The paper studies dynamical properties in semigroups modulo ideals.
problem Analyzing shadowing, expansivity, and stability in semigroups with ideals.
method Investigates shadowing, expansivity, and stability properties in uniform transformation semigroups modulo an ideal.
result Establishes that if a semigroup exhibits shadowing and expansivity modulo an ideal, it is also topologically stable modulo that ideal.
Method validates periodic and singular orbits in fast-slow systems with one-dimensional slow variable.
problem Validation of periodic, homoclinic, and heteroclinic orbits in fast-slow systems with one-dimensional slow variable.
method Topological tools (isolating blocks, cone condition, covering relations) and additional techniques (slow shadowing, m-cones) for rigorous numerics.
result Validation of global orbits for fast-slow systems across a wide range of ε.
We study the geometry and topology of immersed surfaces in Euclidean 3-space whose Gauss map satisfies a certain two-piece-property, and solve the ``shadow problem" formulated by H. Wente.
An asset network systemic risk (ANWSER) model is presented to investigate the impact of how shadow banks are intermingled in a financial system on the severity of financial contagion. Particularly, the focus of this study is the impact of the following three representative topologies of an interbank loan network betwee…
We introduce a topological combinatorial game called the Link Smoothing Game. The game is played on the shadow of a link diagram and legal moves consist of smoothing precrossings. One player's goal is to keep the diagram connected while the other player's goal is to disconnect the shadow. We make significant progress t…
We discuss the concept of the shadow boundary of a centrally symmetric convex ball K (actually being the unit ball of a Minkowski normed space) with respect to a direction x of the Euclidean n-space Rn. We introduce the concept of general parameter spheres of K corresponding to this direction and prove t…
PhD thesis on quantum invariants and knots in 3-manifolds.
problem Quantum invariants and knots in 3-manifolds.
method Skein theory and connected sums of S^1xS^2.
result Extended Tait conjecture and Eisermann's theorem to connected sums of S^1xS^2.
To any utility maximization problem under transaction costs one can assign a frictionless model with a price process S∗, lying in the bid/ask price interval [S,Sˉ]. Such process S∗ is called a \emph{shadow price} if it provides the same optimal utility value as in the original model with bid-as…
Geometric proof confirms link volume conjecture.
problem Volume conjecture for hyperbolic 3-manifolds.
method Topological and skein theoretic techniques.
result Complements of octahedral fully augmented links are isometric to fundamental shadow links.
It is well known that the description of topological and geometric properties of bisectors in normed spaces is a non-trivial subject. In this paper we introduce the concept of bounded representation of bisectors in finite dimensional real Banach spaces. This useful notion combines the concepts of bisector and shadow bo…
New geometric methods solve a conjecture for hyperbolic 3-manifolds.
problem Verifying the 1-loop conjecture for hyperbolic 3-manifolds.
method Constructing geometric ideal triangulations and solving gluing equations.
result Proves the 1-loop conjecture for a large class of hyperbolic 3-manifolds.
New invariants for singular knots and links defined using shadow structures.
problem Defining invariants for singular knots and links.
method Introducing action of singquandles on sets and defining shadow counting and polynomial invariants.
result Enhanced shadow counting invariant for singular knots and links.
Researchers create links in 3-sphere satisfying volume conjecture.
problem Proving hyperbolic links in 3-sphere satisfy volume conjecture.
method Topological tools, homeomorphic complement property, fundamental shadow links.
result Links in 3-sphere homeomorphic to fundamental shadow links satisfy volume conjecture.
The study of shadow of Vietoris-Rips complexes and their homotopy properties.
problem Understanding the geometric/topological behavior of the shadow projection map p. method Inverse system techniques from shape theory to study systems of shadow complexes.
result The limit map limp exhibits favorable homotopy-theoretic properties when X is an ANR. Special shadow-complexity equals k+1 for k copies of S1×S3.
problem Calculating the special shadow-complexity of a specific 4-manifold.
method Defined by Costantino using Turaev's shadows, proved for connected sums of S1×S3.
result The special shadow-complexity of k copies of S1×S3 is k+1.
New invariant measures complexity of 2-knots in 4D space.
problem Measuring complexity of 2-knots in 4D space.
method Introduced shadow-complexity based on Turaev shadows.
result Characterized 2-knots with shadow-complexity up to 1.
