Positive braids minimize knot untangling steps.
problem Finding the minimum number of steps to untangle knots.
method Analyzing positive braids and their knot closures, comparing ascending number to unknotting number.
result Ascending number equals unknotting number for knots from positive braids.
Ascending numbers are determined for 64 knots with at most n=10 crossings. After proving the theorem about the signature of alternating knot families, we distinguished all families of knots obtained from generating alternating knots with at most 10 crossings, for which the unknotting number can be confirmed by using th…
We introduce a new numerical invariant of knots and links from the descending diagrams. It is considered to live between the unknotting number and the bridge number.
Robot untangles knots by walking and switching crossings.
problem Untangling knots using a robot with limited memory.
method The robot walks a knot diagram, switching crossings, and combinatorially proves knot transformations.
result Minimal moves to transform knots into an unknot are bounded by (7C+1)C.
ASCEND discovers causal relationships in multi-omics data by leveraging known hierarchical structure.
problem Causal inference in high-dimensional multi-omics data, especially when ignoring the hierarchical structure.
method Two-tiered divide-and-conquer strategy with ancestral conditioning sets.
result Achieves polynomial-time complexity and accurately recovers ancestral relationships.
The study proves stabilizing of ascending chains in specific groups.
problem Stabilization of ascending chains in bounded rank subgroups of 3-manifold groups.
method Reduction to hyperbolic 3-manifolds and use of geometrization.
result Ascending chains in toral relatively hyperbolic groups stabilize.
No hyperbolic group can have an infinite chain of free subgroups of fixed rank.
problem Infinite ascending chains of free subgroups in hyperbolic groups.
method Proof by contradiction and properties of hyperbolic groups.
result Hyperbolic groups do not contain strictly ascending chains of free quasiconvex subgroups of constant rank.
Generic groups satisfy a chain condition for subgroups.
problem Understanding subgroup structures in generic groups.
method Proving for fixed integers m,t,k in generic m-generator t-relator groups. result Generic groups satisfy the Ascending Chain Condition for k-generated subgroups. The paper proves an ascending chain condition for subgroups in hyperbolic and graph 3-manifolds.
problem Proving an ascending chain condition for subgroups in specific types of 3-manifolds.
method Uses profinite techniques and geometric proofs for hyperbolic and graph manifolds.
result Established the ascending chain condition for free subgroups of constant rank in closed hyperbolic and graph 3-manifolds.
Hyperbolicity proven for a specific type of group extension.
problem Proving hyperbolicity of a specific group extension.
method Analyzing ascending HNN extension of groups with a free factor system and an injective endomorphism.
result Ascending HNN extension of a group is hyperbolic relative to a collection of maximal parabolic subgroups.
We obtain several restrictions on the terms of the ascending central series of a nilpotent Lie algebra g under the presence of a complex structure J. In particular, we find a bound for the dimension of the center of g when it does not contain any non-trivial J-invariant ideal. Thanks to thes…
Minimal crossing virtual links have minimal supporting genus.
problem Understanding the relationship between the number of crossings and the genus of virtual links.
method Developed a new parity theory for virtual links to prove minimal crossing implies minimal genus.
result Minimal crossing virtual links have minimal supporting genus.
The study connects knot crossing numbers to surface properties and tunnel numbers.
problem Understanding the relationship between knot crossing numbers and surface properties.
method Combines surface ascending-number estimates, bridge-number estimates, and amalgamation arguments for Heegaard splittings.
result Establishes a linear relationship between the crossing number and the Heegaard deficiency of the surface.
We give an example of a subgroup of SL(2,C) which is a strictly ascending HNN extension of a non-abelian finitely generated free group F. In particular, we exhibit a free group F in SL(2,C) of rank 6 which is conjugate to a proper subgroup of itself. This answers positively a question of Drutu and Sapir. The main ingre…
Classifies 3-braids from choreographic motions on Lissajous curves, linking them to mapping classes and geodesics.
problem Classifying 3-braids from choreographic motions on Lissajous curves.
method Parametrization in terms of levels and slopes, using dilatation and geodesic cutting sequences.
result Dilatation of pseudo-Anosov mapping classes increases with level or slope.
Paper solves isomorphism problem for specific Baumslag-Solitar groups.
problem Isomorphism problem for small rose non-ascending generalized Baumslag-Solitar groups.
method Analyzed group actions on trees with specific stabilizers.
result Isomorphism problem solvable for the specified groups.
