Graceful Signed Graphs

A (p, q)-sigraph S is an ordered pair (G, s) where G = (V, E) is a (p, q)-graph and s is a function which assigns to each edge of G a positive or a negative sign. Let the sets E ⁺ and E ^– consist of m positive and n negative edges of G, respectively, where m + n = q. Given positive integers k and d, S is said to be (k, d)-graceful if the vertices of G can be labeled with distinct integers from the set {0, 1, ..., k + (q – 1)d such that when each edge uv of G is assigned the product of its sign and the absolute difference of the integers assigned to u and v the edges in E ⁺ and E ^– are labeled k, k + d, k + 2d, ..., k + (m – 1)d and –k, – (k + d), – (k + 2d), ..., – (k + (n – 1)d), respectively.In this paper, we report results of our preliminary investigation on the above new notion, which indeed generalises the well-known concept of (k, d)-graceful graphs due to B. D. Acharya and S. M. Hegde.     

On super edge-antimagic total labelings of

A graph G of order p and size q is called (a,d)-edge-antimagic total if there exists a bijective function f:V(G)E(G)→{1,2,…,p+q} such that the edge-weights w(uv)=f(u)+f(v)+f(uv), uvE(G), form an arithmetic sequence with first term a and common difference d. The graph G is said to be super (a,d)-edge-antimagic total if the vertex labels are 1,2,…,p. In this paper we study super (a,d)-edge-antimagic properties of mK_n, that is, of the graph formed by the disjoint union of m copies of K_n.     

Some complete metrics on spaces of fuzzy subsets

The classes of the L_p,∞- and L_p-metrics play an important role to develop a probability theory in fuzzy sample spaces. All of these metrics are known to be separable, but not complete. The classes are closely related as for each L_p,∞-metric there exists some L_p-metric which induces the same topology. This paper deals with the completion of the L_p,∞- and L_p-metrics. We can also show that the relationship between the classes of L_p,∞- and L_p-metrics still holds for the obtained respective classes of their completions.     

Conditions for -graphic sequences to be potentially -graphic

An r-graph is a loopless undirected graph in which no two vertices are joined by more than r edges. An r-complete graph on m+1 vertices, denoted by , is an r-graph on m+1 vertices in which each pair of vertices is joined by exactly r edges. A non-increasing sequence π=(d₁,d₂,…,d_n) of nonnegative integers is said to be r-graphic if it is realizable by an r-graph on n vertices. An r-graphic sequence π is said to be potentially -graphic if it has a realization containing as a subgraph. In this paper, some conditions for r-graphic sequences to be potentially -graphic are given. These are generalizations from 1-graphs to r-graphs of four theorems due to Rao [A.R. Rao, The clique number of a graph with given degree sequence, in: A.R. Rao (Ed.), Proc. Symposium on Graph Theory, in: I.S.I. Lecture Notes Series, vol. 4, MacMillan and Co. India Ltd., (1979), 251–267; A.R. Rao, An Erdös-Gallai type result on the clique number of a realization of a degree sequence (unpublished)] and Kézdy and Lehel [A.E. Kézdy, J. Lehel, Degree sequences of graphs with prescribed clique size, in: Y. Alavi et al., (Eds.), in: Combinatorics, Graph Theory, and Algorithms, vol. 2, New Issues Press, Kalamazoo Michigan, 1999, 535–544].     

Characterizations of spaces in the unit ball of

Let B denote the unit ball of . For 0<p<∞, the holomorphic function spaces Q_p and Q_p,0 on the unit ball of are defined as

and

In this paper, we give some derivative-free, mixture and oscillation characterizations for Q_p and Q_p,0 spaces in the unit ball of .     

