We prove the local motivic monodromy conjecture for singularities that are nondegenerate with respect to a simplicial Newton polyhedron. It follows that all poles of the local topological zeta functions of such singularities correspond to eigenvalues of monodromy acting on the cohomology of the Milnor fiber of some nearby point, as do the poles of Igusa's local $p$-adic zeta functions for large primes $p$.

This study provides new evidence on how medical care mitigates the economic consequences of health shocks for individuals and their partners. To identify causal effects, I focus on medical scientific discoveries and exploit longitudinal administrative data for Sweden, using a triple differences design. The results indicate that medical innovation strongly mitigates the negative economic consequences of health shocks for individuals and have spillover effects on their partners. These spillovers are relatively large because medical innovation compensates for partners wage losses in conditions when welfare support for caregiving is insufficient. Overall, the findings indicate that medical innovation not only produces substantial economic gains but also reduces disease-related economic inequalities.

We introduce a new class of balanced allocation processes which are primarily characterized by ``filling'' underloaded bins. A prototypical example is the Packing process: At each round we only take one bin sample, if the load is below the average load, then we place as many balls until the average load is reached; otherwise, we place only one ball. We prove that for any process in this class the gap between the maximum and average load is $\mathcal{O}(\log n)$ w.h.p. for any number of balls $m\geq 1$. For the Packing process, we also provide a matching lower bound. Additionally, we prove that the Packing process is sample-efficient in the sense that the expected number of balls allocated per sample is strictly greater than one. Finally, we also demonstrate that the upper bound of $\mathcal{O}(\log n)$ on the gap can be extended to the Memory process studied by Mitzenmacher, Prabhakar and Shah (2002).

To each finitely generated group $G$, we associate a quasi-isometric invariant called the \emph{Dehn spectrum} of $G$. If $G$ is finitely presented, our invariant is closely related to the Dehn function of $G$, but provides more information by encoding the isoperimetric behavior of $G$ at various scales. The main goal of this paper is to initiate the study of the Dehn spectrum of finitely generated (but not necessarily finitely presented) groups. In particular, we compute the Dehn spectrum of small cancellation groups, certain wreath products, and free Burnside groups of sufficiently large odd exponent. We also address several natural questions concerning the structure of the poset of Dehn spectra. As an application, we show that there exist $2^{\aleph_0}$ pairwise non-quasi-isometric finitely generated groups of finite exponent.

We revisit recent developments for the Maximum Weight Independent Set problem in graphs excluding a subdivided claw $S_{t,t,t}$ as an induced subgraph [Chudnovsky, Pilipczuk, Pilipczuk, Thomassé, SODA 2020] and provide a subexponential-time algorithm with improved running time $2^{\mathcal{O}(\sqrt{n}\log n)}$ and a quasipolynomial-time approximation scheme with improved running time $2^{\mathcal{O}(\varepsilon^{-1} \log^{5} n)}$. The Gyárfás' path argument, a powerful tool that is the main building block for many algorithms in $P_t$-free graphs, ensures that given an $n$-vertex $P_t$-free graph, in polynomial time we can find a set $P$ of at most $t-1$ vertices, such that every connected component of $G-N[P]$ has at most $n/2$ vertices. Our main technical contribution is an analog of this result for $S_{t,t,t}$-free graphs: given an $n$-vertex $S_{t,t,t}$-free graph, in polynomial time we can find a set $P$ of $\mathcal{O}(t \log n)$ vertices and an extended strip decomposition (an appropriate analog of the decomposition into connected components) of $G-N[P]$ such that every particle (an appropriate analog of a connected component to recurse on) of the said extended strip decomposition has at most $n/2$ vertices.

The NP-complete graph problem Cluster Editing seeks to transform a static graph into a disjoint union of cliques by making the fewest possible edits to the edges. We introduce a natural interpretation of this problem in temporal graphs, whose edge sets change over time. This problem is NP-complete even when restricted to temporal graphs whose underlying graph is a path, but we obtain two polynomial-time algorithms for restricted cases. In the static setting, it is well-known that a graph is a disjoint union of cliques if and only if it contains no induced copy of $P_3$; we demonstrate that no general characterisation involving sets of at most four vertices can exist in the temporal setting, but obtain a complete characterisation involving forbidden configurations on at most five vertices. This characterisation gives rise to an FPT algorithm parameterised simultaneously by the permitted number of modifications and the lifetime of the temporal graph.