Study minimax estimation of stratified structure from i.i.d. samples.
problem Estimating stratified structure from i.i.d. samples of stratified mixtures of immersed manifolds.
method Ascending hierarchical co-detection of points belonging to different layers, identifying number of layers and their dimensions, assigning points to layers accurately, estimating tangent spaces optimally.
result Achieves optimal estimation of mixture components at their optimal dimension-specific rates adaptively.
Determinant modulo 8 classifies virtual knots based on polynomial coefficients.
problem Classifying virtual knots using determinant modulo 8.
method Introduced a determinant for checkerboard colorable virtual knots and proved its classification by the coefficient of z2 in the ascending polynomial. result Determinant modulo 8 classifies virtual knots based on polynomial coefficients.
This paper examines from an experimental perspective random forests, the increasingly used statistical method for classification and regression problems introduced by Leo Breiman in 2001. It first aims at confirming, known but sparse, advice for using random forests and at proposing some complementary remarks for both …
It seems to be a pearl of conventional wisdom that parameter learning in deep sum-product networks is surprisingly fast compared to shallow mixture models. This paper examines the effects of overparameterization in sum-product networks on the speed of parameter optimisation. Using theoretical analysis and empirical exp…
New techniques in Khovanov homology help distinguish exotic surfaces in 4-ball.
problem Distinguishing exotic surfaces in the 4-ball that are not diffeomorphic.
method Developed new techniques for distinguishing cobordism maps on Khovanov homology using knot symmetries and braid factorizations.
result Distinguishes smooth surfaces in the 4-ball that are exotically knotted.
New surfaces in 4-ball differ topologically but not diffeomorphically.
problem Distinguishing surfaces in 4-ball that are topologically equivalent but not diffeomorphic.
method 1-twist rim surgery, sutured Floer homology, cobordism map, knot Floer homology.
result Infinitely many surfaces are topologically isotopic but not diffeomorphic.
Study geometric actions of groups on horocyclic products.
problem Understanding geometric actions of groups on horocyclic products.
method Analyzing geometric actions of groups on horocyclic products of CAT(-κ) spaces.
result Groups acting on horocyclic products are either ascending HNN extensions of finitely-generated virtually nilpotent groups or not finitely presented.
Given a Kaehler group G and a primitive class φ∈H1(G;Z), we show that the rank gradient of (G;φ) is zero if and only if Ker φ is finitely generated. Using this approach, we give a quick proof of the fact (originally due to Napier and Ramachandran) that Kaehler groups are not properly ascending or descending…
Empirical evidence is given for a significant difference in the collective trend of the share prices during the stock index rising and falling periods. Data on the Dow Jones Industrial Average and its stock components are studied between 1991 and 2008. Pearson-type correlations are computed between the stocks and avera…
We prove a generalized version of Kazhdan's theorem for canonical forms on Riemann surfaces. In the classical version, one starts with an ascending sequence {Sn→S} of finite Galois covers of a hyperbolic Riemann Surface S, converging to the universal cover. The theorem states that the sequence of for…
Paper shows DMS as an EM algorithm with improved convergence.
problem Improving the convergence of DMS algorithm.
method Shows DMS as a generalized EM algorithm and provides new proofs.
result Demonstrates global convergence and linear convergence of DMS.
In this article we dwell into the class of so called ill posed Linear Inverse Problems (LIP) in machine learning, which has become almost a classic in recent times. The fundamental task in an LIP is to recover the entire signal / data from its relatively few random linear measurements. Such problems arise in variety of…
New manifolds help understand group actions on complex chains.
problem Understanding compact Lie group actions on Morse and Floer chains.
method Introduced new manifolds called forest biassociahedra and bimultiplihedra.
result Derived algebraic structures like bialgebras and bimodules.
The paper characterizes and examines nilpotent complex structures on stratified Lie algebras.
problem Characterizing and understanding nilpotent complex structures on stratified Lie algebras.
method Introduced a new descending series pj to prove a new characterization of nilpotent complex structures and examined whether these structures preserve the strata. result Found that there exists a J-invariant stratification on a step 2 nilpotent Lie algebra with a complex structure. We solve three open problems concerning infinite-dimensional Lie groups posed in a recent survey article by K.-H. Neeb: (1) There exists a subgroup of some infinite-dimensional Lie group G which does not admit an initial Lie subgroup structure; (2) The pathology cannot occur if G is a direct limit of an ascending seque…
Develops a method to compute Morse homology for clean but not necessarily transverse intersections.
problem Computing Morse homology for clean but not necessarily transversely intersecting manifolds.
method Constructs minimal semi-global Kuranishi structures for moduli spaces of Morse trajectories, generalizing obstruction bundle gluing.
result Obtains iterated gluing equals simultaneous gluing, maintaining computability.
