Introduces pro-rata rationing for groundwater markets to resolve supply-demand imbalances.
problem Resolving supply-demand imbalances in groundwater markets.
method Introduces a pro-rata rationing mechanism and analyzes its properties in markets with exogenous restrictions and a leader-follower setting.
result Pro-rata approach provides a unique and fair rationing device.
A new model calculates optimal clearing payments in dynamic financial networks.
problem Determining fair clearing payments in networks with potential defaults.
method Extends Eisenberg-Noe model to multiple time periods, solving linear programs for optimal payments.
result Proves the model satisfies the priority of debt claims requirement and finds unique optimal payments.
We propose a framework to study optimal trading policies in a one-tick pro-rata limit order book, as typically arises in short-term interest rate futures contracts. The high-frequency trader has the choice to trade via market orders or limit orders, which are represented respectively by impulse controls and regular con…
We approximate the distribution of total expenditure of a retail company over warranty claims incurred in a fixed period [0, T], say the following quarter. We consider two kinds of warranty policies, namely, the non-renewing free replacement warranty policy and the non-renewing pro-rata warranty policy. Our approximati…
The paper calculates ruin probabilities for insurers with phase-type distributed claims.
problem Calculating ruin probabilities for insurers with specific claim distributions.
method Change-of-measure technique applied to phase-type distributed claim amounts.
result The mixture of Erlangs best fits real-world loss data, improving risk assessment.
Paper uses K-NN resampling to simulate and evaluate LOB markets.
problem Simulating and evaluating limit order book (LOB) markets.
method Applies K-nearest neighbor (K-NN) resampling to LOB simulation and evaluation. result Demonstrates the effectiveness and efficiency of K-NN resampling in LOB simulation and evaluation. The paper constructs new rational homology 3-spheres bounding rational homology 4-balls.
problem Constructing rational homology 3-spheres that bound rational homology 4-balls.
method Exploring plumbed 3-manifolds and using rational homology circles.
result Infinite families of rational homology 3-spheres that bound rational homology 4-balls.
The present paper attempts to show an alternative approach with regards to rational Pythagorean-hodograph (PH) curves and especially more natural approach for rational PH helices (i.e. rational helices). It exploits geometric features of rational helices to obtain a simpler construction of these curves and apply this t…
Classifies real rational knots and curves in a specific quadric space.
problem Classifying real rational knots and curves in a quadric space of signature (3,2). method Classification through a study of real rational curves of low degree in the quadric.
result Provides representatives of all real rational knots of degree ≤5 in the quadric. Rational knots and links in solid torus characterized by continued fractions.
problem Characterizing rational knots and links in solid torus.
method Using rational tangles and continued fractions, and generalizing to skein module invariants.
result Rational links in solid torus fully characterized by rational tangles and continued fractions.
New method for simplifying knots with specific properties.
problem Understanding knots with a specific unknotting number.
method Derive and apply the Montesinos trick for proper rational tangle replacement.
result Prove that knots with proper rational unknotting number one are prime and classify certain types.
The study calculates the average genus of rational knots and links.
problem Finding the average genus of rational knots and links.
method Enumerating and calculating the number of rational knots and links with a given crossing number.
result A precise formula for the average minimal genus of rational knots and links.
We note that a rational 3-tangle diagram is obtained from a combination of four generators. There is an algorithm to distinguish two rational 3-tangle diagrams up to isotopy. However, there is no perfect classification about rational 3-tangle diagrams such as the classification of rational 2-tangle diagrams cor…
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.
Classifies 3-manifolds bounding rational homology balls.
problem Identifying 3-manifolds that bound rational homology balls.
method Used constraints from Donaldson's diagonalization theorem and Heegaard Floer correction terms.
result Complete classification of spherical 3-manifolds bounding rational homology balls.
Study rational homology spheres with two legs, classifying those bounding rational balls.
problem Classifying Seifert rational homology spheres with two complementary legs.
method Complete classification of Seifert manifolds with 3 exceptional fibers and two complementary legs.
result Classification of Seifert manifolds bounding rational homology balls.
Classifies surgeries on torus knots and cables that bound rational homology balls.
problem Which surgeries on torus knots and cables bound rational homology balls?
method Classification based on integral surgeries and rational numbers q/p for cables.
result Set of rational numbers q/p for cables of a given knot K is bounded.
