In this paper we formulate a geometric theory of the mechanics of growing solids. Bulk growth is modeled by a material manifold with an evolving metric. Time dependence of metric represents the evolution of the stress-free (natural) configuration of the body in response to changes in mass density and "shape". We show t…
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48 results for “linear energy growth”
Study on existence of ground states on curved spaces with conditions on potential growth.
problem Existence of ground states for aggregation-diffusion models on Cartan-Hadamard manifolds.
method Investigation of a free energy functional on Cartan-Hadamard manifolds, considering entropy and interaction energies.
result Necessary and sufficient conditions for existence of ground states are found, depending on the growth of the attractive potential.
Finite Morse index implies finitely many ends and linear energy growth for solutions to the Allen-Cahn equation.
problem Proving finite ends and linear energy growth for solutions to the Allen-Cahn equation.
method Curvature decay estimate on level sets, indirect blow-up technique, Toda system analysis.
result Finite Morse index implies finitely many ends and linear energy growth for solutions to the Allen-Cahn equation.
Study on infinite energy maps from surfaces to CAT(0) spaces.
problem Harmonic maps with infinite energy from Riemann surfaces to CAT(0) spaces.
method Estimates of energy growth near punctures, proof of uniqueness.
result Precise estimates of energy growth near punctures and proof of uniqueness of harmonic maps.
In this paper we study the energy function associated to fourth order equations of critical growth on smooth compact conformally flat manifolds of dimension greater or equal than 5.
Paper proves positivity of energy function on Riemannian manifolds.
problem Investigating positivity of energy function on Riemannian manifolds.
method Using energy function to prove positivity of initial energy.
result Simple method to obtain growth of eigen-solutions.
New inequality shows energy growth and decay in geometric problems.
problem Understanding energy behavior in geometric problems.
method Introduced a symmetric (log-)epiperimetric inequality.
result Energy growth and decay observed in geometric problems.
Study on -surfaces in negatively curved 3-manifolds, focusing on energy and entropy.
problem Understanding the growth rate and asymptotic behavior of -surfaces in negatively curved 3-manifolds.
method Proved results on the asymptotic behavior of high energy -surfaces, including upper bounds and rigidity theorems.
result Determined a rigid upper bound for the growth rate of quasi-Fuchsian -surfaces in negatively curved 3-manifolds.
This article derives prognostic expressions for the evolution of globally aggregated economic wealth, productivity, inflation, technological change, innovation and growth. The approach is to treat civilization as an open, non-equilibrium thermodynamic system that dissipates energy and diffuses matter in order to sustai…
Polyconvex energies with conformal invariance have smooth stationary points outside a discrete set.
problem Stationary points of conformally invariant polyconvex energies
method Proving smoothness of stationary points
result Smooth stationary points outside a discrete set
This paper reviews monetary standards using physics concepts.
problem The outdated gold-standard and its relevance to modern macroeconomics.
method A physics-based review of monetary standards.
result A new monetary standard based on energy supply capacity promotes sustainable growth.
The aim of this paper is to construct a natural Riemann-Lagrange differential geometry on 1-jet spaces, in the sense of nonlinear connections, generalized Cartan connections, d-torsions, d-curvatures, jet electromagnetic fields and jet Yang-Mills energies, starting from some given non-linear evolution DEs systems model…
Why is GDP growth linear?q-fin.GN
In many European countries the growth of the real GDP per capita has been linear since 1950. An explanation for this linearity is still missing. We propose that in artificial intelligence we may find models for a linear growth of performance. We also discuss possible consequences of the fact that in systems with linear…
Classifies positive solutions to critical p-Laplace equation.
problem Classifying positive solutions to a specific type of partial differential equation.
method Analyzes solutions on \(\mathbb{R}^n\) with various energy growth conditions and infinity behavior.
result Provides classification under different conditions, including rigidity in some cases.
Limiting measures and energy growth for sequences of solutions to Taubes's Seiberg-Witten equationsmath.AP
Study sequences of solutions to Taubes's Seiberg-Witten equations with unbounded energy.
problem Behavior of solutions with unbounded energy and their limiting nodal sets.
method Novel maximum principle for unbounded energy solutions, connection to vector field dynamics.
result Limiting nodal set converges to invariant set of vector field for slow energy growth.
