Papers

For positive integers $n$ and $m$, let $f(n,m)$ be the least integer $h\ge0$ such that $(m,m+h]$ contains distinct integers $a_1,\ldots,a_n$ satisfying $i\mid a_i$ for $1\le i\le n$, and put $F(n)=\max_{m\in\mathbb{N}} f(n,m)$. A recent theorem of van Doorn [INTEGERS, 2026; arXiv:2601.16972] gives $F(n)-f(n,n)>0.36\,n\log n/\log\log n$ for sufficiently large $n$. We prove \[ \liminf_{n\to\infty} \frac{F(n)-f(n,n)}{n\log n} \ge \frac{1}{\mathrm{e}}. \] Thus, for every fixed $c<1/\mathrm{e}$ and all sufficiently large $n$, some interval of length $c\,n\log n$ contains no system of pairwise distinct multiples of $1,2,\ldots,n$. The proof applies an Erdős–Pomerance smooth-number obstruction at starting points $m\asymp n\log n$, using local saddle-point estimates of Hildebrand and Tenenbaum.

Dimensionality reduction has proven powerful for identifying neural manifolds, which are low-dimensional structures underlying high-dimensional neural activity. These low-dimensional representations have improved the interpretability of population-level coding. Yet whether such low-dimensional representations are biologically relevant and confer functional advantages in learning systems, or merely reflect neuron-level activity, remains contested in neuroscience. We show that an explicit information bottleneck forcing a recurrent neural network to learn a low-dimensional representation is necessary for rotational and out-of-distribution generalisation in a time-series prediction task. Using information-theoretic measures of causal emergence, we characterise the dynamics of this representation across the memorisation-to-generalisation transition, finding a non-monotonic trajectory which shows an initial decrease, a minimum, and a subsequent rise to a maximum, even as prediction loss falls monotonically. This trajectory scales with task complexity, and the magnitude of emergent structure reliably predicts generalisation performance. Analysis of hippocampal CA1 and medial prefrontal activity in rats learning an alternating maze task reveals analogous non-monotonic emergence dynamics that track behavioural performance. Together, these findings indicate that the ability of neural networks to learn compact, distributed and emergent representations confers a functional advantage for generalisation, supporting a causal role for learned representations in cognition.

We develop a weak formulation of spacelike mean curvature flow in pseudo-Euclidean space. The framework is based on spacelike integer rectifiable varifolds and a pseudo-Euclidean version of first variation. Particular attention is paid to the presence of a fixed spacelike boundary. We introduce a generalized mean curvature vector and a weak spacelike conormal along the boundary. Under natural uniform spacelikeness and curvature assumptions, we prove a closure theorem for spacelike varifolds with boundary. We then define spacelike Brakke flows by means of the corresponding pseudo-Euclidean Brakke inequality, and prove a compactness theorem for sequences of spacelike Brakke flows with fixed boundary. Basic monotonicity properties are established, reflecting the sign structure of the ambient indefinite metric. Finally, we adapt Ilmanen's elliptic regularization procedure to prove existence of spacelike Brakke flows, and the local regularity theorem of White to the pseudo-Euclidean setting. The results provide a varifold setting for studying weak spacelike mean curvature flow beyond singularities.

In this paper we investigate sharp upper bounds for the first derivative of rational Bézier curves. A long-standing conjecture posited that the linear bound $\|\mathbf{R}'(t)\| \le n\Omega \max\|\Delta_i\|$ holds for all degrees. We prove that the bound is indeed valid for $n \leq 6$, thus resolving the last open low-degree case. The problem is reformulated as maximizing a variance-like function over a compact box. Using a block argument we show that optima can only appear on one-dimensional faces, reducing the task to a finite family of polynomial inequalities, which are verified exactly via real quantifier elimination. A notable practical feature is that the bound can be evaluated in linear time with respect to the degree, making it attractive for real-time geometric processing. The same structural analysis illustrates the failure for $n=7$ and outlines how the true worst-case constant can be computed.

