The study proves bounds on hypersurface areas using macroscopic stability and Gromov simplicial norms.
problem Bounding areas of hypersurfaces in manifolds with lower scalar curvature bounds.
method Introduced a macroscopic stability inequality and used it to derive area bounds via Gromov simplicial norms.
result Lower bounds on areas of hypersurfaces in terms of Gromov simplicial norms of their homology classes.
We derive a class of macroscopic differential equations that describe collective adaptation, starting from a discrete-time stochastic microscopic model. The behavior of each agent is a dynamic balance between adaptation that locally achieves the best action and memory loss that leads to randomized behavior. We show tha…
The study shows a 3D manifold's macroscopic dimension is 1 under specific curvature constraints.
problem Understanding the macroscopic dimension of 3D Riemannian manifolds with curvature restrictions.
method Analyzing the volume and homology of balls in the manifold.
result A 3D manifold with the specified curvature constraints has macroscopic dimension 1.
Macroscopic price evolution models are commonly used for investment strategies. There are first promising achievements in defining microscopic agent based models for the same purpose. Microscopic models allow a deeper understanding of mechanisms in the market than the purely phenomenological macroscopic models, and thu…
Study macroscopic equity market properties affecting active strategies.
problem Lack of adequate models for active equity strategies.
method Empirical study using CRSP Database, focusing on market capitalizations and returns.
result Highlight stylized facts and open questions in equity markets.
In this note we construct a closed 4-manifold having torsion-free fundamental group and whose universal covering is of macroscopic dimension 3. This yields a counterexample to Gromov's conjecture about the falling of macroscopic dimension.
This study compares microscopic and macroscopic models for commodity index derivatives pricing.
problem Lack of accurate futures curve dynamics in macroscopic models for real scenarios.
method Calibrated both microscopic and macroscopic models using S\&P GSCI Crude Oil excess-return index derivatives.
result Macroscopic models struggle to capture futures curve dynamics, affecting pricing and sensitivities.
The paper proves conditions for the existence of small Urysohn width hypersurfaces in manifolds with positive scalar curvature.
problem Conditions for the existence of small Urysohn width hypersurfaces in manifolds with positive scalar curvature.
method Adaptation of Guth's macroscopic version of the Schoen-Yau descent argument.
result A complete Riemannian manifold with positive macroscopic scalar curvature contains a non-nullhomologous hypersurface of small Urysohn width.
Data-driven framework learns coarse-scale PDEs from fine-scale observations.
problem Deriving macroscopic PDEs from microscopic observations is challenging.
method Machine learning algorithms (Gaussian Processes, Artificial Neural Networks, Diffusion Maps) to uncover macroscopic fields and their evolution.
result Identifies multiple macroscopic PDEs approximating fine-scale microscopic models.
The macroscopic version of Urysohn width for scalar curvature is disproven in high dimensions.
problem Disproving the macroscopic version of Gromov's Urysohn width conjecture for scalar curvature.
method Novel estimate on Urysohn width of circle bundles and a new notion of ruling for Riemannian manifolds.
result The macroscopic version of Gromov's Urysohn width conjecture for scalar curvature is false in dimensions four and above.
We construct a counterexamples in dimensions n>3 to Gromov's conjecture \cite{Gr1} that the macroscopic dimension of rationally essential n-dimensional manifolds equals n.
Study on GD and SGD over diagonal networks, focusing on stepsizes and regularisation.
problem Understanding the impact of stochasticity and large stepsizes on gradient descent and SGD solutions.
method Investigation of GD and SGD over diagonal linear networks with macroscopic stepsizes, proving convergence and characterizing solutions.
result Large stepsizes consistently benefit SGD for sparse regression problems, but can hinder GD recovery of sparse solutions, especially in the edge of stability regime.
The paper proves macroscopic versions of conjectures about scalar curvature and volume bounds.
problem Bounding simplicial volume and L2-Betti numbers with scalar curvature constraints. method Using upper bounds on volumes of 1-balls in universal covers.
result Macroscopic versions of conjectures about scalar curvature and volume bounds are proven.
Paper uses Ricci curvature to measure and forecast China's stock market stability.
problem Measuring and predicting systemic stability of China's stock market.
method Geometric measure derived from discrete Ricci curvature applied to financial networks.
result Ricci curvature effectively captures market stability and predicts future trends.
