Some limit results for longest common subsequences

The length l_n of a longest common subsequence before time n sequences (B₁₁, B₁₂, …) (B₂₁, B₂₂, …) is the cardinality of the largest increasing set of pairs of integers {(j_1α, j_2α)}

$\text{l?}\text{j}{}_{11}\text{<}\text{j}{}_{12}\text{<·<}\text{j}{}_{11}{}_{\mathrm{n}}\text{?}\text{n}\text{l?}\text{j}{}_{21}\text{<}\text{j}{}_{22}\text{<·<}\text{j}{}_{21}{}_{\mathrm{n}}\text{?}\text{n}$

such that ?1?α?l_n, (B_1j1α=B_2j2α). If B₁ and B₂ are independent random sequences with co-ordinates i.i.d. uniform on {1, 2, …, k}, it follows from Kingman's subadditive ergodic theorem that the ratio l_n/n converges to a constant c_k a.s. A method of deriving lower bounds for the constants c_k is given, the bounds obtained improving known lower bounds, for k>2. The rate of decrease of c_k with k is shown to be no faster than 1/√k, contrasting with P{B_1i?B_2i}=1/k. Finally, an alternative method of deriving lower bounds is given and used to improve the lower bound for c₂.     

An extremal problem for sets with applications to graph theory

Let X₁, …, X_n be n disjoint sets. For 1 ? i ? n and 1 ? j ? h let A_ij and B_ij be subsets of X_i that satisfy |A_ij| ? r_i and |B_ij| ? s_i for 1 ? i ? n, 1 ? j ? h, $\text{(∪}{}_{\mathrm{i}}\text{A}{}_{\mathrm{ij}}\text{) ∩ (∪}{}_{\mathrm{i}}\text{B}{}_{\mathrm{ij}}\text{) = ?}$ for 1 ? j ? h, $\text{(∪}{}_{\mathrm{i}}\text{A}{}_{\mathrm{ij}}\text{) ∩ (∪}{}_{\mathrm{i}}\text{B}{}_{\mathrm{il}}\text{) ≠ ?}$ for 1 ? j < l ? h. We prove that $\text{h?}\text{Π}\text{i=1}\text{n}\text{r}{}_{\mathrm{i}}\text{+s}{}_{\mathrm{i}}\text{r}{}_{\mathrm{i}}$. This result is best possible and has some interesting consequences. Its proof uses multilinear techniques (exterior algebra).     

A complete set of invariants for finite groups and other results

For a finite group G and a set I ? {1, 2,…, n} let

${}_{\mathrm{G}}\text{(n,I) = ∑}{}_{\mathrm{g\; \in \; G}}\text{ε}{}_1\text{(g)?ε}{}_2\text{(g)???ε}{}_{\mathrm{n}}\text{(g)}$

,where

$\text{ε}{}_{\mathrm{i}}\text{(g)=g}\text{if}\text{i=∈ I,}$

$\text{ε}{}_{\mathrm{l}}\text{(g)=l}\text{if}\text{i=∈ I.}$

We prove, among other results, that the positive integers

$\text{tr}\text{(e}{}_{\mathrm{G}}\text{(n,I}{}_1\text{)+?+e}{}_{\mathrm{G}}\text{(n,I}{}_{\mathrm{r}}\text{))}{}^{\mathrm{k}}\text{:n,r,k,?1, I}{}_{\mathrm{j}}\text{?{1,…,n}, 1?|i}{}_{\mathrm{j}}\text{|?3}$

for 1 ? j ? r, I_j1 ∩ I_j2 ∩ I_j3 ∩ I_j4 = Ø for any 1 ? j₁ <j₂ <j₃ <j₄ ? r, determine G up to isomorphism. We also show that under certain assumptions finite groups are determined up to isomorphism by the number of their subgroups.     

