We prove an explicit formula for the number of n × n upper triangular matrices, over GF (q), whose square is the zero matrix. Theorem. The number of n × n upper-triangular matrices over GF (q) (the finite field with q elements), whose square is the zero matrix, is given by the polynomial C n (q), where, C 2n (q) = j 2n n − 3j − 2n n − 3j − 1 · q n 2 −3j 2 −j , C 2n+1 (q) = j 2n + 1 n − 3j − 2n ...

In this letter, the (q, h)–analogue of Newton’s binomial formula is obtained in the (q, h)–deformed quantum plane which reduces for h = 0 to the q–analogue. For (q = 1, h = 0), this is just the usual one as it should be. Moreover, the h–analogue is recovered for q = 1. Some properties of the (q, h)–binomial coefficients are also given. This result will contribute to an introduction of the (q, h...

Our paper provides a complete characterization of leverage and default in binomial economies with financial assets serving as collateral. First, our Binomial No-Default Theorem states that any equilibrium is equivalent (in real allocations and prices) to another equilibrium in which there is no default. Thus actual default is irrelevant, though the potential for default drives the equilibrium a...

We obtain connection coefficients between q-binomial and q-trinomial coefficients. Using these, one can transform q-binomial identities into a q-trinomial identities and back again. To demonstrate the usefulness of this procedure we rederive some known trinomial identities related to partition theory and prove many of the conjectures of Berkovich, McCoy and Pearce, which have recently arisen in...

Contents 1. Introduction 1 2. The central limit theorem 1 3. The normal approximation of the binomial distribution 2 4. The normal approximation of the mixed binomial distribution 4 5. The law of large numbers approximation of a mixed binomial distribution 6 6. Finding the mixed binomial distribution by using Monte Carlo simulations 6 6.

J. Marcinkiewicz and A. Zygmund proved in 1936 that, for all functions f points x, the existence of nth Peano derivative f(n)(x) is equivalent to both f(n−1)(x) generalized Riemann $${{\tilde D}_n}f\left(x \right)$$ , based at x,x + h,x 2h,x 22{h,…,x} 2n−1h. For q ≠ 0, ±1, we introduce: two q-analogues n-th Dnf(x) Gaussian derivatives qDnf(x) $$_q{{\bar are h, x+qh, x+q2h,…, x qn−1h x+h,x qh, q...

Wendt's binomial circulant determinant, W„ , is the determinant of an m by m circulant matrix of integers, with {i, ;')th entry (i,TM.i) whenever 2 divides m but 3 does not. We explain how we found the prime factors of Wm for each even m < 200 by implementing a new method for computations in algebraic number fields that uses only modular arithmetic. As a consequence we prove that if p and q = m...