LLM agents have made rapid progress on software engineering and ML research tasks, but these advances often assume access to a working runnable environment. For research artifacts released alongside published papers, setting up such an environment from a fresh machine remains a major bottleneck. Existing environment setup benchmarks do not cover the full scope of research artifact deployment, which involves multi-language toolchains, system-level dependencies beyond containers (e.g. GPU/CUDA and kernel configurations), and legacy artifact compatibility. We introduce DeployBench, a multi-domain benchmark of 51 research-artifact deployment tasks spanning AI/ML, computer systems, and scientific computing, covering all these dimensions. Each task is verified by a hidden pipeline that executes the paper's designated experiment and checks its outputs. Evaluating four state-of-the-art LLMs with OpenHands yields pass-rates from 7.8% - 51.0% . Failures are dominated by a completion-judgment problem: 97 of 154 are agent-terminated self-stops, where the agent's pre-finish checks validate a different or weaker target than the paper-specific task requires. DeployBench highlights the gap between current agents and autonomous deployment, and offers a realistic testbed for scientific research agents.

GREX-PLUS (Galaxy Reionization EXplorer and PLanetary Universe Spectrometer) is a mission candidate for a JAXA strategic L-class mission to be launched in the 2030s. Its primary science goals are two-fold: galaxy formation and evolution, and planetary system formation and evolution. The GREX-PLUS spacecraft will carry a telescope with a 1 m primary mirror aperture cooled down to 50 K. The two science instruments will be onboard: a wide-field camera in the 2--8 $μ$m wavelength band and a high-resolution spectrometer with a wavelength resolution of 30,000 in the 10--18 $μ$m band. The GREX-PLUS wide-field camera aims to detect the first generation of galaxies at redshift $z>15$. The GREX-PLUS high-resolution spectrometer aims to identify the location of the water ``snowline'' in protoplanetary disks. Both instruments will provide unique datasets for a broad range of scientific topics, including galaxy mass assembly, the origin of supermassive blackholes, infrared background radiation, molecular spectroscopy in the interstellar medium, transit spectroscopy of exoplanet atmospheres, planetary atmospheres in the Solar System, and so on. This document is the second version of a collection of scientific themes that can be achieved with GREX-PLUS. Each section in Chapters~2 and 3 is based on presentations at several GREX-PLUS Science Workshops.

We investigate the evolution of circuit complexity for a quantum scalar field in a non-singular bouncing cosmological background. Unlike previous perturbative approaches, we utilize the Lewis-Riesenfeld invariant method to derive an exact non-perturbative framework for the complexity evolution across the bounce. We demonstrate that the complexity remains finite at the bounce, dominated by the squeezing of the quantum state due to spacetime contraction. Crucially, we find that the post-bounce growth of complexity exhibits a strong correlation with cosmological particle production. Our analysis reveals that the "chirping" contribution to complexity acts as a geometric memory of the transition, effectively quantifying the information cost of particle creation in non-trivial spacetime geometries.

In open nonequilibrium systems, the thermodynamic entropy of a subsystem is not generally a Lyapunov function. Even during relaxation toward equilibrium, it may decrease temporarily because of exchanges with external reservoirs. This raises a basic question: what thermodynamic quantity, if any, organizes irreversible relaxation in an open system? We address this question using an explicit open-piston model coupled to both a pressure reservoir and a heat bath. The reversible sector is formulated as a Nambu rotational flow generated by the extended energy and the subsystem entropy, while the irreversible sector is written as a gradient flow generated by a dissipation potential $S_{NB}$. In the adiabatic reversible limit, the Nambu bracket produces the oscillatory piston motion on the intersection of conserved level surfaces. After coupling to a heat bath and adding friction, the subsystem entropy $S$ can exhibit nonmonotonic oscillations, whereas $S_{NB}=S-H_{1}/T_{b}$ increases monotonically under the proposed positive-semidefinite dissipative structure. We show that this monotonicity is not a consequence of identifying $S_{NB}$ with thermodynamic entropy. Rather, it follows from two geometric conditions: the reversible Nambu flow preserves $S_{NB}$, and the irreversible dynamics can be written as a positive-semidefinite gradient flow generated by $S_{NB}$. The open-piston model therefore provides a minimal macroscopic realization in which thermodynamic entropy, dissipation potential, reversible temporal order, and irreversible relaxation can be separated explicitly.