Unitary interpolants,factorization indices and infinite Hankel block matrices

The existence, uniqueness, and construction of unitary n × n matrix valued functions $\text{?(ζ) = ∑}{}_{\mathrm{j\; =\; ?\infty }}{}^{\mathrm{\infty }}\text{?}{}_{\mathrm{j}}\text{ζ}{}^{\mathrm{j}}$ in Wiener-like algebras on the circle with prescribed matrix Fourier coefficients $\text{?}{}_{\mathrm{j}}\text{= γ}{}_{\mathrm{j}}$ for j ? 0 are studied. In particular, if $\text{Σ ¦γ}{}_{\mathrm{j}}\text{¦ < ∞}$, then such an ? exists with $\text{Σ ¦?}{}_{\mathrm{j}}\text{¦ < ∞}$ if and only if ∥Γ₀∥ ? 1, where Γ_v, denotes the infinite block Hankel matrix (γ_{j + k + v}), j, k = 0, 1,…, acting in the sequence space l_n². One of the main results is that the nonnegative factorization indices of every such ? are uniquely determined by the given data in terms of the dimensions of the kernels of $\text{I ? Γ}{}_{\mathrm{v}}{}^1\text{Γ}{}_{\mathrm{v}}$, whereas the negative factorization indices are arbitrary. It is also shown that there is a unique such ? if and only if the data forces all the factorization indices to be nonnegative and simple conditions for that and a formula for ? in terms of certain Schmidt pairs of Γ₀ are given. The results depend upon a fine analysis of the structure of the kernels of $\text{I ? Γ}{}_{\mathrm{v}}{}^1\text{Γ}{}_{\mathrm{v}}$ and of the one step extension problem of Adamjan, Arov, and Krein (Funct. Anal. Appl.2 (1968), 1–18). Isometric interpolants for the nonsquare case are also considered.     

An asymptotic evaluation of the cycle index of a symmetric group

Let $\text{Z(S}{}_{\mathrm{n}}\text{;?(x))}$ denote the polynomial obtained from the cycle index of the symmetric group Z(S_n) by replacing each variable s_i by f(x¹). Let f(x) have a Taylor series with radius of convergence ? of the form f(x)=x^k + a_k+1x^k+1 + a_k+2x^k+2+? with every a₁?0. Finally, let 0<x<1 and let x??. We prove that

This limit is used to estimate the probability (for n and p both large) that a point chosen at random from a random p-point tree has degree n + 1. These limiting probabilities are independent of p and decrease geometrically in n, contrasting with the labeled limiting probabilities of $\text{1}\text{n!e}$.In order to prove the main theorem, an appealing generalization of the principle of inclusion and exclusion is presented.     

On a conjecture by D. Ž. D̄okovic

It is shown that if $\text{A?Ω}{}_{\mathrm{n}}\text{?{J}{}_{\mathrm{n}}\text{}}$ satisfies

$\text{nkσ}{}_{\mathrm{k}}\text{(A)?(n?k+1)}{}^2\text{σ}{}_{\mathrm{k?1}}\text{(A)}$

$\text{(k=1,2,…,n)}$

, where σ_k(A) denotes the sum of all kth order subpermanent of A, then Per[λJ_n+(1?λ)A] is strictly decreasing in the interval 0<λ<1.     

The Dirichlet problem for the Kohn Laplacian on the Heisenberg group,I

For (x,y,t)∈$\text{R}$ⁿ × $\text{R}$ⁿ × $\text{R}$, denote $\text{X}{}_{\mathrm{j}}\text{=}\text{?}\text{?x}{}_{\mathrm{j}}\text{+ 2y}{}_{\mathrm{j}}\text{?}\text{?t}\text{, y}{}_{\mathrm{j}}\text{=}\text{?}\text{?y}{}_{\mathrm{j}}\text{? 2x}{}_{\mathrm{j}}\text{?}\text{?t}$ and $\text{L}{}_{\mathrm{\alpha }}\text{=?}\text{1}\text{4}\text{∑}\text{j=1}\text{n}\text{X}{}_{\mathrm{j}}{}^2\text{+ Y}{}_{\mathrm{j}}{}^2\text{+}\text{?}\text{?t}$. When α = n ? 2q, $\text{L}$_a represents the action of the Kohn Laplacian □_b on q-forms on the Heisenberg group. For ?n < α < n, we construct a parametrix for the Dirichlet problem in smooth domains D near non-characteristic points of ?D. A point w of ?D is non-characteristic if one of X₁,…, X_n, Y₁,…, Y_n is transverse to ?D at w. This yields sharp local estimates in the Dirichlet problem in the appropriate non-isotropic Lipschitz classes. The main new tool is a “convolution calculus” of pseudo-differential operators that can be applied to the relevant layer potentials, for which the usual asymptotic composition formula is false. Characteristic points are treated in Part II.     

