Knotting occurs in polymers and affects polymer properties.Physical understanding of polymer knots is limited due to the complex confo rmational space of knotted structures.The knotting problem can be handled by the t...Knotting occurs in polymers and affects polymer properties.Physical understanding of polymer knots is limited due to the complex confo rmational space of knotted structures.The knotting problem can be handled by the tube model,which assumes that knotted polymer segments are confined in a virtual tube.Recently,we quantified this virtual tube using a computational algorithm.The algorithm was limited to the simplest knot:3_(1)knot.It remains unclear how the tube model and computational algorithm are applied to more complex knots.In this work,we apply the tube model to 4_(1),5_(1),and 5_(2) knots,resulting in several findings.First,the computational algorithm developed for 3_(1) knot cannot be directly applied to 4_(1) knot.After modifying the algorithm,we quantify the tubes for 4_(1) knot.Second,we find that,for all four knot types,the knotcore region have less average bending energy density than unknotted regions when the chain bending stiffness is small.This counterintuitive result is explained by the tube model.Third,for all four knot types,polymer segments at the boundaries of knot cores adopt nearly straight conformations(almost zero bending)and exhibit lower local bending compared to other knot-core regions and unknotting regions.This local behavior is also consistent with prediction from the tube model.This counterintuitive result is also explained by the tube model.Fourth,for all four knot types,when a polymer has non-uniform bending stiffness,a knot prefers certain chain positions such that the knot boundary locates at one stiff segment.Overall,our work paves the way for applying the tube model to complex polymer knots and obtains many common results for different knot types,which can be useful in understanding many knotting systems,such as DNA knots in vivo.展开更多
Knots are discovered in a wide range of systems,from DNA and proteins to catheters and umbilical cords,and have thus attracted much attention from physicists and biophysicists.Langevin dynamics simulations were perfor...Knots are discovered in a wide range of systems,from DNA and proteins to catheters and umbilical cords,and have thus attracted much attention from physicists and biophysicists.Langevin dynamics simulations were performed to study the knotting properties of coarsegrained knotted circular semiflexible polyelectrolyte (PE) in solutions of different concentrations of trivalent salt.We find that the length and position of the knotted region can be controlled by tuning the bending rigidity b of the PE and the salt concentration C_(S).We find that the knot length varies nonmonotonically with b in the presence of salt,and the knot localizes and is the tightest at b=5.As b>5,the knot swells with b increase.In addition,similar modulations of the knot size and position can be achieved by varying the salt concentration C_(S).The knot length varies nonmonotonically with C_(S) for b>0.The knot localizes and becomes tightest at C_(S)=1.5×10^(-4) mol/L in the range of C_(S)≤1.5×10^(-4) mol/L.As C_(S)>1.5×10^(-4) mol/L,the knot of the circular semiflexible PE swells at the expense of the overall size of the PE.Our results lay the foundation for achieving broader and more precise external adjustability of knotted PE size and knot length.展开更多
Tree knots are generally considered defects in wood,but how the surrounding structures of the defects affects strength of wood has not been studied.Here the mechanical properties of static compression and hole bearing...Tree knots are generally considered defects in wood,but how the surrounding structures of the defects affects strength of wood has not been studied.Here the mechanical properties of static compression and hole bearing tests were designed for encased knots and intergrown knots,and the strengthening mechanism of streamline tissue and connecting interface was analyzed by finite element modeling.And the two reinforced structures were applied to composite structural holes and connecting holes,which significantly improved open hole compressive strength and hole bearing strength.And the finite element models for two kinds of composite hole were created to analyze how the stress field around the reinforced structure strengthens the composite.Both the experimental results and the finite analysis results show that the streamline structure could effectively improve the compressive properties of composite structural holes,and the connecting interface provided a stable constraint for giving full play to the hole bearing properties of stronger materials.These two structures will provide reference for the structural design of lightweight composites.展开更多
