Linear programming(LP)decoding is a classic decoding method for linear block codes,and has attracted recent researches because its potential in joint channel processing.However,for polar codes,LP decoders has long bee...Linear programming(LP)decoding is a classic decoding method for linear block codes,and has attracted recent researches because its potential in joint channel processing.However,for polar codes,LP decoders has long been outperformed by CRCaided successive cancellation list(CA-SCL)decoders.To increase the competitiveness of 5G NR LP polar decoding,it is possible to gain performance improvements by exploiting the cyclic redundancy check(CRC)setup.In this paper,we propose a combined scheme of reduced sparsified factor graph-sparsified CRC(RSFG-SCRC)and augmented generator matrix-CRC(AGM-CRC),for polytope generation in adaptive linear programming(ALP)decoder for 5G polar codes.Augmented generator matrix(AGM)polytope and improved maximum cycle strategy-auxiliary node pairs 4(MCS-ANP-4)algorithm are proposed,to make efficient use of CRC constraints and minimize the constraint size for the decoder.Numerical simulations show that adaptive linear programming decoders with our proposed RSFG-SCRC and AGM-CRC polytopes can achieve significantly better block error rate(BLER)performance than a benchmark CA-SCL-8 decoder especially in harsh low-to-medium SNR regions.展开更多
Minkowski sums are of theoretical interest and have applications in fields related to industrial backgrounds. In this paper we focus on the specific case of summing polytopes as we want to solve the tolerance analysis...Minkowski sums are of theoretical interest and have applications in fields related to industrial backgrounds. In this paper we focus on the specific case of summing polytopes as we want to solve the tolerance analysis problem described in [1]. Our approach is based on the use of linear programming and is solvable in polynomial time. The algorithm we developed can be implemented and parallelized in a very easy way.展开更多
The research deals with the methodology intended to root robust quality indices in the interval control system, the parameters of which are affinely included in the coefficients of a characteristic polynomial. To dete...The research deals with the methodology intended to root robust quality indices in the interval control system, the parameters of which are affinely included in the coefficients of a characteristic polynomial. To determine the root quality indices we propose to depict on the root plane not all edges of the interval parametric polytope(as the edge theorem says), but its particular vertex-edge route. In order to define this route we need to know the angle sequence at which the edge branches depart from any integrated pole on the allocation area. It is revealed that the edge branches can integrate into the route both fully or partially due to intersection with other branches. The conditions which determine the intersection of one-face edge images have been proven. It is shown that the root quality indices can be determined by its ends or by any other internal point depending on a type of edge branch. The conditions which allow determining the edge branch type have been identified. On the basis of these studies we developed the algorithm intended to construct a boundary vertex-edge route on the polytope with the interval parameters of the system. As an illustration of how the algorithm can be implemented, we determined and introduced the root indices reflecting the robust quality of the system used to stabilize the position of an underwater charging station for autonomous unmanned vehicles.展开更多
To study the Schneider＇s projection problem, Lutwak, Yang and Zhang recently introduced a new .affine invariant functional U（P） for convex polytopes in R^n. In the paper, we obtain the analytic expression of the af...To study the Schneider＇s projection problem, Lutwak, Yang and Zhang recently introduced a new .affine invariant functional U（P） for convex polytopes in R^n. In the paper, we obtain the analytic expression of the affine-invariant U（P） defined on a specific subclass of origin-symmetric convex polytopes in Rn and give an application of U（P） to the Lp-Minkowski problem.展开更多
The problem of the robust D-stability analysis for linear systems with parametric uncertainties is addressed. For matrix polytopes, new conditions via the affine parameter-dependent Lyapunov function of uncertain syst...The problem of the robust D-stability analysis for linear systems with parametric uncertainties is addressed. For matrix polytopes, new conditions via the affine parameter-dependent Lyapunov function of uncertain systems are developed with the benefit of the scalar multi-convex function. To be convenient for applications, such conditions are simplified into new linear matrix inequality （LMI） conditions, which can be solved by the powerful LMI toolbox. Numerical examples are provided to indicate that this new approach is less conservative than previous results for Hurwitz stability, Schur stability and D-stability of uncertain systems under certain circumstances.展开更多
