In this paper, we prove that the Hilbert scheme $\mathbf{H}_4(X_5)$ of rational quartic curves on the quintic del Pezzo threefold $X_5$ is isomorphic to a Grassmannian bundle over the Hilbert scheme of lines on $X_5$. In particular, $\mathbf{H}_4(X_5)$ is smooth and irreducible. Our approach builds upon the geometry of rational quartic curves on $X_5$ studied by Fanelli-Gruson-Perrin in their work on the moduli space of stable maps to $X_5$.

We extend the holographic V-QCD model by introducing a charged scalar field sector to represent the condensation of paired quark matter in the deconfined phase. By incorporating this new sector into the previously established framework for nuclear and quark matter, we obtain a phase diagram that, in addition to the first-order deconfinement transition and its critical end-point, also features a second-order transition between paired and unpaired quark matter. The critical temperature for quark pairing exhibits only a mild dependence on the chemical potential and can reach values as high as $T_\mathrm{crit} \approx 30~\rm MeV$. Comparison of the growth rate for the formation of homogeneous paired phases to the growth rate of previously discovered modulated phases suggests that the former is subdominant to the latter.

For a generalized step reinforced random walk, starting from the origin, the first step is taken according to the first element of an innovation sequence. Then in subsequent epochs, it recalls a past epoch with probability proportional to a regularly varying sequence $\{μ_n\}$ of index $γ>-1$; recalls and repeats the step taken with probability $p$, or with probability $1-p$ takes a fresh step from the innovation sequence. The innovation sequence is assumed to be i.i.d.\ with mean zero. We study the corresponding step reinforced random walk process with linearly scaled time as an r.c.l.l.\ function on $[0, \infty)$. We prove law of large numbers for the linearly scaled process almost surely and in $L^1$ for all possible values of $p$ and $γ$. Assuming finite second moments for the innovation sequence, we obtain interesting phase transitions based on the boundedness of a sequence associated with $\{μ_n\}$. The random walk suitably scaled converges almost surely to a process, which may not be Gaussian, when the sequence is bounded and the convergence is in distribution to a Gaussian process otherwise. This phase transition introduces the point of criticality at $p_c=\frac{γ+1/2}{γ+1}$ for $γ>-\frac12$. For the subcritical regime, the process is diffusive, while it is superdiffusive otherwise. However, for the critical regime, the scaled process can converge almost surely or in distribution depending on the choice of sequence $\{μ_n\}$. Almost sure convergence in the critical regime is new. In the critical regime, the scaling can include many more novel choices in addition to $\sqrt{n \log n}$. Further, we use linear time scale and time independent scales even for the critical regime. We argue the exponential time scale for the critical regime is not natural. All the convergences in all the regimes are obtained for the process as an r.c.l.l.\ function.

The dependence of quantum algorithms on state fidelity is difficult to characterize analytically and is best explored experimentally as hardware scales and noisy simulations become intractable. While low fidelity states are often disregarded, they may still retain valuable information, as long as their dominant eigenvector approximates the target state. Through classical purification, we demonstrate the ability to recover resources specific to quantum computing such as magic and entanglement from noisy states generated by Pauli product formulas, which are common subroutines of many quantum algorithms. Additionally we show that the fidelity of the purified state is dependent on both the magnitude and order in which magic and entanglement are generated, which can be used to inform the order of operators within an ansatz. Consistent across simulation and experiment on IonQ's Aria quantum device, correlations within a state are found to be much more robust to noise than magic, and we show the advantage of designing algorithms targeting these low error states. This study uses quantum informatic tools for analyzing and optimizing quantum algorithms in a noisy framework.

In this article, we focus on the sampling of the configurational Gibbs-Boltzmann distribution, that is, the calculation of averages of functions of the position coordinates of a molecular $N$-body system modelled at constant temperature. We show how a formal series expansion of the invariant measure of a Langevin dynamics numerical method can be obtained in a straightforward way using the Baker-Campbell-Hausdorff lemma. We then compare Langevin dynamics integrators in terms of their invariant distributions and demonstrate a superconvergence property (4th order accuracy where only 2nd order would be expected) of one method in the high friction limit; this method, moreover, can be reduced to a simple modification of the Euler-Maruyama method for Brownian dynamics involving a non-Markovian (coloured noise) random process. In the Brownian dynamics case, 2nd order accuracy of the invariant density is achieved. All methods considered are efficient for molecular applications (requiring one force evaluation per timestep) and of a simple form. In fully resolved (long run) molecular dynamics simulations, for our favoured method, we observe up to two orders of magnitude improvement in configurational sampling accuracy for given stepsize with no evident reduction in the size of the largest usable timestep compared to common alternative methods.

