Joel Hamkins asks whether there is a $Π^0_1$-formula $ρ(x)$ such that $ρ({\ulcorner ϕ\urcorner})$ is independent over ${\sf PA}+ϕ$, if this theory is consistent, where this construction is extensional in $ϕ$ with respect to {\sf PA}-provable equivalence. We show that there can be no such extensional Rosser formula of any complexity. We give a positive answer to Hamkins' question for the case where we replace Extensionality by a weaker demand that we call \emph{Conditional Extensionality}. For this case, we prove an even stronger result, to wit, there is a $Π^0_1$-formula $ρ(x)$ that is extensional and $Π^0_1$-flexible. We leave one important question open: what happens when we weaken Extensionality to Consistent Extensionality, i.e., Extensionality for consistent extensions?

Ultralight bosonic (ULB) fields with mass $m_ϕ \ll 1$~eV often arise in theories beyond the Standard Model (SM). If such fields exist, violent astrophysical events that result in emission of gravitational wave, photon, or neutrino signals could also produce bursts of high-density relativistic ULB fields. Detection of such ULB fields in terrestrial or space-based laboratories correlated with other signals from transient astrophysical events opens a novel avenue for multimessenger astronomy. We show that quantum sensors are particularly well-suited to observe emitted scalar and pseudoscalar axion-like ULB fields coupled to SM. We demonstrate that multimessenger astronomy with ULB fields is possible even when accounting for matter screening effects.

We report on the determination of the gradient flow scales in $N_f=2+1$ QCD using highly improved staggered quark (HISQ) ensembles generated by the HotQCD Collaboration for bare gauge couplings ranging from $β= 6.423$ to $8.400$. Using bottomonium splittings, kaon decay constant, the decay constant of unmixed $η_s$ meson and the $ϕ$ meson mass we obtained the values of the gradient flow scales in physical units, $\sqrt{t_0} = 0.14428(48)$~fm and $w_0 = 0.17391(52)$~fm. Using the same physical inputs we revisit the determination of the potential $r_1$ scale and find $r_1 = 0.3112(24)$~fm. As a byproduct of our study we obtain the running of the gauge coupling in the gradient flow scheme. We find that within the uncertainties the running of the gradient flow coupling obtained on the lattice is compatible with the perturbative results up to flow radius $\sqrt{8 τ_F}=0.15$ fm.

We report Dirac Landau polaritons observed by terahertz (THz) magnetoreflectivity spectroscopy, demonstrating strong coupling between cyclotron transitions of two-dimensional (2D) Dirac fermions in HgTe quantum wells and optical cavity modes. Under pulsed electrical injection we observe efficient nonlinear electroluminescence, with a strongly out-of-equilibrium polariton distribution dominated by emission from the upper polariton branches. Model analysis of the bias-dependent emission intensity and spectral narrowing indicates a polariton occupancy per mode approaching unity, with a possible contribution from stimulated polariton emission in the spectral region of the upper anticrossing. These results open prospects toward Dirac Landau polariton condensates and low-threshold, tunable THz polariton lasers based on cyclotron emission.

This paper focuses on the problem of automatic link selection in multi-channel multiple access control using bandit feedback. In particular, a controller assigns multiple users to multiple channels in a time-slotted system, where in each time slot, at most one user can be assigned to a given channel, and at most one channel can be assigned to a given user. Given that user $i$ is assigned to channel $j$, the transmission fails with a fixed unknown probability $1-q_{i,j}$. The assignments are made dynamically using success/failure feedback. The goal is to maximize the time-average utility, where we consider an arbitrary (possibly nonsmooth) concave, entrywise nondecreasing utility function. The first proposed algorithm has fast $\mathcal{O}(\sqrt{\log(T)/T})$ convergence. However, this algorithm requires solving a convex optimization problem within each iteration, which can be computationally expensive. The second algorithm has slower $\mathcal{O}(\sqrt[3]{\log(T)/T})$ convergence, while avoiding the costly inner optimization. Both of these algorithms are adaptive. In particular, the convergence guarantee holds for any interval of $T$ consecutive slots during which the success probabilities do not change. We further study several special cases. In the single-channel setting, we obtain both fast $\mathcal{O}(\sqrt{\log(T)/T})$ convergence and efficient implementation via a simpler adaptive mechanism. We also consider a UCB-based non-adaptive algorithm with max-weight-type decisions. Simulations highlight intriguing performance trade-offs and demonstrate rapid adaptation of the proposed adaptive schemes.

