New invariants show stronger virtual knot sets.
problem Classifying virtual knots using invariants.
method Defining -order invariants using forbidden moves.
result Set of -order invariants is strictly stronger.
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Topologically Directed Simulations Reveal the Impact of Geometric Constraints on Knotted Proteinsmath.GT
Method optimizes knotting pathways in constrained polymers.
problem Understanding how geometric constraints affect knot formation in polymers.
method Topological steering using knotoid spectrum and mean unravelling number.
result Geometric constraints increase the frequency of twist knots in polymers.
This paper analyzes Airbnb listings in the city of San Francisco to better understand how different attributes such as bedrooms, location, house type amongst others can be used to accurately predict the price of a new listing that optimal in terms of the host's profitability yet affordable to their guests. This model i…
Local Writhing Dynamicscond-mat.soft
We present an alternative local definition of the writhe of a self-avoiding closed loop which differs from the traditional non-local definition by an integer. When studying dynamics this difference is immaterial. We employ a formula due to Aldinger, Klapper and Tabor for the change in writhe and propose a set of local,…
Market Fluctuations: multiplicative and percolation models, size effects and predictionscond-mat.stat-mech
We present a set of models of the main stylized facts of market price fluctuations. These models comprise dynamical evolution with threshold dynamics and Langevin price equation with multiplicative noise, percolation models to describe the interaction between traders and hierarchical cascade models to unravel the possi…
The hyperbolic dodecahedral space of Weber and Seifert has a natural non-positively curved cubulation obtained by subdividing the dodecahedron into cubes. We show that the hyperbolic dodecahedral space has a 6-sheeted irregular cover with the property that the canonical hypersurfaces made up of the mid-cubes give a ver…
Some natural hidden symmetries in the Verma modules over the Virasoro algebra are constructed in terms of geometric quantization. Their differential geometric meaning is established and their expression via -conformal symmetries in the Verma modules over the Lie algebra is found. The analysis and the unr…
EDD uses entropy of distance distributions to cluster unlabeled data.
problem Challenges in clustering unlabeled high-dimensional data.
method EDD employs Shannon entropy to quantify distance distribution peaks.
result EDD detects varying degrees of clustering sensitivity.
Analysis of the phenomenon of speculative trading in one of its basic manifestations: postage stamp bubblescond-mat.stat-mech
We document and analyze the empirical facts concerning one of the clearest evidence of speculation in financial trading as observed in the postage collection stamp market. We unravel some of the mechanisms of speculative behavior which emphasize the role of fancy and collective behavior. In our conclusion, we propose a…
Clusters asset classes to identify lead-lag relationships in market regimes.
problem Understanding lead-lag relationships between different asset classes.
method Defining macroeconomic regimes by clustering indices and investigating lead-lag relationships.
result Unravels market features and highlights informative market trends or risks.
A new knot move preserves pass-move equivalence and differs in count.
problem Defining and analyzing a new knot move.
method Introducing the --move and comparing it to pass-move and -move.
result Equivalence under --move for knots is equivalent to pass-move equivalence.
We invoke a Gaussian mixture model (GMM) to jointly analyse two traditional emission-line classification schemes of galaxy ionization sources: the Baldwin-Phillips-Terlevich (BPT) and vs. [NII]/H (WHAN) diagrams, using spectroscopic data from the Sloan Digital Sky Survey Data Release 7 and SEAGal/STARLI…
It is well known that any two diagrams representing the same oriented link are related by a finite sequence of Reidemeister moves O1, O2 and O3. Depending on orientations of fragments involved in the moves, one may distinguish 4 different versions of each of the O1 and O2 moves, and 8 versions of the O3 move. We introd…
In this paper, we introduce an equivalence relation on the set of local moves and classify local moves, called the extended -moves, up to the equivalence. Moreover, by inducing a binary relation on the set of equivalence classes of local moves, we show that an extended -move realizes the crossing change or the …
Minimal sets of moves for isotopic knots and trivalent graphs identified.
problem Identifying minimal sets of moves for isotopic knots and trivalent graphs.
method Provided and proved the existence of minimal generating sets of oriented Reidemeister moves for isotopic knots and spatial trivalent graphs.
result Twelve minimal generating sets of oriented Reidemeister moves for isotopic knots and ten for spatial trivalent graphs identified.
