Using the stress energy tensor, we establish some monotonicity formulae for vector bundle-valued p-forms satisfying the conservation law, provided that the base Riemannian (resp. Kähler) manifolds poss some real (resp. complex) p-exhaustion functions. Vanishing theorems follow immediately from the monotonicity formulae…
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48 results for “monotonic growth”
In this paper, we derive a new monotonicity formula for the plurisuhbarmonic functions on complete Kähler manifolds with nonnegative bisectional curvature. As applications we derive the sharp estimates for the dimension of the spaces of holomorphic functions (sections) with polynomial growth, which in particular, parti…
The paper proves a Minkowski inequality on specific Riemannian manifolds.
problem Establishing a Minkowski inequality on manifolds with nonnegative Ricci curvature.
method Analyzing Riemannian manifolds with nonnegative Ricci curvature and Euclidean Volume Growth.
result Validated an optimal Minkowski inequality for certain subsets.
We study volume growth, entropy and stability for translating solitons of mean curvature flow. First, we prove that every complete properly immersed translator has at least linear volume growth. Then, by using Huisken's monotonicity formula, we compute the entropy of the grim reaper and the bowl solitons. We also give …
New groups found in hyperbolic 4D and 5D space have minimal growth rate.
problem Finding minimal growth rates in hyperbolic Coxeter groups.
method Combinatorial properties of hyperbolic Coxeter polyhedra, partial classification results, monotonicity properties of growth rates.
result Coxeter groups and in and have the smallest growth rate.
Monotonicity formula and stratification of the singular set of perimeter minimizers in RCD spacesmath.DG
Paper finds formulas for minimizing perimeter in special spaces, proving key dimensions and existence.
problem Understanding the structure of perimeter minimizing sets in specific metric spaces.
method Established a monotonicity formula and proved rigidity for perimeter minimizers in RCD(0,N) spaces.
result Sharp Hausdorff dimension estimates for singular strata and existence of blow-down cones.
Study finds non-monotonic Value of Information in dynamic multi-market monopoly.
problem Investigates non-monotonicity in Value of Information for a price-setting monopolist.
method Uses a Bayesian inverse problem with Kalman-Bucy-Stratonovich filter in a dynamic discrete model.
result Non-monotonic relationship between signal variance and Value of Information.
Mathematical properties of the historical GDP/cap distributions are discussed and explained. These distributions are frequently incorrectly interpreted and the Unified Growth Theory is an outstanding example of such common misconceptions. It is shown here that the fundamental postulates of this theory are contradicted …
Uniqueness of asymptotic limits for Ricci-flat manifolds with linear volume growth is proven.
problem Proving uniqueness of asymptotic limits for noncollapsed Ricci flat manifolds with linear volume growth.
method Relating uniqueness to the existence of a harmonic function asymptotic to a Busemann function, proving uniqueness via a monotone quantity.
result Proves uniqueness of the asymptotic limit and establishes a polynomial convergence rate.
We introduce a spectrum of monotone coarse invariants for metric measure spaces called Poincaré profiles. The two extremes of this spectrum determine the growth of the space, and the separation profile as defined by Benjamini--Schramm--Timár. In this paper we focus on properties of the Poincaré profiles of groups with …
Singapore's cooling measures did not increase housing wealth overall.
problem The impact of cooling measures on housing wealth distribution.
method Examined Singapore's cooling measures over ten rounds, analyzing welfare from housing wealth.
result Welfare from housing wealth in the last round might not be higher than before 2009, depending on the deflator.
We investigate a class of quadratic-exponential growth BSDEs with jumps. The quadratic structure introduced by Barrieu & El Karoui (2013) yields the universal bounds on the possible solutions. With local Lipschitz continuity and the so-called A_gamma-condition for the comparison principle to hold, we prove the existenc…
Data describing historical growth of income per capita [Gross Domestic Product per capita (GDP/cap)] for the world economic growth and for the growth in Western Europe, Eastern Europe, Asia, former USSR, Africa and Latin America are analysed. They follow closely the linearly-modulated hyperbolic distributions represent…
Holomorphic maps are a special case of Hermitian pluriharmonic maps between almost Hermitian manifolds.
problem Characterizing maps between almost Hermitian manifolds.
method Introducing Hermitian pluriharmonic maps and proving their properties.
result Holomorphic or anti-holomorphic maps are Hermitian pluriharmonic.
Study of Thurston's Master Teapot and its properties.
problem Characterize geometric and topological properties of Thurston's Master Teapot.
method Establish basic geometric and topological properties through analysis of self-maps and intersections.
result The Master Teapot is connected, contains the unit cylinder, and its intersection with grows monotonically with .
