Papers

We introduce a new geometric construction of Prym–Tyurin varieties of odd prime exponent $p$ arising from intermediate coverings of $\mathbb Z_p\times\mathbb Z_p$–Galois covers of smooth curves. We determine precisely when these coverings give rise to Prym–Tyurin varieties, namely when the base curve is elliptic and in the case of isotropic étale covers over curves of genus $2$. The explicit nature of the construction makes it possible to analyze the geometry of the associated Prym–Tyurin varieties and to prove that the corresponding Prym–Tyurin maps, indexed by the choice of a generating vector, are generically finite onto their images.

We develop homological Real trace methods, an approach to understanding the continuous mod two Bredon homology of Real topological periodic homology and Real topological negative cyclic homology, generalizing prior work of Bruner and Rognes. We use this new approach to do several computations, including the continuous mod two Bredon homology of the Real topological negative cyclic homology of Real bordism.

Bounded ratios of products of minors of positive definite matrices have a long history, starting with Hadamard's inequality in 1893. It states that for every positive semidefinite matrix $A$ $$ \det A \le A_{11} \cdots A_{nn}. $$ This inequality was subsequently generalized by Fisher and then further by Koteljanskii. The latter states that for every positive semidefinite matrix $A$ and any index sets $\alpha_1, \alpha_2 \subseteq \{1,\ldots, n\}$ one has $$ \det A[\alpha_1 \cup \alpha_2] \det A[\alpha_1 \cap \alpha_2] \le \det A[\alpha_1] \det A[\alpha_2], $$ where $A[\alpha]$ denotes the principal submatrix determined by the indexes in $\alpha$. In a manuscript published only on the arXiv in 2008, Hall and Johnson made three conjectures about ratios of products of principal minors of $4\times4$ positive definite matrices, denoted by $R_i$, $i=1,2,3$, see (2) and (3). They hypothesized that the supremum of $R_1$ was $27/16$, while the supremum of the other two ratios was $1$. Such ratios are called absolutely bounded. The conjecture for $R_1$ was affirmed in [17] and it is the only known bounded determinantal ratio with supremum bigger than one. The goal of this paper is to affirm the conjecture for $R_2$ and $R_3$. It is known that the upper bound for the ratios $R_i$, $i=1,2,3$, does not follow from repeated applications of Koteljanskii's inequality. In addition, Hall and Johnson showed that $R_i$ is bounded above by $4$, for $i=1,2,3$.

In this paper, we propose a series of theoretical results for prescribed-time control to rigorously prove that the closed-loop system will not reach the equilibrium point before the prescribed time. Thus, it is convenient to prove that the settling time is exactly equal to the prescribed time instant. In other words, conservative estimation of the settling time, which often accompanies classical finite-/fixed-time or prescribed-time control results, is theoretically avoided in this paper. Specifically, we introduce a local Lipschitz condition, a local homogeneous linear-growth condition, and an Osgood condition in sequence. Both reduction to absurdity and the Lyapunov method are applied to demonstrate that the system cannot converge to zero before the prescribed time instant. Finally, the proposed framework is applied to prescribed-time controllers based on adaptive backstepping, parametric Lyapunov equations, and distributed leader-following consensus, demonstrating the generality of the proposed results.

In many industrial applications one aims for the design of efficient chemical reactions. This is often done using experimental analysis of the reactors in combination with numerical twins. However, the simultaneous modeling of flow, mixing and chemical reaction remains a complicated task because the computational costs are still significant and oftentimes prohibitive. Here, we model and analyze mixing processes of chemical substances by a transfer operator method using the Lagrangian trajectories of passive particles. In practice, we represent the spatial distributions of the chemical species concentrations by density vectors. These are evolved by means of a stochastic transition matrix, which is obtained as a numerical approximation of a transfer operator. In a further step, we model the reaction of the fluids by specific update rules of the density vectors. The method is applicable for closed and open example systems and different chemical processes such as competitive consecutive reactions. We demonstrate that the dynamic processes are reflected well by this method. Furthermore, we evaluate the interdependence of the reaction fronts with Lagrangian coherent structures which we define using set-oriented methods that also work well on three dimensional data.

