This work investigates the long-time asymptotic behavior of a diffusing passive scalar advected by fluid flow in a straight channel with a periodically varying cross-section. The goal is to derive an asymptotic expansion for the scalar field and estimate the timescale over which this expansion remains valid, thereby generalizing Taylor dispersion theory to periodically modulated channels. By reformulating the eigenvalue problem for the advection-diffusion operator on a unit cell using a Floquet-Bloch-type eigenfunction expansion, we extend the classical Fourier integral of the flat-channel problem to a periodic setting, yielding an integral representation of the scalar field. This representation reveals a slow manifold that governs the algebraically decaying dynamics, while the difference between the scalar field and the slow manifold decays exponentially in time. Building on this, we derive a long-time asymptotic expansion of the scalar field. We show that the validity timescale of the expansion is determined by the real part of the eigenvalues of a modified advection-diffusion operator, which depends solely on the flow and geometry within a single unit cell. This framework offers a rigorous and systematic method for estimating mixing timescales in channels with complex geometries. We show that non-flat channel boundaries tend to increase the timescale, while transverse velocity components tend to decrease it. The approach developed here is broadly applicable and can be extended to derive long-time asymptotics for other systems with periodic coefficients or periodic microstructures.

Photon antibunching in resonance fluorescence - the emission from a single, resonantly driven two-level quantum emitter - is a paradigmatic signature of nonclassical light. Photon entanglement, by contrast, manifests as correlations that can defy any classical description and is typically regarded as a distinct quantum effect. Here, we experimentally extract pairs of narrowband, time-bin-entangled photons from the antibunched resonance fluorescence of a single trapped atom. We verify entanglement via violation of the CHSH Bell inequality and by reconstructing the two-photon density matrix. The observed correlations vanish when the coincidence time window exceeds the antibunching timescale, revealing underlying multimode entanglement in the emitted field. Our results establish a direct link between photon antibunching and photon-photon entanglement, unifying two canonical signatures of nonclassical light.

Ensuring accurate violation detection in power systems is paramount for operational reliability. This paper introduces an enhanced voltage recovery violation index (EVRVI), a comprehensive index designed to quantify fault-induced delayed voltage recovery (FIDVR). EVRVI enhances traditional entropy-based methods by leveraging Empirical Mode Decomposition (EMD) to extract key features from the voltage signal, which are then used to quantify over-voltage (OV) and under-voltage (UV) events. Our simulations on the Nordic system, involving over 245k scenarios, demonstrate EVRVI's superior ability to identify and categorize voltage recovery issues compared to the traditional entropy-based measure. EVRVI not only significantly reduces false negatives in violation detection but also provides a reliable framework for over-voltage detection, making it an invaluable tool for modern power system studies.

This paper presents a novel Short-Term Voltage Stability Index (STVSI), which leverages Lyapunov Exponent-based detection to assess and quantify short-term stability triggered by Over Excitation Limiters (OELs) or undamped oscillations in voltage. The proposed method is measurement-based and decomposes the voltage trajectory into two key components using Empirical Mode Decomposition (EMD): a residual part, which indicates delayed voltage recovery, and an oscillatory part, which captures oscillations. The residual component is critical, as it can detect activation of OELs in synchronous generators or Low Voltage Ride-Through (LVRT) relays in inverter-based resources, potentially leading to instability within the quasisteady-state time frame. Meanwhile, the oscillatory component may indicate either a stable or unstable state in the short term. To accurately assess stability, STVSI employs an entropy-based metric to measure the proximity of the system to instability, with specific indices for short-term voltage stability based on oscillations and recovery. Simulations on the Nordic power system demonstrate that STVSI effectively identifies and categorizes voltage stability issues. Moreover, STVSI not only detects voltage stability conditions but also qualitatively assesses the extent of stability, providing a nuanced measure of stability.

The increasing integration of Distributed Energy Resources (DERs) in distribution networks presents new challenges for voltage regulation and reactive power support. This paper extends a sensitivity-aware reactive power dispatch algorithm tailored to manage smart inverters operating under different control modes, including PQ, PV, and Volt-Var (VV). The proposed approach dynamically optimizes reactive power dispatch and voltage setpoints, enabling effective coordination among distribution systems as a virtual power plant (VPP) to support the transmission network. The algorithm is applied to the IEEE 13-bus and IEEE-123 bus test systems, and its performance is validated by comparing results with OpenDSS simulations across various operating scenarios. Results show that the maximum error in the voltages is less than 0.015 pu.