Mechanically interlocked molecules (MIMs) have unique properties with broad applications, yet constructing both knotted and linked topologies from the same ligand remains challenging due to their distinct geometric de...Mechanically interlocked molecules (MIMs) have unique properties with broad applications, yet constructing both knotted and linked topologies from the same ligand remains challenging due to their distinct geometric demands. To address this, we design and synthesize a conformationally adaptive ligand 4,7-bis(3-(pyridin-4-yl) phenyl) benzo[c][1,2,5]thiadiazole (L1) with a tunable torsional angle θ of N1C1C2N2 ranging from 7.5° to 108.9°. Utilizing coordination-driven self-assembly at ambient temperature, L1 selectively assembles with binuclear half-sandwich units RhB1, RhB2, RhB3, and RhB4 featuring Cp*^(Rh^(Ⅲ)) (Cp* = η^(5)-pentam-ethylcyclopentadienyl) into distinct topologies: Solomon links Rh-1, trefoil knots Rh-2, molecular tweezers Rh 3, and Rh-4, respectively. Crucially, the self-adaptability of ligand L1 directs topology formation through pro-gramming different combination of noncovalent interactions (π-x stacking, CH..π interaction, and lone pair-π interaction), thus navigating divergent assembly pathways by conformational switching, as evidenced by X-ray crystallography analysis, independent gradient model (IGM) analysis, detailed nuclear magnetic resonance (NMR) spectroscopy and electrospray ionization time-of-flight/mass spectrometry (ESI-TOF/MS). This strategy can also be extended to construct Cp*^(Irl^(Ⅲ)) analogs (Solomon links Ir-1, trefoil knots Ir-2, molecular tweezers Ir-3 and Ir-4), demonstrating metal-independent control and achieving intricate topologies in a high yield.展开更多
In this work,Langevin dynamics simulations were carried out to thoroughly investigate the swapping process of composite knots under tension in a cuboid nanochannel.From our analysis,the free energy profiles of knot sw...In this work,Langevin dynamics simulations were carried out to thoroughly investigate the swapping process of composite knots under tension in a cuboid nanochannel.From our analysis,the free energy profiles of knot swapping under different conditions were extracted from the overall probability distribution of the relative distance between the centers of composite knots.In addition,the impact of the stretching force,confinement size,and bending stiffness on the free energy profiles was directly identified.Especially,the influence of topology structure is for the first time reported.The increasing stretching force in a fixed confinement or the confinement size under a constant stretching force does not alter their respective equilibrium populations at the separate state and the entangled state.In contrast,a bending stiffness larger than 15 enhanced the formation of the entangled state.The topology structure of the 51knot,which was different from the 52knot,resulted in forming a metastable state in the free energy profiles.The increasing stretching forces yielded an enhancement of the following free energy barrier.展开更多
We deal with the properties of incompressible and pairwise incompressible surfaces in knot complements through the application of relevant properties of almost simple topological graphs.We analyze the topological grap...We deal with the properties of incompressible and pairwise incompressible surfaces in knot complements through the application of relevant properties of almost simple topological graphs.We analyze the topological graph invariants associated with surfaces embedded in the complements of alternating and almost alternating knots.Specifically,we prove that the characteristic numbers of these graphs remain invariant under two fundamental transformations(R-move and S^(2)-move).Leveraging the interplay between characteristic numbers and Euler characteristics,and further connecting Euler characteristics to surface genus,we derive novel results regarding the genus of incompressible pairwise incompressible surfaces.Additionally,we establish a discriminant criterion to determine when such surfaces in knot complements admit genus zero.展开更多
The boundary knot method(BKM)is a simple boundary-type meshless method.Due to the use of non-singular general solutions rather than singular fundamental solutions,BKM does not need to consider the artificial boundary....The boundary knot method(BKM)is a simple boundary-type meshless method.Due to the use of non-singular general solutions rather than singular fundamental solutions,BKM does not need to consider the artificial boundary.Therefore,this method has the merits of purely meshless,easy to program,high solution accuracy and so on.In this paper,we investigate the effectiveness of the BKM for solving Helmholtz-type problems under various conditions through a series of novel numerical experiments.The results demonstrate that the BKM is efficient and achieves high computational accuracy for problems with smooth or continuous boundary conditions.However,when applied to discontinuous boundary problems,the method exhibits significant numerical instability,potentially leading to substantial deviations in the computed results.Finally,three potential improvement strategies are proposed to mitigate this limitation.展开更多
We discuss the properties of incompressible pairwise incompressible surfaces in a knot complement by using twist crossing number. Let K be a pretzel knot or rational knot that its twistindex is less than 6, and l...We discuss the properties of incompressible pairwise incompressible surfaces in a knot complement by using twist crossing number. Let K be a pretzel knot or rational knot that its twistindex is less than 6, and let F be an incompressible pairwise incompressible surface in S 3-K. Then F is a punctured sphere.展开更多
