This work unlocks the manufacturing of nanophotonic quantum systems that exploit the unique material properties of single-crystal diamond (SCD). We achieve this by introducing a semiconductor-compatible process for the direct bonding of multiple high-quality, ultrathin diamond films onto a carrier wafer, enabling the subsequent parallel nanofabrication of optoelectronic integrated circuits. Central to this approach is a new diamond surface-preparation method that avoids boiling tri-acid mixtures while producing exceptionally clean 20 um thin single crystals. These platelets are bonded side-by-side to 100 mm silica wafers and exhibit a record shear strength of 45.1 MPa for (100)-oriented diamond, surpassing all previously reported bonding attempts. Evidence indicates that the bonding is dominated by van der Waals interactions, likely arising from mismatched protonation mechanisms between Si-OH and C-OH surface terminations, rather than from covalent-bond-driven mechanisms. Despite this non-molecular nature, the heterostructures remain stable through liquid immersions and standard nanofabrication steps. Because the method depends primarily on surface cleanliness and roughness rather than specific chemistries, it is broadly transferable across wafer materials. This capability to parallel-bond ultrathin SCD films onto large-area substrates provides a scalable route to high-performance platforms spanning nanophotonic quantum technologies, high-power electronics, MEMS, and biotechnology.

This paper presents a new dynamic integral quadratic constraint (IQC) for the repeated Rectified Linear Unit (ReLU). These dynamic IQCs can be used to analyze stability and induced $\ell_2$-gain performance of discrete-time, recurrent neural networks (RNNs) with ReLU activation functions. These analysis conditions can be incorporated into learning-based controller synthesis methods, which currently rely on static IQCs. We show that our proposed dynamic IQCs for repeated ReLU form a superset of the dynamic IQCs for repeated, slope-restricted nonlinearities. We also prove that the $\ell_2$-gain bounds are nonincreasing with respect to the horizon used in the dynamic IQC filter. A numerical example using a simple (academic) RNN shows that our proposed IQCs lead to less conservative bounds than existing IQCs.

FeO (wüstite), which exhibits complex electronic and structural properties with increasing pressure and temperature, is a key mineralogical phase for understanding deep planetary interiors. However, direct measurements of its spin state at high-pressure and temperature remain challenging in static compression experiments. Here, we employ laser-driven shock compression to extend the FeO principal Hugoniot up to $\sim$900 GPa and perform in situ X-ray diffraction and X-ray emission spectroscopy up to 250 GPa, probing FeO's crystal structure and spin state. We demonstrate a continuous spin crossover of iron in FeO over a broad pressure range, with the high-spin state persisting beyond Earth's core-mantle boundary (CMB) conditions. These observations provide new experimental constraints on iron spin state at extreme conditions essential for geophysical models of (exo)planetary interiors.

Converter-based generators and loads are growing in prevalence on power grids across the globe. The rise of these resources necessitates controllers that handle the power electronic devices' strict current limits without jeopardizing stability or overly constraining behavior. Existing controllers often employ complex, cascaded control loop architecture to saturate currents, but these controllers are challenging to tune properly and can destabilize following large disturbances. In this paper, we extend previous analysis to prove the feasible output region of a grid-connected converter is convex regardless of filter topology. We then formulate a convex optimal control problem from which we derive a projected gradient descent-based controller with convergence guarantees. This approach drives the converter toward optimality in real-time and differs from conventional control strategies that regulate converter outputs around predefined references regardless of surrounding grid conditions. Simulation results demonstrate safe and stabilizing behavior of the proposed controller, in both the single-converter-infinite-bus systems and multi-converter networks.

