Both bilevel and robust optimization are established fields of mathematical optimization and operations research. However, only until recently, the similarities in their mathematical structure has neither been studied theoretically nor exploited computationally. Based on the recent results by Goerigk et al. (2025), this paper is the first one that provides an extensive computational study for solving strictly robust optimization problems with decision-dependent uncertainty sets as equivalent bilevel optimization problems. If the uncertainty set can be dualized, the respective bilevel techniques to obtain a single-level reformulation are very similar compared with the classic dualization techniques used in robust optimization but lead to larger single-level problems to be solved. Our numerical study shows that this usually leads to larger computation times. For the more challenging case of decision-dependent uncertainty sets represented by mixed-integer linear models, one cannot apply classic dualization techniques from robust optimization. Thus, we compare the presented bilevel approach with an established method from the literature, which is based on quantified mixed-integer linear programs. Our numerical results indicate that, for the problem class of decision-dependent robust optimization problems with mixed-integer linear uncertainty sets, the bilevel approach performs better in terms of computation times.

Thermal radiative transfer models physical phenomena ranging from supernovas in astrophysics to radiation from a hohlraum striking a fusion target in plasma physics. Transport and absorption of particles in radiative transfer at different rates lead to a complex interaction between the material and particles that involves highly varying time scales. Resolving these effects can require prohibitively small step sizes, which, combined with nonlinear effects and the particle density's high-dimensional phase space, render conventional numerical methods computationally expensive. This work presents an asymptotic--preserving, mass conservative, rank-adaptive, and parallel integrator for a macro--micro decomposition-based dynamical low-rank approximation of the thermal radiative transfer equations. The proposed integrator efficiently incorporates reflection-transmission type boundary conditions in the low-rank factors. It captures the nonlinear effects of thermal radiation and is energy stable with the step size restriction capturing both hyperbolic and parabolic CFL conditions. The efficacy of the proposed integrator is demonstrated with numerical experiments.

The Merino-Welsh conjecture states that for a graph $G$ without loops and bridges the Tutte polynomial $T_G(x,y)$ satisfies the inequality $$\max(T_G(2,0),T_G(0,2))\geqslant T_G(1,1).$$ Later Jackson proved that for any matroid $M$ without loops and coloops we have $$T_M(3,0)T_M(0,3)\geqslant T_M(1,1)^2.$$ The value $3$ in this statement was improved to $2.9243$ by Beke, Csáji, Csikvári and Pituk. In this paper, we further improve on this result by showing that $$T_M(2.355,0)T_M(0,2.355)\geqslant T_M(1,1)^2.$$ We also prove that the Merino--Welsh conjecture is true for matroids $M$, where all circuits of $M$ and its dual $M^*$ have length between $\ell$ and $(\ell-2)^2(\ell^2-4\ell+2)$ for some $\ell\geqslant 4$.

We introduce the Eggbox Ising model, a tunable construction of rugged energy landscapes defined by distances to a prescribed set of patterns. Correlated pattern ensembles realize arbitrary k-step replica-symmetry-breaking structures and controllable Parisi overlap distributions p(q), consistent with the hierarchical overlap structure observed in a simple word-embedding example from empirical data. A softened variant allows a systematic expansion leading to Hopfield-type couplings (and higher-body terms). We analyze the density of states and show that suitable potentials induce discontinuous finite-temperature transitions with metastability and hysteresis.

When modelling spacetime and classical physical fields, one typically assumes smoothness (infinite differentiability). But this assumption and its philosophical implications have not been sufficiently scrutinized. For example, we can appeal to analytic functions instead, which are also often used by physicists. Doing so leads to very different philosophical interpretations of a theory. For instance, our world would be 'hyperdeterministic' with analytic functions, in the sense that every field configuration is uniquely determined by its restriction to an arbitrarily small region. Relatedly, the hole argument of general relativity does not get off the ground. We argue that such an appeal to analytic functions is technically feasible and, conceptually, not obviously objectionable. The moral is to warn against rushing to draw philosophical conclusions from physical theories, given their drastic sensitivity to mathematical formalisms.