Little q-Legendre Polynomials and Irrationality of Certain Lambert Series

Certain q-analogs h _p(1) of the harmonic series, with p = 1/q an integer greater than one, were shown to be irrational by Erds (J. Indiana Math. Soc. 12, 1948, 63–66). In 1991–1992 Peter Borwein (J. Number Theory 37, 1991, 253–259; Proc. Cambridge Philos. Soc. 112, 1992, 141–146) used Padé approximation and complex analysis to prove the irrationality of these q-harmonic series and of q-analogs ln _p (2) of the natural logarithm of 2. Recently Amdeberhan and Zeilberger (Adv. Appl. Math. 20, 1998, 275–283) used the qEKHAD symbolic package to find q-WZ pairs that provide a proof of irrationality similar to Apéry's proof of irrationality of (2) and (3). They also obtain an upper bound for the measure of irrationality, but better upper bounds were earlier given by Bundschuh and Väänänen (Compositio Math. 91, 1994, 175–199) and recently also by Matala-aho and Väänänen (Bull. Australian Math. Soc. 58, 1998, 15–31) (for ln _p (2)). In this paper we show how one can obtain rational approximants for h _p(1) and ln _p (2) (and many other similar quantities) by Padé approximation using little q-Legendre polynomials and we show that properties of these orthogonal polynomials indeed prove the irrationality, with an upper bound of the measure of irrationality which is as sharp as the upper bound given by Bundschuh and Väänänen for h _p(1) and a better upper bound as the one given by Matala-aho and Väänänen for ln _p (2).     

On Thompson's Theorem

Let p be a prime divisor of the order of a finite group G. Thompson (1970, J. Algebra14, 129–134) has proved the following remarkable result: a finite group G is p-nilpotent if the degrees of all its nonlinear irreducible characters are divisible by p (in fact, in that case G is solvable). In this note, we prove that a group G, having only one nonlinear irreducible character of p′-degree is a cyclic extension of Thompson's group. This result is a consequence of the following theorem: A nonabelian simple group possesses two nonlinear irreducible characters χ₁ and χ₂ of distinct degrees such that p does not divide χ₁(1)χ₂(1) (here p is arbitrary but fixed). Our proof depends on the classification of finite simple groups. Some properties of solvable groups possessing exactly two nonlinear irreducible characters of p′-degree are proved. Some open questions are posed.     

The asymptotic behavior of quadratic forms in -mixing random variables

Let {Z_i,i≥1} be a linear process defined by with {d_j,j≥0} being a regular varying sequence of real numbers and {ξ_t,−∞<t<∞} being a sequence of -mixing random variables. The present paper studies the asymptotic behavior of the quadratic form under some mild assumptions on d_j and ξ_t. Meanwhile, the similar results of α-mixing random variables are presented.     

A Variation of an Extremal Theorem Due to Woodall

We consider a variation of an extremal theorem due to Woodall [12, or 1, Chapter 3] as follows: Determine the smallest even integer (₃C₁,n), such that every n-term graphic sequence = (d₁, d₂,..., d_n) with term sum () = d₁ + d₂ + ... + d_n (₃C₁,n) has a realization G containing a cycle of length r for each r = 3,4,...,l. In this paper, the values of (₃C_l,n) are determined for l = 2m – 1,n 3m – 4 and for l = 2m,n 5m – 7, where m 4.AMS Mathematics subject classification (1991) 05C35Project supported by the National Natural Science Foundation of China (Grant No. 19971086) and the Doctoral Program Foundation of National Education Department of China     

Superapproximation and Commutator Properties of Discrete Orthogonal Projections for Continuous Splines

This paper builds upon the L_p-stability results for discrete orthogonal projections on the spaces S_h of continuous splines of order r obtained by R. D. Grigorieff and I. H. Sloan in (1998, Bull. Austral. Math. Soc.58, 307–332). Properties of such projections were proved with a minimum of assumptions on the mesh and on the quadrature rule defining the discrete inner product. The present results, which include superapproximation and commutator properties, are similar to those derived by I. H. Sloan and W. Wendland (1999, J. Approx. Theory97, 254–281) for smoothest splines on uniform meshes. They are expected to have applications (as in I. H. Sloan and W. Wendland, Numer. Math. (1999, 83, 497–533)) to qualocation methods for non-constant-coefficient boundary integral equations, as well as to the wide range of other numerical methods in which quadrature is used to evaluate L₂-inner products. As a first application, we consider the most basic variable-coefficient boundary integral equation, in which the constant-coefficient operator is the identity. The results are also extended to the case of periodic boundary conditions, in order to allow appplication to boundary integral equations on closed curves.     