Identity-based software signing tools aim to make software artifact provenance verifiable while reducing the operational burden of long-lived key management. However, there is limited cross-tool longitudinal evidence about which usability problems arise in practice and how those problems evolve as tools mature. This gap matters because unusable signing and verification workflows can lead to incomplete adoption, misconfiguration, or skipped verification, undermining intended integrity guarantees. We conducted the first mining-software-repositories study of five open-source identity-based signing ecosystems: Sigstore, OpenPubKey, HashiCorp Vault, Keyfactor, and Notary v2. We analyzed approximately 3,900 GitHub issues from Nov. 2021 to Nov. 2025. We coded each issue for the reported usability concern and the implicated architectural component, and compared patterns across tools and over time. Across ecosystems, reported concerns concentrate in verification workflows, policy and configuration surfaces, and integration boundaries. Longitudinal Poisson trend analysis shows substantial declines in reported issues for most ecosystems. However, across usability themes, workflow- and documentation-related concerns decline unevenly across tools and concern types, and verification workflows and configuration surfaces remain persistent friction points. These results indicate that identity-based signing reduces some usability burdens while relocating complexity to verification semantics, policy configuration, and deployment integration. Designing future signing ecosystems therefore requires treating verification semantics and release workflows as first-class usability targets rather than peripheral integration concerns.

The purpose of this paper is twofold: 1) Applications of Gallagher's larger sieve modulo prime squares do not work. In some relevant cases we can transform the residue class information modulo $p^2$ to more suitable residue information modulo $p$, so that we can successfully apply the sieve. 2) The applications to Hilbert cubes are of interest in their own right: We study the maximal dimension of Hilbert cubes in various multiplicatively defined sets. For the squareful numbers in $[1,N]$ we achieve an upper bound of the dimension of $d=O(\log N)$. The same upper bounds follow for multiplicative semigroups of integers defined by a positive proportion of the primes, and the set of integers representable by an irreducible positive definite binary quadratic form. Eventually, making use of the sun flower lemma we give an improvement on the maximal dimension $d$ of subset sums in the set of pure powers in $[1,N]$.

In 1999, Heath, Pemmaraju, and Trenk [SIAM J. Comput. 28(4), 1999] extended the classic notion of book embeddings to digraphs, introducing the concept of upward book embeddings, in which the vertices must appear along the spine in a topological order and the edges are partitioned into pages, so that no two edges in the same page cross. For a partitioned digraph $G=(V,\bigcup^k_{i=1} E_i)$, that is, a digraph whose edge set is partitioned into $k$ subsets, an upward book embedding is required to assign edges to pages as prescribed by the given partition. In a companion paper, Heath and Pemmaraju [SIAM J. Comput 28(5), 1999] proved that the problem of testing the existence of an upward book embedding of a partitioned digraph is linear-time solvable for $k=1$ and recently Akitaya, Demaine, Hesterberg, and Liu [GD, 2017] have shown the problem NP-complete for $k\geq 3$. In this paper, we study upward book embeddings of partitioned digraphs and focus on the unsolved case $k=2$. Our first main result is a novel characterization of the upward embeddings that support an upward book embedding in two pages. We exploit this characterization in several ways, and obtain a rich picture of the complexity landscape of the problem. First, we show that the problem remains NP-complete when $k=2$, thus closing the complexity gap for the problem. Second, we show that, for an $n$-vertex partitioned digraph $G$ with a prescribed planar embedding, the existence of an upward book embedding of $G$ that respects the given planar embedding can be tested in $O(n \log^3 n)$ time. Finally, leveraging the SPQ(R)-tree decomposition of biconnected graphs into triconnected components, we present a cubic-time testing algorithm for biconnected directed partial $2$-trees.

We propose and simulate a laboratory platform to study the effects of positrons in magnetic reconnection using laser-driven capacitor coils. Using particle-in-cell simulations, we show that externally injected MeV electron-positron pairs are trapped in the coil current sheet, significantly modifying the reconnection dynamics and particle acceleration. These pairs increase the reconnection rate by a factor of approximately 8, which Ohm's law decomposition reveals to be driven by the divergence of the generalized pressure tensor. Based on their high energy and magnetization, the pairs also substantially broaden the diffusion region. Particle tracking simulations in realistic coil magnetic fields further demonstrate that injected pairs can remain confined for several picoseconds, providing conditions for sustained interaction with the reconnection region. These results establish a near-term pathway to laboratory studies of positron-influenced reconnection, bridging high-energy-density experiments with pair-dominated astrophysical environments.

