Papers

Based on the theory of Muckenhoupt weights, short and conceptually simpler proofs are provided for the following two implications: (1) if $(X, d, \mu)$ is a space of homogeneous type where $d$-balls are open sets and the Lebesgue differentiation theorem holds true and if $E \subset X$ is a weakly porous set whose maximal $E$-free hole function $\rho_{d, E}$ is doubling, then $\dist{\cdot, E}^{-\alpha} \in A_1(X, d, \mu)$ for some $\alpha > 0$; and (2) now without the assumption on the validity of Lebesgue's differentiation theorem, if $\dist{\cdot, E}^{-\alpha} \in A_1(X, d, \mu)$ for some $\alpha > 0$, then $\rho_{d, E}$ is doubling.

We study convex optimization over a compact convex set when the objective is smooth and convex only on an open domain. Under a boundary-blow-up condition, every feasible initialization yields a compact invariant sublevel set separated from the complement of the objective domain, and an optimal solution exists. For a domain-aware Armijo projected-gradient method, a safe-neighborhood analysis establishes well-defined objective evaluations, finite backtracking, sufficient decrease, and a run-specific but iteration-independent positive lower bound on the accepted step sizes. These properties yield explicit sublinear objective and stationarity guarantees, together with convergence of the full iterate sequence. Positive curvature restricted to feasible displacement directions further guarantees uniqueness and linear convergence. We apply the framework to controllability scoring with prescribed input directions and compact convex allocation constraints. Feasibility is characterized exactly by controllability of the input directions eligible for positive allocation, while restricted injectivity of the Gramian map guarantees uniqueness of the optimal allocation and provides explicit strong-convexity bounds. A directed-network example illustrates how candidate exclusion can preserve or destroy feasibility and alter the optimal allocation in a criterion- and horizon-dependent manner.

Right-angled Artin groups (RAAGs) and right-angled Coxeter groups (RACGs) associated with finite simple graphs are fundamental objects in geometric group theory. Their one-variable growth series with respect to the standard generating sets was classically expressed by Chiswell in terms of the one-variable independence polynomial of the defining graph [2] with suitable substitutions of the variable. In this paper, we investigate the multivariate growth series of graph products of groups and derive explicit formulas in terms of the multivariate independence polynomial of the underlying graph through suitable substitutions of variables. As special cases, we obtain multivariate growth series formulas for RAAGs and RACGs, thereby extending the classical one-variable identities. We further show that the coefficients of the multivariate growth series of RAAGs and RACGs admit explicit descriptions in terms of the double-marked and marked chromatic polynomials of graphs. This connection reveals a rich interplay between growth series and graph coloring invariants. In particular, we obtain completely explicit formulas for all the coefficients in the case of chordal graphs, which include, for example, trees and complete graphs.

We test whether an architecture that succeeds on a simple flow regime also succeeds on a richer one, with each trained separately on each regime. We explore two transient flows under time-varying boundary conditions: the three-dimensional slurry film in chemical-mechanical planarisation (CMP), central to semiconductor manufacturing, and the two-dimensional Kármán vortex street (KVS). Eight surrogate models on one shared pipeline differ in whether they learn the full field or a latent representation, and in whether they predict in one shot or step by step. No single architecture wins both regimes. On the film, a one-shot full-field model reconstructs the cumulative wall shear stress to 2.7% relative error. On the wake, a latent autoregressive DeepONet retains 90% of the shedding power that direct and one-shot models damp to almost zero. The treatment of time decides the outcome. The self-sustained wake calls for autoregressive feedback and the boundary-driven film for a direct map. Pointwise RMSE hides the damped oscillation on the wake, compresses the sixfold lead on the film's process target, and picks the damped model under wake extrapolation. The evaluation scores five physical questions. Trained surrogates answer queries 10^3 to 10^4 times faster than the finite-element solver and pay off from the first query beyond the training set on the film and from the third on the wake. Neither the winning architecture nor its validation holds across regimes. The choice of surrogate should follow the dynamical character of the target flow, and its validation should resolve the failure modes.

