The paper characterizes ∗-Ricci-Bourguignon solitons on Kenmotsu manifolds.
problem Characterizing ∗-Ricci-Bourguignon solitons on Kenmotsu manifolds. method Analyzing conditions for compressing, balancing, or enlarging ∗-Ricci-Bourguignon on Kenmotsu manifolds; estimating curvature properties; featuring with torse-forming vector fields; providing an example. result Found conditions and curvature properties for ∗-Ricci-Bourguignon solitons on Kenmotsu manifolds. Estimates parameters in a deviated Gaussian mixture model.
problem Testing goodness-of-fit between a known function and a mixture of experts.
method Constructs novel Voronoi-based loss functions to estimate parameters.
result Characterizes local convergence rates of parameter estimation more accurately.
The paper characterizes contact metric manifolds with specific solitons.
problem Characterizing contact metric manifolds with ∗-conformal Ricci solitons. method Analyzing properties of (2n+1)-dimensional N(k)-contact metric manifolds. result The manifold is locally isometric to a flat (n+1)-dimensional manifold and an n-dimensional manifold of constant curvature 4. Study on Schouten solitons on Kenmotsu manifolds, focusing on torse-forming vector fields.
problem Characterizing ∗-η-Schouten solitons on Kenmotsu manifolds. method Investigation of ∗-η-Schouten solitons on Kenmotsu manifolds with torse-forming potential vector fields. result Characterization of the soliton and derivation of scalar curvature for Kenmotsu manifolds.
The paper develops a local index formula for complex manifolds with C∗-action.
problem Analyzing the m-index on complex manifolds with C∗-action. method Applying the method of transversal heat kernel asymptotics.
result Obtained a local index formula for the m-index. The paper studies special solitons on specific contact metric manifolds.
problem Characterizing solitons on N(k)-contact metric manifolds.
method Analyzing ∗-conformal Einstein solitons and gradient solitons on N(k)-contact metric manifolds. result Conditions for solitons to be expanding, steady, or shrinking are determined.
Every infinitely edge-connected graph has a minor of Farey graph or Tℵ0∗t.
problem Characterizing edge-connected graphs with specific minor properties.
method Analyzing the minor structure of infinitely edge-connected graphs.
result Infinitely edge-connected graphs contain Farey graph or Tℵ0∗t as a minor. In this paper, we explore dynamics of the nonholonomic system called vakonomic mechanics in the context of Lagrange-Dirac dynamical systems using a Dirac structure and its associated Hamilton-Pontryagin variational principle. We first show the link between vakonomic mechanics and nonholonomic mechanics from the viewpoi…
Let G be a simply connected solvable Lie group with a lattice Γ and the Lie algebra $\g$ and a representation ρ:G→GL(Vρ) whose restriction on the nilradical is unipotent. Consider the flat bundle Eρ given by ρ. By using "many" characters {α} of G and "many" flat line bundles {Eα} over G/Γ, w…
The paper studies a new soliton on Kenmotsu manifolds and derives its scalar curvature.
problem Characterizing a new soliton on Kenmotsu manifolds.
method Analyzing the ∗−κ-Ricci-Bourguignon almost soliton on Kenmotsu structure manifolds. result Derivation of the scalar curvature for a Kenmotsu manifold with the ∗−κ-Ricci-Bourguignon soliton. IIn this article, we study the instanton equation on the cylinder over a closed manifold X which admits non-zero smooth 3-form P and 4-form Q. Our results are (1) if X is a \textbf{good} manifold, i.e., P,Q satisfying d∗XP=d∗XQ=0, then the instanton with integrable curvature decays exponenti…
In this paper we show that a parallel differential form Ψ of even degree on a Riemannian manifold allows to define a natural differential both on Ω∗(M) and Ω∗(M,TM), defined via the Frölicher-Nijenhuis bracket. For instance, on a Kähler manifold, these operators are the complex differential and the Dolbe…
Determining possible failure scenarios is a critical step in the evaluation of autonomous vehicle systems. Real-world vehicle testing is commonly employed for autonomous vehicle validation, but the costs and time requirements are high. Consequently, simulation-driven methods such as Adaptive Stress Testing (AST) have b…
Constructs Dirac generating operators for split Courant algebroids.
problem Defines Dirac generating operators for split Courant algebroids.
method Explicit construction of Dirac generating operators.
result Square of Dirac generating operator yields invariant.
The study introduces a new soliton concept to classify Sasakian 3-manifolds.
problem Classifying Sasakian 3-manifolds under specific conditions.
method Introducing and studying ∗-Ricci-Yamabe solitons on contact metric manifolds. result Sasakian 3-manifolds admitting ∗-Ricci-Yamabe solitons are ∗-Ricci flat, positive Sasakian, and have Fano transverse geometry. Defines and studies Clairaut Riemannian maps between manifolds and Ricci solitons.
problem Characterizing and analyzing Clairaut Riemannian maps.
method Using geodesic curves, necessary and sufficient conditions for harmonicity and Clairaut maps are derived.
result Necessary conditions for various properties of Clairaut Riemannian maps are established.