We develop new algorithmic techniques for VLSI detailed routing. First, we improve the goal-oriented version of Dijkstra's algorithm to find shortest paths in huge incomplete grid graphs with edge costs depending on the direction and the layer, and possibly on rectangular regions. We devise estimates of the distance to the targets that offer better trade-offs between running time and quality than previously known methods, leading to an overall speed-up. Second, we combine the advantages of the two classical detailed routing approaches - global shortest path search and track assignment with local corrections - by treating input wires (such as the output of track assignment) as reservations that can be used at a discount by the respective net. We show how to implement this new approach efficiently.

Multiple hypothesis testing practices vary widely, without consensus on which are appropriate when. This paper provides an economic foundation for these practices designed to capture leading examples, such as regulatory approval on the basis of clinical trials. MHT adjustments are appropriate in our framework to the extent that research costs are invariant to the number of hypotheses. Control of average size, as for example via a Bonferroni correction, emerges in the limit case where all costs are fixed; in the opposite limit, where costs vary in proportion to the hypothesis count, no correction is needed. We illustrate implications by calculating explicit critical values using data on actual costs in the drug approval process and in program evaluation research; these suggest that some MHT adjustment is warranted in these applications, but not as much as implied by standard practice.

Let $Λ(T)$ denote the set of leaves in a tree $T$. One natural problem is to look for a spanning tree $T$ of a given graph $G$ such that $Λ(T)$ is as large as possible. This problem is called maximum leaf number, and it is a well-known NP-hard problem. Throughout recent decades, this problem has received considerable attention, ranging from pure graph theoretic questions to practical problems related to the construction of wireless networks. Recently, a similar but stronger notion was defined by Bradshaw, Masařík, and Stacho [Flexible List Colorings in Graphs with Special Degeneracy Conditions, ISAAC 2020]. They introduced a new invariant for a graph $G$, called the robust connectivity and written $κ_ρ(G)$, defined as the minimum value $\frac{|R \cap Λ(T)|}{|R|}$ taken over all nonempty subsets $R\subseteq V(G)$, where $T = T(R)$ is a spanning tree on $G$ chosen to maximize $|R \cap Λ(T)|$. Large robust connectivity was originally used to show flexible choosability in non-regular graphs. In this paper, we investigate some interesting properties of robust connectivity for graphs embedded in surfaces. We prove a tight asymptotic bound of $Ω(γ^{-\frac{1}{r}})$ for the robust connectivity of $r$-connected graphs of Euler genus $γ$. Moreover, we give a surprising connection between the robust connectivity of graphs with an edge-maximal embedding in a surface and the surface connectivity of that surface, which describes to what extent large induced subgraphs of embedded graphs can be cut out from the surface without splitting the surface into multiple parts. For planar graphs, this connection provides an equivalent formulation of a long-standing conjecture of Albertson and Berman [A conjecture on planar graphs, 1979], which states that every planar graph on $n$ vertices contains an induced forest of size at least $n/2$.

The Directed Grid Theorem, stating that there is a function $f$ such that a directed graphs of directed treewidth at least $f(k)$ contains a directed grid of size at least $k$ as a butterfly minor, after being a conjecture for nearly 20 years, has been proven in 2015 by Kawarabayashi and Kreutzer. However, the function $f$ obtained in the proof is very fast growing. In this work, we show that if one relaxes directed grid to bramble of constant congestion, one can obtain a polynomial bound. More precisely, we show that for every $k \geq 1$ there exists $t = \mathcal{O}(k^{48} \log^{13} k)$ such that every directed graph of directed treewidth at least $t$ contains a bramble of congestion at most $8$ and size at least $k$.

