We present RTLScout, an autonomous system that combines LLM-driven agentic design with circuit-level synthesis optimization and arithmetic architecture sweeps. An LLM agent iteratively writes, evaluates, and refines RTL designs using tool calls, guided by quantitative PPA (power, performance, area) feedback from Yosys and OpenROAD. We introduce a multi-run elite pool framework, where the best designs and lessons learned seed subsequent agent runs. The pipeline comprises four complementary phases: agentic code optimization, agentic gate-level rewriting, arithmetic architecture sweeps, and an optional high-effort gate-level refinement pass. On an IEEE-754-compliant 16-bit floating-point multiplier with subnormal support, RTLScout reduces area by 35% and delay by 45% relative to a starting design synthesized in ASAP7 technology. Each phase provides distinct improvements, and high-effort gate-level optimization is most effective as a refinement of already well-optimized designs rather than a substitute for earlier stages. The resulting Pareto front outperforms a commercial-tool reference design on the same technology.

In resource-constrained small-satellite settings, AI inference must operate under tight size, power, and payload budgets, which tend to limit onboard compute capability and data handling. These conditions motivate establishing a clear baseline for quantized AI inference under bounded compute and memory resources. To instantiate this baseline, a representative embedded-vision neural-network workload serves as the reference case. With this motivation, this paper presents a measurement-based characterization of quantized execution for this AI workload on highly constrained embedded platforms (for instance, Cortex-M), grounded as a lower-bound operating point. In this regime, scaling tends to rely on explicit orchestration rather than OS-managed, transparent multicore scheduling, and timing behavior is shaped by instruction efficiency and memory movement. As a result, the characterization provides a structured reference for estimating execution time across orchestrated configurations (e.g., multiple cores and/or devices), treating orchestration and architectural variation as explicit design choices. We report latency metrics alongside data-movement observations, and interpret these measurements in light of ALU/SIMD utilization under quantized arithmetic for the Cortex-M. Finally, we outline how this baseline provides a reference point for positioning the results against more space-typical embedded processor classes (e.g., LEON/NOEL-V).

Obtaining intersection-freeness or global injectivity is important in computer graphics. However, it is challenging, especially for the non-oriented deformation primitives. Most methods often rely on an intersection-free initialization and track the continuous trajectory to keep the legitimacy and cannot be used for the task without such an initialization. For the latter one in 3D space, we introduce a novel second-order differentiable energy defined from the 3D triangle-triangle intersection testing, and a GPU-based inexact Newton optimization route. We show that intersection can be efficiently resolved integrated with our method, requiring no user interaction, history information or a valid initialization.

Geant4 Monte Carlo simulation relies on the Constructive Solid Geometry (CSG) method for complex geometric modeling. This method has low efficiency and a high application threshold. Importing triangular facet formats such as STL/OBJ via CADmesh is a promising alternative, but systematic evaluations of format compatibility, geometric accuracy, and physical simulation deviations are lacking. Construct open-source experimental environment based on Geant4 11.0, CADmesh 1.3.0 and FreeCAD 1.0. We design high and low precision gradient test cases using simple geometric bodies and complex engineering models, and systematically evaluate the import success rate, facet loss rate, volume error, and particle transport dose deviation for STL and OBJ formats.The results show a 100% import success rate for both formats; the volume error rate is <= 0.018% for high-precision models and <= 0.288% for low-precision models. The two formats share the same vertex facet data structure. This study designs a general adaptive interface. The interface reduces the number of parsing code lines by about 70% and maintains geometric accuracy.Furthermore, the tetrahedral mesh loading takes 3.1 times longer than tessellated solids, but the simulation time can be reduced from 15194.3 s to 77.28 s.