Lennard-Jones clusters, while an easy system, have a significant number of non equivalent configurations that increases rapidly with the number of atoms in the cluster. Here, we aim at determining the cluster partition function; we use the nested sampling algorithm, which transforms the multidimensional integral into a one-dimensional one, to perform this task. In particular, we use the nested_fit program, which implements slice sampling as search algorithm. We study here the 7-atom and 36-atom clusters to benchmark nested_fit for the exploration of potential energy surfaces. We find that nested_fit is able to recover phase transitions and find different stable configurations of the cluster. Furthermore, the implementation of the slice sampling algorithm has a clear impact on the computational cost.

Violations of Lorentz Invariance, a cornerstone of modern physics, are predicted by theories of quantum gravity and by extensions of General Relativity involving new vector or tensor fields. In the weak-field limit, such a violation would primarily manifest as a non-zero value for the post-Newtonian parameter $α_1$, which is identically zero in General Relativity. We present a new test of Local Lorentz Invariance by searching for this signature in the orbits of the LAGEOS and LAGEOS II satellites. By applying a Phase Sensitive Detection technique to the mean argument of latitude, derived from about 30 years of Satellite Laser Ranging data, we isolate the periodic signal potentially induced by a preferred reference frame aligned with the Cosmic Microwave Background. {Our analysis yields a new constraint $|α_1| \sim 2 \times 10^{-5}$. This result improves upon the previous best limit from Lunar Laser Ranging and provides the most stringent constraint to date on preferred-frame effects in Earth's gravity.}

We study entropic fluctuations in the Spin-Fermion model describing an $N$-level quantum system coupled to several independent thermal free Fermi gas reservoirs. We establish the quantum Evans-Searles and Gallavotti-Cohen fluctuation theorems and identify their link with entropic ancilla state tomography and quantum phase space contraction of non-equilibrium steady state. The method of proof involves the spectral resonance theory of quantum transfer operators developed by the authors in previous works.

We study the maximum ratio of the Euclidean norm to the comass norm of p-covectors in Euclidean n-space and improve the known upper bound found in the standard references by Whitney and Federer. We go on to prove stable systolic inequalities when the fundamental cohomology class of the manifold is a cup product of forms of lower degree.

We derive an exact solution for a simple non-autonomous delay differential equation (DDE) over the entire real-time axis, representing it as a sum of Gaussian-shaped dynamics with distinct peak positions. This marks the first explicit solution for non-autonomous DDEs and is a rare example even among general DDEs. The constructed solution offers key physical insights and facilitates the analysis of system properties, such as the envelope profile of the dynamics.

I review the twistor theory construction of stationary and axisymmetric, Lorentzian signature solutions of the Einstein vacuum equations and the related toric Ricci-flat metrics of Riemannian signature, \cite{W,MW,F,FW}. The construction arises from the Ward construction \cite{W2} of anti-self-dual Yang-Mills fields as holomorphic vector bundles on twistor space, with the observation of Witten \cite{LW} that the Einstein equations for these metrics include the anti-self-dual Yang-Mills equations. The principal datum for a solution is the holomorphic patching matrix $P$ for a holomorphic vector bundle on a reduced twistor space, and $P$ is typically simpler than the corresponding metric to write down. I give a catalogue of examples, building on earlier collections \cite{F,AG}, and consider the inverse problem: how far does the rod structure of such a metric, together with its asymptotics, determine $P$?

Let $G$ be a plane graph and let $C$ be a cycle in $G$. For each finite face of $G$, count the number of edges of $C$ the face contains. We call this the Slitherlink signature of $C$. The symmetric difference $A$ of two cycles with the same signature is totally even, meaning every vertex is incident to an even number of edges in $A$ and every face contains an even number of edges in $A$. In this paper, we completely characterize totally even subsets in the triangular grid, and count the number of edges in any totally even subset of the triangular grid. We also show that the size of the symmetric difference of two cycles with the same signature in the triangular grid is divisible by $12$; this is best possible since 12 is the greatest common divisor of all the sizes of the symmetric difference between two cycles with the same signature in a triangular grid.