We present an isometry and parametrisation invariant of embeddings of $S^1$ into Euclidean space. We do so by representing the distance between pairs of points on the embedded circle as a function on a Möbius band, the two-point finite subset space of $S^1$. We call this function the chordal distance transform of the embedding. We show that the sublevel set persistent homology of the chordal distance transform satisfies the desired isometry and parametrisation invariance, and is a continuous transform with respect to the Whitney topology on the space of circle embeddings and the bottleneck distance in the space of persistence diagrams. We then considered the generic behaviour of the chordal distance transform for $C^2$ and finite piecewise linear embeddings. In the $C^2$-case, we show that non-boundary critical points of the chordal distance transform are finite and non-degenerate on an open and dense subset of circle embeddings. Consequently, its persistent homology is pointwise finite dimensional for generic $C^2$-embeddings. In the finite piecewise linear case, we also find piecewise-continuous analogues of non-degenerate critical points, and give generic conditions for the homological critical points of the chordal distance transform to be non-degenerate. In order to gain a geometric interpretation of the chordal distance transform and its persistent homology, we give a geometric characterisation of the $C^2$ and finite piecewise linear non-degenerate critical points. Finally, we consider how the chordal distance transform can be generalised to capture geometric features involving $n\geq 2$ points on an embedded shape, as a function on the $n$-point finite subset space.

Lorentz violation in hadronic systems is related to Lorentz-violating operators of quarks and gluons. Due to the nonperturbative nature of quantum chromodynamics (QCD) at low energies, establishing these relationships is complex. Chiral perturbation theory (ChPT) is an effective theory that provides one method of connecting quark- and gluon-level operators to those at the hadronic level, which can be used to calculate hadronic observables.

Topological phases of matter are commonly understood as emerging either from crystalline symmetry and intrinsic spin-orbit coupling or from disorder-driven electronic renormalization. In realistic materials, however, structural defects naturally combine both ingredients. Here, we demonstrate a general and material-independent mechanism by which atomic vacancies can induce topological phase transitions in two-dimensional semiconductors that are otherwise topologically trivial. Vacancies generate locally ordered dangling-bond states governed by well-defined hopping and spin-orbit interactions, while their spatial distribution and mutual coupling introduce long-range disorder. As vacancy concentration increases, the hybridization of these defect states forms an emergent electronic subspace that undergoes a topological transition. Using a tight-binding framework supported by large-scale density functional theory calculations, we show that this vacancy-induced electronic reconstruction can robustly stabilize quantum spin Hall, quantum anomalous Hall, and Weyl semimetal phases, depending on symmetry breaking and spin polarization. Our results establish vacancies not merely as perturbations, but as active design elements capable of transforming trivial insulators into topological quantum matter, opening realistic routes for defect-engineered topological devices.

Free-space optical communication links can enable high-capacity wireless connectivity in urban areas. We discuss the feasibility, challenges, and recent developments for high-capacity urban free-space optical links at kilometer scale.

We demonstrate a new magneto-optical trap (MOT) configuration using a simple pair of crossed wires rotated at 45 deg and an appropriate bias field to generate a MOT of >10^8 atoms. The same pair of wires, with slightly adjusted control parameters, is then used to magnetically trap the atoms and cool them via forced evaporative cooling into a Bose-Einstein condensate (BEC) with >10^4 atoms. We present the theoretical framework for generating a quadrupole field using a pair of crossed wires with arbitrary rotation angle, along with the atom chip design and fabrication. Finally, we describe the experimental protocols required for BEC production using only a single crossed-wire atom chip.