We introduce and study a one-parameter family of probability measures $\{\mu_t\}_{t>0}$ defined by the Boolean additive convolution $\mu_t=\beta_t\uplus U_t$, where $\beta_t$ is the symmetric Bernoulli distribution with variance $t$ and $U_t$ is the uniform distribution on $[-t/2,t/2]$. Exploiting the linearization property of the Boolean energy transform, we derive an explicit formula for the reciprocal Cauchy transform and obtain its Laurent expansion. As applications, we compute the Boolean cumulants and moment sequence of $\mu_t$, establish asymptotic estimates for the moments, and investigate analytic properties including support criteria and a differential equation satisfied by the Boolean energy transform. We further analyze scaling limits, prove stability under suitable normalizations, and show that the family does not form a Boolean convolution semigroup.

Persistent Homology (PH) is an important tool in Topological Data Analysis for point clouds, but can be prohibitively expensive for large datasets. A practical approximation is to compute PH on a smaller subset of representative points, known as landmarks. Existing landmark selection methods mainly address challenges such as outlier contamination rather than restoring the topological features of the full dataset. To select topologically relevant landmarks, we propose a landmark selection method based on Hodge Laplacian. It assigns each point a topological relevance score based on its harmonic participation and selects points with high scores as landmarks. By quantifying each point's participation in global homological structures, our method restores the PH of the original point cloud more accurately than geometric baselines and a method built upon local PH on both synthetic and real-world datasets.

A Strassen-type algorithm has many realizations with the same exact product and multiplication count yet different fp8 error because basis changes reshape coefficient geometry, posing the question of which to run. No current account settles this: classical stability controls worst-case $\ell_1$ growth, not the expected-error magnitude, and the Dumas–Pernet–Sedoglavic optimizer could only be called probably optimal, its global optimality unproved. To settle this, we attach to each realization a coefficient functional $\Phi$, a scalar summary of its coefficient geometry, which we minimize over the change-of-basis orbit. This Kempf–Ness problem on a Hadamard manifold lets us certify the global $\Phi$ optimum rather than merely search for it: an exact moment-map zero fixes $\Phi_{\min} = 200/9$, and de Groote's classification extends that optimality to every exact real rank-7 $2\times2$ decomposition. Every exact real rank-7 realization therefore has a $\Phi$-predicted RMS constant at least $5/3$ times that of the cubic algorithm, at fixed noise coefficient. We then introduce an explicit block-scaled e4m3 model in which $\Phi$ is the leading-order coefficient of relative expected mean-squared error, and we test the resulting $\Phi$-predicted ordering against realized fp8 error. Ordering and re-basing experiments support that prediction within tested fused block-scaled regimes, and on real matmul tiles from four architecture families the $\Phi$-optimal realization falls in the fp8 low-error region. Across two $\sim$70B models on real deep_gemm kernels, the same realization removes 10 to 55% of classic Strassen's excess NLL over the clean model. Algorithm realization thus becomes a mathematically certified design problem rather than a tuning choice: an independent low-precision axis with a global $\Phi$ optimum and measured fp8 relevance.

We comment on the optimally-transported generalized method of moments (OTGMM) of Schennach and Starck (2026) and give counterexamples to Theorems 2-6. In the small-error analysis, the assumptions do not ensure consistency or asymptotic normality (Theorems 2-3). In the large-error analysis, Theorem 4 states the equivalence between the OTGMM estimator and the GMM estimator with modified moments, and Theorems 5-6 assert consistency and asymptotic normality. Through counterexamples, we show that the equivalence in Theorem 4 fails and a component of the Lagrange multiplier has different limits under the two estimators (Theorems 5-6). We also point out that, under misspecification, the population value selected by OTGMM depends on the metric. We give sufficient conditions and establish results following the original motivation for OTGMM. Our corrections to Theorems 4-6 change their assumptions or interpretation more substantially than those to Theorems 2 and 3. Furthermore, we correct Assumption 16 and the test's null distribution.

We study the functorial and categorical structure of the canonical rigid direct-system representation of finite local-unit-aligned totally ordered monoids. The local-unit map $\tau$ induces a canonical $\tau$-multiplication-coherent decomposition into component monoids, and the associated representation reconstructs the original ordered monoid from a finite chain-indexed rigid direct system whose proper transition maps are unit-constant. The present paper identifies the morphism classes for which this representation is categorical. First, we prove an equivalence between finite local-unit-aligned totally ordered monoids with strict block morphisms and rigid direct systems with directed-order-compatible system morphisms. Second, we prove an intrinsic equivalence for $\tau$-compatible homomorphisms, that is, isotone unital homomorphisms commuting with the local-unit map. In this second setting, distinct positive idempotents and hence distinct canonical components may collapse to a single target component; on the direct-system side this is represented by non-injective isotone index maps together with component maps satisfying the corresponding collapse and absorption axioms. Thus the canonical rigid direct-system representation is functorial both for strict block morphisms and for intrinsic $\tau$-compatible homomorphisms.