We obtain a Chern-Osserman type equality of a complete properly immersed surface in Euclidean space, provided the L^2-norm of the second fundamental form is finite. Also, by using a monotonicity formula, we prove that if the L^2-norm of mean curvature of a noncompact surface is finite, then it has at least quadratic ar…
Improved analysis of extragradient methods for structured VIPs.
problem Efficiently solving large-scale VIPs with weaker conditions.
method Single-call stochastic extragradient methods with expected residual condition.
result Convergence guarantees for quasi-strongly monotone and weak Minty VIPs.
Perelman has discovered two integral quantities, the shrinker entropy $\cW$ and the (backward) reduced volume, that are monotone under the Ricci flow $\pa g_{ij}/\pa t=-2R_{ij}$ and constant on shrinking solitons. Tweaking some signs, we find similar formulae corresponding to the expanding case. The {\it expanding entr…
Study optimal portfolio for households with two goals: random and fixed deadlines.
problem Optimal portfolio choice for households managing random and fixed deadlines.
method Maximizes weighted sum of probabilities of funding both goals in a Black-Scholes market.
result Non-monotonic value function due to interaction between goals under forced funding.
Both analytic and geometric forms of an optimal monotone principle for -integral of the Green function of a simply-connected planar domain with rectifiable simple curve as boundary are established through a sharp one-dimensional power integral estimate of Riemann-Stieltjes type and the Huber analytic and geome…
A new ML algorithm solves complex economic control problems.
problem Solving high-dimensional, finite-horizon stochastic control problems in economics.
method Deep neural network representation of optimal policy functions with three key features.
result Efficiently solves various economic control problems including recursive utility and growth models.
Growth rate of real GDP per capita is represented as a sum of two components -- a monotonically decreasing economic trend and fluctuations related to a specific age population change. The economic trend is modeled by an inverse function of real GDP per capita with a numerator potentially constant for the largest develo…
The classical Hadamard three circle theorem is generalized to complete Kähler manifolds. More precisely, we show that the nonnegativity of the holomorphic sectional curvature is a necessary and sufficient condition for the three circle theorem. As corollaries, two sharp monotonicity formulae for holomorphic functions a…
The paper classifies and characterizes special surfaces in a 3D space.
problem Understanding surfaces in a vertical force field.
method Analyzing -minimal surfaces with specific properties.
result A full classification of complete flat embedded -minimal surfaces.
Study shows how large neural networks avoid overfitting through decoupling of feature learning and complexity growth.
problem Understanding inductive bias and generalization in large neural networks.
method Dynamical mean field theory applied to large two-layer networks.
result Training dynamics of large networks exhibit a separation of timescales, decoupling feature learning and overfitting.
Growth Estimates for Generalized Harmonic Forms on Noncompact Manifolds with Geometric Applicationsmath.DG
The paper establishes conditions for harmonic forms on noncompact manifolds.
problem Conditions for harmonic forms on noncompact manifolds.
method Introducing Condition W and proving conditions for harmonic forms.
result Conditions for harmonic forms on noncompact manifolds are established.
Study on manifolds with density using modified Hessians for curvature comparison.
problem Developing comparison geometry on manifolds with density.
method Modified Hessian approach based on weighted sectional curvature framework.
result Derivation of Hessian comparison and shape operator comparison theorems.
The stable crossing number of a twist family of knots and the satellite crossing number conjecturemath.GT
The paper studies knots formed by twisting a circle around a base knot and conjectures a linear growth in crossing numbers.
problem Understanding the growth rate of crossing numbers in twist families of knots.
method Introduced the stable crossing number and used geometric wrapping and algebraic winding to establish conjectures.
result The crossing number of grows like as for coherent twist families.
New dimension concept for groups based on percolation probability.
problem Defining a new dimension for groups using percolation probability.
method Introducing percolation dimension for groups using symmetric probability measures.
result The percolation dimension has natural properties like monotonicity and coincides with growth rate exponents for various groups.
The paper addresses monotonicity in machine learning models for fairness and accountability.
problem Ensuring fairness and accountability in transparent machine learning models.
method Study of three types of monotonicity (individual, weak pairwise, strong pairwise) and propose monotonic groves of neural additive models.
result Monotonic groves of neural additive models maintain transparency, accountability, and fairness.
Entropy defined for submanifolds; applies to mean curvature flow limits of surfaces.
problem Entropy for submanifolds in Riemannian manifolds.
method Entropy defined and shown to be monotone along mean curvature flow.
result Partial regularity of mean curvature flow limits of surfaces.