Jones polynomial coincidences explored for rational knots.
problem Identifying coincidences in Jones polynomial of rational knots.
method Moves on continued fraction expansion of rational knots, conjectured to generate all coincidences.
result Conjectured moves are sufficient to generate all Jones rational coincidences.
Lower bounds on rational slice genus using Heegaard Floer invariants.
problem Measuring complexity of homology classes in 4-manifolds.
method Introducing rational slice genus, bounding Heegaard Floer τ invariants, using satellite links and closed braids.
result Lower bounds on rational slice genus in terms of Heegaard Floer τ invariants.
New rational homology balls found in specific knot 2-handlebodies.
problem Finding rational homology balls in 2-handlebodies.
method Proving existence and properties of rational homology balls through smooth embeddings and rational blow-ups.
result Rationally blown-up 4-manifolds cannot be obtained by ordinary blow-ups under certain conditions.
This paper gives two new combinatorial topological proofs of the classification of rational tangles. Each proof rests on an elegant lemma showing that rational tangles are isotopic to canonical alternating rational tangles. The first proof defines the tangle fraction from the canonical form and uses flyping to prove in…
The article explores toric spaces of regular polyhedra, highlighting rational and non-rational cases.
problem Exploring toric spaces associated with regular convex polyhedra.
method Symplectic and complex toric spaces associated with five regular convex polyhedra.
result The regular dodecahedron and icosahedron cannot be treated via standard toric geometry.
Study connects taffy pulling, fractions, and rational tangles.
problem Understanding the relationship between taffy pulling, fractions, and rational tangles.
method Developed a taffy analogue for Conway's characterization of rational tangles and gave a geometric connection.
result Direct geometric connection between rational tangles and taffy pulls.
New knots show linear independence in slice concordance.
problem Understanding the structure of rationally slice knots.
method Provided an infinite family of knots that are linearly independent.
result Found knots that are linearly independent and infinite order.
The paper proves that rational concordance of double twist knots is reciprocal.
problem Determining when double twist knots are rationally slice.
method Using Donaldson's diagonalization theorem, the paper constructs rational ball obstructions for non-rational sliceness.
result Infinitely many prime power-fold cyclic branched covers do not bound a rational ball, proving the converse of rational concordance.
New Brieskorn spheres found to bound rational homology balls but not integral ones.
problem Identifying Brieskorn spheres that bound rational homology balls but not integral ones.
method Using modifications of Fintushel and Stern's argument and handlebody diagrams.
result Found new families of Brieskorn spheres (Σ(2,4n+1,12n+5) and Σ(3,3n+1,12n+5)) that bound rational homology balls but not integral ones. The study explores pinwheels in symplectic surfaces and non-squeezing of rational homology balls.
problem Understanding when Lagrangian pinwheels embed in symplectic rational and ruled surfaces.
method Almost toric fibrations and symplectic rational blow-up.
result A rational homology ball embeds into a rational homology cylinder if and only if the parameter is greater than or equal to 1.
New findings on knots that are both topologically and rationally slice.
problem Understanding knots that are both topologically and rationally slice.
method Analyzing the concordance group of knots in S3. result There are infinitely many topologically slice knots that are strongly rationally slice but not slice.
Paper introduces a new invariant for knots in contact 3-manifolds.
problem Bounding rational Thurston-Bennequin invariants for knots.
method Introduces a rational τ invariant for rationally null-homologous knots.
result Provides an upper bound for the sum of rational Thurston-Bennequin invariant and rational rotation number.
This paper is an introduction to rational tangles, rational knots and links and their applications to DNA. The paper can be read as an introduction to our more technical papers on rational tangles (math.GT/0311499) and on rational knots (math.GT/0212011). The present paper includes a self-contained account of the tangl…
Classifies torus bundles bounding 4-manifolds with rational homology.
problem Classifying torus bundles over the circle that bound 4-manifolds with rational homology.
method Completely classified torus bundles over the circle that bound 4-manifolds with rational homology.
result Completely classified torus bundles over the circle that bound 4-manifolds with rational homology.
New varieties found without smooth curves.
problem Finding varieties without smooth rational curves.
method Constructing normal rationally connected varieties.
result Found varieties of arbitrary large dimensions without smooth rational curves.