The H(n)-move simplifies virtual and welded knots and links.
problem Tackling the unknotting of virtual and welded links.
method Extending the H(n)-move to virtual and welded links and showing their equivalence to Reidemeister moves.
result Virtualization and forbidden move can be realized by a finite sequence of generalized Reidemeister moves and H(n)-moves.
We prove that the classical set of moves for standard spines of 3-manifolds (i.e. the MP-move and the V-move) does not suffice to relate to each other any two standard skeleta of a 3-manifold with marked boundary. We also describe a condition on the 3-manifold with marked boundary that tells whether the generalised set…
The statistical properties of the multipliers of the absolute returns are investigated using one-minute high-frequency data of financial time series. The multiplier distribution is found to be independent of the box size when is larger than some crossover scale, providing direct evidence of the existence of sca…
The (ordinary) unknotting-number of 1-dimensional knots, which is defined by using the crossing-change, is a very basic and important invariant. It is very natural to consider the `unknotting-number' associated with other local-moves on n-dimensional knots, where n is a natural number. In this paper we prove the follow…
Classifies virtual links up to a specific move.
problem Characterizing -component virtual links using -moves.
method Refined odd writhe and linking numbers.
result Classifies -component virtual links up to -moves.
New rational band moves simplify knot classification.
problem Classifying knots using rational moves.
method Introduced oriented rational band moves and proved their effectiveness.
result Knots that can be unlinkified by rational moves are rationally slice.
We prove that the crossing changes, Delta moves, and sharp moves are unknotting operations on welded knots.
New methods for delta-moves on algebraically split links identified.
problem Understanding delta-moves on algebraically split links.
method Introducing self and mixed delta-moves, proving equivalence, and calculating delta-splitting numbers.
result Two links are mixed delta-equivalent if they have the same pairwise linking number and components.
Minimal moves for surfaces in 4D discovered, linking planar and spatial moves.
problem Finding the minimal set of moves for surfaces in 4D.
method Derived minimal generating set of spatial moves, translated into planar moves.
result Minimal generating set of spatial moves for surfaces in 4D.
Study virtualized Delta, Sharp, and Pass moves for oriented virtual knots and links.
problem Conditions for unknotting oriented virtual knots and links.
method Local moves (Delta, Sharp, Pass) for oriented virtual links.
result Lower bounds for the unknotting numbers are proven and shown to be best possible.
Minimal generating sets of Reidemeister moves identified and classified.
problem Classifying minimal generating sets of Reidemeister moves.
method Determined minimal generating sets, provided classifications, and identified candidates.
result 12 out of 16 candidates for minimal generating sets were proven minimal.
We start a systematic analysis of links up to 5-move equivalence. Our motivation is to develop tools which later can be used to study skein modules based on the skein relation being deformation of a 5-move (in an analogous way as the Kauffman skein module is a deformation of a 2-move, i.e. a crossing change). Our main …
New diagonal move simplifies knots and links efficiently.
problem Efficiently unknotting knots and links.
method Introduces diagonal move, proving its effectiveness for classical and welded knots.
result Diagonal move reduces any knot or link to the unknot or unlink with fewer operations.
New moves help untangle complex knots.
problem Deforming twisted knots into simpler forms.
method Finite sequences of extended Reidemeister moves and three forbidden moves.
result Any twisted knot can be simplified.
In this paper, we unravel a fundamental connection between weighted finite automata~(WFAs) and second-order recurrent neural networks~(2-RNNs): in the case of sequences of discrete symbols, WFAs and 2-RNNs with linear activation functions are expressively equivalent. Motivated by this result, we build upon a recent ext…
Minimal moves for surfaces in 4D identified.
problem Classifying surfaces embedded in 4D space.
method Derived minimal generating set of planar moves.
result Identified minimal moves for surfaces in 4D.
