AI for Science

In 1977, Chung, Chung and Liu generalized the definition of the Ramsey number. They introduced the $s$-chromatic Ramsey number as follows. Let $1\leq s< t$ be integers and let $A_{1}, A_{2}, \dots, A_{c}$ be subsets with size $s$ of $[t]$, where $c= {t\choose s}$. For given graphs $G_{1}, G_{2}, \dots, G_{c}$, the {\it $s$-chromatic Ramsey number} $r^{s, t}(G_{1}, G_{2}, \dots, G_{c})$, is the minimum positive integer $N$ such that every $t$-coloring of $E(K_{N})$ yields a copy of $G_{i}$ whose edges are colored by colors in the color set $A_{i}$ for some $i\in [c]$. The {\it star-critical $s$-chromatic Ramsey number} $r_{*}^{s, t}(G_{1}, G_{2}, \dots, G_{c})$, is the minimum integer $\ell$ such that every $t$-coloring of the edges in $K_{N}- E(K_{1, N- 1- \ell})$ yields a copy of $G_{i}$ whose edges are colored by colors in the color set $A_{i}$ for some $i\in [c]$, where $N= r^{s, t}(G_{1}, G_{2}, \dots, G_{c})$. If $G_{1}= G_{2}= \dots= G_{c}= G$, then we simplify them to $r^{s, t}(G)$ (also called the {\it weakened Ramsey number}) and $r^{s, t}_{*}(G)$, respectively. In this paper, we determine all the values of $r^{s, t}(K_{1, m})$ and $r_{*}^{s, t}(K_{1, m})$, and part of the value of $r^{s, t}(K_{1, m_{1}}, K_{1, m_{2}}, \dots, K_{1, m_{c}})$.

Over an arbitrary field, we conduct a comprehensive study of the polynomial identities and codimensions of two- and three-dimensional metabelian non-Lie Leibniz algebras. In addition, we compute the images of multihomogeneous polynomials on two-dimensional Leibniz algebras and, as a consequence, prove that the image of any multilinear polynomial evaluated on such algebras is always a vector space. Our analysis includes the three nontrivial isomorphism classes in dimension two and the ten isomorphism classes in dimension three, all of which are metabelian. In particular, we determine finite bases for their corresponding $T$-ideals and provide explicit bases for the associated relatively free graded algebras.

The outlines of a "Galois theory" for bimeromorphic geometry is here developed, via the study of model-theoretic definable binding groups in the theory CCM of compact complex spaces. As an application, a structure theorem about principal meromorphic bundles with algebraic structure group, and admitting no horizontal subvarieties, is deduced. Examples of algebraic groups arising as binding groups are provided, as is a characterisation of when they are linear. Using binding groups in CCM it is shown that, in contrast to the situation in differentially closed fields, there are many algebraic groups which admit nontrivial definable torsors over acl-closed sets in the theory DCCM of existentially closed differential CCM-structures. A self-contained exposition of the binding group theorem in totally transcendental theories, that emphasises the bitorsorial nature of the construction, is also included.

This article investigates the phenomenon of maximal rigidity in spatial processes, where perfect interpolation of the process is possible from partial information, specifically, from its restriction to a strict subdomain, often resulting in a trivial tail $σ$algebra. A classical example known since the 1930's is that a time series is fully determined by its values on the negative integers if its spectrum has a gap, or at least a sufficiently deep zero. We extend such results to higher dimensions and continuous settings by establishing a connection with the concept of uniqueness pairs, rooted in the uncertainty principle of harmonic analysis. We present several other manifestations of this principle, unify and strengthen seemingly unrelated results across different models: quasicrystals and stealthy processes are shown to be maximally rigid on cones, and discrete integer-valued processes are necessarily periodic when they have a simply connected spectrum. Finally, we identify a surprising class of continuous fields with seemingly standard behavior, such as linear variance and finite dependency range, that undergo a phase transition: they are perfectly interpolable on B(0, $ρ$) for $ρ$ ___ 2 $π$ but exhibit no rigidity for $ρ$ > 2.

A fundamental property of Segre classes is their birational invariance. This invariance implies that the Segre class of a closed subscheme only depends on the integral closure of the defining ideal sheaf. In this paper, we show that, conversely, the Segre class of a closed subscheme encodes an integral dependence criterion for its defining ideal sheaf. As an application, we prove that Aluffi's Segre zeta function provides an integral dependence criterion for homogeneous ideals in polynomial rings.