As large language models (LLMs) are increasingly trained on scraped web corpora without authorisation, content owners require forensic methods to prove that their documents were included in a model's training set. We propose \textbf{SIGIL} (\textbf{S}ubtle \textbf{I}njection for \textbf{G}round-truth \textbf{I}nference of \textbf{L}LM training data), a framework that embeds imperceptible \emph{canary sequences} into protected text and code such that any LLM trained on those documents exhibits statistically detectable behavioural signatures when probed with targeted queries. SIGIL defines five canary strategies -- lexical-rare, lexical-phrase, syntactic, semantic, and code-pattern -- and a \emph{Membership Inference Score} (MIS) grounded in the Neyman-Pearson hypothesis testing framework with formal false-positive rate (FPR) control. Simulator parameters are calibrated against the empirical membership inference literature, yielding realistic heterogeneous results across $36{,}000$ trials: overall AUC $= 0.892$, rising from $0.831$ at $0.1\%$ injection to $0.947$ at $10\%$. Detection rates range from $33\%$ to $78\%$ across model-size and injection-rate conditions. Code Pattern canaries achieve the highest AUC ($0.903$, Cohen's $d = 1.84$); Syntactic Structure the lowest ($0.875$, $d = 1.63$). All four experimental factors -- injection rate, model size, canary strategy, and mixture ratio -- have significant independent effects on MIS ($p < 0.001$). SIGIL maintains AUC $> 0.86$ even under $100\%$ paraphrasing ($\text{AUC} = 0.864$), confirming robustness through semantic leakage that survives surface-level rewriting.

The advancement of malware poses obstacles for cybersecurity, necessitating the development of advanced detection techniques. This paper proposes an approach to enhance malware detection through the use of a generative artificial intelligence model. Specifically, variational autoencoders (VAEs) are used with the random forest, XGBoost and sequential model machine learning classifiers. Generated synthetic malware samples are used to address the critical issue of data scarcity for new or less common malware types. This approach can be used to augment datasets to improve classifier robustness. The proposed methodology uses VAEs to create high-quality diverse synthetic datasets that closely mimic real-world malware data. The effectiveness of these augmented datasets is evaluated by comparing the performance of the machine learning classifiers when they are trained with the original data and when they are trained with the synthetic data-augmented datasets. The results demonstrate a notable improvement in the accuracy, precision, recall and F1-scores of the classifiers, when they are trained using the augmented datasets. The enhanced performance for detecting various malware classes shows the potential of this approach to facilitate adaptation to evolving malware threats effectively. This work demonstrates the utility of generative AI for cybersecurity. It also provides a foundation for future research aimed at developing more resilient and adaptive malware detection systems.

Lattice structures are of growing importance in additive manufacturing, where complex internal geometries are increasingly required for lightweight, high surface-to-volume ratios, multifunctionality, and other superior mechanical properties. Conventional lattice modeling methods typically represent struts with simple primitives, such as cylinders or cones, limiting geometric diversity and the design space. Although recent efforts have increased strut-shape complexity to address this issue, they often do so at the expense of computational efficiency and modeling robustness. As a result, achieving both rich geometric expressiveness and efficient computation remains a challenging problem. In this paper, we present an implicit modeling method that expands the design and optimization space of lattice structures while preserving the modeling robustness and efficiency of implicit representations. In our method, each strut is defined as a convolution surface over a skeletal graph, and its profile shape is controlled by a cubic Hermite curve. By exploiting the polynomial structure of both the convolution kernel and the cubic Hermite curve-controlled profile, we derive analytical expressions for efficient field evaluation, avoiding costly and unstable numerical computation. Four case studies have been conducted to validate the proposed method in terms of profile shape diversity, graded lattice modeling, as well as slicing robustness and efficiency.

Springer fibers are subvarieties of the flag variety parameterized by nilpotent matrices. They are central objects of study in geometry representation theory. This paper focuses on two-row Springer fibers, those corresponding to nilpotent matrices with two Jordan blocks. Irreducible components of two-row Springer fibers are in bijection with two-row standard Young tableaux and also with noncrossing matchings. Inspired by the combinatorial commutative algebra of matrix Schubert varieties, we define a polynomial ideal for each noncrossing matching and prove that these ideals define the corresponding components of the Springer fiber. Our proofs leverage geometric descriptions of Springer fibers established by Fung, Stroppel--Webster, Fresse, and Goldwasser--Nadeem--Sun--Tymoczko. Using these ideals to compute examples, we give two conjectural formulas for the cohomology class of each component of a two-row Springer fiber. We apply commutative algebra techniques to prove these conjectures for a specific family of two-row tableaux.

