Extremely large-scale reconfigurable intelligent surfaces (XL-RISs) have emerged as a promising technology for millimeter-wave (mmWave) communications. However, the exceedingly large aperture of XL-RISs renders the RIS-user links likely to operate in the near-field region, where the conventional planar-wave assumption and angular-domain sparse representation become invalid, thus making channel estimation significantly more challenging. In this paper, we investigate cascaded channel estimation for an XL-RIS-aided multi-user multiple-input multiple-output (MU-MIMO) system, in which the BS-RIS channel is modeled in the far field, while the RIS-user channels exhibit near-field spherical-wave characteristics. To tackle the resulting hybrid-field estimation problem, we propose a low-overhead two-stage channel estimation scheme by jointly exploiting the common BS-RIS link shared by all users and the polar-domain sparsity of the RIS-user channels. Specifically, the multi-antenna users are firstly decomposed into multiple virtual single-antenna users, based on which the common BS-RIS parameters are extracted from a typical virtual user and the RIS-user channels are initialized via compensated polar-domain sparse recovery. Then, an alternating least-squares refinement procedure is developed to jointly improve the common BS-RIS operator and the user-specific RIS-side channels. Simulation results show that the proposed scheme achieves competitive channel estimation performance with substantially reduced pilot overhead compared with the existing near-field benchmarks.

We introduce a weak version of the classical length function, termed the weak length function, defined on subsets of $R$-modules over a unital ring $R$, and further consider the concept of mean weak length for $RΓ$-modules associated with an amenable group $Γ$. Under an appropriate upgrading condition together with certain mild assumptions, we establish that the mean weak length function is additive with respect to short exact sequences. This result has two consequences. First, we provide a purely algebraic proof of the additivity of algebraic entropy, which is a property originally established via topological entropy methods. Second, within our unified framework, we give an alternative and conceptual proof of the additivity of mean length, previously obtained by Li-Liang and Virilli using different approaches.

In this paper, we build upon the work of Gluck and Warner who showed in 1983 that the set of positively oriented fibrations of a 3-sphere by oriented great circles is in bijection with the set of distance-decreasing maps from the 2-sphere to itself. One approach to generalizing their result to higher-dimensional spheres involves understanding when exactly two Hopf fibrations of $S^{2n-1}$ are guaranteed to agree on a fiber. We give a complete characterization of this phenomenon, and we discuss the barriers which prevent us from obtaining a fully general version of Gluck and Warner's result.

Modern online platforms such as marketplaces, ride-hailing services, and food-delivery systems serve a dual role: they are both markets where participants interact and transact, and operators that design and govern how these markets function. These platforms connect multiple sides, for example buyers, sellers, and couriers, facilitating access that would otherwise be difficult to achieve. By setting the rules of the market, platforms determine who participates, how interactions take place, and how value is created and distributed. In response to these rules, participants may behave strategically, deciding whether to join the platform and which transactions to pursue. This thesis studies how platform design affects market outcomes through three key levers: pricing that determines participants' gains when operating on a platform; matching that governs which interactions are feasible among participants; and bundling that shapes the structure of supply when the platform itself acts as a market participant. Across these levers, the goal in this thesis is to understand how platforms can be designed to balance platform profitability with overall market welfare. The first part of this thesis studies pricing, including both the commission fees that participants pay to a platform and the prices associated with each transaction. The second part of this thesis studies matching. By shaping recommendation systems and consumer search, platforms influence which transactions take place. The third part of this thesis analyzes bundling. As a marketplace operator, a platform may be able to source products from sellers and offer them as bundled packages to buyers. Collectively, this thesis shows how pricing, matching, and bundling serve as complementary design levers through which platforms can shape market outcomes.

We propose a new integral based on Taylor measures, study its properties extensively, and we illustrate that it includes many concepts from mathematics as special cases. In particular, the new integral emerges as a generalization of the discrete Fourier transform, and we identify general conditions for it to be invertible when applied to any real or complex sequence. Applications to the mathematical sciences are also presented.