Hairpin vortices are fundamental structures within turbulent boundary layers, playing a crucial role in energy dissipation, mixing, and momentum transport. However, accurately extracting these structures remains challenging due to their irregular shapes, varying scales, and entanglement with surrounding vortical structures. This paper presents a novel framework for the extraction of hairpin vortices from turbulent boundary layers. The method begins by identifying vortical regions and decomposing them into smaller segments using merge tree based segmentation. A novel bottom up rejoining approach is then introduced to group candidate segments according to the geometric and physical characteristics of hairpin vortices, resulting in regions that encompass complete hairpin vortex structures. These regions are subsequently refined and validated through skeleton analysis to detect the characteristic hairpin shape and are further confirmed using additional scalar based criteria. Finally, smooth enclosing surfaces are generated for effective visualization. To enable quantitative evaluation, reference hairpin vortices are extracted from several flow datasets and used as ground truth. Compared with existing approaches, the proposed method eliminates manual parameter tuning, reduces under and over segmentation, and significantly improves both accuracy and computational efficiency. Demonstrations on multiple turbulent flow cases show that the method is robust and effective for hairpin vortex extraction under varying boundary layer conditions.

Large language models (LLMs) demonstrate increasing proficiency on competitive programming benchmarks, yet technical reports predominantly publish aggregate pass rates, obscuring domain-specific vulnerabilities. We present a systematic empirical study of LLM failure patterns using a balanced taxonomy of 315 Codeforces problems across seven algorithm categories and three difficulty tiers. We evaluate GPT-4o and Claude Sonnet 4.6 under strict execution-based conditions, controlling for temperature (T = 0.2). To isolate the impact of reasoning frameworks on algorithmic correctness, we conduct an ablation study comparing direct zero-shot generation against zero-shot Chain-of-Thought (CoT). Our findings reveal a severe divergence from standard NLP benchmarks: forcing CoT aggressively penalizes GPT-4o, dropping its pass rate from 46.0% to 36.8% and exacerbating a critical weakness in Greedy logic. Conversely, while Claude maintains a higher logical baseline (63.5% under CoT), the expanded text generation severely degrades its markdown instruction adherence, causing its Compile Errors to more than triple (from 9 to 31, a 244% increase). Furthermore, failure-mode analysis indicates that Wrong Answer (WA) is the dominant verdict for both models--accounting for over 90% of GPT-4o's and roughly 70% of Claude's unaccepted solutions. These findings empirically demonstrate that standard prompt engineering techniques fail to bridge the algorithmic reasoning gap in competitive programming environments.

We propose a controlled momentum-space dual of the Fock--Lorentz (FL) transformations and use it to derive a deformed relativistic mass shell. Restricting the FL conformal factor to the cosmological-frame world line $\vx=0$, the invariant relation takes the form $(E^{2}-\vp^{2}c^{2})(1+ct/R)^{2}=m_{0}^{2}c^{4}$, which is equivalent to the standard special-relativistic dispersion law with a time-dependent apparent mass $\mapp(t)=m_{0}/(1+ct/R)$. Canonical quantization then yields Klein--Gordon (KG) and Dirac equations containing a slowly varying mass scale. We show explicitly that squaring the Dirac equation reproduces the KG operator, modulo first-order corrections proportional to $\dot\mapp$ that are suppressed by the ratio of the Compton wavelength to the FL scale. The construction is not presented as a unique covariant phase-space theory; rather, it is a world-line ansatz designed to isolate the spectral consequences of the FL conformal factor. As applications, we study the one-dimensional KG and Dirac oscillators. In the adiabatic regime, governed by the small parameter $ε=c/(Rω)\ll1$, closed-form instantaneous spectra are obtained. The Dirac-oscillator calculation is carried out in component form and then reduced to the physical spinor spectrum, thereby avoiding the double counting of the upper and lower component ladders. Dimensionless plots illustrate the apparent-mass drift, the induced spectral evolution, and the domain of adiabatic validity. For cosmological values of $R$, non-adiabatic corrections are entirely negligible; in the formal limit $t\to\infty$ the apparent mass tends to zero and, for fixed quantum number, the instantaneous levels collapse toward $E=0$.