In this paper we construct numerically a new class of unstable self-similar solutions of the incompressible Navier-Stokes equations in $\mathbb{R}^3$. Our solutions are axially symmetric and homogeneous of degree $-1$ at $\infty$, and are unstable in the sense that the linearization around these solutions contains unstable modes. Solutions of this type have been discovered numerically by Guillod and Šverák and Hou, Wang, and Yang, and have applications to proving non-uniqueness results. The main novelty in this paper is that we discover the existence of such solutions in the space of axially symmetric swirl-free (ASSF) vector fields. These approximate solutions are defined on all of $\mathbb R^3$ and achieve global pointwise residuals of order $10^{-10}$. We discuss the numerical construction of these solutions in detail, as well as their relevance to the problem of non-uniqueness of solutions of the incompressible Navier-Stokes equations in 3D, in the space of ASSF solutions.

We derive asymptotic multivariate normal limits and explicit non-asymptotic normal approximation bounds for group sequential quasi-maximum likelihood estimators under possible model misspecification and within-group dependence. The bounds, obtained using Stein's method, have known constants and apply to a class of dependent-data estimating problems in which the likelihood used for estimation may differ from the true data-generating mechanism. We compute the limiting covariance structure and finite-sample bound explicitly for a Poisson generalized linear mixed model with random group effects and illustrate the results using data from an epilepsy clinical trial.

We compare two cluster algebras related to a positive element $\mathtt{b}$ in the braid group of finite $ADE$ type. One is the localized bosonic extension ${\widetilde{\mathbb{A}}}_\mathbb{C}(\mathtt{b})$ equipped with an initial seed arising from an admissible chain $\mathfrak{C}$ of $i$-boxes, which is deeply connected to monoidal categorification. The other is the coordinate ring $\mathbb{C}[X({\underlineΔ} {\boldsymbol{i}})]$ of the braid variety $X({\underlineΔ} {\boldsymbol{i}})$ equipped with an initial seed arising from a Demazure weave $\mathfrak{W}$, where ${\boldsymbol{i}}$ and ${\underlineΔ}$ are expression sequences of $\mathtt{b}$ and the half twist $Δ$, respectively. We explicitly construct a Demazure weave $\mathfrak{W}_{\underlineΔ}(\mathfrak{C})$ for each admissible chain $\mathfrak{C}$ associated with ${\boldsymbol{i}}$, and prove that there exists an algebra isomorphism $φ_{\boldsymbol{i}}\colon {\widetilde{\mathbb{A}}}_\mathbb{C}(\mathtt{b})\to\mathfrak{C}[X({\underlineΔ} {\boldsymbol{i}})]$ which is compatible with the two seeds arising from $\mathfrak{C}$ and $\mathfrak{W}_{\underlineΔ}(\mathfrak{C})$. Moreover, the isomorphism $φ_{\boldsymbol{i}}$ sends the PBW vectors ${\overline{\mathsf{p}}}_{\boldsymbol{i},k} \in {\widetilde{\mathbb{A}}}_\mathbb{C}(\mathtt{b})$ to the coordinates $z_k \in \mathfrak{C}[X({\underlineΔ} {\boldsymbol{i}})]$ indexed by the letters of ${\boldsymbol{i}}$. As applications, we investigate a connection between Demazure weaves and signed words via the $i$-boxes and interpret the isomorphism $φ_{\boldsymbol{i}}$ from the viewpoint of monoidal categorification using Hernandez--Leclerc categories.

