The square-peg problem is solved using configuration spaces and multijet transversality.
problem Proving that every simple closed curve in the plane has an odd number of inscribed squares.
method Using the multijet transversality theorem and configuration spaces, we find a dense set of smooth embeddings for which the configuration space of points is transverse to any submanifold.
result A dense family of smoothly embedded circles in the plane and in Rn have an odd number of inscribed square-like quadrilaterals. Square-like quadrilaterals inscribed in space curves proven for finite total curvature.
problem Square-peg problem in space curves.
method Local curvature analysis and limiting argument on approximating curves.
result Every embedded curve with finite total curvature has an inscribed square-like quadrilateral.
Proves a generalized table theorem for odd Euler characteristic surfaces.
problem Proving a generalized table theorem for surfaces with odd Euler characteristic.
method Using the square peg problem for smooth curves, the result is generalized to real valued functions on Riemannian surfaces with odd Euler characteristic.
result Proves the table conjecture for even functions on the two sphere.
Square can fit inside curves close to smooth ones.
problem Finding inscribed squares in nearly smooth curves.
method Using curvature and a map to relate curves, proving existence of inscribed squares.
result Curves close to smooth ones contain inscribed squares.
The paper analyzes liquidity in decentralized finance, deriving impact functions and de-pegging risks.
problem Understanding and quantifying market impact and de-pegging risk in decentralized finance.
method Derives market impact functions for optimal-growth liquidity providers, views Constant Product Market Maker as a Carnot engine, and links de-pegging risks to catastrophe bonds.
result New insights into liquidity models and de-pegging risks in decentralized finance.
Rectangular peg problem solved for many curves.
problem Rectangular peg problem for continuous Jordan curves.
method Microlocal sheaf theory and recent work of Greene and Lobb.
result Affirmative answer for a large class of rectifiable curves.
Floer homology applied to inscribing rectangles into curves.
problem Determining if a Jordan curve can inscribe a square.
method Constructing Floer homology from inscribed rectangles and using spectral invariants.
result A Jordan curve inscribes a square if its enclosed area exceeds half a circle's area.
The paper proves that any smooth curve can have two similar inscribed rectangles.
problem Finding two similar inscribed rectangles in a smooth Jordan curve.
method Lagrangian Floer homology and differential topological computation.
result Generic doubling of inscribed rectangles in smooth Jordan curves.
This paper uses a mean-field game to model stablecoin market dynamics and recovery.
problem Understanding who restores the peg during de-pegging events of stablecoins.
method Dynamic, agent-based mean-field game framework for fiat-collateralized stablecoins.
result The equilibrium formulation endogenously maps market frictions into a price path and order flows, allowing for stress testing and attribution of peg-reverting pressure.
Square inscribed in a curve made of two graph functions.
problem Finding inscribed squares in curves formed by graph functions.
method Analysis of spectral invariants of Jordan Floer homology under curve perturbations.
result Existence of inscribed squares in curves with specific Lipschitz constants.
Rectangles can fit on smooth curves, proving a theorem about Klein bottles.
problem Fitting rectangles on smooth curves.
method Shevchishin's theorem about Klein bottle embeddings.
result Similar rectangles can be placed on smooth Jordan curves.
We consider a stochastic game between a trader and a central bank in a target zone market with a lower currency peg. This currency peg is maintained by the central bank through the generation of permanent price impact, thereby aggregating an ever increasing risky position in foreign reserves. We describe this situation…
Novel AMM model for pegged cryptoassets using nested OU processes.
problem Liquidity and risk management in markets for pegged cryptoassets.
method Multi-level nested Ornstein-Uhlenbeck (OU) processes for exchange rate dynamics, calibrated and filtered AMM model.
result Consistent efficient quotes and improved liquidity provision for pegged cryptoassets.
Optimal control of reserve assets for stablecoins to maintain peg stability.
problem Balancing immediate liquidity and yield on reserve assets for stablecoin peg maintenance.
method Developed a stochastic model predictive control framework with moment closure for event intensities, incorporating a soft-thresholding structure for rebalancing.
result Optimal policy shifts predictably toward cash as expected outflows intensify or windows lengthen, preserving most bill carry in calm markets and quickly building cash during stress.
This paper compares hedging strategies for pegged FX markets using a RS model.
problem Hedging performance in pegged foreign exchange markets.
method Regime switching model, Fourier approach for calibration, exact and approximated delta hedging.
result Approximated RS delta hedge is a viable alternative to the exact RS delta hedge and significantly faster.
