The Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data

This paper considers the stability and convergence results for the Euler implicit/explicit scheme applied to the spatially discretized two-dimensional (2D) time-dependent Navier-Stokes equations. A Galerkin finite element spatial discretization is assumed, and the temporal treatment is implicit/explict scheme, which is implicit for the linear terms and explicit for the nonlinear term. Here the stability condition depends on the smoothness of the initial data , i.e., the time step condition is in the case of , in the case of and in the case of for mesh size and some positive constant . We provide the -stability of the scheme under the stability condition with and obtain the optimal error estimate of the numerical velocity and the optimal error estimate of the numerical pressure under the stability condition with .

     

On irregular prime power divisors of the Bernoulli numbers

Let ( ) denote the usual th Bernoulli number. Let be a positive even integer where or . It is well known that the numerator of the reduced quotient is a product of powers of irregular primes. Let be an irregular pair with . We show that for every the congruence has a unique solution where and . The sequence defines a -adic integer which is a zero of a certain -adic zeta function originally defined by T. Kubota and H. W. Leopoldt. We show some properties of these functions and give some applications. Subsequently we give several computations of the (truncated) -adic expansion of for irregular pairs with below 1000.

     

New expansions of numerical eigenvalues for by nonconforming elements

The paper explores new expansions of the eigenvalues for in with Dirichlet boundary conditions by the bilinear element (denoted ) and three nonconforming elements, the rotated bilinear element (denoted ), the extension of (denoted ) and Wilson's elements. The expansions indicate that and provide upper bounds of the eigenvalues, and that and Wilson's elements provide lower bounds of the eigenvalues. By extrapolation, the convergence rate can be obtained, where is the maximal boundary length of uniform rectangles. Numerical experiments are carried out to verify the theoretical analysis made.

     

On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations

We give an error analysis of Strang-type splitting integrators for nonlinear Schrödinger equations. For Schrödinger-Poisson equations with an -regular solution, a first-order error bound in the norm is shown and used to derive a second-order error bound in the norm. For the cubic Schrödinger equation with an -regular solution, first-order convergence in the norm is used to obtain second-order convergence in the norm. Basic tools in the error analysis are Lie-commutator bounds for estimating the local error and -conditional stability for error propagation, where for the Schrödinger-Poisson system and for the cubic Schrödinger equation.

     

Identifying minimal and dominant solutions for Kummer recursions

We identify minimal and dominant solutions of three-term recurrence relations for the confluent hypergeometric functions and , where (not both equal to 0). The results are obtained by applying Perron's theorem, together with uniform asymptotic estimates derived by T. M. Dunster for Whittaker functions with large parameter values. The approximations are valid for complex values of , and , with .

     

The hyperdeterminant and triangulations of the 4-cube

The hyperdeterminant of format is a polynomial of degree in unknowns which has terms. We compute the Newton polytope of this polynomial and the secondary polytope of the -cube. The regular triangulations of the -cube are classified into -equivalence classes, one for each vertex of the Newton polytope. The -cube has coarsest regular subdivisions, one for each facet of the secondary polytope, but only of them come from the hyperdeterminant.

     

Monotonicity of some functions involving the gamma and psi functions

Let , where is Euler's gamma function. We determine conditions for the numbers so that the function is strongly completely monotonic on . Through this result we obtain some inequalities involving the ratio of gamma functions and provide some applications in the context of trigonometric sum estimation. We also give several other examples of strongly completely monotonic functions defined in terms of and functions. Some limiting and particular cases are also considered.

     

A convergence and stability study of the iterated Lubkin transformation and the -algorithm

In this paper we analyze the convergence and stability of the iterated Lubkin transformation and the -algorithm as these are being applied to sequences whose members behave like as , where and are complex scalars and is a nonnegative integer. We study the three different cases in which (i) , , and (logarithmic sequences), (ii) and (linear sequences), and (iii) (factorial sequences). We show that both methods accelerate the convergence of all three types of sequences. We show also that both methods are stable on linear and factorial sequences, and they are unstable on logarithmic sequences. On the basis of this analysis we propose ways of improving accuracy and stability in problematic cases. Finally, we provide a comparison of these results with analogous results corresponding to the Levin -transformation.

     

On the irreducibility of Hecke polynomials

Let be the characteristic polynomial of the th Hecke operator acting on the space of cusp forms of weight for the full modular group. We record a simple criterion which can be used to check the irreducibility of the polynomials . Using this criterion with some machine computation, we show that if there exists such that is irreducible and has the full symmetric group as Galois group, then the same is true of for each prime .

     

Computations of Eisenstein series on Fuchsian groups

We present numerical investigations of the value distribution and distribution of Fourier coefficients of the Eisenstein series on arithmetic and non-arithmetic Fuchsian groups. Our numerics indicate a Gaussian limit value distribution for a real-valued rotation of as , and also, on non-arithmetic groups, a complex Gaussian limit distribution for when near and , at least if we allow at some rate. Furthermore, on non-arithmetic groups and for fixed with near , our numerics indicate a Gaussian limit distribution for the appropriately normalized Fourier coefficients.

     

A search for Fibonacci-Wieferich and Wolstenholme primes

A prime is called a Fibonacci-Wieferich prime if , where is the th Fibonacci number. We report that there exist no such primes . A prime is called a Wolstenholme prime if . To date the only known Wolstenholme primes are 16843 and 2124679. We report that there exist no new Wolstenholme primes . Wolstenholme, in 1862, proved that for all primes . It is estimated by a heuristic argument that the ``probability' that is Fibonacci-Wieferich (independently: that is Wolstenholme) is about . We provide some statistical data relevant to occurrences of small values of the Fibonacci-Wieferich quotient modulo .