Knot theory is a branch of topology in pure mathematics, however, it has been increasingly used in different sciences such as chemistry. Mathematically, a knot is a subset of three-dimensional space which is homeomorp...Knot theory is a branch of topology in pure mathematics, however, it has been increasingly used in different sciences such as chemistry. Mathematically, a knot is a subset of three-dimensional space which is homeomorphic to a circle and it is only defined in a closed loop. In chemistry, knots have been applied to synthetic molecular design. Mathematics and chemistry together can work to determine, characterize and create knots which help to understand different molecular designs and then forecast their physical features. In this study, we provide an introduction to the knot theory and its topological concepts, and then we extend it to the context of chemistry. We present parametric representations for several synthetic knots. The main goal of this paper is to develop a geometric and topological intuition for molecular knots using parametric equations. Since parameterizations are non-unique;there is more than one set of parametric equations to specify the same molecular knots. This parametric representation can be used easily to express geometrically molecular knots and would be helpful to find out more complicated molecular models.展开更多
Variable and unpredictable food resources at stopover sites bring severe challenges to migrating shorebirds. Opportunistic foraging strategies, referring to shorebirds consuming prey in proportion to their availabilit...Variable and unpredictable food resources at stopover sites bring severe challenges to migrating shorebirds. Opportunistic foraging strategies, referring to shorebirds consuming prey in proportion to their availability, allow shorebirds to replenish fuel and nutrient reserves efficiently for continuing their migration. Chongming Dongtan, located in the Yangtze River estuary of eastern China, is the first major stopover site of shorebirds on the Chinese mainland during their northward migration. We investigated the diet of Great Knots (Calidris tenuirostris) at Chongming Dongtan during the spring stopovers of 2009 and 2010 through benthos sampling and dropping analysis. The benthos samples were categorized into gastropods, bivalves, polychaetes, crustaceans and insect larvae. Dropping analysis indicated that gastropods and bivalves constituted more than 70% of the diet of the Great Knot, with Assiminea violacea and Corbicula fluminea being the most frequently consumed. Chi-square tests indicated that for each prey category, there was no significant difference between the frequency of its occurrence in the benthos samples and dropping samples during the early stopover periods of 2009 and 2010 and during the late stopover periods of 2010. Although there was a statistically significant difference between the frequency of occurrence of prey in the total macrobenthos and in the droppings of the Great Knots during the late stopover period in 2009, the more abundant prey were more frequently consumed by the Great Knots. This suggests that Great Knots adopted an opportunistic foraging strategy during their stopover at Chongming Dongtan.展开更多
We study zeros of the Jones polynomial and their distributions for torus knots and 2-bridge knots. We prove that e(2m+1)πi/2and e(2m+1)πi/4(m is a positive integer)can not be the zeros of Jones polynomial for torus ...We study zeros of the Jones polynomial and their distributions for torus knots and 2-bridge knots. We prove that e(2m+1)πi/2and e(2m+1)πi/4(m is a positive integer)can not be the zeros of Jones polynomial for torus knots T p,q by the knowledge of the trigonometric function. We elicit the normal form of Jones polynomials of the 2-bridge knot C(-2, 2, ···,(-1)r2) by the recursive form and discuss the distribution of their zeros.展开更多
Topological methods are rapidly developing and are becoming more used in physics, biology and chemistry. One area of topology has showed its immense potential in explaining potential financial contagion and financial ...Topological methods are rapidly developing and are becoming more used in physics, biology and chemistry. One area of topology has showed its immense potential in explaining potential financial contagion and financial crisis in financial markets. The aforementioned method is knot theory. The movement of stock price has been marked and braids and knots have been noted. By analysing the knots and braids using Jones polynomial, it is tried to find if there exists an untrivial knot equal to unknot? After thorough analysis, possible financial contagion and financial crisis prediction are analysed by using instruments of knot theory pertaining in that sense to Jones, Laurent and Alexander polynomial. It is proved that it is possible to predict financial disruptions by observing possible knots in the graphs and finding appropriate polynomials. In order to analyse knot formation, the following approach is used: “Knot formation in three-dimensional space is considered and the equations about knot forming and its disentangling are considered”. After having defined the equations in three-dimensional space, the definition of Brownian bridge concerning formation of knots in three-dimensional space is defined. Using analogy method, the notion of Brownian bridge is translated into 2-dimensional space and the foundations for the application of knot theory in 2-dimensional space have been set up. At the same time, the aforementioned approach is innovative and it could be used in accordance with stochastic analysis and quantum finance.展开更多
