Phase transitions are divided into first-order phase transitions and continuous ones in current classification. While the latter shows striking phenomena of scaling and universality, the former is generically characterized by discontinuous jumps in extensive variables and pronounced hysteresis. Recent studies have demonstrated universal scaling behavior controlled by a cubic fixed point in first-order phase transitions. However, more recent investigations into the hysteresis in a dynamic mean-field quartic model driven through its first-order phase transitions have revealed new scaling exponents for different driving rates. Here, we discover a new exponent for large driving rates arising surprisingly from critical phenomena and show that, depending on the magnitude of the driving rates and on the absence or presence of noise, the same mean-field model remarkably exhibits several universality classes with definite universal scaling exponents governed by their corresponding fixed points through a systematic scaling analysis based on renormalization group theory. The theories and their various crossovers between different fixed points along with complete universal scaling of full curve collapse are verified by numerical results. This further confirms universal scaling in first-order phase transitions.

We study the "set parameterized matching" problem, a generalization of the classical parameterized matching problem introduced by Baker. In set parameterized matching, both the pattern and text are sequences where each position contains a set of characters rather than a single character. Two set-strings parameterized match if there exists a bijection between their alphabets that maps one to the other set-wise. Boussidan introduced this problem for the case of equal-length set-strings. We present a randomized algorithm running in $O(N + M)$ time with high probability, where $N$ is the text size and $M$ is the pattern size. Our approach employs a novel three-layer hashing scheme based on Karp-Rabin fingerprinting that addresses the challenges of (1) the size blowup in representations of the problem, (2) set-to-set matching, and (3) the dynamic nature of encodings of text substrings during pattern scanning.

Black holes are expected to end their lifetime in a burst of Hawking radiation, emitting all Standard Model particles at ultra-high energies. The evaporation of a nearby primordial black hole (PBH) has been proposed as an explanation for the high-energy neutrino-like event reported by KM3NeT. Such a scenario requires the source to be extremely close to Earth, implying detectable gamma-ray and cosmic-ray emission. Accounting for the time-dependent field of view of gamma-ray observatories, we find that current experiments should have observed a pre-burst signal, while neutrino telescopes would also detect lower-energy events before the burst. The absence of such multimessenger signals strongly disfavors a PBH origin of the KM3-230213A event in the minimal 4D Schwarzschild scenario.

In this work, we report on the high-resolution infrared spectrum of CCH$^+$ ($^3Π$) recorded in the range $3066-3184$~cm$^{-1}$ by means of leak-out spectroscopy. This spectral range covers the fundamental of the CH stretching mode and a highly excited bending vibrational mode. Based on this data (385 ro-vibrational lines), accurate spectroscopic descriptions of the ground and the two vibrationally excited states of CCH$^+$ were obtained. Besides the band origins, spin-orbit coupling constants, rotational constants, centrifugal distortion constants and $Λ$-doubling constants for the ground and excited vibrational states have been derived. This effective Hamiltonian analysis allowed a search for pure rotational lines of CCH$^+$ in its electronic and vibrational ground state using a two-color millimeterwave - infrared scheme. We observed all rotational transitions from $J^{\prime\prime} = 2$ up to $J^{\prime\prime} = 6$ within the $Ω= 2$ lowest energy fine structure component with resolved hyperfine splittings. This data has already guided the first detection of CCH$^+$ in space toward the Orion Bar photo-dissociation region, and has the potential to support further astronomical searches for CCH$^+$ either through radio or infrared spectroscopy, for example with the James Webb Space Telescope.

