... This also implies that thenumberof periodic points is bounded by an exponential function ofthe period The notion of a flat critical point used in [MMS] is a nonstandard one from the point of ... perturbation of any element of Diffr (M ) Another justification for considering diffeomorphisms in Euclidean space is that the problem of exponential/superexponential growth ofthenumberof periodic points ... of initial points ofI n ˜ × Measure of periodicity × Measure of hyperbolicity −1 The first term onthe right-hand side of (3.24) is of order γn (up to an exponential function in n) In Section...
... aeration can be improved by increasing thenumberofthe pipes in the pile’s bottom zone Furthermore, the distance between the pipes seems to facilitate the air penetration into the composting pile ... natural convection 2) The composting rate was increased in increasing in thenumberofthe perforated pipes placed vertically in the composting bed within certain limits for air delivery in the composting ... effect ofthenumberofthe vertical pipes for air supply in the composting pile onthe composting rate was investigated by measuring the bed temperature variations and the carbon content during the...
... geodesics on X and their relationship with the Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces Simple closed geodesics Let cX (L) be thenumberof primitive closed geodesics of ... branches A lamination γ on S is carried by τ if there is a differentiable map f : S → S homotopic to the identity taking γ to τ such that the restriction of df to a tangent line of γ is nonsingular Every ... is the geodesic length ofthe measured lamination λ on X For more details see [Th] Counting integral multi-curves In this section, we study the growth ofthenumberof integral multi-curves of...
... From Figure 4.3, we know that the θ-direction of B (i, j;k) is mapped by s j into the θdirection of B( j,∗;∗) if I( i, j;k) > 0; the θ-direction of B (i, j;k ) is mapped by s j into opposition ofthe ... {B(ik ,i1 ; j1 ) ,B (i1 ,i2 ; j2 ) ,B (i2 ,i3 ; j3 ) , ,B(ik−1 ,ik ; jk ) } ofthe components ofthe intersection of sliding sets Since B(ik ,i1 ; j1 ) ranges in t-direction from one component of ... Mi ), where Mi = i Mi − Int(Di ) for ≤ i ≤ n and Mi = S2 × I for n + ≤ i ≤ n + n (see [8]) Each Mi admits the orientation coincident with that of M, and each ∂Mi inherits the orientation of Mi...
... that there are exactly ϕ maximal (k, )-sum-free arithmetic progressions with the difference ρ Precisely ϕr of them have length n/ρ and ϕ − ϕr are of length n/ρ Since these progressions are pairwise ... subsets of [1, n] are contained in arithmetic progression r mod ρ, where (k − )r ≡ mod ρ If d > ρ then every progression r mod d consists of at most n/(ρ + 1) elements hence it contains no more ... assumptions (1), (2) implies d |(k − )m Since A is a (k, )-sum-free set, it follows that (k − )m > td ≥ ( + − 2κ)Λ, which gives the required inequality Theorem Assume that k > ≥ are positive integers...
... differential equation, with no appearance ofthe function itself, only the first and third derivatives The differential equation (14) is solvable: its explicit form (abridged) is the one given in the ... numberof passes in quickselect, as given in the work by Kirschenhofer et al [17], and thenumberof ascendants in LBSTs relies onthe initial conditions The reason is that in the mentioned paper only ... ] The next theorem gives the variance, which is now easy to obtain THE ELECTRONIC JOURNAL OF COMBINATORICS (1998), #R20 14 the insertion hits one of these leaves Figure The fringe heuristic Theorem...
... mentions the “stronger conjecture” that for every σ, thelimitof F (n, σ)1/n exists and is finite According to Wilf (private communication, 1999) the original conjecture was of this latter form In ... this short note we give, as Theorem 1, a simple argument, involving subadditivity, which shows that the two versions ofthe conjecture are equivalent Here is some background information onthe ... k Theorem There exists an n-permutation, with n = k2 , containing every k-permutation as a subpattern; i. e m(k) ≤ k the electronic journal of combinatorics (1999), #N1 Proof Consider the lexicographic...
... see this, observe that the edge (x, x ) may be chosen in ni ways (minimality of x fixes the orientation ofthe edge), and that the choice of (x, x ) combined with the information provided by the ... assuming we consider the partition A, B distinct from B, A There is slack in the calculation in [2] and this will absorb the − o(1) term and so (5) proves the lower bound in Theorem For the upper ... spanning tree of G by (i) deleting the lexicographically first edge of each cycle of Df and then (ignoring orientation) extending the electronic journal of combinatorics (2000), #R57 10 √ the kf...
... even if the graph is very dense, the extreme case being an acyclic orientation of a complete bipartite graph) The most interesting and natural constraint is the requirement that the digraph ... must be biregular and the cardinality of each vertex class of G must be divisible by For any pair m and n both divisible by it is easy to construct a biregular Eulerian bipartite digraph with edge ... + 1, the electronic journal of combinatorics (2001), #N6 giving inequality (a) To prove inequality (b), let M be the submatrix of M consisting of rows βm+1, , m and all columns of M Since...
... extended only to all divisors ofthe quadratfrei radical q of m Indeed, only the quadratfrei divisors of m have a non trivial contribution 3.2 The contribution ofthe rectangular m-th powers to the ... ofthe ring some positive power of which belongs to I; in the present situation, q(m) is the positive generator ofthe radical ofthe ideal of Z generated by m the electronic journal of combinatorics ... permutations is stable under conjugacy in n This characterisation, already mentioned in [Be] is established in section The computation ofthe exponential generating function (EGF) Pm of these...