We introduce a reinforcement learning method for a class of non-Markov systems; our approach extends the actor-critic framework given by Rose et al. [New J. Phys. 23 013013 (2021)] for obtaining scaled cumulant generating functions characterizing the fluctuations. The actor-critic is implemented using neural networks; a particular innovation in our method is the use of an additional neural policy for processing memory variables. We demonstrate results for current fluctuations in various memory-dependent models with special focus on semi-Markov systems where the dynamics is controlled by nonexponential interevent waiting time distributions.

Noncollinear magnets are notoriously difficult to describe within first-principles approaches based on density-functional theory (DFT) because of the presence of low-lying spin excitations. At the level of ground-state calculations, several methods exist to constrain the magnetic moments to a predetermined configuration, and thereby accelerate convergence towards self-consistency. Their use in a perturbative context, however, remains very limited. Here we present a general methodological framework to achieve parametric control over the local spin moments at the linear-response level. Our strategy builds on the concept of Legendre transform to switch between various flavors of magnetic functionals, and to relate their second derivatives via simple linear-algebra operations. Thereby, we can address an arbitrary response function at the time-dependent DFT level of theory with optimal accuracy and minimal computational effort. In the low frequency limit, we identify the leading correction to the existing adiabatic formulation of the problem [S. Ren \emph{et al.}, Phys. Rev. X {\bf 14}, 011041 (2024)], consisting in a renormalization of the phonon and magnon masses due to electron inertia. As a demonstration, we apply our methodology to the THz optical response of bulk CrI$_3$ and Cr$_2$O$_3$, where we identify contributions from hybrid (electro)magnons with mixed spin-lattice character.

Biharmonic and conformal-biharmonic maps are two fourth-order generalizations of the well-studied notion of harmonic maps in Riemannian geometry. In this article we consider maps into the Euclidean sphere and investigate a geometric algorithm that aims at rendering a given harmonic map either biharmonic or conformally biharmonic. For biharmonic maps we find that in the case of a closed domain the maximum principle imposes strong restrictions on our approach, whereas there is more flexibility when we have a non-compact domain and we highlight this difference by a number of examples. Concerning conformal-biharmonic maps we show that our algorithm produces explicit critical points for maps between spheres. Moreover, it turns out that we do not get strong restrictions as we obtain for biharmonic maps, such that our algorithm might produce additional conformal-biharmonic maps between spheres beyond the ones found in this article.

The augmentation scheme provides a nontraditional approach to nonlinear integer programming by iteratively refining incumbent solutions along objective-improving directions from the Graver basis. Its main computational bottleneck, however, lies in the practical difficulty of accessing such directions. To address this challenge, we develop a massively parallel heuristic for approximating Graver basis, extracting promising directions by optimizing nonconvex continuous problems using parallelizable first-order methods. Experiments on QPLIB and MINLPLib instances show that our method achieves comparable performance to advanced solvers.

The paper is devoted to proving Allard-Michael-Simon-type $L^p$-Sobolev inequalities $(p>1)$ with explicit constants in the setting of Euclidean minimal submanifolds of arbitrary codimension. Our results require separate discussions for the cases $p\geq 2$ and $1<p<2$, respectively. In particular, for $p\geq 2$, we obtain an asymptotically sharp and codimension-free Sobolev constant. Our argument is based on optimal mass transport theory on Euclidean submanifolds and also provides an alternative, unified proof of the recent isoperimetric inequalities of Brendle (J. Amer. Math. Soc., 2021) and Brendle and Eichmair (Notices Amer. Math. Soc., 2024).

This paper introduces fast R updating algorithms specifically designed for statistical applications, including regression, filtering, and model selection, where data structures change frequently. Although traditional QR decomposition is essential for matrix operations, it becomes computationally intensive when dynamically updating the design matrix in statistical models. The proposed algorithms efficiently update the R matrix without the need for recalculation of Q, thereby significantly reducing computational costs in practical computational scenarios. The provision of scalable solutions for high-dimensional regression models is a key strength of these algorithms, enhancing the feasibility of large-scale statistical analyses and model selection in data-intensive fields. A thorough simulation study and the analysis of real-world data demonstrate that the methods achieve a substantial reduction in computational time without compromising accuracy. The discussion illustrates the benefits of these algorithms across a wide range of models and applications in statistics and machine learning.