We develop a variational thermodynamic framework for statistical systems governed by a self-referential nonlinear operator Omega characterized by structural exponents alpha > 0, beta >= 0, a symmetric kernel K, and a self-coupling constant kappa >= 0. The central object is the self-consistency entropy S[Psi] = -D_KL(Psi || Omega Psi), which vanishes at the fixed points of Omega and serves as a natural Lyapunov functional. Within the local kernel (mean-field) approximation, minimization of the free energy F = U - T S admits the Tsallis q-exponential distribution as an equilibrium state, with the entropic index q = alpha + beta emerging directly from the fixed-point structure of the operator rather than being postulated. The framework yields a consistent thermodynamic description, including a generalized equation of state PV = (2 - q) T, response functions, and a critical temperature associated with spontaneous symmetry breaking. The relation q = alpha + beta connects independently measurable structural exponents of the feedback mechanism to the observed tail index, providing a parameter-free criterion that distinguishes this approach from superstatistics, constrained entropy maximization, and q-deformed formalisms. This work establishes an operator-theoretic foundation for nonextensive statistical mechanics in which nonlinear self-referential feedback naturally generates Tsallis statistics in the mean-field limit.

Using Cartan equivalence method, invariant coframes are constructed for two branches of rank one and zero, which characterize linearizable third-order ODEs under contact transformations with four- and five-dimensional Lie symmetry algebras, respectively. A procedure for deriving the corresponding contact transformations is also presented, along with illustrative examples.

Enterprise technology modernization programs fail at a documented and costly rate, yet the dominant explanation -- inadequate engineering capability -- is incorrect. The primary failure mode is a governance deficit: the absence of structured, repeatable operating routines for how organizations plan, execute, validate, and hand off complex technology change. Existing frameworks -- ITIL, COBIT, TOGAF, scaled agile methodologies, and cloud provider well-architected frameworks -- address adjacent concerns but do not provide an integrated, portable institutional operating model for controlled modernization across migrations, data platforms, and AI-enabled automation. This article presents the Enterprise Modernization Reliability and Governance Framework (EMRGF), a practitioner-developed governance operating model derived from 24 years of applied delivery experience across financial services, industrial manufacturing, and retail enterprises. EMRGF comprises four interlocking modules -- Cloud and Legacy Modernization Governance, Data Platform Reliability and Evidence Integrity, AI-Enabled Automation Governance, and Mission-Critical Reliability and Root-Cause Routines -- operationalized through five implementation tools and a training-of-trainers institutionalization model. Empirical application at scale has produced a 30% reduction in development effort, a 35% reduction in testing cycles, zero-disruption migrations across high-volume data estates, and 99.9% data reliability in mission-critical analytics pipelines. The framework is explicitly aligned with U.S. national policy mandates including NIST CSF 2.0, NIST AI RMF, and Executive Orders 14028 and 14110, and is designed for institutional adoption without ongoing external dependency.

We study the dynamics of a single-particle wave packet on a one-dimensional lattice subject to periodic random phase kicks with finite spatial correlation length. This stroboscopic setting provides a controllable model of dephasing in driven quantum systems. Using a momentum-space formulation, we show that the evolution is governed by an accumulated phase whose structure determines the spreading of the wave packet. We find that the phase kicks strongly suppress ballistic transport and induce diffusion at long times. We derive an explicit analytical expression for the diffusion coefficient as a function of the correlation length, in excellent agreement with numerical simulations. Our results uncover a simple mechanism by which spatially correlated phase noise controls quantum transport, and provide a quantitatively testable prediction for diffusion in periodically driven lattice systems. Possible experimental realizations in cold-atom platforms are discussed.

The rotation of the casing and rotor of a gyroscope is studied by considering frictional effects. Friction causes the casing to rotate, and over time, air dissipation and friction at the touchpoint gradually stop this rotation. Several models for air friction, friction between the rotor and casing, and friction at the touchpoint are analyzed. Fit results demonstrate that while some of these models can describe the primary motion, certain effects require further study to yield more precise results. These findings can aid in developing improved models for the rotation of satellites.

We show that infinitely many three-term arithmetic progressions $N, N+d, N+2d$ of powerful numbers exist with $d = 2\sqrt{N} + 1$. We further conjecture that infinitely many of these progressions consist of three consecutive terms in the sequence of powerful numbers, which would answer a question of Erdős in the negative.