We aim to characterize the thermodynamic and dynamical conditions leading to condensation in cluster cores, and to assess the role of magnetic fields. We implement a Monte-Carlo tracer particle algorithm in the GPU-accelerated code AthenaPK, and run a purely hydrodynamical and a magnetohydrodynamical (MHD) simulations of an idealized cool-core cluster. We identify the subset of hot ICM tracers that undergo a transition to the cold phase and reconstruct their histories over a lookback time of $300\,\mathrm{Myr}$ prior to condensation. In both runs, the large majority of tracers transitioning to the cold phase follow a thermodynamic pathway driven by mixing, whereby hot ambient gas is entrained onto low-entropy seed clumps that subsequently grow into larger clouds and filaments. In the hydrodynamical run, these seeds form mainly via in-situ cooling at the edges of AGN cavities. In the MHD run, the cold gas cycle is more complex: AGN outflows occasionally shred portions of existing filaments into fragments which are then uplifted, seeding further condensation. In the MHD run, the properties of condensing tracers begin to diverge from the background ICM significantly earlier than in the hydrodynamical run (${\sim}150\,\rm Myr$ before the cooling transition versus ${\sim}30\,\rm Myr$), with vorticity and magnetic energy growing together. The turbulent Mach number at condensation is also systematically lower than in the hydrodynamical run. We examine the post-condensation evolution of individual cold structures in the MHD run, namely a massive core filament and two isolated clouds in quiescent regions. We find that magnetic tension dominates over ram pressure as the primary drag force, significantly reducing the clouds' terminal velocity. Our results demonstrate that magnetic fields substantially impact the assembly history and kinematic properties of the cold phase in cool-core clusters.

The mass spectra of $ρ$ mesons ($ρ_{Q=\pm 1}^{s_z=0,\pm 1}$ and $ρ_{Q=0}^{s_z=0,\pm 1}$) at finite magnetic field and temperature are studied in frame of the two-flavor Nambu-Jona-Lasinio model. Fully considering the breaking of translational invariance induced by external magnetic field, the analytical form of $ρ$ meson propagators have been derived in the Ritus scheme and Schwinger scheme, which gives the same algebraic formula. When solving the pole equation of $ρ$ meson propagators, multiple solutions of the meson mass appear due to the dimension reduction of their constituent quarks in magnetic fields. At vanishing temperature, we focus on the $ρ$ meson masses $M_ρ$ corresponding to the lowest value solution of the pole equation. $M_{ρ^{-}_+}$, $M_{ρ^{0}_+}$ and $M_{ρ^{\pm}_0}$ increase with magnetic field. $M_{ρ^{+}_+}$ firstly decreases and then becomes saturated with increasing magnetic field. $M_{ρ^0_0}$ is not sensitive to magnetic field. These results are consistent with the available LQCD simulations. At finite temperature, we discuss the lowest four/five solutions of $ρ$ meson masses $M^{i=0,1,2,3,4}_ρ$. With fixed magnetic field, they decrease with temperature, and approach the mass sum of their constituent quarks at high temperature. The mass solution $M^{i}_ρ$ for different mesons $ρ_+^{0,\pm}$ and $ρ_0^{0,\pm}$ may become degenerate at finite magnetic field and temperature.

Modern retrieval pipelines increasingly rely on query reformulation and neural reranking to improve effectiveness, but this comes at a significant computational cost and introduces a fundamental tradeoff between recall and query drift. Generating many reformulated queries can substantially increase recall, yet naively merging or exhaustively reranking their results is prohibitively expensive. In this work, we argue that the core challenge is not reformulation generation itself, but the adaptive selection of reformulations and their retrieved documents under a strict inference budget. We propose ReformIR, a budget-aware retrieval framework that treats query reformulations as first-class features and performs online relevance estimation using a strong reranker as a teacher. Given multiple reformulated queries, ReformIR constructs a large candidate pool and learns a lightweight surrogate model that estimates document utility from reformulation-specific retrieval signals. Under a fixed reranking budget, the surrogate adaptively prioritizes both reformulations and documents, selectively querying a teacher reranker anchored to the original query. This process increases recall while actively suppressing drift through online feature selection over reformulations. We conduct extensive experiments on the MSMARCO passage corpora and TREC Deep Learning benchmarks (DL19-DL22). Our results show that ReformIR consistently outperforms existing reformulation strategies, particularly as the number of reformulations increases, where prior methods suffer from severe quality degradation due to drift. Our findings also suggest a shift in retrieval system design, rather than using large language models as rerankers, their capacity is more effectively leveraged in the reformulation stage with feedback-driven optimization.

