Waiting Time Problems in a Two-State Markov Chain

Let F ₀ be the event that l ₀ 0-runs of length k ₀ occur and F ₁ be the event that l ₁ 1-runs of length k ₁ occur in a two-state Markov chain. In this paper using a combinatorial method and the Markov chain imbedding method, we obtained explicit formulas of the probability generating functions of the sooner and later waiting time between F ₀ and F ₁ by the non-overlapping, overlapping and "greater than or equal" enumeration scheme. These formulas are convenient for evaluating the distributions of the sooner and later waiting time problems.     

The sequence of codimensions of PI-algebras

Bounds and asymptotic formulas are given for the size of the irreducible representations of the symmetric groups. These are applied to obtain information on the identities and codimension sequencec _n(R) of a PI-algebraR, of a PI-algebraR of characteristic zero, e.g., the “ultimate” width of the hook in which the diagrams of the cocharacters ofR lies is <=(lim c _n (R)^1/n ) ² , and lim c_n(R)^1/n≦ 2(lim c_n(R)^1/n)² for rings with no right (or left) total annihilators.     

Reflection of Long Game Formulas

We study game formulas the truth of which is determined by a semantical game of uncountable length. The main theme is the study of principles stating reflection of these formulas in various admissible sets. This investigation leads to two weak forms of strict-II¹₁ reflection (or ∑₁-compactness). We show that admissible sets such as H(ω₂) and L_ω2 which fail to have strict-II¹₁ reflection, may or may not, depending on set-theoretic hypotheses satisfy one or both of these weaker forms. Mathematics Subject Classification : 03C70, 03C75.     

The domination numbers of the 5 × n and 6 × n grid graphs

The k × n grid graph is the product P_k × P_n of a path of length k ? 1 and a path of length n ? 1. We prove here formulas found by E. O. Hare for the domination numbers of P₅ × P_n and P₆ × P_n. © 1993 John Wiley & Sons, Inc.     

The complexity of the Specht modules corresponding to hook partitions

We show that the complexity of the Specht module corresponding to any hook partition is the p-weight of the partition. We calculate the variety and the complexity of the signed permutation modules. Let E _s be a representative of the conjugacy class containing an elementary abelian p-subgroup of a symmetric group generated by s disjoint p-cycles. We give formulae for the generic Jordan types of signed permutation modules restricted to E _s and of Specht modules corresponding to hook partitions μ restricted to E _s where s is the p-weight of μ.      

An exponential lower bound to the size of bounded depth frege proofs of the pigeonhole principle

We prove lower bounds of the form exp(n^{ε d}), ε_d > 0, on the length of proofs of an explicit sequence of tautologies, based on the Pigeonhole Principle, in proof systems using formulas of depth d, for any constant d. This is the largest lower bound for the strongest proof system, for which any superpolynomial lower bounds are known.     

Probabilities in the (<Emphasis Type="Italic">k</Emphasis>, ℓ) hook

We consider particular (k, ℓ)-hook probability measures on the space of the infinite standard Young tableaux, and calculate the probability that the entry at the (1, 2) cell is odd. As n goes to infinity, this, approximately, is the corresponding probability on tableaux of size n in the (k, ℓ) hook. In few cases of small k and ℓ we find exact formulas for the corresponding numbers of such standard tableaux.     

Index Reduction Formulas for Twisted Flag Varieties, II

This is a continuation of the determination begun in K-Theory 10 (1996), 517–596, of explicit index reduction formulas for function fields of twisted flag varieties of adjoint semisimple algebraic groups. We give index reduction formulas for the varieties associated to the classical simple groups of outer type A _n-1 and D _n, and the exceptional simple groups of type E ₆ and E ₇. We also give formulas for the varieties associated to transfers and direct products of algebraic groups. This allows one to compute recursively the index reduction formulas for the twisted flag varieties of any semi-simple algebraic group.     

Expected Sums of General Parking Functions

A (u₁, u₂, . . . )-parking function of length n is a sequence (x₁, x₂, . . . , x_n) whose order statistics (the sequence (x₍₁₎, x₍₂₎, . . . , x_(n)) obtained by rearranging the original sequence in non-decreasing order) satisfy x_(i) u_(i). The Gonarov polynomials g _n (x; a₀, a ₁, . . . , a _n-1) are polynomials biorthogonal to the linear functionals (a _i) Dⁱ, where (a) is evaluation at a and D is differentiation. In this paper, we give explicit formulas for the first and second moments of sums of u-parking functions using Gonarov polynomials by solving a linear recursion based on a decomposition of the set of sequences of positive integers. We also give a combinatorial proof of one of the formulas for the expected sum. We specialize these formulas to the classical case when u _i=a+ (i-1)b and obtain, by transformations with Abel identities, different but equivalent formulas for expected sums. These formulas are used to verify the classical case of the conjecture that the expected sums are increasing functions of the gaps u_i+1 - u_i. Finally, we give analogues of our results for real-valued parking functions.AMS Subject Classification: 05A15, 05A19, 05A20, 05E35.     