The authors present a new simple algorithm to approximate weakly stochastic differential equations in the spirit of [1] and [2]. They apply it to the problem of pricing Asian options under the Heston stochastic volatility model, and compare it with other known methods. It is shown that the combination of the suggested algorithm and quasi-Monte Carlo methods makes computations extremely fast. [1] Shigeo Kusuoka, ``Approximation of Expectation of Diffusion Process and Mathematical Finance,'' Advanced Studies in Pure Mathematics, Proceedings of Final Taniguchi Symposium, Nara 1998 (T. Sunada, ed.), vol. 31 2001, pp. 147--165. [2] Terry Lyons and Nicolas Victoir, ``Cubature on Wiener Space,'' Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 460 (2004), pp. 169--198.

We give explicit, highly symmetric equations for the versal deformation of the singularity $L_{n+1}^n$ consisting of n+1 lines through the origin in n-dimensional affine space in generic position. These make evident that the base space of the versal deformation of $L_{n+1}^n$ is isomorphic to the total space for $L_{n}^{n-1}$, if n>4. By induction it follows that the base space is irreducible and Gorenstein. We discuss the known connection to a modular compactification of the moduli space of (n+1)-pointed curves of genus 1. For other elliptic partition curves it seems unfeasable to compute the versal deformation in general. It is doubtful whether the base space is Gorenstein. For rational partition curves we show that the base space in general has components of different dimensions.

Driven net- and no-net-flux MHD turbulence simulations up to $10,\!368^3$ reveal sign-mixed velocity-magnetic, velocity-vorticity, and magnetic-current aligned patches below the energy equipartition scale. The first two angles scale as $λ^{1/8}$ and $λ^{1/16}$, while magnetic-current alignment varies weakly with scale. We develop and test a constant-flux transport model for departures from relaxed states, which predicts both exponents. These findings affect eddy anisotropy, reconnection-mediated turbulence onset, large-scale dynamos, and the nature of magnetized turbulence.

We present a new simulation-secure quantum oblivious transfer (QOT) protocol based on one-way functions in the plain model. With a focus on practical implementation, our protocol surpasses prior works in efficiency, promising feasible experimental realization. We address potential experimental errors and their correction, offering analytical expressions to facilitate the analysis of the required quantum resources. Technically, we achieve simulation security for QOT through an equivocal and relaxed-extractable quantum bit commitment.

In this paper, we present an independent proof of the Cauchy identities for staircase matrices, originally discovered in arXiv:2411.03117, using the combinatorics of the Bruhat poset and the bubble-sort procedure. Additionally, we derive new insights into certain coefficients appearing in one of these identities. The first part of the paper focuses on combinatorial aspects. It is self-contained, of independent interest, and introduces a generalization of parabolic Bruhat graphs for monotone functions on an arborescent poset. The second part examines the intersections of Demazure modules within a given integrable representation. Finally, we propose a generalization of the classical Howe duality for staircase matrices in terms of the corresponding distributive lattice of Demazure submodules. Computing the associated character yields the desired Cauchy identities for staircase matrices.

The G-Wishart distribution is a core component for the Bayesian analysis of Gaussian graphical models as the conjugate prior for the precision matrix. Evaluating the marginal likelihood of such models usually requires computing high-dimensional integrals to determine the G-Wishart normalising constant. Closed-form results are known for decomposable or chordal graphs, while an explicit representation as a formal series expansion has been derived recently for general graphs. The nested infinite sums, however, do not lend themselves to computation, remaining of limited practical value. Borrowing techniques from random matrix theory and Fourier analysis, we provide novel exact results well suited to the numerical evaluation of the normalising constant for classes of graphs beyond chordal graphs. We additionally develop a Monte Carlo scheme for general graphs, which can be orders of magnitude more efficient than current approaches.

A cholera transmission model is formulated that incorporates water-borne and horizontal transmissions as well as infectivity of deceased individuals. The model includes an Allee effect for the bacteria in the environment and imperfect and waning vaccination. Mathematical properties of the model are investigated, with an environmental bistability shown to combine with a vaccine-driven one, although a computational search for the latter fails to detect its presence in realistic parameter ranges. The computational analysis also considers the interplay between vaccination strategy, vaccine efficacy and waning, as well as the effect of transmission of the disease during funeral rites. The effect of control scenarios such as WASH or Safe and dignified burials are assessed.