We present a magnetic trapping scheme for cold 87Rb atoms based on light-induced fictitious magnetic fields generated by the evanescent field of an optical nanofiber (ONF) integrated with an optical tweezers. We calculate and compare the trapping potentials for both Gaussian and Laguerre-Gaussian modes of the tweezers beam, combined with a quasi-linearly polarized ONF-guided field. Based on the optical powers in the tweezers and ONF modes, we analyze the trap depths and the positions of the potential minima from the nanofiber surface. We show that, by varying the optical powers in the two fields, the trap position can be tuned over several hundred nanometers, while simultaneously influencing the trap depth and trap frequencies. Such control over atom-surface position is essential for studying distance-dependent effects on atoms trapped near a dielectric surface and optimizing atom-photon interfaces for quantum technology applications.

We obtain an upper bound for the sum $\sum_{n\leq N} (a_{n}/φ(a_{n}))^{s}$, where $φ$ is Euler's totient function, $s\in \mathbb{N}$, and $a_{1},\ldots, a_{N}$ are positive integers (not necessarily distinct) with some restrictions. As applications, for any $t>0$, we obtain an upper bound for the number of $n\in [1,N]$ such that $a_{n}/ φ(a_{n})> t$.

We consider a system of two stochastic differential equations (SDEs) with competing two-way interactions driven by Brownian motions and spectrally positive $α$-stable random measures. Such a SDE system can be identified as a Lotka-Volterra type population model. We find nearly sharp conditions for one of the population to become extinct or extinguished.

This book is about conformal prediction and related inferential techniques that build on permutation tests and exchangeability. These techniques are useful in a diverse array of tasks, including hypothesis testing and providing uncertainty quantification guarantees for machine learning systems. Much of the current interest in conformal prediction is due to its ability to integrate into complex machine learning workflows, solving the problem of forming prediction sets without any assumptions on the form of the data generating distribution. Since contemporary machine learning algorithms have generally proven difficult to analyze directly, conformal prediction's main appeal is its ability to provide formal, finite-sample guarantees when paired with such methods. The goal of this book is to teach the reader about the fundamental technical arguments that arise when researching conformal prediction and related questions in distribution-free inference. Many of these proof strategies, especially the more recent ones, are scattered among research papers, making it difficult for researchers to understand where to look, which results are important, and how exactly the proofs work. We hope to bridge this gap by curating what we believe to be some of the most important results in the literature and presenting their proofs in a unified language, with illustrations, and with an eye towards pedagogy.

The perturbative contribution to the correlator of two HQET heavy-light currents expanded in light-quark masses up to quadratic terms is calculated up to 4 loops. The leading large-$β_0$ limit is also considered, so that terms with the highest degrees of $n_f$ are calculated to all orders in $α_s$. Borel images of coefficient functions in this limit contain renormalon poles. Naive nonabelianization works surprisingly poorly for the coefficient functions considered here.

We consider a class of external potentials on the complex plane $\mathbb{C}$ for which the coincidence set to the obstacle problem contains a Jordan curve in the exterior of the droplet. We refer to this curve as a spectral outpost. We study the corresponding Coulomb gas at $β=2$. Generalizing recent work in the radially symmetric case, we prove that the number of particles which fall near the spectral outpost has an asymptotic Heine distribution, as the number of particles $n\to\infty$. We also consider a class of potentials with disconnected droplets whose connected components are separated by a ring-shaped spectral gap. We prove that the fluctuations of the number of particles that fall near a given component has an asymptotic discrete normal distribution, which depends on $n$. For the case of disconnected droplets we also consider fluctuations of general smooth linear statistics and show that they tend to distribute as the sum of a Gaussian field and an independent, oscillatory, discrete Gaussian field. Our techniques involve a new asymptotic formula on the norm of monic orthogonal polynomials in the bifurcation regime and a variant of the method of limit Ward identities of Ameur, Hedenmalm, and Makarov.

We investigate (3+1)d topological orders in fermionic systems with an anomalous $\mathbb{Z}_{2N}^{\mathrm{F}}$ symmetry, where its $\mathbb{Z}_2^{\mathrm{F}}$ subgroup is the fermion parity. Such an anomalous symmetry arises as the discrete subgroup of the chiral U(1) symmetry of $ν$ copies of Weyl fermions of the same chirality. Inspired by the crystalline correspondence principle, we deform the anomalous $\mathbb{Z}_{2N}^\mathrm{F}$ symmetry of (3+1)d Weyl fermion to the anomalous $C_N \times \mathbb{Z}_2^\mathrm{F}$ symmetry. Then we microscopically construct symmetry-preserving gapped boundary states of the closely-related (4+1)d $C_N\times \mathbb{Z}_2^{\mathrm{F}}$ symmetry-protected topological (SPT) state (with $C_N$ being the $N$-fold rotation), whenever it is possible. In particular, for $ν=N$, we show that the (3+1)d symmetric gapped state admits a topological $\mathbb{Z}_4$ gauge theory description at low energy, and propose that a similar theory saturates the corresponding $\mathbb{Z}_{2N}^\mathrm{F}$ anomaly. For $N \nmid ν$, our construction admits no topological quantum field theory (TQFT) symmetric gapped state; while for $ν=N/2$, we find a non-TQFT symmetric gapped state via stacking lower-dimensional (2+1)d non-discrete-gauge-theory topological order inhomogeneously. For other values of $ν$, no symmetric gapped state is possible within our construction, which is consistent with the no-go theorem by Cordova-Ohmori.