Determining the steady state of an open quantum system is crucial for characterizing quantum devices and studying various physical phenomena. Often, computing a single steady state is insufficient, and it is necessary to explore its dependence on multiple external parameters. In such cases, calculating the steady state independently for each combination of parameters quickly becomes intractable. Perturbation theory (PT) can mitigate this challenge by expanding steady states around reference parameters, minimizing redundant computations across neighboring parameter values. However, PT has two significant limitations: it relies on the pseudo-inverse -- a numerically costly operation -- and has a limited radius of convergence. In this work, we remove both of these roadblocks. First, we introduce a variational perturbation theory (VPT) and its multipoint generalization that significantly extends the radius of convergence even in the presence of non-analytic effects such as dissipative phase transitions. Then, we develop two numerical strategies that eliminate the need to compute pseudo-inverses. The first relies on a single LU decomposition to efficiently construct the steady state within the convergence region, while the second reformulates VPT as a Krylov space recycling problem and uses preconditioned iterative methods. We benchmark these approaches across various models, demonstrating their broad applicability and significant improvements over standard PT.

It is known that the truncated Brown--Peterson spectra can be equipped with a certain nice algebra structure, by the work of J. Hahn and D. Wilson, and these ring spectra can be viewed as rings of integers of local fields in chromatic homotopy theory. Furthermore, they satisfy both Rognes' redshift conjecture and the Lichtenbaum--Quillen property. For lower-height cases, the K-theory of the truncated polynomial algebras over these ring spectra is well understood through the work of L. Hesselholt, I. Madsen, and others. In this paper, we demonstrate that the Segal conjecture fails for truncated polynomial algebras over higher chromatic local fields, and consequently, the Lichtenbaum--Quillen property fails. However, the weak redshift conjecture remains valid. Additionally, we provide some other examples where Segal conjecture holds.

A group is called decomposable if it can be expressed as a direct product of two proper subgroups, and indecomposable otherwise. This paper explores the decomposability of virtual Artin groups, which were introduced by Bellingeri, Paris, and Thiel as a generalization of classical Artin groups within the framework of virtual braid theory. We establish that for any connected Coxeter graph Γ, the associated virtual Artin group VA[Γ] is indecomposable. Specifically, virtual braid groups are indecomposable. As a consequence of the indecomposability result, we deduce that studying the automorphism group of a virtual Artin group reduces to analyzing the automorphism groups of its irreducible components.

Due to the transformation of the power system, the effective use of flexibility from the distribution system (DS) is becoming crucial for efficient network management. Leveraging this flexibility requires interoperability among stakeholders, including Transmission System Operators (TSOs) and Distribution System Operators (DSOs). However, data privacy concerns among stakeholders present significant challenges for utilizing this flexibility effectively. To address these challenges, we propose a machine learning (ML)-based method in which the technical constraints of the DSs are represented by ML models trained exclusively on non-sensitive data. Using these models, the TSO can solve the optimal power flow (OPF) problem and directly determine the dispatch of flexibility-providing units (FPUs), in our case, distributed generators (DGs), in a single round of communication. To achieve this, we introduce a novel neural network (NN) architecture specifically designed to efficiently represent the feasible region of the DSs, ensuring computational effectiveness. Furthermore, we incorporate various PQ charts rather than idealized ones, demonstrating that the proposed method is adaptable to a wide range of FPU characteristics. To assess the effectiveness of the proposed method, we benchmark it against the standard AC-OPF on multiple DSs with meshed connections and multiple points of common coupling (PCCs) with varying voltage magnitudes. The numerical results indicate that the proposed method achieves performant results while prioritizing data privacy. Additionally, since this method directly determines the dispatch of FPUs, it eliminates the need for an additional disaggregation step. By representing the DSs technical constraints through ML models trained exclusively on non-sensitive data, the transfer of sensitive information between stakeholders is prevented.

Strongly correlated topological phases of matter are central to modern condensed matter physics and quantum information technology but often challenging to probe and control in material systems. The experimental difficulty of accessing these phases has motivated the use of engineered quantum platforms for simulation and manipulation of exotic topological states. Among these, the Laughlin state stands as a cornerstone for topological matter, embodying fractionalization, anyonic excitations, and incompressibility. Although its bosonic analogs have been realized on programmable quantum simulators, a genuine fermionic Laughlin state has yet to be demonstrated on a quantum processor. Here, we realize the ν = 1/3 fermionic Laughlin state on IonQ's Aria-1 trapped-ion quantum computer using an efficient and scalable Hamiltonian variational ansatz with 369 two-qubit gates on a 16-qubit circuit. Employing symmetry-verification error mitigation, we extract key observables that characterize the Laughlin state, including correlation hole and chiral edge modes, with strong agreement to exact diagonalization benchmarks. This work establishes a scalable quantum framework to simulate material-intrinsic topological orders and provides a starting point to explore its dynamics and excitations on digital quantum processors.