A central limit theorem for generalized multilinear forms

Let X₁, …, X_n be independent random variables and define for each finite subset I {1, …, n} the σ-algebra = σ{X_i : i ε I}. In this paper -measurable random variables W_I are considered, subject to the centering condition E(W_I ) = 0 a.s. unless I J. A central limit theorem is proven for d-homogeneous sums W(n) = Σ_{I = d}W_I, with var W(n) = 1, where the summation extends over all (_n^d) subsets I {1, …, n} of size I = d, under the condition that the normed fourth moment of W(n) tends to 3. Under some extra conditions the condition is also necessary.     

Summability of orthogonal expansions of several variables

Summability of spherical h-harmonic expansions with respect to the weight function ∏_j=1^d |x_j|^2κ_j (κ_j0) on the unit sphere S^d−1 is studied. The main result characterizes the critical index of summability of the Cesàro (C,δ) means of the h-harmonic expansion; it is proved that the (C,δ) means of any continuous function converge uniformly in the norm of C(S^d−1) if and only if δ>(d−2)/2+∑_j=1^d κ_j−min_1jd κ_j. Moreover, it is shown that for each point not on the great circles defined by the intersection of the coordinate planes and S^d−1, the (C,δ) means of the h-harmonic expansion of a continuous function f converges pointwisely to f if δ>(d−2)/2. Similar results are established for the orthogonal expansions with respect to the weight functions ∏_j=1^d |x_j|^2κ_j(1−|x|²)^μ−1/2 on the unit ball B^d and ∏_j=1^d x_j^κ_j−1/2(1−|x|₁)^μ−1/2 on the simplex T^d. As a related result, the Cesàro summability of the generalized Gegenbauer expansions associated to the weight function |t|^2μ(1−t²)^λ−1/2 on [−1,1] is studied, which is of interest in itself.     

Comparison Theorems on Convex Hypersurfaces in Hadamard Manifolds

In a Hadamard manifold with sectional curvaturebounded from below by –k ₂ ², we give sharp upper estimates for the difference circumradius minus inradiusof a compact k ₂-convex domain, and we getalso estimates for the quotient (Total d-mean curvature)/Area of a convex domain.     

A new extension theorem for linear codes

For an [n,k,d]_q code with k3, gcd(d,q)=1, the diversity of is defined as the pair (Φ₀,Φ₁) with

All the diversities for [n,k,d]_q codes with k3, d−2 (mod q) such that A_i=0 for all i0,−1,−2 (mod q) are found and characterized with their spectra geometrically, which yields that such codes are extendable for all odd q5. Double extendability is also investigated.     

Vertex Set Partitions Preserving Conservativeness

Let G be an undirected graph and ={X₁, …, X_n} be a partition of V(G). Denote by G/ the graph which has vertex set {X₁, …, X_n}, edge set E, and is obtained from G by identifying vertices in each class X_i of the partition . Given a conservative graph (G, w), we study vertex set partitions preserving conservativeness, i.e., those for which (G/ , w) is also a conservative graph. We characterize the conservative graphs (G/ , w), where is a terminal partition of V(G) (a partition preserving conservativeness which is not a refinement of any other partition of this kind). We prove that many conservative graphs admit terminal partitions with some additional properties. The results obtained are then used in new unified short proofs for a co-NP characterization of Seymour graphs by A. A. Ageev, A. V. Kostochka, and Z. Szigeti (1997, J. Graph Theory34, 357–364), a theorem of E. Korach and M. Penn (1992, Math. Programming55, 183–191), a theorem of E. Korach (1994, J. Combin. Theory Ser. B62, 1–10), and a theorem of A. V. Kostochka (1994, in “Discrete Analysis and Operations Research. Mathematics and its Applications (A. D. Korshunov, Ed.), Vol. 355, pp. 109–123, Kluwer Academic, Dordrecht).     