We give a homological characterization of n-manifolds whose universal covering $\Wi M$ has Gromov's macroscopic dimension $\dim_{mc}\Wi M<n$. As the result we distinguish dimmc from the macroscopic dimension dimMC defined by the author \cite{Dr}. We prove the inequality $\dim_{mc}\Wi M<\dim_{MC}\Wi M=n$ f…
Study curvature and symplectic properties of symmetric products of surfaces.
problem Distinguishing between macroscopic dimensions in Riemannian manifolds.
method Detailed study of curvature and symplectic properties using symmetric products of surfaces.
result Symmetric products of surfaces sharply distinguish between two macroscopic dimensions.
Bayesian networks link pore-scale to continuum-scale properties of porous media.
problem Understanding macroscopic properties from microscopic ones in porous media.
method Bayesian networks to model causal relationships and joint probability distributions.
result Causal relationships impact predictions of macroscopic properties from microscopic ones.
We present examples of agent-based and stochastic models of competition and business processes in economics and finance. We start from as simple as possible models, which have microscopic, agent-based, versions and macroscopic treatment in behavior. Microscopic and macroscopic versions of herding model proposed by Kirm…
Trained MLPs' weights are exchangeable, leading to stable kernel behavior.
problem Assumptions of IID parameters in trained models are violated.
method Showed weights in MLPs are exchangeable and identified kernel stability.
result Layer-wise kernel of fully-connected layers remains approximately constant during training.
A fast, approximate method for variable selection in GLMs tackles correlated data.
problem Variable selection in generalized linear models with correlated data.
method Replica method of statistical mechanics and vector approximate message passing.
result The proposed algorithm provides fast convergence and high approximation accuracy.
The paper extends macroscopic market making to stochastic games, revealing properties and solving equations.
problem Price competition among market makers in a stochastic game setting.
method Extension of macroscopic market making framework to stochastic games, introducing multidimensional characteristic equations.
result New well-posedness results for forward-backward stochastic differential equations.
We prove a conjecture of Gromov's to the effect that manifolds with isotropic curvature bounded below by 1 (after possibly rescaling) are macroscopically 1-dimensional on the scales greater than 1. As a consequence we prove that compact manifolds with positive isotropic curvature have virtually free fundamental groups.…
Paper connects micro to macro models of limit order books using SPDEs.
problem Modeling high-frequency trading dynamics in limit order books.
method Microscopic to mesoscopic to macroscopic limit analysis, SPDEs.
result Macroscopic limit described by reflected SPDEs.
Proves Gromov's conjecture for a specific type of groups.
problem Gromov's conjecture for right-angled Artin groups.
method Analyzes universal covering spaces of manifolds with specific fundamental groups.
result Confirms Gromov's conjecture for right-angled Artin groups.
We give the first examples of rationally inessential but macroscopically large manifolds. Our manifolds are counterexamples to the Dranishnikov rationality conjecture. For some of them we prove that they do not admit a metric of positive scalar curvature, thus satisfy the Gromov positive scalar curvature conjecture. Fu…
Take a torus with a Riemannian metric. Lift the metric on its universal cover. You get a distance which in turn yields balls. On these balls you can look at the Laplacian. Focus on the spectrum for the Dirichlet or Neumann problem. We describe the asymptotic behaviour of the eigenvalues as the radius of the balls goes …
A new macroscopic market making model connects market making and optimal execution.
problem Connecting market making and optimal execution problems.
method Using continuous processes for orders, the model bridges the gap between market making and optimal execution.
result Demonstrates the model's effectiveness through various noise and intensity function scenarios.
Global stability proved for Navier-Stokes equations on hyperbolic space.
problem Stability of the Navier-Stokes equations on hyperbolic space.
method Proved global stability with exponential decay rate for small initial data.
result Exponential decay rate of $μλ_\Def^{(3)}$ for Navier-Stokes equations on hyperbolic space.
We show that for a rationally inessential orientable closed n-manifold M whose fundamental group π is a duality group the macroscopic dimension of its universal cover is strictly less than n:$$ \dim_{MC}\Wi M<n.$$ As a corollary we obtain the following 0.1 Theorem. The inequality $ \dim_{MC}\Wi M<n$ holds for t…
Constructs infinitely many examples of large manifolds with circle bundles of positive scalar curvature.
problem Existence of circle bundles over large manifolds with positive scalar curvature metrics.
method Symplectic geometry techniques.
result Infinitely many examples of macroscopically large manifolds with circle bundles of positive scalar curvature.
We consider an original problem that arises from the issue of security analysis of a power system and that we name optimal discovery with probabilistic expert advice. We address it with an algorithm based on the optimistic paradigm and on the Good-Turing missing mass estimator. We prove two different regret bounds on t…
Meta-materials simulation sped up with energy surrogates.
problem Challenging simulation of complex meta-materials due to high-fidelity PDEs.
method Learned component-level surrogates using neural networks to model stored potential energy.
result Surrogates enable accurate macroscopic behavior simulation without full structure simulation.
