关于图的带宽的一些定理

引言 设G是有N个顶点的图,V(G)是G的全体顶点的集合,称任一个1—1对应的函数f:V(G)→{1,2,…,N}为G(或V(G))上的一个标号,记 B(f)=max{f(u)-f(v):u与v是G上相邻顶点},称B(f)为标号f的带宽.又记 B(G)=min{B(f):f是V(G)上的标号},称 B(G)为图G的带宽.若f是V(G)上的一个标号且B(f)=B(G),则称f为V(G)     

关于奇强协调图的一些结果

对于一个(p,q)-图G,如果存在一个单射f:V(G)→{0,1,…,2q-1},使得边标号集合{f(uv)|uv∈E(G)}={1,3,5,…,2q-1},其中边标号为f(uv)=f(u)+f(v),那么称G是奇强协调图,并称f是G的一个奇强协调标号.通过研究若干奇强协调图,得出一些奇强协调图的性质.     

优美图的一些性质

对于一个(p,q)-图G,如果存在一个单射.f:V(G)→{0,1,…,q},使得边标号集合{f(uv)| uv∈E(G)}={1,2,…,q},其中边标号为f(uv)=|f(u)-f(v)|,那么称G是优美图,并称.f是G的一个优美标号.通过研究若干优美图,得出一些优美图的性质.     

两类图的L(2.1)-标号数

图G的一个L(2.1)-标号是从顶点集V(G)到非负整数的一个函数f,使得若d(u,v)=1时,有|f(u)-f(v)|≥2;若d(u,v)=2时,有|f(u)-f(v)|≥1.图G的L(2.1)-标号数λ(G)是G的所有L(2.1)-标号下的跨度max{f(v):v∈V(G)}的最小数.图Fn+1*为扇图的路上每个顶点增加一个悬挂边得到的图.图Hn为轮图的圈上每个顶点增加一个悬挂边得到的图.本文确定了图Fn+1*与Hn的L(2.1)-标号数.     

几类有趣图的奇优美性和奇强协调性

对于简单图G=〈V,E〉,如果存在一个映射f:V(G)→{0,1,2,…,2|E|-1}满足:1)对任意的u,v∈V,若u≠v,则f(u)≠f(v);2)max{f(v)|v∈V}=2|E|-1;3)对任意的e_1,e_2∈E,若e_1≠e_2,则g(e_1)≠g(e_2),此处g(e)=|f(u)+f(v)|,e=uv;4)|g(e)|e∈E}={1,3,5,…,2|E|-1},则称G为奇优美图,f称为G的奇优美标号.设G=〈V,E〉是一个无向简单图.如果存在一个映射f:V(G)→{0,1,2,…,2|E|-1},满足:1)f是单射;2)■uv∈E(G),令f(uv)=f(u)+f(v),有{f(uv)|uv∈E(G)}={1,3,5,…,2|E|-1},则称G是奇强协调图,f称为G的.奇强协调标号或奇强协调值.给出了链图、升降梯等几类有趣图的奇优美标号和奇强协调标号.     

竖梯的局部替换图的L(d,1,1)-标号

图的L(d,1,1)-标号定义为顶点集V(G)到非负整数集的映射f,且当d(u,v)=1时,均有|f(u)-f(v)|≥d,当d(u,v)=2,3时,均有|f(u)-f(v)|≥1.不妨设0为最小标号,则称图G的所有L(d,1,1)-标号中的最大跨度max{f(v):v∈V(G)}的最小数为图的L(d,1,1)-标号数,记为λ_d(G).基本给出了竖梯的局部替换图的L(d,1,1)-标号数的确切值或界.     

Pkn(k≡2(mod 3))的邻点可区别的强全染色

对简单图G(V,E),V(Gk)=V(G),E(Gk)＝E(G)U{uv|d(u,v)=k},称Gk为G的k次方图,其中d(u,v)表示u,v在G中的距离.设f为用k色时G的正常全染色法,对 uv∈E(G),满足C(u)≠C(v),其中C(u)＝{f(u)}U{f(v)|uv∈E(G)}U{f(uv)|uv∈E(G)},则称f为G的k邻点可区别的强全染色法,简记作k-ASVDTC,且称Xast(G)＝min{k|k-ASVDTC ofG}为G的邻点可区别的强全色数.本文得到了k≡2(mod 3)时的Xast(Pkn),其中Pn为n阶路.     