Given any $r$-edge coloring of $K_{n,n}$, how large is the maximum (over all $r$ colors) sized monochromatic subgraph guaranteed to be? We give answers to this problem for $r \leq 8$, when $r$ is a perfect square, and when $r$ is one less than a perfect square all up to a constant additive term that depends on $r$. We give a lower bound on this quantity that holds for all $r$ and is sharp when $r$ is a perfect square up to a constant additive term that depends on $r$. Finally, we give a construction for all $r$ which provides an upper bound on this quantity up to a constant additive term that depends on $r$, and which we conjecture is also a lower bound.

Auction is applied for trade with various mechanisms. A simple but practical question is which mechanism, typically first-price or second-price auctions, is preferred from the perspective of bidders or sellers. A celebrated answer is revenue equivalence, where each bidder's equilibrium payoff is proven to be independent of auction mechanisms (and a seller's revenue, too). In reality, however, auction environments like the value distribution of items would vary over time, and such equilibrium bidding cannot always be achieved. Indeed, bidders must continue to track their equilibrium bidding by learning in first-price auctions, but they can keep their equilibrium bidding in second-price auctions. This study discusses whether and how revenue equivalence is violated in the long run by comparing the time series of non-equilibrium bidding in first-price auctions with those of equilibrium bidding in second-price auctions. We characterize the value distribution by two parameters: its basis value, which means the lowest price to bid, and its value interval, which means the width of possible values. Surprisingly, our theorems and experiments find that revenue equivalence is broken by the correlation between the basis value and the value interval, uncovering a novel phenomenon that could occur in the real world.

In this paper we study realizations of highest weight modules for the complex Lie algebra $\mathfrak{gl}_n$ with respect to non-standard Gelfand-Tsetlin subalgebras. We also provide sufficient conditions for such subalgebras to have a diagonalizable action on these realizations.

We consider a scalar conservation law with linear and nonlinear flux function on a bounded domain $Ω\subset{\R}^2$ with Lipschitz boundary $\partialΩ.$ We discretize the spatial variable with the standard finite element method where we use a local extremum diminishing flux limiter which is linearity preserving. For temporal discretization, we use the second order explicit strong stability preserving Runge--Kutta method. It is known that the resulting fully-discrete scheme satisfies the discrete maximum principle. Under the sufficiently regularity of the weak solution and the CFL condition $k = \mathcal{O}(h^2)$, we derive error estimates in $L^{2}-$ norm for the algebraic flux correction scheme in space and in $\ell^\infty$ in time. We also present numerical experiments that validate that the fully-discrete scheme satisfies the temporal order of convergence of the fully-discrete scheme that we proved in the theoretical analysis.

In this work we investigate how quantum expanders (i.e. quantum channels with few Kraus operators but a large spectral gap) can be constructed from unitary designs. Concretely, we prove that a random quantum channel whose Kraus operators are independent unitaries sampled from a $2$-design measure is with high probability an optimal expander (in the sense that its spectral gap is as large as possible). More generally, we show that, if these Kraus operators are independent unitaries of the form $U^{\otimes k}$, with $U$ sampled from a $2k$-design measure, then the corresponding random quantum channel is typically an optimal $k$-copy tensor product expander, a concept introduced by Harrow and Hastings (Quant. Inf. Comput. 2009).