Let M be a matroid defined on a finite set E and L?⊂?E?. L is locked in M if??and ?are 2-connected, and . In this paper, we prove that the nontrivial facets of the bases polytope of M are described by the lo...Let M be a matroid defined on a finite set E and L?⊂?E?. L is locked in M if??and ?are 2-connected, and . In this paper, we prove that the nontrivial facets of the bases polytope of M are described by the locked subsets. We deduce that finding the maximum-weight basis of M is a polynomial time problem for matroids with a polynomial number of locked subsets. This class of matroids is closed under 2-sums and contains the class of uniform matroids, the Vámos matroid and all the excluded minors of 2-sums of uniform matroids. We deduce also a matroid oracle for testing uniformity of matroids after one call of this oracle.展开更多
Given two disjoint 3-dimensional convex polytopes P and Q and a straight direction along Which P moves in translation, this paper presents a linear algorithm for determining Whether P collides with Q, and the possible...Given two disjoint 3-dimensional convex polytopes P and Q and a straight direction along Which P moves in translation, this paper presents a linear algorithm for determining Whether P collides with Q, and the possible collision positions on P and Q. This result is achieved by using the hierarchicat representation of polytopes, of which the preprocessing time is linear with space.展开更多
The paper studies reachability problem of autonomous ai^ine systems on n-dimensional polytopes.Our goal is to obtain both the largest positive invariant set in the polytope and the backward reachable set(the attractio...The paper studies reachability problem of autonomous ai^ine systems on n-dimensional polytopes.Our goal is to obtain both the largest positive invariant set in the polytope and the backward reachable set(the attraction domain)of each facet.Special attention is paid to the largest stable invariant affine subspace.After presenting several useful properties of those sets,a partition procedure is given to determine the largest positive invariant set in the polytope and all the attraction domains of facets.展开更多
In this paper,we provide a sufficient condition,in the case of 0<p<1,for the existence of solutions to the general L_(p) Minkowski problem for polytopes.
The logarithmic capacitary Minkowski problem asks for necessary and sufficient conditions on a finite Borel measure on the unit sphere so that it is the logarithmic capacitary measure of a convex body.This article com...The logarithmic capacitary Minkowski problem asks for necessary and sufficient conditions on a finite Borel measure on the unit sphere so that it is the logarithmic capacitary measure of a convex body.This article completely solves the case of discrete measures whose support sets are in general position.展开更多
The recent Polytope ARTMAP（PTAM） suggests that irregular polytopes are more flexible than the predefined category geometries to approximate the borders among the desired output predictions.However,category expansion...The recent Polytope ARTMAP（PTAM） suggests that irregular polytopes are more flexible than the predefined category geometries to approximate the borders among the desired output predictions.However,category expansion and adjustment steps without statistical information make PTAM not robust to noise and category overlap.In order to push the learning problem towards Structural Risk Minimization（SRM）,this paper proposes Hierarchical Polytope ARTMAP （HPTAM） to use a hierarchical structure with different levels,which are determined by the complexity of regions incorporating the input pattern.Besides,overlapping of simplexes from the same desired prediction is designed to reduce category proliferation.Although HPTAM is still inevitably sensible to noisy outliers in the presence of noise,main experimental results show that HPTAM can achieve a balance between representation error and approximation error,which ameliorates the overall generalization capabilities.展开更多
In this paper we study the cubical stuctures and fundamental groups of real toric spaces.We give an explicit presentation of the fundamental group of the real toric space over a simple polytope.Then using this present...In this paper we study the cubical stuctures and fundamental groups of real toric spaces.We give an explicit presentation of the fundamental group of the real toric space over a simple polytope.Then using this presentation,we give a description of the existence of non-degenerate colourings on a simple polytope from a homotopy point of view.展开更多