In this paper we develop an algebraic theory to study the problem of finding the minimum distance point from an algebraic variety with respect to the Hermitian distance function. The theory generalizes the Euclidean Distance degree introduced in arXiv:1309.0049, replacing a positive symmetric bilinear form by a Hermitian form. Various examples are presented to show the robustness of the machinery.

Max-plus algebra is a semiring with addition $a\oplus b = \max(a,b)$ and multiplication $a\otimes b = a+b$. It is applied in cases, such as combinatorial optimization and discrete event systems. We consider the power of max-plus square matrices, which is equivalent to obtaining the all-pair maximum weight paths with a fixed length in the corresponding weighted digraph. Each $n$-by-$n$ matrix admits the CSR expansion that decomposes the matrix into a sum of at most $n$ periodic terms after $O(n^{2})$ times of powers. In this study, we propose an $O(n(m+n \log n))$ time algorithm for the CSR expansion, where $m$ is the number of nonzero entries in the matrix, which improves the $O(n^{4} \log n)$ algorithm known for this problem. Our algorithm is based on finding the roots of the characteristic polynomial of the max-plus matrix. These roots play a similar role to the eigenvalues of the matrix, and become the growth rates of the terms in the CSR expansion.

We derive converse Lyapunov theorems for input-to-state stability (ISS) of linear infinite-dimensional analytic systems. While we show that ISS in general does not imply the existence of a coercive quadratic ISS Lyapunov function, even if the input operator is bounded, we prove that indeed quadratic ISS Lyapunov functions always exist for $p$-admissible input operators with $p<2$, provided the semigroup is similar to a contraction on a Hilbert space. The constructions are semi-explicit and rely on classical results on analytic semigroups and similarity to contractive ones. In the case of self-adjoint generators, they coincide with the canonical Lyapunov function being the norm squared.

In this paper, we develop the mathematical framework for filtering problems arising from biophysical applications where data is collected from confocal laser scanning microscopy recordings of the space-time evolution of intracellular wave dynamics of biophysical quantities. In these applications, signals are described by stochastic partial differential equations (SPDEs) and observations can be modelled as functionals of marked point processes whose intensities depend on the underlying signal. We derive both the unnormalized and normalized filtering equations for these systems, demonstrate the asymptotic consistency and approximations of finite dimensional observation schemes respectively partial observations. Our theoretical results are validated through extensive simulations using synthetic and real data. These findings contribute to a deeper understanding of filtering with point process observations and provide a robust framework for future research in this area.

This is the third work in a series on the (local in time) well-posedness of the initial boundary value problem (IBVP) for the vacuum Einstein equations in general relativity with geometric boundary conditions. Here we study the conformal-mean curvature boundary conditions, consisting of the conformal class of the boundary metric and mean curvature of the boundary. We prove that at metrics of uniformly bounded geometry to all orders, the linearized problem has a solution space with dense range in $C^{\infty}$ and establish a Holmgren-type uniqueness theorem valid for general smooth linearized solutions. These results require the addition of an arbitrary corner angle term at the intersection of the Cauchy surface and the timelike boundary.

We present a novel approach to implementing all-optical Rectified Linear Unit (ReLU) activation functions using compact doubly-resonant cavities with dimensions of approximately $10\,μ\mathrm{m}$. Our design leverages $χ^{(2)}$ nonlinear processes within carefully engineered photonic structures that simultaneously resonate at both fundamental and second-harmonic frequencies. By exploiting the phase-sensitive nature of second-harmonic generation, we demonstrate an optical analog to the ReLU function, achieving femtojoule-level activation energy-comparable to state-of-the-art approaches-while reducing device footprint by two orders of magnitude compared to previous implementations. We develop the theoretical framework using coupled-mode theory and validate it through rigorous finite-difference time-domain simulations. Beyond ReLU, we show that the same physical structure can implement alternative activation functions such as ELU and GELU through simple adjustments to input conditions. Neural network simulations demonstrate that our proposed optical activation functions achieve classification accuracy within $0.4\%$ of ideal electronic implementations while offering significant advantages in energy efficiency and processing speed. This work represents a significant advancement toward realizing energy-efficient, high-density optical neural networks for next-generation artificial intelligence hardware.