The fuzzy Sombor index applies the classical Sombor index to fuzzy graphs, incorporating both edge membership values and fuzzy vertex degrees. For $α>1$, the general fuzzy Sombor index it is defined as \[ \mathrm{SO}^μ_α(Γ)=\sum_{uv\in V(Γ)} \left( μ(u,v)\, \sqrt{μ_u^2+μ_v^2} \right)^α. \] This paper analyses extremal features of $\mathrm{SO}^μ$ across different types of fuzzy graphs. We determine the maximum value (resp. minimum value) of $\mathrm{SO}^μ$ characterise in regular fuzzy graph. We established significant inequality between the fuzzy Sombor index and other well-known fuzzy topological indices.

The aim of this paper is to generalize some fixed point theorems in the class of convex contraction of order $m$ on a complete suprametric space. Then, we will prove that the class of convex contraction of order m is strong enough to generate a fixed point on a complete suprametric spaces but do not force the mapping to be continuous at the fixed point, and it can be replaced by relatively weaker conditions of $k$-continuity or $T$-orbitally lower semi-continuous. On this way a new and distinct solution to the open problem of Rhoades (Contemp Math 72:233-245,1988) is found. In sequel, we will prove some fixed point results in the setting suprametric spaces which are generalizations of the results regarding Sehgal, Ćirić and Fisher's quasi-contraction. Some examples and application will be approved our results.

This paper introduces a class of extended central factorial numbers generated by a parity-dependent recurrence relation, termed the "flickering operator". We demonstrate that the resulting triangular structure, now indexed as OEIS A395021, provides a unified recursive framework for alternating bit sequences (A000975) and normalized tangent-secant coefficients (A036969). This study provides an alternative integer-based expansion for power sums. While similar to the central factorial methods explored by Knuth (1993), our flickering basis offers an integrated computational scheme that avoids fractional Bernoulli numbers by construction. We provide explicit closed-form expressions, discuss its geometric derivation from finite difference tables, and present a full Python implementation. Structural Synthesis. A key contribution of this work is the unification of previously disparate combinatorial sequences into a single coherent framework. While certain columns of the flickering triangle T(n, k) (such as A008957) could be partially retrieved from the diagonals of existing central factorial arrays, our structure provides a complete representation including previously unindexed even-positioned terms. Furthermore, the row-wise analysis reveals that the flickering operator generates full integer sequences where previously only the odd-indexed elements (e.g., A002451) were identified. This synthesis bridges the gap between these sequences, positioning A395021 as the underlying master structure.

In American options, the early exercise feature allows the option to be exercised at any time prior to expiration. However, this flexibility introduces a challenge: the pricing model must value the option while simultaneously determining an unknown, time-varying exercise boundary. The Heston model is one of the most popular ways to model real market behavior because it allows volatility to change over time. However, unlike European options, there is no closed-form solution for American options under the Heston model, so we have to use numerical methods. In this paper, we propose a novel approach to solving the stochastic Heston partial differential equation for American options, using coupled physics-informed neural networks (PINNs) to predict both the option price and the free boundary, while employing curriculum learning and adaptive resampling to stabilize model training. Our work builds on recent deep learning methods but introduces a more effective training strategy to address the limitations of these approaches. The numerical results demonstrate the effectiveness of the proposed learning framework, providing a robust and efficient alternative to pricing American options, enabling rapid inference and accurate estimation under stochastic volatility.

The resummation of superfactorially divergent series represents a significant computational challenge in mathematical physics. In the present paper the resummation of a specific class of Stieltjes series characterized by a moment sequence growing as $(2n)!$ will be addressed. Despite the fact that Carleman's condition is satisfied for these series, the convergence rate of Padé approximants is severely hindered by the logarithmic divergence of the associated Carleman series. Weniger's $δ$ transformation is proposed as a highly efficient alternative resummation tool. By employing recently established results on the converging factors of superfactorially divergent Stieltjes series, an exact integral representation for the truncation error is obtained. This representation enables the rigorous derivation of the leading-order asymptotic behavior of the transformation error, as well as the estimation of the related convergence rate, for real positive arguments. Numerical experiments strongly support the theoretical findings, suggesting that the $δ$ transformation offers a robust and computationally efficient framework for decoding this class of wildly divergent expansions

