Temporal networks consist of timestamped directed interactions that may appear continuously in time, yet few studies have directly tackled the continuous-time modeling of networks. Here, we introduce a maximum-entropy approach to temporal networks and with basic assumptions on constraints, the corresponding network ensembles admit a modular and interpretable representation: a set of global time processes and a static maximum-entropy edge, e.g. node pair, probability. This time-edge labels factorization yields closed-form log-likelihoods, degree, clustering and motif expectations, and yields a whole class of effective generative models. We provide the maximum-entropy derivation for the non-homogeneous Poisson Process (NHPP) intensities governing the probability of directed edges in temporal networks via the functional optimization over path entropy, connecting NHPP modeling to maximum-entropy network ensembles. NHPPs consistently improve log-likelihood over generic Poisson processes, while the maximum-entropy edge labels recover strength constraints and reproduce expected unique-degree curves. We discuss the limitations of this framework and how it can be integrated with multivariate Hawkes calibration procedures, renewal theory, and neural kernel estimation in graph neural networks.

In this paper, we study the geometry and topology of complete gradient shrinking Sasaki-Ricci solitons. We first prove that they must be connected at infinity. This is a Sasaki analogue of gradient shrinking Kähler-Ricci solitons. Secondly, with the positive sectional curvature or positive transverse holomorphic bisectional curvature, we show that they must be compact. All results are served as a generalization of Perelman in dimension three, of Naber in dimension four, and of Munteanu-Wang in all dimensions, respectively.

The emergence of Mixture-of-Experts (MoE) has transformed the scaling of large language models by enabling vast model capacity through sparse activation. Yet, converting these performance gains into practical edge deployment remains difficult, as the massive memory footprint and communication demands often overwhelm resource-limited environments. While centralized cloud-based solutions are available, they are frequently plagued by prohibitive infrastructure costs, latency issues, and privacy concerns. Moreover, existing edge-oriented optimizations largely overlook the complexities of heterogeneous hardware, focusing instead on isolated or uniform device setups. In response, this paper proposes Prism, an inference framework engineered for collaborative MoE serving across diverse GPU-equipped edge servers. By leveraging the intrinsic sparsity and input locality of MoE workloads, Prism minimizes inter-server communication and optimizes expert placement within diverse resource constraints. The framework integrates an activation-aware placement strategy that balances local request coverage with memory utilization, supplemented by a runtime migration mechanism to adapt expert distribution to dynamic workload changes. Experiments on contemporary MoE models and datasets demonstrate that Prism reduces inference latency by up to 30.6% and significantly lowers communication costs compared to state-of-the-art baselines, confirming the effectiveness of cooperative edge-based MoE serving.

Axion inflation is a well-motivated model of cosmic inflation with a rich phenomenology. The abundant production of gauge fields during axion inflation notably sources a stochastic gravitational-wave (GW) background signal, which nourishes the hope that future GW searches might have a chance to probe the model. In this paper, we scrutinize GW production during axion inflation in the gradient expansion formalism (GEF), a powerful numerical technique that captures the nonlinear dynamics of the system in the limit of vanishing axion gradients. We focus on {single-field} axion inflation coupled to a pure Abelian gauge sector, i.e., pure axion inflation (PAI), and perform the first-ever {detailed} parameter scan of GW production in the Abelian PAI model close to the onset of strong backreaction. {We approximate the axion potential around its minimum by a quadratic mass term and study the tensor modes that exit the Hubble horizon as the axion rolls down this potential.} Remarkably enough, we find that GW signals within the reach of future GW interferometers can only be realized in parameter regions that also lead to strong backreaction and that are in conflict with the upper limit on $ΔN_{\rm eff}$, i.e., the allowed energy density of dark radiation. This observation defines a clear target for future lattice studies of axion inflation that may confirm or improve the predictions of our GEF benchmark.

This work presents a novel cosmic-ray scattering experiment employing a Resistive Plate Chambers (RPC) muon tomography system. By introducing the scattering angle between incident and outgoing cosmic-ray tracks as a key observable, this approach enables simultaneous studies of secondary cosmic-ray composition and searching for new physics. During a 63-day campaign, 1.18 million cosmic ray scattering events were recorded and analyzed. By performing combined template fits to the observed angular distribution, particle abundances are measured -- for example, resolving the electron component at $\sim 2\%$ precision. Furthermore, constraints are established on elastic muon dark matter (DM) scattering cross-sections for muon-philic dark matter. At the $95\%$ confidence level, the limit reaches 1.61 $\times$ $10^{-17}$ $\rm{cm}^{2}$ for 1 GeV slow DM, demonstrating sensitivity limit to light muon-coupled slow DM, in scenarios where a strongly interacting dark matter component is captured and thermalized within the Earth, leading to large surface densities.