AST provides a method to validate safe autonomy without unsafe simplifications.
problem Validation of safe autonomy in complex systems.
method Adaptive Stress Testing (AST) approach.
result AST can find failures without unsafe simplifications.
The nullity distributions of the two curvature tensors \, $\overast{R}$ and $\overast{P}$ of the Chern connection of a Finsler manifold are investigated. The completeness of the nullity foliation associated with the nullity distribution NR∗ is proved. Two counterexamples are given: the first shows that $\N_{R…
An n-dimensional manifold M is said to be rationally 4-periodic if there is an element e∈H4(M;Q) with the property that cupping with e, ⋅∪e:H∗(M;Q)→H∗+4(M;Q) is injective for 0<∗≤dimM−4 and surjective when 0≤∗<dimM−4. W…
Let G→P→M be a flat principal bundle over a closed and oriented manifold M of dimension m=2d. We construct a map of Lie algebras $Ψ: \H_{2\ast} (L M) \to ø(\Mc)$, where $\H_{2\ast} (LM)$ is the even dimensional part of the equivariant homology of LM, the free loop space of M, and $\Mc$ is the Maurer-C…
The paper provides infinite presentations for surface groups.
problem Understanding fundamental groups of surfaces.
method Infinite presentations of fundamental groups using simple loops.
result Infinite presentations for fundamental groups of surfaces and non-orientable surfaces.
Optimal scaling found to depend on operator norm across large models and datasets.
problem Lack of unifying principle for optimal hyperparameter scaling across models and datasets.
method Discovered that optimal scaling is conditioned on the operator norm of the output layer.
result The optimal learning rate/batch size pair (η∗,B∗) consistently has the same operator norm value. We study Jacobi structures on the dual bundle A∗ to a vector bundle A such that the Jacobi bracket of linear functions is again linear and the Jacobi bracket of a linear function and the constant function 1 is a basic function. We prove that a Lie algebroid structure on A and a 1-cocycle φ∈Γ(A∗) indu…
The dimension algebra of graded groups is introduced. With the help of known geometric results of extension theory that algebra induces all known results of the cohomological dimension theory. Elements of the algebra are equivalence classes dim(A) of graded groups A. There are two geometric interpretations of thos…
Study on Ricci solitons and Einstein metrics in weak β-Kenmotsu manifolds.
problem Characterizing Einstein metrics in weak β-Kenmotsu manifolds.
method Adapted ∗-Ricci tensor to weak almost contact manifolds and studied its interaction with weak β-Kenmotsu structures. result New characteristics of Einstein metrics obtained.
It is inconceivable how chaotic the world would look to humans, faced with innumerable decisions a day to be made under uncertainty, had they been lacking the capacity to distinguish the relevant from the irrelevant---a capacity which computationally amounts to handling probabilistic independence relations. The highly …
Let M be an n−dimensional differentiable manifold equipped with a torsion-free linear connection ∇ and T∗M its cotangent bundle. The present paper aims to study a metric connection $\widetilde{% \nabla }$ with nonvanishing torsion on T∗M with modified Riemannian extension ${}\bar{g}_{\nabl…
The problem of prescribing Gaussian curvature on Riemann surface with conical singularity is considered. Let (Σ,β) be a closed Riemann surface with a divisor β, and Kλ=K+λ, where K:Σ→R is a Hölder continuous function satisfying maxΣK=0, K≡0, and λ∈R. If the Eule…
Study ∗-η-Ricci solitons on weak Kenmotsu f-manifolds.
problem Characterize ∗-η-Ricci solitons on weak Kenmotsu f-manifolds. method Adapted ∗-Ricci tensor to weak metric f-manifolds, studied the interaction with weak βf-Kenmotsu structure. result Obtained new characteristics of η-Einstein metrics. We establish some fundamental relations between Dirac subbundles L for the generalized Courant algebroid (A⊕A∗,φ+W) over a differentiable manifold M and the associated Dirac subbubndles L~ for the corresponding Courant algebroid A~⊕A~∗ over M×R.
A new method learns the optimal pricing map for semiparametric dynamic pricing problems.
problem Optimizing pricing strategies in a semiparametric valuation model with unknown utility and noise.
method Developed a modular policy called ORBIT that uses a scalar pilot index, localizes a benchmark price, and learns a local polynomial approximation of the oracle price map.
result Achieves regret bound of \( \widetilde{O}\big(T^{\frac{2β-1}{4β-3}}+\sqrt{dT}\big) \) for the linear utility model and minimax sharp lower bound.
Let M be an n−dimensional differentiable manifold with a symmetric connection ∇ and T∗M be its cotangent bundle. In this paper, we study some properties of the modified Riemannian extension on T∗M defined by means of a symmetric -tensor field c on M.…
Estimates network structure from correlated node outputs of wide-sense stationary processes.
problem Learning edge connectivity from node outputs of latent inputs.
method Wide-sense stationary stochastic processes, Laplacian matrix estimation, ℓ1-regularized Whittle's MLE.
result The MLE recovers the sparsity pattern of the Laplacian matrix with high probability.
