Approximating the diagonal of a Hessian: which sample set of points should be used

Numerical Algorithms - An explicit formula based on matrix algebra to approximate the diagonal entries of a Hessian matrix with any number of sample points is introduced. When the derivative-free...     

Polynomials That Force a Unital Ring to be Commutative

We characterize polynomials f with integer coefficients such that a ring with unity R is necessarily commutative if f(R) = 0, in the sense that f(x) = 0 for all ${x \in R}$ . Such a polynomial must be primitive, and for primitive polynomials the condition f(R) = 0 forces R to have nonzero characteristic. The task is then reduced to considering rings of prime power characteristic and the main step towards the full characterization is a characterization of polynomials f such that R is necessarily commutative if f(R) = 0 and R is a unital ring of characteristic some power of a fixed prime p.     

Polynomials which take Gaussian integer values at Gaussian integers

A factorial set for the Gaussian integers is a set G = {g₁, g₂ … g_n} of Gaussian integers such that $\text{G(z) =}\text{Π}{}_{\mathrm{k}}\text{(z ? g}{}_{\mathrm{k}}\text{)}\text{g}{}_{\mathrm{k}}$ takes Gaussian integer values at Gaussian integers. We characterize factorial sets and give a lower bound for $\text{max}{}_{\mathrm{<mtext>\parallel z\parallel 2=</mtext><mtext>n</mtext><mtext>\pi </mtext>}}\text{∥ G(z)∥}$. It is conjectured that there are infinitely many factorial sets. A Gaussian integer valued polynomial (GIP) is a polynomial with the title property. A bound similar to the above is given for max_∥z∥2=n ∥ G(z)∥ if G(z) is a GIP. There is a relation between factorial sets and testing for GIP's. We discuss this and close with some examples of factorial sets, and speculate on how to find more.     

Who should be given priority in a queue?

We consider a memoryless single server queue with two classes of customers, each having its fixed entry fee. We show that profit and social welfare may benefit from a service discipline based on relative priorities.     

Zero-Mean Cosine Polynomials which are Non-Negative for as Long as Possible

For a given integer n, all zero-mean cosine polynomials of orderat most n which are non-negative on [0,(n/(n+1))] are found,and it is shown that this is the longest interval [0,] on whichsuch cosine polynomials exist. Also, the longest interval [0,]on which there is a non-negative zero-mean cosine polynomialwith non-negative coefficients is found. As an immediate consequence of these results, the correspondingproblems of the longest intervals [,] on which there are non-positivecosine polynomials of degree n are solved. For both of these problems, all extremal polynomials are found.Applications of these polynomials to Diophantine approximationare suggested.     

A relation algebra which is not a cylindric reduct

There is an incomplete atomic relation algebra which is not the reduct of any 4-dimensional cylindric algebra. This completes the answer to a problem in [Mo61].Presented by Bjarni Jónsson.     

A Lie grading which is not a semigroup grading

This note is devoted to the construction of a graded Lie algebra, whose grading is not given by a semigroup; thus providing a counterexample to an assertion by Patera and Zassenhaus.     

Tire track geometry: Variations on a theme

We study closed smooth convex plane curves Λ enjoying the following property: a pair of pointsx, y can traverse Λ so that the distances betweenx andy along the curve and in the ambient plane do not change; such curves are calledbicycle curves. Motivation for this study comes from the problem how to determine the direction of the bicycle motion by the tire tracks of the bicycle wheels; bicycle curves arise in the (rare) situation when one cannot determine which way the bicycle went. We discuss existence and non-existence of bicycle curves, other than circles; in particular, we obtain restrictions on bicycle curves in terms of the ratio of the length of the arcxy to the perimeter, length of Λ, the number and location of their vertices, etc. We also study polygonal analogs of bicycle curves, convex equilateraln-gonsP whosek-diagonals all have equal lengths. For some values ofn andk we prove the rigidity result thatP is a regular polygon, and for some we construct flexible bicycle polygons. Partially supported by an NSF grant.     

Polynomials in a matrix

Let A be a matrixp(x) a polynomial. Put B=p(A). It is shown that necessary and sufficient conditions for A to be a polynomial in B are (i) if λ is any eigenvalue of A, and if some elementary divisor of A corresponding to λ is nonlinear, thenp ^′(λ)≠0;and (ii) if λ,μ are distinct eigenvalues of A, then p(λ)p(μ) are also distinct. Here all computations are over some algebraically closed field.     

Polynomials in a matrix

Let A be a matrixp(x) a polynomial. Put B=p(A). It is shown that necessary and sufficient conditions for A to be a polynomial in B are (i) if λ is any eigenvalue of A, and if some elementary divisor of A corresponding to λ is nonlinear, thenp^'(λ)≠0;and (ii) if λ,μ are distinct eigenvalues of A, then p(λ)p(μ) are also distinct. Here all computations are over some algebraically closed field.     

On a partial integral which can be derived from poisson matrix

Consider a surface which is a common level of some functions. Suppose that this surface is invariant under a Hamiltonian system. The question is if a partial integral can be derived explicitly from the Poisson matrix of these functions. In some cases such an integral is equal to the determinant of the matrix. This paper establishes a necessary and sufficient condition for this to hold true. The partial integral that results is not trivial if the induced Poisson structure is non-degenerate at one point at least. Therefore, the invariant surface must be even-dimensional.     

On a construction of critical graphs which are not sensitive

Let ?(G) be theclosed-set lattice of a graphG. G issensitive if the following implication is always true for any graphG′: ?(G)??(G′)?(G)?G′G iscritical if ?(G)??(G-e) for anye inE(G) and ?(G)??(G+e) for anye in \(\left( {\bar G} \right)\) where \(\bar G\) is the complement ofG. Every sensitive graph is, a fortiori, critical. Is every critical graph sensitive? A negative answer to this question is given in this note.     

To be or not to be constructive,that is not the question

In the early twentieth century, L.E.J. Brouwer pioneered a new philosophy of mathematics, called intuitionism. Intuitionism was revolutionary in many respects but stands out – mathematically speaking – for its challenge of Hilbert’s formalist philosophy of mathematics and rejection of the law of excluded middle from the ‘classical’ logic used in mainstream mathematics. Out of intuitionism grew intuitionistic logic and the associated Brouwer–Heyting–Kolmogorov interpretation by which ‘there exists $x$’ intuitively means ‘an algorithm to compute $x$ is given’. A number of schools of constructive mathematics were developed, inspired by Brouwer’s intuitionism and invariably based on intuitionistic logic, but with varying interpretations of what constitutes an algorithm. This paper deals with the dichotomy between constructive and non-constructive mathematics, or rather the absence of such an ‘excluded middle’. In particular, we challenge the ‘binary’ view that mathematics is either constructive or not. To this end, we identify a part of classical mathematics, namely classical Nonstandard Analysis, and show it inhabits the twilight-zone between the constructive and non-constructive. Intuitively, the predicate ‘$x$ is standard’ typical of Nonstandard Analysis can be interpreted as ‘$x$ is computable’, giving rise to computable (and sometimes constructive) mathematics obtained directly from classical Nonstandard Analysis. Our results formalise Osswald’s longstanding conjecture that classical Nonstandard Analysis is locally constructive. Finally, an alternative explanation of our results is provided by Brouwer’s thesis that logic depends upon mathematics.