Papers

Let $(X_1,\ldots,X_n)$ be independent nonnegative random variables with $\mathbb{E} X_i\le1$, and write $S=\sum_iX_i$. For $\delta>0$, we prove that \[ \mathbb{P}\left(S<\mathbb{E} S+\delta\right)\ge b_{n,\delta}, \] where $b_{n,\delta}=\delta(n/(n+\delta))^n$ for $0<\delta<1$ and $b_{n,\delta}=(1-1/(n+\delta))^n$ for $\delta\ge1$. The bound is sharp for every $n$ and $\delta\ge 1$. In particular, since $b_{n,\delta} \ge e^{-1}$ for $\delta \ge 1$, our result proves Feige's conjecture [Feige, 2004] in the affirmative for $\delta\ge 1$. The proof is found by ChatGPT 5.6 Pro. It combines the exact Dirichlet calibration theorem of Vlassis and Thomas [Vlassis and Thomas, 2026], which resolves Gaffke's conjecture in statistics, with results in convex geometry including Grünbaum's centroid theorem [Grünbaum, 1960] and its generalization by Letwin and Yaskin [Letwin and Yaskin, 2024].

We establish sharp minimax limits for two-sample testing of Hölder-smooth densities under central differential privacy. Given two independent samples, the goal is to decide whether the underlying distributions are identical or separated in $L_1$ distance, while releasing only an $\varepsilon$-differentially private decision. We show that privacy changes the classical smooth-testing boundary through multiple regimes: the optimal separation radius is the maximum of four terms, consisting of the classical nonprivate rate and three distinct privacy-induced barriers. Which barrier is active depends on the privacy budget and the smoothness-to-dimension ratio, yielding a sharp phase diagram. Our upper bound discretizes the samples, applies a private discrete two-sample test to the resulting histograms, and chooses the bin resolution to balance approximation bias, sampling fluctuations, and privacy noise. The procedure also admits a permutation-calibrated implementation with finite-sample type~I error control. For the lower bounds, we combine smooth perturbation constructions with privacy-specific coupling and transport inequalities, showing that all four terms are unavoidable. Finally, when the smoothness is unknown, we develop a multiscale private test that attains the optimal adaptive rate and prove a matching lower bound. Adaptation costs exactly an iterated-logarithmic factor, and this cost appears only in the classical nonprivate term.

Economy-wide electrification is driving steep load growth, while renewables and batteries are increasing pressure at the distribution grid edge often requiring costly upgrades. Local Electricity Markets (LEMs) can coordinate distributed energy resources (DERs) more efficiently than conventional designs. This paper extends the LEM framework by modeling photovoltaic (PV) inflow as an Ornstein-Uhlenbeck (OU) process to capture uncertainty and temporal correlation. Using rolling-horizon alternating optimal power flow on the IEEE 123-node feeder, we compare operator, customer, and system costs plus storage value under three market designs: flat tariff (FT), real-time pricing (RTP), and LEM. Results show that the LEM achieves the lowest customer and system costs, under uncertainty, while RTP performs worst. It outperforms the other markets across most storage and solar penetration levels. However, it may worsen customer outcomes at very low PV penetration, indicating that LEM adoption should accompany PV expansion and precede storage expansion. We examine two approaches to estimating battery storage value. Storage value depends on storage penetration, imbalance between storage and PV penetrations, location, and whether batteries are colocated with renewables and/or load. Poorly placed units can worsen congestion, whereas well-sited units benefit owners and the system. Both the marginal value of new storage and the stored energy value of existing capacity decline as PV penetration rises. However, they respond oppositely to battery penetration: marginal value falls because of diminishing returns as the fleet grows, while stored energy value rises as more batteries compete for the same local PV generation. These metrics capture different dimensions of storage value under a LEM: one measures the benefit of expanding the fleet, while the other measures the scarcity premium on energy already stored.

In 1977, Karl F. Barth posed the following problem: given countable dense sets $A\subset\mathbb{R}$ and $B\subset\mathbb{C}$, does there exist a transcendental entire function $f$ such that $f(A)=B$ and $f(\mathbb{R}\setminus A)\subset\mathbb{C}\setminus B$? We review results related to this question and prove that there exist transcendental entire functions $f$ such that $f\restriction_A\colon A\to B$ is bijective, $f^{-1}(B)\cap\mathbb{R}=A$, and $f'(a)\neq0$ for every $a\in A$. In fact, the set of such functions has the cardinality of the continuum. At the end, we give two extensions of the result, one for countably many pairwise disjoint pairs of dense sets and one with $\mathbb{R}$ replaced by a closed unbounded subset of $\mathbb{C}$ of planar Lebesgue measure zero.