Let $\|M\|_{\Delta}$ denote the simplicial volume of $M$, $V_r(X,h)=\sup_{x\in X}\operatorname{Vol}_h\big(B_h(x,r)\big)$, and $\mathbb{H}^n$ denotes hyperbolic $n$-space. We prove that, if a closed oriented $n$-manifold $M$ admits a hyperbolic metric, then there is a dimensional constant $\delta_n>0$ such that every Riemannian metric $g$ on $M$ with \[ \frac{\operatorname{Vol}_g(M)}{\|M\|_{\Delta}}<\delta_n \] satisfies \[ V_r(\widetilde M,\widetilde g)\ge V_r(\mathbb{H}^n) \quad\text{for every }r\ge 1. \]

The purpose of this note is to establish further properties of the variable Lorentz spaces $\mathfrak{L}^{p(\cdot), q(\cdot)}(\mathbb{R}^n)$ introduced by L. Ephremidze, V. Kokilashvili and S. Samko, which will allow us to apply the theory of Hardy spaces associated with ball quasi-Banach function spaces, and so define the variable Hardy-Lorentz spaces associated with $\mathfrak{L}^{p(\cdot), q(\cdot)}(\mathbb{R}^n)$. Then, the finite and infinite atomic decompositions for these spaces will be deduced immediately. We also obtain the boundedness of singular integrals and fractional type operators in variable Hardy-Lorentz spaces, and provide a Fefferman-Stein vector-valued inequality for the fractional maximal operator on the $r$-convexification of variable Lorentz spaces.

We provide a definitive analysis of the number of correct guesses in the complete-feedback card guessing game after an asymmetric single-shelf shuffle with parameter $p\in (0, 1)$. We explicitly describe the optimal strategy that maximizes the expected number of correct guesses. We study the number of correct guesses, under an optimal strategy, using methods from analytic combinatorics. In addition, we find the explicit distribution for the number of correct guesses, and thus find the mean and the variance for the number of correct guesses. We show that the distribution of the number of correct guesses is log-concave. We also study the limiting behaviour of the number of correct guesses as the number of cards goes to infinity. In particular, we prove a (local) central limit theorem and a large deviation principle with an explicit rate function. We also prove phase transitions for the number of correct guesses near $p=0$ and $p=1$. Prior to this work, only the optimal strategy and the expected number of correct guesses under the optimal strategy were known for the special case of $p=1/2$.

In this paper, we develop a direction adaptation evolution strategy (DAES) – a new MAES-type method – for noisy derivative-free optimization, designed to reconcile the population-based search mechanisms of evolution strategies with rigorous complexity analysis. Unlike standard MAES schemes, DAES fixes the adaptation matrix to the identity and replaces matrix adaptation with a structured direction-generation mechanism based on symmetric sampling, joint sorting-selection of noisy function values, three-group recombination, and a new triangular search direction. Specifically, candidates are sampled along paired positive and negative directions, their inexact function values are jointly ranked, and the reordered directions are partitioned into three groups to construct three recombination points whose geometry defines the triangular direction. A signed sufficient-decrease search and extrapolation mechanism is then applied along this direction. This structure yields a population-based MAES-type algorithm that retains competitive practical behavior while being amenable to nonasymptotic analysis under noisy evaluations. We establish high-probability complexity bounds for nonconvex, convex, and strongly convex objective functions and derive corresponding guarantees at the noise-limited accuracy level. To the best of our knowledge, these results provide the first high-probability complexity guarantees for a noisy MAES-type derivative-free method with this direction-adaptation structure. Finally, numerical experiments on the 655 prince test problems from the BARON collection compare DAES with the advanced MAES-type solver MADFO and show a favorable trade-off between evaluation efficiency and ultimate robustness.