Topological orders in 2+1d are spontaneous symmetry-breaking (SSB) phases of 1-form symmetries in pure states. The notion of symmetry is further enriched in the context of mixed states, where a symmetry can be either ``strong" or ``weak". In this work, we apply a Rényi-2 version of the proposed equivalence relation in [Sang, Lessa, Mong, Grover, Wang, & Hsieh, to appear] on density matrices that is slightly finer than two-way channel connectivity. This equivalence relation distinguishes general 1-form strong-to-weak SSB (SW-SSB) states from phases containing pure states, and therefore labels SW-SSB states as ``intrinsically mixed". According to our equivalence relation, two states are equivalent if and only if they are connected to each other by finite Lindbladian evolution that maintains continuously varying, finite Rényi-2 Markov length. We then examine a natural setting for finding such density matrices: disordered ensembles. Specifically, we study the toric code with various types of disorders and show that in each case, the ensemble of ground states corresponding to different disorder realizations form a density matrix with different strong and weak SSB patterns of 1-form symmetries, including SW-SSB. Furthermore we show by perturbative calculations that these disordered ensembles form stable ``phases" in the sense that they exist over a finite parameter range, according to our equivalence relation.

In this work we investigate several phenomenological and technical aspects related to electroweak (EW) corrections at a high-energy muon collider, focusing on direct production processes (no VBF configurations). We study in detail the accuracy of the Sudakov approximation, in particular the Denner-Pozzorini algorithm, comparing it with exact calculations at NLO EW accuracy. We also assess the relevance of resumming EW Sudakov logarithms (EWSL) at 3 and 10 TeV collisions. Furthermore, we scrutinise the impact of additional Heavy Boson Radiation (HBR), namely the weak emission of $W, Z$, and Higgs bosons in inclusive and semi-inclusive configurations. All results are obtained via the fully automated and publicly available code MadGraph5_aMC@NLO.

The focus of this paper is a key component of a methodology for understanding, interpolating, and predicting fish movement patterns based on spatiotemporal data recorded by spatially static acoustic receivers. Unlike GPS trackers which emit satellite signals from the animal's location, acoustic receivers are akin to stationary motion sensors that record movements within their detection range. Thus, for periods of time, fish may be far from the receivers, resulting in the absence of observations. The lack of information on the fish's location for extended time periods poses challenges to the understanding of fish movement patterns, and hence, the identification of proper statistical inference frameworks for modeling the trajectories. As the initial step in our methodology, in this paper, we devise and implement a simulation-based imputation strategy that relies on both Markov chain and random-walk principles to enhance our dataset over time. This methodology will be generalizable and applicable to all fish species with similar migration patterns or data with similar structures due to the use of static acoustic receivers.

Heavy-tailed models are used as a way to gain robustness against outliers in Bayesian analyses. In frequentist analyses, M-estimators are often employed. In this paper, the two approaches are tentatively reconciled by considering M-estimators as maximum likelihood estimators of heavy-tailed models. From this perspective, it is realized that a fundamental difference exists as frequentists, contrarily to Bayesians, do not require these heavy-tailed models to be proper. For instance, a popular robust estimator in linear regression, Tukey's biweight M-estimator, does not correspond to a proper heavy-tailed model. Thus, a Bayesian practitioner does not have access to the same range of tools as a frequentist practitioner. It is shown through two real-data linear regression analyses that the former may in consequence obtain significantly different estimation results than the latter, where the difference is due to a more pronounced influence by the outliers in the former case. It is highlighted that a way to give these practitioners access to the same range of tools is for the Bayesian to adopt the generalized Bayesian framework of Bissiri et al. (2016) which allows the use of improper models (Jewson and Rossell, 2022), in combination with proper prior distributions yielding proper generalized posterior distributions. A complete reconciliation of the Bayesian and frequentist approaches to robustness is then achieved. An extensive theoretical study of the generalized Bayesian counterpart of Tukey's biweight M-estimator is provided, which includes a robustness characterization result and a Bernstein--von Mises result, the latter allowing to calibrate the generalized posterior distribution for meaningful uncertainty quantification. After adopting the generalized Bayesian framework, the Bayesian practitioner obtains similar results as the frequentist practitioner in the aforementioned examples.

A Stokes wave is a traveling free-surface periodic water wave that is constant in the direction transverse to the direction of propagation. In 1981 McLean discovered via numerical methods that Stokes waves are unstable with respect to transverse perturbations. In \cite{CreNguStr} for the case of infinite depth we proved rigorously that the spectrum of the water wave system linearized at small Stokes waves, with respect to transverse perturbations, contains unstable eigenvalues lying approximately on an ellipse. In this paper we consider the case of finite depth and prove that the same spectral instability result holds for all but finitely many values of the depth. The computations and some aspects of the theory are considerably more complicated in the finite depth case.