Let $X = [ \xi_1, \,\, \xi_2,...\,\, ,\xi_d]^\top$ be a zero-mean random vector of large dimension $d$ ($d \rightarrow \infty$) with (hidden) covariance matrix $M = (m_{ij})_{1 \leq i, j \leq d},$ where $m_{ij} = m_{ji} = \textbf{Cov}(\xi_i, \xi_j).$ Let $X_1, X_2, \dots, X_n$ be $n$ iid samples of $X$. Consider the sample covariance matrix $$\textstyle \tilde{M} := \frac{1}{n} \sum_{i=1}^{n} X_i X_i^\top.$$ In practice, one frequently uses the eigenvectors and eigenspaces of $\tilde M$ as estimators for those of $M$. A central task is to provide an error analysis for these estimators. In this paper, we provide an optimal error analysis, obtaining upper and lower bounds of matching order of magnitude, for a wide range of parameters $d$ and $n$, under mild assumptions on $M$. As corollaries, we obtain new necessary and sufficient conditions for the consistency of the estimators. In these conditions, we only require the number of samples $n$ to depend linearly on the effective rank of $M$, which can be much smaller than the dimension $d$.

Given graphs $H$ and $F$, the generalized Turán number ${\rm ex}(n,H,F)$ is the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. Alon and Shikhelman (J. Combin. Theory Ser. B, 2016) initiated the systematic study of generalized Turán problems. Let $T$ be a tree on $k$ vertices, and write $n=a(k-1)+b$, where $0\leq b<k-1$. Recently, Gerbner and Palmer (Electron. J. Combin., 2026) proposed the following conjecture: for every $r\geq3$, the graph $aK_{k-1}\cup K_b$ maximizes the number of copies of $K_r$ among all $n$-vertex $T$-free graphs. In this paper, we verify their conjecture when $r=k-2$ or $r=k-3\geq5$. More precisely, we show that ${\rm ex}(n,K_r,T)=a\binom{k-1}{r}+\binom{b}{r}$ and characterize all extremal graphs.

For a graph $H$ with $3\mid e(H)$, the zero-sum Ramsey number $R(H,\Z_3)$ is the least integer $N$ such that every labeling of the edges of $K_N$ by elements of $\Z_3$ contains a copy of $H$ whose edge labels sum to zero. We determine the last previously unresolved infinite family in the complete-graph case modulo $3$. More precisely, we prove that \(R(K_n,\Z_3)=n+3\) for every $n\ge 10$ satisfying $n\equiv 1\pmod 3$. Consequently, for $k\ge 1$, \(R(K_{9k+7},\Z_3)=9k+10\), resolving a problem of Caro and Mifsud.

We construct explicit wave packet frames adapted to Euclidean balls and use them to obtain quantitative eigenvalue estimates for spatio–spectral limiting operators. Let \(d\geq 2\), let \(B_d(R)\subset \R^d\) be the Euclidean ball of radius \(R\), and let \(S\subset \R^d\) be a measurable set such that $\partial S$ has finite $(d-\eta)$-upper Minkowski content for $0 < \eta \leq 1$. We construct a unit-norm frame for \(L^2(B_d(R))\), with frame bounds depending only on the dimension $d$, whose elements are adapted to the radial and angular geometry of the ball. We prove quantitative Fourier localization estimates for this frame: Relative to \(S\), the frame decomposes into packets concentrated in \(S\), packets concentrated in \(\R^d\setminus S\), and an exceptional family whose cardinality is bounded explicitly in terms of \(R\), and the Minkowski content of \(\partial S\). As an application, we derive an upper bound for the plunge region of the spatio-spectral limiting operator associated to the sets $B_d(R)$ and $S$.