Traditional significant wave height (SWH) estima- tion from HF radar typically relies on spectral analysis of the received radar signals. This process was previously simplified by establishing a linear relationship between SWH and the standard deviation of received HF radar voltages under first- order scattering. Building on this approach, this paper presents a physics-informed regression model that incorporates second- order scattering effects through a quadratic formulation derived from a Neumann expansion. The proposed method is evaluated using HF radar data collected in July 2018 at Argentia, New- foundland, with collocated buoy measurements as ground truth. The model achieves a minimum root-mean-square error (RMSE) of approximately 19 cm.

We develop a family-based route to unicyclic graphs whose independence polynomials are unimodal but not log-concave. The paper is organized around one flagship statement: for the explicit KL-closure family $U_{k,r}$, with $r\in\{0,1,2\}$ and admissible $k$, the independence polynomial is unimodal but not log-concave. The proof separates the closure polynomial into a dominant convolution term and a real-rooted correction term. On the non-log-concavity side, we prove symbolically that the penultimate log-concavity inequality fails for every admissible parameter. On the unimodality side, we prove that the main convolution term $H_{k,r}=G_kF_{k+r}$ is unimodal with a controlled mode, using a combination of exact coefficient formulas, Ibragimov's strong-unimodality principle, and a residue-class growth argument. Darroch localization and an adjacent-mode bridge lemma then transfer that mode statement to the full KL closure polynomial. This yields an explicit infinite family of unicyclic graphs with unimodal but non-log-concave independence polynomials. In the exact range $k\le 400$, we further verify that the penultimate break is unique and determine exact mode formulas for $H_{k,r}$, the binomial correction term, and $I(U_{k,r};x)$ itself. The paper also places the KL family inside a broader reservoir program involving Galvin, Ramos-Sun, and Bautista-Ramos trees, from which we obtain substantial universal exact theorems for finite ranges.

For a graph $G$ with at least two vertices, the maximum local edge-connectivity of $G$ is the maximum number of edge-disjoint $(u,v)$-paths over all distinct pairs of vertices $(u,v)$ in $G$. Stiebitz and Toft (2018) proved a Brooks-type theorem for graphs with maximum local edge-connectivity $k$, showing that a graph with maximum local edge-connectivity $k$ is not $k$-colourable if and only if it has a block in $\mathcal{H}_k$, which is the class of graphs that can be obtained by taking Hajós joins of copies of $K_{k+1}$ and, when $k=3$, odd wheels. We prove that a $2$-connected graph with maximum local edge-connectivity $k$ is $k$-choosable if and only if it is not in $\mathcal{H}_k$. On the other hand, deciding $k$-choosability when restricted to graphs with maximum local edge-connectivity $k$ (that might not be $2$-connected) is $Π_2$-complete. To prove the former result, we first prove several generalisations of a well-known characterisation of degree-choosability; these may be of independent interest.

A standard approach to solving optimistic bilevel linear programs (BLPs) is to replace the lower-level problem with its Karush-Kuhn-Tucker (KKT) optimality conditions and reformulate the resulting complementarity constraints using auxiliary binary variables. This yields a single-level mixed-integer linear programming (MILP) model involving big-$M$ parameters. While sufficiently large and bilevel-correct big-$M$s can be computed in polynomial time, verifying a priori that given big-$M$s do not cut off any feasible or optimal lower-level solutions is known to be computationally difficult. In this paper, we establish two complementary hardness results. First, we show that, even with a single potentially incorrect big-$M$ parameter, it is $coNP$-complete to verify a posteriori whether the optimal solution of the resulting MILP model is bilevel optimal. In particular, this negative result persists for min-max problems without coupling constraints and applies to strong-duality-based reformulations of mixed-integer BLPs. Second, we show that verifying global big-$M$ correctness remains computationally difficult a posteriori, even when an optimal solution of the MILP model is available.

