Approximation of the Jacobian matrix in (m, 3)-methods of solving stiff systems

A theorem is proved which states that the maximal order of accuracy in L-stable (m, k)-methods with Jacobian matrix freezing is equal to four.     

Infinite families of 2- and 3-designs with parameters v = p + 1, k = (p − 1)/2i + 1, where p odd prime, 2e T (p − 1), e ≥ 2, 1 ≤ i ≤ e

Let p be an odd prime number such that p − 1 = 2^em for some odd m and e ≥ 2. In this article, by using the special linear fractional group PSL(2, p), for each i, 1 ≤ i ≤ e, except particular cases, we construct a 2-design with parameters v = p + 1, k = (p − 1)/2ⁱ + 1 and λ = ((p − 1)/2ⁱ+1)(p − 1)/2 = k(p − 1)/2, and in the case i = e we show that some of these 2-designs are 3-designs. Likewise, by using the linear fractional group PGL(2,p) we construct an infinite family of 3-designs with the same v k and λ = k(k − 2). These supplement a part of [4], in which we gave an infinite family of 3-designs with parameters v = q + 1, k = (q + 1)/2 = (q − 1)/2 + 1 and λ = (q + 1)(q − 3)/8 = k(k − 2)/2, where q is a prime power such that q − 1 = 2m for some odd m and q > 7. Some of the designs given in this article and in [4] fill in a few blanks in the table of Chee, Colbourn, and Kreher [2]. © 1997 John Wiley & Sons, Inc.     

An Algorithm to Construct <Emphasis Type="BoldItalic">k</Emphasis>-Regular <Emphasis Type="BoldItalic">k</Emphasis>-Connected Graphs with the Maximum <Emphasis Type="BoldItalic">k</Emphasis>-Diameter

 Let f(2m,k) be the Maximum k-diameter of k-regular k-connected graphs on 2m vertices. In this paper we give an algorithm and prove that we can construct k-regular k-connected graphs on 2m vertices with the maximum k-diameter using it. We also prove some known results about f(2m,k) and verify that we can get some unknown values of f(2m,k) by our algorithm. Received: December 1, 2000 Final version received: March 12, 2002 Acknowledgments. We thank the referee for many useful suggestions.     

Quasi‐symmetric designs with fixed difference of block intersection numbers

The following results for proper quasi‐symmetric designs with non‐zero intersection numbers x,y and λ > 1 are proved.

• (1) Let D be a quasi‐symmetric design with z = y ? x and v ≥ 2k. If x ≥ 1 + z + z³ then λ < x + 1 + z + z³.
• (2) Let D be a quasi‐symmetric design with intersection numbers x, y and y ? x = 1. Then D is a design with parameters v = (1 + m) (2 + m)/2, b = (2 + m) (3 + m)/2, r = m + 3, k = m + 1, λ = 2, x = 1, y = 2 and m = 2,3,… or complement of one of these design or D is a design with parameters v = 5, b = 10, r = 6, k = 3, λ = 3, and x = 1, y = 2.
• (3) Let D be a triangle free quasi‐symmetric design with z = y ? x and v ≥ 2k, then x ≤ z + z².
• (4) For fixed z ≥ 1 there exist finitely many triangle free quasi‐symmetric designs non‐zero intersection numbers x, y = x + z.
• (5) There do not exist triangle free quasi‐symmetric designs with non‐zero intersection numbers x, y = x + 2.

© 2006 Wiley Periodicals, Inc. J Combin Designs 15: 49–60, 2007     

On ε‐biased generators in NC0

Cryan and Miltersen (Proceedings of the 26th Mathematical Foundations of Computer Science, 2001, pp. 272–284) recently considered the question of whether there can be a pseudorandom generator in NC⁰, that is, a pseudorandom generator that maps n‐bit strings to m‐bit strings such that every bit of the output depends on a constant number k of bits of the seed. They show that for k = 3, if m ≥ 4n + 1, there is a distinguisher; in fact, they show that in this case it is possible to break the generator with a linear test, that is, there is a subset of bits of the output whose XOR has a noticeable bias. They leave the question open for k ≥ 4. In fact, they ask whether every NC⁰ generator can be broken by a statistical test that simply XORs some bits of the input. Equivalently, is it the case that no NC⁰ generator can sample an ε‐biased space with negligible ε? We give a generator for k = 5 that maps n bits into cn bits, so that every bit of the output depends on 5 bits of the seed, and the XOR of every subset of the bits of the output has bias 2. For large values of k, we construct generators that map n bits to bits such that every XOR of outputs has bias . We also present a polynomial‐time distinguisher for k = 4,m ≥ 24n having constant distinguishing probability. For large values of k we show that a linear distinguisher with a constant distinguishing probability exists once m ≥ Ω(2^kn^?k/2?). Finally, we consider a variant of the problem where each of the output bits is a degree k polynomial in the inputs. We show there exists a degree k = 2 pseudorandom generator for which the XOR of every subset of the outputs has bias 2^?Ω(n) and which maps n bits to Ω(n²) bits. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Jacobians among abelian threefolds: A formula of klein and a question of serre

In this paper we give a criterion when an indecomposable principally polarized abelian threefold (A, a) defined over a field k = ℂ is a Jacobian over k. More precisely, (A, a) is a Jacobian over k if and only if the value of a certain geometric Siegel modular form χ₁₈(A, a) is a square over k. This answers a question of J.-P. Serre.     