Generative AI relies on finding reverse time models for a discrete-time forward diffusion with non-Gaussian initial state, but uses indirect approaches as there is no theory for direct reversal in discrete time. This paper develops a theory for directly finding reverse diffusions for discrete time nonlinear processes with non-Gaussian states and process noise. We also give a necessary and sufficient condition for the reverse model to be input-affine when the forward process is linear and the process noise Gaussian, and show that for a wide variety of state densities an input-affine reverse diffusion does not exist. This is among several differences between the reversal of stochastic difference equations and their continuous time counterparts.

The matching codewords framework is a key tool in recent algorithms for solving the Linear Code Equivalence (LCE) problem and in security analyses of LCE-based cryptographic schemes such as LESS. These analyses often rely on the assumption that a random $q$-ary linear code has no monomial automorphisms other than scalar multiples of the identity. For binary codes, Lefmann, Phelps, and Rödl established the corresponding rigidity phenomenon in the relevant logarithmic dimension range. For general $q$, Hou established an averaged result over all dimensions, whereas the recent prescribed-dimension result of Di Giusto and Ravagnani applies only in a restricted regime near $n/2$. For every fixed prime power $q$ and every fixed real number $\varepsilon>0$, we prove that a uniformly random $k$-dimensional code $\mathcal{C}\subseteq\mathbb{F}_q^n$ has a trivial monomial automorphism group with probability tending to $1$ as $n\to\infty$, provided that $m:=\min\{k,n-k\}\geq(2+\varepsilon)\log_q n$. Furthermore, when $m \le 2 \log_q n + C$, where $C$ is a constant independent of $n$, we also show that the probability that the automorphism group of $\mathcal{C}$ is nontrivial is at least $\frac{1}{2} - \varepsilon$ for large enough $n$.

A class of bipartite graphs is said to have the strong Erdős-Hajnal property if there exists $\varepsilon > 0$ such that every graph $((A, B), E)$ in the class contains a complete or empty induced subgraph with parts $X \subseteq A$, $Y \subseteq B$ where $|X| \ge \varepsilon|A|$ and $|Y| \ge \varepsilon|B|$. Scott, Seymour and Spirkl proved that it is enough to forbid a forest and the bipartite complement of a forest. In this paper, we provide quantitative bounds on $\varepsilon$ when we restrict to matchings.

We determine the exact extremal balance data of all 1,338,193,159,771 unlabeled posets on fourteen elements. The least balance constant exceeding $1/3$ is $37/106$. The least over posets that are not nontrivial ordinal sums is $254/725$, attained by a ladder with broken rungs; this confirms a conjecture of Peczarski at order 14, while orders 12 and 13 reproduce De Loof, De Baets, and De Meyer. No balance constant lies in the gap above $1/3$ that Peczarski conjectures to be empty. Exactly 128 classes attain $1/3$, and every one is an ordinal sum of singletons and copies of the three-element poset with one relation, a family whose non-chain members are counted by $a(n)-1$, where $a(n)=a(n-1)+a(n-3)$. In the linear-extension-majority digraph the longest simple cycle has length 8, against 7 at order 13, and exactly 30 classes attain it; of the thirteen such classes whose witnesses the census retains, nine have a cycle spectrum containing no odd cycle at all. A second exhaustive pass over the same classes verifies Peczarski's Gold Partition Conjecture through fourteen elements, extending his order-11 frontier and implying in particular that the $1/3$-$2/3$ Conjecture holds through order 14. All arithmetic is exact and every extremal witness is recomputed by an independent program.

We present several remarks on cardinal coefficients associated with the generalized Baire space for an inaccessible cardinal kappa. Our inquiry originates in the work of van der Vlugt arXiv:2307.14118 . We provide evidence that the appropriate coefficients to consider are those determined by binary relations restricted to a club. In other words, we make a case for the systematic use of clubs.

A battery-swapping station must provide every arriving vehicle with a charged battery while minimizing the time-of-use cost of recharging returned units. Coordinating heterogeneous compatibility, vehicle-specific return times, and finite charger capacity requires service-aware recharge decisions across the planning horizon. We formulate a per-battery mixed-integer linear program that captures these operational features under a hard no-stockout constraint and derive a provably equivalent reduced form with fewer explicit binary variables. In the synthetic scaling study, a price-guided battery-path heuristic returned a full-service schedule for every instance; regime-level median solve times ranged from 0.24 to 8.0 seconds. Its median cost premiums were 7-8% over certified reference costs for small- and medium-scale instances, and its certified ex post optimality-gap upper bounds were 9-12% for large- and extra-large-scale instances. For each operational baseline, the certified reference schedules reduced charging-energy cost by 50-60% on instances that the baseline fully served and for which a certified reference was available. In a 30-day replay of 1,002 swaps recorded at a commercial station, the reduced-model and heuristic rolling controllers served every swap and reduced charging-energy cost by approximately 50% relative to immediate charging.