Barrier derivatives depend on extrema and first-passage events and are therefore highly sensitive to volatility dynamics -- especially to the instantaneous return-volatility correlation $ρ$, often called ``leverage''. This sensitivity makes accurate and fast pricing under realistic stochastic-volatility specifications difficult: two-dimensional PDE solvers are expensive inside calibration loops, while Monte Carlo methods converge slowly when barrier hits are rare and discretely monitored. In equity markets in particular, the pronounced implied-volatility skew motivates factoring in a negative return-volatility correlation. We study a class of continuous-path stochastic-clock volatility models in which the log-price is represented as a Brownian motion run on a random increasing clock. In the baseline independent-clock case (ρ=0), a broad family of barrier-relevant objects-maximum distributions, survival probabilities, and killed joint laws-reduces to one-dimensional quantities determined by the Laplace transform of the terminal clock. This yields transform-only pricing formulas for single- and double-barrier contracts that are fast and numerically stable once the clock transform is available, notably for affine and quadratic clocks. To incorporate leverage without forfeiting tractability, we develop a systematic small-ρexpansion around the ρ=0 backbone. The expansion produces a hierarchy of forced problems whose forcing terms are semi-analytic and computable from baseline barrier objects. We provide two implementable leverage-correction routes\,: forced PDEs and a Duhamel-type Monte Carlo representation, and we show how Pad{é} acceleration can extend practical accuracy to equity-like correlations. Calibration then proceeds by\,: (i) fitting clock parameters from vanillas using only one-dimensional transforms, (ii) precomputing the ρ=0 barrier backbone once, and (iii) iterating on ρ(and any remaining parameters) using the fast semi-analytic corrections-optionally Pad{é}-accelerated-inside a standard least-squares loop.

We correct the statements and proofs of the (auxiliary) Propositions 4.1 and 4.2 of our paper `Evaluation of motivic functions, non-nullity, and integrability in fibers' in Advances in Mathematics, Vol. 409, Part A, Paper No. 108635, 29 pages (2022), and we explain how the proofs of the main results can be adapted to work with those corrected propositions.

The multidimensional Uncertain Volatility Model leads to robust option pricing problems under joint volatility and correlation uncertainty. Their numerical resolution quickly becomes challenging because the associated stochastic control problem is high-dimensional. We propose a backward actor-critic stochastic policy gradient scheme tailored to this setting. The method combines a discrete dynamic programming principle with Proximal Policy Optimization and shallow neural-network approximations of both the value function and the control policy. A key ingredient is the policy parameterization: continuous controls are represented through a squashed Gaussian policy built on a C-vine representation of correlation matrices, which enforces positive semidefiniteness by construction. Numerical experiments on a range of multidimensional derivatives show that the method yields accurate prices, remains computationally efficient, and compares favorably with existing Monte Carlo and machine-learning-based benchmarks for robust pricing in the Uncertain Volatility Model.

Integer partitions have fascinated people for centuries, from Ramanujan's groundbreaking congruences to the modern theory of modular forms. This paper investigates the statistical properties of odd unimodal sequences--a natural refinement where sequences rise to a peak and then fall, but with the constraint that all parts must be odd, and develops a comprehensive statistical theory for their rank and shape parameters. We establish the asymptotic distribution of the rank statistic and demonstrate that, when properly normalized, it converges to the hyperbolic secant distribution. Beyond the rank distribution, limiting distributions of the peak, the largest parts on either side of the peak, and the joint behavior of small parts are also proved. These results reveal a rich probabilistic structure that parallels the classical theory of integer partitions while exhibiting distinctive new features arising from the odd-part constraint. The analysis employs a synthesis of modular transformation theory, false theta function asymptotics, and conditioned Boltzmann models. This extends the probabilistic machinery previously developed for unimodal sequences into a more general and analytically demanding setting, offering a unified approach that bridges modular forms and probability.