Self‐dual and self‐petrie‐dual regular maps

Regular maps are cellular decompositions of surfaces with the “highest level of symmetry”, not necessarily orientation‐preserving. Such maps can be identified with three‐generator presentations of groups G of the form G = 〈a, b, c|a² = b² = c² = (ab)^k = (bc)^m = (ca)² = … = 1〉; the positive integers k and m are the face length and the vertex degree of the map. A regular map (G;a, b, c) is self‐dual if the assignment b?b, c?a and a?c extends to an automorphism of G, and self‐Petrie‐dual if G admits an automorphism fixing b and c and interchanging a with ca. In this note we show that for infinitely many numbers k there exist finite, self‐dual and self‐Petrie‐dual regular maps of vertex degree and face length equal to k. We also prove that no such map with odd vertex degree is a normal Cayley map. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory 69:152‐159, 2012     

<Emphasis Type="Italic">k</Emphasis>-Free-like groups

The following results are proved. In Theorem 1, it is stated that there exist both finitely presented and not finitely presented 2-generated nonfree groups which are k-free-like for any k ⩾ 2. In Theorem 2, it is claimed that every nonvirtually cyclic (resp., noncyclic and torsion-free) hyperbolic m-generated group is k-free-like for every k ⩾ m + 1 (resp., k ⩾ m). Finally, Theorem 3 asserts that there exists a 2-generated periodic group G which is k-free-like for every k ⩾ 3. Supported by NSF (grant Nos. DMS 0455881 and DMS-0700811). (A. Yu. Olshanskii, M. V. Sapir) Supported by RFBR project No. 08-01-00573. (A. Yu. Olshanskii) Supported by BSF grant (USA–Israel). (M. V. Sapir) Translated from Algebra i Logika, Vol. 48, No. 2, pp. 245–257, March–April, 2009.     

Solution of a covering problem related to labelled tournaments

Suppose we have a tournament with edges labelled so that the edges incident with any vertex have at most k distinct labels (and no vertex has outdegree 0). Let m be the minimal size of a subset of labels such that for any vertex there exists an outgoing edge labelled by one of the labels in the subset. It was known that m ≤ (^k+1₂) for any tournament. We show that this bound is almost best possible, by a probabilistic construction of tournaments with m = O(k²/log k). We give explicit tournaments with m = k^2−o(1). If the number of vertices is bounded by N < 2^k₁ we have a better upper bound of m = O(k log N), which is again almost optimal. We also consider a relaxation of this problem in which instead of the size of a subset of labels we minimize the total weight of a fractional set with analogous properties. In that case the optimal bound is 2k − 1. © 1996 John Wiley & Sons, Inc.     

GENERALISED MAXIMAL INDEPENDENCE AND CLIQUE NUMBERS OF GRAPHS

Abstract

A set B of vertices of a graph G = (V,E) is a k-maximal independent set (kMIS) if B is independent but for all ?-subsets X of B, where ? ? k—1, and all (? + 1)-subsets Y of V—B, the set (B—X) u Y is dependent. A set S of vertices of C is a k-maximal clique (kMc) of G iff S is a kMIS of [Gbar]. Let β_k, (G) (w_k(G) respectively) denote the smallest cardinality of a kMIS (kMC) of G—obviously β_k(G) = w_k([Gbar]). For the sequence m₁ ? m₂ ?…? m_n = r of positive integers, necessary and sufficient conditions are found for a graph G to exist such that w_k(G) = m_k for k = 1,2,…,n and w(G) = r (equivalently, β_k(G) = m_k for k = 1,2,…,n and β(G) = r). Define s_k(?,m) to be the largest integer such that for every graph G with at most s_k(?,m) vertices, β_k(G) ? ? or w_k(G) ? m. Exact values for s_k(?,m) if k ≥ 2 and upper and lower bounds for s₁(?,m) are de termined.     