We prove a transversality "lifting property" for compactified configuration spaces as an application of the multijet transversality theorem: the submanifold of configurations of points on an arbitrary submanifold of Euclidean space may be made transverse to any submanifold of the configuration space of points in Euclid…
Silkswap models stablecoin trading with minimal price impact.
problem Efficient trading of fiat-pegged stablecoins with minimal price impact.
method Silkswap uses an invariant price impact curve for asymmetric trading, derived from a hybrid function.
result Silkswap outperforms Curve Finance in price impact for stablecoin trading.
Central bank strategy to maintain currency exchange rate within limits.
problem Maintaining a currency exchange rate within a target zone despite adverse economic trends.
method Modeling the problem with a continuous-time market impact model and solving it as a stochastic control problem.
result Optimal strategy minimizes accumulated inventory of foreign currency.
Similar simplices can be inscribed in most smoothly embedded spheres.
problem Inscribing families of similar simplices in spheres.
method Diffeomorphic mapping and techniques from previous work on inscribing triangles.
result A dense family of spheres allows inscribing similar simplices of every pose.
Proves existence of equivariant maps avoiding diagonal in n-space.
problem Existence of equivariant maps between spaces.
method Different approach to vanishing all obstructions.
result Vanishing of all equivariant obstructions.
Stablecoin system improves resilience to extreme market events.
problem Vulnerability of stablecoins to extreme volatility and adversarial attacks.
method MVF-Composer uses multi-agent simulations to stress-test and down-weight manipulative signals.
result Reduces peak peg deviation by 57% and mean recovery time by 3.1x under adversarial conditions.
New method learns time-invariant rewards from demonstrations.
problem Learning robust rewards for tasks with varying execution times.
method Model-based inverse reinforcement learning with time-invariant costs.
result Approach enables learning from misaligned demonstrations and generalizes spatially.
Model simulates Perpetual Futures market with agent behavior.
problem Reproduce Perpetual Futures market dynamics.
method Agent-based model with heterogeneous agents trading via a central limit order book.
result Simulation accurately reproduces Perpetual Futures price pegging to Spot price.
Study develops hybrid model to mitigate stablecoin liquidity risk.
problem Increasing integration of stablecoins introduces liquidity risk during market stress.
method Hybrid monetary architecture with 100% reserve backing and liquidity facilities.
result Demonstrates significant reduction in peg deviations and stress persistence.
Stablecoins are reshaping global monetary systems, offering hybrid structures with public and private monies.
problem The evolution of stablecoins from crypto innovation to a global monetary component.
method Econometric analysis and hybrid system design modeling.
result Stablecoins maintain strong peg stability, and a hybrid system design ensures financial resilience.
Model predicts depegging dynamics of stablecoins like Tether and Bitcoin.
problem Understanding depegging effects of stablecoins on cryptocurrencies.
method Multivariate Hawkes process model.
result Numerical example shows model's effectiveness.
HAL accelerates the generation of training sets for accurate interatomic potentials.
problem Generating accurate and transferable interatomic potentials is time-consuming and requires expert input.
method HAL framework using a physically motivated sampler with a biasing term to drive high uncertainty configurations.
result HAL-generated training databases for alloys and polymers predict macroscopic properties with high accuracy.
Study option pricing in sideways markets and target zones.
problem Option pricing in sideways markets and target zones.
method Closed-form option pricing formulas for sideways markets and target zones.
result Closed-form option pricing formulas for sideways markets and target zones.
This paper addresses the optimal scheduling of the liquidation of a portfolio using a new angle. Instead of focusing only on the scheduling aspect like Almgren and Chriss, or only on the liquidity-consuming orders like Obizhaeva and Wang, we link the optimal trade-schedule to the price of the limit orders that have to …
The nature of monetary arrangements is often discussed without any reference to its detailed construction. We present a graph representation that allows for a clear understanding of modern monetary systems. First, we show that systems based on commodity money are incompatible with credit. We then study the current char…
This paper designs a new on-chain option that amortizes perpetual options for blockchain environments.
problem No equivalent standard for on-chain options exists, leading to high-frequency oracles and liquidation engines failures.
method Develops an amortizing perpetual option contract tailored to blockchain constraints, introducing a decentralized market framework.
result Demonstrates that the new contract functions as a risk primitive for DeFi, enabling applications like endogenous collateralization and de-peg insurance.