A binary $(n,m)$ quantum random access code (QRAC) compresses an $n$-bit classical string into an $m$-qubit quantum state, from which a decoder attempts to recover a randomly selected target bit. Of particular interest is the optimal average probability of success, $P^{Q,\mathrm{avg},\mathrm{opt}}_{n,m}$, which was numerically conjectured to satisfy the bound $P^{Q,\mathrm{avg},\mathrm{opt}}_{n,m}\leq \frac{1}{2}+\frac{1}{2}\sqrt{\frac{m}{n}}$. Recent constructions of $(n,n-1)$ QRACs by Suzuki and $(n,n-2)$ QRACs by Akibue et al. meet this bound exactly, raising the question of their strict optimality. In this work, we settle this question by proving the conjectured upper bound for $m\in\{n-1,n-2\}$, thereby precisely determining $P^{Q,\mathrm{avg},\mathrm{opt}}_{n,n-1}$ and $P^{Q,\mathrm{avg},\mathrm{opt}}_{n,n-2}$. The proof utilizes a translation recently studied by Lin and de Wolf from local to global reconstruction via pretty good measurement, along with dimensional and positive-semidefinite constraints on an induced channel.

We study ball-covering properties of dual Banach spaces and their connections with the geometry of predual unit balls. One of our main results shows that, for every separable Banach space $X$, the unit ball $B_X$ is a slicely countably determined set if and only if $\operatorname{bc}(X^*)=1$, where $\operatorname{bc}(\cdot)$ is the ball-covering index introduced by A. J. Guirao, A. Lissitsin, and V. Montesinos. We obtain several sufficient conditions for the uniform ball-covering property in dual spaces, including duals of spaces with a $K$-unconditional basis for $K<2$, and duals of separable spaces whose unit ball is the closed convex hull of a set of uniformly strongly exposed points. The constant $2$ is sharp: there is a space with a $2$-unconditional basis whose dual fails the ball-covering property. Applications are given to spaces of operators and to Lipschitz spaces. In particular, $\mathcal L(L_p[0,1])$ has the uniform ball-covering property for every $1<p<\infty$, which answers a question posed by Q. Bao, R. Liu, and J. Shen. As an application to Lipschitz spaces, we prove that $\operatorname{Lip}_0(M)$ has the uniform ball-covering property whenever $M$ is a separable complete ultrametric or Hölder metric space.

We show that a finite volume cusped hyperbolic 3-manifold admits at most finitely many veering triangulations, and in fact this number is bounded above by the number of Giroux torsion free tight contact structures. This resolves Kirby problem K3 3.21e. More generally, for any 3-manifold $M$ and any link $L$, we show finiteness for the family of pseudo-Anosov flows which admits a \emph{strictly positive Birkhoff section relative to $L$}. Combined with work of Li, this implies that a fixed closed $3$-manifold admits at most finitely many pseudo-Anosov flows without perfect fits. This resolves Kirby problem K3 3.21d.

Let $(R,\frak m)$ and $(S, \frak n)$ be commutative Noetherian local rings, let $\varphi: R\to S$ be a flat local homomorphism. In this paper, we first characterize the ascent and descent of Buchsbaum modules and generalized Cohen-Macaulay modules via the flat extension $\varphi$. Then, we provide relationships between the parameter ideals of a finitely generated $R$-module $M$ and those of $M\otimes_RS$ under the assumption that $\frak{n} = \frak{m} S.$

This paper studies the Cauchy problem for the fully parabolic Keller-Segel system. The main results show that there exists a critical threshold $A_{\rm crit}>0$ for steady states $(A,A)$ such that the steady states are nonlinearly stable when $A\le A_{\rm crit}$ and nonlinearly unstable when $A>A_{\rm crit}$. We discuss asymptotic convergence rates as well. In the subcritical case $A<A_{\rm crit}$, the rates correspond to those of the heat equation, and in the critical case $A=A_{\rm crit}$, the rates correspond to half those of the heat equation.