We consider the relation between simplicial volume and two of its variants: the stable integral simplicial volume and the integral foliated simplicial volume. The definition of the latter depends on a choice of a measure preserving action of the fundamental group on a probability space. We show that integral foliated s…
The paper tackles non-monotonic learning performance and proposes algorithms to make models more monotone.
problem Non-monotonic learning performance where more data does not always improve model quality.
method Proposes three algorithms to make supervised learning models more monotone, proving consistency and monotonicity with high probability.
result The algorithm MT-HT reduces less than 1% non-monotonic decisions on MNIST while maintaining competitive error rates.
The study revises GDPpc trends and redistributes economic power among countries.
problem Analyzing global economic power shifts over time.
method Modeling GDPpc as a trend and fluctuations, analyzing historical data.
result Revised GDPpc trends show significant shifts in global economic power.
Probit Monotone BARTstat.ML
Probit Monotone BART estimates binary outcomes using monotonic functions.
problem Estimating conditional mean functions for binary outcomes with monotonicity constraints.
method Proposes a new BART variant that incorporates monotonicity constraints for binary outcomes.
result Allows for more precise estimation of monotonic functions in binary outcome models.
Monotone neural networks can approximate and interpolate functions efficiently.
problem Understanding the efficiency and expressiveness of monotone neural networks.
method Solving the monotone interpolation problem using depth-4 networks and comparing size bounds with arbitrary networks.
result Monotone neural networks can approximate and interpolate functions efficiently, but may require exponential size in high dimensions.
Improves k-NN for monotonic data with robustness against noise.
problem Class noise in real-life data violates monotonic constraints in k-NN.
method Monotonic Fuzzy k-NN (MonFkNN) with new fuzzy membership calculation.
result Significant accuracy improvements and robustness against monotonic noise.
Study examines explainable machine learning for monotonic models, finding Integrated gradients better for strong monotonicity.
problem Applying explainable machine learning to science-informed models.
method Proposed axioms for monotonicity, tested Shapley value and Integrated gradients methods.
result Integrated gradients provides better explanations for strong monotonicity.
Formula proves monotonicity for anisotropic minimal hypersurfaces.
problem Understanding anisotropic minimal hypersurfaces.
method Proved a monotonicity formula under a sign assumption on the Minkowski norm.
result Monotonicity formula for anisotropic minimal hypersurfaces.
The paper develops algorithms to restore monotonicity in non-monotone functions.
problem Non-monotone solutions from heuristic algorithms need to be corrected.
method Develops algorithms to restore monotonicity with limited queries.
result Restores monotonicity while degrading the function value by at most ε.
New example of manifolds with monotonic heat kernels found.
problem Understanding monotonicity of heat kernels on manifolds.
method Analyzing new examples and classifying flat tori.
result Generic metrics fail monotonicity at large times.
Proposes a Bayesian nonparametric model for monotonic functions.
problem Imposing monotonicity constraints in Bayesian nonparametric models.
method Numerical solutions of stochastic differential equations for nonparametric model of monotonic functions.
result Demonstrates competitive results on benchmark functions and utility in temporal alignment of time-series data.
Nonnegative matrix factorization (NMF) factorizes a non-negative matrix into product of two non-negative matrices, namely a signal matrix and a mixing matrix. NMF suffers from the scale and ordering ambiguities. Often, the source signals can be monotonous in nature. For example, in source separation problem, the source…
Monotone functions and mapsmath.LO
In [S. Basu, A. Gabrielov, N. Vorobjov, Semi-monotone sets. arXiv:1004.5047v2 (2011)] we defined semi-monotone sets, as open bounded sets, definable in an o-minimal structure over the reals, and having connected intersections with all translated coordinate cones in R^n. In this paper we develop this theory further by d…
COMET learns monotonic neural networks by incorporating counterexamples.
problem Enforcing monotonicity constraints in neural networks for real-world tasks.
method Counterexample-guided learning technique for ReLU neural networks.
result COMET achieves state-of-the-art results and improves model quality.
A macroeconomic model based on the economic variables (i) assets, (ii) leverage (defined as debt over asset) and (iii) trust (defined as the maximum sustainable leverage) is proposed to investigate the role of credit in the dynamics of economic growth, and how credit may be associated with both economic performance and…
Monotonic differentiable sorting networks improve upon previous methods.
problem Non-monotonicity in differentiable sorting networks.
method Relaxation of conditional swap operations using sigmoid functions to ensure monotonicity.
result Monotonic differentiable sorting networks improve upon previous methods.
We prove three new monotonicity formulas for manifolds with a lower Ricci curvature bound and show that they are connected to rate of convergence to tangent cones. In fact, we show that the derivative of each of these three monotone quantities is bounded from below in terms of the Gromov-Hausdorff distance to the neare…