Expand all Collapse all | Results 1 - 11 of 11 |
1. CMB 2014 (vol 57 pp. 495)
JeÅmanowicz' Conjecture with Congruence Relations. II Let $a,b$ and $c$ be primitive Pythagorean numbers such that
$a^{2}+b^{2}=c^{2}$ with $b$ even.
In this paper, we show that if $b_0 \equiv \epsilon \pmod{a}$
with $\epsilon \in \{\pm1\}$
for certain positive divisors $b_0$ of $b$,
then the Diophantine equation $a^{x}+b^{y}=c^z$ has only the
positive solution $(x,y,z)=(2,2,2)$.
Keywords:exponential Diophantine equations, Pythagorean triples, Pell equations Categories:11D61, 11D09 |
2. CMB Online first
On an Exponential Functional Inequality and its Distributional Version Let $G$ be a group and $\mathbb K=\mathbb C$ or $\mathbb
R$.
In this article, as a generalization of the result of Albert
and Baker,
we investigate the behavior of bounded
and unbounded functions $f\colon G\to \mathbb K$ satisfying the inequality
$
\Bigl|f
\Bigl(\sum_{k=1}^n x_k
\Bigr)-\prod_{k=1}^n f(x_k)
\Bigr|\le \phi(x_2, \dots, x_n),\quad \forall\, x_1, \dots,
x_n\in G,
$
where $\phi\colon G^{n-1}\to [0, \infty)$. Also, as a distributional
version of the above inequality we consider the stability of
the functional equation
\begin{equation*}
u\circ S - \overbrace{u\otimes \cdots \otimes u}^{n-\text {times}}=0,
\end{equation*}
where $u$ is a Schwartz distribution or Gelfand hyperfunction,
$\circ$ and $\otimes$ are the pullback and tensor product of
distributions, respectively, and $S(x_1, \dots, x_n)=x_1+ \dots
+x_n$.
Keywords:distribution, exponential functional equation, Gelfand hyperfunction, stability Categories:46F99, 39B82 |
3. CMB 2011 (vol 54 pp. 464)
A Characterization of the Compound-Exponential Type Distributions In this paper, a fixed point equation of the
compound-exponential type distributions is derived, and under some
regular conditions,
both the existence and uniqueness of
this fixed point equation are investigated.
A question posed by Pitman and Yor
can be partially answered by using our approach.
Keywords:fixed point equation, compound-exponential type distributions Categories:62E10, 60G50 |
4. CMB 2011 (vol 55 pp. 882)
Equivalence of $L_p$ Stability and Exponential Stability of Nonlinear Lipschitzian Semigroups
$L_p$ stability and exponential stability are two important concepts
for nonlinear dynamic systems. In this paper, we prove that a
nonlinear exponentially bounded Lipschitzian semigroup is
exponentially stable if and only if the semigroup is $L_p$ stable
for some $p>0$. Based on the equivalence, we derive two sufficient
conditions for exponential stability of the nonlinear semigroup. The
results obtained extend and improve some existing ones.
Keywords:exponentially stable, $L_p$ stable, nonlinear Lipschitzian semigroups Categories:34D05, 47H20 |
5. CMB 2011 (vol 55 pp. 339)
From Matrix to Operator Inequalities We generalize LÃ¶wner's method for proving that matrix monotone
functions are operator monotone. The relation $x\leq y$ on bounded
operators is our model for a definition of $C^{*}$-relations
being residually finite dimensional.
Our main result is a meta-theorem about theorems involving relations
on bounded operators. If we can show there are residually finite dimensional
relations involved and verify a technical condition, then such a
theorem will follow from its restriction to matrices.
Applications are shown regarding norms of exponentials, the norms
of commutators, and "positive" noncommutative $*$-polynomials.
Keywords:$C*$-algebras, matrices, bounded operators, relations, operator norm, order, commutator, exponential, residually finite dimensional Categories:46L05, 47B99 |
6. CMB 2010 (vol 54 pp. 527)
On the Dichotomy of the Evolution Families: A Discrete-Argument Approach
We establish a discrete-time criteria guaranteeing the existence of an
exponential dichotomy in the continuous-time
behavior of an abstract evolution family. We prove that an evolution
family ${\cal U}=\{U(t,s)\}_{t
\geq s\geq 0}$ acting on a Banach space $X$ is uniformly
exponentially dichotomic (with respect to its continuous-time
behavior) if and only if the
corresponding difference equation with the inhomogeneous term from
a vector-valued Orlicz sequence space $l^\Phi(\mathbb{N}, X)$
admits
a solution in the same $l^\Phi(\mathbb{N},X)$. The technique of
proof effectively eliminates the continuity hypothesis on the
evolution family (\emph{i.e.,} we do not assume that $U(\,\cdot\,,s)x$
or $U(t,\,\cdot\,)x$ is continuous on $[s,\infty)$, and respectively
$[0,t]$). Thus, some known results given by
Coffman and Schaffer, Perron, and Ta Li are extended.