In this paper, the knot strengths of the seven strands which were made of polyethleneterephthalate(PET), nylon 6, polyvinyl formal fibre, polypropylene and polyethylene with differentspecification used for industrial ...In this paper, the knot strengths of the seven strands which were made of polyethleneterephthalate(PET), nylon 6, polyvinyl formal fibre, polypropylene and polyethylene with differentspecification used for industrial purposes were tested and discussed. The results of experimentshow: the knot strength loss does not only depend on the breaking elongation and the diameter ofsample, but also on the shape of the load-extension curve and twist factor of sample and other fac-tors; the range of the knot strength loss and the breaking energy loss vary in a wide range, in thepresent case. the former is from 2.7% to 58.14% and the latter from 16.36% to 78.76%. Thestrength loss of the polypropylene filament is the least among the samples investigated.展开更多
[ Objective ] The paper was to study the effects of anti-nematode preparations with different mechanisms on changes of enzyme systems and membrane permeability of tomato leaves, so as to provide reference basis for ef...[ Objective ] The paper was to study the effects of anti-nematode preparations with different mechanisms on changes of enzyme systems and membrane permeability of tomato leaves, so as to provide reference basis for effective control of soil root-knot nematode in greenhouse. [ Method] With tomato seedlings af- fected by root-knot nematode as material, changes of superoxide dismutase（SOD）, peroxidase（ POD）, relative conductivity and malondialdehyde （MDA） in toma- toes were tested after the seedling soil was treated by preparations of Wuxianmei, Hailvsu, Duxiandna and Avermectin. [ Result] After treated by different prepara- tions, SOD and POD activity of tomato leaves were higher than control, and that treated by Wuxianmei was the highest. In addition to Duxiandna, the relative con- ductivity and MDA content of other treatments were significantly lower than control. When tomatoes were planted for 70 d, the effect of Avermectin against reot-knot nematode Was the best of 66.3%. [ Conclusion] After tomatoes were infected by root-knot nematode, different preparation treatments all had certain control effect, which made the physical indicators of tomato have obvious change. Integrated control of multiple preparations in greenhouse was beneficial to control soil root-knot nematode.展开更多
By means of the method of torus knot theory, this paper gives the complete processes of obtaining the knotted pictures of four Bell b^es from the knotted pictures of four basic two qubit states.
The relationship between a link diagram and its corresponding planar graph is briefly reviewed. A necessary and sufficient condition is given to detect when a planar graph corresponds to a knot. The relationship betwe...The relationship between a link diagram and its corresponding planar graph is briefly reviewed. A necessary and sufficient condition is given to detect when a planar graph corresponds to a knot. The relationship between planar graph and almost planar Seifert surface is discussed. Using planar graph, we construct an alternating amphicheiral prime knot with crossing number n for any even number n 〉 4. This gives an affirmative answer to problem 1.66（B） on Kirby＇s problem list .展开更多
By means of the torus knot theory method, this paper presents the complete process of obtaining the knotted pictures of eight GHZ states on the surface of a trivial torus from the knotted pictures of eight basic three...By means of the torus knot theory method, this paper presents the complete process of obtaining the knotted pictures of eight GHZ states on the surface of a trivial torus from the knotted pictures of eight basic three-qubit states on the surface of a trivial torus. Thus, we obtain eight knotted pictures 121 linkage on the ordinary plane.展开更多
We show that certain satellite knots of every strongly negative-amphicheiral rational knot are rational-slice knots. This proof also shows that the O-surgery manifold of a certain strongly negative amphicheiral knot s...We show that certain satellite knots of every strongly negative-amphicheiral rational knot are rational-slice knots. This proof also shows that the O-surgery manifold of a certain strongly negative amphicheiral knot such as the figure-eight knot bounds a compact oriented smooth 4-manifold homotopy equivalent to the 2-sphere such that a second homology class of the 4-manifold is represented by a smoothly embedded 2-sphere if and only if the modulo two reduction of it is zero.展开更多
Knots are discovered in biophysical systems,such as DNA and proteins.Knotted portions in knotted DNA are significantly bent and their corresponding bending angles are comparable with or larger than the sharp bending a...Knots are discovered in biophysical systems,such as DNA and proteins.Knotted portions in knotted DNA are significantly bent and their corresponding bending angles are comparable with or larger than the sharp bending angle resulting in flexible defects.The role of flexible defects in the interplay of supercoiling and knotting of circular DNA were predicted by a Monte Carlo simulation.In knotted DNA with a particular knot type,a flexible defect noticeably enhances the supercoiling of the knotted DNA and the decreasing excitation energy makes the knotted portion more compact.A reduction in twist rigidity and unwinding of flexible defects are incorporated into the numerical simulations,so that interplay of supercoiling and knotting of circular DNA is studied under torsional conditions.Increasing unwinding not only results in a wider linking number distribution,but also leads to a drift of the distribution to lower values.A flexible defect has obvious effects on knotting probability.The summation of equilibrium distribution probability for nontrivial knotted DNA with different contour length does not change with excitation energy monotonically and has a maximum at an intermediate value of excitation energy around 5 kBT.In the phase space of knot length and gyration radius of knotted DNA,knot length does not anticorrelate with its gyration radius,which is attributed to the flexible defect in the knotted portion,which leads to the release of bending energy and inhibited the competition between entropy and bending energy.展开更多
