Study of metrics on positive-definite matrices from power potential, linking to power means.
problem Understanding metrics on positive-definite matrices derived from power potential.
method Explicit expressions for geodesics and distance function derived from Hessian of power potential.
result Geodesics and distance function converge to weighted matrix geometric mean as β tends to zero.
The paper calculates asymptotic expansions for specific types of oscillatory integrals.
problem Analyzing oscillatory integrals with complex phase functions.
method Using asymptotic expansions of simpler phase functions to derive results for more complex cases.
result Explicit computation of coefficients in asymptotic expansions for certain integrals.
The paper studies surface contraction in hyperbolic and spherical spaces.
problem Surface contraction in hyperbolic and spherical spaces.
method Contracting curvature flow of smooth closed surfaces in hyperbolic and spherical spaces.
result The evolving surface has positive scalar curvature for hyperbolic case and contracts to a point in finite time for specific powers of mean curvature.
We study the evolution of convex complete non-compact graphs by positive powers of Gauss curvature. We show that if the initial complete graph has a local uniform convexity, then the graph evolves by any positive power of Gauss curvature for all time. In particular, the initial graph is not necessarily differentiable.
Study asymptotics of extension and orthogonal Bergman kernels for high tensor powers of positive line bundles.
problem Asymptotic behavior of Bergman kernels for high tensor powers of positive line bundles.
method Analyzing the Schwartz kernel of the Ohsawa-Takegoshi extension operator and orthogonal Bergman projector, proving exponential estimates and asymptotic expansions.
result Explicit asymptotic expansions for the Ohsawa-Takegoshi extension operator and orthogonal Bergman projector.
We study convex entire graphs evolving with normal velocity equal to a positive power of the mean curvature. Under mild assumptions we prove longtime existence.
Derives spectral density function for symplectic manifolds.
problem Calculating spectral density functions on symplectic manifolds.
method Explicit local formula derivation for spectral density function.
result Explicit formula for spectral density function.
In this paper, we study the power of Gaussian curvature flow of a compact convex hypersurface and establish its Harnack inequality when the power is negative. In the Harnack inequality, we require that the absolute value of the power is strictly positive and strictly less than the inverse of the dimension of the hypers…
New method clusters signed graphs using matrix power means.
problem Clustering signed graphs with positive and negative relations.
method Signed Power Mean Laplacian, defined as matrix power mean of normalized standard and signless Laplacians.
result Signed power mean Laplacian captures ground truth clusters under reasonable settings.
Prove top wedge power of Ricci form has finite integral for Kähler manifolds with positive sectional curvature
problem Prove top wedge power of Ricci form has finite integral for Kähler manifolds with positive sectional curvature
method Use Bézout estimates and a Lipschitz weight with finite Monge-Ampère mass
result Prove top wedge power of Ricci form has finite integral for Kähler manifolds with positive sectional curvature
Approximations to utility indifference prices are provided for a contingent claim in the large position size limit. Results are valid for general utility functions on the real line and semi-martingale models. It is shown that as the position size approaches infinity, the utility function's decay rate for large negative…
New findings show almost all strongly quasipositive braids can be fibered.
problem Understanding which strongly quasipositive braids can be fibered.
method Using the Birman-Ko-Lee presentation and geometric plumbing techniques.
result All closures of strongly quasipositive braids with certain conditions are fibered.
A new method for multiple testing reduces false discoveries while maximizing power.
problem Maximizing statistical power while controlling false discoveries in multiple testing scenarios.
method Adaptive sampling approach inspired by multi-armed bandits to minimize sample size.
result The method achieves sample complexity close to information theoretic lower bounds and outperforms uniform sampling.
We prove that the L2 metric on the direct image of an adjoint positive line bundle by a locally trivial submersion between projective manifolds is Nakano positive, under the assumption that the typical fiber has zero first Betti number. As a consequence, we get that the symmetric powers of an ample vector bundle ten…
The study generalizes Bochner Laplacian results to Riemann surfaces.
problem Analyzing curvature vanishing line bundles on Riemann surfaces.
method Exploiting the relation of Bochner Laplacian on tensor powers with sR Laplacian.
result Bergman kernel expansion for semi-positive line bundles.
New method estimates log-determinant using trace powers, avoiding classical limitations.
problem Estimating log-determinant of large matrices efficiently and accurately.
method Interpolating moment-generating function and its derivative at zero using trace powers.
result No continuous estimator using finite moments can be uniformly accurate over unbounded conditioning.
