Euclid,Fibonacci, and Pascal—recursed!

Selected algorithms of three famous mathematicians are expressed in recursive computer programmes using a programming language (APL). Specifically: (a) Euclid's greatest common divisior algorithm; (b) Fibonacci's ‘ golden ‘ number series; (c) Pascal's triangle.

The programmes are designed to be viewed directly by students in order to elucidate the algorithms. Such programmes can be stimulants for learning mathematics.     

n-tribonacci triangles and applications

In this note we introduce n-tribonacci triangle, which is similar to Pascal's triangle, to derive an explicit formula for the tribonacci numbers by using some properties of our triangle.     

On some methods of deriving geometric expectation

Formulas presented for the calculation of ∑ _j=1 ⁿ j^k (n, k ∈ N do not have a closed form; they are in the form of recursive or complex formulas. Here an attempt is made to present a simple formula in which it is only necessary to compute the numerical coefficients in a recursive form, and the coefficients in turn follow a simple pattern (almost similar to Pascal's Triangle). Although the pattern for calculating numerical coefficients based on forming a table is easy, non-recursive formulas are presented to determine the numerical coefficients.     

The Mathematical Basis of Phenotype Ratios

Certain classical experiments in genetics provide interesting ratios amongst the offspring population which relate to the pattern of the inheritance of the parents’ features. In this paper these ratios are analysed.

The occurrence of each class in each ratio is found to be 3 ^n‐x and the sum of the classes of a ratio to be 4ⁿ where nis the number of genes in the system and xthe number of features which are recessive. The classes when grouped show group coefficients to be in accordance with Pascal's Triangle, and the ratios themselves are shown to be a particular case of the binomial expansion (p+q)ⁿ where pis 3 and qis 1.

Some aspects of the value of this analysis to a biologist are considered.

     

Unavoidable sequences in constructive analysis

Five recursively axiomatizable theories extending Kleene's intuitionistic theory FIM of numbers and numbertheoretic (choice) sequences are introduced and shown to be consistent, by a modified relative realizability interpretation which verifies that every sequence classically defined by a Π¹₁ formula is unavoidable (cannot fail to exist) and that no sequence can fail to be classically Δ¹₁. The analytical form of Markov's Principle fails under the interpretation. The notion of strongly inadmissible rule of inference is introduced, with examples (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

M-bonomial coefficients and their identities

In this note, we introduce M-bonomial coefficients or (M-bonacci binomial coefficients). These are similar to the binomial and the Fibonomial (or Fibonacci–binomial) coefficients and can be displayed in a triangle similar to Pascal's triangle from which some identities become obvious.     

New large sets of t‐designs

Abstact: We introduce generalizations of earlier direct methods for constructing large sets of t‐designs. These are based on assembling systematically orbits of t‐homogeneous permutation groups in their induced actions on k‐subsets. By means of these techniques and the known recursive methods we construct an extensive number of new large sets, including new infinite families. In particular, a new series of LS[3](2(2 + m), 8·3^m ? 2, 16·3^m ? 3) is obtained. This also provides the smallest known ν for a t‐(ν, k, λ) design when t ≥ 16. We present our results compactly for ν ≤ 61, in tables derived from Pascal's triangle modulo appropriate primes. © 2000 John Wiley & Sons, Inc. J Combin Designs 9: 40–59, 2001     

Score sequences of oriented graphs

We extend Landau's concept of the score structure of a tournament to that of the score sequence of an oriented graph, and give a condition for an arbitrary integer sequence to be a score sequence. The proof is by construction of a specific oriented graph Δ(S) with given score sequence S. It is shown that Δ(S) is transitive and has the minimum number of arcs among the oriented graphs with score sequence S.     

Type,Cotype Constants and Clarkson's Inequalities for Banach Spaces

It is shown that a Banach space satisfies Clarkson's inequalities if and only if its “type or cotype constant” is 1, which implies in particular that the notions of G_o - and G_n - Fourier type by Milman [16] are equivalent. A sequence of related results is also given.     

A Structure Preserving Embedding of Semigroups Into Regular Semigroups

For any semigroup S a regular semigroup 𝒞(S) that embeds S can be constructed as the direct limit of a sequence of semigroups each of which contains a copy of its predecessor as a subsemigroup whose elements are regular. The construction is modified here to obtain an embedding of S into a regular semigroup R such that the nontrivial maximal subgroups of R are isomorphic to the Schützenberger groups of S and such that the restriction to S of any of Green's relations on R is the corresponding Green's relation on S.     

Approximation of the vth Root of N

We present a class of functions g_K(w), K ≥ 2, for which the recursive sequences w_{n + 1} = g_K(w_n) converge to N^1/v with relative error . Newton's method results when K = 2. The coefficients of the g_K(w) form a triangle, which is Pascal's for v = 2. In this case, if w₁ = x₁/y₁, where x₁, y₁ is the first positive solution of Pell's equation x² ? Ny² = 1, then w_{n + 1} = x_{n + 1}/y_{n + 1} is the Kⁿpth or 2Kⁿpth convergent of the continued fraction for , its period p being even or odd.     