Let $\alpha'(G)$ be the matching number of a graph $G$, and let its Randić index be $R(G)=\sum_{uv\in E(G)}(d(u)d(v))^{-1/2}$. In 2006, Aouchiche, Hansen, and Zheng conjectured that the maximum of $R(G)-\alpha'(G)$ over all $n$-vertex graphs is attained by the complete bipartite graph whose smaller part has $\lfloor\frac{n+4}{7}\rfloor$ vertices; the conjecture has remained open since then. In this paper, we prove that every $n$-vertex König–Egerváry graph, and in particular every bipartite graph, satisfies \[ R(G)\le\sqrt{\alpha'(G)\left(n-\alpha'(G)\right)}, \] and we characterize the graphs attaining equality as the bipartite graphs all of whose components are semiregular with a common degree ratio. The König–Egerváry hypothesis cannot be dropped, but the Berge–Tutte formula reduces the general case to it, and in this way we determine the maximum of $R(G)-\alpha'(G)$ for every $n\ge4$, together with all extremal graphs. The conjecture is therefore false, and it fails for infinitely many orders: the optimal part size is governed by the proportion $\frac{2-\sqrt2}{4}$ rather than by $\frac17$. The two proportions give asymptotic slopes differing by less than $3.7\cdot10^{-5}$, which is why a search over graphs of small order does not distinguish them. The equality statement fails as well, since the extremal graphs are not only the complete bipartite ones.

As financial institutions transition from traditional predictive models to autonomous agentic systems, the static model inventory requirements of traditional model risk management (MRM) face structural obsolescence. This paper proposes a dynamic Inventory-as-Code (IaC) governance loop that treats the model inventory as a living architectural component rather than a periodic documentation artifact. We make four principal contributions. First, we introduce a calibrated Degree of Autonomy (DoA) materiality score with an explicit, taxonomized tool-complexity weighting scheme that addresses the dominance problem of naive additive risk formulations. Second, we construct the agent inventory as a Directed Acyclic Graph (DAG) and define a formal Composite Risk Propagation algorithm under which upstream validation failures induce risk penalties on all reachable descendants. Third, we develop a Trajectory Monitoring protocol based on distributional cosine drift across ensembled Chain-of-Thought (CoT) embeddings, with an explicit procedure for constructing and certifying the Golden Path baseline, a matched-bootstrap calibration that we show is necessary to avoid a severe false-positive artifact in the naive alternative, and a two-stage response that separates legitimate reasoning variation from detrimental drift without demanding that a single exceedance event trigger an irreversible action. Fourth, we address practical complications largely absent from the prior literature: LLM base-model version changes, latent feedback loops in nominally acyclic agent graphs, and a two-pass execution structure that stages validation ahead of risk propagation.

High dimensional statistical theory has established the importance of constant aspect ratio, when the number of dimensions ($d$) and samples ($n$) satisfy $n,d\to\infty$ with $n/d\to \gamma\in(0,\infty)$, in understanding the limits of canonical estimation problems. In particular, for estimating the top eigenvector of a $d\times d$ population covariance matrix from $n$ iid samples, the BBP phase transition gives a precise threshold – a simple functional of the aspect ratio – such that the top sample principal component attains nonzero asymptotic correlation with the truth only when the leading population eigenvalue exceeds it. In this paper, we show that for online / streaming algorithms the story is very different, and constant aspect ratio is insufficient for nonzero overlap. We study Oja's algorithm, the most popular method for online PCA. Let $\Sigma=\theta^2 v_0v_0^\top+I\in\mathbb{R}^{d\times d}$, and run Oja's algorithm with step size $\delta/d$ on $n$ iid samples $X_k\sim\mathcal{N}(0,\Sigma)$, with output $\hat v_n$. Then, as $n,d\to\infty$ with $n/d\log d\to\gamma\in(0,\infty)$, we establish a phase transition: $|\langle\hat v_n,v_0\rangle|\to 0$ when $\gamma<\gamma_*$, and $\to\rho_*$ when $\gamma>\gamma_*$. Here $\rho_*=\rho_*(\theta,\delta)=\sqrt{(\theta^2-\delta/2)_+/\theta^2(1+\delta/2)}$ and $\gamma_*=\gamma_*(\theta,\delta)=1/2\delta(\theta^2-\delta/2)_+$. Further, at criticality, when $n=[\gamma_*d\log d+\eta d]$ and $d\to\infty$, $\eta\in\mathbb{R}$, the correlation is random: $|\langle\hat v_n,v_0\rangle|\stackrel{w}{\to}\rho_*|G|\exp(\eta/2\gamma_*)/\sqrt{\rho_*^4+G^2\exp(\eta/\gamma_*)}$ where $G\sim\mathcal{N}(0,1)$. This is in stark contrast to ordinary high dimensional PCA, where nonzero overlap is possible at constant $n/d$ and improves as $n/d$ increases.