基金financially supported by the National Natural Science Foundation of China(No.22273080)Research Grants Council of Hong Kong(Nos.11313322 and 11307224)。
文摘Knotting occurs in polymers and affects polymer properties.Physical understanding of polymer knots is limited due to the complex confo rmational space of knotted structures.The knotting problem can be handled by the tube model,which assumes that knotted polymer segments are confined in a virtual tube.Recently,we quantified this virtual tube using a computational algorithm.The algorithm was limited to the simplest knot:3_(1)knot.It remains unclear how the tube model and computational algorithm are applied to more complex knots.In this work,we apply the tube model to 4_(1),5_(1),and 5_(2) knots,resulting in several findings.First,the computational algorithm developed for 3_(1) knot cannot be directly applied to 4_(1) knot.After modifying the algorithm,we quantify the tubes for 4_(1) knot.Second,we find that,for all four knot types,the knotcore region have less average bending energy density than unknotted regions when the chain bending stiffness is small.This counterintuitive result is explained by the tube model.Third,for all four knot types,polymer segments at the boundaries of knot cores adopt nearly straight conformations(almost zero bending)and exhibit lower local bending compared to other knot-core regions and unknotting regions.This local behavior is also consistent with prediction from the tube model.This counterintuitive result is also explained by the tube model.Fourth,for all four knot types,when a polymer has non-uniform bending stiffness,a knot prefers certain chain positions such that the knot boundary locates at one stiff segment.Overall,our work paves the way for applying the tube model to complex polymer knots and obtains many common results for different knot types,which can be useful in understanding many knotting systems,such as DNA knots in vivo.
基金financially supported by the National Natural Science Foundation of China (No. 22363005)Jiangxi Provincial Natural Science Foundation (Nos. GJJ2200416 and 20202BABL203015)。
文摘Knots are discovered in a wide range of systems,from DNA and proteins to catheters and umbilical cords,and have thus attracted much attention from physicists and biophysicists.Langevin dynamics simulations were performed to study the knotting properties of coarsegrained knotted circular semiflexible polyelectrolyte (PE) in solutions of different concentrations of trivalent salt.We find that the length and position of the knotted region can be controlled by tuning the bending rigidity b of the PE and the salt concentration C_(S).We find that the knot length varies nonmonotonically with b in the presence of salt,and the knot localizes and is the tightest at b=5.As b>5,the knot swells with b increase.In addition,similar modulations of the knot size and position can be achieved by varying the salt concentration C_(S).The knot length varies nonmonotonically with C_(S) for b>0.The knot localizes and becomes tightest at C_(S)=1.5×10^(-4) mol/L in the range of C_(S)≤1.5×10^(-4) mol/L.As C_(S)>1.5×10^(-4) mol/L,the knot of the circular semiflexible PE swells at the expense of the overall size of the PE.Our results lay the foundation for achieving broader and more precise external adjustability of knotted PE size and knot length.
文摘Tree knots are generally considered defects in wood,but how the surrounding structures of the defects affects strength of wood has not been studied.Here the mechanical properties of static compression and hole bearing tests were designed for encased knots and intergrown knots,and the strengthening mechanism of streamline tissue and connecting interface was analyzed by finite element modeling.And the two reinforced structures were applied to composite structural holes and connecting holes,which significantly improved open hole compressive strength and hole bearing strength.And the finite element models for two kinds of composite hole were created to analyze how the stress field around the reinforced structure strengthens the composite.Both the experimental results and the finite analysis results show that the streamline structure could effectively improve the compressive properties of composite structural holes,and the connecting interface provided a stable constraint for giving full play to the hole bearing properties of stronger materials.These two structures will provide reference for the structural design of lightweight composites.
基金Department of Chemistry,Fudan Uni-versity,the National Natural Science Foundation of China(22031003,21720102004)the Shanghai Science Technology Committee(19DZ227010O)the Alexander von Humboldt Foundation for a Humboldt Research Award.