We present a data-driven framework for reconstructing band structures using Koopman operator analysis and dynamic mode decomposition (Koopman-DMD). Instead of deriving spectra from an explicit Hamiltonian, the approach reconstructs band dispersion and modal dynamics directly from spatiotemporal data, including wavefunctions and observables. This framework establishes a correspondence between Hamiltonian Floquet-Bloch decomposition and Koopman-DMD, whereby the extracted DMD modes encode frequencies, decay or growth rates, spatial profiles and projection weights. These quantities allow the reconstruction of spectral functions, local density of states, and delocalized-to-localized measures such as the inverse participation ratio. Also, these extended DMD modes enable inference of quantum-geometric and topological properties, including the quantum metric, Berry curvature and geometric phases. Applications to prototypical one- and two-dimensional tight-binding models, including disordered Su-Schrieffer-Heeger model and its Floquet and non-Hermitian variants, graphene and Haldane models, demonstrate that Koopman-DMD provides a unified route for the data-driven analysis of wave propagation, localization, and topological phases in condensed matter, photonics, and related fields.

Background: Open source software ecosystems exhibit dense dependency networks in which maintenance degradation of structurally central packages can propagate widely. Despite increasing attention to open source sustainability, existing support mechanisms lack an explicit, dependencyaware notion of ecosystem-level impact to guide support decisions. Aims: In this paper, we introduce a dependency-aware model of ecosystem impact that captures how changes in maintenance activities propagate through the Python Package Index (PyPI) ecosystem and affect its overall state. Based on this model, we prioritize packages for ecosystem support using our dependency-propagated notion of ecosystem impact. Method: Applying this framework to a snapshot of 718,750 PyPI packages and over 2 million dependencies, we compare our impact-driven support strategy with existing support mechanisms (Tidelift, Ecosyste$.$ms, and GitHub Sponsors) and with PageRank as a baseline measure of structural importance. Results: Our results show that a large share of the modeled ecosystem impact (approximately 80%) can be attributed to just 0.1% of all PyPI packages when prioritized based on dependency-propagated impact. In contrast, externally defined support sets vary substantially in their alignment with ecosystem impact. We further analyze maintainer reach and metadata accessibility, revealing that ecosystem impact, social footprint, and operational feasibility represent distinct but complementary dimensions of ecosystem support. Conclusions: Dependencyaware ecosystem impact modeling provides a transparent and systematic basis for prioritizing support in large-scale software ecosystems. Our findings suggest that effective support strategies, driven by ecosystem stewards, funding bodies, and organizations operating support programs, should complement existing allocation logic with impact-informed decision making.

We prove small energy regularity for a parabolic boundary reaction Ginzburg-Landau problem in the full range $s\in (0,1)$, answering a question posed by Hyder, Segatti, Sire and Wang. We also obtain a similar small energy regularity result for fractional harmonic maps to spheres. Both results are uniform as $s\to 1$.

Data-parallel (DP) load balancing has emerged as a first-order bottleneck in large-scale LLM serving. When a model is sharded across devices via tensor parallelism (TP) or expert parallelism (EP) and replicated across many DP workers, every decode step ends in a synchronization barrier whose latency is set by the most heavily loaded worker; even modest persistent imbalance across DP workers compounds, step after step, into a substantial fraction of wasted compute. The problem is hard for reasons specific to LLM decoding: assignments are sticky (migrating KV caches has a high cost), per-request loads grow over time, arrivals are non-stationary, and the router must decide within a sub-100\,ms decode budget over hundreds of waiting requests and tens of workers. We present \textbf{BalanceRoute}, a family of practical online routing algorithms that target this bottleneck. The first, \textbf{BR-0}, requires no prediction infrastructure and uses a piecewise-linear F-score that captures the sharp asymmetry between admissions that fill safe margin and those that overflow into the envelope; a two-stage decomposition keeps per-step cost compatible with millisecond-scale scheduling. The second, \textbf{BR-H}, generalizes BR-0 with a short, constant lookahead $H$ and a lightweight termination-classifier interface, extending the F-score to a horizon-discounted form. We deploy BalanceRoute on a 144-NPU cluster and evaluate against vLLM baselines on both a proprietary production trace and the public Azure-2024 trace. Across both workloads, BalanceRoute substantially reduces average DP imbalance and improves end-to-end serving throughput.