Proxy-based race inference is increasingly used to conduct fairness assessments when protected-class data are unavailable or legally restricted -- most prominently in U.S. fair-lending enforcement, and now explicitly contemplated in emerging insurance regulation, including Colorado's draft SB21-169 testing framework and New York's Insurance Circular Letter No. 7. Despite this growing regulatory relevance, little is known about how standard regression-based discrimination analyses behave when race is measured with error through proxies such as Bayesian Improved Surname Geocoding (BISG) or Bayesian Improved First Name and Surname Geocoding (BIFSG). This paper studies the consequences of using proxy-imputed race as a categorical regressor in regression-based fairness assessments. Treating proxy race as a categorical covariate subject to misclassification, we show that proxy-based coefficients become weighted mixtures of true group effects, systematically shrinking estimated disparities toward the majority group -- even when overall classification accuracy is high. Empirically, using a linked North Carolina voter-insurance dataset with self-reported race and ZIP-level auto insurance premiums, we demonstrate two mechanisms through which it distorts inference: (i) the intrinsic mixing of group effects implied by misclassification, and (ii) structured errors that vary with ZIP-level racial composition and socioeconomic conditions and remain correlated with pricing residuals after controls. As a result, regression-based disparity estimates can be attenuated or amplified relative to analogous analyses based on self-reported race. Our findings caution against treating proxy race as a plug-in substitute in regulatory testing and highlight design implications for proxy-based audit frameworks in insurance and other high-stakes domains.

We apply proof mining techniques to obtain quantitative and qualitative results on asymptotic and T-asymptotic regularity for the inexact generalized Halpern iteration, a viscosity-type extension of an iteration recently studied by Kanzow and Shehu. Specializing our results to the Kanzow-Shehu iteration and the sequential averaging method (SAM) yields analogous results for these iterations. Furthermore, we compute rates of (T-)asymptotic regularity for particular choices of the parameter sequences, and for one of them, we obtain linear rates as an application of a lemma due to Sabach and Shtern.

Quantum reservoir computing (QRC) is a hardware-implementation-friendly quantum neural network scheme with minimal physical system requirements and a proven advantage over classical counterparts. We use an extension of the positive-P phase space method to efficiently simulate a bosonic, linear silicon-chip based QRC system excited with a single nonclassical state, a "kitten" state. In combination with input-encoding coherent states, our method allows to obtain exact results for all correlation functions without Hilbert space cutoff. Surprisingly, we find that such a setting - where the only "quantumness'' derives from a single input mode, is sufficient to obtain significant (over 9-fold) reduction of classification error over the classical counterpart. Our work provides a promising direction toward efficient quantum computation with accessible optical hardware.

The interplay between superconductivity and ferromagnetism has long been pursued as a route to unconventional Josephson effects, yet suitable material platforms remain limited. Here we report Josephson junctions based on epitaxial Al/InAs/(Ga,Fe)Sb heterostructures grown by low-temperature molecular beam epitaxy, achieving atomically abrupt superconductor/semiconductor/ferromagnetic interfaces. The devices exhibit clear proximity-induced superconductivity, including multiple Andreev reflections and gate-tunable supercurrents, confirming transparent coupling across the hybrid structure. Under perpendicular magnetic fields, the junctions reveal highly unconventional Fraunhofer interference patterns with hysteresis, flux jumps, asymmetric lobe evolution, and clear nonreciprocity, providing strong evidence of induced ferromagnetism and broken time-reversal symmetry in the superconducting channel. Gate control further modulates the critical current, highlighting the semiconducting nature of the system. Our results demonstrate that ferromagnetic semiconductor heterostructures can serve as a highly tunable platform for exploring proximity-induced superconductivity and superconducting diode effects, and for advancing device concepts at the intersection of magnetism and quantum electronics.