and uses it imply that application In this paper the author gives a method of constructing characteristic matrices, to determine the Buchstaber invariants of all simple convex 3-polytopes, which each simple convex 3-p...and uses it imply that application In this paper the author gives a method of constructing characteristic matrices, to determine the Buchstaber invariants of all simple convex 3-polytopes, which each simple convex 3-polytope admits a characteristic function. As a further of the method, the author also gives a simple new proof of five-color theorem.展开更多
In a previous work, we described the Minimal Model Program in the family of Q-Gorenstein projective horospherical varieties, by studying certain continuous changes of moment polytopes of polarized horospherical variet...In a previous work, we described the Minimal Model Program in the family of Q-Gorenstein projective horospherical varieties, by studying certain continuous changes of moment polytopes of polarized horospherical varieties. Here, we summarize the results of the previous work and we explain how to generalize them in order to describe the Log Minimal Model Program for pairs（X, Δ） when X is a projective horospherical variety.展开更多
In this paper a triangulation of continuous and arbitrary refinement of grid sizes is proposed for simplicial homotopy algorithms to compute zero points on a polytope P. The proposed algorithm generates a piecewise li...In this paper a triangulation of continuous and arbitrary refinement of grid sizes is proposed for simplicial homotopy algorithms to compute zero points on a polytope P. The proposed algorithm generates a piecewise linear path in P × [1,∞) from any chosen interior point x0 of P on level {1} to a solution of the underlying problem. The path is followed by making linear programming pivot steps in a linear system and replacement steps in the triangnlation.The starting point x0 is left in a direction to one vertex of P. The direction in which x0 leaves depends on the function value at x0 and the polytope P. Moreover, we also give a new equivalent form of the Brouwer fixed point theorem on polytopes. This form has many important applications in mathematical programming and the theory of differential equations.展开更多
We show that the volume of the projection bodyΠ(Z)of an n-dimensional zonotope Z with n+1 generators and of volume 1 is always exactly 2^(n).Moroever,we point out that an upper bound on the volume ofΠ(K)of a central...We show that the volume of the projection bodyΠ(Z)of an n-dimensional zonotope Z with n+1 generators and of volume 1 is always exactly 2^(n).Moroever,we point out that an upper bound on the volume ofΠ(K)of a centrally symmetric n-dimensional convex body of volume 1 is at least 2^(n)(9/8)^([n/3]).展开更多
Game theory can be applied to the air combat decision-making problem of multiple unmanned combat air vehicles(UCAVs).However,it is difficult to have satisfactory decision-making results completely relying on air comba...Game theory can be applied to the air combat decision-making problem of multiple unmanned combat air vehicles(UCAVs).However,it is difficult to have satisfactory decision-making results completely relying on air combat situation information,because there is a lot of time-sensitive information in a complex air combat environment.In this paper,a constraint strategy game approach is developed to generate intelligent decision-making for multiple UCAVs in complex air combat environment with air combat situation information and time-sensitive information.Initially,a constraint strategy game is employed to model attack-defense decision-making problem in complex air combat environment.Then,an algorithm is proposed for solving the constraint strategy game based on linear programming and linear inequality(CSG-LL).Finally,an example is given to illustrate the effectiveness of the proposed approach.展开更多
Based on the linear parameter-varying （LPV） adaptive observer, the robust fault diagnosis for a class of LPV systems with external disturbances is studied. Since the flight control system （FCS） is nonlinear and ti...Based on the linear parameter-varying （LPV） adaptive observer, the robust fault diagnosis for a class of LPV systems with external disturbances is studied. Since the flight control system （FCS） is nonlinear and time-varying, the LPV technique is used for FCS. And then the adaptive fault estimation algorithm based on the LPV adaptive observer is proposed to estimate the fault. To minimize the effect of disturbances on the fault estimation, the H~ robust performance index is introduced to design the LPV adaptive fault diagnosis observer and the fault estimation algorithm. The result shows that the method has good estimation performance and is robust to external disturbances. The design method is presented in terms of linear matrix inequalities （LMIs）. Finally, a helicopter LPV FCS model with the actuator fault is used to illustrate the effectiveness of the proposed method.展开更多
基金supported by China Postdoctoral Science Foundation(No.2020M670469)National Key Research and Development Program of China(No.2019YFB1803303,No.2020YFB1806702).