In this paper we investigate the global existence, blow-up and standing waves for the $L^{2}$-supercritical nonlinear Klein-Gordon equations with quadratic asymmetric interaction (LSNKG). First, by introducing a suitable auxiliary functional, using concavity analysis and virial estimates, we obtain a finite time blow-up result for solutions to the Cauchy problem of (LSNKG) when the initial energy is negative. Next, by defining appropriate functionals, manifolds and a constrained variational problem, we employ variational method and the Lagrange multiplier method to derive the existence of ground state solutions for the corresponding nonlinear elliptic (steady-state) system, thereby to establish the existence of standing wave with the ground state for (LSNKG). Then, using the variational characterization of the ground state solutions and constructing invariant sets under the flow generated by the Cauchy problem for (LSNKG), we combine the potential well argument with concavity analysis to establish a sharp threshold between blow-up in finite time and global existence. Finally, by exploiting the variational characterization of the ground state, introducing appropriate scalings, and choosing suitable initial data based on the ground state, we justify the instability of standing wave with the ground state for (LSNKG).

Given a holomorphic matrix-valued function, the poles of its sketched resolvent are generically its eigenvalues. Once a good rational approximation of the sketched resolvent is obtained, the poles of this rational approximation typically lie close to those eigenvalues, thus providing a flexible framework for solving both linear and nonlinear eigenvalue problems. However, the accuracy of the computed eigenvalues is limited and remains poorly understood. This paper analyzes the convergence of this approach and demonstrates the effectiveness of two techniques to improve accuracy: block probing and zooming in. We also establish the backward and forward stability of polefinding for a barycentric rational form via a generalized eigenproblem. Numerical experiments demonstrate the sharpness of our theoretical results.

When a complex surface $X$ admits a nowhere vanishing holomorphic 2-form, it determines a (holomorphic) symplectic structure on $X$. We study the symplectic geometry of such a symplectic structure when $X$ is an elliptic surface. When the elliptic fibration is nonisotrivial, we define a factorization of Kodaira's functional invariant, called the symplecto-functional invariant and prove that the symplecto-functional invariant determines the symplectic geometry of a nonisotrivial elliptic fibration. This leads to a classification of isogenies of nonisotrivial symplectic elliptic fibrations with a fixed source. We also classify isogenies of symplectic elliptic fibrations with a fixed target by studying symplectic automorphisms of germs of singular fibers. As an application, we prove that a symplecto-biholomorphic map between germs of fibers of nonisotrivial elliptic K3 surfaces can be extended to compositions of isogenies of K3 surfaces.

Let $X_1,\ldots,X_N$ be independent random vectors in $\mathbb{R}^n$ with common isotropic log-concave distribution $\mu$ and set $P_{N,n}^{\mu}:=\operatorname{conv}\{\pm X_i:1\leqslant i\leqslant N\}$. Assume that $N/n=\gamma\geqslant \gamma_0$ where $\gamma_0>1$ is an absolute constant. We prove that with probability at least $1-C\gamma\exp(-c n^{1/4})$ every $k$-dimensional subspace $E$ of $(\mathbb{R}^n,\|\cdot\|_{P_{N,n}^{\mu}})$ satisfies $d_{\mathrm{BM}} (E,\ell_\infty^k) \geqslant c\gamma^{-C}k^\alpha$ for every $1\leqslant k\leqslant n$ where $c,C,\alpha>0$ are absolute constants. Consequently, with the same probability, $(\mathbb{R}^n,\|\cdot\|_{P_{N,n}^{\mu}})$ has cotype $q(\gamma)<\infty$ with cotype constant depending only on $\gamma$, in particular the cotype exponent and the cotype constant are independent of $n$ and of $\mu$. The proof adapts the deterministic coefficient scheme of Huang-Tikhomirov replacing the Gaussian estimates in their argument by estimates for isotropic log-concave random matrices. As an application, using the log-concave extension of Gluskin's theorem, we obtain a separable Banach space of finite cotype for which the Banach-Mazur diameter of its $k$-dimensional subspaces is of order $k$ and whose finite-dimensional building blocks are generated by isotropic log-concave random polytopes.

