Heilbronn's conjecture on Waring's number (mod p)

Let p be a prime k|p−1, t=(p−1)/k and γ(k,p) be the minimal value of s such that every number is a sum of s kth powers . We prove Heilbronn's conjecture that γ(k,p)?k1/2 for t>2. More generally we show that for any positive integer q, γ(k,p)?C(q)k1/q for ?(t)?q. A comparable lower bound is also given. We also establish exact values for γ(k,p) when ?(t)=2. For instance, when t=3, γ(k,p)=a+b−1 where a>b>0 are the unique integers with a²+b²+ab=p, and when t=4, γ(k,p)=a−1 where a>b>0 are the unique integers with a²+b²=p.     

ESTIMATES ON EIGENVALUES FOR THE BIHARMONIC OPERATOR ON A BOUNDED DOMAIN IN H~n（-1）

In this paper, we consider eigenvalues of the Dirichlet biharmonic operator on a bounded domain in a hyperbolic space. We obtain universal bounds on the (k + 1)th eigenvalue in terms of the first kth eigenvalues independent of the domains.     

Sums and k-sums in abelian groups of order k

Let G be an abelian group of order k. How is the problem of minimizing the number of sums from a sequence of given length in G related to the problem of minimizing the number of k-sums? In this paper we show that the minimum number of k-sums for a sequence a₁,…,ar that does not have 0 as a k-sum is attained at the sequence b₁,…,br−k+1,0,…,0, where b₁,…,br−k+1 is chosen to minimise the number of sums without 0 being a sum. Equivalently, to minimise the number of k-sums one should repeat some value k−1 times. This proves a conjecture of Bollobás and Leader, and extends results of Gao and of Bollobás and Leader.     

Solutions of the equation zez = a(z + b)

The theory of complex variables is used to develop exact closed-form solutions for the, in general, complex zeros of the exponential polynomial F(z) = z exp z ? a(z + b), a complex and b real. The established zeros are related to canonical solutions of suitably posed Riemann problems and are expressed ultimately in terms of elementary quadratures.     

On the diophantine equation x(x + 1)(x + 2)…(x + (m − 1)) =g(y)

Let g(y) ? Q[Y] be an irreducible polynomial of degree n ≥ 3. We prove that there are only finitely many rational numbers x, y with bounded denominator and an integer m ≥ 3 satisfying the equation x(x + 1) (x + 2)…(x + (m − 1) ) = g(y). We also obtain certain finiteness results when g(y) is not an irreducible polynomial.     

Characterization of the graphs of the Johnson schemes G(3k, k) and G(3k + 1, k)

The graphs of the Johnson schemes G(3k, k) and G(3k + 1, k) are characterized by their parameters. In particular this finishes the characterization of the tetrahedral graphs G(n, 3).     

Laplacian integral graphs in S(a, b)

Let G_n,m be the family of graphs with n vertices and m edges, when n and m are previously given. It is well-known that there is a subset of G_n,m constituted by graphs G such that the vertex connectivity, the edge connectivity, and the minimum degree are all equal. In this paper, S(a, b)-classes of connected (a, b)-linear graphs with n vertices and m edges are described, where m is given as a function of a,b∈N/2. Some of them have extremal graphs for which the equalities above are extended to algebraic connectivity. These graphs are Laplacian integral although they are not threshold graphs. However, we do build threshold graphs in S(a, b).     

Polynomial algorithms for m × (m + 1) integer programs and m × (m + k) diophantine systems

Recent results of Kannan and Bachem (on computing the Smith Normal Form of a matrix) and Lenstra (on solving integer inequality systems) are used with classical results by Smith to obtain polynomial-time algorithms for solving m × (m + 1) equality constrained integer programs and m × (m + k) systems of diophantine equations for fixed k.     