We show the uniqueness of the cylindrical tangent cone $C(\mathbb{S}^2 \times \mathbb{S}^4) \times \mathbb{R}$ for area-minimizing hypersurfaces in $\mathbb{R}^9$, completing the uniqueness of all tangent cones of the form $C_{p,q} \times \mathbb{R}$ proved by Simon for dimensions at least 10 and Székelyhidi for the Simons cone.

In the vicinity of the glass transition, the characteristic relaxation time (e.g., the alpha-relaxation time in dielectric spectroscopy) of a glass-former exhibits a strongly super-Arrhenius temperature dependence, as compared to the classical Arrhenius behavior at high temperatures. A comprehensive description of both regions thus requires five parameters. Here, we demonstrate that many glass-formers exhibit a universal scaling, with only two material-specific parameters setting the timescale and the temperature scale; the other three being universal constants. Furthermore, we show that the master curve can be described by the recently developed two-state, two-(time) scale (TS2) theory (Soft Matter 2020, 16, 810) and regress the universal TS2 parameters. We also show the connection between the TS2 model and the Hall-Wolynes elastic relaxation theory.

The Electron Cyclotron Resonance Ion Plasma ACcelerator (ECRIPAC) is an original concept for a plasma-based particle accelerator able to generate pulsed ion beams with adjustable energy, targeting mostly medical applications. This paper thoroughly reviews the working principle and physical theory behind the ECRIPAC accelerator concept, incorporating significant corrections to the existing limited literature on the subject, making it a suitable reference for future studies. Mathematical derivations for several physical formulas are also included. Moreover, a detailed theoretical investigation of the stability condition for the ion acceleration is presented, highlighting more stringent limitations than previously anticipated. Next, the impact of several physical parameters on the accelerator design is analyzed, providing an overview of achievable external fields and plasma characteristics allowing a stable ion acceleration.

The Electron Cyclotron Resonance Ion Plasma ACcelerator (ECRIPAC) is an original accelerator concept proposed in the nineties for the generation of highly energetic pulsed ion beams, suitable for a wide array of applications. The initial studies on the subject were characterized by an important calculation mistake, leading to an incomplete and erroneous literature on the topic. Nevertheless, the simple and well mastered techniques involved in the system (radio frequency and magnetic field), together with the device compactness, are strong motivations for further studies on ECRIPAC. This work proposes a comprehensive introduction to the ECRIPAC accelerator physics, including a summary of its corrected theory. The designs of several compact demonstrator devices, able to accelerate different ion species to energies up to 100 MeV, are presented. A particular focus is devoted to a He2+ accelerator, capable of generating 9.5 MeV/nucleon ions inside a 1.8 m long accelerating cavity. This device has been simulated using a Monte-Carlo (MC) code, developed to model the electron dynamics inside this system. The MC results show an excellent agreement with the updated theory, which validates the new theoretical framework of ECRIPAC. Finally, some estimations for the beam parameters of the ion bunch extracted from the accelerator are provided.

Muonium (Mu), a bound state of a positively charged muon and an electron, can diffuse through crystal lattices and interact with defect centers in insulators and semiconductors. We demonstrate that this Mu's diffusive property can be used to probe defects in a diamond crystal lattice; specifically, substitutional nitrogen atoms (N$_\text{s}^0$) and nitrogen-vacancy (NV) centers in type-Ib diamond. Upon interaction with these defects, Mu can exchange its electron's spin or change its charge state, which result in muon spin relaxation. However, muons in diamond (and semiconductors in general) can be in a few distinctive muonium states, with each state contributing to the muon signal. In addition, these states can undergo site and charge exchange interaction, forming a dynamic network. Hence, to study the Mu interaction with point defects, the muon data have to be deconvoluted to isolate signals from the diffusing species. To achieve this goal, we have modeled the Mu state exchange dynamics and numerically simulated the time evolution of muon spin polarization by the density matrix method. With a global curve fit to a set of longitudinal field scan data, one can extract the Mu transition rates that involve interaction with the defect centers. The diffusing tetrahedral interstitial Mu was found to interact with the paramagnetic N$_\text{s}^0$ center via electron spin exchange. In contrast, they are converted to form a diamagnetic center upon interaction with the negatively charged NV center.

