Spectra and tilings play an important role in analysis and geometry respectively.The relations between spectra and tilings have bafied the mathematicians for a long time.Many conjectures,such as the Fuglede conjecture...Spectra and tilings play an important role in analysis and geometry respectively.The relations between spectra and tilings have bafied the mathematicians for a long time.Many conjectures,such as the Fuglede conjecture,are placed on the establishment of relations between spectra and tilings,although there are no desired results.In the present paper we derive some characteristic properties of spectra and tilings which highlight certain duality properties between them.展开更多
In geometry, there are several challenging problems studying numbers associated to convex bodies. For example, the packing density problem, the kissing number problem, the covering density problem, the packing-coverin...In geometry, there are several challenging problems studying numbers associated to convex bodies. For example, the packing density problem, the kissing number problem, the covering density problem, the packing-covering constant problem, Hadwiger＇s covering conjecture and Borsuk＇s partition conjecture. They are flmdamental and fascinating problems about the same objects. However, up to now, both the methodology and the technique applied to them are essentially different. Therefore, a common foundation for them has been much expected. By treating problems of these types as functionals defined on the spaces of n-dimensional convex bodies, this paper tries to create such a foundation. In particular, supderivatives for these functionals will be studied.展开更多
Let G<sub>1</sub> and G<sub>2</sub> be finite digraphs,both with vertex set V.Suppose that each vertexv of V has nonnegative integers f（v） and g（v） with f（v）≤g（v）,and each arc e of G&l...Let G<sub>1</sub> and G<sub>2</sub> be finite digraphs,both with vertex set V.Suppose that each vertexv of V has nonnegative integers f（v） and g（v） with f（v）≤g（v）,and each arc e of G<sub>4</sub> hasnonnegative integers a<sub>i</sub>（e） and b<sub>i</sub>（e） with a<sub>i</sub>（e）≤b<sub>i</sub>（e）,i=1,2.In this paper we give anecessary and sufficient condition for the existence of k arborescences in G<sub>4</sub> covering each are（?） of G<sub>i</sub> at least a<sub>i</sub>（e） and at most b<sub>i</sub>（e） times,i=1,2,and satisfying the condition that foreach v in Vf（v）≤r<sub>1</sub>（v）=r<sub>2</sub>（v）≤g（v）where r<sub>4</sub>（v） denote the number of the arborescences in G<sub>?</sub> rooted at v.展开更多
Let G be a finite simple graph. A G-design (G-packing design, G-covering design) of λKv, denoted by (v, G,λ)-GD ((v, G, λ)-PD, (v, G,λ)-CD), is a pair (X,B) where X is the vertex set of Kv and B is a collection of...Let G be a finite simple graph. A G-design (G-packing design, G-covering design) of λKv, denoted by (v, G,λ)-GD ((v, G, λ)-PD, (v, G,λ)-CD), is a pair (X,B) where X is the vertex set of Kv and B is a collection of subgraphs of Kv, called blocks, such that each block is isomorphic to G and any two distinct vertices in Kv are joined in exactly (at most, at least) λ blocks of B. A packing (covering) design is said to be maximum (minimum) if no other such packing (covering) design has more (fewer) blocks. In this paper, we determine the existence spectrum for the K2,3-designs of λKv,λ> 1, and construct the maximum packing designs and the minimum covering designs of λKv with K2,3 for any integer λ.展开更多
According to the compound packing problem in ammunition supply system in our army, the non-standard pallet series design model is proposed, and the original problem that can be solved as a set cover problem with a nes...According to the compound packing problem in ammunition supply system in our army, the non-standard pallet series design model is proposed, and the original problem that can be solved as a set cover problem with a nested bin-packing problem, is analyzed, then two heuristic algorithms are applied to solve the problem.展开更多
A maximum （v, G, λ）-PD and a minimum （v, G, λ）-CD axe studied for 2 graphs of 6 vertices and 7 edges. By means of ＂difference method＂ and ＂holey graph design＂, we obtain the result： there exists a （v, Gi,...A maximum （v, G, λ）-PD and a minimum （v, G, λ）-CD axe studied for 2 graphs of 6 vertices and 7 edges. By means of ＂difference method＂ and ＂holey graph design＂, we obtain the result： there exists a （v, Gi, λ）-OPD （OCD） for v ≡ 2, 3, 4, 5, 6 （mod 7）, λ ≥ 1, i = 1, 2.展开更多
Let λKv be the complete multigraph with v vertices and G a finite simple graph. A G-design (G-packing design, G-covering design) of λKv, denoted by (v,G,λ)-GD ((v,G,λ)-PD, (v,G,λ)-CD), is a pair (X,B) where X is ...Let λKv be the complete multigraph with v vertices and G a finite simple graph. A G-design (G-packing design, G-covering design) of λKv, denoted by (v,G,λ)-GD ((v,G,λ)-PD, (v,G,λ)-CD), is a pair (X,B) where X is the vertex set of Kv and B is a collection of subgraphs of Kv, called blocks, such that each block is isomorphic to G and any two distinct vertices in Kv are joined in exactly (at most, at least) λ blocks of B. A packing (covering) design is said to be maximum (minimum) if no other such packing (cov...展开更多
Let S belong to R^2 be the attractor of the iterated function system {f1, f2, f3 } iterating on the unit equilateral triangle So. where fi（x） =λix ＋ bi, i = 1,2, 3, x =（x1, x2）, b1=（0, 0）, b3=（1-λ3 /2,√3...Let S belong to R^2 be the attractor of the iterated function system {f1, f2, f3 } iterating on the unit equilateral triangle So. where fi（x） =λix ＋ bi, i = 1,2, 3, x =（x1, x2）, b1=（0, 0）, b3=（1-λ3 /2,√3/2 （1-λ3）） This paper determines the exact Hausdorff measure, centred covering measure and packing measure of S under some conditions relating to the contraction parameter.展开更多
基金supported by the Key Project of Chinese Ministry of Education (Grant No.108117)National Natural Science Foundation of China (Grant No.10871123)
文摘Spectra and tilings play an important role in analysis and geometry respectively.The relations between spectra and tilings have bafied the mathematicians for a long time.Many conjectures,such as the Fuglede conjecture,are placed on the establishment of relations between spectra and tilings,although there are no desired results.In the present paper we derive some characteristic properties of spectra and tilings which highlight certain duality properties between them.