The singular cohomology ring of a matroid is an algebraic invariant which generalizes the Chow ring of a matroid. We study combinatorial and Lefschetz properties of the singular cohomology ring of a uniform matroid. Combinatorially, we construct an explicit basis for the singular cohomology ring in terms of Koszul homology. From this basis we derive multiple formulas for the Hodge numbers of the cohomology ring that recover and extend known formulas for the Chow polynomial of a uniform matroid. We also use this basis to show that the singular cohomology ring of a uniform matroid satisfies the "quasi-projective strong Lefschetz property" -- a slight weakening of the Hard Lefschetz property found in the Chow ring of a matroid.

The classical Erdős-Kac theorem gives a central limit theorem for the number of prime divisors of a random integer. We prove an analog for the number of ramified primes in a random $G$-extension of a number field when $G$ is abelian. This builds on previous work of Lemke Oliver and Thorne in the cases $G = S_d$ ($2 \le d \le 5$), and provides the first examples where local ramification events at distinct primes are not independent. We develop probability results that can be used "out of the box" to prove Erdős-Kac theorems for sequences of ideals in a number field, subject to Tauberian hypotheses involving finite sums of Euler products.

We introduce and analyze a mixed Virtual Element Method for the $p$-Laplace equation in a non-Hilbertian setting, covering the full range $p \in (1, \infty)$. The discrete framework combines standard mixed Virtual Element spaces with a novel non-linear stabilization term designed to mimic the power-law structure of the continuous operator. We establish discrete inf-sup stability under non-Hilbertian norms and rigorously prove the continuity and coercivity of the discrete form. This guarantees the well-posedness of the problem and allows us to derive a priori error estimates for the primal variable and the flux. A set of numerical tests supports the theoretical derivations.

We show that, for every $\varepsilon>0$ and all sufficiently large $k$, any spanning subgraph of the $k$th power of a cycle with minimum degree at least $(1+\varepsilon)k$ contains a Hamilton cycle. This asymptotically settles a conjecture of Espuny Díaz, Lichev, and Wesolek.

We construct locally minimizing $(1,2)$-clusters whose exterior interfaces are asymptotic to various prescribed singular area-minimizing cones. For $n+1 \leq 7$, Bronsard & Novack characterized all minimizing $(1,2)$-clusters as standard lenses, whose exterior interface is planar. For $n+1 \in [8,2700]$, the authors together with Bronsard showed the existence of a locally minimizing $(1,2)$-cluster whose exterior interface blows down to some (unknown, possibly non-unique) singular area-minimizing hypercone. For $n+1=8$, this was shown independently by Novaga, Paolini & Tortorelli. Here we develop a refined construction using the Hardt-Simon foliation that realizes prescribed cones. For a singular area-minimizing hypercone $C$ that has an isolated singularity or is cylindrical, we show that if $C$ satisfies an explicit energy bound, then there is a locally minimizing $(1,2)$-cluster whose exterior interface is asymptotic to $C$ with quantitative rates. In fact, if $C$ is an area minimizing Lawson cone satisfying this energy bound, we produce a countably infinite family of distinct locally minimizing clusters asymptotic to $C$, distinguished by their prescribed asymptotic decay to leading order. We verify this energy bound for the generalized Simons cones $C_{k,k}$ in every even ambient dimension $n+1 = 2k+2\geq 8$, and for the cylindrical cone $C_{3,3}\times\mathbb{R}$ in $\mathbb{R}^9$, where $C_{3,3}$ is the Simons cone, therefore answering the cone realization problem in these cases. This in particular removes the upper bound of 2700 on the ambient dimension when $n+1$ is even in our preceding work.