Study optimal holomorphic extensions for jets along submanifolds as tensor powers increase.
problem Optimal holomorphic extensions of jets along submanifolds for high tensor powers.
method Careful study of Schwartz kernels and Bergman projectors for asymptotic analysis.
result Explicit asymptotic formula for the extension operator as tensor power tends to infinity.
CR invariant differential operators on densities with leading part a power of the sub-Laplacian are derived. One family of such operators is constructed from the ``conformally invariant powers of the Laplacian'' via the Fefferman metric; the powers which arise for these operators are bounded in terms of the dimension. …
The paper uses g-adic representations to connect number theory and graph theory.
problem Relating metric properties of Cayley graphs to additive number theory problems.
method Using g-adic representations to study Cayley graphs of integers with various generating sets. result Explicit formulas for the smallest positive integer of a given length and bounds on Cayley graph diameter.
Study of torsion forms for positive line bundles.
problem Analyse of torsion forms for line bundles.
method Investigation of asymptotic behaviour of equivariant holomorphic torsion forms.
result Equivariant extension of Puchol's result.
Researchers prove Transformers are Turing-complete and analyze their components.
problem Understanding the computational power and limitations of Transformers.
method Analyzed Turing-completeness of vanilla and modified Transformers, and necessity of components.
result Transformers with positional masking and positional encodings are Turing-complete.
Improved reasoning model by sampling from power distribution without additional training.
problem Efficiently sampling from a sharpened distribution to improve reasoning models.
method Entropy-Cut Metropolis-Hastings algorithm that identifies key decision points for resampling.
result The method consistently improves reasoning models across various datasets.
Let Gamma be a group generated by two positive multi-twists. We give some sufficient conditions for Gamma to be free or have no `unexpectedly reducible' elements. For a group Gamma generated by two Dehn twists, we classify the elements in Gamma which are multi-twists. As a consequence we are able to list all the lanter…
New balanced metrics introduced for SPD matrices, improving metric choice.
problem Lack of principles for choosing SPD matrix metrics.
method Introducing balanced metrics that relate existing metrics.
result Two new balanced metric families introduced: mixed-power-Euclidean and mixed-power-affine.
Quandle cocycle invariants form a powerful and well developed tool in knot theory. This paper treats their variations - namely, positive and twisted quandle cocycle invariants, and shadow invariants. We interpret the former as particular cases of the latter. As an application, several constructions from the shadow worl…
By combining (i) the economic theory of rational expectation bubbles, (ii) behavioral finance on imitation and herding of investors and traders and (iii) the mathematical and statistical physics of bifurcations and phase transitions, the log-periodic power law (LPPL) model has been developed as a flexible tool to detec…
Proportional transaction costs present difficult theoretical problems in trading algorithm design, on account of their lack of analytical tractability. The author derives a solution of DT-NT-DT form for an arbitrary model in which the the traded asset has diffusive dynamics described by one or more stochastic risk fact…
The paper proves the concavity of entropy power for diffusion equations and applies it to new inequalities.
problem Proving concavity of p-Rényi entropy power for diffusion equations. method Analyzing positive solutions to doubly nonlinear diffusion equations and applying Lp-Sobolev and Gagliardo-Nirenberg inequalities. result New proofs and improvements of Lp-Gagliardo-Nirenberg inequalities. Paper proves long-term investor behavior based on power utility coefficient.
problem Understanding long-term behavior of optimal strategies in financial markets.
method Bayesian financial market model with power utility maximization.
result Optimal strategy behavior depends on power utility coefficient sign.
The role of kernels is central to machine learning. Motivated by the importance of power-law distributions in statistical modeling, in this paper, we propose the notion of power-law kernels to investigate power-laws in learning problem. We propose two power-law kernels by generalizing Gaussian and Laplacian kernels. Th…
Upper diameter bound for manifolds with positive scalar curvature.
problem Estimating the maximum size of manifolds with positive scalar curvature.
method Proving an upper diameter bound using scalar curvature integral, Yamabe constant, and manifold dimension.
result The power of scalar curvature integral in diameter estimates is sharp and occurs at round spheres with canonical metric.