Keywords:evolution families, exponential dichotomy, Orlicz sequence spaces, admissibility Categories:34D05, 47D06, 93D20 |
7. CMB 2010 (vol 54 pp. 364)
Lyapunov Theorems for the Asymptotic Behavior of Evolution Families on the Half-Line
Two theorems regarding the
asymptotic behavior of evolution families are established in
terms of the solutions of a certain Lyapunov operator equation.
Keywords:evolution families, exponential instability, Lyapunov equation Categories:34D05, 47D06 |
8. CMB 2010 (vol 53 pp. 327)
Multidimensional Exponential Inequalities with Weights We establish sufficient conditions on the weight functions $u$ and $v$ for the validity of the multidimensional weighted inequality $$ \Bigl(\int_E \Phi(T_k f(x))^q u(x)\,dx\Bigr)^{1/q} \le C \Bigl (\int_E \Phi(f(x))^p v(x)\,dx\Bigr )^{1/p}, $$
where 0<$p$, $q$<$\infty$, $\Phi$ is a logarithmically convex function, and $T_k$ is an integral operator over star-shaped regions. The condition is also necessary for the exponential integral inequality. Moreover, the estimation of $C$ is given and we apply the obtained results to generalize some multidimensional Levin--Cochran-Lee type inequalities.
Keywords:multidimensional inequalities, geometric mean operators, exponential inequalities, star-shaped regions Categories:26D15, 26D10 |
9. CMB 2000 (vol 43 pp. 239)
On the Number of Divisors of the Quadratic Form $m^2+n^2$ For an integer $n$, let $d(n)$ denote the ordinary divisor function.
This paper studies the asymptotic behavior of the sum
$$
S(x) := \sum_{m\leq x, n\leq x} d(m^2 + n^2).
$$
It is proved in the paper that, as $x \to \infty$,
$$
S(x) := A_1 x^2 \log x + A_2 x^2 + O_\epsilon (x^{\frac32 +
\epsilon}),
$$
where $A_1$ and $A_2$ are certain constants and $\epsilon$ is any
fixed positive real number.
The result corrects a false formula given in a paper of Gafurov
concerning the same problem, and improves the error $O \bigl(
x^{\frac53} (\log x)^9 \bigr)$ claimed by Gafurov.
Keywords:divisor, large sieve, exponential sums Categories:11G05, 14H52 |
10. CMB 1998 (vol 41 pp. 398)
Band-limited wavelets with subexponential decay It is well known that the compactly supported wavelets cannot belong to
the class $C^\infty({\bf R})\cap L^2({\bf R})$. This is also true for
wavelets with exponential decay. We show that one can construct
wavelets in the class $C^\infty({\bf R})\cap L^2({\bf R})$ that are
``almost'' of exponential decay and, moreover, they are
band-limited. We do this by showing that we can adapt the
construction of the Lemari\'e-Meyer wavelets \cite{LM} that
is found in \cite{BSW} so that we obtain band-limited,
$C^\infty$-wavelets on $\bf R$ that have subexponential decay,
that is, for every $0<\varepsilon<1$, there exits $C_\varepsilon>0$
such that $|\psi(x)|\leq C_\varepsilon e^{-|x|^{1-\varepsilon}}$,
$x\in\bf R$. Moreover, all of its derivatives have also
subexponential decay. The proof is constructive and uses the
Gevrey classes of functions.
Keywords:Wavelet, Gevrey classes, subexponential decay Category:42C15 |
11. CMB 1998 (vol 41 pp. 86)
On \lowercase{$q$}-exponential functions for \lowercase{$|q| =1$} We discuss the $q$-exponential functions $e_q$, $E_q$ for $q$ on the unit
circle, especially their continuity in $q$, and analogues of the limit
relation $ \lim_{q\rightarrow 1}e_q((1-q)z)=e^z$.
Keywords:$q$-series, $q$-exponentials Categories:33D05, 11A55, 11K70 |