A decomposition theorem for an analytic function

Let f(z), an analytic function with radius of convergence R (0 < R < ∞) be represented by the gap series ∑_{k = 0}^∞c_kz^λ_k. Set and define the growth constants ?, λ, T, t by

$\text{?}\text{λ}\text{=}\text{lim sup}\text{r→R}\text{inf}\text{{}\text{log}\text{[Rr /(R?r)]}{}^{\mathrm{?1}}\text{log}{}^{\mathrm{+}}\text{log}{}^{\mathrm{+}}\text{M(r)}}$

, and if 0 < ? < ∞,

$\text{T}\text{t}\text{=}\text{lim sup}\text{r→R}\text{inf}\text{{[Rr /(R?r)]}{}^{\mathrm{??}}\text{log}{}^{\mathrm{+}}\text{M(r)}}$

. Then, assuming 0 < t < T < ∞, we obtain a decomposition theorem for f(z).     

A partition identity

Let n₁+n₂+?+n_m=n where the n_i's are integers (possibly negative or greater than n). Let p=(k₁,…,k_m), where k₁+k₂+?+k_m=k, be a partition of the nonnegative integer k into m nonnegative integers and let P denote the set of all such partitions. For m?2, we prove the combinatorial identity

$\text{∑}\text{p∈P}\text{∏}\text{i=1}\text{m}\text{n}{}_{\mathrm{i}}\text{+1?k}{}_{\mathrm{i}}\text{k}{}_{\mathrm{i}}\text{=}\text{∑}\text{i?0}\text{j+m?2}\text{m?2}\text{n+1?k?2}{}_{\mathrm{j}}\text{k?2}{}_{\mathrm{j}}$

which implies the surprising result that the left side of the above equation depends on n but not on the n_i's.     

Asymptotic properties of general symmetric hyperbolic systems

Results on partition of energy and on energy decay are derived for solutions of the Cauchy problem $\text{?u}\text{?t}\text{+ ∑}{}_{\mathrm{j\; =\; 1}}{}^{\mathrm{n}}\text{A}{}_{\mathrm{j}}\text{?u}\text{?x}{}_{\mathrm{j}}\text{= 0, u(0, x) = ?(x)}$. Here the A_j's are constant, k × k Hermitian matrices, x = (x₁,…, x_n), t represents time, and u = u(t, x) is a k-vector. It is shown that the energy of Mu approaches a limit $\text{E}{}_{\mathrm{M}}\text{(?)}\text{as}\text{¦ t ¦ → ∞}$, where M is an arbitrary matrix; that there exists a sufficiently large subspace of data ?, which is invariant under the solution group U₀(t) and such that $\text{U}{}_0\text{(t)? = 0}\text{for}\text{¦ x ¦ ? a ¦ t ¦ ? R, a}\text{and}\text{R}$ depending on ? and that the local energy of nonstatic solutions decays as $\text{¦ t ¦ → ∞}$. More refined results on energy decay are also given and the existence of wave operators is established, considering a perturbed equation $\text{E(x)}\text{?u}\text{?t}\text{+ ∑}{}_{\mathrm{j\; =\; 1}}{}^{\mathrm{n}}\text{A}{}_{\mathrm{j}}\text{?u}\text{?x}{}_{\mathrm{j}}\text{= 0,}\text{where}\text{¦ E(x) ? I ¦ = O(¦ x ¦}{}^{\mathrm{?1\; ?\; ?}}\text{)}$ at infinity.     