We introduce a class of flat currents with fractal properties, called fractional currents, which satisfy a compactness theorem and remain stable under pushforwards by Hölder continuous maps. In top dimension, fractional currents are the currents represented by functions belonging to a fractional Sobolev space. The space of $α$-fractional currents is in duality with a class of cochains, $α$-fractional charges, that extend both Whitney's flat cochains and $α$-Hölder continuous forms. We construct a partially defined wedge product between fractional charges, enabling a generalization of the Young integral to arbitrary dimensions and codimensions. This helps us identify $α$-fractional $m$-currents as metric currents of the snowflaked metric space $(\mathbb{R}^d, \mathrm{d}_{\mathrm{Eucl}}^{(m+α)/(m+1)})$.

Three-body resonances are ubiquitous in quantum few-body physics and are characterized by a finite lifetime before decaying into continuum states of their composing subsystems. In this work we present a theoretical study on the possibility to stabilize three-body resonances to so-called bound states in a continuum: resonances with vanishing width that do not decay. Within a two-channel approach we unveil the underlying mechanism and show how the lifetime can be made infinitely long by a continuous tuning of system parameters. The validity of our theory is illustrated in two different examples: a mass-imbalanced system in one dimension and a system of three identical bosons in three dimensions, relevant to Efimov physics. Crucially, for the latter we find that one of the parameters that can be tuned to achieve a three-body bound state in a continuum is an external magnetic field, a common tunable variable in cold-atom experiments. Due to the generality of this stabilization effect, it is expected to be applicable to a wide range of unstable few-body systems, opening new perspectives for fundamental studies as well as technical applications.

In this paper, we study the dynamics of a two-dimensional viscous fluid evolving through a porous medium or a Hele-Shaw cell, driven by gravity and surface tension. A key feature of this study is that the fluid is confined within a vessel with vertical walls and below a dry region. Consequently, the dynamics of the contact points between the vessel, the fluid and the dry region are inherently coupled with the surface evolution. A similar contact scenario was recently analyzed for more regular viscous flows, modeled by the Stokes [GuoTice2018] and Navier-Stokes [GuoTice2024] equations. Here, we adopt the same framework but use the more singular Darcy's law for modeling the flow. We prove global-in-time a priori estimates for solutions initially close to equilibrium. Taking advantage of the Neumann problem solved by the velocity potential, the analysis is carried out in non-weighted $L^2$-based Sobolev spaces and without imposing restrictions on the contact angles.

The classical Hilbert specialization property is a field-theoretic tool ensuring that polynomial irreducibility over a field is preserved under specialization of some of the variables. We develop an integral counterpart by introducing the notion of {integrally Hilbertian rings}, where specialization takes place inside a ring and irreducibility is required over the ring. A core part shows how new obstacles to irreducibility such as coefficient divisors or fixed divisors can be dealt with over Krull domains, a large class of rings including UFDs, Dedekind domains, etc. As a result, we obtain a general criterion for integral hilbertianity, along with many examples, \hbox{e.g.} all rings of integers of number fields. Polynomial rings over arbitrary domains are other examples. As an application, we prove a polynomial variant of the Schinzel Hypothesis on prime values of polynomials with integer coefficients: if $\mathcal{Z}$ is an integrally Hilbertian ring, the hypothesis becomes a true statement if the ring of integers ${\mathbb Z}$ is replaced by the polynomial ring $\mathcal{Z}[U]$ and ``prime'' by ``irreducible''. This result generalizes previous works and fits in a unified framework for Schinzel-type phenomena that we introduce. We further obtain an additional conclusion that has some noteworthy consequences for the classical Schinzel Hypothesis itself.