Kernel ridge regression is a standard method for functional data analysis, but its exact behavior is less understood. We study tensor-product kernel ridge regression for estimating the $r$-th moment function of a random function based on noisy discrete observations. The formulation includes mean estimation, covariance estimation, and higher-order moment estimation in a single framework. Our main result gives a precise $1+o_{\mathbb{P}}(1)$ expansion for the $L^2$ error at each admissible regularization parameter. The expansion consists of bias and three variance terms corresponding respectively to variation across the independent sample paths, latent signal variation at each sample point, and variation from measurement errors, identifying the refined error structure underlying functional data. As applications, we show that KRR attains the minimax rate for source smoothness $s \leq 2$ but becomes suboptimal in the sparse regime for $s>2$ due to saturation. A technical ingredient is a set of concentration inequalities for $U$-statistics suited to the dependent product structure of functional observations.

We exhibit a connected graph on 24 vertices with maximum degree 3, independence number 9 and zero forcing number 11, refuting a 2017 conjecture of TxGraffiti recorded as Conjecture 2 of the survey of Davila, Brimkov and Pepper. The same construction with a different gadget gives a connected cubic graph on 36 vertices with independence number 15 and zero forcing number 17; the conjecture therefore fails also in the cubic form in which the survey's Lean 4 appendix states it. In particular Z <= alpha + 1 is not a universal bound for connected cubic graphs, and the value Z = alpha + 2 is attained.

Let $\mathcal{F}$ be a $k$-uniform hypergraph. The famous Erdős–Ko–Rado (1961) theorem determines the maximum size and extremal structure for $\mathcal{F}$ being $t$-intersecting, that is, $|F_1 \cap F_2| \ge t$ for any two edges $F_1, F_2$ of $\mathcal F$. The codegree squared sum $\mathrm{co}_2(\mathcal{F})$ is the square of the $\ell_2$-norm of the codegree vector of all $(k-1)$-sets in $\mathcal{F}$, which was initially introduced for Turán problems of hypergraphs. Recently, Brooks and Linz (2026), as well as Wu and Zhang (2026) investigated the maximum value of $\mathrm{co}_2(\mathcal{F})$ and corresponding extremal structures for $\mathcal{F}$ being $t$-intersecting. Moreover, Brooks and Linz asked if the classical results on intersecting families can be extended to $\mathrm{co}_2(\mathcal{F})$. In this paper, by developing the spectral techniques for incidence matrices, we study the extremal problems of $\mathrm{co}_2(\mathcal{F})$ for $\mathcal{F}$ being intersecting families in finite vector spaces, affine spaces, and attenuated spaces, and establish the Erdős–Ko–Rado theorems in $\ell_2$-norm for the three finite spaces.

For $c>1$ and an integer radix $b\ge2$, we study the positive integers $m$ for which $mb^k\le c^m<(m+1)b^k$ for some $k\ge0$; for integer $c$, this is the self-prefix leading-digit condition. We derive an exact shrinking-target criterion; for $c\ge2$, an exact signed-discrepancy identity isolates both infinitude and the conjectural logarithmic count. For $c\ge2$ with nonintegral logarithmic slope, Lambert $W_{-1}$ inversion produces a candidate sequence with an eventual two-gap law and an exact counting formula; for $(c,b)=(2,10)$ all consecutive candidate gaps are $3$ or $4$. For algebraic $c$ with irrational $\log_b c$, the Lambert-root phases satisfy deterministic moving-target asymptotics in an explicit nontrivial power range strictly below the critical scale. For irrational logarithmic slope, actual hits obey fixed-difference and arithmetic-chain rigidity; for multiplicatively independent integer parameters, coherent endpoint hits at floor resonance centers force every intermediate term. Finally, set $\rho=\{\log_b c\}$. For fixed multiplicatively independent integers $c,b$, an interpolated continued-fraction locator has bit complexity $O(N^{1-1/\nu}\operatorname{polylog}N)$ for every $\nu>\mu(\rho)$. We give an explicit certified instance for $(2,10)$, whose infinitude remains open.