Spielman-Srivastava spectral sparsification preserves Laplacian quadratic forms to within (1 +/- epsilon), but does not directly control the adjacency spectral radius lambda_1, which governs the NIMFA epidemic threshold and arises in spectral clustering. We prove |lambda_1(A_H) - lambda_1(A_G)| <= epsilon(2 Delta - lambda_1) deterministically, with a sharp epsilon*lambda_1 bound for reweighting sparsifiers via Perron-Frobenius monotonicity. Under effective-resistance sampling, Matrix Bernstein gives O(epsilon Delta / sqrt(c)) with high probability. Combining eigenvector delocalization with resolvent perturbation theory, we establish that for graphs with delocalized Perron eigenvectors and spectral gap = Omega(Delta), the distortion is O(epsilon Delta sqrt(log n) / sqrt(n)) + O(epsilon^2 Delta^2 / delta_gap), with corollaries for Erdos-Renyi graphs, regular expanders, and stochastic block models. Lower bounds establish tightness for regular graphs.

We study optimal design of $\varepsilon$-locally differentially private mechanisms for binary hypothesis testing. Each observation is drawn from one of two known distributions $P_0,P_1$ on a finite alphabet of size $k$, privatized by a mechanism $Q$, and then used to infer which distribution generated the data. We measure testing utility using an $f$-divergence, including total variation, KL, and hockey-stick divergences, between the two induced output distributions. Previous work established structural properties of optimal mechanisms, but only yielded exponential-time algorithms. We prove a sharp structure: for every $\varepsilon$ and every $f$-divergence objective, after sorting the alphabet by likelihood ratio, there exists an optimal mechanism that partitions the sorted alphabet into contiguous blocks and applies randomized response to the block label. We call this class Sort-Partition-Randomize (SPR). This characterization yields an exact dynamic program that computes an optimal mechanism in $O(k^3)$ time, and more generally in $O(\ell k^2)$ time with an $\ell$-output budget. Our results make it possible to efficiently compute and characterize the exact optimum across the full privacy range, beyond asymptotic privacy regimes.

We consider local optimization of a weak distance between two compact surfaces embedded in the three-dimensional Euclidean space on its special Euclidean group. Identifying those objects with the associated surface measures, their discrepancy is quantified in terms of the inhomogeneous Sobolev norm of negative order via the Plancherel theorem. Then, applying an isometry to one surface corresponds to pushforwarding its surface measure and the distance can be regarded as a function on the Lie group. For appropriate exponents of the Sobolev norm, the second power of the function acquires sufficient differentiability that allows to search for its local minima in a derivative-based framework, and the gradient of the objective function has a favorable structure for efficient implementations using the nonuniform fast Fourier transform. In numerical experiments, we observe convergence of the SR1 trust-region method applied to a few root-finding problems and discuss its connection to a more geometric quantity.

Given a 3-manifold $M$ with a network of line and point defects in its boundary, we define the skein module of this configuration, generalizing the well-studied case of 3-manifolds which only admit point defects in the boundary. We prove that when all defects are labelled by semisimple data, our skein module is isomorphic to the state space of $\partial M$ in the defect version of the Reshetikhin-Turaev TQFT constructed by Carqueville-Runkel-Schaumann. Our defect skein modules follow naturally by globalizing the graphical calculus of module categories and functors thereof, and generalize the possible defect data considered in the defect TQFT beyond the semisimple case.

We study the spectral properties of Laplacians on the Schreier graphs $Γ_n$ of the basilica group, the iterated monodromy group of the polynomial $z^2 - 1$, which is an important example in the theory of self-similar, amenable but not elementarily amenable, automaton groups. Building heavily on results by Brzoska, Jarvis, George, Rogers and Teplyaev about certain subgraphs of the basilica graphs, we develop a new recursive framework for computing the characteristic polynomials of these Laplacians. Our analysis reveals a simple underlying dynamical system and proves approximation results for the Kesten-von Neumann-Serre (KNS) spectral measure.

With $ρ= \frac{13-\sqrt{69}}{10} \approx 0.47$, it was recently established that there exist multiplicative Sidon sets (sets without any non-trivial solutions to $ab = cd$) in $\{1, 2, \ldots, n\}$ with maximal gap size $\ll_{\varepsilon} n^{ρ+ \varepsilon}$. Here we improve upon this result and show that one can take $ρ= \frac{10}{33} \approx 0.303$ instead.

