... Moreover, there is also a variant ofthe BCP for self-maps ofa non-Archimedean metric space proved by Prieß-Crampe [10] (also see Petalas and Vidalis [9]) It turns out that, in general, the contraction ... obtain only some particular cases ofthe contraction principle via a discrete argument Finally, we discuss a variant of ET given by Rus [11] (cf Remarks 2.1 and 4.3) Proofof Eilenberg’s theorem ... Actually, a minor modification ofthe above proof shows that the assumptions of Theorem 1.1 could be weakened: the relations Rn need not be transitive if we assume the uniqueness ofa point x...
... nonempty cancel out ofthe above sum, because they are counted once for each subset of S, with alternating signs Thus what survives is the set of placements of d non-taking rooks on the extended board ... for all x Acknowledgments This work was supported in part by a National Science Foundation Graduate Fellowship and a National Science Foundation Postdoctoral Fellowship References [1] T Chow, The ... is, and the purpose of this note is to provide such aproofThe knowledgeable reader will recognize that the main idea is borrowed from [4] Proof Observe that d (−1) R(B; − x −...
... counting semistandard tableaux of shape µ/ν satisfying certain properties, and it is a not-toodifficult exercise to show that these two formulations count the same tableaux Via the specialization ν = ... representations ofthe qanalogue of classical Lie algebras, J Algebra 165 (1994), 295–345 [L] P Littelmann, A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras, Invent Math 116 (1994), ... Math., to appear [vL] M AA van Leeuwen, The Littlewood-Richardson rule, and related combinatorics, in “Interactions of Combinatorics and Representation Theory,” MSJ Memoirs 11, Math Soc Japan,...
... of φ−1 when there is a primed integer in the image, we consider only the value ofthe integer and ignore the prime We leave the details of verifying that φ and φ−1 are inverse maps to the reader ... combinatorial properties ofthe classical formula, one might hope for a simpler formula For this reason it may be of interest to investigate other q-analogues ofthe sum of cubes formula In addition, as ... about aproof described to him by N.M Ferrers The Ferrers shape ofa partition is an array of boxes, left justified, in which the number of boxes in the first row is equal to the size of the...
... side ofa hive (read left-toright) Then a is convex, i.e ≥ (ai−1 + ai+1) Put another way, the list (a1 − a0 , a2 − a1 , , an − an−1) is a weakly decreasing list of integers Proof There are two ... of h’s entries at the two obtuse angles of p is greater than or equal to the sum of h’s entries at the two acute angles of p Proof Add up all the rhombus inequalities from the rhombi inside and ... satisfies the det−1 and Pieri rules Lemma Let p be a lattice parallelogram in the hive triangle trin, with edges parallel to the edges in the triangular lattice, and h a hive of size n Then the...
... adjacency matrix ofthe graph is less than ε as long as k ≥ (cε + o(1)) log |G| Their proof involves a clever combinatorial argument that controls the behavior of random walks taken on the ... space V For A ∈ A( V ), we let A denote the operator norm ofA equal to the largest absolute value obtained by an eigenvalue ofAThe cone of positive operators P(V ) = {A ∈ A( V ) | ∀v, Av, v ≥ 0} ... element ofthe multiset of absolute values of eigenvalues of A( G) As mentioned above, a strong relationship between λ2 (·) and expansion has been achieved In particular, if G is a d-regular graph...
... and decrease the entry of p that was one larger than p1 by one Let the obtained permutation be p Similarly, decrease q1 by 1, and increase the entry of q that was one larger than q1 by Let the ... = 1, then remove these entries, to get the permutations p∗ and q∗ (After natural relabeling, these are both permutations of length n − 1.) Because ofthe extreme values of at least one ofthe ... we know of in which a sequence whose length is quadratic in terms ofthe length ofthe input objects is shown to be log-concave Theproofof our claim 2.1 The outline oftheproof It is easy to...
... functions, to evaluate the determinant ofa Hankel matrix of large Schr¨der numbers Here we use a combinatorial approach that simplifies the o evaluation ofthe Hankel determinants of large and small Schr¨der ... proof by considering a matrix of order n(n + 1) the determinant of which gives an Their proof is reduced to the computation ofthe determinant ofa Hankel matrix of order n that involves large ... tilings ofthe Aztec diamond of order n and the set of n-tuples (π1 , , πn ) of large Schr¨der paths satisfying o conditions (A1 ) and (A2 ) Proof: Given a tiling T of ADn , we associate T with an...
... lower than Cenibra in all parameters Votorantim, Suzano Bahia Sul, Ripasa and International paper are integrated producers ofthe writing and printing paper By calculating the average ofthe three ... Klabin and Aracruz that present data for years 2003, 2004 and 2005 and Cenibra which data are available only for 2005 ** The companies not have this value calculated All plants, excepting Aracruz and ... Brazilian companies and comparing with the international Paper performance, the Brazilian group reaches an average lower than foreign subsidiary in five ofthe six parameters The values reached...
... special cases of this argument, the reader can consult theproofof Lemma 5.2 of [A2 ] and the discussion at the end of Section 10 of [AC].) The terms in the expansion therefore vanish The remaining ... slightly at odds with that of [A7 ] and [A1 0].) The trace formula applies to the case ofa global field, and to a finite set of valuations V of F that contains Vram (G, ζ) We recall that Vram (G, ... global theorems, which together amount to a stabilization of each ofthe terms in the trace formula In the second paper [II], we established a key reduction in theproofof one ofthe global theorems...