A combinatorial interpretation of the inverse kostka matrix

The Kostka matrix K relates the. homogeneous and the Schur bases in the ring of symmetric functions where K _λ,μenumerates the number of column strict tableaux of shape λ and type μ. We make use of the Jacobi -Trudi identity to give a combinatorial interpretation for the inverse of the Kostka matrix in terms of certain types of signed rim hook tabloids. Using this interpretation, the matrix identity KK ^?1=Iis given a purely combinatorial proof. The generalized Jacobi-Trudi identity itself is also shown to admit a combinatorial proof via these rim hook tabloids. A further application of our combinatorial interpretation is a simple rule for the evaluation of a specialization of skew Schur functions that arises in the computation of plethysms.     

Self-dual codes over the Kleinian four group

We introduce self-dual codes over the Kleinian four group K=Z ₂×Z ₂ for a natural quadratic form on K ⁿ and develop the theory. Topics studied are: weight enumerators, mass formulas, classification up to length 8, neighbourhood graphs, extremal codes, shadows, generalized t-designs, lexicographic codes, the Hexacode and its odd and shorter cousin, automorphism groups, marked codes. Kleinian codes form a new and natural fourth step in a series of analogies between binary codes, lattices and vertex operator algebras. This analogy will be emphasized and explained in detail.     

On the Hochschild Cohomology of Triangular Algebras

Abstract

In order to study the Hochschild cohomology of an n-triangular algebra 𝒯 _n , we construct a spectral sequence, whose terms are parametrized by the length of the trajectories of the quiver associated with 𝒯 _n , and which converges to the Hochschild cohomology of 𝒯 _n . We describe explicitly its components and its differentials which are sums of cup products. In case n = 3 we study some properties of the differential at level 2. We give some examples of use of the spectral sequence and recover formulas for the dimension of the cohomology groups of particular cases of triangular algebras.     

Some Riemann boundary value problems in Clifford analysis

In this paper, we mainly study the R_m (m>0) Riemann boundary value problems for functions with values in a Clifford algebra C?(V_{3, 3}). We prove a generalized Liouville‐type theorem for harmonic functions and biharmonic functions by combining the growth behaviour estimates with the series expansions for k‐monogenic functions. We obtain the result under only one growth condition at infinity by using the integral representation formulas for harmonic functions and biharmonic functions. By using the Plemelj formula and the integral representation formulas, a more generalized Liouville theorem for harmonic functions and biharmonic functions are presented. Combining the Plemelj formula and the integral representation formulas with the above generalized Liouville theorem, we prove that the R_m (m>0) Riemann boundary value problems for monogenic functions, harmonic functions and biharmonic functions are solvable. Explicit representation formulas of the solutions are given. Copyright © 2009 John Wiley & Sons, Ltd.     

A multiset hook length formula and some applications

A multiset hook length formula for integer partitions is established by using combinatorial manipulation. As special cases, we rederive three hook length formulas, two of them obtained by Nekrasov–Okounkov, the third one by Iqbal, Nazir, Raza and Saleem, who have made use of the cyclic symmetry of the topological vertex. A multiset hook-content formula is also proved.     

On the eigenvalue problem for toeplitz matrices generated by rational functions

Formulas are given for the characteristic polynomials {p_n (λ)}and the eigenvectors of the family {T_n }of Toeplitz matrices generated by a formal Laurent series of rational function R(z). The formulas are in terms of the zeros of a certain fixed polynomial with coefficients which are simple functions of λ and the coefficients of R(z). The complexity of the formulas is independent ofn.     

Normalized local theta correspondence and the duality of inner product formulas

We conjecture that local theta correspondence can be normalized by the leading coefficient of a weighted local period integral, and that there exists a duality of local and global inner product formulas. The conjecture is verified for the pair (, PGL ₂) and (SL ₂, SO(2, 2)). As an application, global inner product formulas are obtained for liftings in the directions PGL ₂ → , GSO(2, 2) → GL ₂.     

Polynomiality of some hook-length statistics

We prove a conjecture of Okada giving an exact formula for a certain statistic for hook-lengths of partitions:

$\frac{1}{n!} \sum_{\lambda \vdash n} f_{\lambda}^2 \sum_{u \in \lambda} \prod_{i=1}^{r}\bigl(h_u^2 - i^2\bigr) = \frac{1}{2(r+1)^2} \binom{2r}{r}\binom{2r+2}{ r+1} \prod_{j=0}^{r} (n-j),$

where f _λ is the number of standard Young tableaux of shape λ and h _u is the hook length of the square u of the Young diagram of λ. We also obtain other similar formulas.

     

Weak Arithmetics and Kripke Models

In the first section of this paper we show that i Π₁ ≡ W⌝⌝lΠ₁ and that a Kripke model which decides bounded formulas forces iΠ₁ if and only if the union of the worlds in any (complete) path in it satisflies IΠ₁. In particular, the union of the worlds in any path of a Kripke model of HA models IΠ₁. In the second section of the paper, we show that for equivalence of forcing and satisfaction of Π_m‐formulas in a linear Kripke model deciding Δ₀‐formulas it is necessary and sufficient that the model be Σ_m‐elementary. This implies that if a linear Kripke model forces PEM_prenex, then it forces PEM. We also show that, for each n ≥ 1, iΦ_n does not prove ℋ︁(IΠ_n's are Burr's fragments of HA.     

A note on the sum of uniform random variables

An inductive procedure is used to obtain distributions and probability densities for the sum S_n of independent, non-equally uniform random variables. Some known results are then shown to follow immediately as special cases. Under the assumption of equally uniform random variables some new formulas are obtained for probabilities and means related to S_n. Finally, some new recursive formulas involving distributions are derived.