We formulate a local corner-factor conjecture for the determinant of the Neumann jump operator on a piecewise real-analytic cutting curve. For the mirror double of a simply connected geodesic polygon with interior angles $\pi\alpha_1,\ldots,\pi\alpha_N$, the conjectural determinant is \[ \Det_{\angle}'\cN = \frac{\length(\partial P)}2 \prod_{j=1}^N\alpha_j^{-1/2}. \] Here $\Det_{\angle}'$ denotes an intrinsic determinant, still to be constructed, that is required to satisfy a Burghelea–Friedlander–Kappeler gluing formula; only its quotient by $\length(\partial P)$ is purely angle-dependent. Three model calculations support the conjecture: a flat polygon and its double give $\frac12\prod_j\alpha_j^{-1/2}$; a spherical spindle split into congruent lunes gives $1/(2\alpha)$ for two angles $\pi\alpha$; and every spherical Coxeter triangle with angles $(\pi/p,\pi/q,\pi/r)$ gives $\frac12\sqrt{pqr}$, including $\sqrt2$ for the octant. The conjecture also reduces the Dirichlet determinant of a constant-curvature polygon to that of its closed double. Although the singular anomaly formula applies to the double in general, an explicit evaluation in nonzero curvature requires solving its uniformization problem. We discuss connections with the Neumann jump determinants in the work of Wiegmann–Zabrodin and Wang and with the recent Grunsky-operator approach to Coulomb gases on domains with corners.

We study the weak stability of the kinetic Cucker-Smale (in short, KCS) model in a phase-spatially extended setting, which can be formally derived from the infinite Cucker-Smale model in the mean-field limit. For a bounded Lipschitz communication weight function, we derive finite-time Osgood-type weak stability for measure-valued solutions with exponential velocity tails and finite spatial second moments. Unlike the phase-spatially confined setting, the solution operator to the KCS model is not Lipschitz continuous with respect to initial data. This is due to the unbounded velocity tail and the corresponding absence of a uniform Lipschitz bound for the alignment force. As an application of weak stability, we obtain an i.i.d. sampling consequence: empirical measures generated from independent initial samples converge to the measure-value solution for the corresponding kinetic model in any finite time interval, in expectation.

We obtain explicit formulas for the expected number and height distribution of critical points of smooth isotropic Gaussian random fields on $\mathbb{R}^d$. The expected number density formula is expressed in terms of at most one-dimensional integrals, regardless of the dimension $d$. To obtain the formulas, we provide a variant of de Bruijn's theorem, as well as Weierstrass' convolution formula with a Gaussian random variable and its inversion.

Physical reservoirs, neuromorphic devices, and wave-mediated systems often possess memory, feedback, and rich state-dependent dynamics, but these properties do not by themselves establish autonomous computation. Here we develop a closure criterion for autonomous physical computation, motivated by the wave–particle walker. We formulate the walker as a stroboscopic reservoir with state, where the wave field stores an exponentially decaying trace of previous droplet impacts and guides future motion through local slope coupling. This model separates physical writing, storage, reading, feedback, and externally triggered erasure. We then define computation as robust coarse-grained transition preservation: a physical map implements an abstract transition only when a coarse-graining satisfies compositionality, with abstract states realized by separated physical basins and transitions stable under noise. Autonomous physical computation requires a further closure condition: an internal physical readout state must select the next physical operation. This criterion classifies the wave–particle walker as a wave-memory machine with genuine Turing-like primitives, but not as a closed autonomous physical computer, because the erasing phase shift is externally imposed. The framework turns this distinction into a design principle: memory becomes autonomous computation when physical readout basins are coupled back to operation selection.