Hybrid two-dimensional electron gas-superconductor (2DEG-S) structures in a quantized magnetic field offer a promising platform for realizing new topological phases. While recent experiments reveal chiral Andreev edge states, their charge conductance is not integer quantized and is disorder sensitive, raising the question of whether topological protection survives. We argue that it does, but manifests in the spin transport channel. The 2DEG-S system belongs to symmetry class C of the Altland-Zirnbauer classification, which supports an even-integer quantized transverse spin conductivity - the spin quantum Hall effect, so far unobserved experimentally. We demonstrate that 2DEG-S hybrids host topologically protected edge states carrying a spin current with an even-integer quantized spin conductance robust against disorder. Finally, we propose an experimental setup to probe this protection via electrical measurements, establishing a concrete route to detect the class C origin of the chiral Andreev edge states.

An equivariant characteristic quasi-polynomial is a quasi-polynomial in $q$ consisting of the permutation character on the mod $q$ complement of the corresponding Coxeter arrangement. This concept is a refinement of the conventional characteristic quasi-polynomials of root systems. In this paper, we will show equivariant-theoretic refinements of the some properties of characteristic quasi-polynomials of root systems. Furthermore, we will explicitly compute equivariant characteristic quasi-polynomials of all irreducible reduced root systems and discuss the relationship with root systems constructed by the folding of the extended Dynkin diagrams.

A photonic time crystal usually writes a clock into a medium. Here one clock does more than program the medium: it seeds a quasiperiodic temporal order that the nonlinear medium selects for itself. In a guided-wave lattice of nonlinear dipoles, a single-tone pump modulates the polarization sector, while Maxwell--polarization back-action selects two response frequencies whose only resolved low-order relation is the pump-locked sum condition. Their sum phase locks to the pump and the complementary phase winds, producing a photonic discrete time quasicrystal with torus-like phase dynamics and a discrete combination spectrum. Site-resolved measurements show locked-phase coherence across the measured lattice sites over a finite control-parameter window. These results establish a route from externally programmed time-varying media to self-organized temporal order in nonlinear photonic systems.

We investigate trilinear structures as a natural extension of the Hirota bilinear formalism in integrable systems. While bilinear equations are associated with Grassmannian geometry and Plücker relations, trilinear equations suggest a higher algebraic structure involving three-slot couplings of tau functions. Focusing on the stationary axisymmetric Einstein equations, we show that when the Ernst potential is written in a tau-ratio form, the nonlinear equation decomposes into a cubic sector containing all second-derivative terms and a quartic gradient envelope. The cubic sector is identified with a YTSF-type trilinear kernel. We formulate a general trilinear kernel criterion and apply it to the Tomimatsu--Sato solutions. In particular, we demonstrate that the $δ=3$ solution possesses the same trilinear kernel structure as the $δ=2$ case, with a universal normalization up to a constant factor. These results suggest that the trilinear kernel represents a universal structure governing the highest-derivative sector of the Ernst system, providing a new perspective on integrability beyond the bilinear hierarchy.

We present the quantization of all symmetry reductions of the Einstein--Hilbert Lagrangian that correctly reproduce the reduced Einstein's field equations -- i.e., characterized by the infinitesimal group actions obeying the principle of symmetric criticality. These correspond to the spacetime symmetries of spherical/hyperbolic/planar Schwarzschild/Taub--NUT, BI/BII/BIII-metrics, near-horizon extreme Kerr geometry, swirling universe, closed/open/flat FLRW cosmologies, other FLRW-type metrics, and Bianchi type I, II, VIII, and IX spacetimes. We derive the Hamiltonian and the conformal symmetries of the superspace metrics (the conditional symmetries), promote them to operators, and solve the Wheeler--DeWitt equation with and without imposing these symmetries.

The classification of general probabilistic theories (GPTs) whose CHSH value is stable under arbitrary rounds of teleportation and entanglement swapping was obtained in Dmello and Gross work and consists of seven families, indexed by characters of the Klein four-group $K_4$, the cyclic group $\mathbb{Z}_4$, and the dihedral group $D_4$. The question of which of these families admits a realization within standard quantum mechanics was left open. In this work we resolve this question completely. Using elementary representation theory, we prove that exactly two families are quantum-realizable, namely $χ^{K_4}_{1234}$ and $χ^{D_4}_{125}$, while the remaining five admit no quantum realization.