Soft and active condensed matter represent a class of fascinating materials that we encounter in our everyday lives -- and constitute life itself. Control signals interact with the dynamics of these systems, and this influence is formalized in control theory and optimal control. Recent advances have employed various control-theoretical methods to design desired dynamics, properties, and functionality. Here we provide an introduction to optimal control aimed at physicists working with soft and active matter. We describe two main categories of control, feedforward control and feedback control, and their corresponding optimal control methods. We emphasize their parallels to Lagrangian and Hamiltonian mechanics, and provide a worked example problem. Finally, we review recent studies of control in soft, active, and related systems. Applying control theory to soft, active, and living systems will lead to an improved understanding of the signal processing, information flows, and actuation that underlie the physics of life.

In this paper, we prove a Heintze-Karcher type inequality for capillary hypersurfaces supported on various hypersurfaces in the hyperbolic space. The equality case only occurs on capillary totally umbilical hypersurfaces. Then we apply this result to prove the Alexandrov type theorem for embedded capillary hypersurfaces in the hyperbolic space. In addition, we prove some other rigidity results for capillary hypersurfaces supported on totally geodesic plane in $\mathbb B^{n+1}_+$.

A persistence diagram is a finite multiset of birth-death pairs representing the lifetimes of topological features across a filtration. Existing functional and kernel representations of persistence diagrams are typically constructed extrinsically through embeddings into auxiliary spaces. For filtrations with finite indexing sets, the associated virtual persistence diagram group obtained by Grothendieck completion of the persistence diagram monoid is a finitely generated lattice. We define a phase map sending each persistence interval to a circular coordinate and a character map aggregating the phases of intervals in a virtual persistence diagram. We introduce heat damping on characters of virtual persistence diagram groups to suppress the unstable frequencies. We derive Lipschitz bounds for the resulting kernels and apply them in a synthetic segmentation experiment.

We show that the canonical actions of the Thompson group V and its generalizations on the Cantor set are not strongly ergodic. This implies that the associated crossed product von Neumann algebras are not full. This also yields a non-embedding result for the Thompson groups.

In this paper, we consider a contextual stochastic optimization problem in which unknown parameters follow distributions that depend on contextual covariates and decisions. The problem is motivated by transportation infrastructure decisions such as facility location or network design. In such high-stakes settings, decisions must often be communicated, justified, and reconsidered under alternative stakeholder requirements. To this end, we propose a framework for computing relative counterfactual explanations. These explanations identify the smallest changes in the covariates required for a solution to satisfy prescribed constraints while limiting the performance loss to a controlled level. Whereas relative explanations have been introduced in prior literature, to the best of our knowledge, this is the first work focusing on problems with binary decision variables and endogenous uncertainty. We propose a methodology that uses the Wasserstein distance as a regularization term in the objective. Beyond improving tractability, this regularization yields explanations with desirable structural properties: it produces sparser counterfactuals, induces smoother transitions in the underlying choice distributions, and keeps the counterfactual behavior close to realistic demand patterns. We illustrate the method using a choice-based competitive facility location problem and present numerical experiments that demonstrate its ability to efficiently compute sparse, plausible, and interpretable explanations. We further validate the framework on a real-world case study of electric vehicle charging station planning in Montreal, where the explanations reveal the minimal capacity investments and environmental conditions required to justify including a candidate location in the charging network.

In this survey, we describe recent progress on asymptotic properties of various automorphic orbits in free groups. In particular, we address the problem of counting potentially positive elements of a given length. We also discuss complexity (worst-case, average-case, and generic-case) of Whitehead's automorphism problem and relevant properties of automorphic orbits, including orbit-blocking words.

A family of compact n-manifolds is locally combinatorially defined (LCD) if it can be specified by a finite number of local triangulations. We show that LCD is equivalent to the existence of a compact branched n-manifold W, such that the family is precisely those manifolds that immerse into W. In subsequent papers, the equivalence will be used to show that, for each of the eight Thurston geometries, the family of closed 3-manifolds admitting that geometry is LCD.

A complex hyperplane arrangement $\mathcal{A}$ is said to be decomposable if there are no elements in the degree 3 part of its holonomy Lie algebra besides those coming from the rank 2 flats. When this purely combinatorial condition is satisfied, it is known that the associated graded Lie algebra of the arrangement group $G$ decomposes (in degrees greater than 1) as a direct product of free Lie algebras. It follows that the $I$-adic completion of the Alexander invariant $B(G)$ also decomposes as a direct sum of "local" invariants and the Chen ranks of $G$ are the sums of the local contributions. Moreover, if $B(G)$ is separated, then the degree 1 cohomology jump loci of the complement of $\mathcal{A}$ have only local components, and the algebraic monodromy of the Milnor fibration is trivial in degree 1.