We investigate the long-time behaviour of a one-dimensional compressible viscous fluid with general capillarity and density dependent viscosity, driven by a stochastic additive noise. In particular, we prove the existence of invariant measures by applying the Krylov-Bogoliubov method in a setting where the dynamics is supported on a non-complete phase space. This analysis is further enhanced by the derivation of a refined stability result determining the continuous dependence with respect to the initial data. The present paper exhibits some properties and results for Korteweg fluids which are not known in absence of the capillarity tensor. In particular, we prove that the Markov semigroup associated with strong solutions is Feller and we can consider ranges of the adiabatic and viscosity exponents $γ$ and $α$ larger than those available in the current ergodic literature for compressible fluids. Also the interplay between the choice of the physical domain and the use of a damping term is discussed.

For a Fano manifold $X$, Gamma conjecture II aims to use $\mathcal{D}_{\rm{coh}}^b(X)$ to describe the asymptotic behavior of its Dubrovin connection via $\widehatΓ$-integral structure. It was proposed by Galkin, Golyshev and Iritani, and can be regarded as a quantitative refinement of Dubrovin's conjecture on Fano manifolds with semisimple big quantum cohomology. As a step toward Gamma conjecture II, we define the Gamma-I property at points satisfying the (SR) condition, arising from the original Gamma conjecture I. We prove that the property holds globally in the following sense: if it holds at one such point, then it holds throughout the connected component of the (SR)-region containing that point. Based on this global Gamma-I property, we establish a strategy-type theorem relating Gamma conjecture II to the Gamma-I property at a possibly non-semisimple point, together with an analysis of small quantum cohomology. We further apply this theorem to prove Gamma conjecture II for del Pezzo surfaces; the proof combines Iritani's Galois action with addtional elementary operations on exceptional collections, and its most technically involved step consists in verifying the required global Gamma-I property.

We study local well-posedness for a class of stochastic reaction-diffusion equations driven by multiplicative, possibly colored, noise. The interaction between rough stochastic forcing and polynomial nonlinearities naturally leads to solutions with low spatial regularity, making the treatment of the nonlinear terms delicate. Our main contribution is a general local existence and uniqueness theory for SPDEs with rough noise and highly irregular initial data. The framework also yields new results in standard noise regimes, including trace-class noise and space-time white noise. We identify the critical initial-data spaces for a wide range of nonlinearities, and we establish instantaneous parabolic regularization, general blow-up criteria, and sufficient conditions for positivity preservation. We apply the abstract theory to several prototypical models, including the stochastic Allen-Cahn, Burgers, Fisher-KPP, and coupled Gray-Scott equations. Finally, in the one-dimensional space-time white-noise setting, we combine our local theory with existing global a priori results in a highly singular regime.

We develop a nine-compartment nonlinear epidemic model incorporating two co-circulating viral strains (ancestral I1 and the Alpha variant B.1.1.7 I2, which is 43-90% more transmissible, c2=1.5), a super-spreader subpopulation, partial vaccine-induced immunity with waning, and explicit hospitalization dynamics with differentiated mortality. Transmission and vaccination rates are treated as time-varying control inputs and identified from Italian COVID-19 data (January-May 2021) via a Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) control-node parameterization, reducing calibration to a fourteen-variable Sequential Quadratic Programming (SQP) problem with monotonicity and box constraints. A parametric bootstrap (n=1000) quantifies parameter uncertainty. The calibrated model achieves R^2=0.966 for active hospitalizations, R^2=0.987 for cumulative fatalities, and R^2=0.999 for cumulative vaccinations. Well-posedness, the basic reproduction number in closed form, and local and global stability of the disease-free equilibrium are established analytically. An L-infinity approximation error bound shows that the PCHIP control-node parameterization converges to the true time-varying parameters at rate O(h^2) as the node spacing vanishes. Local identifiability and a noise stability bound are established via the Fisher information matrix. A sufficient threshold condition proves epidemic decay under time-varying suppression whenever the effective reproduction number remains persistently below one. Sensitivity analyses consistently rank hospital throughput parameters above the transmission rate, providing a mathematical basis for the observation that reactive containment measures cannot prevent a hospitalization peak already driven by the pre-existing latent viral load.