In radiation transfer simulations, an M1 method achieves substantial computational savings by replacing the full angular transport equation with a low-order moment system. Because this reduced system is not closed, a closure model is required to represent the unknown higher-order moments using lower-order moments. While machine learning (ML)-based closures can improve accuracy beyond classical analytic closures, unconstrained learned closures may produce non-real characteristic speeds and consequently cause numerical solver breakdown. To guarantee real eigenvalues of the Jacobian associated with ML closures, we propose a hyperbolic neural closure for the M1 radiative transfer system. Rather than directly predicting closure terms, we parameterize the Jacobian through two neural networks: (i) a symmetric matrix network and (ii) a strictly convex entropy network whose Hessian defines a positive definite symmetrizer. These components are combined to yield a Jacobian that is similar to a symmetric matrix, thereby ensuring real eigenvalues. The closure is then reconstructed by numerical integration of the learned Jacobian field along a prescribed integration path. Numerical experiments show that the proposed closure not only achieves higher closure accuracy than classical analytic closures, but also improves solution accuracy and remains stable in discontinuous Galerkin simulations for radiative transfer problems.

The thrust of this article is to show that on all Banach spaces, stochastic approximation holds when the noise sequence is an i.i.d. sequence with mean 0, without imposing any condition on the geometry of the space. Also, the same is true when the noise is a sequence of independent random variables under appropriate conditions on the moment. In this case, we need to require that the noise sequence is tight.

We develop a sketch-and-restart framework for computing the action of a matrix function on a vector, $f(A) b$, where $A$ is large, sparse, and non-Hermitian. The framework combines quadrature-based restarting with Arnoldi-like decompositions generated by sketched or truncated Arnoldi processes. Within this framework, we develop two classes of restarted algorithms. The first uses a fixed Krylov subspace dimension and is based either on the sketched Arnoldi process or on a new sketched harmonic Arnoldi process proposed in this work. The second class chooses the Krylov subspace dimension adaptively by running the truncated Arnoldi process until the condition number of the generated basis, estimated from its sketch, exceeds a prescribed threshold. We also establish the convergence of the restarted sketched harmonic Arnoldi method for Stieltjes functions under the assumption that $A$ is positive real. Numerical experiments demonstrate the effectiveness of the proposed framework, including the computational savings achieved through sketching, the storage reduction enabled by adaptive truncation, and the acceleration obtained from thick restarting.

We investigate the Ball Covering Property (BCP) of Banach spaces through the lens of Birkhoff-James orthogonality, yielding a new geometric characterization of the property. Applying this framework, we characterize the BCP of the bounded linear operator space $\mathbb{L}(\mathbb{X}, \mathbb{Y})$ under specific conditions on $\mathbb{X}$ and $\mathbb{Y}$, proving that an earlier necessary condition is sufficient. As an application, we provide a complete affirmative answer to an open question concerning the BCP of $\mathbb{L}(L^p[0,1])$. We further apply our results to vector-valued Lipschitz spaces $\operatorname{Lip}_0(M,\mathbb{Y})$, obtaining a characterization of the BCP in this setting under the assumption of the Radon-Nikodým property. We also study the stability of the BCP under $p$-norm direct sums. In finite dimensions, we provide a sufficient condition for an $n$-dimensional Banach space to have a minimal covering by $n+1$ balls. Furthermore, we find an upper bound for the minimal ball covering number of $\mathbb{L}(\mathbb{X}, \mathbb{Y})$ in the finite-dimensional setting and prove that this number is exactly $mn+1$ when $\mathbb{X}$ is an $m$-dimensional strictly convex space and $\mathbb{Y}$ is an $n$-dimensional smooth space.

Let $K$ be a field and put $L_d=L_K(R_d)\cong L_K(1,d)$. Ordered leaf sets in the rooted $d$-ary tree determine copies of general linear groups over $K$ inside $L_d^\times$. We prove that these copies generate $L_d^\times$ for every $d\geq2$. In the binary case, $L_2^\times=\langle 1+eaf^*,1+fbe^*:a,b\in L_2\rangle$. We characterize finite generation of $L_d^\times$, determine the subgroup represented by monomial matrices, and embed $\GL_\infty(K)$ in $L_2^\times$. Over a finite field, finite presentability of $L_d^\times$ is equivalent to finite generation of the unstable $K_2$-group $K_2(n,L_d)$ for every $n=1+r(d-1)\geq5$, where $r\geq0$; we also compute $K_2(L_d)$.