Random walks on graphs are an essential primitive for many randomised algorithms and stochastic processes. It is natural to ask how much can be gained by running $k$ multiple random walks independently and in parallel. Although the cover time of multiple walks has been investigated for many natural networks, the problem of finding a general characterisation of multiple cover times for worst-case start vertices (posed by Alon, Avin, Koucký, Kozma, Lotker, and Tuttle~in 2008) remains an open problem. First, we improve and tighten various bounds on the stationary cover time when $k$ random walks start from vertices sampled from the stationary distribution. For example, we prove an unconditional lower bound of $Ω((n/k) \log n)$ on the stationary cover time, holding for any $n$-vertex graph $G$ and any $1 \leq k =o(n\log n )$. Secondly, we establish the stationary cover times of multiple walks on several fundamental networks up to constant factors. Thirdly, we present a framework characterising worst-case cover times in terms of stationary cover times and a novel, relaxed notion of mixing time for multiple walks called the partial mixing time. Roughly speaking, the partial mixing time only requires a specific portion of all random walks to be mixed. Using these new concepts, we can establish (or recover) the worst-case cover times for many networks including expanders, preferential attachment graphs, grids, binary trees and hypercubes.

Perverse schobers are categorifications of perverse sheaves. In prior work we constructed a perverse schober on a partial compactification of the stringy Kähler moduli space (SKMS) associated by Halpern-Leistner and Sam to a quasi-symmetric representation of a reductive group. When the group is a torus the SKMS corresponds to the complement of the GKZ discriminant locus (which is a hyperplane arrangement in the quasi-symmetric case shown by Kite). We show here that a suitable variation of the perverse schober we constructed provides a categorification of the associated GKZ hypergeometric system in the case of non-resonant parameters. As an intermediate result we give a description of the monodromy of such "quasi-symmetric" GKZ hypergeometric systems.

Many NP-complete graph problems are polynomial-time solvable on graph classes of bounded clique-width. Several of these problems are polynomial-time solvable on a hereditary graph class ${\cal G}$ if they are so on the atoms (graphs with no clique cut-set) of ${\cal G}$. Hence, we initiate a systematic study into boundedness of clique-width of atoms of hereditary graph classes. A graph $G$ is $H$-free if $H$ is not an induced subgraph of $G$, and it is $(H_1,H_2)$-free if it is both $H_1$-free and $H_2$-free. A class of $H$-free graphs has bounded clique-width if and only if its atoms have this property. This is no longer true for $(H_1,H_2)$-free graphs, as evidenced by one known example. We prove the existence of another such pair $(H_1,H_2)$ and classify the boundedness of clique-width on $(H_1,H_2)$-free atoms for all but 18 cases.

Interval graphs, intersection graphs of segments on a real line (intervals), play a key role in the study of algorithms and special structural properties. Unit interval graphs, their proper subclass, where each interval has a unit length, has also been extensively studied. We study mixed unit interval graphs---a generalization of unit interval graphs where each interval has still a unit length, but intervals of more than one type (open, closed, semi-closed) are allowed. This small modification captures a much richer class of graphs. In particular, mixed unit interval graphs are not claw-free, compared to unit interval graphs. Heggernes, Meister, and Papadopoulos defined a representation of unit interval graphs called the bubble model which turned out to be useful in algorithm design. We extend this model to the class of mixed unit interval graphs and demonstrate the advantages of this generalized model by providing a subexponential-time algorithm for solving the MaxCut problem on mixed unit interval graphs. In addition, we derive a polynomial-time algorithm for certain subclasses of mixed unit interval graphs. We point out a substantial mistake in the proof of the polynomiality of the MaxCut problem on unit interval graphs by Boyaci, Ekim, and Shalom (2017). Hence, the time complexity of this problem on unit interval graphs remains open. We further provide a better algorithmic upper-bound on the clique-width of mixed unit interval graphs. Clique-width is one of the most general structural graph parameters, where a large group of natural problems is still solvable in the tractable time when an efficient representation is given. Unfortunately, the exact computation of the clique-width representation is \NP-hard. Therefore, good upper-bounds on clique-width are highly appreciated, in particular, when such a bound is algorithmic.

We prove that the image of the lifted period map on the universal cover lies in a complex Euclidean space. We also prove that the Teichmüller spaces of a class of polarized manifolds have complex affine structures.