直径为4的树的奇强协调性

对简单图G=〈V,E〉,如果存在一个映射f:V→{0,1,2,…,2 E-1}满足1)对任意的u,v∈V,若u≠v,则f(u)≠f(v);2)对任意的e1,e2∈E,若e1≠e2,则g(e1)≠g(e2),此处g(e)=f(u)+f(v),e=uv;3){g(e)e∈E}={1,3,5,…,2 E-1},则称G为奇强协调图,f称为G的奇强协调标号.给出了直径为4的树的奇强协调标号.     

点接手镯图的L(1,1,1)-标号

图的L(1,1,1)-标号定义为顶点集V(G)到非负整数集的映射f,且当d(u,v)=1,2,3时,均有|f(u)-f(v)|≥1.不妨设0为最小标号,则称图G的所有L(1,1,1)-标号中的最大跨度f(v)的最小数为图的L(1,1,1)-标号数,记为λ(G).基本给出了点接手镯图的L(1,1,1)-标号数的确切值.     

度为奇数的正则图的上负全控制数

f:v(G)→{一1,0,1}称为图G的负全控制函数,如果对任意点V∈V,均有f[v]≥1,其中 f[v]= ∑,f(u).如果对每个点v∈V,不存在负全控制函数g:V(G)→{-l,0,1),g≠f,满u∈N(v)足g(v)≤f(v),则称f是-个极小负全控制函数.图的上负全控制数F-t(G)=max{w(f)|f,是G的极小负全控制函数},其中w(f)=∑/v∈V(G)f(v).本文研究正则图的上负全控制数,证明了:令G是-个v∈V(G)n阶r-正则图.若r为奇数,则Γt-(G)＜＝r2 1/r2 2r-1n.     

Excluded–Minor Characterizations of Antimatroids arisen from Posets and Graph Searches

An antimatroid is a family of sets such that it contains an empty set, and it is accessible and closed under union of sets. An antimatroid is a ’dual’ or ‘antipodal’ concept of matroid.We shall show that an antimatroid is derived from shelling of a poset if and only if it does not contain a minor isomorphic to S₇ where S₇ is the smallest semimodular lattice that is not modular (See Fig. 1). It is also shown that an antimatroid is a node-search antimatroid of a digraph if and only if it does not contain a minor isomorphic to D₅ where D₅ is a lattice consisting of five elements Ø {x},{y}, {x, y} and {x, y, z}. Furthermore, an antimatroid is shown to be a node-search antimatroid of an undirected graph if and only if it does not contain D₅ nor S₁₀ as a minor: S₁₀ is a locally free lattice consisting of ten elements shown in Fig. 2.     

Erratum Observer design for nonlinear systems using linear approximations

The publisher regrets that in the above article published inIMA Journal of Mathematical Control and Information, Volume20 Number 3, September 2003, pp. 359–370, Fig. 2 wasprinted three times and Fig. 3 and Fig. 4 were omitted. Figures3 and 4 are now reproduced correctly, on the following pages.     

General stability of composite panels reinforced with flexible rods taking account of the side boundary conditions

Reinforced panels are the basic load-bearing elements of various structures. Optimization of massive structures requires consideration of deformation of the panel cross-sections. This is particularly important in determining the bearing strength at buckling. The load scheme, conditions for fixation of the panel cross-section, and bend-torsional stiffness taking account of the deformation of the rod cross-section affect the buckling load in real structures. The stress distribution prior to buckling must be known to solve the buckling problem properly. The stress in the panel is proportional to the active load. The stress distribution is assumed to be known according to our previous method [1]. The load scheme and panel dimensions are shown in Fig. 1. The stress distribution in the panel prior to buckling can be found using Eqs. (1)-(3). A view of the cross-section is given in Fig. 1. The displacements in the panel at buckling for the boundary area are found using Eqs. (4)-(6), while the stresses in the skin and stiffness are found using Eq. (7). Roots k₁ and k₂ are those of the characteristic equation and is a dimensionless coordinate. The problem was solved using variational theory. The potential energy is given by Eqs. (8) and (9) by orihogonalization of Eqs. (5). The basic equations are converted to Eqs. (10) by evaluation of the components in Eqs. (8) and (9). Its calculation (11) gives the compression load. Optimization of parameter gives the critical strength P₁ = 6.93 kN (without taking account of the boundary area) and P₂ = 5.31 kN (taking account of the boundary area).Translated from Mekhanika Kompozitnikh Materialov, Vol. 30, No. 4, pp. 540–546, July–August, 1994.     