Types of superregular matrices and the number of n‐arcs and complete n‐arcs in PG (r,q)

Based on the classification of superregular matrices, the numbers of non‐equivalent n‐arcs and complete n‐arcs in PG(r, q) are determined (i) for 4 ≤ q ≤ 19, 2 ≤ r ≤ q ? 2 and arbitrary n, (ii) for 23 ≤ q ≤ 32, r = 2 and n ≥ q ? 8<$>. The equivalence classes over both PGL (k, q) and PΓL(k, q) are considered throughout the examinations and computations. For the classification, an n‐arc is represented by the systematic generator matrix of the corresponding MDS code, without the identity matrix part of it. A rectangular matrix like this is superregular, i.e., it has only non‐singular square submatrices. Four types of superregular matrices are studied and the non‐equivalent superregular matrices of different types are stored in databases. Some particular results on t(r, q) and m′(r, q)—the smallest and the second largest size for complete arcs in PG(r, q)—are also reported, stating that m′(2, 31) = 22, m′(2, 32) = 24, t(3, 23) = 10, and m′(3, 23) = 16. © 2006 Wiley Periodicals, Inc. J Combin Designs 14: 363–390, 2006     

SCORE LISTS IN （h, k）-BIPARTITE HYPERTOURNAMENTS

Given non-negative integers m,n,h and k with m ≥ h ＞ 1 and n ≥ k ＞ 1, an (h, k)-bipartite hypertournament on m n vertices is a triple (U, V, A), where U and V are two sets of vertices with |U| = m and |V| = n, and A is a set of (h k)-tuples of vertices,called arcs, with at most h vertices from U and at most k vertices from V, such that for any h k subsets U1 ∪ V1 of U ∪ V, A contains exactly one of the (h k)! (h k)-tuples whose entries belong to U1 ∪ V1. Necessary and sufficient conditions for a pair of non-decreasing sequences of non-negative integers to be the losing score lists or score lists of some(h, k)-bipartite hypertournament are obtained.     

Generalizing Pancyclic and <Emphasis Type="Italic">k</Emphasis>-Ordered Graphs

Given positive integers kmn, a graph G of order n is (k,m)-pancyclic if for any set of k vertices of G and any integer r with mrn, there is a cycle of length r containing the k vertices. Minimum degree conditions and minimum sum of degree conditions of nonadjacent vertices that imply a graph is (k,m)-pancylic are proved. If the additional property that the k vertices must appear on the cycle in a specified order is required, then the graph is said to be (k,m)-pancyclic ordered. Minimum degree conditions and minimum sum of degree conditions for nonadjacent vertices that imply a graph is (k,m)-pancylic ordered are also proved. Examples showing that these constraints are best possible are provided.Acknowledgments. The authors would like to thank the referees for their careful reading of the paper and their useful suggestions.Final version received: January 26, 2004     

A note on the girth‐doubling construction for polygonal graphs

A near‐polygonal graph is a graph Γ which has a set ?? of m‐cycles for some positive integer m such that each 2‐path of Γ is contained in exactly one cycle in ??. If m is the girth of Γ then the graph is called polygonal. Given a polygonal graph Γ of valency r and girth m, Archdeacon and Perkel proved the existence of a polygonal graph Γ₂ of valency r and girth 2m. We will show that this construction can be extended to one that yields a polygonal graph Γ₃ of valency r and girth 3m, but that making the cycles any longer with this construction does not yield a polygonal graph. We also show that if Aut(Γ) is 2‐arc transitive, so is Aut(Γ_k) for k = 2, 3. © 2010 Wiley Periodicals, Inc. J Graph Theory 68: 246‐254, 2011     

New cases of Reay’s conjecture on partitions of points into simplices with -dimensional intersection

Reay’s conjecture asserts that every set of (m−1)(d+1)+k+1 points in general position in (with 0≤k≤d) has a partition X₁,X₂,…,X_m such that is at least k-dimensional. We prove this conjecture in several cases: when m≤8 (for arbitrary d and k), when d=6,d=7 and d=8 (for arbitrary m and k), and when k=1 and d≤24 (for arbitrary m).     

On the critical exponents of random k‐SAT

There has been much recent interest in the satisfiability of random Boolean formulas. A random k‐SAT formula is the conjunction of m random clauses, each of which is the disjunction of k literals (a variable or its negation). It is known that when the number of variables n is large, there is a sharp transition from satisfiability to unsatisfiability; in the case of 2‐SAT this happens when m/n → 1, for 3‐SAT the critical ratio is thought to be m/n ≈ 4.2. The sharpness of this transition is characterized by a critical exponent, sometimes called ν = ν_k (the smaller the value of ν the sharper the transition). Experiments have suggested that ν₃ = 1.5 ± 0.1. ν₄ = 1.25 ± 0.05, ν₅ = 1.1 ± 0.05, ν₆ = 1.05 ± 0.05, and heuristics have suggested that ν_k → 1 as k → ∞. We give here a simple proof that each of these exponents is at least 2 (provided the exponent is well defined). This result holds for each of the three standard ensembles of random k‐SAT formulas: m clauses selected uniformly at random without replacement, m clauses selected uniformly at random with replacement, and each clause selected with probability p independent of the other clauses. We also obtain similar results for q‐colorability and the appearance of a q‐core in a random graph. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 21: 182–195, 2002     