New virtualized Δ-move simplifies virtual knots and links.
problem Simplifying virtual knots and links.
method Introducing a new local deformation called the virtualized Δ-move.
result Virtualized Δ-move is an unknotting operation for virtual knots.
Explains the difference between EMA and moving EMA, focusing on market trend indicators.
problem Understanding the difference between exponential moving average and moving exponential average.
method Explains the mathematical tools and definitions of trend indicators.
result Discusses the properties of the MACD indicator and its use in market trend analysis.
New algorithm explains neural network extrema for better model generalization.
problem Understanding and improving neural network generalization.
method Dynamic extremum mapping and solving linear equations.
result Number of extrema correlates with model parameters, and new algorithm simplifies gradient issues.
Neural networks simplify SDR in regression tasks.
problem Sufficient dimension reduction in regression problems.
method Applying neural networks with rank regularization to estimate the central mean subspace.
result Neural networks effectively perform SDR, consistent with theoretical estimations.
Simplified equation predicts model sensitivity to data.
problem Understanding model sensitivity to training data is challenging and costly.
method Derived using Bayesian principles, the Memory-Perturbation Equation (MPE) unifies and generalizes existing sensitivity measures.
result Empirical results show sensitivity estimates during training can predict generalization on unseen test data.
Algorithm for recognizing and performing Reidemeister moves in Gauss diagrams.
problem Recognizing and performing Reidemeister moves in Gauss diagrams.
method Simple algorithm for recognizing and performing Reidemeister moves in Gauss diagrams.
result Simple algorithm for recognizing and performing Reidemeister moves in Gauss diagrams.
Minimal sets of moves for rotational Reidemeister diagrams are identified.
problem Understanding the minimal sets of moves for rotational Reidemeister diagrams.
method Detailed description and proof of minimal generating sets for rotational Reidemeister moves.
result Minimal generating sets for oriented, framed links contain 5 moves.
4-move kills Alexander polynomial
problem Whether the 4-move is an unknotting operation
method showing every knot can be reduced via 4-moves and isotopies
result every knot can be reduced to one with a trivial Alexander polynomial
Complete classification of links up to specific moves.
problem Classifying links using specific local moves.
method Clasper theory.
result Complete classification of links up to clasp-pass moves.
We show Vector Autoregressive Moving Average models with scalar Moving Average components could be estimated by generalized least square (GLS) for each fixed moving average polynomial. The conditional variance of the GLS model is the concentrated covariant matrix of the moving average process. Under GLS the likelihood …
Roseman moves are seven types of local modification for surface-link diagrams in -space which generate ambient isotopies of surface-links in -space. In this paper, we focus on Roseman moves involving triple points, one of which is the famous tetrahedral move, and discuss their independence. For each diagram of an…
In this paper, we formulate a new local move on virtual knot diagram, called arc shift move. Further, we extend it to another local move called region arc shift defined on a region of a virtual knot diagram. We establish that these arc shift and region arc shift moves are unknotting operations by showing that any virtu…
We describe various handle moves in contact surgery diagrams, notably contact analogues of the Kirby moves. As an application of these handle moves, we discuss the respective classifications of long and loose Legendrian knots.
Deep residual networks (ResNets) and their variants are widely used in many computer vision applications and natural language processing tasks. However, the theoretical principles for designing and training ResNets are still not fully understood. Recently, several points of view have emerged to try to interpret ResNet …
In the present paper, we consider local moves on classical and welded diagrams: (self-)crossing change, (self-)virtualization, virtual conjugation, Delta, fused, band-pass and welded band-pass moves. Interrelationship between these moves is discussed and, for each of these move, we provide an algebraic classification. …
New moves prove crossing number sum for knots.
problem Proving the additivity of crossing number for knots.
method Defined restricted Reidemeister moves to prove additivity.
result Crossing number of combined knots is at least sum of individual knots.
Analyses cohomology relations for moving frames and coframes.
problem Relating Hopf cyclic cohomology of moving frames and coframes.
method Uses van Est analogy for DG Hopf algebras.
result Establishes cohomology isomorphism for DG Hopf algebras.