We perform a complete one-loop renormalization analysis of CPT-odd Lorentz-violating scalar quantum chromodynamics with adjoint scalar matter. Working to first order in the preferred background vector and treating the corresponding operators as perturbative insertions, we compute the ultraviolet-divergent parts of the relevant two-, three-, and four-point Green's functions for the gauge, scalar, and ghost fields. We show that the gauge sector develops the expected Carroll-Field-Jackiw-type correction, which generically turns out to be divergent in our theory, although the divergence vanishes in a certain gauge, while the scalar sector displays the corresponding CPT-odd single-derivative term proportional to the background vector. We further demonstrate that several of the Lorentz-violating corrections the three- and four-point function are ultraviolet finite. All one-loop divergences may be absorbed into counterterms already allowed by the classical Lagrangian, providing an explicit proof of the multiplicative renormalizability of the theory at this order. We also obtain the associated renormalization constants and one-loop $β$-functions for the gauge coupling, the Lorentz violation parameters, and the scalar self-interaction.

We propose a graphical authentication scheme that follows a simple ``Pick and Sort'' design in which users choose visual elements and arrange them within a grid. Both the number of selected elements and the grid size are configurable, and the visual elements can be customized for specific user groups, such as children. A preliminary study with a prototype implementation indicated that the scheme is easy to learn and flexible to deploy. Although login times are longer than those of conventional authentication methods, the additional interaction may be acceptable in scenarios that are not time-critical, such as infrequent-access use cases or as a secondary authentication mechanism.

To efficiently support Large Language Models (LLMs), modern GPGPU architectures have introduced new features and programming paradigms, such as warp specialization. These features enable temporal overlap between the producer and consumer, as well as between matrix multiplication and activation function operations, substantially improving performance. To conduct effective AI infrastructure and computer architecture research, cycle-accurate simulators that support these new features, together with analytical models that faithfully capture workload characteristics, are essential. However, existing academic tools provide limited support for these emerging requirements. Existing cycle-accurate simulators do not incorporate new NVIDIA GPU features, such as the Tensor Memory Accelerator (TMA), in a timely manner. Moreover, existing analytical models can misestimate DRAM traffic under certain configurations. In this paper, we build a simulation pipeline from FlashAttention-3 kernel instrumentation to cycle-accurate simulation. The simulator achieves a mean absolute percentage error (MAPE) of 5.7\% and a maximum absolute percentage error of 12.7\% against H800. We also provide a theoretical analysis of FlashAttention-3 and explain why existing analytical models can produce inaccurate traffic estimates.

The nuclear charge radius is one of the most fundamental quantities of the atomic nucleus. It can be deduced from a combination of experimental measurements of muonicX-raytransitionenergieswithmodellingofthoseX-raytransitionenergies. In thisworkwepresentMuDirac (1.3.0), whichisanopen, publiclyavailable, sustainable and computationally efficient software tool that will be at put the disposal of the negative muon community. With MuDirac (1.3.0), the community will be able to accurately and efficiently estimate nuclear properties, such as the nuclear charge radius, by assuming a 2-parameter Fermi distribution of the nuclear charge.