文摘Linear programming(LP)decoding is a classic decoding method for linear block codes,and has attracted recent researches because its potential in joint channel processing.However,for polar codes,LP decoders has long been outperformed by CRCaided successive cancellation list(CA-SCL)decoders.To increase the competitiveness of 5G NR LP polar decoding,it is possible to gain performance improvements by exploiting the cyclic redundancy check(CRC)setup.In this paper,we propose a combined scheme of reduced sparsified factor graph-sparsified CRC(RSFG-SCRC)and augmented generator matrix-CRC(AGM-CRC),for polytope generation in adaptive linear programming(ALP)decoder for 5G polar codes.Augmented generator matrix(AGM)polytope and improved maximum cycle strategy-auxiliary node pairs 4(MCS-ANP-4)algorithm are proposed,to make efficient use of CRC constraints and minimize the constraint size for the decoder.Numerical simulations show that adaptive linear programming decoders with our proposed RSFG-SCRC and AGM-CRC polytopes can achieve significantly better block error rate(BLER)performance than a benchmark CA-SCL-8 decoder especially in harsh low-to-medium SNR regions.
文摘Minkowski sums are of theoretical interest and have applications in fields related to industrial backgrounds. In this paper we focus on the specific case of summing polytopes as we want to solve the tolerance analysis problem described in [1]. Our approach is based on the use of linear programming and is solvable in polynomial time. The algorithm we developed can be implemented and parallelized in a very easy way.
基金supported by the Ministry of Education and Science of the Russian Federation (No. 2.3649. 2017/PCh)
文摘The research deals with the methodology intended to root robust quality indices in the interval control system, the parameters of which are affinely included in the coefficients of a characteristic polynomial. To determine the root quality indices we propose to depict on the root plane not all edges of the interval parametric polytope(as the edge theorem says), but its particular vertex-edge route. In order to define this route we need to know the angle sequence at which the edge branches depart from any integrated pole on the allocation area. It is revealed that the edge branches can integrate into the route both fully or partially due to intersection with other branches. The conditions which determine the intersection of one-face edge images have been proven. It is shown that the root quality indices can be determined by its ends or by any other internal point depending on a type of edge branch. The conditions which allow determining the edge branch type have been identified. On the basis of these studies we developed the algorithm intended to construct a boundary vertex-edge route on the polytope with the interval parameters of the system. As an illustration of how the algorithm can be implemented, we determined and introduced the root indices reflecting the robust quality of the system used to stabilize the position of an underwater charging station for autonomous unmanned vehicles.
基金Project supported by the National Natural Science Foundation of China (No.10671119)
文摘To study the Schneider＇s projection problem, Lutwak, Yang and Zhang recently introduced a new .affine invariant functional U（P） for convex polytopes in R^n. In the paper, we obtain the analytic expression of the affine-invariant U（P） defined on a specific subclass of origin-symmetric convex polytopes in Rn and give an application of U（P） to the Lp-Minkowski problem.
基金supported by the National Natural Science Foundation of China (6090405161021002)
文摘The problem of the robust D-stability analysis for linear systems with parametric uncertainties is addressed. For matrix polytopes, new conditions via the affine parameter-dependent Lyapunov function of uncertain systems are developed with the benefit of the scalar multi-convex function. To be convenient for applications, such conditions are simplified into new linear matrix inequality （LMI） conditions, which can be solved by the powerful LMI toolbox. Numerical examples are provided to indicate that this new approach is less conservative than previous results for Hurwitz stability, Schur stability and D-stability of uncertain systems under certain circumstances.