Successful implementation of California's Renewable Portfolio Standard (RPS) mandating 33 percent renewable energy generation by 2020 requires inclusion of a robust strategy to mitigate increased risk of energy deficits (blackouts) due to short time-scale (sub 1 hour) intermittencies in renewable energy sources. Of the…
Asymptotic behavior of energy of a harmonic map defined on an asymptotically hyperbolic manifold is considered. Using the growth of energy, we show that a harmonic map defined on some asymptotically hyperbolic manifolds has to be constant if the total energy is finite, or if the map approaches a point fast enough, in t…
Study on RCD(0,N) spaces with small linear diameter growth.
problem Understanding structure properties of RCD(0,N) spaces.
method Analyzing the (revised) fundamental group of RCD(0,N) spaces.
result Proved that the revised fundamental group is finitely generated for RCD(0,N) spaces with small linear diameter growth.
Study noncompact RCD(0,N) spaces with linear volume growth, proving diameter bounds and a splitting theorem.
problem Understanding non-compact RCD(0, N) spaces with linear volume growth.
method Analyzing properties of level sets and applying geometric inequalities.
result Diameter of level sets of a Busemann function grows at most linearly.
Study harmonic function growth on curved spaces, proving inequalities.
problem Understanding growth rates of harmonic functions on curved manifolds.
method Applied a double-sided Price inequality to estimate growth rates.
result Effective estimates for harmonic function growth rates on curved manifolds.
FPDeep accelerates CNN training on FPGA clusters with high parallelism and energy efficiency.
problem Scaling DNN training to large clusters with high utilization and balanced workload.
method Hybrid model and layer parallelism, fine-grained pipeline, balanced workload partitioning.
result FPDeep achieves high parallelism and utilization, reducing storage demand to on-chip memory.
We prove that the semistability growth of hyperbolic groups is linear, which implies that hyperbolic groups which are sci (simply connected at infinity) have linear sci growth. Based on the linearity of the end-depth of finitely presented groups we show that the linear sci is preserved under amalgamated products over f…
The paper studies energy levels between non-homotopic maps on manifolds, finding sharp growth rates as the energy parameter approaches the manifold's dimension.
problem Finding energy levels between non-homotopic maps on Riemannian manifolds.
method Constructing paths and using homotopy classes to estimate energy levels, proving sharp growth rates as the energy parameter approaches the manifold's dimension.
result Sharp growth rates of energy levels as the energy parameter approaches the manifold's dimension, with lower bounds established.
A singularity theorem based on asymptotic volume growth
problem Proving singularity theorems
method Introducing asymptotic volume-expansion invariants
result Proving an explicit upper bound on the time-separation from a hypersurface to its chronological past
We prove that simply connected open Riemannian manifolds of bounded geometry, linear growth and sublinear filling growth (e.g. finite filling area) are simply connected at infinity.
Hamiltonian Descent Methodsmath.OC
New methods achieve linear convergence on broader convex functions.
problem Optimization on broader convex functions with singular or unbounded second derivatives.
method Discretizations of conformal Hamiltonian dynamics.
result Linear convergence on convex functions with singular or unbounded second derivatives.
Let be a virtually special group. Then the residual finiteness growth of is at most linear. This result cannot be found by embedding into a special linear group. Indeed, the special linear group , for , has residual finiteness growth .
Willmore flow preserves low energy surfaces to planes.
problem Preserving low energy surfaces to planes under Willmore flow.
method Willmore flow equation for complete, properly immersed surfaces in Rn.
result Complete Willmore surfaces with low energy converge to planes.
The paper analyzes MENA region's energy consumption and policy needs for renewable energy.
problem High dependency on oil and low renewable energy penetration in MENA region.
method Analysis of World Bank datasets and policy portfolio in MENA countries.
result MENA region has high potential for solar energy but faces challenges in decoupling economic growth from energy consumption.
New model predicts grain boundary migration in metals.
problem Anisotropic grain boundary migration in polycrystals.
method Level set-finite element formulation based on thermodynamics and mechanics.
result First analytical solution for anisotropic grain boundary configurations.
Stable solutions to a specific equation are one-dimensional.
problem Stability and dimensionality of solutions to the Allen-Cahn equation.
method Analysis of stable solutions with bounded energy density.
result Stable solutions to the Allen-Cahn equation are one-dimensional.
Symmetric log-epiperimetric inequality for harmonic maps with analytic target and applicationsmath.DG
New proof of harmonic map uniqueness with analytic targets.
problem Uniqueness of energy-minimizing harmonic maps with analytic targets.
method Symmetric (log)-epiperimetric inequality for harmonic maps with analytic targets.
result Tangents at infinity of energy-minimizing harmonic maps are unique.
