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1.
There is a 1941 conjecture of Erdos and Turán on what is now called additive basis that we restate:
Suppose that
If 0$"> for all , then is unbounded.
Conjecture 0.1(Erdos and Turán). Suppose that is an increasing sequence of integers and
Suppose that
If 0$"> for all , then is unbounded.
Our main purpose is to show that the sequence cannot be bounded by . There is a surprisingly simple, though computationally very intensive, algorithm that establishes this.
2.
We study the kernels of the remainder term of Gauss-Turán quadrature formulas
for classes of analytic functions on elliptical contours with foci at , when the weight is one of the special Jacobi weights ; ; , ; , . We investigate the location on the contour where the modulus of the kernel attains its maximum value. Some numerical examples are included.
for classes of analytic functions on elliptical contours with foci at , when the weight is one of the special Jacobi weights ; ; , ; , . We investigate the location on the contour where the modulus of the kernel attains its maximum value. Some numerical examples are included.
3.
S. Gurak. 《Mathematics of Computation》2006,75(256):2021-2035
For a positive integer , set and let denote the group of reduced residues modulo . Fix a congruence group of conductor and of order . Choose integers to represent the cosets of in . The Gauss periods corresponding to are conjugate and distinct over with minimal polynomial To determine the coefficients of the period polynomial (or equivalently, its reciprocal polynomial is a classical problem dating back to Gauss. Previous work of the author, and Gupta and Zagier, primarily treated the case , an odd prime, with fixed. In this setting, it is known for certain integral power series and , that for any positive integer holds in for all primes except those in an effectively determinable finite set. Here we describe an analogous result for the case , a prime power ( ). The methods extend for odd prime powers to give a similar result for certain twisted Gauss periods of the form where denotes the usual Legendre symbol and .
4.
Paul Alton Hagelstein 《Proceedings of the American Mathematical Society》2005,133(8):2327-2334
Let denote a sequence of measurable functions on , and let denote the weak norm. It is shown that
where is a sequence of independent random variables taking on values and with equal probability. Moreover, it is shown that
The paper concludes by providing an example indicating that, if , then the estimate
is the best possible.
where is a sequence of independent random variables taking on values and with equal probability. Moreover, it is shown that
The paper concludes by providing an example indicating that, if , then the estimate
is the best possible.
5.
We prove that for integers 1,m\geq 1$"> and positive rationals the series
is irrational. Furthermore, if all the positive rationals are less than then the series
is also irrational.
is irrational. Furthermore, if all the positive rationals are less than then the series
is also irrational.
6.
Leping Sun. 《Mathematics of Computation》2006,75(253):151-165
In this paper we are concerned with the asymptotic stability of the delay differential equation
where are constant complex matrices, and 0$"> stand for constant delays . We obtain two criteria for stability through the evaluation of a harmonic function on the boundary of a certain region. We also get similar results for the neutral delay differential equation
where and are constant complex matrices and 0$"> stands for constant delays , . Numerical examples on various circumstances are shown to check our results which are more general than those already reported.
where are constant complex matrices, and 0$"> stand for constant delays . We obtain two criteria for stability through the evaluation of a harmonic function on the boundary of a certain region. We also get similar results for the neutral delay differential equation
where and are constant complex matrices and 0$"> stands for constant delays , . Numerical examples on various circumstances are shown to check our results which are more general than those already reported.
7.
On some inequalities for the incomplete gamma function 总被引:5,自引:0,他引:5
Horst Alzer. 《Mathematics of Computation》1997,66(218):771-778
Let be a positive real number. We determine all real numbers and such that the inequalities
are valid for all . And, we determine all real numbers and such that
hold for all .
8.
Let be a row diagonally dominant matrix, i.e.,
where with We show that no pivoting is necessary when Gaussian elimination is applied to Moreover, the growth factor for does not exceed The same results are true with row diagonal dominance being replaced by column diagonal dominance.
where with We show that no pivoting is necessary when Gaussian elimination is applied to Moreover, the growth factor for does not exceed The same results are true with row diagonal dominance being replaced by column diagonal dominance.
9.
Nakao Hayashi Elena I. Kaikina Pavel I. Naumkin 《Transactions of the American Mathematical Society》2006,358(3):1165-1185
We study large time asymptotics of small solutions to the Cauchy problem for nonlinear damped wave equations with a critical nonlinearity
where 0,$"> and space dimensions . Assume that the initial data
where \frac{n}{2},$"> weighted Sobolev spaces are
Also we suppose that
where
Then we prove that there exists a positive such that the Cauchy problem above has a unique global solution satisfying the time decay property
for all 0,$"> where
where 0,$"> and space dimensions . Assume that the initial data
where \frac{n}{2},$"> weighted Sobolev spaces are
Also we suppose that
0,\int u_{0}\left( x\right) dx>0, \end{displaymath}">
where
Then we prove that there exists a positive such that the Cauchy problem above has a unique global solution satisfying the time decay property
for all 0,$"> where
10.
Tapani Matala-aho Keijo Vä ä nä nen Wadim Zudilin. 《Mathematics of Computation》2006,75(254):879-889
The three main methods used in diophantine analysis of -series are combined to obtain new upper bounds for irrationality measures of the values of the -logarithm function when and .