The detection of CP violation in neutrino oscillations is one of the most important goals of the next generation of neutrino experiments. Here we study the detectability of the CP-violating phase in the oscillation of atmospheric neutrinos. Liquid scintillator detectors of a few kilotons can probe the low-energy range of the atmospheric neutrino flux. We calculate the expected rate, spectrum, and zenith angle distribution for a typical liquid scintillator detector for different detector sites. We include a typical detector response with different capabilities for flavour identification and a background model. The sensitivity is estimated using a Poisson likelihood analysis.

We explore the growth of structure in wave-like dark matter models, where the field and density spectra are peaked at sub-horizon wavenumbers. Starting with the Schrödinger-Poisson system, we derive the scale-dependent evolution of the matter power spectrum during radiation and matter domination. We find a suppression of adiabatic perturbations during radiation domination, controlled by a free-streaming length, and scale-dependent growth of the initially white-noise isocurvature power, controlled by a Jeans scale during matter domination. The results are in qualitative, and in some regimes quantitative, agreement with the quasi-particle picture. We verify the analytic results of the power spectrum with 3+1-dimensional cosmological Schrödinger-Poisson simulations. We propose an analytic formula for the halo mass function, which is in rough agreement with the simulation results at early times after matter-radiation equality. Our simulations show that early halos typically host a soliton.

Hypergraphs model complex, non-binary relationships like co-authorships, social group memberships, and recommendations. Like traditional graphs, hypergraphs can grow large, posing challenges for storage, transmission, and query performance. We propose HyperCSA, a novel compression method for hypergraphs that maintains support for standard queries over the succinct representation. HyperCSA achieves compression ratios of 26% to 79% of the original file size on real-world hypergraphs - outperforming existing methods on all large hypergraphs in our experiments. Additionally, HyperCSA scales to larger datasets than existing approaches. Furthermore, for common real-world hypergraphs, HyperCSA evaluates neighbor queries 6 to 40 times faster than both standard data structures and other hypergraph compression approaches.

We present a concrete string-theoretic mechanism that generates four-dimensional de Sitter vacua from non-geometric R-flux compactifications of heterotic string theory. The construction rests on three pillars: the Malcev algebra generated by the R-flux phase-space brackets; its universal Sabinin envelope, which ensures a consistent non-associative gauge structure in doubled geometry; and the leading alpha-prime torsion-squared correction to the heterotic action, whose strictly positive contribution uplifts the scalar potential. Positivity, guaranteed by Sabinin-algebra identities, stabilizes the overall breathing mode at positive energy, yielding a controlled metastable de Sitter scenario within heterotic effective field theory when supplemented by a standard hidden-sector gaugino-condensation uplift.

We present a universal thermodynamic framework for quantum systems that may be strongly coupled to thermal environments. Unlike previous approaches, our method enables a clear definition of thermostatic properties while preserving the same gauge freedoms as in the standard weak-coupling regime and retaining the von Neumann expression for thermodynamic entropy. Furthermore, it provides a formulation of general first and second laws using only variables accessible through microscopic control of the system, thereby enhancing experimental feasibility. We validate the framework by applying it to a paradigmatic model of strong coupling with a structured bosonic reservoir.

In online auctions, fraudulent behaviors such as shill bidding pose significant risks. This paper presents a conceptual framework that applies dynamic, behavior-based penalties to deter auction fraud using blockchain smart contracts. Unlike traditional post-auction detection methods, this approach prevents manipulation in real-time by introducing an economic disincentive system where penalty severity scales with suspicious bidding patterns. The framework employs the proposed Bid Shill Score (BSS) to evaluate nine distinct bidding behaviors, dynamically adjusting the penalty fees to make fraudulent activity financially unaffordable while providing fair competition. The system is implemented within a decentralized English auction on the Ethereum blockchain, demonstrating how smart contracts enforce transparent auction rules without trusted intermediaries. Simulations confirm the effectiveness of the proposed model: the dynamic penalty mechanism reduces the profitability of shill bidding while keeping penalties low for honest bidders. Performance evaluation shows that the system introduces only moderate gas and latency overhead, keeping transaction costs and response times within practical bounds for real-world use. The approach provides a practical method for behaviour-based fraud prevention in decentralised systems where trust cannot be assumed.