When modeling an application of practical relevance as an instance of a combinatorial problem X, we are often interested not merely in finding one optimal solution for that instance, but in finding a sufficiently diverse collection of good solutions. In this work we initiate a systematic study of diversity from the point of view of fixed-parameter tractability theory. First, we consider an intuitive notion of diversity of a collection of solutions which suits a large variety of combinatorial problems of practical interest. We then present an algorithmic framework which --automatically-- converts a tree-decomposition-based dynamic programming algorithm for a given combinatorial problem X into a dynamic programming algorithm for the diverse version of X. Surprisingly, our algorithm has a polynomial dependence on the diversity parameter.

We consider the eigenvalues of the Laplacian on an open, bounded, connected set in $\mathbb{R}^n$ with $C^2$ boundary, with a Neumann boundary condition or a Robin boundary condition. We obtain upper bounds for those eigenvalues that have a corresponding eigenfunction which achieves equality in Courant's Nodal Domain theorem. In the case where the set is also assumed to be convex, we obtain explicit upper bounds in terms of some of the geometric quantities of the set. Corrigendum. A previous version of this work was accepted and published by the "Bulletin de la Société Mathématique de France" (see [2] in the bibliography of Appendix B). It contained a gap: the classical (Euclidean) Faber-Krahn inequality was applied in a setting where it might not hold. This version reproduces the previous one with the addition of a corrigendum in Appendix B that addresses the issue. All the results in Sections 2--8 and most of those in Section 9 are thus preserved.

In this short survey article we collect the current state of the art in the nascent field of \textit{quantum enhancements}, a type of knot invariant defined by collecting values of quantum invariants of knots with colorings by various algebraic objects over the set of such colorings. This class of invariants includes classical skein invariants and quandle and biquandle cocycle invariants as well as new invariants.

The $k$-Colouring problem is to decide if the vertices of a graph can be coloured with at most $k$ colours for a fixed integer $k$ such that no two adjacent vertices are coloured alike. If each vertex u must be assigned a colour from a prescribed list $L(u) \subseteq \{1,\cdots, k\}$, then we obtain the List $k$-Colouring problem. A graph $G$ is $H$-free if $G$ does not contain $H$ as an induced subgraph. We continue an extensive study into the complexity of these two problems for $H$-free graphs. The graph $P_r+P_s$ is the disjoint union of the $r$-vertex path $P_r$ and the $s$-vertex path $P_s$. We prove that List $3$-Colouring is polynomial-time solvable for $(P_2+P_5)$-free graphs and for $(P_3+P_4)$-free graphs. Combining our results with known results yields complete complexity classifications of $3$-Colouring and List $3$-Colouring on $H$-free graphs for all graphs $H$ up to seven vertices.

We study the Steiner Tree problem, in which a set of terminal vertices needs to be connected in the cheapest possible way in an edge-weighted graph. This problem has been extensively studied from the viewpoint of approximation and also parametrization. In particular, on one hand Steiner Tree is known to be APX-hard, and W[2]-hard on the other, if parameterized by the number of non-terminals (Steiner vertices) in the optimum solution. In contrast to this we give an efficient parameterized approximation scheme (EPAS), which circumvents both hardness results. Moreover, our methods imply the existence of a polynomial size approximate kernelization scheme (PSAKS) for the considered parameter. We further study the parameterized approximability of other variants of Steiner Tree, such as Directed Steiner Tree and Steiner Forest. For neither of these an EPAS is likely to exist for the studied parameter: for Steiner Forest an easy observation shows that the problem is APX-hard, even if the input graph contains no Steiner vertices. For Directed Steiner Tree we prove that approximating within any function of the studied parameter is W[1]-hard. Nevertheless, we show that an EPAS exists for Unweighted Directed Steiner Tree, but a PSAKS does not. We also prove that there is an EPAS and a PSAKS for Steiner Forest if in addition to the number of Steiner vertices, the number of connected components of an optimal solution is considered to be a parameter.