Kerckhoff proved that the geodesic length function $\ell_\Omega$ of a filling $\Omega$ on $S_g$ attains a unique minimum in Teichmüller space. Recent work of Ernesto Girondo et al. computed these minima for uniform fillings using the algebraic machinery of dessins d'enfants and Grothendieck-Belyi surfaces. We present an elementary optimization approach for $4$-regular topological uniform fillings, bypassing this framework. Furthermore, we analyze two special classes of non-uniform $4$-regular fillings using fat graphs and optimization techniques. We explicitly compute their minima and prove that in both classes, the minimum of these length functions is attained at a triangle surface.

We study the additive structure of dense subset sum in multi-dimension, and use the structure to develop efficient algorithms for the dense subset sum problem. More precisely, given a set $A$ of $n$ vectors in the $d$-dimensional hyperrectangle $[N_1]\times [N_2]\times\cdots\times [N_d]$, we study the structure of $\mathcal{S}(A)$, which is the set of all subset sums of $A$. We focus on the dense regime of the problem where $n \gg \sqrt{\Phi}$ and $\Phi = N_1 \times \cdots \times N_d$. We show that for any constant $d\geq 1$, if $n \gg \sqrt{\Phi}$, then $\mathcal{S}(A)$ contains a long generalized progression in multi-dimension. If we further have that no non-trivial lattice can contain the majority of $A$, then $\mathcal{S}(A)$ contains all the integer points in the zonotope $\{x_1\vec{a}_1 + \cdots + x_n\vec{a}_n: o(1)\leq x_j \leq 1-o(1), x_j \in \mathbb{R}\}$. Compared to the previous results for $d \geq 2$, our result significantly reduces the density threshold and enlarges the region inside which all the integer points belong to $\mathcal{S}(A)$. Also, it matches the bound for the 1-dimensional case. Using our combinatorics result, we also develop an $\tilde{O}(n)$-time algorithm for the dense subset sum problem in multi-dimension.

Let $n\geqslant 1, \Omega\subset\mathbb{H}^n $ be a domain. We construct a Koszul-type complex for the ideal sheaf $\mathcal{I}_X^{(k)}$ of $k$-regular functions vanishing on $X=\{(q_0, q_1, \cdots, q_{n-1})\in \Omega: q_0=0\}$ in several quaternionic variables: $$0\to \mathcal{R}^{(k+2)}\xrightarrow{\widetilde{\mathscr{L}}^{(k)}} \mathcal{R}^{(k+1)}\oplus\mathcal{R}^{(k+1)}\xrightarrow{\mathscr{L}^{(k)}} \mathcal{I}_X^{(k)}\to 0,$$ where $k\geqslant 0$, $\mathcal{R}^{(k)}$ is the sheaf of $k$-regular functions on $\Omega$, $\widetilde{\mathscr{L}}^{(k)}=(-L_1^{(k+2)},L_0^{(k+2)})^{T}$, $\mathscr{L}^{(k)}=(L_0^{(k+1)},L_1^{(k+1)})$, and $L_0^{(k)},L_1^{(k)}$ are multiplication-like operators on $k$-regular functions. This gives the quaternionic analogue of the classical Koszul complex. And we present the long exact sequence in cohomology for the case $\Omega\cap\{q_0=0\}=\emptyset$ with explicit differential connecting maps, by applying the Cauchy-Fueter complex and cohomological methods. As an application, in the special case $n=1, k=1$, the operator pair $(L_0^{(1)}, L_1^{(1)})$ is shown to be surjective if and only if $H^3(\Omega, \mathbb{R})=0$. Furthermore, a cohomological vanishing criterion is given for $H^1(\Omega,\mathcal{I}_X^{(k)})$; under this criterion, every $k$-regular function on $\{q_0=0\}\cap\Omega$ extends to a $k$-regular function on $\Omega$.