文摘Mechanically interlocked molecules (MIMs) have unique properties with broad applications, yet constructing both knotted and linked topologies from the same ligand remains challenging due to their distinct geometric demands. To address this, we design and synthesize a conformationally adaptive ligand 4,7-bis(3-(pyridin-4-yl) phenyl) benzo[c][1,2,5]thiadiazole (L1) with a tunable torsional angle θ of N1C1C2N2 ranging from 7.5° to 108.9°. Utilizing coordination-driven self-assembly at ambient temperature, L1 selectively assembles with binuclear half-sandwich units RhB1, RhB2, RhB3, and RhB4 featuring Cp*^(Rh^(Ⅲ)) (Cp* = η^(5)-pentam-ethylcyclopentadienyl) into distinct topologies: Solomon links Rh-1, trefoil knots Rh-2, molecular tweezers Rh 3, and Rh-4, respectively. Crucially, the self-adaptability of ligand L1 directs topology formation through pro-gramming different combination of noncovalent interactions (π-x stacking, CH..π interaction, and lone pair-π interaction), thus navigating divergent assembly pathways by conformational switching, as evidenced by X-ray crystallography analysis, independent gradient model (IGM) analysis, detailed nuclear magnetic resonance (NMR) spectroscopy and electrospray ionization time-of-flight/mass spectrometry (ESI-TOF/MS). This strategy can also be extended to construct Cp*^(Irl^(Ⅲ)) analogs (Solomon links Ir-1, trefoil knots Ir-2, molecular tweezers Ir-3 and Ir-4), demonstrating metal-independent control and achieving intricate topologies in a high yield.
基金The National Natural Science Foundation of China under Grant Nos.11864006,11874309,12164007,and 12204118。
文摘In this work,Langevin dynamics simulations were carried out to thoroughly investigate the swapping process of composite knots under tension in a cuboid nanochannel.From our analysis,the free energy profiles of knot swapping under different conditions were extracted from the overall probability distribution of the relative distance between the centers of composite knots.In addition,the impact of the stretching force,confinement size,and bending stiffness on the free energy profiles was directly identified.Especially,the influence of topology structure is for the first time reported.The increasing stretching force in a fixed confinement or the confinement size under a constant stretching force does not alter their respective equilibrium populations at the separate state and the entangled state.In contrast,a bending stiffness larger than 15 enhanced the formation of the entangled state.The topology structure of the 51knot,which was different from the 52knot,resulted in forming a metastable state in the free energy profiles.The increasing stretching forces yielded an enhancement of the following free energy barrier.
基金Supported by the National Natural Science Foundation of China(Grant No.12026411)。
文摘We deal with the properties of incompressible and pairwise incompressible surfaces in knot complements through the application of relevant properties of almost simple topological graphs.We analyze the topological graph invariants associated with surfaces embedded in the complements of alternating and almost alternating knots.Specifically,we prove that the characteristic numbers of these graphs remain invariant under two fundamental transformations(R-move and S^(2)-move).Leveraging the interplay between characteristic numbers and Euler characteristics,and further connecting Euler characteristics to surface genus,we derive novel results regarding the genus of incompressible pairwise incompressible surfaces.Additionally,we establish a discriminant criterion to determine when such surfaces in knot complements admit genus zero.
基金Supported by the Key Scientific Research Plan of Colleges and Universities in Henan Province(23B140006)。
文摘The boundary knot method(BKM)is a simple boundary-type meshless method.Due to the use of non-singular general solutions rather than singular fundamental solutions,BKM does not need to consider the artificial boundary.Therefore,this method has the merits of purely meshless,easy to program,high solution accuracy and so on.In this paper,we investigate the effectiveness of the BKM for solving Helmholtz-type problems under various conditions through a series of novel numerical experiments.The results demonstrate that the BKM is efficient and achieves high computational accuracy for problems with smooth or continuous boundary conditions.However,when applied to discontinuous boundary problems,the method exhibits significant numerical instability,potentially leading to substantial deviations in the computed results.Finally,three potential improvement strategies are proposed to mitigate this limitation.
文摘We discuss the properties of incompressible pairwise incompressible surfaces in a knot complement by using twist crossing number. Let K be a pretzel knot or rational knot that its twistindex is less than 6, and let F be an incompressible pairwise incompressible surface in S 3-K. Then F is a punctured sphere.