Non-linear autonomous systems of differential equations and Carleman linearization procedure

The non-linear autonomous of differential equations

$\text{x}\text{?}{}_{\mathrm{i}}\text{=}\text{∑}\text{j}\text{a}{}_{\mathrm{ij}}\text{x}{}_{\mathrm{j}}\text{+}\text{∑}\text{j,k}\text{b}{}_{\mathrm{ijk}}\text{x}{}_{\mathrm{j}}\text{x}{}_{\mathrm{k}}\text{(}\text{x}\text{?}{}_{\mathrm{i}}\text{=dx}{}_{\mathrm{i}}\text{/dt, i, j, k= 1,2,…n)}$

which plays an important role in chemical kinetics and other fields of physics (turbulence and plasma physics) is investigated using the Carleman linearization procedure.     

Pairs of additive equations II. Large odd degree

Let k be an odd positive integer. Davenport and Lewis have shown that the equations

$\text{a}{}_1\text{x}{}_1{}^{\mathrm{k}}\text{+…+a}{}_{\mathrm{n}}\text{x}{}_{\mathrm{n}}{}^{\mathrm{k}}\text{=0}$

with integer coefficients, have a nontrivial solution in integers x₁,…, x_N provided that

$\text{N?[36klog6k]}$

Here it is shown that for any ? > 0 and k > k₀(?) the equations have a nontrivial solution provided that

$\text{N?}\text{8}\text{log 2}\text{+?}\text{k log k.}$

     

Pancyclic graphs and a conjecture of Bondy and Chvátal

A p-vertex graph is called pancyclic if it contains cycles of every length l, 3 ≤ l ≤ p. In this paper we prove the following conjecture of Bondy and Chvátal: If a graph G has vertex degree sequence d₁ ≤ d₂ ≤ … ≤ d_ν, and if $\text{d}{}_{\mathrm{k}}\text{≤ k <}\text{p}\text{2}$ implies d_ν?k ≥ p ? k, then G is pancyclic or bipartite.     

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].     

Toeplitz matrices commuting with tridiagonal matrices

we prove that if R is a nonscalar Toeplitz matrix R_{i, j}=r_?i?j? which commutes with a tridiagonal matrix with simple spectrum, then

$\text{r}{}_{\mathrm{k}}\text{r}{}_1\text{=}\text{u}{}_{\mathrm{k-1}}\text{r}{}_2\text{r}{}_1\text{cos p}\text{u}{}_{\mathrm{k-1}}\text{(cos p)}$

, k=4, 5,…, with U_k the Chebychev polynomial of the second kind, where p is determined from

$\text{cos p=}\text{1}\text{2}\text{r}{}^2{}_1\text{?r}{}_1\text{r}{}_3\text{r}{}^2{}_2\text{?r}{}_1\text{r}{}_3$

.     