High-fidelity logical \emph{T}-gate realization constitutes a core prerequisite for large-scale fault-tolerant quantum computing. However, conventional magic state distillation requires massive physical qubit overhead across successive distillation rounds, alongside sophisticated measurement and feedback control, thereby inducing considerable spatial and temporal resource consumption. Herein, we propose a controlled superconducting geometric logical \emph{T} gate scheme that achieves high-order suppression of universal errors, by integrating decoherence-free subspace encoding with multi-loop optimized composite geometric pulse engineering. Guided by tailored trajectory design, we systematically establish unified gate construction frameworks for conventional geometric, composite geometric, and optimized composite geometric protocols. By flexibly controling additional parametric degrees of freedom, the proposed scheme achieves substantially enhanced robustness against diverse noise sources. Numerical simulations reveal that, within tunable superconducting quantum circuits, our geometric logical \emph{T} gate outperforms both conventional composite geometric and dynamical gates in suppressing Rabi frequency, detuning, and residual inter-qubit crosstalk errors that can all be suppressed to the fourth order, while additionally providing inherent suppression of collective dephasing errors. The present strategy alleviates intrinsic limitations of mainstream approaches and opens a promising avenue toward robust high-fidelity logical \emph{T} gate construction.

Limits on spin-0 axion-like-particles (ALPs) coupling to photons are reinterpreted as constraints on massive spin-2 graviton-like-particles (GLPs) with universal coupling $α_\text{G}/M_\text{P}$ (where $M_\text{P}$ is the reduced Planck mass) to the Standard Model fields. A minimally model-dependent recasting is performed, exploiting the formally analogous production and detection mechanisms for both particle types, based on the Primakoff and Gertsenshtein effects, i.e., photon-axion/graviton conversion. Constraints originally derived in the ALP mass vs. photon-coupling plane ($m_\text{a}, g_{\text{a}γ}$) are translated into the corresponding bounds in the GLP ($m_\text{G}, α_\text{G}/M_\text{P}$) parameter space over the full mass range, $m_\text{a,G} \approx 10^{-20}$--$10^{14}$eV probed in current and future experimental setups including cavity-based detectors (haloscopes and resonant upconversion devices), helioscopes, magnetometers, optical interferometers, beam dumps, fixed-target, and collider experiments, as well as astrophysical and cosmological constraints. Generic scenarios are considered in which GLPs are a dark matter candidate and not. Whereas current ALP searches do not set stronger bounds on massive spin-2 particles than fifth-force tests, future magnetometers, two-beam interferometers, and upconversion experiments have the potential to provide very strong sensitivity, down to $α_\text{G}/M_\text{P} \approx 10^{-32} \text{GeV}^{-1}$, for light graviton-like particles with $m_\text{G}\lesssim 10^{-8}$eV. These future detectors exhibit comparatively greater sensitivity to massive gravitons than to axions. For massive gravitons at the TeV scale, exclusive diphoton decay searches, employed in ALP studies, offer a complementary approach to standard searches for spin-2 resonances in other inclusive final states.

A modified gravitational model whose action is given by an arbitrary function of the Ricci scalar, the matter Lagrangian density, a scalar field, and its kinetic term is investigated as an extension of the gravitational sector including an additional dynamical degree of freedom. Within this framework, the causal structure of rotating cosmological solutions is analyzed by considering Gödel and Gödel-type spacetimes as background geometries used as theoretical probes of the model consistency. Different matter sources are examined, including a perfect fluid and scalar-field configurations. It is found that the standard Gödel metric is not compatible with the scalar sector of the theory unless the model reduces to the General Relativity limit. In contrast, Gödel-type geometries admit a wider class of solutions whose causal properties depend on the model parameters and on the matter content. In particular, perfect-fluid sources may lead to either causal or noncausal configurations, whereas scalar-field configurations constrain the geometry to the causal limit, preventing the formation of closed timelike curves, highlighting that the scalar field plays a nontrivial dynamical role distinct from that of a cosmological constant.

In this investigation, we provide an exact analytical vortex solution for a Bose liquid in two dimensions with beyond mean-field correction (BMF). Analytical solutions in two-dimensional systems with BMF corrections are rarely found in the literature. The present result provides a clear framework for understanding vortex structures in low-dimensional quantum fluids and serves as a reliable benchmark for future theoretical and experimental studies.