We construct a family of bounded, smooth, simply connected, nonconvex planar domains $\Omega_{a,\epsilon}$. Let $u_{a,\epsilon}$ be the corresponding torsion function satisfying \[ -\Delta u_{a,\epsilon}=1\quad\text{in }\Omega_{a,\epsilon},\qquad u_{a,\epsilon}=0\quad\text{on }\partial\Omega_{a,\epsilon}. \] There exists \(a^*\in(0,1)\) such that, for every \(a\in(a^*,1)\) and all sufficiently small \(\epsilon>0\), the origin is a unique global maximizer of $u_{a,\epsilon}$. Moreover, \[ \lim_{a\downarrow a^*}\lim_{\epsilon\to0} \lambda_{\max}\bigl(D^2u_{a,\epsilon}(0)\bigr)=0. \] In addition, the ratios $\text{diam}(\Omega_{a,\epsilon}) / \text{inrad}(\Omega_{a,\epsilon})$ are uniformly bounded. Hence the Hessian estimate proved by Steinerberger (J. Funct. Anal. 274, 1611–1630, 2018) for convex planar domains cannot be extended to smooth, simply connected, nonconvex planar domains. \vskip0.2cm Our domains are star-shaped, symmetric with respect to both coordinate axes and convex in the horizontal direction. When the limiting slit is sufficiently long, we further show that certain superlevel sets of the torsion function are not star-shaped. This strengthens the counterexample to the star-shapedness question raised by Gladiali and Grossi (Amer. J. Math. 144, 1221–1240, 2022) under stronger geometric assumptions.

Communication complexity provides a natural framework for quantifying the classical resources required to reproduce quantum statistics. In the qubit prepare-and-measure scenario, two classical bits have been shown to be necessary and sufficient to simulate arbitrary qubit states and arbi- trary quantum measurements exactly. However, this result does not exclude the possibility that restricted families of measurements may admit accurate 1-bit classical approximations. We use a neural network procedure to demonstrate that a single bit can achieve high average accuracy for specific measurement families. A performance analysis of our neural network reveals that symmet- ric measurements with uniformly weighted elements, such as those forming regular polyhedra, are particularly amenable to this restricted communication. By analyzing the patterns learned by the neural network, we derive an analytical protocol that is extremely accurate for finite information- ally complete symmetric configurations and becomes exact in the limit of a continuous isotropic measurement.

Discrete optimization algorithms are often analyzed through continuous-time limiting ODEs, but a convergence certificate for the ODE is not automatically one for the discrete algorithm. We develop contact Hamiltonian systems as a setting where the transfer can be made precise. A contact Hamiltonian $H$ on $J^1(\mathbb{R}^n)$ obeys the intrinsic decay identity $\dot H = -H\,\partial_s H$, so an augmented energy $\mathcal{E}$ built from $H$, together with the conformal rate $\partial_s H$, is a continuous-time rate certificate whenever $\mathcal{E}$ controls the objective gap. Our main theorem states, under three named and independently checkable hypotheses, that an order-$r$ contact splitting with step $h$ transfers this certificate over the finite horizon set by backward error analysis. The discrete decay envelope is governed by the modified conformal factor up to $O(h^r)$ perturbations plus a backward-error shadowing defect, and the mechanism is inherited exactly because the modified Hamiltonian is itself a contact Hamiltonian. Quadratic heavy ball is a fully solvable example: its projected dissipative-leapfrog spectrum agrees with established conformal-symplectic optimization theory, while the augmented contact Hamiltonian yields a sharp objective-to-certificate comparison that verifies the transfer hypotheses. For strongly convex objectives with state-dependent damping, an explicit Bregman-type Lyapunov certificate instead transfers by an auxiliary-shadowing corollary. The decomposition $H=K+V+D$ into kinetic, objective-encoding potential, and dissipation terms serves as a design template, with a catalogue of closed-form sub-flows including contact-specific damping families. Numerical experiments confirm the predicted conformal-factor tracking orders and show competitive performance on ill-conditioned benchmarks and deep-learning tasks.

We study the non-equilibrium density fluctuations of a one-dimensional two-species symmetric simple exclusion process in contact with critical slow boundary reservoirs. The boundary rates are of order $1/n$, which leads to Robin boundary conditions at the macroscopic level. We prove that the coupled fluctuation field associated with the empirical densities of two species converges to a generalized Ornstein-Uhlenbeck process. The limiting process is characterized by a linear martingale problem whose coefficients reflect the combined effects of bulk diffusion, species conversion, and the boundary reservoirs.