New method maps high-dimensional image spaces using MCMC to reveal patterns.
problem Characterizing complex probability densities in high-dimensional image spaces.
method Attraction-Diffusion (AD) MCMC tool to map metastable regions.
result AD efficiently maps highly non-convex probability densities.
Study on functions computed by deep-layered machines finds same distribution in neural networks and Boolean circuits.
problem Understanding the space of functions computed by deep-layered machines.
method Investigation of Boolean functions on random-layered machines, including neural networks and Boolean circuits.
result The space of functions computed at large depth limit is characterized and the macroscopic entropy of Boolean functions is either monotonically increasing or decreasing with depth.
A new model for defective media using two scales.
problem Modeling defects in media with two scales.
method Generalization of Riemann-Cartan manifolds and fibre bundle theory, constructing a first-order placement map.
result Emergent behaviors like dislocations and disclinations arise from the interaction of macroscopic and microscopic scales.
Take a riemanniann nilmanifold, lift its metric on its universal cover. In that way one obtains a metric invariant under the action of some co-compact subgroup. We use it to define metric balls and then study the spectrum of the laplacian for the dirichlet problem on them. We describe the asymptotic behaviour of the sp…
Unified model for market dynamics, linking price and order flow.
problem Modeling market dynamics and order flow in a unified framework.
method Markovian market model driven by a hidden Brownian efficient price, signal-driven and queue-reactive models.
result Stability of mid-price around efficient price at macroscopic scale, behavior as diffusion.
Hybrid model assesses flash crash contagion and systemic risk.
problem Understanding conditions for flash crash contagion and systemic risk.
method Micro-macro agent-based model with endogenous price impact.
result Systemic risk depends on algorithmic trader behavior, leverage, and network topology.
Entropy helps explain disorder in both macro and micro systems.
problem Connecting macro and micro systems with entropy analysis.
method Analyzing entropy from both macroscopic and microscopic perspectives.
result Entropy measures disorder in both macroscopic and microscopic systems.
We proposed a market simulation model (micro model) which displays multifractality and reproduces many important stylized facts of speculative markets. From this model we analytically extracted the MMAR model (Multifractal Model of Asset Returns) for the macroscopic limit.
New methods connect low-loss points on neural network surfaces.
problem Connecting low-loss points on neural network loss surfaces.
method Macroscopic distributional assumptions and global connection models.
result Accuracy correlates with complexity and sensitivity.
The definition of the covariant space-time averaging scheme for the objects (tensors, geometric objects, etc.) on differentiable metric manifolds with a volume n-form, which has been proposed for the formulation of macroscopic gravity, is analyzed. An overview of the space-time averaging procedure in Minkowski spacetim…
Because of their tractability and their natural interpretations in term of market quantities, Hawkes processes are nowadays widely used in high-frequency finance. However, in practice, the statistical estimation results seem to show that very often, only nearly unstable Hawkes processes are able to fit the data properl…
Study essentiality and simplicial volume of manifolds fibered over spheres.
problem When manifolds fibered over spheres are essential or have positive simplicial volume.
method Analyzing mapping tori and fiber bundles over spheres, using results on macroscopic dimension and characteristic classes.
result Mapping tori of odd-dimensional manifolds with non-zero simplicial volume are essential, while fiber bundles over spheres of dimension d > 1 have zero simplicial volume.
Proves a conjecture about hyperbolic manifolds and their volume.
problem Proving a conjecture about the volume of hyperbolic manifolds.
method Uses a non-sharp macroscopic version of a conjecture by R. Schoen, involving smoothing techniques and simplicial volume.
result Shows that for certain hyperbolic manifolds, there exists a ball with a larger volume than the hyperbolic one.
Analyzes unsupervised neural networks using statistical mechanics and Monte Carlo simulations.
problem Understanding computational capabilities of unsupervised neural networks.
method Statistical mechanics approach and Monte Carlo simulations.
result Obtained a phase diagram summarizing network performance.
We are looking for the agent-based treatment of the financial markets considering necessity to build bridges between microscopic, agent based, and macroscopic, phenomenological modeling. The acknowledgment that agent-based modeling framework, which may provide qualitative and quantitative understanding of the financial…
We analyze the stability properties of equilibrium solutions and periodicity of orbits in a two-dimensional dynamical system whose orbits mimic the evolution of the price of an asset and the excess demand for that asset. The construction of the system is grounded upon a heterogeneous interacting agent model for a singl…