文摘Knot theory is a branch of topology in pure mathematics, however, it has been increasingly used in different sciences such as chemistry. Mathematically, a knot is a subset of three-dimensional space which is homeomorphic to a circle and it is only defined in a closed loop. In chemistry, knots have been applied to synthetic molecular design. Mathematics and chemistry together can work to determine, characterize and create knots which help to understand different molecular designs and then forecast their physical features. In this study, we provide an introduction to the knot theory and its topological concepts, and then we extend it to the context of chemistry. We present parametric representations for several synthetic knots. The main goal of this paper is to develop a geometric and topological intuition for molecular knots using parametric equations. Since parameterizations are non-unique;there is more than one set of parametric equations to specify the same molecular knots. This parametric representation can be used easily to express geometrically molecular knots and would be helpful to find out more complicated molecular models.
基金supported by the National Natural Science Foundation of China(Grant No.30670269,31071939)
文摘Variable and unpredictable food resources at stopover sites bring severe challenges to migrating shorebirds. Opportunistic foraging strategies, referring to shorebirds consuming prey in proportion to their availability, allow shorebirds to replenish fuel and nutrient reserves efficiently for continuing their migration. Chongming Dongtan, located in the Yangtze River estuary of eastern China, is the first major stopover site of shorebirds on the Chinese mainland during their northward migration. We investigated the diet of Great Knots (Calidris tenuirostris) at Chongming Dongtan during the spring stopovers of 2009 and 2010 through benthos sampling and dropping analysis. The benthos samples were categorized into gastropods, bivalves, polychaetes, crustaceans and insect larvae. Dropping analysis indicated that gastropods and bivalves constituted more than 70% of the diet of the Great Knot, with Assiminea violacea and Corbicula fluminea being the most frequently consumed. Chi-square tests indicated that for each prey category, there was no significant difference between the frequency of its occurrence in the benthos samples and dropping samples during the early stopover periods of 2009 and 2010 and during the late stopover periods of 2010. Although there was a statistically significant difference between the frequency of occurrence of prey in the total macrobenthos and in the droppings of the Great Knots during the late stopover period in 2009, the more abundant prey were more frequently consumed by the Great Knots. This suggests that Great Knots adopted an opportunistic foraging strategy during their stopover at Chongming Dongtan.
基金Supported by the National Science Foundation of China（11471151） Supported by Program for Liaoning Excellent Talents in University（LR2011031）
Acknowledgment The authors would like to thank the referees for kind suggestions and many useful comments
文摘We study zeros of the Jones polynomial and their distributions for torus knots and 2-bridge knots. We prove that e(2m+1)πi/2and e(2m+1)πi/4(m is a positive integer)can not be the zeros of Jones polynomial for torus knots T p,q by the knowledge of the trigonometric function. We elicit the normal form of Jones polynomials of the 2-bridge knot C(-2, 2, ···,(-1)r2) by the recursive form and discuss the distribution of their zeros.
文摘Topological methods are rapidly developing and are becoming more used in physics, biology and chemistry. One area of topology has showed its immense potential in explaining potential financial contagion and financial crisis in financial markets. The aforementioned method is knot theory. The movement of stock price has been marked and braids and knots have been noted. By analysing the knots and braids using Jones polynomial, it is tried to find if there exists an untrivial knot equal to unknot? After thorough analysis, possible financial contagion and financial crisis prediction are analysed by using instruments of knot theory pertaining in that sense to Jones, Laurent and Alexander polynomial. It is proved that it is possible to predict financial disruptions by observing possible knots in the graphs and finding appropriate polynomials. In order to analyse knot formation, the following approach is used: “Knot formation in three-dimensional space is considered and the equations about knot forming and its disentangling are considered”. After having defined the equations in three-dimensional space, the definition of Brownian bridge concerning formation of knots in three-dimensional space is defined. Using analogy method, the notion of Brownian bridge is translated into 2-dimensional space and the foundations for the application of knot theory in 2-dimensional space have been set up. At the same time, the aforementioned approach is innovative and it could be used in accordance with stochastic analysis and quantum finance.
文摘In this paper, the knot strengths of the seven strands which were made of polyethleneterephthalate(PET), nylon 6, polyvinyl formal fibre, polypropylene and polyethylene with differentspecification used for industrial purposes were tested and discussed. The results of experimentshow: the knot strength loss does not only depend on the breaking elongation and the diameter ofsample, but also on the shape of the load-extension curve and twist factor of sample and other fac-tors; the range of the knot strength loss and the breaking energy loss vary in a wide range, in thepresent case. the former is from 2.7% to 58.14% and the latter from 16.36% to 78.76%. Thestrength loss of the polypropylene filament is the least among the samples investigated.