New flow expands hypersurfaces in hyperbolic space, showing round limiting shape for certain powers.
problem Understanding the limiting shape of hypersurfaces expanding in hyperbolic space.
method Introduced shifted inverse curvature flow with positive power p for a smooth curvature function. result For 0<p≤1, limiting shape is always round as maximal existence time is approached. J. Hempel's definition of the distance of a Heegaard surface generalizes to a complexity for a knot which is in bridge position with respect to a Heegaard surface. Our main result is that the distance of a knot in bridge position is bounded above by twice the genus, plus the number of boundary components, of an essenti…
Fold maps associated to geodesic random walks on curved spaces.
problem Understanding the behavior of geodesic random walks on curved surfaces.
method Analyzing mappings from the unit tangent sphere to a manifold with non-positive curvature.
result For odd powers of the unit tangent sphere, these mappings are fold maps.
New metrics defined on SPD matrices link to divergences and curvature.
problem Defining and characterizing metrics on SPD matrices.
method Developed a principle of deformed metrics and introduced balanced bilinear forms.
result Introduce Mixed-Euclidean metrics with negative sectional curvature.
In this paper, we study the existence and non-existence result of positive solutions to a singular elliptic equation with negative power on the bounded smooth domain or in the whole Euclidean space. Our model arises in the study of the steady states of thin films and other applied physics. We can get some useful local …
Cheap permutation tests speed up distribution testing without sacrificing accuracy.
problem Efficiently testing distribution differences and independence.
method Group datapoints into bins and permute only these bins, using stored sufficient statistics.
result Cheap permutation tests maintain the accuracy and optimality of standard tests but are significantly faster.
We show the existence of uniformly bounded sequences of increasing numbers of orthonormal sections of powers Lk of a positive holomorphic line bundle L on a compact Kähler manifold M. In particular, we construct for each positive integer k, orthonormal sections s1k,…,snkk in H0(M,Lk), $n_k\geβ…
Study eigenvalues of Bochner Laplacian on symplectic manifolds.
problem Understanding low-lying eigenvalues of Bochner Laplacian on symplectic manifolds.
method Analyzes high tensor powers of positive line bundles on symplectic manifolds.
result Asymptotic expansions for low-lying eigenvalues.
Study reveals Transformer's expressive power and mechanisms.
problem Understanding the approximation properties of Transformer for sequence modeling.
method Systematic study of Transformer's components and their combined effects, establishing approximation rates.
result Reveals roles of critical parameters in Transformer, such as number of layers and attention heads.
Paper analyzes privacy risks of learning complex models.
problem Privacy leakage from learning high-dimensional graphical models.
method Theoretical analysis of tracing attacks on Bayesian networks.
result Provides a tight upper bound on tracing attack power.
Estimates cohomology dimensions for Nakano q-semipositive line bundles.
problem Estimating cohomology dimensions for Nakano q-semipositive line bundles.
method Asymptotic estimates for high tensor powers of semipositive line bundles over q-convex manifolds and various complex manifolds.
result Optimal order estimates for cohomology dimensions.
While invasively recorded brain activity is known to provide detailed information on motor commands, it is an open question at what level of detail information about positions of body parts can be decoded from non-invasively acquired signals. In this work it is shown that index finger positions can be differentiated fr…
Generalizes prediction-powered inference for binary classifier evaluation.
problem Evaluation of binary classifiers with partially observed outcomes.
method Generalizes PPI to any regular asymptotically linear estimator and proposes modified estimators for covariate shift.
result PPI can be a computationally-simple alternative to existing methods, achieving no greater than the semi-parametric efficiency lower bound in certain scenarios.
The paper proves structures for complex projective varieties with certain tangent bundle properties.
problem Characterizing complex projective varieties based on properties of their tangent bundles.
method Analyzing the positivity of exterior powers of tangent bundles and using étale covers.
result Complex projective varieties with nef exterior powers of tangent bundles are Fano fiber spaces over Abelian varieties.
Classifies surfaces translating under specific curvature flows.
problem Classifying surfaces translating under flows by sub-affine-critical powers of Gauss curvature.
method Analyzes entire graphs of surfaces translating under flows by sub-affine-critical powers of the Gauss curvature.
result Lists all translating solitons possibly model Type II singularities for convex closed solutions in all positive powers.
The study finds complete translating solitons for certain powers of Gaussian curvature in Riemannian products.
problem Exploring translating solitons in Riemannian products with powers of Gaussian curvature.
method Investigating Kα-flows in Riemannian products MimesR for M=Rn,Sn,HFm. result Existence of complete rotational translating solitons for certain values of α in MimesR. The purpose of this paper is first to give an asymptotic formula for the holomorphic analytic torsion forms of a fibration associated with increasing powers of a given line bundle. Secondly, we generalize this formula, thanks to the theory of Toeplitz operators, in the case where the powers of the line bundle is replac…