We consider the problem of \textit{rate loss} - a strict penalty suffered in achievable rates due to the lack of channel state information at the receiver (Rx) of a classical-quantum (CQ) channel. First, we identify non-commutative CQ channels and analytically prove a rate loss. Building on this, we next prove that coset-code-based strategies can strictly outperform conventional unstructured IID-code-based strategies for non-commutative 3-user CQ broadcast channels.

Every (resp. a generic) complex $n \times n$ matrix can be expressed as a product of $2n+5$ (resp. $\lfloor n/2 \rfloor +1$) Toeplitz matrices. Motivated by this result, it is natural to ask the following question: what is the minimum number of Toeplitz matrices required to factor a given matrix? We generalize this question from Toeplitz structure to more general structures. In this paper, we introduce the notion of structured matrix factorization length when the set of matrices with a given structure is an affine variety $X \subseteq \mathbb{C}^{n \times n}$. Then we introduce the $r$-th $X$-factorization variety, defined as the Zariski closure of the set of products of $r$ matrices in $X$, and use it to define the border structured matrix factorization length. In particular, we study the cases in which $X$ is the affine variety of Toeplitz, Hankel, bidiagonal, tridiagonal, skew-symmetric or companion matrices. We calculate the dimension of the $X$-factorization varieties for all these cases, and discuss how numerical algebraic geometry can be used to obtain computational evidence for the degrees of $X$-factorization varieties with an example. In addition, we propose methods for deriving lower and upper bounds for (border) structured matrix factorization length. For lower bounds, we develop a method based on displacement rank, which can also be used to obtain some defining equations of the $r$-th $X$-factorization variety; for upper bounds, we suggest an approach using alternating minimization.

Icart's encoding is a classical deterministic map from finite fields to elliptic curves and a basic ingredient in early hash-to-curve constructions. We determine the exact one-output distribution of this map in the exceptional \(j=0\) case. More precisely, for \[ E_{0,b}:Y^2=X^3+b,\ q\equiv2\pmod3, \] we compute the complete fibre distribution of \(f_{0,b}:\mathbb F_q\to E_{0,b}(\mathbb F_q)\). This gives closed formulae for the image size, total variation distance from uniform, collision probability, power sums, entropy measures and basic batch statistics. We also derive the exact second moment of all nontrivial character sums of the output distribution. Via the Weil pairing, this becomes an exact energy formula for pairing-character tests on the supersingular \(j=0\) family whose odd prime order subgroups have embedding degree two.

In a recent article, Cheng, Jiang and Yuan gave an affirmative answer to Klee's convex bodies problem of Banach spaces in the sense of strict convexity and Gâteaux smoothness. In this paper, we continue to study this problem in strong senses, such as local uniform convexity, uniform convexity, Fréchet smoothness and uniform smoothness. As a result, we show (1) Every convex body in a Banach space $X$ is approximated by locally uniformly convex bodies with respect to the Hausdorff metric if and only if $X$ admits an equivalent locally uniformly convex norm; (2) Every convex body in $X$ can be approximated by Fréchet smooth convex bodies if $X$ admits an equivalent norm so that its dual norm is locally uniformly convex on $X^*$; 3. Every convex body in $X$ can be approximated by both locally uniformly convex and Fréchet smooth convex bodies if $X$ is reflexive; 4. If $X$ is separable, then every convex body in $X$ can be approximated by both locally uniformly convex and Fréchet smooth convex bodies if and only if $X$ is an Asplund space; (5) the following statements are equivalent: A. $X$ is super reflexive; B. Every convex body in $X$ can be uniformly approximated by uniformly convex bodies; C. Every convex body in $X$ can be uniformly approximated by uniformly smooth convex bodies; D. Every convex body in $X$ can be uniformly approximated by both uniformly convex and uniformly smooth convex bodies.