Estimating nonlinear functionals of probability distributions from samples is a fundamental statistical problem. The "plug-in" estimator obtained by applying the target functional to the empirical distribution of samples is biased. Resampling methods such as the bootstrap derive artificial datasets from the original one by resampling. Comparing the outcome of the plug-in estimator in the original and resampled datasets allows estimating and thus correcting the bias. In the asymptotic setting, iterations of this procedure attain an arbitrarily high order of bias correction, but finite sample results are scarce. This work develops a new theoretical understanding of bootstrap bias correction by viewing it as an iterative linear solver for the combinatorial operation of Möbius inversion. It sharply characterizes the regime of linear convergence of the bootstrap bias reduction for moment polynomials. It uses these results to show its superalgebraic convergence rate for band-limited functionals. Finally, it derives a modified bootstrap iteration enabling the unbiased estimation of unknown order-$m$ moment polynomials in $m$ bootstrap iterations.

Given two (hereditary) complete cotorsion pairs $(\mathcal{X}_1,\mathcal{Y}_1)$ and $(\mathcal{X}_2,\mathcal{Y}_2)$ in an exact category with $\mathcal{X}_1\subseteq \mathcal{Y}_2$, we prove that $\left({\rm Smd}\langle \mathcal{X}_1,\mathcal{X}_2 \rangle,\mathcal{Y}_1\cap \mathcal{Y}_2\right)$ is also a (hereditary) complete cotorsion pair, where ${\rm Smd}\langle \mathcal{X}_1,\mathcal{X}_2 \rangle$ is the class of direct summands of extension of $\mathcal{X}_1$ and $\mathcal{X}_2$. As an application, we construct complete cotorsion pairs, such as $(^\perp\mathcal{GI}^{\leqslant n},\mathcal{GI}^{\leqslant n})$, where $\mathcal{GI}^{\leqslant n}$ is the class of modules of Gorenstein injective dimension at most $n$. And we also characterize the left orthogonal class of exact complexes of injective modules and the classes of modules with finite Gorenstein projective, Gorenstein flat, and PGF dimensions.

Real-time monitoring of dynamic biological processes in the body is critical to understanding disease progression and treatment response. This data, for instance, can help address the lower than 50% response rates to cancer immunotherapy. However, current clinical imaging modalities lack the molecular contrast, resolution, and chronic usability for rapid and accurate response assessments. Here, we present a fully wireless image sensor featuring a 2.5$\times$5 mm$^2$ CMOS integrated circuit for multicolor fluorescence imaging deep in tissue. The sensor operates wirelessly via ultrasound (US) at 5 cm depth in oil, harvesting energy with 221 mW/cm$^{2}$ incident US power density (31% of FDA limits) and backscattering data at 13 kbps with a bit error rate <$10^{-6}$. In-situ fluorescence excitation is provided by micro-laser diodes controlled with a programmable on-chip driver. An optical frontend combining a multi-bandpass interference filter and a fiber optic plate provides >6 OD excitation blocking and enables three-color imaging for detecting multiple cell types. A 36$\times$40-pixel array captures images with <125 $μ$m resolution. We demonstrate wireless, dual-color fluorescence imaging of both effector and suppressor immune cells in ex vivo mouse tumor samples with and without immunotherapy. These results show promise for providing rapid insight into therapeutic response and resistance, guiding personalized medicine.