Foreman and Magidor showed that the continuum hypothesis implies the existence of a countably-closed $\aleph_2$-cc forcing notion $\mathbb P$ for adding $\square_{\aleph_1}$. Here, we show that $\mathbb P$ may consistently be realized as an $\aleph_2$-Souslin tree. More generally, we prove that $\square_\lambda$ may be added by a $\lambda^+$-Souslin tree, providing the first analog of the Foreman–Magidor forcing at the level of successors of singular cardinals. Our construction is uniform and extends to inaccessible cardinals as well.

In this paper, we propose and analyze a class of high-order numerical schemes within a fully discrete finite element framework for the incompressible Navier-Stokes equations with no-slip boundary conditions. The temporal discretization employs a kth-order (k=1,...,6) implicit-explicit backward difference formula (IMEX-BDFk), in which the nonlinear convection term is treated explicitly and the linear Stokes part implicitly, whereas the spatial discretization utilizes Taylor-Hood finite elements. We establish the stability and uniform boundedness of the numerical solution. We further establish optimal order error estimates in both space and time without any CFL-type condition, in the sense that the time step is independent of the spatial mesh size. In three dimensions, these include L2- and H1-norm error estimates for the velocity and L2-norm error estimates for the pressure, with temporal convergence rates up to sixth order for all variables. Numerical experiments are presented to demonstrate the effectiveness of the scheme and to confirm the theoretical convergence rates.

Raw-Hermite sensing and collision on a fixed discrete-velocity set can convert a uniform translation into artificial coupling between nominally distinct nonequilibrium orders. We develop a central-Hermite formulation for a D3Q125 kinetic model with order-resolved log-Gaussian relaxation and compare three variants: raw sensing/raw collision (A), central sensing/raw collision (B), and central sensing/central collision (C). In homogeneous translated second-order perturbations, model C preserves third- and fourth-order modal purity to machine precision, whereas A and B develop boost-dependent cross-order content. Across a grid-CFL-boost matrix, model C reduces the post-transport collision frame discrepancy relative to A by 65.342-98.102% (median 81.131%) in the total relative L-infinity measure. Long-time calculations remain positive and conservative to numerical precision, although the accumulated benefit is configuration dependent because transport continually re-injects frame error. A transport study further reveals a clear trade-off: central-Hermite interface reconstruction strongly suppresses the third-order discrepancy but amplifies the fourth-order discrepancy. The fully central-Hermite collision therefore substantially reduces collision-induced cross-order frame discrepancy, while residual dependence remains due to discrete transport and finite velocity-space representation. This moment-space improvement does not by itself establish a comparable reduction in macroscopic Galilean transport error.

Brauer tree algebras are important and fundamental blocks in the modular representation theory of groups. Aihara develped an algorithm, which we call a mutation reduction, for getting from a Brauer tree algebra to the simpler Brauer star algebra using a sequence of mutations centered on edges. Schaps and Zvi, using the Schaps-Zakay theory of pointing the tree, showed that different algorithms for the sequence of mutations give permutations of the edges. We give a modification of Aihara's algorithm and tested it against the original algorithm for computational efficiency. We prove that all versions of Aihara's algorithm are the fastest possible in the sense of requiring the least number steps to reach the Brauer star, when compared to all possible complete mutation reduction algorithms.

A set $D$ of vertices graph $G$ is a total outer-independent dominating set (TOIDS) of $G$ if every vertex of $G$ is adjacent to at least one vertex in $D$, and $V(G)\setminus D$ is an independent set of $G$. A TOI-coalition in $G$ comprises two disjoint sets of vertices $A$ and $B$ of $G$, neither of which is a TOIDS but whose union $A\cup B$ is a TOIDS of $G$. We say that the sets $A$ and $B$ form a TOI-coalition, and are TOI-coalition partners. A TOI-coalition partition in $G$ is a vertex partition $\Psi =\{V_1, V_2,..., V_k\}$ in which every set forms a TOI-coalition with another set in $\Psi$. The TOI-coalition number $C_t^{oi}(G)$ is the maximum cardinality among all TOI-coalition partitions of $G$. In this work, the above-mentioned concepts are introduced and studied. The existence of a TOI-coalition partition is investigated. Several sharp upper bounds on $C_t^{oi}(G)$ are established. Finally, the exact values of $C_t^{oi}(G)$ for some graph classes are obtained.