For $1 \leq a \leq 2$, we say Proposition $(1+a)$ holds if every sufficiently large even integer $N$ can be written as $$N = p + rq, \quad r \leq q^{a-1},$$ where $r$ is either $1$ or prime, and $p,q$ are primes. Thus Proposition $(1+1)$ is essentially the binary Goldbach Conjecture, and Proposition $(1+2)$ is Chen's theorem. We prove unconditionally that Proposition $(1+1.9)$ is true. Assuming the Elliott--Halberstam Conjecture, the exponent $1.9$ can be improved to $1.4$. Analogously, Proposition $(1-a)$ is formulated for the Twin Prime Conjecture. Unconditionally, we prove Proposition $(1-1.75)$, and under the Elliott--Halberstam Conjecture, Proposition $(1-1.4)$. For six decades, a substantial theoretical divide has persisted between Propositions $(1+2)$ and $(1+1)$, and likewise between Propositions $(1-2)$ and $(1-1)$. By constructing new weighted sieves and adopting new analytic tools, this paper establishes a connecting pathway between them and achieves breakthroughs in this line of research.

In this note we provide corrections to Theorem 5.4 of the paper ``Isomorphism classes of Drinfeld modules over finite fields'', arXiv:2209.15033. The main theorems of this paper, Theorem A and B in its introduction, are valid as stated; in the proof of Theorem B the argument needs to be modified by replacing the erroneous Theorem 5.4 by the theorem of this note.

The Analytic Hierarchy Process (AHP) and other classic Multi-Criteria Decision Making (MCDM) techniques excel when decision makers can provide consistent pair wise judgments. Real world problems, however, often involve inconsistent, n-wise, or non-linear preference structures that render traditional methods inadequate. The α-Discounting MCDM (α-D MCDM) extends AHP by embedding a global discounting parameter α that transforms an inconsistent system of preference equations into a solvable algebraic system.

A multiset is a 'set' in which elements may occur multiple times. These structures are ideal for expressing the outcome of an election, for instance of the form 60 'yes' and 40 'no'. Moreover, multisets are a useful datatype in vote counting algorithms. This will be illustrated in three different forms of vote counting, known as: 'instant-runoff', 'De Borda', and 'single transferrable vote'. The relevant abstract properties of multisets are: (1) they form a (free) commutative monoid, and (2) they form a functor, and (3) also a monad. This paper illustrates how such categorical properties can be put to good use in deriving and expressing election outcomes. The emphasis is not on the (elementary) category theory involved, but on its application in voting systems.

The sixth-generation wireless networks are envisioned to deliver ubiquitous, seamless, and intelligent connectivity that reaches far beyond the limits of terrestrial infrastructure. Non-terrestrial networks (NTNs) are central to this vision, extending coverage to underserved regions, remote terrain, and disaster zones that terrestrial deployment cannot economically reach. However, NTN architecture faces numerous limitations: severe path loss over long distances, long propagation delays, large and time-varying Doppler shifts, limited visibility windows, and tight on-board energy and computing budgets. Semantic communication (SemCom), which conveys the meaning of data rather than its raw bit-level representation, is unusually well matched to these conditions: extreme compression rate for task-oriented eases bandwidth scarcity, deep joint source-channel coding prevents the cliff effect due to low signal-to-noise ratio, and generative-AI reconstructs content from sparse cues that survive rain-faded or blocked links. This observation, that each NTN limitation maps onto a SemCom property that addresses it, motivates our survey. We first walk through the NTN limitations one by one, pairing each with the SemCom design choices that complement it, then we organize the literature along three axes: the NTN platform, the semantic methodology, and the supporting techniques, and follow this with platform-by-platform deep dives on satellite-centric, UAV/HAPS-centric, and integrated SAGIN systems. The survey concludes by identifying open research problems, gaps in existing standards, and future directions, including the application of foundation models, energy-aware scheduling, and quantum-assisted SemCom for deep space communication.