Fabrication process-induced performance variability remains a formidable barrier in the high-volume manufacturing of semiconductor chips. With skyrocketing Artificial Intelligence (AI) workload, demand for non-volatile and computational memories is growing exponentially. As embedded non-volatile memory, ferroelectric Hf0.5Zr0.5O2 emerged as a strong candidate due to their CMOS back-end-of-line (BEOL) compatibility, scalability and high performance. However, their sensitive crystallization kinetics leads to significant device-to-device (D2D) non-uniformity leading to unpredictability of performance over wafer scale. In this work, we demonstrate unsupervised machine learning can analyze intra-die D2D variations and predict performance of "unseen" dies efficiently. We present a framework utilizing Principal Component Analysis (PCA) and K-Means clustering to analyze D2D performance variations in HZO capacitors and building on data from multiple dies, we move beyond traditional descriptive statistics to a predictive "Virtual Metrology" approach that separates performance categories, defined by key parameters like remanent polarization (Pr) and coercive voltage (Vc). The analysis further extends to comparing uniformity across different dies across the wafer showing the proposed methodology can accurately predict device performance on untested dies with a low Mean Absolute Percentage Error (MAPE) in the range of 5-10%, suggesting a robust path for accelerated yield improvement and reduced metrology overhead.

We investigate reentrant topological transitions in a one-dimensional Su--Schrieffer--Heeger chain with power-law quasiperiodically modulated intracell hopping. The modulation is characterized by a positive integer exponent $n$ and a tunable parameter $β$, which continuously interpolates between the smooth power-law quasiperiodic limit and a sign-function limit that becomes square-wave-like for odd $n$ and uniform for even $n$. By combining analytical calculations of the zero-mode inverse localization length with numerical evaluations of a real-space topological indicator, we determine the topological phase diagrams in the $β\to 0$, $β\to\infty$, and finite-$β$ regimes. We show that deterministic quasiperiodic modulation can induce TAI-like reentrant topological phases within finite parameter windows. The formation of these phases depends crucially on the parity of $n$: for positive modulation strength, odd-power modulations can induce reentrant topology from the clean trivial regime $|t_1|>1$, whereas even-power modulations allow such reentrance only from the negative clean trivial regime $t_1<-1$. Exact analytical expressions for the zero-mode inverse localization length are obtained for $n=1,2,3,4$, yielding explicit or implicit transition conditions. The finite-$β$ results demonstrate that the parity-dependent structure remains robust throughout the interpolation between the two limiting cases. This parity effect originates from whether the modulation preserves or removes the sign structure of $\cos x$. We further propose an electrical-circuit implementation and discuss experimentally accessible signatures of the reentrant trivial--topological--trivial transition.

We investigate the saturation regime of the condensing symmetric inclusion process on the discrete one-dimensional torus in the thermodynamical limit. In this regime, the total mass concentrates on a finite number of sites, forming condensates. Our main result establishes that, under appropriate scaling, the positions of the condensates converge to a system of coalescing Brownian motions on the continuum torus. In particular, condensates perform diffusive motion until they meet, at which point they merge and their masses coagulate. This provides a rigorous derivation of a macroscopic coalescing diffusion from an underlying interacting particle system with condensation. The main technical difficulty arises from the complicated coalescence mechanism of two condensates of particles, whose trajectories are very difficult to track completely. The key idea is to control the coalescing time instead and prove that it is negligible compared to the time-scale of condensate movement. By combining this with precise estimates of movements without coalescence, we can prove its convergence to coalescing Brownian motions.

The Solomon-Tits theorem says that the poset of proper non-trivial subspaces of a finite-dimensional vector space has realisation equivalent to a wedge of spheres. In this paper we prove a variant of this result for collections of geodesic subspaces of Euclidean, hyperbolic, or spherical geometry, assuming the collection is generated either by points or by hyperplanes. In the third paper of this series of papers, we will combine this with the homological stability theorems from the first paper to compute the homology of groups of scissors automorphisms in these geometries.