Let $C^1[0,1]$ be the complex linear space of all continuously differentiable complex-valued functions $f$ on the unit interval $[0,1]$ with respect to the norm $ \|f\|_{\sigma} = |f(0)| +\|f'\|_\infty$. Let $P_1, P_2: C^1[0,1] \rightarrow C^1[0,1]$ be distinct, nonzero idempotent maps, which are not necessarily linear, such that $P_1P_2 = P_2P_1 = 0$ and $P_1+P_2 = I$, where $I$ denotes the identity operator. It is proved that $\lambda_1P_1 + \lambda_2P_2$ is an isometry on $C^1[0,1]$, for some distinct unit modulus complex numbers $\lambda_1, \lambda_2$, if and only if either $\lambda_1 + \lambda_2 = 0$, or $\lambda_1P_1 + \lambda_2P_2$ is an isometry for all such complex numbers $\lambda_1, \lambda_2$. In the former case, the collection $\{P_1, P_2\}$ is called a family of generalized bi-circular idempotents; in the latter case, it is called a family of bi-circular idempotents. The structure of isometric reflection on $C^1[0,1]$, that is, an isometry $T$ such that $T^2 = I$, is characterized, and its relationship with the above class of idempotents maps is also discussed.

Decision-making in complex systems often involves dealing with imprecise or uncertain information, frequently represented using fuzzy sets, particularly Triangular Fuzzy Numbers (TFNs). A crucial aspect of many fuzzy methods is the quantification of distance between TFNs. Many distance measures assume that all values are in the same scale, requiring a preliminary normalization stage when applied to heterogeneous attributes with different scales or units. This paper proposes the Triangular Fuzzy Rescaling Distance (d_{TR}), a metric designed to address this challenge. The d_{TR} uniquely integrates Linear Rescaling (LRE) directly into the distance calculation, ensuring normalization during the comparison of fuzzy numbers. We formally prove that d_{TR} satisfies the properties of a metric, including non-negativity, identity, symmetry, and the triangle inequality. Furthermore, we demonstrate that d_{TR} is bounded, scale-invariant, and origin-invariant. These properties, combined with a weighting vector for prioritizing dimensions, make d_{TR} suitable for applications involving heterogeneous fuzzy data, such as the construction of synthetic indicators, distance-based machine learning algorithms or multicriteria-decision aiding.

We present a novel microlocal analysis of beam hardening artifacts arising in cone-beam X-ray CT, where the set of X-ray sources is restricted to a 1D smooth curve $ \gamma\subset \mathbb{R}^3$. We show that, when the CT data is modeled in the standard way using the Beer Lambert law, the exponential term in the model creates singularities in the data that are not present in the linear X-ray transform. We assume that the attenuation coefficient $\mu$ (the reconstruction target) has a jump discontinuity across a surface $\mathcal{S}\subset\mathbb{R}^3$, and is smooth otherwise. We prove that these additional singularities in the data occur when the X-ray beam is tangent to $\mathcal{S}$ at two points simultaneously. To investigate how the singularities in the data propagate to the reconstruction space, we apply Filtered Back Projection (FBP) type reconstruction. We prove that the artifacts due to beam hardening are locally of conormal type, and lie on a 2-D surface which is the union of all double tangent rays which intersect $\gamma$. The artifacts are notably weaker than the reconstructed jumps of $\mu$ (i.e., the desired singularities), and we quantify this using the order of the corresponding conormal distributions. While our primary theory applies to the regions of $\mathcal{S}$ that are smooth, we also extend our theory to non-smooth $\mathcal{S}$ with "ridges." These arise where $\mathcal{S}$ is locally the intersection of two smooth surface patches meeting transversely along a curve (e.g., the edge of a cuboid). In addition, we present simulated reconstructions of metal objects in circular cone-beam CT to validate our theory.