Graph symmetries identify structural regularities and reduce the computational complexity of network analysis. In weighted graphs, however, exact automorphisms are rare because real-valued weights seldom coincide. We introduce a general framework for detecting approximate symmetries by aggregating weights into discrete categories, generating a sequence of coarser graphs on which classical automorphism analysis applies. The approximation path is fully configurable, based on interaction magnitudes, and can be matched to the empirical weight distribution. Applied to 250 empirical food webs using logarithmic aggregation, the method reveals that automorphisms emerge even at low approximation levels and almost always form small orbits. Orbit sizes rarely exceed two or three vertices, reflecting the combinatorial fragility of larger symmetric sets. Even so, symmetric vertices occupy diverse structural positions in the network and high connectivity does not imply asymmetry. The observation of just local permutations confirms the conclusions of trophic species and niche analysis. A case study demonstrates that automorphisms can also recover latent ecological structure. The minimal aggregation level at which two vertices become substitutable provides a quantitative measure of role similarity. The framework offers a principled, automorphism-based approach for quantifying similarity and redundancy in weighted complex networks.

Let $\mathcal{I} = \{I_k\}_{k \in \mathbb{N}}$ be a graded family of monomial ideal. We use the Newton-Okounkov body of $\mathcal{I}$ to: (a) give a characterization for the Noetherian property of the Rees algebra of the family; and (b) present a combinatorial interpretation for the analytic spread of the family. We also apply these results to investigate the generation type and the Veronese degree of the symbolic Rees algebra of a monomial ideal.

The starting point of this paper is a duality for sequences of natural numbers which, under mild hypotheses, interchanges subadditive and superadditive sequences and inverts their asymptotic growth constants. We are motivated to explore this sequence duality since it arises naturally in at least two important algebraic-geometric contexts. The first context is Macaulay-Matlis duality, where the sequence of initial degrees of the family of symbolic powers of a radical ideal is dual to the sequence of Castelnuovo-Mumford regularity values of a quotient by ideals generated by powers of linear forms. This philosophy is drawn from an influential paper of Emsalem and Iarrobino. We generalize this duality to differentially closed graded filtrations of ideals. In a different direction, we establish a duality between the sequence of Castelnuovo-Mumford regularity values of the symbolic powers of certain ideals and a geometrically inspired sequence we term the jet separation sequence. We show that this duality underpins the reciprocity between two important geometric invariants: the multipoint Seshadri constant and the asymptotic regularity of a set of points in projective space.

We extend discrete Morse-Bott theory to the setting of loop-free (or acyclic) categories. First of all, we state a homological version of Quillen's Theorem A in this context and introduce the notion of cellular categories. Second, we present a notion of vector field for loop-free categories. Third, we prove a homological collapsing theorem in the absence of critical objects in order to obtain the Morse inequalities. Examples are provided through the exposition. This answers partially a question by T. John: whether there is a Morse theory for loop-free (or acyclic) categories? [14].

Elementary fuzzy Cellular Automata (CA) are known as continuous counterpart of elementary CA, which are 2-state CA, via the polynomial representation of local rules. In this paper, we first develop a new fuzzification methodology for $q$-state CA. It is based on the vector representation of $q$-state CA, that is, the $q$-states are assigned to the standard basis vectors of the $q$-dimensional real space and the local rule can be expressed by a tuple of $q$ polynomials. Then, the $q$-state vector-valued fuzzy CA are defined by expanding the set of the states to the convex hull of the standard basis vectors in the $q$-dimensional real space. The vector representation of states enables us to enumerate the number-conserving rules of 3-state vector-valued fuzzy CA in a systematic way.

We define the eñe product for the multiplicative group of polynomials and formal power series with coefficients on a commutative ring and unitary constant coefficient. This defines a commutative ring structure where multiplication is the additive structure and the eñe product is the multiplicative one. For polynomials with complex coefficients, the eñe product acts as a multiplicative convolution of their divisor. We study its algebraic properties, its relation to symmetric functions on an infinite number of variables, to tensor products, and Hecke operators. The exponential linearizes also the eñe product. The eñe product extends to rational functions and formal meromorphic functions. We also study the analytic properties over the complex numbers, and for entire functions. The eñe product respects Hadamard-Weierstrass factorization and is related to the Hadamard product. The eñe product plays a central role in predicting the phenomenon of the "statistics on Riemann zeros" for Riemann zeta function and general Dirichlet $L$-functions discovered by the author. It also gives reasons to believe in the Riemann Hypothesis as explained in the survey "Notes on the Riemann Hypothesis".