A charged particle in a magnetic field possesses discrete energy levels associated with particle's rotation around the field lines. The radiative transitions between these levels are the well-known cyclotron transitions. We show that a bound complex of particles with a non-zero net charge displays analogous transitions between the states of confined motion of the entire complex in the field. The latter {\em bound-ion cyclotron transitions} are affected by a coupling between the collective and internal motions of the complex and, as a result, differ from the transitions of a "reference" bare ion with the same mass and charge. We analyze the cyclotron transitions for complex ions by including the coupling within a rigorous quantum approach. Particular attention is paid to comparison of the transition energies and oscillator strengths to those of the bare ion. Selection rules based on integrals of collective motion are derived for the bound-ion cyclotron transitions analytically, and the perturbation and coupled-channel approaches are developed to study the transitions quantitatively. Representative examples are considered and discussed for positive and negative atomic and cluster ions.

We prove a conjecture of Griffiths on simultaneous normalization of all periods which asserts that the image of the lifted period map on the universal cover lies in a bounded domain in a complex Euclidean space. As an application we prove that the Teichmüller spaces of a large class of projective manifolds have complex affine structures.

We prove a conjecture of Griffiths on simultaneous normalization of all periods which asserts that the image of the lifted period map on the universal cover lies in a bounded domain in complex Euclidean space.

Models for the production of knowledge and systems of innovation and science are key elements for characterizing a country in view of its scientific thematic profile. With regard to scientific output and publication in journals of international visibility, the countries of the world may be classified into three main groups according to their thematic bias. This paper aims to classify the countries of the world in several broad groups, described in terms of behavioural models that attempt to sum up the characteristics of their systems of knowledge and innovation. We perceive three clusters in our analysis: 1) the biomedical cluster, 2) the basic science & engineering cluster, and 3) the agricultural cluster. The countries are conceptually associated with the clusters via Principal Component Analysis (PCA), and a Multidimensional Scaling (MDS) map with all the countries is presented.

We present period-luminosity diagrams for nearby Miras and semiregulars, selecting stars with parallaxes better than 20 per cent and well-determined periods. Using K-band magnitudes, we find two well-defined P-L sequences, one corresponding to the standard Mira P-L relation and the second shifted to shorter periods by a factor of about 1.9. The second sequence only contains semiregular variables, while the Mira sequence contains both Miras and semiregulars. Several semiregular stars show double periods in agreement with both relations. The Whitelock evolutionary track is shown to fit the data, indicating that the semiregulars are Mira progenitors. The transition between the two sequences may correspond to a change in pulsation mode or to a change in the stellar structure. Large amplitude pulsations leading to classical Mira classification occur mainly near the tip of the local AGB luminosity function.

Inverse Compton Scattering (ICS) by a relativistic electron beam jet at GeV energies (emitted by a compact object as a NS, BH,...), a NSJ, onto thermal BBR photons (from a nearby stellar companion) may originate a collinear gamma jet (GJ). Due to the binary system interaction the GJ precession would blaze suddenly toward the observer leading to a GRB event. The internal GJ cone structure is ruled by relativistic kinematics into a concentric onion-like sequence of photon rings, the softer in the external boundaries, the harder in the inner cone. The pointing and the crossing of such different GJ photon rings to the detector lead to a GRB hardness spectra evolution nearly corresponding to most observed ones. Moreover expected time integral spectra are also comparable with known GRB spectra. The total energy input of tens of thousands of such NSJ in an extended galactic halo, mainly cosmic rays electrons, should be reflected into the recent observational evidence (COMPTEL) of a diffused relic extended halo. Evidences of such precessing jets are offered by the discover of galactic superluminal sources, recent HH jets, SN1987A outer rings, Hourglass Nebula, planetary Egg Nebula, GROJ1744-28 binary X-rays pulsar.