We generalize known results on normalized symmetric coordinates and their dual differential operators, established for the symmetric group, to the complex monomial reflection group $G(r,n):=\mu_r\wr S_n$ (with $G(1,n)=S_n$). The central tool, proved in detail, is a transfer principle: the substitution $y_i=x_i^r$ identifies the invariant ring of $G(r,n)$ with that of $S_n$ and transports, term by term, the corresponding operators and coordinates into explicit rational objects in the original variables $x_i$. From this we deduce the $G(r,n)$-analogues of the known results for $S_n$ existence and uniqueness of the dual coordinates $U_k$, and a Weyl algebra structure localized at the discriminant of $G(r,n)$ with complete and self-contained proofs for the points that do not follow directly from the transfer (Leibniz rule, main theorem). We then treat the total diagonal: its preimage splits into $r^{n-1}$ lines permuted transitively by the group, and there, unlike the transferred operators $\Delta_i$, the raw derivatives $\partial_{x_i}$ exhibit a phenomenon specific to $r\ge2$ that we describe completely via a Fa di Bruno-type structure formula. Finally, we give a closed formula for the constants of this structure formula (via Bell polynomials), completely resolve the degeneracy at an isolated point $x_i=0$ (the operator $\Delta_i$ extends holomorphically there, with $\Delta_i\phi=\partial_i^r\phi/r!$), and deduce from this a partial analogue of the description of the tangent space to the GIT quotient $\CC^n/G(r,n)$; the case of several coordinates vanishing simultaneously remains open and is precisely delineated. Full proofs of all new results are given in detail.

Physics-informed and hybrid models integrate data and physics, and read learned coefficients as physical. Yet a small prediction error guarantees neither that the postulated operator is right, nor that the design identifies the coefficient, nor that training reached its minimum. On one parabolic benchmark where a sensor aggregates several diffusion rates and the truth is known, a correct operator recovers the coefficient to machine precision. A wrong one leaves it $30\%$ off while the in-domain error is 1.36 times the noise, and drifts $27\%$ across windows. At $7\%$ noise the error falls below the noise; the coefficient stays $30\%$ off. Without the truth, a post-training coefficient diagnostic questions the trained model with three classical statistics, without oracle; the first two need no refit. Does it explain the data? Can the design resolve the named physics? Does the coefficient depend on the window? The specification test keeps its pre-registered false-alarm ceiling and rejects every replicate of both wrong operators tested; a network frozen at its minimum returns the same verdicts. Where the design is blind, a wrong operator confined to the unseen rate leaves the test mute; the design reading flags it beforehand. A network stopped short of its minimum mimics a wrong operator; an analytic second fit separates the two. The test has a notch against a missing reaction term, miscalibrates under correlated noise and loses power beyond $7\%$ noise. Each warning names its remedy (operator, experiment or training); silence certifies nothing.

We construct a $4$-parameter family of properly embedded genus-zero surfaces with at most two boundary components in a Riemannian $3$-ball. The construction is a free boundary analog of the canonical $5$-parameter family of surfaces on the $3$-sphere by Marques-Neves in the proof of Willmore conjecture. In the Euclidean unit ball, this family realizes the critical catenoid as a Simon-Smith min-max limit. As applications, we give an upper bound of Almgren-Pitts $4$-width $\omega_4(\mathbb{B}^3)$ by the area of the critical catenoid. Moreover, we prove the existence of at least three free boundary minimal annuli in any compact $3$-manifold with nonnegative Ricci curvature and strictly convex boundary.

With the progress of the semiconductor industry toward increasingly complex compute devices and tighter process tolerances, advanced process control has become crucial. This work explores a novel framework to infer the underlying dynamics of semiconductor processes, directly from raw equipment log-file time-series data. By modelling the tool dynamics as a stochastic dynamical system comprising (a) a deterministic component and (b) a stochastic component, we estimate entropy transfer rates between variables through the Liang-Kleeman and Pires formalism. Preliminary results indicated that 7.5% of the inferred dependencies were known, 36.0% were plausible, 17.5% represented previously uncharacterised relationships, and 39.0% were inconsistent with established process knowledge. These findings demonstrate the framework's capability to uncover novel causal insights, while motivating further improvements to reduce inconsistent findings.