The largest known biangular lines systems in dimension 7 through 20 are due to Ganzhinov and Szöllősi, and the lines make angles $\arccos(\frac{1}{5})$ and $\arccos(\frac{3}{5})$. They are constructed by making use of the nice connection with integral lattices. We explore further the connection between biangular lines and integral lattices to classify the largest biangular line systems with these angles, in dimension 7 through 10. We also give a new biangular line system in dimension 15, that matches the size of the largest known in this dimension.

Let $F$ be the set of positive cube-full integers. We prove that, for every $\varepsilon>0$, there is a reduced residue class in which $F+F+F$ has upper relative density at most $\varepsilon$. Consequently, the positive integers not representable as sums of at most three cube-full numbers have positive lower natural density. This proves the infinitude assertion in Erdős Problem \#940 for $r=3$. The argument combines a cubic-character estimate with a truncation of the parametrisation by the canonical 4-full part.

We derive a fast factorization of the exact Karhunen–Loève transform (KLT) of an AR(1) source by mapping it onto the Discrete Cosine Transform. For even $N$ and every $\rho\in(0,1)$, the KLT factors exactly into Chen's fast DCT-II structure—butterfly plus fixed half-size DCT-II and DCT-IV cores—completed by two orthogonal corrections: eigenvector matrices of diagonal-plus-rank-one matrices carrying the entire $\rho$-dependence. Fast DCT factorizations are reused unchanged; as $\rho\to1$ the corrections become identities, recovering Chen's algorithm. At short lengths the design is explicit: at $N=4$ the transform is one butterfly and two rotations with $\tan2\psi_s=-2/(1-\rho)$, $\tan2\psi_a=2/(1+\rho)$; at $N=8$ every correction entry is in radicals, via one quartic serving both branches, and factors into six Givens rotations. Complete and scaled realizations run the exact 8-point KLT in 32 and 24 multiplications, about twice the fixed DCT-II; at $N=4$, a single extra multiplication. At large $N$, fast-multipole application of the corrections yields the exact KLT in $O(N\log N)$ operations to prescribed accuracy; this overhead thus peaks at intermediate sizes and vanishes in both limits.

For a finite-dimensional algebra $A$ over a field we organize the Hochschild cohomology of $A$ with coefficients in all derived tensor powers of the Serre bimodule $\omega=\D A$ into a single bigraded derived invariant, the Serre–Hochschild plane $\T^{p,m}(A)$, defined over all of $\mathbb Z^2$ without inverting $\omega$. The column $m=0$ is the Tamarkin–Tsygan calculus, $m=1$ is dual Hochschild homology (the column governing Han's conjecture), $m=2$ recovers the $\tau$-Hochschild shadow, and $m=-1$ Keller's Calabi–Yau completions. We prove a ladder theorem making every column a symmetric module over the calculus, a coefficient spectral sequence, a Serre reflection fixing the Han column, a dictionary identifying the plane of a geometric algebra with twisted polyvector cohomology of a smooth projective variety, finite generation of the homology column under the condition $\Fg$ with a growth trichotomy for $\dim\HH_n(A)$, and a complete graded Tate criterion for Han's conjecture on periodic algebras. Han's conjecture follows for symmetric periodic algebras in every characteristic and for local periodic algebras in characteristic zero. We exhibit the first example of a stably traceless periodic algebra, showing that the graded criterion is sharp.

A maximum principle for certain strictly convex functions belonging to $W^{2,1}_{\rm loc}(\Omega)\cap C^1(\Omega)$ is established and then applied to obtain a $W^{2,1 +\eps}_{\rm loc}(\Omega)$-estimate.