These notes provide a rigorous and accessible introduction to measure and integration theory, with emphasis on the conceptual transition from the Riemann integral to the Lebesgue integral and the role played by limiting processes in modern analysis. The manuscript develops the basic theory of measurable sets, measurable functions, measures on $σ$-algebras, Lebesgue integration, convergence theorems, and $L^p$ spaces. Particular attention is devoted to the interaction between integration and convergence, as well as to the limitations of the Riemann integral that motivate the development of measure theory. The exposition seeks to balance mathematical rigor with pedagogical clarity through detailed proofs, examples, exercises, and supplementary projects. These notes are intended primarily for undergraduate students in mathematics and related areas encountering measure theory for the first time, although they may also serve as a reference for introductory graduate courses in analysis.

We develop an analytic umbral approach to Bessel moments, using them as a concrete testbed justifying the passage from formal indicial umbral calculus to Mellin--Barnes umbral transmutation theory. [...] While the formal procedure reproduces the correct results in suitable convergence chambers, it may lead to non-admissible hypergeometric expansions at physically relevant parameter values. The cubic moment provides the basic example [...] We show that this obstruction is removed by replacing the purely formal expansion with an analytic umbral transmutation. In this setting, exponential umbral pairings are interpreted through Mellin--Barnes integrals, and Ramanujan's Master Theorem acts as an inverse selection principle for the spectral ground state, or clock, associated with a given Bessel product. The factorisation \(J_0^3=J_0J_0^2\) produces two distinct clocks and reduces the cubic full-line moment to a one-dimensional Barnes integral, equivalently to a Meijer \(G\)-function. This gives the classical value of the cubic Bessel moment and clarifies why the divergent Appell realisation is only a local representation of a globally meaningful umbral identity. The same mechanism is then applied to scaled cubic products and to the fourth Bessel moment. [...] The fifth moment marks the first genuinely higher-rank case: the natural umbral grouping leads to a bivariate Barnes transmutation rather than to an ordinary Meijer \(G\)-function. Finally, we discuss real fractional powers \(J_0^α\), \(α>2\), showing that the same interpretation persists beyond integer moments. [...] The resulting picture identifies Bessel moments as values of effective umbral transmutations and separates the global analytic meaning of the umbral representation from the local convergence properties of its hypergeometric residue expansions.

We propose a systematic way to construct trigonometric-like functions beyond the classical sine--cosine pair by factorising rational umbral operators. The guiding idea is simple: the usual trigonometric functions may be viewed as cyclic components arising from a finite factorisation, and the same principle can be extended to an $n$-fold decomposition of rational umbral expressions. For each integer $n\geq 2$, the construction produces $n$ functions which play the role of higher-order trigonometric components: their sum reconstructs the corresponding umbral function, while the individual components isolate the different cyclic sectors of its expansion. The construction is developed first in the formal umbral setting. The quadratic case $n=2$ gives the Gaussian trigonometric functions, in which the cosine-like component is a Gaussian and the sine-like component is its natural umbral companion. The cubic case $n=3$ yields a three-component cyclic system and shows how the same idea extends beyond the usual even--odd decomposition. These examples suggest that trigonometric factorisation is not restricted to ordinary rotations, but belongs to a broader cyclic principle in umbral calculus. We then reinterpret the same formal identities through the recently developed analytic umbral framework. In this second step, the cyclic components are realised by Mellin--Barnes pairings, and the root-of-unity decomposition is related to the splitting of the corresponding spectral kernel. This analytic formulation provides contour representations, local expansions, and sectorial asymptotics for the functions obtained formally. Finally, we indicate how the same cyclic kernels act on Fourier transforms. The resulting framework presents higher-order umbral trigonometric functions as natural cyclic components of factorised rational or exponential umbral operators.