Two widely-used non-local extensions of dynamical mean field theory (DMFT), cellular DMFT (CDMFT) and the dynamical cluster approximation (DCA), both yield self-energies marred by having some unphysical properties: CDMFT yields real-space self-energies that are not translationally invariant, and DCA yields momentum-space self-energies with discontinuities in their momentum dependence. It is often desirable to remove these flaws by post-processing cluster DMFT results, using strategies called periodization for CDMFT and interpolation for DCA -- for brevity, we refer to both cases as interpolation. However, traditional interpolation approaches struggle to capture intricate structures such as hole pockets in the hole-doped square-lattice Hubbard model, as highlighted in Phys. Rev. B 105, 35117 (2022). Further, these approaches interpolate frequency-dependent functions, which may lead to causality violations. Here, we propose Liouvillian interpolation, a novel, intuitive, and robust scheme for interpolating cluster DMFT results. Our key idea is to interpolate frequency-independent matrix elements of the single-particle irreducible part of the Liouvillian, obtained from a continued-fraction expansion of the cDMFT self-energy. We demonstrate that the ingredients of such an expansion possess a more local Fourier expansion than the functions involved in traditional interpolation schemes, and that Liouvillian interpolation inherently conserves causality. We illustrate our method for the one-dimensional Hubbard model using CDMFT, and for the two-dimensional Hubbard model using four-patch DCA. For the latter, we find that L-interpolation can (depending on doping) yield Fermi and Luttinger arcs which together form a closed surface.

In this paper, we study the Cauchy problem for a heat equation governed by a mixed local--nonlocal diffusion operator with spatially irregular coefficients. We first establish classical well-posedness in an energy framework for bounded, measurable coefficients that satisfy uniform positivity, and we derive an a priori estimate ensuring uniqueness and continuous dependence on the initial data. We then extend the notion of solution to distributional coefficients and initial data by a Friedrichs-type regularisation procedure. Within this very weak framework, we establish the existence and uniqueness of solution nets and prove consistency with the classical weak solution whenever the coefficients are regular.

Future vehicular networks require continuous connectivity to serve highly mobile users in urban environments. To mitigate the coverage limitations of fixed terrestrial macro base stations (MBS) under non line-of-sight (NLoS) conditions, fleets of unmanned aerial base stations (UABSs) can be deployed as aerial base stations, dynamically repositioning to track vehicular users and traffic hotspots in coordination with the terrestrial network. This paper addresses cooperative multi-agent trajectory design under different service areas and takeoff configurations, where rapid and safe adaptation across scenarios is essential. We formulate the problem as a multi-task decentralized partially observable Markov decision process and solve it using centralized training and decentralized execution with double dueling deep Q-network (3DQN), enabling online training for real-world deployments. However, efficient exploration remains a bottleneck, with conventional strategies like $ε$-greedy requiring careful tuning. To overcome this, we propose the multi-agent meta-advisor with advisor override (MAMO). This framework guides agent exploration through a meta-policy learned jointly across tasks. It uses a dynamic override mechanism that allows agents to reject misaligned guidance when the advisor fails to generalize to a specific scenario. Simulation results across three realistic urban scenarios and multiple takeoff configurations show that MAMO achieves faster convergence and higher returns than tuned $ε$-greedy baselines, outperforming both an advisor-only ablation and a single generalized policy. Finally, we demonstrate that the learned UABS fleet significantly improves network performance compared to deployments without aerial support.

Transportable all-optical atomic clocks represent the next-generation devices for precision time keeping, ushering a new era in encompassing a wide range of PNT (Positioning, Navigation and Timing) applications in the civil and strategic sectors. Their performance relies on ultra-stable, narrow-linewidth lasers, frequency stabilized to a compact portable optical cavity. Among various designs, the cubic spacer-based ultra-stable cavity is particularly well-suited for transportable applications due to its low sensitivity to vibrations, owing to its symmetric geometry and robust mounting structure. While longer cavities offer a lower fundamental thermal noise floor, one needs to strike a balance between transportability and size. In this aspect, the 7.5 cm dual-axis cubic cavity offers a lower fundamental thermal noise floor in comparison to smaller counterparts, while still retaining a reasonable SWaP (Size, Weight and Power) for terrestrial and aerial PNT applications. Its dual-axis design also enables multi-wavelength laser stabilization, making it a promising candidate for future transportable clock applications. This work presents a detailed study of the 7.5 cm dual-axis cubic cavity using FEM (Finite Element Method) to evaluate its mechanical and thermal stability. We analyze the impact of various geometric factors, mounting forces, and machining imperfections, while also modelling thermal effects such as conduction, radiation, and mirror heating within a vacuum chamber and thermally shielded environment. Our findings provide design insights for developing robust dual-axis optical reference cavities, advancing the deployment of portable atomic clocks for next-generation applications in PNT, geodesy, VLBI (Very Long Baseline Interferometry) and deep space missions.