文摘Let M be a matroid defined on a finite set E and L?⊂?E?. L is locked in M if??and ?are 2-connected, and . In this paper, we prove that the nontrivial facets of the bases polytope of M are described by the locked subsets. We deduce that finding the maximum-weight basis of M is a polynomial time problem for matroids with a polynomial number of locked subsets. This class of matroids is closed under 2-sums and contains the class of uniform matroids, the Vámos matroid and all the excluded minors of 2-sums of uniform matroids. We deduce also a matroid oracle for testing uniformity of matroids after one call of this oracle.
文摘Given two disjoint 3-dimensional convex polytopes P and Q and a straight direction along Which P moves in translation, this paper presents a linear algorithm for determining Whether P collides with Q, and the possible collision positions on P and Q. This result is achieved by using the hierarchicat representation of polytopes, of which the preprocessing time is linear with space.
基金Supported by National Natural Science Foundation of China(60504024)Zhejiang Provincial Natural Science Foundation of China(Y106010)the Specialized Research Fund for the Doctoral Program of Higher Education of China(20060335022)
文摘The paper studies reachability problem of autonomous ai^ine systems on n-dimensional polytopes.Our goal is to obtain both the largest positive invariant set in the polytope and the backward reachable set(the attraction domain)of each facet.Special attention is paid to the largest stable invariant affine subspace.After presenting several useful properties of those sets,a partition procedure is given to determine the largest positive invariant set in the polytope and all the attraction domains of facets.
基金Supported by the National Natural Science Foundation of China(Grant Nos.12371060,12401069)the Scientific Research Startup Fund for PhD Recipients(Grant No.BS25006)+1 种基金the Recruitment Program for Young Professionals of Chinathe Fundamental Research Funds for the Central Universities(Grant No.GK202307001)。
文摘In this paper,we provide a sufficient condition,in the case of 0<p<1,for the existence of solutions to the general L_(p) Minkowski problem for polytopes.
文摘The logarithmic capacitary Minkowski problem asks for necessary and sufficient conditions on a finite Borel measure on the unit sphere so that it is the logarithmic capacitary measure of a convex body.This article completely solves the case of discrete measures whose support sets are in general position.
基金Supported by the National Basic Research 973 Program of China under Grant No.2007CB311006.
文摘The recent Polytope ARTMAP（PTAM） suggests that irregular polytopes are more flexible than the predefined category geometries to approximate the borders among the desired output predictions.However,category expansion and adjustment steps without statistical information make PTAM not robust to noise and category overlap.In order to push the learning problem towards Structural Risk Minimization（SRM）,this paper proposes Hierarchical Polytope ARTMAP （HPTAM） to use a hierarchical structure with different levels,which are determined by the complexity of regions incorporating the input pattern.Besides,overlapping of simplexes from the same desired prediction is designed to reduce category proliferation.Although HPTAM is still inevitably sensible to noisy outliers in the presence of noise,main experimental results show that HPTAM can achieve a balance between representation error and approximation error,which ameliorates the overall generalization capabilities.
基金Partially supported by the NSFC(Grant No.11971112)the China Scholarship Council(Grant No.202106100095)。
文摘In this paper we study the cubical stuctures and fundamental groups of real toric spaces.We give an explicit presentation of the fundamental group of the real toric space over a simple polytope.Then using this presentation,we give a description of the existence of non-degenerate colourings on a simple polytope from a homotopy point of view.
