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1. CMB 2011 (vol 55 pp. 285)
Uniqueness Implies Existence and Uniqueness Conditions for a Class of $(k+j)$-Point Boundary Value Problems for $n$-th Order Differential Equations |
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)
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)
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)
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)
On the Singular Behavior of the Inverse Laplace Transforms of the Functions $\frac{I_n(s)}{s I_n^\prime(s)}$ |
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)
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 |