On the mutual embeddability of (2k, k, k − 1) and (2k − 1, k, k) quasi-residual designs

The complements of blocks containing a given point in a (2k ? 1, k, k) design, enlarged by this point, and the blocks not containing it, form a (2k ? 1, k, k) design. Likewise, the complements of blocks containing a given point in a (2k, k, k ? 1) design and the blocks not containing it, form a (2k ? 1, k, k) design. In this paper we show that if a quasi-residual (2k ? 1, k, k) design is obtained from an embeddable (2k ? 1, k, k) or (2k, k, k ? 1) design, then it is also embeddable, and describe an example of non-embeddable (12, 6, 5) design such that all (11, 6, 6) designs obtained from it are embeddable.     

A lower bound for v in a t − (v,k, λ) design

In this paper it is shown that if v ? k + 1 then $\text{v ? t ? 1 + (k ? t + 1)(k ? t + 2)}\text{λ}$, where v, k, λ and t are the characteristic parameters of a t ? (v, k, λ) design. We compare this bound with the known lower bounds on v.     

On the uniqueness of the graphs G(n, k) of the Johnson schemes

T. A. Dowling (J. Combin. Theory6 (1969), 251–263) proved the uniqueness of the graphs G(n, k) of the Johnson schemes for n > 2k(k ? 1) + 4. We improve this result by showing the uniqueness of G(n, k) for n > 4k.     

On All Fractional （a,b, k）-Critical Graphs

Let a,b,k,r be nonnegative integers with 1≤a≤b and r≥2.LetG be a graph of order n with n(a+b)(r(a+b)-2)+ak/a.In this paper,we first show a characterization for all fractional(a,b,k)-critical graphs.Then using the result,we prove that G is all fractional(a,b,k)-critical if δ(G)≥(r-1)b2/a+k and |NG(x1)∪NG(x2)∪···∪NG(xr)|≥bn+ak/a+b for any independent subset {x1,x2,...,xr} in G.Furthermore,it is shown that the lower bound on the condition|NG(x1)∪NG(x2)∪···∪NG(xr)|≥bn+ak/a+b is best possible in some sense,and it is an extension of Lu's previous result.     

Sign patterns,nonsingularity, and the solvability of Ax = b

We investigate conditions on the sign-pattern class of the (n–1)st compound of a real n-by-n matrix A such that the solvability of Ax = b⁽ⁱ⁾ for i = 1,…,k, k<n, with specific b⁽ⁱ⁾, insures the nonsingularity of A. The number and choice of right-hand sides b⁽ⁱ⁾ sufficient for the task depends only on the sign-pattern class of the (n–1)st compound of A. The result for k = 1 generalizes a known fact about totally nonnegative matrices and an observation about M-matrices, thus providing another unifying result for these two classes of matrices.     

Cubic Thue inequalities with negative discriminant

We give an upper bound for the solutions of the family of cubic Thue inequalities |x³+axy²+by³|?k when a is positive and larger than a certain value depending on b. For the case k=a+|b|+1 and a?360b⁴ we show that these inequalities have only trivial solutions. For the case k=a+|b|+1 and |b|=1,2, we solve these inequalities for all a?1. Our method is based on Padé approximations using Rickert's integrals. We also use a generalization of Legendre's theorem on continued fractions.     

On the order of the error function of the (k,r)-integers

Let k and r be fixed integers such that 1 < r < k. Any positive integer n of the form n = a^kb, where b is r-free, is called a (k, r)-integer. In this paper we prove that if Q_k,r(x) denotes the number of (k, r)-integers ≤ x, then $\text{Q}{}_{\mathrm{k,r}}\text{(x) =}\text{xζ(k)}\text{ζ(r)}\text{+ Δ}{}_{\mathrm{k,r}}\text{(x)}$, where $\text{Δ}{}_{\mathrm{k,r}}\text{(x) = O(x}{}^{\mathrm{<mtext>1</mtext><mtext>r</mtext>}}\text{exp}\text{[?B}\text{log}{}^{\mathrm{<mtext>3</mtext><mtext>5</mtext>}}\text{x (}\text{log log}\text{x)}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>5</mtext>}}\text{])}$, B being a positive constant depending on r and the O-estimate is uniform in k. On the assumption of the Riemann hypothesis, we improve the above order estimate of Δ_k,r(x) and prove that

$\text{1}\text{x}\text{∫}\text{1}\text{α}\text{δ}{}_{\mathrm{k,r}}\text{(t)dt=0(x}\text{1}\text{k}\text{ω(x))}\text{or}\text{0(x}{}^{\mathrm{3/(4r+1)}}\text{ω(x))}$

, according as $\text{k ≤}\text{(4r + 1)}\text{3}$ or $\text{k >}\text{(4r + 1)}\text{3}$, where ω(x) = exp [B log x(log log x)^?1].     