Asymptotic properties of Cramér-Smirnov statistics—A new approach

Let Y₁,…, Y_n be independent identically distributed random variables with distribution function F(x, θ), θ = (θ′₁, θ′₂), where θ_i (i = 1, 2) is a vector of p_i components, p = p₁ + p₂ and for θI, an open interval in ^p, F(x, θ) is continuous. In the present paper the author shows that the asymptotic distribution of modified Cramér-Smirnov statistic under H_n: θ₁ = θ₁₀ + n^−1/2γ, θ₂ unspecified, where γ is a given vector independent of n, is the distribution of a sum of weighted noncentral χ₁² variables whose weights are eigenvalues of a covariance function of a Gaussian process and noncentrality parameters are Fourier coefficients of the mean function of the Gaussian process. Further, the author exploits the special form of the covariance function by using perturbation theory to obtain the noncentrality parameters and the weights. The technique is applicable to other goodness-of-fit statistics such as U² [G. S. Watson, Biometrika 48 (1961), 109–114].     

Robust estimation in the linear model with asymmetric error distributions

In the linear model X^{n × 1} = C^{n × p}θ^{p × 1} + E^{n × 1}, Huber's theory of robust estimation of the regression vector θ^{p × 1} is adapted for two models for the partially specified common distribution F of the i.i.d. components of the error vector E^{n × 1}. In the first model considered, the restriction of F to a set [−a₀, b₀] is a standard normal distribution contaminated, with probability , by an unknown distribution symmetric about 0. In the second model, the restriction of F to [−a₀, b₀] is completely specified (and perhaps asymmetrical). In both models, the distribution of F outside the set [−a₀, b₀] is completely unspecified. For both models, consistent and asymptotically normal M-estimators of θ^{p × 1} are constructed, under mild regularity conditions on the sequence of design matrices {C^{n × p}}. Also, in both models, M-estimators are found which minimize the maximal mean-squared error. The optimal M-estimators have influence curves which vanish off compact sets.     

Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces

Let (E,H,μ) be an abstract Wiener space and let D_V:=VD, where D denotes the Malliavin derivative and V is a closed and densely defined operator from H into another Hilbert space . Given a bounded operator B on , coercive on the range , we consider the operators A:=V^*BV in H and in , as well as the realisations of the operators and in L^p(E,μ) and respectively, where 1<p<∞. Our main result asserts that the following four assertions are equivalent:

(1) with for ;

(2) admits a bounded H^∞-functional calculus on ;

(3) with for ;

(4) admits a bounded H^∞-functional calculus on .

Moreover, if these conditions are satisfied, then . The equivalence (1)–(4) is a non-symmetric generalisation of the classical Meyer inequalities of Malliavin calculus (where , V=I, ). A one-sided version of (1)–(4), giving L^p-boundedness of the Riesz transform in terms of a square function estimate, is also obtained. As an application let −A generate an analytic C₀-contraction semigroup on a Hilbert space H and let −L be the L^p-realisation of the generator of its second quantisation. Our results imply that two-sided bounds for the Riesz transform of L are equivalent with the Kato square root property for A. The boundedness of the Riesz transform is used to obtain an L^p-domain characterisation for the operator L.

Dynamical Spin Susceptibility in the t–J Model in the Superconducting Phase

Dynamical spin susceptibility is calculated for the t–J model in the superconducting phase using the memory function method in terms of the Hubbard operators. The self-consistent system of equations for the memory function is obtained within the mode-coupling approximation. Both itinerant hole excitations and localized spin fluctuations contribute to the memory function. Moreover, the itinerant contribution itself consists of two parts, i.e., the contribution of Bogoliubov quasiparticles and that of Cooper pairs. The spin dynamics is diffusive in the hydrodynamic limit, but the itinerant part does not contribute to the spin diffusion. In the high frequency region, spin–wave-like excitations continue to exist. We discuss our analytic results in the light of neutron scattering experiments performed on the cuprate superconductors.     

On p-rank representations

The p-rank of an algebraic curve X over an algebraically closed field k of characteristic p>0 is the dimension of the vector space H¹(X_et,F_p). We study the representations of finite subgroups GAut(X) induced on H¹(X_et,F_p)k, and obtain two main results.First, the sum of the nonprojective direct summands of the representation, i.e., its core, is determined explicitly by local data given by the fixed point structure of the group acting on the curve. As a corollary, we derive a congruence formula for the p-rank.Secondly, the multiplicities of the projective direct summands of quotient curves, i.e., their Borne invariants, are calculated in terms of the Borne invariants of the original curve and ramification data. In particular, this is a generalization of both Nakajima's equivariant Deuring–Shafarevich formula and a previous result of Borne in the case of free actions.