Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsH. FejzićHAJRUDIN FEJZIć grew up in Bosnia. He received his Ph.D. from Michigan State University and has taught at CSUSB since 1994. He enjoys doing home improvement projects, gardening, and visiting national parks.C. FreilingCHRIS FREILING received his Ph.D. from University of California, Los Angeles, in 1981 and is now professor of mathematics at CSUSB. He also does consulting work for the Center for Communications Research in San Diego, CA. He loves God, family, friends, surfing (in water), and ultimate frisbee.D. RinneDAN RINNE received his Ph.D. from University of California, Santa Barbara, in 1979 and has taught at CSUSB since 1982. He reads a few pages of P. G. Wodehouse every day.

Functional differences that lead to generalized Riemann derivatives were studied by Ash and Jones in (1987). They gave a partial answer as to when these differences satisfy an analog of the Mean Value Theorem. Here we give a complete classification.

Theorem 1 If a finite number of rectangles, every one of which has at least one integer side, perfectly tile a big rectangle, then the big rectangle also has at least one integer side. Fourteen proofs of theorem 1 were published by Wagon : Wagon, S. (1987) Fourteen proofs of a result about tiling a rectangle.The American Mathematical Monthly 94 (7) : 601-617. In the following we establish two proofs of this theorem. These proofs generalize to other situations. In particular Theorem 1 generalizes to the two following statements

A decomposition of a real functionfof one variable is a system of closed setsANand derivativesvNsuch that ⋃N=1AN=Randf=vNonAN. It is known that Peano derivatives and approximate derivatives have decompositions. We extend the notion of decomposition to functions ofmvariables and provem-dimensional analogs of these results. We also show that them-dimensional approximate derivative is Baire 1.

It is shown that n times Peano differentiable functions defined on a closed subset of \(\mathbb{R}^m \) and satisfying a certain condition on that set can be extended to n times Peano differentiable functions defined on \(\mathbb{R}^m \) if and only if the nth order Peano derivatives are Baire class one functions.

We show that the following are equivalent: (i) A rectangle of eccentricity v can be tiled using rectangles of eccentricity u. (ii) There is a rational function with rational coefficients, Q(z) , such that v = Q(u) and Q maps each of the half-planes { z ¦ Re( z ) < 0} and { z ¦ Re( z ) > 0 into itself, (iii) There is an odd rational function with rational coefficients, Q(z) , such that v = Q(u) and all roots of v = Q( z ) have a positive real part. All rectangles in this article have sides parallel to the coordinate axes and all tilings are finite. We let R(x, y) denote a rectangle with base x and height y. In 1903 Dehn [1 ] proved his famous result that R(x, y) can be tiled by squares if and only if y/x is a rational number. Dehn actually proved the following result. (See [4] for a generalization to tilings using triangles.)

In this paper we introduce Peano path derivatives as a natural extension of the notion of path derivatives. We give a sufficient condition on a system of paths to ensure the corresponding Peano path derivative is Baire 1. As consequences, we obtain that unilateral approximate and unilateral I-approximate Peano derivatives are Baire one.

An integral of the generalized Riemann type is developed which inverts the Schwarz derivative of a continuous function.

A procedure for the analysis of vinclozoline in tap and surface water by GC and Hall detector has been developed. Each step — the analysis of vinclozoline in acetone standard solution, in deionized water and in tap water after solid-phase extraction — was checked by statistical tests (variance homogeneity, linearity test, residue analysis, runaway tests, F-test, t-test). The detection limits and determination limits were calculated from the calibration curve and its prediction interval (according to the DFG). The detection limit for vinclozoline in tap water was found to be 0.03 μg/l and the determination limit is 0.06 μg/l by a recovery rate of 89%.

In 1903 M. Dehn proved that a rectangle can be tiled (or partitioned) into finitely many squares if and only if the ratio of its base and height is rational. In this article we show that a square can be tiled with finitely many similar rectangles of eccentricity r if and only if r is an algebraic number and each of its conjugate roots has positive real part.

The main result of this paper is a generalization of the property that, for smooth u u , u x y = 0 {u_{xy}} = 0 implies ( ∗ ) (\ast ) \[ u ( x , y ) = a ( x ) + b ( y ) . u(x,y) = a(x) + b(y). \] Any function having generalized unsymmetric mixed partial derivative identically zero is of the form ( ∗ ) (\ast ) . There is a function with generalized symmetric mixed partial derivative identically zero not of the form ( ∗ ) (\ast ) , but ( ∗ ) (\ast ) does follow here with the additional assumption of continuity. These results connect to the theory of uniqueness for multiple trigonometric series. For example, a double trigonometric series is the L 2 {L^2} generalized symmetric mixed partial derivative of its formal ( x , y ) (x,y) -integral.

Generalized differential operators are ones which agree with differential operators when applied to sufficiently smooth functions but have special symmetry properties which allow them to be defined on less smooth functions. Such operators were used by Cantor [4] in his proof of the uniqueness of representation by trigonometric series and have been an integral part of all extensions of Cantor’s theorem to higher dimensions.

The following is established: Let W W and B B be open sets of real numbers whose union has full measure. If for each x x , the set { h > 0 | x − h ∈ W , x + h ∈ B } \{ h > 0|x - h \in W,x + h \in B\} has density zero at zero, then these sets are all empty. This is then used to prove the following: If f f is a continuous real valued function with a nonnegative lower approximate symmetric derivative, then f f is nondecreasing.