We revisit the partial $\mathrm{C}^{1,α}$ regularity theory for minimizers of non-parametric integrals with emphasis on sharp dependence of the Hölder exponent $α$ on structural assumptions for general zero-order terms. A particular case of our conclusions carries over to the parametric setting of Massari's regularity theorem for prescribed-mean-curvature hypersurfaces and there confirms optimal regularity up to the limit exponent.

Over real numbers, Backhouse classified all four-dimensional Lie superalgebras. From this list, we will investigate those Lie superalgebras that can be obtained as Lagrangian extensions. Moreover, we investigate left-symmetric structures on these Lie superalgebras. Furthermore, except for two of them, they are all Novikov superalgebras.

The holographic superconductor is the holographic dual of superconductors. We recently identified the dual Ginzburg-Landau (GL) theory for a class of bulk 5-dimensional holographic superconductors (arXiv:2207.07182 [hep-th]). However, the result is the strong coupling limit or the large-$N_c$ limit. A natural question is how the dual GL theory changes at finite coupling. We identify the dual GL theory for a minimal holographic superconductor at finite coupling (Gauss-Bonnet holographic superconductor), where numerical coefficients are obtained exactly. The GL parameter $κ$ increases at finite coupling, namely the system approaches a more Type-II superconductor like material. We also point out two potential problems in previous works: (1) the "naive" AdS/CFT dictionary, and (2) the condensate determined only from the GL potential terms. As a result, the condensate increases at finite coupling unlike common folklore.

We give axiomatic characterisations of generalized $ψ$-estimators and (usual) $ψ$-estimators (also called $Z$-estimators), respectively. The key properties of estimators that come into play in the characterisation theorems are the symmetry, the (strong) internality and the asymptotic idempotency. In the proofs, a separation theorem for Abelian subsemigroups plays a crucial role.

Colour algebras over fields of odd characteristic are well-known noncommutative Jordan algebras. We define colour algebras more generally over a unital commutative associative ring with $\frac{1}{2}\in R$, and show that colour algebras can be constructed canonically by employing nondegenerate ternary hermitian forms with trivial determinant. We investigate their structure, automorphism group and derivations. As over fields, colour algebras over $R$ are closely related to octonion algebras over $R$.

We study the distribution of partial sums of Rademacher random multiplicative functions $(f(n))_n$ evaluated at polynomial arguments. We show that for a polynomial $P\in \mathbb Z[x]$ that is a product of at least two distinct linear factors or an irreducible quadratic satisfying a natural condition, there exists a constant $κ_P>0$ such that \[ \frac{1}{\sqrt{κ_P N}}\sum_{n\leq N}f(P(n))\xrightarrow{d}\mathcal{N}(0,1), \] as $N\rightarrow\infty$, where convergence is in distribution to a standard (real) Gaussian. This confirms a conjecture of Najnudel and addresses a question of Klurman-Shkredov-Xu. We also study large fluctuations of $\sum_{n\leq N}f(n^2+1)$ and show that there almost surely exist arbitrarily large values of $N$ such that \[ \Big|\sum_{n\leq N}f(n^2+1)\Big|\gg \sqrt{N \log\log N}. \] This matches the bound one expects from the law of iterated logarithm.

We study a lattice model describing the non-equilibrium dynamics emerging from the pulling of a tracer particle through a disordered medium occupied by randomly placed obstacles. The model is considered in a restricted geometry pertinent for the investigation of confinement-induced effects. We analytically derive exact results for the characteristic function of the moments valid to first order in the obstacle density. By calculating the velocity autocorrelation function and its long-time tail we find that already in equilibrium the system exhibits a dimensional crossover. This picture is further confirmed by the approach of the drift velocity to its terminal value attained in the non-equilibrium stationary state. At large times the diffusion coefficient is affected by both the driving and confinement in a way that we quantify analytically. The force-induced diffusion coefficient depends sensitively on the presence of confinement. The latter is able to modify qualitatively the non-analytic behavior in the force observed for the unbounded model. We then examine the fluctuations of the tracer particle along the driving force. We show that in the intermediate regime superdiffusive anomalous behavior persists even in the presence of confinement. Stochastic simulations are employed in order to test the validity of the analytic results, exact to first order in the obstacle density and valid for arbitrary force and confinement.