We study the problem of lossless text compression, motivated by the rapid growth in the collection and storage of digital textual data - including plain text, source code, and structured formats such as XML - and by recent advances in neural language model-based compression. In particular, recent LLM-based approaches, whether built on symbol-ranking pipelines or paired with a statistical compressor, have demonstrated compression ratios significantly superior to general-purpose compressors such as zstd, gzip, or bzip on text and code. However, these neural approaches suffer from severe throughput limitations, making them not yet practically usable. For the first time in the context of lossless neural text compression, we introduce Diffusion Language Models (DLMs) as an alternative inference paradigm to autoregressive LLM-based approaches. We argue that replacing autoregressive LLMs with DLMs within the same compression framework could overcome the throughput bottleneck caused by their one-symbol-per-step limitation. However, achieving these improvements requires addressing algorithmic challenges introduced by applying DLMs to lossless compression, where the architecture allows the number and positions of symbols encoded at each forward pass to be decided independently. We design efficient and effective strategies to solve these challenges and evaluate them experimentally against LLM-based and general-purpose compressors on enwik8, a well-established textual benchmark. Our results show that the newly proposed DLM-based framework advances the state of the art in lossless text compression. Moreover, as DLMs are still a relatively young paradigm, recent advances toward increasingly capable and efficient models suggest substantial room for further improvements.

Gödel's ontological argument is modal by design. Possibility and necessity are among its central concerns, and connected interpretation decisions – extensions or intensions for the positive properties, constant or varying quantifier domains, the choice of modal logic, etc. – subtly shape the argument and, in particular, whether the modal collapse is implied or not. It is tempting to abstract the modality away and to read the positive properties as merely forming an ultrafilter – which, however, may replace Gödel's argument with a perhaps more manageable alternative that may have little to do with Gödel's original intention. In this spirit, Odifreddi and Gomes argue that modal collapse is an intrinsic structural feature of any theory that characterises positive properties as an ultrafilter and identifies God as its principal generator. This comment shows that the structural claim is false. There is an explicit S5 model, with constant domains, in which positivity is at every world a modal ultrafilter principally generated by a unique and necessarily existing Godlike individual, and in which the modal collapse nevertheless fails – already for an atomic formula. What drives the collapse is not the ultrafilter structure but the rigidity of positivity – itself a thoroughly modal condition: given only that positivity is a modal ultrafilter, modal collapse and rigidity are equivalent. Rigidity is thus exactly what the collapse adds to the ultrafilter structure, and the disagreement is whether it is to be counted part of that structure at all. Two further claims of Odifreddi and Gomes are clarified: one on the derivability of Gödel's Theorem IV, one on the extensionality of positivity. All results are machine-checked in Isabelle/HOL, which distinguishes kernel-checked proofs from model-finder output.

Calibration requires a predictor to be unbiased after conditioning on its own predictions. Multicalibration asks for this guarantee simultaneously across a collection of groups. Many prediction tasks ask for several related features of the same conditional outcome distribution: variance is defined relative to the mean, skewness relative to both mean and variance, and conditional value at risk relative to a quantile. We study multicalibration for a sequence of $k$ properties in which each property is identifiable once the preceding properties are fixed. This framework includes Bayes pairs but does not require the properties to arise from a single loss. For every fixed $k\ge2$, we establish matching upper and lower sample-complexity bounds up to logarithmic factors under regularity conditions. Even with only polylogarithmically many binary groups, achieving multicalibration error $\varepsilon$ requires $\widetilde{\Omega}(\varepsilon^{-(k+2)})$ samples. Conversely, for any finite group family $\mathcal G$, we give a randomized learner using $O(\varepsilon^{-(k+2)}+\varepsilon^{-2}\log|\mathcal G|)$ samples. Thus the sample complexity is $\widetilde{\Theta}(\varepsilon^{-(k+2)})$ for polynomial-size group families. We instantiate the theory for three canonical examples.