Large deflections of eccentrically loaded arches

Zusammenfassung Auf Grund der Hypothesen von Ebenbleiben und Normalität der Querschnitte werden die Differentialgleichungen der nichtlinearen Theorie der Bogenträger abgeleitet und im Falle des schlanken, durch Einzellasten belasteten Kreisbogenträgers mit undehnbarer Mittellinie auf die Form der Pendelgleichung gebracht. Diese Gleichung wird dann benutzt, um die grossen Durchbiegungen und die Spannungsresultierenden eines Zweigelenkkreisbogens, der durch eine lotrechte exzentrische Einzellast belastet wird, zu berechnen. In der Nähe der kritischen Last bewirken kleine Exzentrizitäten bedeutende Grössenänderungen der Spannungsresultierenden und der Durchbiegungen.

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Notation A cross-sectional area of curved beam - a radius of centroidal circle - E modulus of elasticity - e eccentricity of the load (Fig. 2) - F an arbitrary function - H horizontal component of the internal forceR acting on a cross section of the arch rib (Fig. 2) - h _P horizontal displacement of the loadP (Fig. 2) - I moment of inertia of the cross-sectional area - k ² =4p ²/(1+4p ² sin²₀) - L span (distance between supports),L=2a sin - M internal bending couple (Figs. 1 and 2) - N internal normal tensile force (Figs. 1 and 2) - n distributed tangential load (Fig. 1) - P downward point load (Fig. 2) - p ² –R a ² /E I - Q internal shearing force (Figs. 1 and 2) - q distributed normal load (Fig. 1) - R internal resultant force (Fig. 2);R ²=H ²+V ²=N ²+Q ² - radius of curvature of the undeformed centroidal curve - s length along the unextended centroidal curve measured from the left support - length along the unextended centroidal curve measured from the right support - u tangential displacement component of the centroidal curve (Fig. 1) - V vertical component ofR (Fig. 2) - v _P vertical displacement of the loadP (Fig. 2) - w normal displacement component (Fig. 1) - x, y rectangular coordinates of the deformed left portion of the centroidal curve (Fig. 2) - Z - z normal distance (positive inward) from centroidal curve (Fig. 1) - half subtending angle of the arch (Fig. 2) - angle of rotation of the centroidal curve (Fig. 1) - extensional strain of the centroidal curve - ^z extensional strain of the linez=constant - y cos–x sin - angle between the tangent to the formed left portion of the centroidal curve and the horizontal (Fig. 2) - (u–w)/r, whereu=du/dø - angle betweenH andR - x cos+y sin - normal stress along the centroidal curve - ^z normal stress along the linez=constant - angle measured from the radius at the left support of the undeformed arch - (–)/2 (Fig. 2) - (+u)/r, where =d/dø A bar over a letter indicates that the entity pertains to the right portion of the arch. Asterisk indicates the deformed configuration. Primes indicate derivatives with respect to ø.     

A simple pumping mechanism in a valveless tube

Summary A torus consisting of an elastic and a rigid piece of tube is considered (Fig. 1). Surprisingly high velocities are generated by suddenly closing the tube in the vicinity of a junction between the elastic and the rigid part and opening again a fraction of a second later. A simple analysis shows (Eqns. 9, 18 and 19) that a velocity maximum of order is reached at a time of order after closing the tube. The lengthsb, c andl are defined in Fig. 1 anda ₀ is the wave speed in the elastic part of the torus.