For an affine Lie algebra $\mathfrak g$ of type $A^{(2)}_{2n}$ we get a special set of partitions for the highest weight modules $V(\Lambda)$, where $\Lambda=\Lambda_n$, and we also get a some results for the canonical basis elements. We demonstrate a symmetry of weights with fixed $n $-content around the weights gotten from the empty partition by adding multiples of the null root. We also find the highest power of the quantum parameter $q$ in an important set of canonical basis elements.

Consider a finite abelian extension $K/k$ of global fields with Galois group $G$. We study the rank-one component of the generalized Stickelberger module associated with $K/k$ and a finite set $S$ of places of $k$. Under explicit splitting conditions on $S$, we compute generalized indices of this module with respect to the torsion-free part of the $S$-unit group of $K$. We also obtain $p$-primary refinements which include the non-semisimple case $p\mid |G|$. As applications, we derive divisibility relations between the $S$-class numbers of $K$ and $k$, both for number fields and for function fields.

S4-subordination algebras are a generalization of the closure algebras. In this paper, we give a topological representation for S4-subordination algebras by means of bitopological spaces $\left<X,\tau,\tau_{S}\right>$, where $\left<X,\tau\right>$ is a Stone space and $\tau_{S}$ is a topology that enables the characterization of the subordination relation. We apply this bitopological representation to give a characterization of S5-subordination algebras and lattice subordinations. We also show that there exists a bijective correspondence between congruence compatible with the subordination and certain closed subsets of the Stone space $\left<X,\tau\right>$ that are also saturated sets of the space $\left<X,\tau_{S}\right>$. Additionally, we explore two types of morphisms between S4-subordination algebras: one based on Boolean homomorphisms and another based on meet-homomorphisms. Finally, we provide a topological representation for each type of morphism.

The viral 2015 photograph known as "The Dress" divides observers into two camps because it is ambiguous: the same image colors can be explained either as a blue-black surface under one illuminant or as a white-gold surface under another. We propose a geometric account in which the ambiguity arises from a singularity in intrinsic image decomposition, the inverse problem of separating an observed image into reflectance and illumination. Our central claim is that the prior-mode section, i.e. the prior-preferred decomposition, switches across an ambiguity boundary in image space, and that any smooth learned model can only approximate this discontinuous switch by forming a thin transition layer. This predicts two observable signatures, where $\Delta$ is the jump between branches of the prior-mode section and $\lambda$ is the regularization strength: for inverse decomposers, an albedo Jacobian scaling as $|\Delta|/\sqrt{\lambda}$; and for forward encoders, the Fernet curvature that blows up on a scale of $1/\sqrt{\lambda}$. On CGIntrinsics ($N=1998$ images, $n=2\times 10^7$ pixels), the color-temperature albedo Jacobian of Careaga DPT has partial Spearman correlation $r=0.41$ with dense ground-truth albedo error, compared with $r=0.087$ and $r=0.021$ for brightness and saturation controls. On "The Dress", CLIP ViT-L/14 exhibits a latent curvature peak of $\kappa=73.03$ at $6473\,\mathrm{K}$, one sampled step from D65 daylight, while a control dress image peaks at $\kappa=34.75$ with no comparable feature near D65. The same characteristic appears across architectures (U-Net inverter, diffusion inverter, ViT encoder) and datasets (rendered indoor scenes, web photograph), each measured with the observable appropriate to its model class.

The metric dimension, which has lots of variants and numerous applications in other fields, is one of the most important and most extensively studied topics in metric graph theory. Results in which resolvability is achieved by considering multisets of distances from a fixed vertex, instead of vectors as in the original version, are surveyed. The concepts discussed are multiset dimension, outer multiset dimension, local multiset dimension, edge multiset dimension, and $k$-multiset antidimension. Along the way, sharp lower bounds on the outer multiset dimension of diameter two graphs and join graphs with edgeless graphs are proved, which solves two open problems from the literature. New results on graphs with local multiset dimension equal to two are also proved. In particular, such graphs are characterized among block graphs. Finally, a list of open problems from the literature is compiled, and several new problems are added to the list for future research.