Maximal initial learning rate for deep ReLU networks identified.
problem Finding the optimal initial learning rate for deep neural networks.
method Simple approach to estimate maximal initial learning rate η∗, analyzing its behavior in constant-width fully-connected ReLU networks. result Maximal initial learning rate η∗ is well predicted as a power of depth × width, with specific conditions for network width and input layer training. In this paper we introduce the notion of Poincaré DGCAs of Hodge type, which is a subclass of Poincaré DGCAs encompassing the de Rham algebras of closed orientable manifolds. Then we introduce the notion of the small algebra and the small quotient algebra of a Poincaré DGCA of Hodge type. Using these concepts, we inves…
Let G=G1∗⋯∗Gk∗F be a countable group which splits as a free product, where all groups Gi are freely indecomposable and not isomorphic to Z, and F is a finitely generated free group. If for all i∈{1,…,k}, both Gi and its outer automorphism group Out(Gi) satisfy t…
The paper studies Hodge structures on contact manifolds and their cohomology.
problem Analyzing Hodge structures transversal to Reeb foliations.
method Applying general results about Hodge structures to contact forms and Reeb vector fields.
result Differential complexes of basic forms are canonically isomorphic under certain conditions.
The paper studies maps between Riemannian and Kähler manifolds, focusing on Clairaut semi-invariant Riemannian maps.
problem Analyzing maps between Riemannian and Kähler manifolds, particularly Clairaut semi-invariant Riemannian maps.
method Recalled and defined Clairaut semi-invariant Riemannian maps, derived necessary and sufficient conditions for geodesic curves and maps, and explored foliations and product manifolds.
result Necessary and sufficient conditions for various properties of Clairaut semi-invariant Riemannian maps were derived.
Paper proposes a new framework to compare trading strategies by accounting for market conditions.
problem Lack of information on how trading strategy performance varies with market conditions.
method Uses a GAMLSS/ZAGA framework to model the Adjusted Information Ratio (IR∗) for a SVMP and BH strategy across 146 folds of the S&P 500. result Dominance of SVMP over BH is conditional on market regime, as shown by differences in expected IR∗ and its variance. Study fundamental groups of RCD spaces without smoothness or curvature bounds.
problem Understanding fundamental groups of RCD spaces without additional conditions.
method Combining tools from RCD spaces, Gromov-Hausdorff convergence, and splitting theorems.
result Fundamental groups of RCD spaces are controlled by a finite number of generators and have specific properties under convergence.
New PCstar algorithm discovers causal structure of max-linear Bayesian networks.
problem Discovering causal structure in max-linear Bayesian networks due to non-faithfulness.
method PC algorithm modified with C∗-separation assumptions. result PCstar algorithm can orient additional edges not possible with standard PC algorithm.
Study of geometric structures on manifolds, focusing on integrability conditions.
problem Understanding the integrability of specific geometric structures.
method Analysis of algebraic types, intrinsic torsions, and distinguished connections.
result Presented first-order integrability conditions and geometric interpretations.
The paper studies critical metrics on a specific type of manifold.
problem Investigating critical metrics on almost Kenmotsu manifolds.
method Introducing and studying the ∗-Miao-Tam critical equation on (2n+1)-dimensional (k,μ)′-almost Kenmotsu manifolds. result If a (2n+1)-dimensional (k,μ)′-almost Kenmotsu manifold satisfies the ∗-Miao-Tam critical equation, it is ∗-Ricci flat and locally isometric to a specific product of manifolds. The paper studies Riemannian maps with Ricci soliton base manifolds.
problem Analyzing Riemannian maps with specific properties of base manifolds.
method Analyzing Riemannian curvature tensor, Ricci tensor, scalar curvature, and necessary conditions for Ricci soliton leaves.
result Necessary and sufficient conditions for harmonicity and biharmonicity of Riemannian maps.
We give a short answer to the question in the title: {\em dendrits}. Precisely we show that the C∗-algebra C(X) of all complex-valued continuous functions on a compactum X is projective in the category C1 of all (not necessarily commutative) unital C∗-algebras if and only if X is a…
Gradient descent-ascent converges to strict local minmax equilibria with a finite timescale separation.
problem Analyzing the convergence of gradient descent-ascent in non-convex, non-concave games with a finite timescale separation.
method Investigates the role of a finite timescale separation parameter τ on gradient descent-ascent in two-player zero-sum games, providing convergence rates and non-convergence results.
result Gradient descent-ascent converges to strict local minmax equilibria for a finite timescale separation parameter τ*.
New method finds failures in high-fidelity simulators with fewer steps.
problem Finding failures in high-fidelity simulators is expensive and impractical.
method Adaptive stress testing with backward algorithm adaptation from low-fidelity to high-fidelity.
result Significantly fewer high-fidelity simulation steps needed to find failures.
Let (S,∗) be a closed oriented surface with a marked point, let G be a fixed group, and let ρ:π1(S)⟶G be a representation such that the orbit of ρ under the action of the mapping class group Mod(S,∗) is finite. We prove that the image of ρ is finite. A similar result holds …