Ambient convergence does not determine a nonlocal energy on a cut body: a cut has zero volume, but a microscopic bond retains which labelled bank its path crosses. We give a direct measure-theoretic resolution of this information and prove the corresponding sharp-interface Gamma-limit. The cut may be a finite or countable family of Borel, countably rectifiable hypersurfaces with finite total surface mass, measurable approximate normals, and measurable binary bank data. No closedness, positive reach, finite atlas, curvature bound, angle condition, or regularity of the branch set is assumed. For heterogeneous Kac interactions generated by a uniformly spanning family of possibly noncommuting flows, the limit is the resolved anisotropic variation plus the outer boundary contact and one coefficient-one contact on each bank. The Gamma-functional contains no compulsory lower-dimensional term where a crack approaches the boundary; constant recovery creates no concentration there. The proof combines an exact spacetime area formula for flow crossings, a one-dimensional split-extension lower bound, and ordered-flow compactness. We also remove compact radial support. A finite first moment yields the same local Griffith functional; at the power-law threshold the limits separate into local, logarithmically renormalized local, and genuinely fractional regimes. Smooth slit and junction detectors are recovered as uniformly equivalent realizations of the direct crossing rule.

Let $\mathcal{M}$ be a von Neumann algebra equipped with a faithful normal semifinite weight, and let $L_p(\mathcal{M})$ denote the associated Haagerup noncommutative $L_p$-space. For $1<p<\infty$, let $q=p/(p-1)$. For $z\in L_q(\mathcal{M})$ and $y\in L_p(\mathcal{M})$, denote by $\langle z,y\rangle=\operatorname{Re}\operatorname{Tr}(z^*y)$ the canonical real duality pairing between $L_q(\mathcal{M})$ and $L_p(\mathcal{M})$. Define the normalized duality map by $J_p(x)=\|x\|_p^{\,2-p}u|x|^{p-1}$ for $x=u|x|\neq0$, and set $J_p(0)=0$. We establish the sharp tangent inequality $$\|x+y\|_p^2\ge\|x\|_p^2+2\langle J_p(x),y\rangle+(p-1)\|y\|_p^2,\qquad x,y\in L_p(\mathcal{M}),\quad 1<p\le2.$$ The reverse inequality holds for $2\le p<\infty$, and the constant $p-1$ is optimal. Our inequality extends Xu's tangent inequality \cite{Xu89} from classical $L_p$-spaces to general noncommutative $L_p$-spaces and provides the sharp tangent formulation of the Ricard–Xu convexity inequality \cite{RX16}. When specialized to Schatten $p$-classes, it also yields the corresponding tangent formulation of the Ball–Carlen–Lieb convexity inequality \cite{BCL94}. Under a suitable positivity assumption, we further establish a sharp complement to Xu's tangent inequality for Schatten $p$-classes in the range $0<p<1$.

The Discrete Fourier Transform (DFT), the Discrete Cosine Transform (DCT), and their block-wise variants underpin most deployed image and video codecs. Their effectiveness rests on three properties: their runtime is near-linear (up to a polylogarithmic factor) in the image size, they are exactly invertible, and they carry few to no parameters. We generalize these transforms to dataset-adaptive isometric multilinear bases that keep all three properties at the cost of a polylogarithmic number of parameters, by parameterizing each basis as an isometric tensor network, inspired by quantum many-body theory, and training one shared basis per image dataset with Riemannian optimization. Because the family contains the DFT and the DCT-IV (a variant of the DCT) as exact points, training can start from these transforms, and it improves on its starting point at the compression level it is trained for. We evaluate the bases on natural photographs and line drawings. On photographs, which the cosine basis already fits well, the best trained basis leads the DCT only under aggressive truncation. On Quick Draw line drawings, under a common codec at matched quality, the best trained basis stores each image in about 20% fewer bytes than the 8x8 block cosine transform of the Joint Photographic Experts Group (JPEG) format, not counting the one-off storage of the basis.

Immersed surfaces in hyperbolic three-space carry several natural Gauss-type maps with distinct geometric roles. The hyperbolic Gauss maps record the ideal endpoints of oriented normal geodesics; the Legendre Gauss lift retains the position-normal data and its contact structure; adjusted Gauss maps arise from gauge normalization and Iwasawa splitting in Weierstrass–Kenmotsu representations; and the conformal Gauss map encodes the mean-curvature sphere congruence in Möbius geometry. We present these constructions in a common framework, emphasizing their target spaces, analytic properties, and mutual relations. Particular attention is given to the generalized DPW method, the necessity of flatness in adjusted rank-one data, and the harmonic-map characterization of Willmore surfaces. The resulting viewpoint distinguishes the asymptotic, contact, integrable, and conformal information carried by an immersed surface in \(\mathbb{H}^{3}(-1)\).