基金Supported by Transformation and Promotion Projects of Agriculture Science and Technology Achievements of Tianjin City"Integration and Demonstration of Integrated Control Technology of Greenhouse Vegetable Fields with Continuous Cropping Obstacles"(0804140)Basic Application and Cutting-edge Technology Research Projects of Tianjin City"Risk Assessment and Regulation Research of Nitrogen and Phosphorus Non-point Source Pollution in Facility Agriculture"(09JCYBJC08600)~~
文摘[ Objective ] The paper was to study the effects of anti-nematode preparations with different mechanisms on changes of enzyme systems and membrane permeability of tomato leaves, so as to provide reference basis for effective control of soil root-knot nematode in greenhouse. [ Method] With tomato seedlings af- fected by root-knot nematode as material, changes of superoxide dismutase（SOD）, peroxidase（ POD）, relative conductivity and malondialdehyde （MDA） in toma- toes were tested after the seedling soil was treated by preparations of Wuxianmei, Hailvsu, Duxiandna and Avermectin. [ Result] After treated by different prepara- tions, SOD and POD activity of tomato leaves were higher than control, and that treated by Wuxianmei was the highest. In addition to Duxiandna, the relative con- ductivity and MDA content of other treatments were significantly lower than control. When tomatoes were planted for 70 d, the effect of Avermectin against reot-knot nematode Was the best of 66.3%. [ Conclusion] After tomatoes were infected by root-knot nematode, different preparation treatments all had certain control effect, which made the physical indicators of tomato have obvious change. Integrated control of multiple preparations in greenhouse was beneficial to control soil root-knot nematode.
文摘By means of the method of torus knot theory, this paper gives the complete processes of obtaining the knotted pictures of four Bell b^es from the knotted pictures of four basic two qubit states.
文摘The relationship between a link diagram and its corresponding planar graph is briefly reviewed. A necessary and sufficient condition is given to detect when a planar graph corresponds to a knot. The relationship between planar graph and almost planar Seifert surface is discussed. Using planar graph, we construct an alternating amphicheiral prime knot with crossing number n for any even number n 〉 4. This gives an affirmative answer to problem 1.66（B） on Kirby＇s problem list .
文摘By means of the torus knot theory method, this paper presents the complete process of obtaining the knotted pictures of eight GHZ states on the surface of a trivial torus from the knotted pictures of eight basic three-qubit states on the surface of a trivial torus. Thus, we obtain eight knotted pictures 121 linkage on the ordinary plane.
文摘We show that certain satellite knots of every strongly negative-amphicheiral rational knot are rational-slice knots. This proof also shows that the O-surgery manifold of a certain strongly negative amphicheiral knot such as the figure-eight knot bounds a compact oriented smooth 4-manifold homotopy equivalent to the 2-sphere such that a second homology class of the 4-manifold is represented by a smoothly embedded 2-sphere if and only if the modulo two reduction of it is zero.
基金The National Natural Science Foundation of China under Grant Nos.11464004,11864006 and 11564015Guizhou Scientific and Technological Program(20185781 and 20185781-15)the Foundation of Key Laboratory of Animal Genetics,Breeding and Reproduction in The Plateau Mountainous Region,Ministry of Education(Guizhou University),KY2020243。
文摘Knots are discovered in biophysical systems,such as DNA and proteins.Knotted portions in knotted DNA are significantly bent and their corresponding bending angles are comparable with or larger than the sharp bending angle resulting in flexible defects.The role of flexible defects in the interplay of supercoiling and knotting of circular DNA were predicted by a Monte Carlo simulation.In knotted DNA with a particular knot type,a flexible defect noticeably enhances the supercoiling of the knotted DNA and the decreasing excitation energy makes the knotted portion more compact.A reduction in twist rigidity and unwinding of flexible defects are incorporated into the numerical simulations,so that interplay of supercoiling and knotting of circular DNA is studied under torsional conditions.Increasing unwinding not only results in a wider linking number distribution,but also leads to a drift of the distribution to lower values.A flexible defect has obvious effects on knotting probability.The summation of equilibrium distribution probability for nontrivial knotted DNA with different contour length does not change with excitation energy monotonically and has a maximum at an intermediate value of excitation energy around 5 kBT.In the phase space of knot length and gyration radius of knotted DNA,knot length does not anticorrelate with its gyration radius,which is attributed to the flexible defect in the knotted portion,which leads to the release of bending energy and inhibited the competition between entropy and bending energy.
