switched OFF and ON. The competing effect of nozzle excitation frequency and acoustic field is observed. At lower nozzle frequencies, the nozzle generates an unstable stream of droplets having different sizes and spacings between them. When the acoustic field is applied at these lower frequencies, the stream of droplets becomes organized, and in some cases, it becomes equispaced and of the same size. However, an opposite behavior is observed at higher frequencies. In these cases, as the acoustic field is applied, an equispaced mono-disperse droplet stream becomes unstable due to the coalescence of droplets within the stream.

Quantifying worst-case and best-case performance under complex market scenarios is a persistent challenge in financial risk management and the verification of path-dependent financial instruments, such as exotic options and structured products. Simulation-based methods are well suited for probabilistic estimation, but they do not directly provide exhaustive guarantees over all admissible scenarios or explicit witnesses for extremal outcomes. To address this, we introduce a quantitative automata-based framework for the exact extremal analysis of financial systems under declarative scenario constraints. At the core of our approach are event history automata (EHAs), a new formal model that integrates regular-expression event patterns with admissible numerical intervals to represent constrained event histories with memory. Quantitative payoffs are represented by weighted finance finite automata (WFFAs), which allow transition weights to depend on observed market values. By computing the synchronized product of EHAs and WFFAs, our framework enables the exact calculation of upper and lower payoff bounds. Furthermore, the method automatically extracts interpretable witness event histories that realize these extremal outcomes. We demonstrate the practical viability of the approach through a case study of an autocallable structured product with path-dependent mechanisms. The case study analyzes how different scenario constraints affect coupon accumulation, early redemption, and protection-loss outcomes. Scalability experiments indicate that the framework's execution remains computationally feasible for practical contract horizons and nontrivial constraint configurations. Overall, this approach provides a mathematically rigorous complement to standard financial simulation methods.

The internet, once celebrated as a decentralized public sphere, is increasingly undermined by practices such as generative search and shadow banning, which divert traffic and suppress visibility. Generative search, powered by Retrieval Augmented Generation RAG, synthesizes content into direct answers, bypassing websites and depriving them of traffic and revenue. This threatens the sustainability of independent content creators, small enterprises, and the open web ecosystem. Shadow banning, a practice that intentionally reduces the visibility of social media posts through algorithmic moderation, exacerbates these issues by chilling free expression and limiting transparency and accountability. This article explores these opaque practices through a legal and regulatory lens. The first part examines the rise of generative search, analyzing its technological and legal implications, including copyright infringement, unfair competition, and unjust enrichment. It also evaluates potential solutions such as licensing agreements and agentic AI. The second part focuses on shadow banning: algorithmic dissuasion, de-ranking, and the reduction of traffic, with specific attention to Chinas Regulation on Algorithmic Recommendations RAR and the EUs Artificial Intelligence Act AIA. Both frameworks offer partial solutions but fall short of ensuring fairness, transparency, and redress mechanisms. Ultimately, the shift toward centralized control by dominant platforms prioritizes profit and risk management over innovation, fairness, and diversity in online expression. To counteract these trends, regulatory interventions, algorithmic transparency, and equitable frameworks are essential. Without such measures, the internet risks losing its character as a democratized public sphere for free expression and innovation.

Biquadratic forms arise naturally in polynomial optimization, tensor analysis, and quantum information theory. A key problem is determining the minimal number of squares needed in a sum-of-squares (SOS) representation of such a form, known as its SOS rank. For fixed dimensions $(m,n)$, the maximum possible SOS rank over all biquadratic forms in $m$ and $n$ variables is denoted $\operatorname{BSR}(m,n)$. Recent advances have established lower bounds on $\operatorname{BSR}(m,n)$ via combinatorial constructions involving bipartite graphs and the orthogonality method. In particular, for the case $(m,n)=(5,4)$, it was shown that $\operatorname{BSR}(5,4)\ge 11$ using only nondegenerate 2-edges. In this paper, we extend this framework by incorporating a degenerate $2$-edge, which introduces a cross term where the two $y$-indices coincide. We construct an explicit $5\times 4$ biquadratic form and apply the orthogonality method to prove that its SOS rank is $12$, thereby improving the lower bound to $\operatorname{BSR}(5,4)\ge12$. This result demonstrates that degenerate $2$-edges yield additional algebraic flexibility beyond purely combinatorial bounds and extends the applicability of the orthogonality method to forms with cross terms involving identical $y$-indices.

This thesis studies the convergence of finite-population stochastic evolutionary dynamics to their deterministic mean-field limit in continuous-time finite state space games. We first develop refined ergodic theorems for Markov chains with a single positive-recurrent class, guaranteeing the existence of a unique invariant distribution and almost-sure convergence of time averages. Next, we prove that the mean-field model, described by a system of Lipschitz-continuous ordinary differential equations, admits a unique solution that depends continuously on its initial condition and that constitutes the almost-sure limit for the empirical distributions with fixed policy. Furthermore, we show that every Mixed Stationary Nash Equilibrium of the mean-field game is approximated by a Nash equilibrium of the corresponding $N$-player game within an error $ε$ for sufficiently large $N$. We finally demonstrate, by Kurtz's theorem, that the empirical state-policy distribution converges in probability to the mean-field trajectory. Numerical simulations conducted in MATLAB confirm the theoretical $\mathcal{O}(N^{-1/2})$ convergence rate in both models across a range of population sizes.