For those concerned with the long-term value of their accounts, it can be a challenge to plan in the present for inflation-adjusted economic growth over coming decades. Here, I argue that there exists an economic constant that carries through time, and that this can help us to anticipate the more distant future: global…
Mathematical study of excess growth rate connects info theory with finance.
problem Understanding the excess growth rate in portfolio theory.
method Axiomatic characterization theorems of excess growth rate in terms of relative entropy, Jensen's inequality gap, and logarithmic divergence.
result Established rich connections between information theory and finance.
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.
Measurement of Economic Growth, Development and Under Development: New Model and Applicationq-fin.GN
Paper presents a model to measure economic growth and development.
problem Measuring relative economic growth of different systems.
method S-Shaped model with linear representation to indicate growth, development, or underdevelopment.
result Model accurately measures economic growth and development of regions and macro regions.
Every production-recycling iteration accumulates an inevitable proportion of its matter-energy in the environment, lest the production process itself would be a system in perpetual motion, violating the second law of Thermodynamics. Such high-entropy matter depletes finite stocks of ecosystem services provided by the e…
The paper proves unique constant solutions for maps with p-Ginzburg-Landau energy.
problem Finding unique constant solutions for maps with p-Ginzburg-Landau energy.
method Assuming growth conditions or asymptotic conditions for the p-Ginzburg-Landau energy, the paper establishes Liouville type theorems.
result Establishes unique constant solutions for constant Dirichlet boundary value problems on starlike domains.
Study on sphere-valued maps, proving energy convergence and current limits.
problem Understanding the behavior of sphere-valued Sobolev maps as their energy grows.
method Proving Gamma-convergence of -energies to the mass of an integral current.
result Jacobian convergence to an area-minimizing current in a cobordism class.
Lower bounds on Ricci curvature limit the volumes of sets and the existence of harmonic functions on Riemannian manifolds. In 1975, Shing Tung Yau proved that a complete noncompact manifold with nonnegative Ricci curvature has no nonconstant harmonic functions of sublinear growth. In the same paper, Yau used this resul…
Subgroup detection in linear growth curve models with generalized linear mixed model (GLMM) treesstat.ME
GLMM trees identify subgroups with different growth patterns in longitudinal data.
problem Identifying subgroups with distinct growth trajectories in longitudinal studies.
method Extended GLMM trees for longitudinal data.
result Extended GLMM trees outperform other methods in accuracy and speed.
On the geometry at infinity of manifolds with linear volume growth and nonnegative Ricci curvaturemath.DG
The study proves manifolds with specific curvature and volume properties always split off a line at infinity.
problem Understanding the geometry at infinity of manifolds with linear volume growth and nonnegative Ricci curvature.
method Analyzing properties of Busemann functions and constructing examples.
result Manifolds with the specified properties always split off a line at infinity, with bounded diameter of level sets of Busemann functions.
Study submanifolds in gradient Ricci solitons with bounded curvature, proving volume growth properties.
problem Volume growth of submanifolds in gradient Ricci solitons with bounded weighted mean curvature.
method Analyzing submanifolds in shrinking gradient Ricci solitons with bounded weighted mean curvature vector.
result Proves polynomial and at least linear volume growth for submanifolds under certain conditions.
Proves effective linear volume growth for 3-manifolds with positive scalar curvature.
problem Volume growth of three-manifolds with positive scalar curvature.
method Utilizes the technique of μ-bubbles and almost-splitting theorem.
result Proves effective linear volume growth for 3-manifolds with non-negative Ricci curvature and uniformly positive scalar curvature.
The paper splits manifolds using infinity harmonic functions with linear growth.
problem Splitting manifolds with specific harmonic functions.
method Analyzes manifolds with non-negative Ricci or sectional curvature, focusing on infinity harmonic functions with linear growth.
result Extends Savin's theorem to surfaces with non-negative sectional curvature.
A new method to value IPOed companies.
problem Valuing companies after IPO.
method Growth Average U1 method.
result Benchmark stocks using linear extrapolation of revenues and profits.
Quantum computing offers energy savings over classical computing.
problem Energy efficiency in computing services.
method Cournot competition model constrained by energy usage.
result Quantum computing firms can outperform classical counterparts in energy efficiency.
Curves in higher dimensions are either affine or have super-Euclidean energy growth.
problem Characterizing entire conformal curves in higher-dimensional spaces.
method Blow-down argument and interaction of generalized Cauchy--Riemann equations with calibrated geometries.
result Entire conformal curves are either affine or have super-Euclidean energy growth.