Modifications in the discount sequence for bandit processes

For discrete–timeK-armed Bandit Processes with discounting, the value V(G,A) represents an observer's optimal expected gain using discount sequence A and prior distribution G on the distributions governing the K arms or processes being observed. In this paper we address the question of which types of modifications in the discount sequence will change the value in a predictable manner. Both positive and negative results are obtained. For example, it is shown that for two-armed bandits with one arm known, permutations of a positive term with a zero term in the discount sequence will change the value in a predictable way, but that this result can not be ex­tended to general two-armed bandits. We also show that there does not exist a ldquo;best information” arm in the sense that if the first term in each of two different discount sequences is zero then under G the same arm will be selected for each discount sequence     

Leap-frogging Newton's method

Using Newton's method as an intermediate step, we introduce an iterative method that approximates numerically the solution of f (x) = 0. The method is essentially a leap-frog Newton's method. The order of convergence of the proposed method at a simple root is cubic and the computational efficiency in general is less, but close to that of Newton's method. Like Newton's method, the new method requires only function and first derivative evaluations. The method can easily be implemented on computer algebra systems where high machine precision is available.     

On the trace problem for Lizorkin–Triebel spaces with mixed norms

The subject is traces of Sobolev spaces with mixed Lebesgue norms on Euclidean space. Specifically, restrictions to the hyperplanes given by x₁ = 0 and x_n = 0 are applied to functions belonging to quasi‐homogeneous, mixed norm Lizorkin–Triebel spaces ; Sobolev spaces are obtained from these as special cases. Spaces admitting traces in the distribution sense are characterised up to the borderline cases; these are also covered in case x₁ = 0. For x₁ the trace spaces are proved to be mixed‐norm Lizorkin–Triebel spaces with a specific sum exponent; for x_n they are similarly defined Besov spaces. The treatment includes continuous right‐inverses and higher order traces. The results rely on a sequence version of Nikol'skij's inequality, Marschall's inequality for pseudodifferential operators (and Fourier multiplier assertions), as well as dyadic ball criteria. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Eigenvalues of Mathieu's Equation and their Branch Points

A comprehensive account is given of the behavior of the eigenvalues of Mathieu's equation as functions of the complex variable q. The convergence of their small-q expansions is limited by an infinite sequence of rings of branch points of square-root type at which adjacent eigenvalues of the same type become equal. New asymptotic formulae are derived that account for how and where the eigenvalues become equal. Known asymptotic series for the eigenvalues apply beyond the rings of branch points; we show how they can now be identified with specific eigenvalues.     

A construction of binary Golay sequence pairs from odd‐length Barker sequences

Binary Golay sequence pairs exist for lengths 2, 10 and 26 and, by Turyn's product construction, for all lengths of the form 2^a10^b26^c where a, b, c are non‐negative integers. Computer search has shown that all inequivalent binary Golay sequence pairs of length less than 100 can be constructed from five “seed” pairs, of length 2, 10, 10, 20 and 26. We give the first complete explanation of the origin of the length 26 binary Golay seed pair, involving a Barker sequence of length 13 and a related Barker sequence of length 11. This is the special case m=1 of a general construction for a length 16m+10 binary Golay pair from a related pair of Barker sequences of length 8m+5 and 8m+3, for integer m≥0. In the case m=0, we obtain an alternative explanation of the origin of one of the length 10 binary Golay seed pairs. The construction cannot produce binary Golay sequence pairs for m>1, having length greater than 26, because there are no Barker sequences of odd length greater than 13. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 478–491, 2009     

Note on the heights of random recursive trees and random m-ary search trees

A process of growing a random recursive tree T_n is studied. The sequence {T_n} is shown to be a sequence of “snapshots” of a Crump–Mode branching process. This connection and a theorem by Kingman are used to show quickly that the height of T_n is asymptotic, with probability one, to c log n. In particular, c = e = 2.718 … for the uniform recursive tree, and c = (2γ)^?1, where γe^1+γ = 1, for the ordered recursive tree. An analogous reduction provides a short proof of Devroye's limit law for the height of a random m-ary search tree. We show finally a close connection between another Devroye's result, on the height of a random union-find tree, and our theorem on the height of the uniform recursive tree. © 1994 John Wiley & Sons, Inc.     

Convergence of a large time step generalization of Godunov's method for conservation laws

A natural generalization of Godunov's method for Courant numbers larger than 1 is obtained by handling interactions between neighboring Riemann problems linearly, i.e., by allowing waves to pass through one another with no change in strength or speed. This method is well defined for arbitrarily large Courant numbers and can be written in conservation form. It follows that if a sequence of approximations converges to a limit u(x,t) as the mesh is refined, then u is a weak solution to the system of conservation laws. For scalar problems the method is total variation diminishing and every sequence contains a convergent subsequence. It is conjectured that in fact every sequence converges to the (unique) entropy solution provided the correct entropy solution is used for each Riemann problem. If the true Riemann solutions are replaced by approximate Riemann solutions which are consistent with the conservation law, then the above convergence results for general systems continue to hold.     

Vertex 2-isomorphism

Whitney's theorem on 2-isomorphism characterizes the set of graphs having the same cycles as a given graph, where a cycle is regarded as a set of edges. In this paper, vertex 2-isomorphism is defined and used to prove a vertex analogue of Whitney's theorem. The main theorem states that two connected graphs have the same set of cycles, where a cycle is now regarded as a set of vertices, if and only if one can be obtained from the other by a sequence of simple operations. © 1992 John Wiley & Sons, Inc.     

The interpolation problem for ridge functions

It is shown that the interpolation problem for ridge functions can be solved if and only if the rank of a certain matrix A equals the number of interpolation points. The elements of the matrix A are either 0 or 1 and can be easilyfound from the arguments of the unknown functions. It is shown that Sun's Characteristic, or incidence matrix C is given by C = AA ^T . From this it follows that the rank condition is equivalent to Sun's positive definite C condition.