Extracting specific items from 10-K reports is challenging due to variations in document formats and item presentation. To improve over traditional rule-based approaches, this study introduces and compares two advanced item segmentation methods: (1) GPT4ItemSeg, using a novel line-ID-based prompting mechanism to utilize a large language model, ChatGPT-4o, for item segmentation, and (2) BERT4ItemSeg, combining a pre-trained language model, BERT, with a Bi-LSTM model in a hierarchical structure to overcome context window constraints. Trained and evaluated on 3,737 annotated 10-K reports, BERT4ItemSeg achieves a macro-F1 of 0.9825, surpassing GPT4ItemSeg (0.9567), conditional random field (0.9818), and rule-based methods (0.9048) for core items (1, 1A, 3, and 7). These approaches enhance item segmentation performance, improving text analytics in accounting and finance. BERT4ItemSeg offers satisfactory item segmentation performance, while GPT4ItemSeg can easily adapt to regulatory changes. Together, they provide an extensible framework for 10-K item segmentation that supports reliable and reproducible results.

Quantum key distribution requires tight and reliable bounds on the secret key rate to ensure robust security. This is particularly so for the regime of finite block sizes, where the optimization of generalized Rényi entropic quantities is known to provide tighter bounds on the key rate. However, such an optimization is often non-trivial, and the non-monotonicity of the key rate in terms of the Rényi parameter demands additional optimization to determine the optimal Rényi parameter as a function of block sizes. In this work, we present a tight analytical bound on the Rényi entropy in terms of the Rényi divergence and derive the analytical gradient of the Rényi divergence. This enables us to generalize existing state-of-the-art numerical frameworks for the optimization of the key rate. With this generalized framework, we show improvements in regimes of high loss and low block sizes, which are particularly relevant for long-distance satellite-based protocols.

Kohnert polynomials and their associated posets are combinatorial objects with deep geometric and representation theoretic connections, generalizing both Schubert polynomials and type A Demazure characters. In this paper, we explore the properties of Kohnert polynomials and their posets indexed by northeast diagrams. We give separate classifications of the bounded, ranked, and multiplicity-free Kohnert posets for northeast diagrams, each of which can be computed in polynomial time with respect to the number of cells in the diagram. As an initial application, we specialize these classifications to simple criteria in the case of lock diagrams.

The Pathwidth Theorem states that if a class of graphs has unbounded pathwidth, then it contains all trees as graph minors. We prove a similar result for dense graphs. More precisely, we give a finite family of tree-like patterns and prove that every graph class of bounded cliquewidth and unbounded linear cliquewidth contains arbitrarily large patterns as induced subgraphs. These patterns mso transduce all trees, and fo transduce subdivisions of all binary trees. In particular, our result provides the missing piece in establishing that the cmso transduction order is total over classes of finite graphs.

We relate group quotients of dg-categories and linear stable $\infty$-categories. Given a group acting on a dg-algebra, we prove that the skew group dg-algebra is Morita equivalent to the dg-categorical homotopy group quotient. We also treat the cases of group actions on dg-categories, with corresponding skew group dg-categories, and of orbit dg-categories. Finally, we describe a version of the skew group algebra in the setting of ring spectra and relate it with $\infty$-categorical group quotients.

The softmax function is a widely used activation function in the output layers of neural networks, responsible for converting raw scores into class probabilities while introducing essential non-linearity. Implementing Softmax efficiently poses challenges on low-end FPGAs due to limited hardware resources and the computational complexity of exponential and division operations. This work evaluates approximate computing techniques for softmax acceleration using Taylor series and interpolation methods using Look-Up Tables (LUTs). These approximations aim to reduce execution time and resource consumption while maintaining acceptable levels of numerical precision. Our findings show that quadratic interpolation with LUTs yields the lowest numerical error. In contrast, Taylor-based approximations offer significantly better performance in terms of execution time and resource efficiency due to their computational simplicity. When applied to real-world deep learning models such as LeNet-5 and MobileNet v2, the first- and second-order Taylor approximations provided substantial trade-offs between accuracy and resource savings, achieving up to 0.2% accuracy degradation and 14% resource reduction compared to exact implementations. These results highlight the effectiveness of approximate Softmax designs on resource-constrained FPGAs and lay the groundwork for their integration into larger models, including large language models (LLMs).

Superconductivity in the bilayer nickelate La$_3$Ni$_2$O$_7$ occurs when the interlayer Ni-O-Ni bond angle becomes straight under pressure, suggesting a strong relationship between the crystal structure and the emergence of superconductivity. In this study, we theoretically propose a way to control the crystal structure of La$_3$Ni$_2$O$_7$ toward the tetragonal symmetry via light irradiation instead of pressure using the idea of nonlinear phononics. Here, resonant optical excitation of an infrared-active (IR) lattice vibration induces a nonlinear Raman-mode displacement through the anharmonic phonon-phonon coupling. We calculate the light-induced phonon dynamics on the anharmonic lattice potential determined by first-principles calculation. We find that the interlayer Ni-O-Ni bond angle gets slightly closer to straight when an appropriate IR mode is selectively excited. Our study suggests that light irradiation can be a promising way for structural control of La$_3$Ni$_2$O$_7$.

