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Search: All articles in the CMB digital archive with keyword exponent

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1. CMB Online first

He, Yubo; Qin, Dongdong; Tang, Xianhua
Ground state and multiple solutions for Kirchhoff type equations with critical exponent
In this paper, we consider the following critical Kirchhoff type equation: \begin{align*} \left\{ \begin{array}{lll} - \left(a+b\int_{\Omega}|\nabla u|^2 \right)\Delta u=Q(x)|u|^4u + \lambda |u|^{q-1}u,~~\mbox{in}~~\Omega, \\ u=0,\quad \text{on}\quad \partial \Omega, \end{array} \right. \end{align*} By using variational methods that are constrained to the Nehari manifold, we prove that the above equation has a ground state solution for the case when $3\lt q\lt 5$. The relation between the number of maxima of $Q$ and the number of positive solutions for the problem is also investigated.

Keywords:Kirchhoff type equation, variational methods, critical exponent, Nehari manifold, ground state
Categories:35J20, 35J60, 35J25

2. CMB 2014 (vol 57 pp. 853)

Pan, Qingfei; Wang, Kun
On the Bound of the $\mathrm{C}^*$ Exponential Length
Let $X$ be a compact Hausdorff space. In this paper, we give an example to show that there is $u\in \mathrm{C}(X)\otimes \mathrm{M}_n$ with $\det (u(x))=1$ for all $x\in X$ and $u\sim_h 1$ such that the $\mathrm{C}^*$ exponential length of $u$ (denoted by $cel(u)$) can not be controlled by $\pi$. Moreover, in simple inductive limit $\mathrm{C}^*$-algebras, similar examples also exist.

Keywords:exponential length

3. CMB 2014 (vol 57 pp. 495)

Fujita, Yasutsugu; Miyazaki, Takafumi
Jeśmanowicz' 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

4. CMB 2014 (vol 58 pp. 30)

Chung, Jaeyoung
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

5. CMB 2012 (vol 57 pp. 113)

Madras, Neal
A Lower Bound for the End-to-End Distance of Self-Avoiding Walk
For an $N$-step self-avoiding walk on the hypercubic lattice ${\bf Z}^d$, we prove that the mean-square end-to-end distance is at least $N^{4/(3d)}$ times a constant. This implies that the associated critical exponent $\nu$ is at least $2/(3d)$, assuming that $\nu$ exists.

Keywords:self-avoiding walk, critical exponent
Categories:82B41, 60D05, 60K35

6. CMB 2011 (vol 55 pp. 271)

Di Vincenzo, M. Onofrio; Nardozza, Vincenzo
On the Existence of the Graded Exponent for Finite Dimensional $\mathbb{Z}_p$-graded Algebras
Let $F$ be an algebraically closed field of characteristic zero, and let $A$ be an associative unitary $F$-algebra graded by a group of prime order. We prove that if $A$ is finite dimensional then the graded exponent of $A$ exists and is an integer.

Keywords:exponent, polynomial identities, graded algebras
Categories:16R50, 16R10, 16W50

7. CMB 2011 (vol 54 pp. 464)

Hwang, Tea-Yuan; Hu, Chin-Yuan
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

8. CMB 2011 (vol 55 pp. 882)

Xueli, Song; Jigen, Peng
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

9. CMB 2011 (vol 55 pp. 339)

Loring, Terry A.
From Matrix to Operator Inequalities
We generalize Löwner'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

10. CMB 2010 (vol 54 pp. 527)

Preda, Ciprian; Sipos, Ciprian
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

11. CMB 2010 (vol 54 pp. 364)

Preda, Ciprian; Preda, Petre
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

12. CMB 2010 (vol 53 pp. 327)

Luor, Dah-Chin
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

13. CMB 2004 (vol 47 pp. 119)

Theriault, Stephen D.
$2$-Primary Exponent Bounds for Lie Groups of Low Rank
Exponent information is proven about the Lie groups $SU(3)$, $SU(4)$, $Sp(2)$, and $G_2$ by showing some power of the $H$-space squaring map (on a suitably looped connected-cover) is null homotopic. The upper bounds obtained are $8$, $32$, $64$, and $2^8$ respectively. This null homotopy is best possible for $SU(3)$ given the number of loops, off by at most one power of~$2$ for $SU(4)$ and $Sp(2)$, and off by at most two powers of $2$ for $G_2$.

Keywords:Lie group, exponent

14. CMB 2001 (vol 44 pp. 346)

Wang, Wei
Positive Solution of a Subelliptic Nonlinear Equation on the Heisenberg Group
In this paper, we establish the existence of positive solution of a nonlinear subelliptic equation involving the critical Sobolev exponent on the Heisenberg group, which generalizes a result of Brezis and Nirenberg in the Euclidean case.

Keywords:Heisenberg group, subLapacian, critical Sobolev exponent, extremals
Categories:35J20, 35J60

15. CMB 2000 (vol 43 pp. 239)

Yu, Gang
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

16. CMB 1998 (vol 41 pp. 398)

Dziubański, Jacek; Hernández, Eugenio
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

17. CMB 1998 (vol 41 pp. 86)

Lubinsky, D. S.
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

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