... White, also, is a name ofa thing, or rather of things Whiteness, again, is the name ofa quality or attribute of those things Man is a name of many things; humanity is a name of an attribute of ... would cutthe matter short by saying, two ideas They would say, that the subject and predicate are both of them names of ideas; the idea of gold, for instance, and the idea of yellow; and that what ... The Calculation Of Chances Chapter XIX OfThe Extension Of Derivative Laws To Adjacent Cases Chapter XX Of Analogy Chapter XXI OfThe Evidence OfThe Law Of Universal Causation Chapter...
... is an anchor of {v1 , v2 } if its distances to v1 and v2 are at most 2t0 The two vertices ofa quarter that are not on the diagonal are anchors ofthe diagonal, and the diagonal may have other ... of Λ (a finite cluster of balls in the packing) that is easier to analyze than the full packing Λ The truncation parameter is the first of many decimal constants that appear Each decimal constant ... Proof Suppose that S and S are adjacent quasi-regular tetrahedra with a common face F By the Lemma 2.22, each ofthe six external faces of this √ pair of quasi-regular tetrahedra has circumradius...
... satisfaction data at and 18 months using principal component analysis with varimax rotation and Kaiser normalization to ascertain whether the eight factors (Care, Staffing, Development, Relationships, ... intentions and working as a nurse Pay was the only factor, apart from staffing on one occasion, to have adirect effect on intentions There was partial evidence ofadirect effect of job satisfaction ... order factor analyse the instrument scores since we could not assume that these factors were the manifestation ofa single underlying latent variable Factor analysis ofthe 6-month data identified...
... proofof Lemma 2, an argument based on linear programming was used It is slightly more powerful than the cloning here In theproofof Lemma 3, a more sophisticated argument due to Kalai4 , based ... that there is a point in hM ofthe M convex sets obtained as in Lemma 2, it is sufficient that either one ofthe original convex sets has at least hM clones, or a proportion of at least i ofthe ... 3, the available bounds yield a bound on the order of 350 on the number of points needed to meet each polytope (We leave this computation as an exercise for the reader.) On the other hand, the...
... The “two-path conjecture” states that if a graph G is the edge-disjoint union of two paths of length n with at least one common vertex, then the graph has a third subgraph that is also a path ... to create a new path of length n Therefore, we may assume that s ≥ and that none of {x0 , xn , y0 , yn } is among the s shared vertices We apply a result of Thomason ([8], Theorem 2.1, pages 263-4): ... union of X and Y Theorem If G decomposes into two paths X and Y , each of length n with n ≥ 2, and X and Y have least one common vertex, then G has a path of length n distinct from X and Y Proof...
... property an exponentially balanced Gray code (cf [10]), as a generalization of (totally) balanced Gray codes Thus, for an exponentially balanced Gray code G(n) one has that the transition count of ... exponentially balanced Gray code, and if n is a power of two, there exists an n-bit totally balanced Gray code Here, we shall present aproof using Theorem 2.1 Theproof is constructive like theproof ... A note on balanced Gray codes”, submitted for publication [10] A. J van Zanten and I N Suparta, “Totally balanced and exponentially balanced Gray codes”, Discrete Analysis and Operation Research,...
... existence ofa familly of scalars (λS )t∈S , such that t S ϕm (P ) = t∈S λt µt (P ) for any polynomial P in Km [T ], and achieves theproofof Lemma Before proceeding with theproofof Theorem 1, ... in P Proofof Theorem For each j in {1, , m}, we may assume that card (Sj ) = nj + (by discarding the extra elements if necessary,) then, using Lemma 2, we find a familly S of scalars (λt j ... ends theproofof Theorem References [1] Alon, N., Combinatorial Nullstellensatz Recent trends in combinatorics (M´traa h´za, 1995) Combin Probab Comput (1999), 7–29 a [2] Shirazi, H and Verstra¨te,...
... log(d(v)!)1/d(v) = v Aproofofthe Kahn-Lov´sz theorem aThe entropy proofofthe Kahn-Lov´sz theorem is complicated by the fact that there can a be edges ofthe graph (and a fixed matching) amongst the neighbors ... that H(X) = H ((XA )A A ) by repeated application of Theorem 3, part d) Thus, by the chain rule, H(X) = H ((XA )A A ) = H(XA |XB , B ≺ A) A We are now ready to present the entropy proofof Theorem ... The original proofof Kahn and Lov´sz was never published, see [3] Friedland’s proofa is based on an extension of Schrijver’s [10] proofofthe Br`gman inequality A short proof, e deducing the...
... with additional restrictions and an auxiliary parameter However, it was not apparent from those proofs why the above result was so simple and compact We will use the block decomposition method of ... exchanging values c and a at positions j and ℓ) that we must have k = b, so that b is a fixed point of π (This is the additional nice property we obtain by considering pattern 321 instead of 123.) ... of Mansour and Vainshtein [2] to make this much more apparent Proof Let π ∈ Sn contain exactly one occurrence of pattern 321 Let (c, b, a) , c > b > a, be the unique occurrence of 321 in π at positions...