We develop an effective framework for inflation in a constrained scalar--Gauss--Bonnet theory motivated by a restricted sector of $f(R,\mathcal{G})$ gravity. Using unified parametrizations of the Hubble expansion rate and the Gauss--Bonnet coupling function within a generalized slow-roll formalism, we derive analytical expressions for the scalar spectral index $n_s$ and tensor-to-scalar ratio $r$, and study their dependence on the model parameters. We show that the Hubble parametrization mainly controls the scalar sector through the slow-roll parameter $ε_1$, while the Gauss--Bonnet-induced contribution $ε_4$ can significantly affect the scalar tilt and strongly suppress primordial tensor modes, naturally leading to very small values of $r$. A representative benchmark solution yields $n_s \simeq 0.958$ and $r \simeq 2.7 \times 10^{-4}$, marginally compatible with current Planck, ACT, and BICEP/Keck constraints. We further investigate the scalar perturbation structure of the exactly constrained theory, where the Lagrange-multiplier constraint forces the lapse perturbation to vanish and, together with the gravitational momentum constraint, implies $\dot{\mathcal{R}}=0$, eliminating the propagating scalar degree of freedom at linear order. This exact result clarifies that the generalized slow-roll treatment should be interpreted as an effective softly constrained description. We also discuss perturbative stability conditions, including the positivity of the relevant kinetic coefficients and propagation speeds. Our results demonstrate that the effective constrained scalar--Gauss--Bonnet framework provides a flexible and observationally viable description of inflation while clarifying the distinction between the exact constrained limit and its effective slow-roll realization.

The anthyphairetic process leads from a pair (a,b) of coprime positive integers to the pair (1,1) by successive subtractions of the smaller number from the bigger one. This process, which is a slow version of Euclid's algorithm applied to the pair (a,b), corresponds naturally to the process of successive blowups leading to the minimal embedded resolution of the plane curve defined by y^a - x^b = 0. This blowup process may be represented graphically by a special two-dimensional simplicial complex called a lotus. This allows to localize the various numbers appearing either during the anthyphairetic process or during the Euclidean algorithm at precise positions inside the lotus. In this introductory article, I recall first the construction of this lotus starting from the sequence of quotients generated by the Euclidean algorithm. I present then an alternative way of constructing it directly from the sequence of pairs of coprime integers generated by the anthyphairetic process, using what I call anthyphairetic rectangles. I conclude by explaining how to reconstruct from a lotus the corresponding sequence of pairs of coprime integers. This is a simple illustration of the way lotuses may serve as computational architectures.

Three lines of second-order phase transitions between the normal phase and two distinct superconducting phases meet at a single point on the phase diagram of UTe$_2$. Contrary to common belief, there are no thermodynamic constraints for such triple points. The phase diagram is interpreted within Landau theory in terms of two superconducting order parameters with different gauge symmetries. Such an interpretation is unique under the assumption of spatial uniformity.

FENE-type conformation-tensor models impose a finite-extensibility constraint that is absent from Oldroyd--B flow: the conformation tensor must satisfy $\CC\succ0$ and $\tr\CC<L^2$. Positive definiteness alone is therefore insufficient, since a numerical state can remain positive while crossing the singular trace barrier. Even a trace-preserving logarithmic parametrization is not enough by itself: high-order reconstruction can remain inside the finite-extensibility domain while injecting artificial FENE entropy. We develop and analyze a barrier-preserving entropy-compatible discretization for FENE-P type flows with polymer center-of-mass molecular diffusion and for trace-singular FENE-family closures with the same entropy structure. The method combines a trace-barrier free energy, a finite-extensibility logarithmic parametrization, a least-damping entropy-compatible barrier-log reconstruction, molecular diffusion paired with the barrier entropy variable, compatible quadrature for polymeric work, and a scaled FENE stress variable for the small-Weissenberg limit. For admissible discrete states we prove finite-extensibility preservation at entropy quadrature points, existence and bisection computability of the maximal entropy-admissible reconstruction parameter, a fully discrete free-energy inequality with relaxation and molecular-diffusion barrier dissipation, a quantitative AP stress closure, and a fixed-discretization Newtonian limit. A conditional relative-entropy estimate is derived on compact subsets of the finite-extensibility domain. Numerical diagnostics verify barrier preservation, entropy-compatible reconstruction, energy decay, AP closure, coupled velocity--pressure--stress accuracy, and high-Weissenberg robustness near the trace constraint.