Engineered Complete Intersections (ECI's) are a class of sparse polynomial systems frequently arising in a number of contexts, both in pure mathematics (e.g. enumerative geometry) and applications (e.g. chemical reaction networks). Based on theoretical results given by the first author, we give several contributions. First we give a new effective technique to tropicalize such systems by generalizing the classical notion of mixed subdivisions introduced by Huber and Sturmfels (1995) to ECI's with the particular goal to efficiently count solutions of square systems of equations in ECI form. We further design a tropical homotopy continuation algorithm for computing such mixed subdivisions, inspired by Jensen (2016), Malajovich (2017) and Daisey and Ren (2024). Our techniques can be used to numerically solve such systems by coupling them with the algorithms introduced by Helminck, Henriksson and Ren (2024). Finally, we give an algorithm to compute Newton polytopes of eliminants of ECI's. This gives a new way to compute, for example, Newton polytopes of so-called $A$-discriminants. Coupled with evaluation-interpolation paradigms our algorithm gives an efficient approach to compute such eliminants. We implemented our algorithms in the form of a software package which we use to demonstrate their practical feasibility on a range of examples.

In this paper we show that the Hall's universal group does not have finite big Ramsey degrees. Our strategy consists of piggybacking on the recent result of Hubička, Konečný, Todorčević and Zucker (announced at EUROCOMB 2025) that the random edge-labelled graph (the Fraïssé limit of the class of all finite complete edge-labelled graphs where the set of labels is countably infinite) does not have finite big Ramsey degrees. We then use the machinery of category theory to transport this result from the context of edge-labelled graphs to the context of groups. The main step in the process is the construction of a functor from the category of edge-labelled graphs and embeddings to the category of groups and group embeddings which takes finite graphs to finite groups. This makes is possible for us to build a subgroup of the Hall's universal group which encodes the random edge-labelled graph. Our main result shows that several families finite groups, including non-cyclic abelian groups, do not have finite big Ramsey degrees in the Hall's universal group. Actually, we show that all potential candidates for groups with finite big Ramsey degrees have to be solvable.

Designing effective reward functions for model-free reinforcement learning under non-holonomic constraints remains a persistent challenge, often resulting in severe local minima such as policy paralysis or over-conservative hazard avoidance. In this work, we present a parameterized reward shaping framework featuring coverage-gated alignment feedback, drive-direction switch regularization, and an aligned episode termination mechanism evaluated on an autonomous parallel parking task. Crucially, we show that environmental reward parameters and algorithmic hyperparameters are deeply co-dependent, requiring joint meta-optimization to achieve stable convergence. By employing surrogate-based Bayesian optimization, our co-optimized Deep Q-Network (DQN) agent resolves characteristic control failure modes, significantly outperforming uncalibrated baselines across both success rate and trajectory smoothness.

In this paper, we provide an explicit method for determining the zero set of monic quadratic general polynomials over Cayley–Dickson algebras with any base field of characteristic not equal to $2$, which reduces the problem to finding the roots of univariate polynomials over the base field. For the special case of locally-complex Cayley–Dickson algebras over the reals, we prove that every such polynomial has a root. We prove the existence of right eigenvalues for any $2 \times 2$ matrix over Cayley's octonions (the real octonion division algebra), and our root-finding method allows the computation of some of the right eigenvalues.