The main goal of this paper is to prove that a connected bounded geometry complete Kahler manifold which has at least 3 filtered ends admits a proper holomorphic mapping onto a Riemann surface. This also provides a different proof of the theorem of Gromov and Schoen that, for a connected compact Kahler manifold whose f…
We show that certain classes of graphs of free groups contain surface subgroups, including groups with positive b2 obtained by doubling free groups along collections of subgroups, and groups obtained by "random" ascending HNN extensions of free groups. A special case is the HNN extension associated to the endomorphi…
AI agents beat previous best on NetHack, but symbolic bots still outperform.
problem Developing AI agents that can ascend in the NetHack game.
method Used a procedurally generated NetHack Learning Environment for reinforcement learning.
result Symbolic bots outperform neural approaches on NetHack.
Polyak proved that the set {Ω1a,Ω1b,Ω2a,Ω3a} is a minimal generating set of oriented Reidemeister moves. One may distinguish between forward and backward moves, obtaining 32 different types of moves, which we call directed oriented Reidemeister moves. In this article we prove that the set of $…
Weather balloons deploy sensors to collect stratospheric data.
problem Limited data collection in the stratosphere.
method Modeling forecast deviation as a Gaussian process to determine sensor release times; novel hardware system for optimal sensor release.
result Data engineering framework effectively collects stratospheric data through real flights and simulations.
This paper proves bi-orderability of two-bridge link groups.
problem Bi-orderability of two-bridge link groups.
method Modified graph theoretic construction of Hirasawa and Murasugi to understand Alexander subgroups.
result Bi-orderability of a large family of two-bridge link groups.
The paper studies kernel smoothing and mean shift for directional data, deriving convergence rates and mode estimation.
problem Statistical and computational problems of kernel smoothing for directional data.
method Generalization of mean shift to directional data, derivation of convergence rates, and investigation of mode estimation.
result Statistical convergence rates of directional KDE and its derivatives, ascending property of directional mean shift, and mode estimation.
MAGE optimizes policies using action gradients from model-based learning.
problem Lack of direct gradient information from critics in actor-critic methods.
method Model-based actor-critic algorithm that learns action-value gradient.
result MAGE outperforms model-free and model-based baselines on continuous control tasks.
Human activity recognition (HAR) in ubiquitous computing is beginning to adopt deep learning to substitute for well-established analysis techniques that rely on hand-crafted feature extraction and classification techniques. From these isolated applications of custom deep architectures it is, however, difficult to gain …
This study assesses how economic shocks affect the efficiency and robustness of international pesticide trade networks.
problem Economic shocks impact the efficiency and robustness of international pesticide trade networks.
method Simulations were used to quantify efficiency and robustness under different economic shocks. Three strategies were tested: descending, random, and ascending node removal.
result The international pesticide trade networks became more efficient and robust except for clustering coefficient. Import-oriented economies were more vulnerable to shocks.
Transformers learn to integrate information from past positions incrementally, specializing heads in distinct patterns.
problem How transformers learn to integrate information from multiple past positions with varying statistical significance.
method High-order Markov chain task, incremental learning, sparse attention patterns, simplified differential equations, stage-wise convergence, early stopping as regularizer.
result Transformers learn to specialize heads in distinct patterns, shifting from competitive to cooperative learning dynamics.
It seems to be very unlikely that all relevant information in the stock market could be fully encoded in a geometrical shape. Still,the present paper will reveal the geometry behind the stock market transactions. The prices of market index (DJIA) stock components are arranged in ascending order from the smallest one in…
We reformulate LIPs as min-max problems for easier solution.
problem Recovering signals from few linear measurements.
method Proposed a min-max reformulation of LIPs.
result Saddle points characterize solutions to LIPs.
New method estimates precision matrices without models, achieving dense, consistent, and model-free properties.
problem Lack of methods that are dense, consistent, and model-free for precision matrix estimation.
method General class of estimators that unify dense, consistent, and model-free properties within a nonasymptotic framework.
result Ridgeless regression exhibits the double descent phenomenon, establishing a precision matrix analogue to linear regression's double descent.
In this work, we investigate the use of three information-theoretic quantities -- entropy, mutual information with the class variable, and a class selectivity measure based on Kullback-Leibler divergence -- to understand and study the behavior of already trained fully-connected feed-forward neural networks. We analyze …
Study on optimal portfolio selection with varying borrowing and saving rates in continuous-time markets.
problem Optimal portfolio selection in markets with different borrowing and saving rates.
method Hamilton-Jacobi-Bellman equation, partial differential equation, verification argument.
result Existence and smoothness of the value function, identification of trading regions and strategies.