Group‐divisible designs with block size k having k + 1 groups,for k = 4, 5

We investigate the spectrum for k‐GDDs having k + 1 groups, where k = 4 or 5. We take advantage of new constructions introduced by R. S. Rees (Two new direct product‐type constructions for resolvable group‐divisible designs, J Combin Designs, 1 (1993), 15–26) to construct many new designs. For example, we show that a resolvable 4‐GDD of type g⁵ exists if and only if g ≡ 0 mod 12 and that a resolvable 5‐GDD of type g⁶ exists if and only if g ≡ 0 mod 20. We also show that a 4‐GDD of type g⁴m¹ exists (with m > 0) if and only if g ≡ m ≡ 0 mod 3 and 0 < m ≤ 3g/2, except possibly when (g,m) = (9,3) or (18,6), and that a 5‐GDD of type g⁵m¹ exists (with m > 0) if and only if g ≡ m ≡ 0 mod 4 and 0 < m ≤ 4g/3, with 32 possible exceptions. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 363–386, 2000     

Upper bounds on the sum of powers of the degrees of a simple planar graph

For a simple planar graph G and a positive integer k, we prove the upper bound 2(n ? 1)^k + 4^k(n ? 4) + 2·3^k ? 2((δ + 1)^k ? δ^k)(3n ? 6 ? m) on the sum of the kth powers of the degrees of G, where n, m, and δ are the order, the size, and the minimum degree of G, respectively. The bound is tight for all m with 0?3n ? 6 ? m≤?n/2? ? 2 and δ = 3. We also present upper bounds in terms of order, minimum degree, and maximum degree of G. © 2010 Wiley Periodicals, Inc. J Graph Theory 67:112‐123, 2011     

The k‐ary Montgomery modular inverse over nonbinary computers

This paper presents a k‐ary Montgomery modular inverse algorithm over nonbinary computers by using Sedjelmaci's right shift k‐ary greatest common divisor scheme. Over traditional binary computers, Kaliski's scheme can output Montgomery modular inverse Q ^{? 1}2ⁿ mod P, where P is coprime to Q and n is the bit length of P. Over k‐ary computers, our algorithm can discover the k‐ary Montgomery inverse Q ^{? 1}k^m mod P, where P, Q, and k are pairwise relatively prime positive integers and m = log _kP. In the worst case, the computational cost of our algorithm is O(m²)k‐ary digit operations. Copyright © 2013 John Wiley & Sons, Ltd.     

Distance graphs with missing multiples in the distance sets

Given positive integers m, k, and s with m > ks, let D_m,k,s represent the set {1, 2, …, m} − {k, 2k, …, sk}. The distance graph G(Z, D_m,k,s) has as vertex set all integers Z and edges connecting i and j whenever |i − j| ∈ D_m,k,s. The chromatic number and the fractional chromatic number of G(Z, D_m,k,s) are denoted by χ(Z, D_m,k,s) and χf(Z, D_m,k,s), respectively. For s = 1, χ(Z, D_m,k,1) was studied by Eggleton, Erdős, and Skilton [6], Kemnitz and Kolberg [12], and Liu [13], and was solved lately by Chang, Liu, and Zhu [2] who also determined χf(Z, D_m,k,1) for any m and k. This article extends the study of χ(Z, D_m,k,s) and χf(Z, D_m,k,s) to general values of s. We prove χf(Z, D_m,k,s) = χ(Z, D_m,k,s) = k if m < (s + 1)k; and χf(Z, D_m,k,s) = (m + sk + 1)/(s + 1) otherwise. The latter result provides a good lower bound for χ(Z, D_m,k,s). A general upper bound for χ(Z, D_m,k,s) is obtained. We prove the upper bound can be improved to ⌈(m + sk + 1)/(s + 1)⌉ + 1 for some values of m, k, and s. In particular, when s + 1 is prime, χ(Z, D_m,k,s) is either ⌈(m + sk + 1)/(s + 1)⌉ or ⌈(m + sk + 1)/(s + 1)⌉ + 1. By using a special coloring method called the precoloring method, many distance graphs G(Z, D_m,k,s) are classified into these two possible values of χ(Z, D_m,k,s). Moreover, complete solutions of χ(Z, D_m,k,s) for several families are determined including the case s = 1 (solved in [2]), the case s = 2, the case (k, s + 1) = 1, and the case that k is a power of a prime. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 245–259, 1999