We study the lift-and-project relaxations of the stable set polytope of graphs generated by $\text{LS}_+$, the SDP lift-and-project operator devised by Lovász and Schrijver. Our focus is on $\ell$-minimal graphs: graphs on $3\ell$ vertices with $\text{LS}_+$-rank $\ell$, i.e., the smallest graphs realizing rank $\ell$. This manuscript makes two complementary contributions. First, we introduce $\text{LS}_+$ certificate packages, a modular framework for certifying membership in $\text{LS}_+$-relaxations using only integer arithmetic and simple, concise calculations, thereby making numerical lower-bound proofs more transparent, reliable, and easier to verify. Second, we apply this framework to a computational search for extremal graphs. We prove that there are at least 49 non-isomorphic 3-minimal graphs and at least 4,107 non-isomorphic 4-minimal graphs, improving the previously known counts of 14 and 588, respectively. Beyond the increase in counts, the new examples sharpen the emerging structural picture: stretched cliques remain central but are not exhaustive, clique number is informative but not decisive, and some extremal graphs exhibit previously unseen graph minor and edge density behaviour. We also determine the smallest vertex-transitive graphs of $\text{LS}_+$-rank $\ell$ for every $\ell \leq 4$.

In a weighted majority voting game, the players' weights are determined based on the constitutional planner's intentions. The weights are challenging to change in numerous cases, as they represent some desired disparity. However, the voting weights and the actual voting power do not necessarily coincide. Changing a decision threshold would offer some remedy. The International Monetary Fund (IMF) is one of the most important international organisations that uses a weighted voting system to make decisions. The voting weights in its Board of Governors depend on the quotas of the 191 member countries, which reflect their economic strengths to some extent. We analyse the connection between the decision threshold and the a priori voting power of the countries by calculating the Banzhaf indices for each threshold between 50% and 87%. The difference between quotas and voting powers is minimised if the decision threshold is 58% or 59%.

By formulating the renormalization group as a quantum channel acting on density matrices in Quantum Field Theories (QFTs), we show that ground-state expectation values of observables supported on slow momentum modes can be approximated by their averages on the fixed-point theories to which the QFTs flow. This is done by studying the fidelity between ground states of different QFTs and arriving at certain hyperscaling relations satisfied at criticality. Our results allow for a clear identification of cases in which one can replace a QFT by its scale-invariant limit in the calculation of expectation values, opening the way for a range of applications, including the improvement of numerical and analytical methods used to tackle the costly computer simulation of critical models.

The HOMFLY-PT polynomial is a two-parameter knot polynomial that admits a character expansion, expressed as a sum of Schur functions over Young diagrams. The Harer-Zagier (HZ) transform, which converts the HOMFLY--PT polynomial into a rational function, can be applied directly to the characters, yielding hence the HZ character expansion. This illuminates the structure of the HZ functions and articulates conditions for their factorisability, including that non-vanishing contributions should come from hook-shaped Young diagrams. An infinite HZ-factorisable family of hyperbolic knots, that can be thought of as a hyperbolic extension of torus knots, is constructed by full twists, partial full twists and Jucys-Murphy twists, which are braid operations that preserve HZ factorisability. Among them, of interest is a family of pretzel links, which are the Coxeter links for E type Dynkin diagrams. Moreover, when the HZ function is non-factorisable, which occurs for the vast majority of knots and links, we conjecture that it can be decomposed into a sum of factorised terms. In the 3-strand case, this is proven using the symmetries of Young diagrams.