The Friedmann--Lemaître--Robertson--Walker (FLRW) spacetime, which was first proposed by Friedmann (1922--1924) and Lemaître (1927) and subsequently developed by Robertson and Walker (1935), is an isotropic and homogeneous cosmological model of the universe. This paper addresses a significant gap in the differential geometry literature by providing a comprehensive examination of the curvature properties of the FLRW spacetime. It is demonstrated that the FLRW spacetime satisfies the curvature condition R \cdot R - Q(S, R)=L_C Q(g, C) alongside several pseudosymmetric-type conditions related to the conformal and conharmonic curvature tensors. Furthermore, the Tachibana tensors Q(g,C) and Q(S, C) are found to exhibit a linear dependence on the tensor $(C \cdot R + R \cdot C)$. Additionally, the spacetime is shown to be a 2-quasi-Einstein manifold, generalized Roter type and Ein(3). The Ricci tensor is shown to be neither cyclic parallel nor of Codazzi type, yet it satisfies several compatibility requirements concerning the R, C, P, K and W curvature tensors. A thorough analysis of Ricci solitons and curvature inheritance properties reveals that the spacetime admits almost Ricci soliton and $η$-Ricci Yamabe soliton structures with respect to the non-Killing soliton vector fields $\frac{\partial}{\partial t}$ and $\frac{\partial}{\partial r}$. Moreover, the spacetime admits generalized curvature inheritance symmetry properties for the Riemann curvature tensor, as well as for the Weyl conformal, concircular, and conharmonic curvature tensors with respect to the coordinate vector field $\frac{\partial}{\partial t}$ and the gradient of $t$. Later, a comparison of the FLRW and Lemaître--Tolman--Bondi (LTB) spacetimes is provided in terms of various curvature-related geometric properties and physical characteristics. Finally, a noteworthy conclusion of the entire study is presented.

We analyze electron dynamics in a plane electromagnetic wave using the Landau-Lifshitz equation with a quantum radiation reaction correction modeled by a Gaunt factor. In this geometry, the quantum parameter $χ$ depends solely on the lightfront momentum, allowing the modified equation of motion to retain the integrable structure of the classical problem. We derive an exact solution for the energy evolution and the four-velocity, which reduces to the known classical result in the appropriate limit. The results provide an analytical and deterministic description of semiclassical radiation reaction in plane-wave fields.

xCPS is an xAct tensor algebra package for symbolic computations within the covariant phase space formalism of field theories. From a generic Lagrangian, xCPS automates the derivation of equations of motion and symplectic currents. It systematically determines whether an infinitesimal transformation in the space of fields is a Noether symmetry and computes the associated Noether charge. Additionally, xCPS can in many cases determine whether a tensorial expression is a divergence and, if so, find its divergence potential. By implementing vertical exterior calculus through a graded, supercommutative wedge product and vertical operators, the package enables efficient computations in gauge theories and higher-derivative models of gravity, including the derivation of thermodynamic quantities like Wald's entropy. xCPS is open-source under the GPL license and available at https://github.com/juanmargalef.

The interaction of charged particles with an intense laser pulse gives rise to a number of characteristic spectral features of emitted radiation, including the generation of harmonics, spectral broadening due to the phase-dependent ponderomotive red shift, and the emergence of intricate sub-harmonic structures. These effects are accumulated over the course of the interaction with the electromagnetic field and are therefore inherently nonlocal in nature. For a deeper understanding of strong-field quantum electrodynamics (SFQED) processes and their practical applications, it is desirable to employ tools that enable simultaneous analysis in the time and energy domains. In time-frequency analysis, such tools are provided by joint distributions (JDs). In this work, we demonstrate how a JD can be devised within the SFQED framework. Specifically, we focus on constructing a non-negative JD, which allows for a clear probabilistic interpretation. We study the properties of the proposed distribution and test its utility by applying it to the nonlinear Compton scattering in complex laser pulse configurations with carrier-envelope phase and variable polarization.

The recent arXiv posting arXiv:2605.20443 by Lohmiller and Slotine attempts to address the omission of the Bohm quantum potential in their proposed exact equivalence between classical action and the Schrödinger equation. They introduce a position-dependent time transformation to argue that the spatial derivatives of the probability density amplitude vanish for the Feynman kernel. A rigorous mathematical examination of this transformation reveals a violation of the multivariable chain rule. The spatial derivatives do not vanish in the physical reference frame. The mathematical framework presented by the authors remains identical to the well-established semiclassical Van Vleck propagator, which is exact exclusively for quadratic potentials.