We investigate the transition probability rate of a Unruh-DeWitt (UD) detector interacting with massless scalar field for a finite duration of proper time, $T$, of the detector. For a UD detector moving at a uniform acceleration, $a$, we explicitly show that the finite-time transition probability rate can be written as a sum of purely thermal terms, and non-thermal transient terms. While the thermal terms are independent of time, $T$, the non-thermal transient terms depend on $(ΔET)$, $(aT)$, and $(ΔE/a)$, where $ΔE$ is the energy gap of the detector. Particularly, the non-thermal terms are oscillatory with respect to the variable $(ΔET)$, so that they may be averaged out to be insignificant in the limit $ΔET \gg 1$, irrespective of the values of $(aT)$ and $(ΔE/a)$. To quantify the contribution of non-thermal transient terms to the transition probability rate of a uniformly accelerating detector, we introduce a parameter, $\varepsilon_{\rm nt}$, called non-thermal parameter. Demanding the contribution of non-thermal terms in the finite-time transition probability rate to be negligibly small, \ie, $\varepsilon_{\rm nt}=δ\ll1$, we calculate the thermalization time -- the time required for the detector to interact with the field to arrive at the required non-thermality, $\varepsilon_{\rm nt}=δ$, and the detector to be (almost) thermalized with the Unruh bath in its comoving frame. Specifically, for small accelerations, $a\llΔE$, we find the thermalization time, $τ_{\rm th}$, to be $τ_{\rm th} \sim (ΔE)^{-1} \times {\rm e}^{2π|ΔE|/a}/δ$; and for large accelerations, $a\gg ΔE$, we find the thermalization time to be $τ_{\rm th} \sim (ΔE)^{-1}/δ$. We comment on the possibilities of bringing down the exponentially large thermalization time at small accelerations, $a\llΔE$.

Lattice gauge theory is an important framework for studying gauge theories that arise in the Standard Model and condensed matter physics. Yet many systems (or regimes of those systems) are difficult to study using conventional techniques, such as action-based Monte Carlo sampling. In this paper, we demonstrate the use of gauged Gaussian projected entangled pair states as an ansatz for a lattice gauge theory involving dynamical physical matter. We study a $\mathbb{Z}_2$ gauge theory on a two dimensional lattice with a single flavor of fermionic matter on each lattice site. For small systems, our results show agreement with results computed by exactly diagonalizing the Hamiltonian, and demonstrate that the approach is computationally feasible for larger system sizes where exact results are unavailable. This is a further step on the road to studying higher dimensions and other gauge groups with manageable computational costs while avoiding the sign problem.

We apply the Kosambi-Cartan-Chern theory to perform an extensive examination of Jacobi stability of geodesics around rotating black hole solutions to dynamical Chern-Simons gravity, a theory that introduces modifications to General Relativity via a scalar field non-minimally coupled to curvature scalars. We present a comparative study between Jacobi and Liapunov stability, pointing out the advantages of the more geometrical method over the usual Liapunov approach.

Positron emission tomography (PET) enables quantification of dynamic physiological processes through time-resolved imaging. In Rb-82 myocardial perfusion PET, kinetic compartment modeling is used to estimate physiological parameters and derive myocardial blood flow. However, conventional nonlinear least squares (NLLS) estimation is sensitive to model misspecification when not all parameters can be reliably estimated and must instead be fixed or initialized using population averages, which can degrade accuracy. This work develops and evaluates two alternative kinetic analysis approaches for Rb-82 PET: a particle smoother-based Expectation-Maximization method (PSEM) and a convolutional neural network (CNN). Both methods were evaluated using simulated Rb-82 dynamic myocardial perfusion studies and compared against NLLS and a Kalman smoother-based Expectation-Maximization (KEM) algorithm across multiple frame durations and noise levels. Across 2-10 s frames, the CNN achieved the lowest relative errors for all parameters (F: 8.78-4.98%, k3: 26.05-25.50%, k4: 34.34-22.76%), significantly outperforming NLLS, KEM, and PSEM (Holm-adjusted p < 1e-15 at 1.0x noise, 2 s frames), although performance degraded under out-of-distribution input-function conditions. Overall, the CNN provided the most accurate and robust in-distribution kinetic parameter estimates across frame durations. In contrast, PSEM exhibited parameter-dependent behavior, improving k3 estimation while underperforming for F, suggesting that further methodological refinement is needed.