On the Diophantine equation ax + by = ck

In this paper we investigate the solvability and the representation of the solutions of the equation ax² +by² = ckⁿ. We extend and improve many known results. In particular, we completely solve the equation (a ± 1)x² + (3a ? 1) = 4aⁿ, 2 ? n.     

On some t−(2k, k, λ) designs

t?(2k, k, λ) designs having a property similar to that of Hadamard 3-designs are studied. We consider conditions (i), (ii), or (iii) for t?(2k, k, λ) designs: (i) The complement of each block is a block. (ii) If A and B are a complementary pair of blocks, then ∥ A ∩ C ∥ = ∥ B ∩ C ∥ ± u holds for any block C distinct from A and B, where u is a positive integer. (iii) if A and B are a complementary pair of blocks, then ∥ A ∩ C ∥ = ∥ B ∩ C ∥ or ∥ A ∩ C ∥ = ∥ B ∩ C ∥ ± u holds for any block C distinct from A and B, where u is a positive integer. We show that a t?(2k, k, λ) design with t ? 2 and with properties (i) and (ii) is a 3?(2u(2u + 1), u(2u + 1), u(2u² + u ? 2)) design, and that a t?(2k, k, λ) design with t ? 4 and with properties (i) and (iii) is the 5-(12, 6, 1) design, the 4-(8, 4, 1) design, a $\text{5?(2u}{}^2\text{, u}{}^2\text{,}\text{1}\text{4}\text{(u}{}^2\text{? 3) (u}{}^2\text{? 4))}$ design, or a $\text{5?(}\text{2}\text{3}\text{u(2u + 1),}\text{1}\text{3}\text{u(2u = 1),}\text{1}\text{5 4}\text{u(2u}{}^2\text{+ u ? 9) (2u}{}^2\text{+ u ? 12))}$ design.     

Enumeration of (p,q)-parking functions

Parking functions are central in many aspects of combinatorics. We define in this communication a generalization of parking functions which we call (p₁,…,p_k)-parking functions. We give a characterization of them in terms of parking functions and we show that they can be interpreted as recurrent configurations in the sandpile model for some graphs. We also establish a correspondence with a Lukasiewicz language, which enables to enumerate (p₁,…,p_k)-parking functions as well as increasing ones.     

PBIB designs with m associate classes induced by full rank matrices modulo pm

This paper presents series of PBIB designs with m associate classes in which the treatment set is a subset of the Z(p^m)-module of n × 1 vectors over the ring of integers modulo p^m, p any prime. The association scheme of this series of designs is determined by the Fuller canonical form under row equivalence of n × 2 matrices [a,b] for vectors a and b in the treatment set. The blocking procedure utilizes full rank s × n matrices over Z(p^m), 1 ? s ? n ? 2, n ? 3. For m = 2, n = 3, s =1 and for each prime p, each PBIB is regular divisible and yields a finite proper uniform projective Hjelmslev plane with parameters j = p and k = p(p + 1).     

Rado Numbers fora(x+y)=bz

In the case of existence the smallest numberN=Ra_kis called a Rado number if it is guaranteed that anyk-coloring of the numbers 1, 2, …, Ncontains a monochromatic solution of a given system of linear equations. We will determine Ra_k(a, b) for the equationa(x+y)=bzifb=2 andb=a+1. Also, the case of monochromatic sequences {x_n} generated bya(x_n+x_n+1)=bx_n+2 is discussed.