... (usually) in bijection with families of non-intersecting lattice paths We shall make use of this bijection in Section 6, together with the main result onthe enumeration of non-intersecting lattice ... thenumberof FPL configurations corresponding to a given matching is invariant under rotation ofthe “positioning” ofthe matching around the square As we mentioned already in the introduction, ... precisely, in Case (2), this region is the reflexion ofthe corresponding cut off part ofthe triangle in the right side of Qn , and in Case (3) it is that region and also the reflexion ofthe corresponding...
... occur in the same order on Ri as they on Pi (resp Qi ) Now H = R1 vR2 is a hamiltonian path of T Furthermore, distinct hamiltonian paths of T [A] (resp T [B]) give distinct hamiltonian paths of ... allow the special case where m = 0; in this case the path Q is a path on vertices, and R = P satisfies the lemma trivially The remainder ofthe proof is by induction on m For m = 1, let i be the minimal ... v1 gives the transitive tournament of order n As noted by Moon [1], there is a bijection between partitions of V \ {v1 , } and hamiltonian paths that include the arc → v1 , and there is a unique...
... we fill in p (i, j) by induction on i, using part (2) if i ≤ j and part (3) if i > j A generalization We mention a mild generalization of Theorem and its corollary Define the polynomial |x| P = ... y1 ofthe vector y We will need a subsidiary function Let N(λ; t) be thenumberof solutions of (i) and (ii) with y1 = t By definition, N(λ) = t≥0 N(λ; t) We need one more definition to state the ... 1)m−n+1 if ≤ n ≤ m Indeed each ofthe n terms in the sum representing p(m, n) is divisible by this quantity A second consequence of part (3) is an efficient algorithm for computing p(m, n) Algorithm...
... research is performed during the author’s visit at the Erwin Schr¨dinger International o Institute for Mathematical Physics The author would like to thank the ESI for hospitality and financial support ... zeros (with Jnq,d is the all one matrix and Inq,d is the identity matrix, both of size nq,d × nq,d ) Therefore, the largest eigenvalue of V is Dq,d and the absolute values of all other eigenvalues ... construction of Alon and Krivelevich [1] Let P G(q, d) denote the projective geometry of dimension d − over finite field q The vertices of P G(q, d) correspond to the equivalence classes of the...
... not different Only the edges of K1 and K viii iii viii iii and edge iii-viii are represented edges of K1 edges of K2 Figure 9.3 Definition 11 Let K2 be the subgraph of C such that it consists of ... goal is to determine the parity ofthenumberof possible constructions of: union of e-graph vi − ii, consisting of a single edge, and all the uncrossed edges connecting to e-graph vi − ii with the ... n−2 is divisible by ) p is equal to thenumberof possible ways of constructing the union of e-graph vi − ii and all the uncrossed edges connecting to e-graph vi − ii with the restriction that they...
... distinguished edge of G An orientation X ofthe edges of G is an e-bipolar orientation of G if it is acyclic, s is the only vertex e-bipolar without incoming edges and t is the only vertex without ... Eulerian orientation of G0 , but there are Eulerian orientations of G0 , which not come from a Eulerian orientation of G Given a Eulerian orientation of G0 we call the orientation ofthe edges incident ... W in every α-orientation of M Such an edge with the same direction in every α-orientation is a rigid edge rigid We denote thenumberof α-orientations of M by rα (M ) Let M be a family of pairs...
... Equalities in (2.12-2.15) hold if and only if G is either a union of copies of C , or a union of copies of C4 and a copy of Ci for i = 5, 6, 7, respectively Equalities in (2.16-2.18) hold if and only ... deduce pi qj − qi pj = pi pj − pi pj + x(pi pj−2 − pi−2 pj ) = (−1 )i 1 xi pj i 1 (3.13) Therefore, if i is odd then qi pj pi qj If i is even then pi qj qi pj These inequalities yield slightly ... Proof Let ≤ i, j and consider the path Pi+j By considering the generating matching polynomial without the match (i, i + 1) and with match (i, i + 1) we get the identity pi+j = pi pj + xpi−1...
... he proves divisibility by powers of for thenumberof matchings ofthe square grid graph Theorem For n odd, thenumberof perfect matchings of Gn is a multiple of n+1 Proof We use the term block ... pair up under the forces of hydrogen bonding When finding a formula for thenumberof perfect matchings of a graph has seemed hard, researchers have focused onthe questions of divisibility ofthe ... Enumeration of Matchings: Problems and Progress, New Perspectives in Geometric Combinatorics, MSRI Publications, 38, (1999) 255-291 [2] P W Kasteleyn, The statistics of dimers on a lattice I: The number...
... An interesting paper of Wagner [5] looks at thenumberof independent sets modulo m Wagner showed that the proportion of trees on n vertices with thenumberof independent sets divisible by ... section, we turn to thenumberof matchings in a graph This is also known as the Hosoya index, or the Z-index in mathematical chemistry For a rooted tree T , let Z(T ) be thenumberof matchings ... covering the root In [5], Wagner also mentioned that for any m ∈ N, the proportion of trees on n vertices with Z(T ) a multiple of m tends to as n tends to infinity The inverse problem in the family...