A common assumption in both grid-following (GFL) and grid-forming (GFM) control systems is that they are open-loop (OL) stable in the vicinity of high-frequency resonances. Hence classical loop-shaping approaches are often used for establishing stability margins and designing active damping (AD) strategies. This paper shows that single-loop GFM (SL-GFM) control schemes incorporating a widely used class of reactive power (RAP) control, referred to as droop-I control, can lead to OL unstable poles. This finding reveals a novel instability mechanism resulting in a reduced stability margin and robustness at high frequencies. The sensitivity of this phenomenon to both RAP and electrical parameters is analyzed in detail. An AD design that explicitly accounts for the newly identified instability mechanism is proposed. We also provide a comparison between such SL-GFM and well-studied GFL control schemes, highlighting quite different resonance features between them. Validation is performed through experiments.

Tool learning has emerged as a powerful auxiliary mechanism that extends the capabilities of large language models (LLMs), enabling them to address complex tasks that demand real-time relevance or high-precision operations. However, beneath this strength lie significant security risks. Prior studies have primarily concentrated on corrupting the outputs of invoked tools, while largely overlooking the vulnerability of the tool selection process itself. To bridge this gap, we introduce a black-box, text-based attack that substantially increases the likelihood of a target tool being selected. We propose SEEM, a two-level coarse-to-fine perturbation method that operates at both the word and character levels. Through comprehensive experiments, we show that merely perturbing the textual information of tools can markedly raise the probability of the target tool being prioritized and ranked higher among candidates. Our findings expose critical weaknesses in the tool selection mechanism and lay the groundwork for developing defenses to secure this essential process.

Permanent electric dipole moments (EDMs) are sensitive probes of the symmetry structure of elementary particles, which in turn is closely tied to the baryon asymmetry in the universe. A meaningful interpretation framework for EDM measurements has to be based on effective quantum field theory. We interpret the measurements performed to date in terms of a hadronic-scale Lagrangian, using the SFitter global analysis framework. We find that part of this Lagrangian is constrained very well, while some of the parameters suffer from too few high-precision measurements. Theory uncertainties lead to weaker model constraints, but can be controlled within the global analysis.

Let $Ψ(\textbf{z},\textbf{a},q)$ a the fundamental solution matrix of the quantum difference equation of a Nakajima variety $X$. In this work, we prove that the operator $$ Ψ(\textbf{z},\textbf{a},q) Ψ\left(\textbf{z}^p,\textbf{a}^p,q^{p^2}\right)^{-1} $$ has no poles at the primitive complex $p$-th roots of unity $q=ζ_p$. As a byproduct, we show that the iterated product of the operators ${\bf M}_{\mathcal{L}}(\textbf{z},\textbf{a},q )$ from the $q$-difference equation on $X$: $$ {\bf M}_{\mathcal{L}} (\textbf{z} q^{(p-1)\mathcal{L}},\textbf{a},q) \cdots {\bf M}_{\mathcal{L}} (\textbf{z} q^{\mathcal{L}},\textbf{a},q) {\bf M}_{\mathcal{L}} (\textbf{z} ,\textbf{a},q) $$ evaluated at $q=ζ_p$ has the same eigenvalues as ${\bf M}_{\mathcal{L}} (\textbf{z}^p,\textbf{a}^p,q^p)$. Upon a reduction of the quantum difference equation of $X$ to the quantum differential equation over the field of finite characteristic, the above iterated product transforms into a Grothendiek-Katz $p$-curvature of the corresponding quantum connection whreas ${\bf M}_{\mathcal{L}} (\textbf{z}^p,\textbf{a}^p,q^p)$ becomes a certain Frobenius twist of that connection. In this way, we give an explicit description of the spectrum of the $p$-curvature of quantum connection for Nakajima varieties.

In this paper, we study the effects of fill probabilities and adverse fills on the trading strategy simulation process. We specifically focus on a stochastic optimal control market-making problem and test the strategy on ES (E-mini S\&P 500), NQ (E-mini Nasdaq 100), CL (Crude Oil) and ZN (10-Year Treasury Note), which are some of the most liquid futures contracts listed on the CME (Chicago Mercantile Exchange). We provide empirical evidence that shows how fill probabilities and adverse fills can significantly affect performance and propose a more prudent simulation framework to deal with this. Many previous works aim to measure different types of adverse selection in the limit order book (LOB), however, they often simulate price processes and market orders independently. This has the ability to largely inflate the performance of a short-term style trading strategy. Our studies show that using more realistic fill probabilities and tracking adverse fills in the strategy simulation process more accurately shows how these types of trading strategies would perform in reality.