The James Webb Space Telescope (JWST) has enabled the discovery of a significant population of galaxies at z > 10. Our understanding of the astrophysical properties of these galaxies relies on fitting templates developed using models predicting the differences between these first galaxies and lower-redshift counterparts. In this work, tests are performed on several of these high-redshift template sets in order to determine how successful they are at predicting both photometric redshifts and full spectral energy distributions (SEDs). Our work shows that the best templates for photometric redshift estimation differ from the best templates for predicting the full SED. Overall, some templates perform adequately at photometric redshift estimation, while all are generally poor predictors of the full SED. A few objects in particular are poorly fit by all the template sets tested. We conclude that although photometric redshifts can be reliable when given a high enough observational depth and adequate filters, models are not yet able to produce robust astrophysical properties for these ultra-high-redshift galaxies.

We introduce a Floquet quasicrystal that simulates the motion of Bloch electrons in a homogeneous magnetic field in discrete time steps. We admit the hopping to be non-reciprocal which, via a generalized Aubry duality, leads us to push the phase that parametrizes the synthetic dimension off of the real axis. This breaks unitarity, but we show that the model is still ``pseudo-unitary''. We unveil a novel mobility edge between a metallic and an insulating phase that sharply divides the parameter space. Moreover, for the first time, we observe a second transition that appears to be unique to the discrete-time setting. We quantify both phase transitions and relate them to properties of the spectrum. If the hopping is reciprocal either in the lattice direction or the synthetic dimension, the model is $\mathcal{PT}$-symmetric, and the spectrum is confined to the unit circle up to a critical point. At this critical point, $\mathcal{PT}$-symmetry is spontaneously broken and the spectrum leaves the unit circle. This transition is topological and measured by a spectral winding number.

The transport of molecules for chemical reactions is critically important in various cellular biological processes. Despite thermal diffusion being prevalent in many biochemical processes, it is unreliable for any sort of directed transport or preferential accumulation of molecules. In this paper we propose a strategy for directed motion in which the molecules are transported by active carriers via polymerization. This transport is facilitated by chemical/activity gradients which generate an effective drift of the polymers. By marginalizing out the active degrees of freedom of the system, we obtain an effective Fokker-Planck equation for the Rouse modes of such active-passive hybrid polymers. In particular, we solve for the steady state distribution of the center of mass and its mean first passage time to reach an intended destination. We focus on how the arrangement of active units within the polymer affect its steady-state and dynamic behaviour and how they can be optimized to achieve high accumulation or rapid motility.

Many statistical applications, such as the Principal Component Analysis, matrix completion, tensor regression and many others, rely on accurate estimation of leading eigenvectors of a matrix. The Davis-Kahan theorem is known to be instrumental for bounding above the distances between matrices $U$ and $\widehat{U}$ of population eigenvectors and their sample versions. While those distances can be measured in various metrics, the recent developments have shown advantages of evaluation of the deviation in the two-to-infinity norm. The purpose of this paper is to develop a toolbox for derivation of upper bounds for the distances between $U$ and $\widehat{U}$ in the two-to-infinity norm for a variety of possible scenarios. Although this problem has been studied by several authors, the difference between this paper and its predecessors is that the upper bounds are obtained under various sets of assumptions. The upper bounds are initially derived with no or mild probabilistic assumptions on the error, and are subsequently refined, when some generic probabilistic assumptions on the errors hold. The paper also provides rectification of the upper bounds in the cases of heavy-tailed or exponentially fast decaying errors. In addition, the paper suggests alternative methods for evaluation of $\widehat{U}$ and, therefore, enables one to compare the resulting accuracies. As an example of an application of the techniques in the paper, we derive sufficient conditions for perfect clustering in a generic setting, and then employ them in various scenarios.

We define the hypergeometric exponential sum associated to a family of representations of a reductive group over a finite field. We introduce the hypergeometric $\ell$-adic sheaf to describe the hypergeometric exponential sum. Motivated by the definition of the hypergeometric sheaf, we introduce the hypergeometric $\mathcal D$-module, prove it is holonomic and estimate its rank. Using the theory of the Fourier transform for vector bundles over a general base developed by Wang, we show how the hypergeometric $\mathcal D$-module controls the general behavior of the hypergeometric sheaf. We apply our results to the estimation of the hypergeometric exponential sum.