Ensuring the safety of control systems often requires the satisfaction of constraints on states (such as position or velocity), control inputs (such as force), and a mixture of states and inputs (such as power that depends on both velocity and force). This paper presents a safety-critical control framework for enforcing mixed state-input constraints through a generalization of backup control barrier functions (backup CBFs). First, we extend the backup CBF approach to maintain multiple decoupled state and input constraints using a single backup set-backup controller pair. Second, we address mixed state-input constraints by converting them into state constraints using a projection from the state-input space to the state space along the backup controller. In the special case of decoupled state and input constraints, the proposed method simplifies the synthesis of backup CBFs by eliminating the need for saturating backup control laws. Finally, we demonstrate the efficacy of the proposed method on an inverted pendulum example, where constraints on the angle (state), torque (input), and power (mixture of state and input) are satisfied simultaneously.

Decentralized optimization on Riemannian manifolds is foundational for many modern machine learning and signal processing applications in which data are non-Euclidean and generated and processed in a distributed manner. Although intrinsic Riemannian methods exploit manifold geometry without relying on Euclidean embeddings, existing decentralized Riemannian optimization algorithms typically use constant step sizes and therefore converge only to a neighborhood of steady-state error. In this paper, we study the decentralized stochastic Riemannian gradient method in the diminishing step-size regime on manifolds with (possibly positive) bounded sectional curvature. We prove an $O(1/T)$ bound for the network consensus error and an $O(\log T/\sqrt{T})$ ergodic bound for the global optimality gap. To the best of our knowledge, this is the first exact, non-asymptotic optimality-gap guarantee for an intrinsic decentralized stochastic Riemannian method in the geodesically convex setting. Furthermore, the diminishing step-size schedule allows substantially larger initial gradient steps than fixed-step baselines, leading to better performance in practice. We illustrate this on the problem of distributed PCA over a Grassmann manifold.

The Galactic Center Excess (GCE) is a compelling signature of dark matter annihilation, but its spectral morphology is difficult to reconcile with the traditional paradigm of a single particle species. In this work, we perform a systematic investigation of multi-component dark matter sectors, exploring scenarios with two ($N=2$) and three ($N=3$) distinct particle species while considering both exclusive and mixed annihilation channels. Using the Akaike Information Criterion (AIC) to rigorously penalize model complexity, we find that the GCE data statistically favors an $N=2$ scenario where each dark matter component annihilates exclusively into a single final state. Our results reveal that the preferred solutions naturally follow a light-plus-heavy mass hierarchy, and that specific final states such as $t\bar{t}$, $ZZ$, and $hh$, which are individually unable to explain the excess are effectively ``resurrected'' by the improved morphological fit provided by the multi-component framework. Furthermore, we show that these scenarios may mitigate the tension with current constraints, reaching compatibility within existing uncertainties. Our results suggest that the GCE may be the first evidence of a diverse dark sector, favoring a multi-scale solution over the minimal WIMP paradigm.

Nowadays, the Standard Model Effective Field Theory (SMEFT) provides a standard framework to parameterize potential deviations from the Standard Model and to combine information from multiple processes in global analyses. This review summarizes dedicated studies that constrain dimension-six Wilson coefficients using three top-quark and four top-quark production processes. We highlight the complementarity of these channels, as well as summarize the main problems and prospects in the area. A concise introduction to the SMEFT formalism and a discussion of the problem of potential perturbative unitarity violation are also provided.

Scalable manufacturing of human induced pluripotent stem cells (iPSCs) is essential for industrial-scale production of cell therapies and regenerative medicines. However, the 3D aggregate cultures used in manufacturing exhibit substantial spatial and metabolic heterogeneity compared with the relatively homogeneous monolayer systems used in laboratory studies, complicating mechanistic understanding and predictive metabolic modeling across culture scales. To address this challenge, we developed a modular multiscale mechanistic foundation model that links molecular, cellular, and macroscopic processes while accounting for spatial and metabolic heterogeneity. The framework integrates extracellular culture dynamics, intracellular metabolic fluxes, and cellular redox states by extending a previously established monolayer kinetic network and coupling it with a biological systems-of-systems (Bio-SoS) multiscale model for aggregate cultures, incorporating explicit redox interactions. Systematic monolayer and aggregate experiments (including multiple isotopic tracers, extracellular metabolite profiling, and two-photon optical redox imaging) were used to improve and validate the model. This integrated framework unifies heterogeneous datasets across culture configurations and enables mechanistic interpretation of metabolic and redox responses across heterogeneous culture scales, providing a quantitative foundation for scalable iPSC biomanufacturing.