A shadow diagram is a knot diagram with under-over information omitted; a shadow movie is a sequence of shadow diagrams related by shadow Reidemeister moves. We show that not every shadow movie arises as the shadow of a Reidemeister movie, meaning a sequence of classical knot diagrams related by classical Reidemeister …
Paper studies knotoid chirality using shadow quandle colorings and invariants.
problem Distinguishing knotoids from their mirrors.
method Shadow quandle colorings and cocycle invariants.
result Knotoid 31 is shown to be chiral. The paper finds infinitely many Mazur type manifolds and corks with shadow complexity one.
problem Finding manifolds and corks with specific shadow complexity.
method Using Turaev's shadow and contractible special polyhedra with one true vertex.
result The discovery of infinitely many Mazur type manifolds and corks with shadow complexity one.
IFS with average shadowing property ensure chain recurrence and have specific examples.
problem Analyzing the average shadowing property in IFS.
method Examining uniformly contracting, conjugacy, and product IFS; proving chain recurrence; introducing examples.
result IFS with average shadowing property ensure chain recurrence.
Every shadow can be a knot or a connected sum of trefoils.
problem Understanding the structure of shadows and their relation to knots.
method Analyzing reduced shadows and their relationship to specific knots.
result Every shadow corresponds to a torus knot, twist knot, or connected sum of trefoils.
Paper optimizes trading strategies by creating shadow prices for markets with transaction costs.
problem Optimizing trading strategies in markets with transaction costs.
method Developed shadow prices to simplify optimization into a frictionless market, considering second-order transaction costs.
result Alternative strategies outperform shadow prices for risk aversion different from one.
We construct elements of the third quandle homology groups of knot quandles, which are called the shadow fundamental classes. They play the same roles for the shadow quandle cocycle invariants of knots as the fundamental classes of knot quandles does for the quandle cocycle invariants. As an application of the shadow f…
The paper constructs corks and exotic 4-manifolds with controlled shadow-complexity.
problem Constructing exotic 4-manifolds with controlled shadow-complexity.
method Using corks of Mazur type, the authors construct exotic pairs of 4-manifolds with controlled shadow-complexity.
result An infinite family of exotic pairs of 4-manifolds with controlled shadow-complexity.
Computes Kauffman bracket polynomial for specific 2-tangle shadows.
problem Calculating Kauffman bracket polynomial for complex tangle structures.
method Computed Kauffman bracket polynomial for specific 2-tangle shadows with up to 4 crossings.
result Computed polynomial for specific 2-tangle shadows.
The paper presents fundamental groups of complements of shadows in 4-balls.
problem Understanding fundamental groups of 4-manifold complements.
method Similar to Wirtinger presentation, focusing on contractible shadows.
result A presentation of fundamental groups of subpolyhedra in 4-balls.
Shadow biquandles and local biquandles have similar homology and invariants.
problem Comparing homology and invariants of shadow biquandles and local biquandles.
method Defined local biquandle structure on a shadow biquandle and showed isomorphic (co)homology groups and invariant equivalence.
result Homology and invariants of shadow biquandles and local biquandles are equivalent.
For A a primitive 2N-root of unity with N odd, the Witten-Reshetikhin-Turaev topological quantum field theory provides a representation of the Kauffman skein algebra of a closed surface. We show that this representation is irreducible and we compute its classical shadow, in the sense of earlier work of the authors (arX…
Study on inflection points of plane curve shadows with fixed embedded shapes.
problem Minimum number of inflection points in plane curves with fixed embedded shadows.
method Finite coorientation problem on building polygons, dynamic programming, universal lower bound, tree-necklace shadows.
result Exact formula for minimum number of normalized inflections for tree-like shadows.
A game on diagrams switches crossing directions to achieve connectedness.
problem Achieving connectedness in diagrams through crossing switches.
method Players switch crossing directions on regions of a diagram to achieve connectedness.
result Connectedness can be achieved through strategic crossing switches.
Proves condition for 4-manifolds with sphere boundary to be standard.
problem Determining when acyclic 4-manifolds with sphere boundary are standard.
method Uses Turaev's shadows to provide a sufficient condition for diffeomorphism to the standard 4-ball.
result If a compact, smooth, acyclic 4-manifold with sphere boundary has shadow-complexity at most 2, it is diffeomorphic to the standard 4-ball.
Method constructs shadowed polyhedra from divides, linking them to Lefschetz fibrations.
problem Linking divides to Lefschetz fibrations.
method Constructing shadowed polyhedra from divides and showing they satisfy the LF-property.
result Links of certain free divides are fibered with positive monodromy.
