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1. CMB 2011 (vol 55 pp. 285)

Eloe, Paul W.; Henderson, Johnny; Khan, Rahmat Ali
Uniqueness Implies Existence and Uniqueness Conditions for a Class of $(k+j)$-Point Boundary Value Problems for $n$-th Order Differential Equations
For the $n$-th order nonlinear differential equation, $y^{(n)} = f(x, y, y', \dots, y^{(n-1)})$, we consider uniqueness implies uniqueness and existence results for solutions satisfying certain $(k+j)$-point boundary conditions for $1\le j \le n-1$ and $1\leq k \leq n-j$. We define $(k;j)$-point unique solvability in analogy to $k$-point disconjugacy and we show that $(n-j_{0};j_{0})$-point unique solvability implies $(k;j)$-point unique solvability for $1\le j \le j_{0}$, and $1\leq k \leq n-j$. This result is analogous to $n$-point disconjugacy implies $k$-point disconjugacy for $2\le k\le n-1$.

Keywords:boundary value problem, uniqueness, existence, unique solvability, nonlinear interpolation
Categories:34B15, 34B10, 65D05

2. CMB 2011 (vol 55 pp. 689)

Berndt, Ryan
A Pointwise Estimate for the Fourier Transform and Maxima of a Function
We show a pointwise estimate for the Fourier transform on the line involving the number of times the function changes monotonicity. The contrapositive of the theorem may be used to find a lower bound to the number of local maxima of a function. We also show two applications of the theorem. The first is the two weight problem for the Fourier transform, and the second is estimating the number of roots of the derivative of a function.

Keywords:Fourier transform, maxima, two weight problem, roots, norm estimates, Dirichlet-Jordan theorem
Categories:42A38, 65T99

3. CMB 2008 (vol 51 pp. 627)

Vidanovi\'{c}, Mirjana V.; Tri\v{c}kovi\'{c}, Slobodan B.; Stankovi\'{c}, Miomir S.
Summation of Series over Bourget Functions
In this paper we derive formulas for summation of series involving J.~Bourget's generalization of Bessel functions of integer order, as well as the analogous generalizations by H.~M.~Srivastava. These series are expressed in terms of the Riemann $\z$ function and Dirichlet functions $\eta$, $\la$, $\b$, and can be brought into closed form in certain cases, which means that the infinite series are represented by finite sums.

Keywords:Riemann zeta function, Bessel functions, Bourget functions, Dirichlet functions
Categories:33C10, 11M06, 65B10

4. CMB 2008 (vol 51 pp. 372)

Ezquerro, J. A.; Hernández, M. A.
Picard's Iterations for Integral Equations of Mixed Hammerstein Type
A new semilocal convergence result for the Picard method is presented, where the main required condition in the contraction mapping principle is relaxed.

Keywords:nonlinear equations in Banach spaces, successive approximations, semilocal convergence theorem, Picard's iteration, Hammerstein integral equations
Categories:45G10, 47H99, 65J15

5. CMB 2002 (vol 45 pp. 399)

Iakovlev, Serguei
On the Singular Behavior of the Inverse Laplace Transforms of the Functions $\frac{I_n(s)}{s I_n^\prime(s)}$
Exact analytical expressions for the inverse Laplace transforms of the functions $\frac{I_n(s)}{s I_n^\prime(s)}$ are obtained in the form of trigonometric series. The convergence of the series is analyzed theoretically, and it is proven that those diverge on an infinite denumerable set of points. Therefore it is shown that the inverse transforms have an infinite number of singular points. This result, to the best of the author's knowledge, is new, as the inverse transforms of $\frac{I_n(s)}{s I_n^\prime(s)}$ have previously been considered to be piecewise smooth and continuous. It is also found that the inverse transforms have an infinite number of points of finite discontinuity with different left- and right-side limits. The points of singularity and points of finite discontinuity alternate, and the sign of the infinity at the singular points also alternates depending on the order $n$. The behavior of the inverse transforms in the proximity of the singular points and the points of finite discontinuity is addressed as well.

Categories:65R32, 44A10, 44A20, 74F10

6. CMB 1999 (vol 42 pp. 359)

Martin, W. J.; Stinson, D. R.
A Generalized Rao Bound for Ordered Orthogonal Arrays and $(t,m,s)$-Nets
In this paper, we provide a generalization of the classical Rao bound for orthogonal arrays, which can be applied to ordered orthogonal arrays and $(t,m,s)$-nets. Application of our new bound leads to improvements in many parameter situations to the strongest bounds (\ie, necessary conditions) for existence of these objects.

Categories:05B15, 65C99

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