The composition of the core of neutron stars (NS) is still under debate. One possibility is that because of the high densities reached in their cores, matter could be deconfined into quark matter. We investigate the existence of hybrid stars, using microscopic models to describe different phases of matter. Within the adopted microscopic models we calculate properties of NS and properties of matter. We want to probe the pQCD calculations influence and analyze properties that identify a transition to deconfined matter. Bayesian approach is applied to generate 8 sets of equations of state (EOS). A Maxwell construction is adopted to describe the deconfinement transition. For the hadron phase, we consider a stiff and a soft EOS obtained from the Relativistic Mean Field model with nonlinear meson terms. For the quark phase, we use 2 different models: the Nambu-Jona-Lasinio model with multiquark interactions and the Mean Field Theory of QCD, a model similar to the vector MIT bag model. Bayesian inference was applied to determine the model parameters that satisfy the X-ray observations from NICER and have phase transition at densities between 0.15 - 0.40 fm$^{-3}$. We also applied restrictions from the pQCD calculations to half of the sets. Hybrid stars are compatible with current observational data. The pQCD restrictions reduce the value of the $M_{max}$. However, even applying this restriction, the models were able to reach values of $2.1 - 2.3 M_\odot$. The conformal limit was still not attained at the center of the most massive stars. The vector interactions are essential to describe hybrid stars with a mass above $2 M_\odot$. The multiquark interactions introduced may affect the limits of some quantities considered as indicators of the presence of a deconfined phase. It is possible to find a set of EOS, that predict that inside NS the renormalized matter trace anomaly is always positive.

The weighted average of inconsistent data is a common and tedious problem that many scientists have encountered. The standard weighted average is not recommended for these cases, and various alternative methods have been proposed. These approaches vary in suitability depending on the nature of the data, which can make selecting the appropriate method difficult without expertise in metrology or statistics. For the analysis of simple data sets presenting inconsistencies, we discuss the method proposed by Sivia in 1996 based on Bayesian statistics. This choice has the intention of maintaining generality while minimising the number of assumptions. In this approach, the uncertainty associated with each input value is considered to be just a lower bound of the true unknown uncertainty. The resulting likelihood function is no longer Gaussian but has smoothly decreasing wings, which allows for a better treatment of scattered data and outliers. To demonstrate the robustness and the generality of the method, we apply it to a series of critical data sets: simulations, CODATA recommended values of the Newtonian gravitational constant, and some particle properties from the Particle Data Group, including the proton charge radius. A freely available Python library is also provided for a simple implementation of the proposed averaging method.

Let $(\mathcal{T}',\mathcal{T},\mathcal{T}'')$ be a recollement of triangulated categories.A complete ideal cotorsion pair in $\mathcal{T}$ induces complete ideal cotorsion pairs in $\mathcal{T}'$ and $\mathcal{T}''$. In addition, if $(\mathcal{I}, \mathcal{I}^\perp )$ and $(\mathcal{J},\mathcal{J}^\perp)$ are two complete ideal cotorsion pairs in a triangulated category, then $(\mathcal{I}\cap\mathcal{J}, \langle\mathcal{I}^\perp,\mathcal{J}^\perp\rangle)$ is also a complete ideal cotorsion pair. By this method, starting from two complete ideal cotorsion pairs in $\mathcal{T}'$ and $\mathcal{T}''$, one can induce a family of complete ideal cotorsion pairs in $\mathcal{T}$.

We establish the Soliton Resolution Conjecture for the radial critical non-linear heat equation in dimension $D\geq 3.$ Thus, every finite energy solution resolves, continuously in time, into a finite superposition of asymptotically decoupled copies of the ground state and free radiation.

We consider a class of optimal portfolio choice problems in continuous time where the agent's transactions create both transient cross-impact driven by a matrix-valued Volterra propagator, as well as temporary price impact. We formulate this problem as the maximization of a revenue-risk functional, where the agent also exploits available information on a progressively measurable price predicting signal. We solve the maximization problem explicitly in terms of operator resolvents, by reducing the corresponding first order condition to a coupled system of stochastic Fredholm equations of the second kind and deriving its solution. We then give sufficient conditions on the matrix-valued propagator so that the model does not permit price manipulation. We also provide an implementation of the solutions to the optimal portfolio choice problem and to the associated optimal execution problem. Our solutions yield financial insights on the influence of cross-impact on the optimal strategies and its interplay with alpha decays.