Data-driven model predictive control based on Willems' fundamental lemma has proven effective for linear systems, but extending stability guarantees to nonlinear systems remains an open challenge. In this paper, we establish conditions under which data-driven MPC, applied directly to input-output data from a nonlinear system, yields practical exponential stability. The key insight is that the existence of an approximate Koopman linear embedding certifies that the nonlinear data can be interpreted as noisy data from a linear time-invariant system, enabling the application of existing robust stability theories. Crucially, the Koopman embedding serves only as a theoretical certificate; the controller itself operates on raw nonlinear data without knowledge of the lifting functions. We further show that the proportional structure of the embedding residual can be exploited to obtain an ultimate bound that depends only on the irreducible offset, rather than the worst-case embedding error. The framework is demonstrated on a synchronous generator connected to an infinite bus, for which we construct an explicit physics-informed embedding with error bounds.

Persistent homology (PH) characterizes the shape of brain networks through persistence features. Group comparison of persistence features from brain networks can be challenging as they are inherently heterogeneous. A recent scale-space representation of persistence diagrams (PDs) through heat diffusion reparameterizes them using a finite number of Fourier coefficients with respect to the Laplace--Beltrami (LB) eigenfunction expansion of the domain, providing a powerful vectorized algebraic representation for group comparisons. In this study, we develop a transposition-based permutation test for comparing multiple groups of PDs using heat-diffusion estimates. We evaluate the empirical performance of the spectral transposition test in capturing within- and between-group similarity and dissimilarity under varying levels of topological noise and cycle location variability. In application, we propose a topological lesion symptom mapping (TLSM) method based on the proposed framework. The method is applied to resting-state functional brain networks of individuals with post-stroke aphasia to identify characteristic cycles associated with varying levels of speech-language impairment.

We present a simple greedy procedure to compute an $(α,β)$-spanner for a graph $G$. We then show that this procedure is useful for building fault-tolerant spanners, as well as spanners for weighted graphs. Our first main result is an algorithm that, given a multigraph $G$, outputs an $f$ edge fault-tolerant $(k,k-1)$-spanner $H$ of size $O(fn^{1+\frac1k})$ which is tight. To our knowledge, this is the first tight result concerning the price of fault tolerance in spanners which are not multiplicative, in any model of faults. Our second main result is a new construction of a spanner for weighted graphs. We show that any weighted graph $G$ has a subgraph $H$ with $O(n^{1+\frac{1}{k}})$ edges such that any path $P$ of hop-length $\ell$ in $G$ has a replacement path $P'$ in $H$ of weighted length $\leq w(P)+(2k-2)w^{(1/2)}(P)$ where $w(P)$ is the total edge weight of $P$, and $w^{(1/2)}$ denotes the sum of the largest $\lceil \frac{\ell}{2} \rceil$ edge weights along $P$. Moreover, we show such approximation is optimal for shortest paths of hop-length $2$. To our knowledge, this is the first construction of a spanner for weighted graphs that strictly improves upon the stretch of multiplicative $(2k-1)$-spanners for all non-adjacent vertex pairs, while maintaining the same size bound. Our technique is based on using clustering and ball-growing, which are methods commonly used in designing spanner algorithms, to analyze simple greedy algorithms. This allows us to combine the flexibility of clustering approaches with the unique properties of the greedy algorithm to get improved bounds. In particular, our methods give a very short proof that the parallel greedy spanner adds $O(kn^{1+\frac{1}{k}})$ edges, improving upon known bounds.

We begin the investigation of the free factor complex of a free group of finite rank. For the case of rank 2 we axiomatize its theory and show that it is $ω$-stable with prime model $AF_2$.