In rank 2, unconditional verification of the Andrews-Curtis conjecture stands at total relator length 12; at length 13 every balanced trivial-group presentation is AC-trivializable or AC-equivalent to the Akbulut-Kirby presentation AK(3), itself open. At length 14, Shehper et al. reduced the Miller-Schupp family to six hard presentations: four stated AC-equivalent to AK(3) with no published move sequences, and two unresolved. We prove four explicit AC-equivalences among these six as machine-checkable elementary-move certificates, replayable in under a second by a small dependency-free verifier: the classical candidate <x,y | x^-1 y^2 x = y^3, y^-1 x^2 y = x^3> (= MS(2, x^-2 y^-1 x^2 y) up to rotation) is equivalent to the unresolved MS(2, y x^2 y^-1 x^-2) (36 moves); MS(2, y x^2 y x^-2) to MS(2, x^-2 y^-1 x^2 y^-1) (85 moves); and MS(3, y x^2 y) and MS(3, y^-1 x^2 y^-1) each to AK(3) (66 and 13 moves). The latter two are, to our knowledge, the first public explicit certificates for any of the four AK(3)-equivalences asserted without proof by Shehper et al., making the MS(3) branch of the length-14 collapse unconditional. The first two realize the automorphism sigma: x -> x, y -> y^-1 on the remaining open classes. We complement the certificates with a computer-assisted exhaustive minimax analysis of the substitution-move graph: the bottleneck distance from MS(3, y x^2 y) to AK(3) is exactly 19, while any such path from either MS(2) representative to AK(3), or between them, must reach total length at least 27. We further analyze the public classification table of the "Two-Hump" campaign, derive a 214-pair class-merger program, and commit an AC-19 membership audit. All certificates, search engines, verifier, and one-command reproduction are archived.

Random Indexing (RI), used almost exclusively in text analysis (notably Temporal Random Indexing, TRI, for tracking word meaning over time), represents a discrete vocabulary with fixed random vectors and accumulates context by vector summation. We explore transplanting this mechanism to change detection in multitemporal images. A naive transplant using k-means to build the visual vocabulary is unstable: pixels are frequently reassigned between acquisition dates due to small radiometric shifts, destroying the pixel-to-vector correspondence RI depends on. We propose a distance-threshold (leader) clustering vocabulary instead, and give a short formal argument – a stability radius from the vocabulary covering/packing properties – for why this is provably more robust to acquisition noise than k-means. Combined with spatial context accumulation, this yields a training-free change detection pipeline, evaluated on four bi-temporal remote-sensing datasets (agriculture, river, urban PolSAR, wildfire) against a Change Vector Analysis baseline, which it consistently approaches but does not surpass. Multi-seed validation exposed and let us fix a degenerate-vector failure mode in a probabilistic-sparsity RI variant, and revealed a second, unresolved source of variability – sensitivity to the leader-clustering visitation order – which we characterize but, despite three attempted corrections, could not eliminate, reporting it as the main open problem. Results are cross-checked against an independent re-implementation. We report an ablation, a hyperparameter sensitivity analysis, an Otsu-vs-Gaussian-mixture thresholding comparison, and discuss RI incremental accumulation for future long time-series monitoring.

Different from most existing literature in the analysis of distributed optimization algorithms that reports sufficient convergence conditions leading to a conservative convergence rate, this work provides the exact worst-case convergence rates for two typical gradient tracking algorithms, DIGing and AugDGM. By eigen-decomposition, we show that two algorithms share the same average-state dynamics, while they differ from each other in the gradient tracking subsystems, which entirely govern algorithm convergence. Exploiting the diagonal structure of this decomposition, we reduce the stability analysis of MIMO systems to that of a set of parameter-varying SISO systems, from which explicit formulas for the exact worst-case convergence rate can be derived. These formulas clearly show that the optimal worst-case convergence rate of DIGing is larger than that of AugDGM under the same assumptions on objective functions and communication networks. Furthermore, we find that there is an inflection point in the graph connectivity, which is $\sigma =1/3$. For graphs with connectivity better than this inflection point, the optimal convergence rate of centralized gradient descent can be achieved by AugDGM provided the condition number of objective functions is worse enough. On the other hand, for graphs with connectivity worse than this inflection point, the centralized optimal rate can never be achieved. Numerical experiments validate the theoretical results.

In this note we show that the well-known stability test proposed by O'Shea, Zames and Falb is not necessary for robust stability of Lur'e systems with slope-restricted nonlinearities and thereby disprove a conjecture by J. Carrasco. Specifically we construct a full-block multiplier which certifies stability of a constructed Lur'e interconnection that does not admit an OZF multiplier.