We introduce a new class of uniformly ergodic MCMC algorithms, termed Diffeomorphic Contraction Sampler (DCS), and provide fast non-asymptotic mixing guarantees for DCS targeting distributions on $\R^d$ with arbitrarily heavy polynomial tails. DCS provides a solution to a well-known problem for MCMC samplers, which typically struggle with the combination of unbounded high-dimensional state space and vanishing gradients. The DCS pulls back a target on $\R^d$ onto a Euclidean ball $B(R)\subset\R^d$ and then samples from the transformed density on the convex set $B(R)$ via algorithms such as the Ball Walk, Hit-and-Run and others. A radial diffeomorphic contraction is chosen so that the pull-back density on $B(R)$ is bounded, implying uniform ergodicity for all targets with a finite polynomial moment. Non-asymptotic bounds for DCS require stronger assumptions such as log-concavity of the pull-back density. In practice, this is achieved approximately by a preconditioned automorphism of the ball $B(R)$, tuned via Variational Inference. Numerical simulation tests demonstrate that the DCS outperforms significantly the No-U-Turns sampler on multi-dimensional heavy-tailed targets arising as real-world posteriors in PosteriorDB benchmark. DCS also numerically outperforms in high-dimensional examples recently developed spherical projection samplers for heavy-tailed target distributions.

Vector kinetic lattice Boltzmann (VKLB) methods have recently emerged as a promising framework for solving hyperbolic partial differential equation (PDE) systems. VKLB discretizes Boltzmann-type equations using a discrete set of lattice velocities, enforces discrete moment constraints, and carefully defines equilibrium distribution functions. In this work, we introduce novel upwinded equilibrium distribution functions constructed from conservation variables and numerical fluxes derived via continuous flux vector splitting based on the eigendecomposition of the flux Jacobian. This formulation enables the weighted-upwind VKLB equilibrium to be applied broadly to general hyperbolic systems. The method is verified on a set of challenging hyperbolic systems that includes the shallow water, Euler and ideal magnetohydrodynamics (MHD) equations. The proposed method demonstrates improved stability, reduced error norms, and sharper shock resolution across increasingly complex verification and benchmark problems.

We develop a concise error analysis for a linear-quadratic elliptic distributed optimal control problem with point tracking and pointwise control constraints, where a standard finite element discretization and a variational discretization are treated in a unified framework. The analysis is self-contained modulo standard techniques in finite element analysis and classical results for elliptic partial differential equations.

In this paper, we revisit the singular asymptotics of the free surface near stagnation points for two-dimensional traveling gravity water waves with vorticity. We prove the nonexistence of full-density degenerate stagnation points beyond the strict two-sided linear growth regime. Our main tools are a modified Weiss-type monotonicity formula and a modified Almgren-type frequency formula. Together, they provide a new approach that completely avoids the use of a Bessel-type differential inequality, which is an essential tool used in the previous literature to prove the nonexistence of full-density degenerate stagnation points (Ann. I. H. Poincaré-AN, 29, 861–885, 2012). As consequences, we obtain uniform bounds for frequency-normalized blow-ups in arbitrary dimension and strong convergence in dimension two. As an application, we extend the Stokes conjecture for rotational waves to a broader class of vorticity distributions.

Hospitals and diagnostic laboratories rely on couriers to collect blood samples from geographically dispersed collection centres and deliver them for analysis before their short viability windows expire; late deliveries force costly re-collection and can delay diagnosis. We study the real-time dispatching of such a courier fleet, in which sample requests arrive stochastically at the centres throughout the day and a central dispatcher must repeatedly decide which vehicles to send, which centres each should visit, and whether to collect urgent samples immediately or consolidate them into later trips, subject to hard delivery deadlines and vehicle capacity limits. Unlike static planning models, which fix routes before demand is known, and reactive heuristics, which respond only to the current backlog, our approach anticipates future arrivals when weighing immediate collection against consolidation. We formulate the problem as a Markov decision process and develop a neural approximate dynamic programming framework for centralized dispatch. The method introduces a dual value function decomposition that separately represents vehicle states and collection-centre states through neural networks trained on post-decision states. These learned estimates are integrated through a matching formulation that selects dispatch actions while balancing supply availability, demand urgency, and downstream opportunity cost. Computational experiments on a realistic Greater Toronto Area network compare the proposed policy with myopic baselines and ablation variants. Results show that the proposed dual value function policy raises the share of sample volume delivered on time by 1 to 9 percentage points over myopic baselines, with the largest gains under tight fleet, capacity, deadline, and routing constraints; these on-time gains, in turn, reduce reliance on costly external couriers.