基金Supported by 973 Programs(Grant Nos.2013CB834201 and 2011CB302401)the National Science Foundation of China(Grant No.11071003)the Chang Jiang Scholars Program of China
文摘In geometry, there are several challenging problems studying numbers associated to convex bodies. For example, the packing density problem, the kissing number problem, the covering density problem, the packing-covering constant problem, Hadwiger＇s covering conjecture and Borsuk＇s partition conjecture. They are flmdamental and fascinating problems about the same objects. However, up to now, both the methodology and the technique applied to them are essentially different. Therefore, a common foundation for them has been much expected. By treating problems of these types as functionals defined on the spaces of n-dimensional convex bodies, this paper tries to create such a foundation. In particular, supderivatives for these functionals will be studied.
基金Work Supported by the exchange program between the Academia Sinica and the Max Planck Society
文摘Let G<sub>1</sub> and G<sub>2</sub> be finite digraphs,both with vertex set V.Suppose that each vertexv of V has nonnegative integers f（v） and g（v） with f（v）≤g（v）,and each arc e of G<sub>4</sub> hasnonnegative integers a<sub>i</sub>（e） and b<sub>i</sub>（e） with a<sub>i</sub>（e）≤b<sub>i</sub>（e）,i=1,2.In this paper we give anecessary and sufficient condition for the existence of k arborescences in G<sub>4</sub> covering each are（?） of G<sub>i</sub> at least a<sub>i</sub>（e） and at most b<sub>i</sub>（e） times,i=1,2,and satisfying the condition that foreach v in Vf（v）≤r<sub>1</sub>（v）=r<sub>2</sub>（v）≤g（v）where r<sub>4</sub>（v） denote the number of the arborescences in G<sub>?</sub> rooted at v.
文摘Let G be a finite simple graph. A G-design (G-packing design, G-covering design) of λKv, denoted by (v, G,λ)-GD ((v, G, λ)-PD, (v, G,λ)-CD), is a pair (X,B) where X is the vertex set of Kv and B is a collection of subgraphs of Kv, called blocks, such that each block is isomorphic to G and any two distinct vertices in Kv are joined in exactly (at most, at least) λ blocks of B. A packing (covering) design is said to be maximum (minimum) if no other such packing (covering) design has more (fewer) blocks. In this paper, we determine the existence spectrum for the K2,3-designs of λKv,λ> 1, and construct the maximum packing designs and the minimum covering designs of λKv with K2,3 for any integer λ.
文摘According to the compound packing problem in ammunition supply system in our army, the non-standard pallet series design model is proposed, and the original problem that can be solved as a set cover problem with a nested bin-packing problem, is analyzed, then two heuristic algorithms are applied to solve the problem.
基金Supported by the National Natural Science Foundation of China (Grant No.10671055)
文摘A maximum （v, G, λ）-PD and a minimum （v, G, λ）-CD axe studied for 2 graphs of 6 vertices and 7 edges. By means of ＂difference method＂ and ＂holey graph design＂, we obtain the result： there exists a （v, Gi, λ）-OPD （OCD） for v ≡ 2, 3, 4, 5, 6 （mod 7）, λ ≥ 1, i = 1, 2.
基金the National Natural Science Foundation of China (No.10671055)
文摘Let λKv be the complete multigraph with v vertices and G a finite simple graph. A G-design (G-packing design, G-covering design) of λKv, denoted by (v,G,λ)-GD ((v,G,λ)-PD, (v,G,λ)-CD), is a pair (X,B) where X is the vertex set of Kv and B is a collection of subgraphs of Kv, called blocks, such that each block is isomorphic to G and any two distinct vertices in Kv are joined in exactly (at most, at least) λ blocks of B. A packing (covering) design is said to be maximum (minimum) if no other such packing (cov...
基金the Foundation of National Natural Science Committee of Chinathe Foundation of the Natural Science of Guangdong Provincethe Foundation of the Advanced Research Center of zhongshan University
文摘Let S belong to R^2 be the attractor of the iterated function system {f1, f2, f3 } iterating on the unit equilateral triangle So. where fi（x） =λix ＋ bi, i = 1,2, 3, x =（x1, x2）, b1=（0, 0）, b3=（1-λ3 /2,√3/2 （1-λ3）） This paper determines the exact Hausdorff measure, centred covering measure and packing measure of S under some conditions relating to the contraction parameter.