We construct an action of the positive part of the Witt algebra on Khovanov--Rozansky $\mathfrak{gl}_N$-link homology and show that link cobordisms induce equivariant maps between twists of the homology. Moreover, the state spaces of simple webs are identified with standard representations of the Witt algebra on polynomials. Some simple relations to Lee homology and genus bounds are derived from the analysis of this presentation.

Recent literature highlights the advantages of implementing social rules via dynamic game forms. We characterize when truth-telling remains a dominant strategy in gradual mechanisms implementing strategy-proof social rules, where agents gradually reveal their private information while acquiring information about others in the process. Our first characterization hinges on the incentive-preservation of a basic transformation on gradual mechanisms called illuminating that partitions information sets. The second relies on a single reaction-proofness condition. We demonstrate the usefulness of both characterizations through applications to second-price auctions and the top-trading cycles algorithm.

We address the enumeration of Eulerian orientations of 4-valent planar maps according to three parameters: the number of vertices, the number of alternating vertices (having in/out/in/out incident edges), and the number of clockwise oriented faces. This is a refinement of the six vertex model studied by Kostov, then Zinn-Justin and Elvey Price, where one only considers the first two parameters. Via a bijection of Ambjorn and Budd, our problem is equivalent to the enumeration of Eulerian partial orientations of general planar maps, counted by the number of edges, the number of undirected edges, and the number of vertices. We first derive from combinatorial arguments a system of functional equations characterising the associated trivariate series $Q(t,ω,v)$. We then derive from this system a compact characterisation of this series. We use it to determine $Q(t,ω,v)$ in three two-parameter cases. The first two cases correspond to setting the variable $ω$ counting alternating vertices (or undirected edges after the AB bijection) to $0$ or $1$: when $ω=0$ we count Eulerian orientations of general planar maps by edges and vertices, and when $ω=1$ we count Eulerian orientations of quartic maps by vertices and clockwise faces. The final forms of these two series, namely $Q(t,0, v)$ and $Q(t,1,v)$, refine those obtained by the authors in an earlier paper for $v=1$. The third case that we solve, namely $v=1$ (but $ω$ arbitrary), is the standard six-vertex model, for which we provide a new proof of the formula of Elvey Price and Zinn-Justin involving Jacobi theta functions. This new derivation remains purely in the world of formal power series, not relying on complex analysis. Our results also use a more direct approach to solving the functional equations, in contrast to the guess and check approaches used in previous work.

In this paper a proof is given of Sugawara's conjecture from 1936, that the ray class field of conductor $\mathfrak{f}$ over an imaginary quadratic field $K$ is generated over $K$ by a single primitive $\mathfrak{f}$-division value of the $τ$-function, first defined by Weber and then modified by Hasse in his 1927 paper giving a new foundation of complex multiplication.

Quantum phase estimation combined with Hamiltonian simulation is the most promising algorithmic framework to computing ground state energies on quantum computers. Its main computational overhead derives from the Hamiltonian simulation subroutine. In this paper we use randomization to speed up product formulas, one of the standard approaches to Hamiltonian simulation. We propose new partially randomized Hamiltonian simulation methods in which some terms are kept deterministically and others are randomly sampled. We perform a detailed resource estimate for single-ancilla phase estimation using partially randomized product formulas for benchmark systems in quantum chemistry and obtain orders-of-magnitude improvements compared to other simulations based on product formulas. When applied to the hydrogen chain, we have numerical evidence that our methods exhibit asymptotic scaling with the system size that is competitive with the best known qubitization approaches.

The field of proof-theoretic semantics (P-tS) offers an alternative approach to meaning in logic that is based on inference and argument (rather than truth in a model). It has been successfully developed for various logics; in particular, Sandqvist has developed such semantics for both classical and intuitionistic logic. In the case of classical logic, P-tS provides a conception of consequence that avoids an a priori commitment to the principle of bivalence, addressing what Dummett identified as a significant foundational challenge in logic. In this paper, we propose an alternative P-tS for classical logic, which essentially extends the P-tS for intuitionistic logic by operating over literals rather than atomic propositions. Importantly, literals are atomic and not defined by negation but are related by a primitive duality encoded inferentially at the atomic level. This semantics illustrates the perspective that classical logic can be understood as intuitionistic logic supplemented by a principle of duality, offering fresh insights into the relationship between these two systems.

The classical and extended deficiency one theorems by Feinberg apply to reaction networks with mass-action kinetics that have independent linkage classes or subnetworks, each with a deficiency of at most one and exactly one absorbing strong component. The theorems assume the existence of a positive equilibrium and guarantee the existence of a unique positive equilibrium in every stoichiometric compatibility class. In our work, we use the $\textit{monomial dependency}$ which extends the concept of deficiency. First, we provide a dependency one theorem for parametrized systems of polynomial equations that are essentially univariate and decomposable. As our main result, we present a corresponding theorem for mass-action systems, which permits subnetworks with arbitrary deficiency and arbitrary number of absorbing strong components. Finally, to complete the picture, we derive the extended deficiency one theorem as a special case of our more general dependency one theorem.