Let $(\sigma_{n,\varepsilon})_{n\in\mathbb{Z},\,0<\varepsilon<1}$ be a summation process in the sense of Brezis, that is: \[ \forall \varepsilon\in(0,1),\quad \sup_{n\in\mathbb{Z}}|n\sigma_{n,\varepsilon}|<\infty \qquad \forall n\in\mathbb{Z},\quad \lim_{\varepsilon\rightarrow0^+}\sigma_{n,\varepsilon}=1. \] We prove that for every $0<\alpha<1/3$ there exists $f\in C^{0,\alpha}(S^1;S^1)$ such that \[ \sum_{n\in\mathbb{Z}}\sigma_{n,\varepsilon}n|\widehat f(n)|^2 \] does not converge to $\mathrm{deg}\,f$ as $\varepsilon\rightarrow 0^+$. This gives a negative answer to Open Problem 5.6 from Brezis's list of favourite open problems for all $p>3$ and $0<\alpha<1/3$, leaving the endpoint $C^{0,1/3}$ unresolved. We also observe that, under their literal formulations, Open Problems 5.7 and 5.8 have immediate positive and negative answers, respectively. The proof of the main result combines degree-zero quotients of Blaschke factors with a Baire category argument.

We construct three counterexamples for the special Lagrangian curvature equation (SLCE). First, in dimension two, we use a post-focal branch of a parallel surface with constant positive Gauss curvature to construct an explicit Lipschitz viscosity solution which is not $C^1$. Second, still in dimension two, we construct a sequence of smooth admissible solutions on a fixed rectangle with uniformly bounded $C^1$-norm but unbounded curvature at one point; furthermore, we show that any uniform $C^{1,\beta}$ estimate fails for $\beta > 1/3$. Third, in dimension three and in the subcritical phase, we construct a Mooney-Savin type Lipschitz viscosity solution whose gradient has a jump discontinuity across an analytic surface. These examples demonstrate the sharpness of the recent a priori estimates by Qiu and Zhou, revealing that their structural assumptions-convexity and the critical phase-are strictly necessary.

Let $G=(\mathbb{Z}/4)^4$. We prove that if $X$ is a rationally connected threefold with a faithful action of $G$, then $X$ is $G$-birational to the Fermat quartic threefold. If $X$ is a terminal $G\mathbb{Q}$-Fano threefold, this birational equivalence is biregular. Consequently, the group $G$ acts faithfully on a rationally connected threefold but does not embed into $\operatorname{Cr}_3(\mathbb{C})$. Combined with earlier results, this yields a complete classification of the pairs $(m,r)$ for which $(\mathbb{Z}/m)^r$ embeds into $\operatorname{Cr}_3(\mathbb{C})$, and of those for which it embeds into $\operatorname{Bir}(X)$ for a rationally connected threefold $X$.

Let $V$ and $U$ be independent standard normal random variables. For each Borel-measurable function $\phi\colon\mathbb{R}\to\mathbb{R}$, let $P_\phi\colon\mathbb{R}\to\mathbb{R}$ be a Borel version of the inverse regression $y\mapsto\mathbb{E}[V\mid\phi(V)+U=y]$. We prove that $\phi(v)\in\operatorname*{arg\,max}_{x\in\mathbb{R}} \mathbb{E}[(v-P_\phi(x+U))x]$ for every $v\in\mathbb{R}$ if and only if $\phi=\operatorname{id}_{\mathbb{R}}$. The rigidity result implies that the one-period Gaussian Kyle (1985) insider-trading model has a unique Borel-measurable equilibrium strategy, namely Kyle's affine strategy. Building on preliminary results of McLennan, Monteiro, and Tourky (2017), the proof establishes that, at equilibrium, the total expected loss of noise traders attains a sharp universal upper bound and that any strategy attaining this bound must be affine almost everywhere.

In this paper, we study three-dimensional combinatorial Yamabe flows on locally finite infinite triangulations in Euclidean and hyperbolic background geometries. Under suitable non-degeneracy and bounded-degree assumptions, we establish the short-time existence and uniqueness of the original flows. We further introduce the extended flows by using the continuous extension of solid angles, and prove long-time existence for both extended flows under suitable initial assumptions.