We develop and analyze layer potential methods to represent harmonic functions on finitely-connected tori (i.e., doubly-periodic harmonic functions). The layer potentials are expressed in terms of a doubly-periodic and non-harmonic Green's function that can be explicitly written in terms of the Jacobi theta function or a modified Weierstrass sigma function. Extending results for finitely-connected Euclidean domains, we prove that the single- and double-layer potential operators are compact linear operators and derive the relevant limiting properties at the boundary. We show that when the boundary has more than one connected component, the Fredholm operator of the second kind associated with the double-layer potential operator has a non-trivial null space, which can be explicitly constructed. Finally, we apply our developed theory to obtain solutions to the Dirichlet and Neumann boundary value problems, as well as the Steklov eigenvalue problem. We present numerical results using Nyström discretizations and find approximate solutions to these problems in several numerical examples. Our method avoids a lattice sum of the free-space Green's function, is shown to be spectrally convergent, and exhibits a faster convergence rate than the method of particular solutions for problems on tori with irregularly shaped holes.

We study the monodromy representations underlying compact Lagrangian fibrations. In the case where the associated period map is generically immersive, we prove that the mondromy representation is irreducible over $\mathbb{C}$. In the alternative case where the fibration is isotrivial, we recover a result of Kim--Laza--Martin, proving that its fibers are isogeneous to a power of an elliptic curve. We show that over $\mathbb{C}$, the monodromy representation underlying an isotrivial Lagrangian fibration is a direct sum of two irreducible $\mathbb{C}$-local systems.

Climate change poses significant challenges to the agricultural and financial sectors, affecting crop productivity and overall financial stability. This study evaluates the robustness of the Actuaries Climate Index$^{TM}$ (ACI), a newer entrant in the field as a tool for measuring climate impacts, by comparing its explanatory power with well-established weather-based indexes (WBIs) across two key sectors. In the agricultural context, the yields of three major crops are predicted using generalized statistical models and advanced machine learning algorithms with climate indexes as explanatory variables. To enhance model reliability and address multicollinearity among weather-related variables, the study also incorporates both principal component analysis and functional principal component analysis. A total of 22 models, each constructed with different sets of explanatory variables, demonstrate the significant impact of wind speed and sea-level changes, alongside temperature and precipitation, on crop yield variability across six regions of the United States. For the financial market application, the analysis adapts the weather derivative framework, as it is a critical instrument for energy companies, insurers, and agribusinesses seeking to hedge against weather-related risks. By analyzing the payoffs of derivative contracts that use WBIs and ACI components as underlying variables, the findings reveal that the ACI framework holds a strong potential as a comprehensive climate risk indicator, not only for the agricultural sector but also for the finance and insurance industries.

Nonstabilizerness or `magic' is a crucial resource for quantum computers which can be distilled from noisy quantum states. However, determining the magic of mixed quantum has been a notoriously difficult task. Here, we provide efficient witnesses of magic based on the stabilizer Rényi entropy which robustly indicate the presence of magic and quantitatively estimate magic monotones. We also design efficient property testing algorithms to reliably distinguish states with high and low magic, assuming the entropy is bounded. We apply our methods to certify the number of noisy T-gates under a wide class of noise models. Additionally, using the IonQ quantum computer, we experimentally verify the magic of noisy random quantum circuits. Surprisingly, we find that magic is highly robust, persisting even under exponentially strong noise. Our witnesses can also be efficiently computed for matrix product states, revealing that subsystems of many-body quantum states can contain extensive magic despite entanglement. Finally, our work also has direct implications for cryptography and pseudomagic: To mimic high magic states with as little magic as possible, one requires an extensive amount of entropy. This implies that entropy is a necessary resource to hide magic from eavesdroppers. Our work uncovers powerful tools to verify and study the complexity of noisy quantum systems.

Guided by the $Q$-shaped derived category framework introduced by Holm and Jorgensen, we provide a differential module analogue of a classical result that characterises when a finitely generated module over a local commutative noetherian ring has finite injective dimension. As an application, we characterise local Cohen-Macaulay rings using the homological algebra of differential modules.

This study presents a 3D version of multiscale approach for investigating magnetization dynamics in multiscale, hybrid micromagnetic-atomistic simulations. The present work introduces engineered discontinuities (i) a double-slit structure, which enables the study of domain wall and spin wave interference, and (ii) a tetrahedron shaped cluster of atoms with tunable anisotropy, which provides insights into how localized anisotropic perturbations influence domain wall pinning and skyrmion stability in fully three-dimensional (3D) hybrid simulations. We considered the dynamics of spin waves, domain walls, as well as 3D skyrmions, in the presence of these defects. The magnonic double-slit experiment demonstrates interference patterns analogous to electronic wave phenomena, offering potential applications in wave-based computing. Additionally, the results reveal the impact of the local anisotropy that leads to distinct transformations, including domain wall deformations, tubular and spherical structures, skyrmion annihilation, and breathing mode. The findings underscore the critical role of defect-induced anisotropic interactions in controlling domain wall motion, skyrmion topology, and spin wave propagation.