The model-free control approach is an advanced control law that requires few information about the process to control. Since its introduction in 2008, numerous applications have been successfully considered, highlighting attractive robustness properties towards tracking efficiency and disturbance rejection. In this work, a variational approach of the model-free control is proposed in order to extend its robustness capabilities. An adaptive formulation of the controller is proposed using the calculus of variations within a symplectic framework, that aims to consider the control law as an optimization problem toward the auto-tuning of its main key parameter. The proposed formulation provides a coupling between the model-free control law and a variational integrator to improve the robustness of the tracking towards process changes and emphasize closed-loop stabilization. Some illustrative examples are discussed to highlight the rightness of the proposed approach.

The global well-posedness of the multi-dimensional compressible Euler equations with damping remains a longstanding open problem. This problem has been partially resolved in the isentropic regime ({\it i.e.}, the adiabatic exponent \(γ>1\)) for small smooth initial data (see \cite{WY, STW}). In this paper, we establish the global well-posedness and asymptotic decay of smooth solutions of the Cauchy problem of the three-dimensional compressible Euler equations with damping for the isentropic regime \(γ>1\) and the isothermal regime \(γ=1\), allowing for partially large initial data. More precisely, the \(L^2\)-norm of the initial data is allowed to be large, while the third-order Sobolev norm of the initial data is assumed to be small. For the isentropic case, we develop a new analytical framework in which all required {\it a priori} estimates of solution $(ρ,u)$ can be derived under the condition that $\int_0^T \big( \|\nablaρ\|_{L^\infty} + \|\nabla u\|_{L^\infty} \big) \, \mathrm{d}t$ remains sufficiently small. Moreover, we obtain the optimal algebraic decay rates of global solutions. Furthermore, we study the isothermal limit of solutions of the isentropic regime as $γ\to 1$, and establish the global well-posedness and asymptotic decay of solutions to the isothermal Euler equations with damping.

In sampling tasks, it is common for target distributions to be known up to a normalizing constant. However, in many situations, even evaluating the unnormalized distribution can be costly or infeasible. This issue arises in scenarios such as sampling from the Bayesian posterior for tall datasets and the `doubly-intractable' distributions. In this paper, we begin by observing that seemingly different Markov chain Monte Carlo (MCMC) algorithms, such as the exchange algorithm, PoissonMH, and TunaMH, can be unified under a simple common procedure. We then extend this procedure into a novel framework that allows the use of auxiliary variables in both the proposal and the acceptance--rejection step. Several new MCMC algorithms emerge from this framework that uses estimated gradients to guide the proposal moves. They have demonstrated significantly better performance than existing methods on both synthetic and real datasets. We also develop theory for the new framework and use it to simplify and extend results for existing algorithms. The code to reproduce the experimental results can be found at https://github.com/ywwes26/Auxiliary-MCMC.

We present the solution of a non-linear magnetic-charged black hole with an anisotropic matter field and further extend it to obtain the corresponding rotating black hole solution using the modified Newman-Janis algorithm. The event horizon and ergosphere of the rotating black hole are studied in terms of the perspective of geometric properties, revealing that the rotating black hole can have up to three horizons. The first law of thermodynamics and the squared-mass formula for the rotating black hole are derived from a thermodynamic perspective, based on which we obtain the thermodynamic quantities and study the thermodynamic stability of the rotating black hole. Additionally, we calculate the Penrose process for the rotating black hole, indicating the influence of various black hole parameters on the maximal efficiency of the Penrose process.

The proposed construction of new particle accelerator-based facilities in the coming decades -- and upgrades to existing facilities -- provides the unique opportunity to embed innovative environmental impact reduction techniques into their design. This living document provides high-level guidelines to improve environmental sustainability in the planning, construction, operational and decommissioning stages of large accelerator facilities. A collection of various resources is provided, with examples of some existing and suggested practices.

We give several formulas for how Iwasawa $μ$-invariants of abelian varieties over unramified $\mathbb{Z}_{p}$-extensions of function fields change under isogeny. These are analogues of Schneider's formula in the number field setting. We also prove that the validity of the Birch--Swinnerton-Dyer conjecture (including the leading coefficient formula) over function fields is invariant under isogeny, without using the result of Kato--Trihan.