Solves relative isoperimetric problem on polygonal domains, focusing on corners.
problem Relative isoperimetric problem on polygonal domains in R2. method Developed techniques for polygonal domains, with special attention to corners.
result Solved the relative isoperimetric problem for a square with a square corner removed.
Shear moves connect square-tiled surfaces in quadratic differentials.
problem Connecting square-tiled surfaces via specific moves.
method Shear moves corresponding to diagonal flips preserving square-tiled properties.
result Connected components of reconfiguration problem are in bijection with moduli space of quadratic differentials.
A new method solves large-scale sparse group square-root Lasso problems efficiently.
problem Large-scale linearly constrained sparse group square-root Lasso problems.
method Dual semismooth Newton based augmented Lagrangian method (ALM).
result The proposed method efficiently solves the problem with numerical experiments demonstrating its effectiveness.
A new algorithm solves nonnegative least squares faster with nonnegative data.
problem Nonnegative least squares problems with nonnegative data.
method Primal-dual perspective accelerated algorithm with adaptive restart.
result Oracle complexity independent of matrix constants, solvable to multiplicative error.
Reduced-rank method improves least-squares regression under output regularity.
problem Least-squares regression with infinite dimensional outputs.
method Reduced-rank method for solving least-squares problems with output regularity assumptions.
result Learning bounds and improved statistical performance compared to full-rank method.
We study randomized sketching methods for approximately solving least-squares problem with a general convex constraint. The quality of a least-squares approximation can be assessed in different ways: either in terms of the value of the quadratic objective function (cost approximation), or in terms of some distance meas…
Privacy-preserving crypto exchanges adjust prices based on Gaussian noise.
problem Ensuring fair pricing in privacy-preserving cryptocurrency exchanges.
method Derive Kyle equilibrium with Gaussian noise perturbation, rescaling price-impact and strategy factors.
result Identify a privacy subsidy as a transfer from LP pool to traders, invariant to noise.
CD converges linearly for MCP/SCAD penalized least squares.
problem Recovering sparse signals from data.
method Coordinate descent for MCP/SCAD penalized least squares.
result CD converges linearly to solutions of MCP/SCAD penalized least squares.
Proposes a partitioned least squares model for feature grouping.
problem Modeling with feature groups to assess variable importance.
method Two methods: alternating least squares and exact reformulation.
result Exact method provides better results in less time.
We find a convex model for traditional nonlinear regression under L2 loss.
problem Nonlinear regression under L2 loss with non-convex optimization.
method Showed a convex nonlinear regression model for least squares problem.
result Existence of a convex model simplifies training complex systems.
Estimation is the computational task of recovering a hidden parameter x associated with a distribution Dx, given a measurement y sampled from the distribution. High dimensional estimation problems arise naturally in statistics, machine learning, and complexity theory. Many high dimensional estimation problems ca…
A fast sketching algorithm solves regularized least squares problems efficiently.
problem Solving large-scale optimization problems with convex or nonconvex regularization.
method Sketching for Regularized Optimization (SRO) algorithm that generates a sketch of the original data matrix and solves the sketched problem.
result General theoretical results for the approximation error between the original and sketched problems, including minimax rates for sparse signal estimation.
Solves non-Abelian Rainich problem for SU(2) gauge fields.
problem Existence of local SU(2) Yang-Mills fields with prescribed stress-energy tensor.
method Canonically identifying tensors with Hermitian forms and defining internal square roots of stress-energy tensors.
result Existence of local SU(2) Yang-Mills field is equivalent to a single differential condition on internal square roots of stress-energy tensor.
The paper analyzes error bounds and KL properties for noisy matrix recovery problems.
problem Noisy low-rank matrix recovery problems.
method Squared F-norm regularization, accelerated alternating minimization method.
result Established error bounds and KL properties for critical points and global minimizers.
In this paper we give an example of a linear group such that its tensor square is not linear. Also, we formulate some sufficient conditions for the linearity of non-abelian tensor products G⊗H and tensor squares G⊗G. Using these results we prove that tensor squares of some groups with one relation a…
Square-tiled surfaces are a class of translation surfaces that are of particular interest in geometry and dynamics because, as covers of the square torus, they share some of its simplicity and structure. In this paper, we study counting problems that result from focusing on properties of the square torus one by one. Af…
New method speeds up solving L0-regularized least-squares problems.
problem Solving L0-regularized least-squares problems efficiently.
method Safe peeling for Branch-and-Bound algorithm.
result Significant gains in solving time and node exploration.