Belief propagation (BP) can be a useful tool to approximately contract a tensor network, provided that the contributions from any closed loops in the network are sufficiently weak. In this manuscript we describe how a loop series expansion can be applied to systematically improve the accuracy of a BP approximation to a tensor network contraction, in principle converging arbitrarily close to the exact result. More generally, our result provides a framework for expanding a tensor network as a sum of component networks in a hierarchy of increasing complexity. We benchmark this proposal for the contraction of iPEPS, either representing the ground state of an AKLT model or with randomly defined tensors, where it is shown to improve in accuracy over standard BP by several orders of magnitude whilst incurring only a minor increase in computational cost. These results indicate that the proposed series expansions could be a useful tool to accurately evaluate tensor networks in cases that otherwise exceed the limits of established contraction routines.

The mechanism of CDW and its 3D structure are important fundamental issues in kagome metals. We have previously shown that, based on a 2D model, $2\times 2$ bond order (BO) emerges due to the paramagnon-interference (PMI) mechanism and that its fluctuations lead to $s$-wave superconductivity. This paper studies these issues based on realistic 3D models of kagome metals AV$_3$Sb$_5$ (A=Cs, Rb, K). We reveal that a commensurate 3D $2\times 2\times 2$ BO is caused by the PMI mechanism, by performing the 3D density-wave (DW) equation analysis for all A=Cs, Rb, K models in detail. Our results indicate a BO transition temperature $T_{\rm BO}\sim 100$K within the regime of moderate electron correlation. The 3D structure of BO is attributed to the three-dimensionality of the Fermi surface, while the 3D structure of BO is sensitively changed, since the Fermi surface is quasi-2D. Based on the analysis of the DW equation, by taking into account a finite third-order Ginzburg-Landau (GL) term, (i) shift stacking $2\times 2\times 2$ BO can be realized via a first-order transition below $T_{\rm BO}$. Here, the in-plane BO pattern (tri-hexagonal or star-of-David) is determined by the sign of the third-order GL term, with hole doping tending to favor the tri-hexagonal state. On the other hand, if the third-order GL term is very small, (ii) alternating vertical stacking BO may instead be realized via a second-order transition. The present study enhances our understanding of the rich variety of BOs observed experimentally. It is confirmed that the PMI mechanism is the essential origin of the 3D CDW of kagome metals.

Stablecoins have historically depegged due from par to large sales, possibly of speculative nature, or poor reserve asset quality. Using a global game which addresses both concerns, we show that the selling pressure on stablecoin holders increases in the presence of a large sale. While precise public knowledge reduces (increases) the probability of a run when fundamentals are strong (weak), interestingly, more precise private signals increase (reduce) the probability of a run when fundamentals are strong (weak), potentially explaining the stability of opaque stablecoins. The total run probability can be decomposed into components representing risks from large sales and poor collateral. By analyzing how these risk components vary with respect to information uncertainty and fundamentals, we can split the fundamental space into regions based on the type of risk a stablecoin issuer is more prone to. We suggest testable implications and connect our model's implications to real-world applications, including depegging events and the no-questions-asked property of money.

The main theme of this paper is the introduction of a new type of polarizations, suited for some open symplectic manifolds, and their applications. These applications include symplectic embedding results that answer a question by Sackel-Song-Varolgunes-Zhu and Brendel, new Lagrangian non-removable intersections at small scales, and a novel phenomenon of Legendrian barriers in contact geometry.

In an era where sustainability is becoming increasingly crucial, we introduce a new Carbon-Aware Ant Colony System (CAACS) Algorithm that addresses the Generalized Traveling Salesman Problem (GTSP) while minimizing carbon emissions. This novel approach leverages the natural efficiency of ant colony pheromone trails to find optimal routes, balancing both environmental and economic objectives. By integrating sustainability into transportation models, CAACS provides a powerful tool for real-world applications, including network design, delivery route planning, and commercial aircraft logistics. Our algorithm's unique bi-objective optimization advances the study of sustainable transportation solutions.