Study shows conditions for rational ellipticity of manifolds with symmetries.
problem Conditions for rational ellipticity of manifolds with symmetries.
method Analyzes conditions on compact simply connected manifolds with G-actions. result Proves rational ellipticity of M/G if M satisfies certain conditions. In this paper, We introduce an invariant of rational n-tangles which is obtained from the Kauffman bracket. It forms a vector with Laurent polynomial entries. We prove that the invariant classifies the rational 2-tangles and the reduced alternating rational 3-tangles. We conjecture that it classifies the rational 3-tan…
Paper connects Conway-Coxeter friezes and rational tangles.
problem Understanding the relationship between Conway-Coxeter friezes and rational tangles.
method Using Kauffman bracket polynomials to compute and connect friezes and rational tangles.
result Provides a complete invariant for Conway-Coxeter friezes of zigzag-type.
Invariants for virtual rational tangles defined using Kauffman's bracket.
problem Defining invariants for virtual rational tangles.
method Recursive formula using Kauffman's bracket polynomial.
result Computed several examples of the invariant.
We give a characterization of closed, simply connected, rationally elliptic 6-manifolds in terms of their rational cohomology rings and a partial classification of their real cohomology rings. We classify rational, real and complex homotopy types of closed, simply connected, rationally elliptic 7-manifolds. We give par…
The study identifies only two rational surfaces with constant scalar curvature Kähler metrics.
problem Finding rational surfaces with constant scalar curvature Kähler metrics.
method Analyzing projective rational surfaces and their Kähler metrics.
result There are only two rational surfaces (projective plane and quadric surface) with constant scalar curvature Kähler metrics.
New 3-manifolds bound rational 4-balls through specific operations.
problem Finding rational homology 3-spheres that bound rational homology 4-balls.
method Two operations that preserve lattice embedding obstruction to bounding rational homology balls.
result Explicit examples of rational surgeries on torus knots that bound rational homology balls.
Study rational homology cobordisms of 3-spheres and lens spaces.
problem When rational homology 3-spheres are cobordant to connected sums of lens spaces.
method Knot Floer homology combined with rational homology cobordism theory.
result Every rational homology cobordism class in the subgroup generated by lens spaces is uniquely represented by a connected sum of lens spaces.
New knot invariant λ bounds rational unknotting.
problem Bounding rational unknotting number.
method Extracted invariant λ from Khovanov homology.
result Invariant λ is lower bound for proper rational unknotting number.
Researchers found all simply connected rational 5-spheres with Sasaki-Einstein metrics.
problem Identifying rational homology 5-spheres with Sasaki-Einstein metrics.
method Comprehensive analysis of simply connected rational homology 5-spheres.
result Determination of which simply connected rational homology 5-spheres admit Sasaki-Einstein metrics.
Neural networks and rational functions can approximate each other efficiently.
problem Approximating neural networks with rational functions and vice versa.
method Demonstrates the existence of rational functions and ReLU networks of polynomial degree and size that approximate each other to a specified error.
result The approximation of neural networks with rational functions and vice versa can be done with polynomial degree and size, with hidden constants depending exponentially on the number of layers.
New proof for curved 3-cohom manifold rational ellipticity.
problem Rational ellipticity of curved manifolds with specific cohomogeneity.
method Proved rationally elliptic for cohomogeneity-three manifolds with positive curvature and no boundary quotient.
result Closed, simply connected, positively curved, cohomogeneity-three manifolds without boundary are rationally elliptic.
New 4-manifold accounts for rationally slice knots.
problem Characterize rationally slice knots.
method Show independence of construction of rational homology ball VK. result Single 4-manifold accounts for all known rationally slice knots.
Proves unicity of q-rational numbers up to conjugacy with new deformations.
problem Proving unicity of q-rational numbers up to conjugacy.
method Using character varieties to prove unicity, and characterizing new deformations.
result Exactly two deformations of q-integers, one new and one old, with new positivity properties.
The paper resolves behavioral finance objections to rational finance theory.
problem Predictability of asset returns, Equity Premium, Volatility Puzzle.
method Statistical models within rational finance theory.
result Offers resolutions to behavioral finance anomalies.
We formulate the equivalence problem, in the sense of E. Cartan, for families of minimal rational curves on uniruled projective manifolds. An important invariant of this equivalence problem is the variety of minimal rational tangents. We study the case when varieties of minimal rational tangents at general points form …