Neutron scattering and muon spin rotation ($μ$SR) measurements on single-crystal SmAu$_3$Al$_7$ reveal magnetically ordered states associated with successive transitions at $T_{\rm N}$ = 2.8 K and $T^*$ = 0.9 K. Magnetic Bragg peaks appear below $T_{\rm N}$ with an incommensurate (IC) propagation vector ${\bf q}$ = (0.30, 0, 1.33). $μ$SR detects spontaneous internal fields below $T_{\rm N}$, and the spectral shape is consistent with the IC magnetic ordering. No anomalies are observed at $T^*$, indicating that the magnetic structure remains essentially unchanged below and above $T^*$. The magnetic order is revealed to be a spatially homogeneous long-range ordered state, rather than a partially disordered state proposed in earlier studies. The possible connection between the IC magnetic order and the enhanced low-temperature magnetic specific heat is discussed.

In this report, we provide a theoretical framework for the magnetic behavior of the Griffiths phase, which, along with three-dimensional spin-1/2 Ising ferromagnetic systems, can be extended to antiferromagnetic as well as ferrimagnetic systems. We find that the magnetic behavior in the Griffiths phase of three-dimensional antiferromagnetic and ferrimagnetic systems is more unusual than that of conventional ferromagnetic systems. However, this study offers a possible framework for the identification of Griffiths phase behavior in three-dimensional antiferromagnetic and ferrimagnetic systems.

This paper studies location privacy in uplink MIMO systems, where a user equipment seeks to spoof the angular signature observed by a single base station performing localization. We propose a blind analog precoder design that manipulates the perceived angle-of-arrival and angle-of-departure configuration without requiring channel-gain knowledge. The method enforces consistency between the received signal and a desired spoofed angular subspace, and is solved using an alternating optimization algorithm under practical amplitude constraints. Simulations in a multipath scenario show that the proposed approach achieves near-perfect angular spoofing and clearly outperforms pilot-only blind spoofing, which exhibits an error floor. The results also show a trade-off between spoofing accuracy and communication rate, depending on the chosen virtual geometry.

Network interference occurs when a unit's outcome depends not only on its own treatment but also on the treatments received by connected units in the network. Experimental designs and analysis methods that ignore such interference can yield biased estimators of causal effects. In this paper, we develop a new experimental design for the estimation and inference of global treatment effect and spillover effect under a model-based framework and ego-cluster randomization. Under this design, the network is partitioned into a collection of ego-clusters, each consisting of a focal unit (the ego) and its network neighbors (the alters), with randomization conducted at the cluster level. We propose model-based estimators for the global treatment effect and spillover effect and establish their consistency and asymptotic normality, with asymptotic variances determined by the ego-cluster structure. Building on these theoretical results, we introduce an ego-clustering algorithm that sequentially selects egos and assigns alters to minimize asymptotic variances. Simulation studies and two empirical applications demonstrate that the proposed procedure yields accurate inference and efficiency improvements over existing network experimental designs.

We revisit Royen's proof of the Gaussian correlation inequality from a supersymmetric point of view. Many key elements in Royen's proof of this inequality have natural geometric interpretations in terms of supersymmetric dimensional reduction from $\mathbb{R}^{3|2}$ to $\mathbb{R}^{1|0}$. In particular, the auxiliary multivariate Gamma distributions appearing in Royen's Laplace-transform argument arise naturally as the body of a supersymmetric radial variable on $\mathbb{R}^{3|2}$. The generalization to the half-integer multivariate Gamma case also follows naturally as a dimensional reduction from $\mathbb{R}^{k+2|2}$ to $\mathbb{R}^{k|0}$. This provides an example in which the supersymmetric localization method is applied to prove correlation inequalities with continuous parameters.