Votes and a half-binomial

In two party elections with popular vote ratio $\text{p}\text{q}\text{,}\text{1}\text{2}\text{≤p=1 ?q}$, a theoretical model suggests replacing the so-called MacMahon cube law approximation $\text{(}\text{p}\text{q}\text{)}{}^3$, for the ratio $\text{P}\text{Q}$ of candidates elected, by the ratio $\text{?}{}_{\mathrm{k}}\text{(p)}\text{?}{}_{\mathrm{k}}\text{(q)}$ of the two half sums in the binomial expansion of (p+q)^2k+1 for some k. This ratio is nearly $\text{(}\text{p}\text{q}\text{)}{}^3$ when k = 6. The success probability $\text{g}{}_{\mathrm{k}}\text{(p)=(}\text{p}{}^{\mathrm{a}}\text{(p}{}^{\mathrm{a}}\text{+q}{}^{\mathrm{a}}\text{)}$ for the power law $\text{(}\text{p}\text{q}\text{)}{}^{\mathrm{a}}\text{?}\text{P}\text{Q}$ is shown to so closely approximate $\text{?}{}_{\mathrm{k}}\text{(p)=Σ}{}_0{}^{\mathrm{k}}\text{(}{}_{\mathrm{r}}{}^{\mathrm{2k+1}}\text{)p}{}^{\mathrm{2k+1?r}}\text{q}{}^{\mathrm{r}}$, if we choose $\text{a = a}{}_{\mathrm{k}}\text{=}\text{(2k+1)!}\text{4}{}^{\mathrm{k}}\text{k!k!}$, that $\text{1≤}\text{?}{}_{\mathrm{k}}\text{(p)}\text{g}{}_{\mathrm{k}}\text{(p)}\text{≤1.01884086}$ for $\text{k≥1}\text{if}\text{1}\text{2}\text{≤p≤1}$. Computationally, we avoid large binomial coefficients in computing $\text{?}{}_{\mathrm{k}}\text{(p)}$ for k>22 by expressing $\text{2?}{}_{\mathrm{k}}\text{(p)?1}$ as the sum $\text{(p?q) Σ}{}_0{}^{\mathrm{k}}\text{(4pq)}{}^{\mathrm{s}}\text{a}{}_{\mathrm{s}}\text{(2s+1)}$, whose terms decrease by the factors $\text{(4pq)(1?}\text{1}\text{2s}\text{)}$. Setting K = 4k+3, we compute a_k for the large k using a continued fraction $\text{πa}{}_{\mathrm{k}}{}^2\text{=}\text{K+1}{}^2\text{(}\text{2K+3}{}^2\text{(}\text{2K+5}{}^2\text{(2K+…)}\text{)}\text{)}$ derived from the ratio of π to the finite Wallis product approximation.     

On the closability of some positive definite symmetric differential forms on c0∞(ω)

Let $\text{S}$ be a Dirichlet form in $\text{L}{}^2\text{(Ω; m)}$, where Ω is an open subset of $\text{R}$ⁿ, n ? 2, and m a Radon measure on Ω; for each integer k with 1 ? k < n, let $\text{S}$_k be a Dirichlet form on some k-dimensional submanifold $\text{Ω}{}_{\mathrm{k}}$ of Ω. The paper is devoted to the study of the closability of the forms E with domain $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ and defined by: $\text{(?,g)=}$$\text{E}$$\text{(?, g)+}\text{∑}\text{i}\text{p}\text{=1}$$\text{E}$k_i where 1 ? k_p < ? < n, and where , g^k_i denote restrictions of ?, g in $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ to . Conditions are given for E to be closable if, for each i = 1,…, p, one has k_i = n ? i. Other conditions are given for E to be nonclosable if, for some i, k_i < n ? i.     

On fair coin-tossing games

Let Ω be a finite set with k elements and for each integer $\text{n ≧ 1}$ let $\text{Ω}{}_{\mathrm{n}}\text{= Ω × Ω × … × Ω}$ (n-tuple) and $\text{Ω}{}_{\mathrm{n}}\text{= {(a}{}_1\text{, a}{}_2\text{,…, a}{}_{\mathrm{n}}\text{) | (a}{}_1\text{, a}{}_2\text{,…, a}{}_{\mathrm{n}}\text{) ∈ Ω}{}_{\mathrm{n}}$ and a_j ≠ a_j+1 for some 1 ≦ j ≦ n ? 1}. Let {Y_m} be a sequence of independent and identically distributed random variables such that P(Y₁ = a) = k^?1 for all a in Ω. In this paper, we obtain some very surprising and interesting results about the first occurrence of elements in $\text{Ω}{}_{\mathrm{n}}$ and in Ω?_n with respect to the stochastic process {Y_m}. The results here provide us with a better and deeper understanding of the fair coin-tossing (k-sided) process.     

A process of runs and its convergence to the brownian motion

Let X₁,X₂,… be i.i.d. random variables with a continuous distribution function. Let R₀=0, R_k=min{j>R_k?1, such that X_j>X_j+1}, k?1. We prove that all finite-dimensional distributions of a process $\text{W}{}^{\mathrm{(n)}}\text{(t)=}\text{(R}{}_{\mathrm{[nt]}}\text{?2[nt])}\text{2}\text{3}\text{n}\text{, t ? [0,1]}$, converge to those of the standard Brownian motion.