Boolean networks are widely used to model gene regulatory systems, where ensembles of Boolean functions serve as null models for assessing structural and dynamical properties. A common approach generates canalizing and nested canalizing functions by sampling their defining parameters uniformly at random. However, because multiple parameterizations can represent the same Boolean function, this induces a non-uniform distribution over distinct functions and systematically biases random ensembles. Here, we develop efficient algorithms for uniform sampling of Boolean functions with prescribed exact or minimal canalizing depth that correct this bias. Our approach combines dynamic programming for sampling canalizing layer structures with rejection-based methods and is implemented in BoolForge. We show that the sampling scheme substantially affects commonly studied function-level metrics. Under traditional parameter-uniform sampling, the expected average sensitivity of nested canalizing functions equals one independent of the number of variables. In contrast, under function-uniform sampling, the expected sensitivity increases with system size and numerically approaches approximately 1.183. This discrepancy arises from an exponential suppression of high-sensitivity functions under parameter-based sampling. These differences propagate to Boolean network models, affecting conclusions about robustness, stability, attractor structure, and baseline dynamical expectations. Revisiting 122 published Boolean gene regulatory network models, we show that function-uniform null models reveal a substantially stronger enrichment of low-sensitivity canalizing architectures than previously inferred. Our results demonstrate that widely used null models systematically underestimate baseline sensitivity and can therefore distort assessments of the stabilizing role of canalization in biological networks.

We study a stochastic functional differential equation (SFDE) with memory driven by a fractional Brownian motion (fBm) with Hurst parameter H>1/2. An Euler-type numerical scheme is proposed and analyzed under suitable regularity conditions on the drift and diffusion coefficients using tools from fractional calculus. We prove the convergence of the scheme and derive the corresponding rate in terms of the discretization step. Numerical simulations illustrate the theoretical results and confirm the accuracy of the proposed method.

In this paper, we establish the Hineva inequality for submanifolds of a real space form endowed with a semi-symmetric non-metric connection. We derive a sharp lower bound for the Ricci curvature of the submanifold in terms of the mean curvature vector and the squared norm of the second fundamental form. We apply this inequality to derive the Hineva inequality for several classes of submanifolds.

We study the Hecke algebra $\mathcal{H}_G(F)$ for $G = \mathrm{GL}_n$ and $n>2$ where $F$ is a non-Archimedean local field of characteristic zero. We show that for $G = \mathrm{GL}_n$ and $n>2$ and any two such fields $E$ and $F$, there is a Morita equivalence $\mathcal{H}_G(E) \sim \mathcal{H}_G(F)$, by using the Bernstein decomposition of the Hecke algebra and by determining the intertwining algebras that yield the Bernstein blocks up to Morita equivalence.

Let $\mathbf{P}$ denote the Riesz projection on the unit circle $\mathbb{T}$ and suppose that $1<p<\infty$. We present a simple proof of the bound $\|\mathbf{P}f\|_p \leq \max(p,q) \|f\|_p$, where $f$ is in $L^p(\mathbb{T})$ and $p^{-1}+q^{-1}=1$. Our proof is a variation of a classical argument due to M. Riesz demonstrating that the Hilbert transform is bounded on $L^p(\mathbb{T})$.

In the present paper, we study the factorization properties of the generalized minors \( Δ(w_{\leq k}Λ,\, v_{\leq k}Λ), \) introduced by Fomin--Zelevinsky, in the coordinate rings of Kac--Moody open Richardson varieties. By analyzing their simple factors in the monoidal category $\mathscr{C}_{w,v}$, we connect the cluster algebra structure of these varieties with the categorical framework developed by Kashiwara--Kim--Oh--Park. In particular, we prove that cluster monomials in the coordinate ring of a Kac--Moody open Richardson variety correspond to isomorphism classes of simple modules in $\mathscr{C}_{w,v}$. As a consequence, we show that the Grothendieck ring $K(\mathscr{C}_{w,v})$ contains the cluster algebra structure on the coordinate ring constructed by Bao--Ye. In finite type, we further prove that Leclerc's seeds coincide with Ménard's seeds for open Richardson varieties, and that the category $\widetilde{\mathscr{C}}_{w,v}$ provides a monoidal categorification of the cluster structure on the open Richardson variety.