In this paper, we give an axiomatic characterization of the cap product in the Hochschild theory of associative unital algebras which are projective over a commutative unital ring. We also give an interpretation of the cap product with coefficients in the algebra via chain maps. We illustrate these results by computing the cap product for truncated polynomial algebras $k[x]/(x^N)$ and for polynomial algebras, where it is identified with the contraction of differential forms by polyvector fields.

Epicormic branches arise from dormant buds patterned during the growth of previous years. Dormant epicormic buds remain on the surface of trees, pushed outward from the pith during secondary growth, but maintaining vascular connections. Epicormic buds can be reactivated, either through natural processes or intentionally, to rejuvenate orchards and control tree architecture. Because epicormic structures are embedded within secondary growth, tomographic approaches are a useful method to study them and understand their development. We apply techniques from image processing to determine the locations of epicormic vascular traces embedded within secondary growth of sweet cherry (Prunus avium L.), revealing the juvenile phyllotactic pattern in the trunk of an adult tree. Techniques include breadth-first search to find the pith of the tree, edge detection to approximate the radius, and a conversion to polar coordinates to threshold and segment phyllotactic features. Intensity values from Magnetic Resonance Imaging (MRI) of the trunk are projected onto the surface of a perfect cylinder to find the locations of traces in the "boundary image". Mathematical phyllotaxy provides a means to capture the patterns in the boundary image by modeling phyllotactic parameters. Our cherry tree specimen has the conspicuous parastichy pair $(2,3)$, phyllotactic fraction 2/5, and divergence angle of approximately 143 degrees. The methods described not only provide a framework to study phyllotaxy, but for image processing of volumetric image data in plants. Our results have practical implications for orchard rejuvenation and directed approaches to influence tree architecture. The study of epicormic structures, which are hidden within secondary growth, using tomographic methods also opens the possibility of studying the genetic and environmental basis of such structures.

There are many metrics available to compare phylogenetic trees since this is a fundamental task in computational biology. In this paper, we focus on one such metric, the $\ell^\infty$-cophenetic metric introduced by Cardona et al. This metric works by representing a phylogenetic tree with $n$ labeled leaves as a point in $\mathbb{R}^{n(n+1)/2}$ known as the cophenetic vector, then comparing the two resulting Euclidean points using the $\ell^\infty$ distance. Meanwhile, the interleaving distance is a formal categorical construction generalized from the definition of Chazal et al., originally introduced to compare persistence modules arising from the field of topological data analysis. We show that the $\ell^\infty$-cophenetic metric is an example of an interleaving distance. To do this, we define phylogenetic trees as a category of merge trees with some additional structure; namely labelings on the leaves plus a requirement that morphisms respect these labels. Then we can use the definition of a flow on this category to give an interleaving distance. Finally, we show that, because of the additional structure given by the categories defined, the map sending a labeled merge tree to the cophenetic vector is, in fact, an isometric embedding, thus proving that the $\ell^\infty$-cophenetic metric is, in fact, an interleaving distance.

We introduce the first unified theory for target tracking using Multiple Hypothesis Tracking, Topological Data Analysis, and machine learning. Our string of innovations are 1) robust topological features are used to encode behavioral information, 2) statistical models are fitted to distributions over these topological features, and 3) the target type classification methods of Wigren and Bar Shalom et al. are employed to exploit the resulting likelihoods for topological features inside of the tracking procedure. To demonstrate the efficacy of our approach, we test our procedure on synthetic vehicular data generated by the Simulation of Urban Mobility package.

We estimate that data centers caused average retail electricity rates to fall modestly in the United States from 2015 to 2024 using an instrumental variables approach. Despite prevailing sentiment, the finding is consistent with economic reasoning: existing large power system fixed costs, economies of scale in transmission and distribution, and declining unit costs for generation imply that durable demand growth lowers average prices. We find patterns of economies of scale for transmission, distribution, and generation costs as well as within and across retail customer classes. We caution that future supply constraints could reverse the effect.