We establish a connection between policy evaluation in Markov decision processes and PageRank in network analysis. For a fixed policy, we show that the value function of a discounted Markov decision process can be obtained, up to an explicit rescaling, from the PageRank vector of a suitably defined time-reversed Markov chain. In this correspondence, the discount factor plays the role of the teleportation parameter, while rewards induce the restart distribution. Beyond the irreducible case, invoking quasi-stationary distributions and Doob $h$-transforms, we prove a general decomposition theorem showing that policy evaluation for arbitrary finite MDPs reduces to a collection of PageRank problems on the recurrent and transient components of the policy-induced Markov chain. This framework naturally extends to undiscounted MDPs with terminal states and to transition-dependent rewards. We conclude by showing efficiency of our approach on a numerical example of a sticky random walk on large deterministic and random graphs.

Autonomous driving systems (ADS) are increasingly deployed in real traffic, yet testing remains fundamentally challenging due to open environments, complex scenarios, and the lack of established processes and metrics. Despite extensive research, a gap persists between academic advances and their applicability in industrial practice. To address this, we conduct an interactive rapid review in collaboration with 21 practitioners from a leading automotive company. Practitioners identified 12 key challenges in ADS testing, and prioritised two as the most critical issues, namely approaches to and completeness of testing for End-to-End (E2E) ADS. We analyzed 17 research studies relevant to these two challenges, most of which focus on generating critical testing scenarios, and subsequently assessed their relevance and applicability in practice. Our study provides the first practitioner-driven review and evaluation of current ADS testing research, reveals practical challenges in ADS testing, offers rapid insights for practitioners, and highlights the need for more context-aware, industry-relevant solutions to bridge the gap between research and practice.

We prove an abstract theorem on keeping the compactness property of a linear operator after interpolation in Banach spaces. Our approach consists of two features. Applying the principle "reductio ad absurdum," we obtain a possibility to carry out all proofs only for some specially constructed subspaces of the given spaces, e.g., having a common Schauder basis. As a second feature, we consider in all assertions only embedding operators obtaining the full result just at the end of the paper. No analytical presentation of operators, spaces and interpolation functors is required and the complex method is admissible as a particular case.

The horocyclic evolutes of spacelike frontals in hyperbolic 2-space have already been defined. Using enveloid theorem, we now define the horocyclic parallel and involute of a spacelike frontal in hyperbolic 2-space as the normal envelopes of its normal and tangent horocycles, respectively. Meanwhile, we investigate the relations among horocyclic evolutes, parallels and involutes.

The inertia bound, introduced by Cvetković in 1971, is a fundamental result in spectral graph theory that provides an upper bound for the independence number of a graph in terms of spectral information about a weighted adjacency matrix of the graph. Recently, this bound has been extended to the socalled inertia-type bounds for estimating the independence and chromatic numbers of graph powers ($k$-independence number and distance-$k$ chromatic number of a graph). These bounds have recently found applications in coding theory and quantum information theory. The inertia-type bounds depend on the choice of a polynomial of degree $k$ and on the eigenvalues of the graph. Currently, optimizing these bounds requires solving several MILPs, which quickly becomes computationally demanding as the graph size or $k$ grows. This computational barrier is a major obstacle to the practical use of these bounds. Moreover, we have a limited theoretical understanding of their performance, even for small $k$. In this paper, we investigate their optimization and complexity. In particular, we improve the MILP formulations, reducing their computational burden and significantly decreasing the running time. Furthermore, we show that the optimization problems associated with the bounds are solvable in polynomial time for fixed $k$ and for small $k$.

We study an intuitionistic version of common knowledge logic (CK), called ICK, which was introduced by Jäger and Marti. ICK extends intuitionistic propositional logic (IPL) by multiple box modalities interpreted as knowledge operators for various agents and a common knowledge operator. Formulae are interpreted over birelational Kripke models satisfying a simple interaction principle between the intuitionistic order and the modal accessibility relations. Furthermore, we consider the restriction to reflexive, S4 and S5 models. We present axiomatizations as well as analytic cyclic sequent calculi for all considered logics and prove them to be sound and complete. Furthermore, we establish the finite model property and decidability, show that proof-search in the cyclic calculi can be automated, provide a translation of CK over S5 into ICK over S5 and establish that the proof-search and validity problems of all considered logics can be solved in exponential time.