基金the National Natural Science Foundation of China(No.10931005)the Shang-hai National Natural Science Foundation(No.10ZR1403600)the Research Fund for the DoctoralProgram of Higher Education of China(No.20100071110001)
文摘and uses it imply that application In this paper the author gives a method of constructing characteristic matrices, to determine the Buchstaber invariants of all simple convex 3-polytopes, which each simple convex 3-polytope admits a characteristic function. As a further of the method, the author also gives a simple new proof of five-color theorem.
文摘In a previous work, we described the Minimal Model Program in the family of Q-Gorenstein projective horospherical varieties, by studying certain continuous changes of moment polytopes of polarized horospherical varieties. Here, we summarize the results of the previous work and we explain how to generalize them in order to describe the Log Minimal Model Program for pairs（X, Δ） when X is a projective horospherical variety.
文摘In this paper a triangulation of continuous and arbitrary refinement of grid sizes is proposed for simplicial homotopy algorithms to compute zero points on a polytope P. The proposed algorithm generates a piecewise linear path in P × [1,∞) from any chosen interior point x0 of P on level {1} to a solution of the underlying problem. The path is followed by making linear programming pivot steps in a linear system and replacement steps in the triangnlation.The starting point x0 is left in a direction to one vertex of P. The direction in which x0 leaves depends on the function value at x0 and the polytope P. Moreover, we also give a new equivalent form of the Brouwer fixed point theorem on polytopes. This form has many important applications in mathematical programming and the theory of differential equations.
文摘We show that the volume of the projection bodyΠ(Z)of an n-dimensional zonotope Z with n+1 generators and of volume 1 is always exactly 2^(n).Moroever,we point out that an upper bound on the volume ofΠ(K)of a centrally symmetric n-dimensional convex body of volume 1 is at least 2^(n)(9/8)^([n/3]).
基金supported by Major Projects for Science and Technology Innovation 2030(Grant No.2018AA0100800)Equipment Pre-research Foundation of Laboratory(Grant No.61425040104)in part by Jiangsu Province“333”project under Grant BRA2019051.
文摘Game theory can be applied to the air combat decision-making problem of multiple unmanned combat air vehicles(UCAVs).However,it is difficult to have satisfactory decision-making results completely relying on air combat situation information,because there is a lot of time-sensitive information in a complex air combat environment.In this paper,a constraint strategy game approach is developed to generate intelligent decision-making for multiple UCAVs in complex air combat environment with air combat situation information and time-sensitive information.Initially,a constraint strategy game is employed to model attack-defense decision-making problem in complex air combat environment.Then,an algorithm is proposed for solving the constraint strategy game based on linear programming and linear inequality(CSG-LL).Finally,an example is given to illustrate the effectiveness of the proposed approach.
基金partly supported by the National Natural Science Foundation of China(12061006)the Science and Technology Project of Education Department of Jiangxi Province(GJJ180414)+1 种基金East China University of Technology Research Foundation for Advanced Talents(DHBK2018050)The second author is supported by the National Natural Science Foundation of China(71762001)。
文摘In this paper,we demonstrate the existence part of the discrete Orlicz-Minkowski problem for p-capacity when 1<p<2.
基金Supported by the National Natural Science Foundation of China(60811120024)Aeronautical Scienceand Technology Innovation Foundation of China(08C52001)~~
文摘Based on the linear parameter-varying （LPV） adaptive observer, the robust fault diagnosis for a class of LPV systems with external disturbances is studied. Since the flight control system （FCS） is nonlinear and time-varying, the LPV technique is used for FCS. And then the adaptive fault estimation algorithm based on the LPV adaptive observer is proposed to estimate the fault. To minimize the effect of disturbances on the fault estimation, the H~ robust performance index is introduced to design the LPV adaptive fault diagnosis observer and the fault estimation algorithm. The result shows that the method has good estimation performance and is robust to external disturbances. The design method is presented in terms of linear matrix inequalities （LMIs）. Finally, a helicopter LPV FCS model with the actuator fault is used to illustrate the effectiveness of the proposed method.