In this work, we propose an efficient two-metric adaptive projection method for solving the $\ell_1$-norm minimization problem. Our approach is inspired by the two-metric projection method, a simple yet elegant algorithm proposed by Bertsekas for bound/box-constrained optimization problems. The low per-iteration cost of this method, combined with the ability to incorporate Hessian information, makes it particularly attractive for large-scale problems, and our proposed method inherits these advantages. Previous attempts to extend the two-metric projection method to $\ell_1$-norm minimization rely on an intermediate reformulation as a bound-constrained problem, which can lead to numerical instabilities in practice, in sharp contrast to our approach. Our algorithm features a refined partition of the index set, an adaptive projection, and a novel linesearch rule. It can accommodate singular Hessians as well as inexact solutions to the Newton linear system for practical implementation. We show that our method is theoretically sound - it has global convergence. Moreover, it is an active-set method capable of manifold identification: the underlying low-dimensional structure can be identified in a finite number of iterations, after which the algorithm reduces to an unconstrained Newton method on the identified subspace. Under an Error Bound condition, the method attains a locally superlinear convergence rate. Hence, when the solution is sparse, it achieves superfast convergence in terms of iterations while maintaining scalability, making it well-suited for large-scale problems. We conduct extensive numerical experiments to demonstrate the practical advantages of the proposed algorithm over several competitive methods from the literature, particularly in large-scale settings. }

We introduce an energetically-optimal method inspired from Shortcut-To-Adiabaticity (STA) processes, named Quantum-Optimal-Shortcut-To-Energetics (QOSTE). QOSTE produces the same transformation as STA for a given protocol used in quantum technologies or thermodynamics, but at the lowest possible energy cost. In the general case of a N- level quantum system, we derive the QOSTE controls using geometrical and optimal control tools, and show that the minimal energy cost is determined by the length of the geodesic in the rotating frame given by the original protocol. For long control times, the scaling of the ratio between the two energy costs of STA and QOSTE is quadratic in time. We benchmark our results with the Landau-Zener protocol for qubits and STIRAP for three-level systems. We observe a drastic reduction in energy with respect to standard STA methods. Finally, using gradient-based optimization algorithms and highlighting the emerging trade-off between robustness and energy cost, we design robust QOSTE outperforming STA both in robustness and energy efficiency.

We present a novel approach for generating collider-quality electron bunches using a plasma photoinjector. The approach leverages recently developed techniques for the spatiotemporal control of laser pulses to produce a moving ionization front in a nonlinear plasma wave. The moving ionization front generates an electron bunch with a current profile that balances the longitudinal electric field of an electron beam-driven plasma wave, creating a uniform accelerating field across the bunch. Particle-in-cell (PIC) simulations of the ionization stage show the formation of an electron bunch with 220 pC charge and low emittance ($\varepsilon_x = 171$ nm rad, $\varepsilon_y = 76$ nm rad). Quasistatic PIC simulations of the acceleration stage show that the bunch is efficiently accelerated to 24 GeV over 2-meters with a final energy spread of less than 1% and emittances of $\varepsilon_x = 189$ nm rad and $\varepsilon_y = 80$ nm rad. This high-quality electron bunch meets the requirements outlined by the Snowmass process for intermediate-energy colliders and compares favorably to the beam quality of proposed and existing accelerator facilities. The results establish the feasibility of plasma photoinjectors for future collider applications making a significant step towards the realization of high-luminosity, compact accelerators for particle physics research.

Let $k$ be a perfect field of characteristic $p > 0$. For a strictly semi-stable scheme over $k[[t]]$, we construct the weight spectral sequence in $p$-adic cohomology using the theory of arithmetic $\mathcal{D}$-modules, whose $E_1$ terms are described by rigid cohomologies of irreducible components of the closed fiber and whose $E_\infty$ terms are conjecturally described by the (unipotent) nearby cycle of Lazda-Pál's rigid cohomology over the bounded Robba ring. We also show its functoriality by pushforward and state the conjecture of its functoriality by pullback and dual.