We introduce an associative algebra Z[X,S] associated to a birack shadow and define enhancements of the birack counting invariant for classical knots and links via representations of Z[X,S] known as shadow modules. We provide examples which demonstrate that the shadow module enhanced invariants are not determined by th…
New forms generalize Whitney forms with rational coefficients for numerical analysis.
problem Numerical problems with singularities near simplex faces.
method Introduce shadow forms and degrees of freedom for integration over faces of blow-up simplices.
result Obtain isomorphism between shadow forms cohomology and cellular cohomology of blow-up simplices.
Study of quandle coloring quivers with dihedral quandles.
problem Link invariants and their enhancements using quandles.
method Introduced shadow quandle coloring quivers and cocycle quivers, studied equivalence with quandle coloring numbers and shadow quandle cocycle invariants.
result Equivalence of quandle coloring quivers with quandle coloring numbers and shadow quandle cocycle quivers with shadow quandle cocycle invariants for specific dihedral quandles.
A proof of a shadow formula for a specific invariant using skein theory.
problem Calculating SU(2)-Reshetikhin-Turaev-Witten invariants for 3-manifolds and graphs.
method Skein theory
result A short proof of Turaev's shadow formula for the specified invariants.
Paper solves shadow problems for spheres and ellipsoids.
problem Finding the minimal number of non-overlapping balls that cover all lines through the center of a sphere or ellipsoid.
method Proposes a new method to solve the shadow problem for spheres and ellipsoids.
result Solves the shadow problem for spheres and ellipsoids, providing a new method.
We define and study branched shadows of 4-manifolds as a combination of branched spines of 3-manifolds and Turaev's shadows. We use these objects to combinatorially represent 4-manifolds equipped with Spinc-structures and homotopy classes of almost complex structures. We then use branched shadows to study complex 4-…
The paper classifies 4-manifolds with shadow complexity zero using combinatorial methods.
problem Classifying acyclic 4-manifolds with shadow complexity zero.
method Combinatorial approach focusing on simple polyhedra and their collapses.
result Any acyclic 4-manifold with shadow complexity zero and boundary is diffeomorphic to a 4-ball.
Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
problem Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
method Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
result Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
For portfolio choice problems with proportional transaction costs, we discuss whether or not there exists a "shadow price", i.e., a least favorable frictionless market extension leading to the same optimal strategy and utility. By means of an explicit counter-example, we show that shadow prices may fail to exist even i…
We give a self-contained introduction to the theory of Turaev's shadows as a tool to study 3 and 4-manifolds. The goal of the present paper twofold: on one side it is intended to be a shortcut to a basic use of the theory of shadows, on the other side it gives a sketchy overview of some of the recent results on shadows…
This paper compares classical shadows and direct quantum measurement for efficient information extraction.
problem Efficiently extracting classical information from quantum states with limited classical post-processing.
method Quantitative resource analysis comparing classical shadows and direct quantum measurement.
result An efficiency frontier between classical shadows and direct quantum measurement is identified.
A rack shadow is a set X with a rack action by a rack R, analogous to a vector space over a field. We use shadow colorings of classical link diagrams to define enhanced rack counting invariants and show that the enhanced invariants are stronger than unenhanced counting invariants.
Study verifies asymptotic expansion for Reshetikhin-Turaev invariants of fundamental shadow link pairs.
problem Verifying the asymptotic expansion conjecture for Reshetikhin-Turaev invariants of fundamental shadow link pairs.
method Using logarithmic holonomies of meridians and hyperbolic cone structures, the study verifies the conjecture for pairs where M∖L is homeomorphic to a fundamental shadow link complement. result The asymptotic expansion conjecture is true for pairs (M,L) with sufficiently small cone angles and M∖L homeomorphic to a fundamental shadow link complement. Paper discusses shadow prices for optimal investment with random endowment and transaction costs.
problem Existence of shadow prices in optimal investment problems with random endowment and transaction costs.
method Utilizes numéraire-based utility maximization under constant proportional transaction costs and random endowment constraints.
result Shadow prices exist in the original market with transaction costs, equivalent to a frictionless shadow market.
We define a subset of an almost complex manifold (M,J) to be a holomorphic shadow if it is the image of a J-holomorphic map from a compact complex manifold. Notice that a J-holomorphic curve is a holomorphic shadow, and so is a complex subvariety of a compact complex manifold. We show that under some conditions on an a…
Quantile regression attacks outperform shadow models in unseen class membership inference attacks.
problem Failure of shadow model attacks on unseen classes due to limited data access.
method Quantile regression attacks that learn features of member examples.
result Quantile regression attacks achieve up to 11x higher TPR than shadow model-based approaches.