Let $G=C_5\times C_5$, and let $S(G)$ denote the set of integer values of its group determinant. Previous work determines the values in $S(G)$ coprime to $5$ and proves that every $5$-divisible value is divisible by $5^8$. We prove the converse inclusion $5^8\mathbb{Z}\subseteq S(G)$. Consequently, $S(G)=\{m\in\mathbb{Z}:m\equiv\pm1\text{ or }\pm7\pmod{25}\}\cup5^8\mathbb{Z}$. The proof uses a general shift criterion and three explicit polynomials whose group determinants are $5^8$, $2\cdot5^8$, and $5^9$. Together with the known classification for $C_{25}$, this completes the Taussky–Todd integer group determinant problem for all groups of order $25$.

We develop a new algorithm for computing (persistent) discrete homology of graphs using reduction to zero differentials and active enumeration. This allows us to compute the fourth homology group of the Greene sphere, along with several previously unknown groups. We also show that persistent discrete homology computes faster than simplicial homology of Vietoris-Rips complex in the high-noise non-metric settings, making it a better choice for noisy data sets.

We completely classify a natural scalar trace-product switching of the Gold almost perfect nonlinear function $x\mapsto x^3$ in every even dimension. Nontrivial switchings occur only for $n=4,6,8$: the admissible coefficients are, respectively, the nonzero trace-zero elements, the six elements of multiplicative order nine, and $\mathbb{F}_4^{*}$. For every even $n\geq10$, no nonzero coefficient is admissible. The infinite range is excluded by additive-character estimates on a Fermat cubic, with exact finite bridges for $n=10,12$. The raw coefficient lists for $n=6,8$ appeared earlier in Arshad's dissertation; our contribution is their intrinsic description, a proof uniform in the dimension, and the resulting dimension-rigidity theorem. We also classify normalized rank-two extensions in dimension eight by $\mathbb{P}^{1}(\mathbb{F}_4)$. A binary trace selector accepts two coefficient values at each non-base projective point, and the eight accepted marked switchings form exactly two extended-affine, hence two CCZ, classes. A centre-independent low-rank derivative criterion reduces each rank-$r$ candidate to $2^r-1$ membership tests in precomputed forbidden sets. The global APN classes reached are known; the results describe their local organization around the Gold centre and rule out this switching mechanism in all larger even dimensions.

In \cite{ABL21}, Autissier, Bonolis and Lamzouri obtained uniform estimates for the distribution function of the maximum of partial sums of a class of families of $m$-periodic complex-valued functions under certain conditions. For example, these conditions are verified by Kloosterman or Birch sums. In this article, under the same assumptions, we obtain an improved estimate for the tail of the distribution of the maximum of partial sums in these families, which gives for the first time an asymptotic formula for the logarithm of the distribution function, in a large uniform range. Furthermore, we prove a "structure theorem" for the maximum of partial sums of a family of complex $m$-periodic functions in our class, which shows that universally over this class, most of the partial sums with large norm are close to their imaginary part, and the maximum is attained around $m/2$. This is in sharp constrast with the results of Bober-Golmaker-Granville-Koukoulopoulos \cite{BGGK18}, Lamzouri \cite{Lam24} and Lamzouri-Nath \cite{LN24} for the distribution of the maximum of character sums in various families of Dirichlet characters.

2-group global symmetries intertwine 0-form and 1-form symmetries in an interesting way. We analyze universal constraints on lines which are charged under the 1-form subgroup of a 2-group, and show that they generically break the 0-form symmetry explicitly. This gives rise to a family of line defects parameterized by the broken symmetry generators, whose existence is invariant under the renormalization group flow of the bulk-defect system. The symmetry breaking is enforced by a `family anomaly,' which is a topological obstruction to a symmetric defect. Our main tools are the Wess-Zumino consistency condition and a generalized anomaly inflow formalism for defects and boundaries. For nonabelian continuous 2-groups the family anomaly stems from a higher Berry connection on the moduli space of defects, and constrains response functions that probe the local action of the broken symmetry. In the continuous abelian case we apply differential cohomology to the 2-group background fields to uncover a subtle generalization of the Wess-Zumino condition, while in the discrete case we recast it in terms of the associativity of symmetry defects. We give numerous illustrative examples, computing when possible explicit forms of the tilt operator (which probes the linear response of the defect to the broken symmetry) and its higher analogs, which are crucial for matching the family anomaly. We also discuss universal features such as how symmetry violation by charged line defects is related to symmetry breaking hierarchies in the bulk, and highlight a number of subtle points including the distinction between simple and non-simple lines, and Postnikov class resolution.