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Zusammenfassung Es wird ein Torus betrachtet, der sich aus einem starren und einem elastischen Rohrstück zusammensetzt. Erstaunlich hohe Geschwindigkeiten werden erzeugt, wenn man das elastische Rohr in der Nähe der Verbindung zum starren Rohr plötzlich schliesst und einen Bruchteil einer Sekunde später wieder öffnet. Einfache Ueberlegungen (Gleichungen 9, 18 und 19) zeigen ein Geschwindigkeitsmaximum der Grössenordnung das ungefähr nach dem Schliessen auftritt. Die Längenb, c undl sind in Fig. 1 definiert unda ₀ ist die Signalgeschwindigkeit im elastischen Rohrteil.

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Dedicated to Professor Dr. J. Ackeret, on the occasion of his 80th birthday     

Operator Ideals,Orthonormal Systems and Lacunary Sets

Given an orthonormal system B in some L²(u) we consider the operator ideals II_B and T_B of B-summing and B-type operators and some related ideals. We characterize by certain weak compactness properties when II_B is equal to the operator ideal II₂ of 2-summing operators. In lose that B consists of characters of a compact abelian group we characterize when II_B coincides with the operator ideal II_γ of Gauss-summing operators and when T_B coincides with the operator ideal II_p of type-2 operators. Moreover, we give a necessary and sufficient condition for Fig to contain the operator ideal II_p of p-summing operators (2 < p < ∞) and for T_B to contain the operator ideal Γ_p of p - factorable operators.     

Quasi-Weyl asymptotics of the spectrum of the vector Dirichlet problem

In a space of vector functions, we consider the spectral problem

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, where

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, and the a _αjk and p _jk are constants, x ∈ Ω, and Ω is a bounded open set. The boundary conditions correspond to the Dirichlet problem. Let N _±(μ) be the positive and negative spectral counting functions. We establish the asymptotics N _±(μ) ～ (mes_mΩ)φ_±(μ) as μ → +0. The functions φ_±(μ) are independent of Ω. In the nonelliptic case, these asymptotics are in general different from the classical (Weyl) asymptotics.

     

Straight-line path method and the eikonal problem

Conclusions Thus, the study of the class of ladder diagrams in the scalar model shows that the eikonal formula corresponds to our allowing in the asymptotic behavior for the -paths that coincide with nucleon lines. In this case, the leading particle, which carries the large momentum, is a nucleon and it does not change its species in the virtual process. The noneikonal contributions to the scattering amplitude are due to processes in which the species of the leading particle changes, i.e., to a transfer of momentum from nucleons to mesons and vice versa. There then arises the important question of the role of twisted graphs corresponding to the original ladder graph with replacement of the final momenta q₁q₂ (compare Fig. 1 and formula (1.2)). The possibility of transferring a large momentum to a meson means that the contribution to the asymptotic behavior of the scattering amplitude may dominate over the eikonal contribution in the same order in the coupling constant. For example, in the fourth order, the twisted graph (see Fig. 16) has the asymptotic behavior 1n s/s.Note that whereas the orthodox eikonal formula corresponds to scattering on a Yukawa quasipotential due to one-meson exchange, allowance for the graph in Fig. 16 leads to the appearance of a correction to the quasipotential of non-Yukawa type. The correction we have found corresponds to the exchange of a nucleon-antinucleon pair and has effective range h/2m, and behaves at short distances like 1n r/r.The example pointed out here demonstrates the importance of the study of the successive corrections to the effective quasipotential at high energies and speaks in favour of the quasipotential in quantum field theory.Joint Institute for Nuclear Research, Dubna. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 18, No. 2, pp. 147–160, February, 1974.     

Experimental observations of the failure of boron-aluminum in the vicinity of a peripherally loaded hole

Conclusions The presented experimental data show that bolted joints of metal composites are fairly effective. The required dimensions can be estimated on the. basis of experiments (and subsequently also of calculation) by the following procedure. First we determine by testing specimens with fairly smalll (point C in Fig. 7b). We point out that may depend on the diameter of the hole. Second, by testing specimens with d/w < (d/w)_A and fairly largel we determine the position of point B (see Fig. 7a, b), point M yields the smallest permissible value ofl_°. Third, we determine the position of point A by testing specimens with reduced width w. We point out that a preliminary estimate of the position of point A can be made according to the position of point A.Translated from Mekhanika Kompozitnykh Materialov, No. 5, pp. 877–881, September–October, 1984.The authors express their gratitude to A. A. Khvostunkov and V. N. Il'in for their assistance in carrying out the experiments.