     

On the minimal polynomial of Gauss periods for prime powers

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 .

     

The convergence of harmonic Ritz values, harmonic Ritz vectors, and refined harmonic Ritz vectors

This paper concerns a harmonic projection method for computing an approximation to an eigenpair of a large matrix . Given a target point and a subspace that contains an approximation to , the harmonic projection method returns an approximation to . Three convergence results are established as the deviation of from approaches zero. First, the harmonic Ritz value converges to if a certain Rayleigh quotient matrix is uniformly nonsingular. Second, the harmonic Ritz vector converges to if the Rayleigh quotient matrix is uniformly nonsingular and remains well separated from the other harmonic Ritz values. Third, better error bounds for the convergence of are derived when converges. However, we show that the harmonic projection method can fail to find the desired eigenvalue --in other words, the method can miss if it is very close to . To this end, we propose to compute the Rayleigh quotient of with respect to and take it as a new approximate eigenvalue. is shown to converge to once tends to , no matter how is close to . Finally, we show that if the Rayleigh quotient matrix is uniformly nonsingular, then the refined harmonic Ritz vector, or more generally the refined eigenvector approximation introduced by the author, converges. We construct examples to illustrate our theory.

     

Two kinds of strong pseudoprimes up to

Let be an odd composite integer. Write with odd. If either mod or mod for some , then we say that is a strong pseudoprime to base , or spsp() for short. Define to be the smallest strong pseudoprime to all the first prime bases. If we know the exact value of , we will have, for integers , a deterministic efficient primality testing algorithm which is easy to implement. Thanks to Pomerance et al. and Jaeschke, the are known for . Conjectured values of were given by us in our previous papers (Math. Comp. 72 (2003), 2085-2097; 74 (2005), 1009-1024).

The main purpose of this paper is to give exact values of for ; to give a lower bound of : ; and to give reasons and numerical evidence of K2- and -spsp's to support the following conjecture: for any , where (resp. ) is the smallest K2- (resp. -) strong pseudoprime to all the first prime bases. For this purpose we describe procedures for computing and enumerating the two kinds of spsp's to the first 9 prime bases. The entire calculation took about 4000 hours on a PC Pentium IV/1.8GHz. (Recall that a K2-spsp is an spsp of the form: with primes and ; and that a -spsp is an spsp and a Carmichael number of the form: with each prime factor mod .)

     

Sign changes in sums of the Liouville function

The Liouville function is the completely multiplicative function whose value is at each prime. We develop some algorithms for computing the sum , and use these methods to determine the smallest positive integer where . This answers a question originating in some work of Turán, who linked the behavior of to questions about the Riemann zeta function. We also study the problem of evaluating Pólya's sum , and we determine some new local extrema for this function, including some new positive values.

     

About the sharpness of the stability estimates in the Kreiss matrix theorem

One of the conditions in the Kreiss matrix theorem involves the resolvent of the matrices under consideration. This so-called resolvent condition is known to imply, for all , the upper bounds and . Here is the spectral norm, is the constant occurring in the resolvent condition, and the order of is equal to .

It is a long-standing problem whether these upper bounds can be sharpened, for all fixed 1$">, to bounds in which the right-hand members grow much slower than linearly with and with , respectively. In this paper it is shown that such a sharpening is impossible. The following result is proved: for each 0$">, there are fixed values 0, K>1$"> and a sequence of matrices , satisfying the resolvent condition, such that for .

The result proved in this paper is also relevant to matrices whose -pseudospectra lie at a distance not exceeding from the unit disk for all 0$">.

     

Short effective intervals containing primes in arithmetic progressions and the seven cubes problem

For any and any non-exceptional modulus , we prove that, for large enough ( ), the interval contains a prime in any of the arithmetic progressions modulo . We apply this result to establish that every integer larger than is a sum of seven cubes.

     

New techniques for bounds on the total number of prime factors of an odd perfect number

Let denote the sum of the positive divisors of . We say that is perfect if . Currently there are no known odd perfect numbers. It is known that if an odd perfect number exists, then it must be of the form , where are distinct primes and . Define the total number of prime factors of as . Sayers showed that . This was later extended by Iannucci and Sorli to show that . This was extended by the author to show that . Using an idea of Carl Pomerance this paper extends these results. The current new bound is .

     

Weak coupling of solutions of first-order least-squares method

A theoretical analysis of a first-order least-squares finite element method for second-order self-adjoint elliptic problems is presented. We investigate the coupling effect of the approximate solutions for the primary function and for the flux . We prove that the accuracy of the approximate solution for the primary function is weakly affected by the flux . That is, the bound for is dependent on , but only through the best approximation for multiplied by a factor of meshsize . Similarly, we provide that the bound for is dependent on , but only through the best approximation for multiplied by a factor of the meshsize . This weak coupling is not true for the non-selfadjoint case. We provide the numerical experiment supporting the theorems in this paper.

     

On the modular curves

Let denote an elliptic curve over and the modular curve classifying the elliptic curves over such that the representations of in the 7-torsion points of and of are symplectically isomorphic. In case is given by a Weierstraß equation such that the invariant is a square, we exhibit here nontrivial points of . From this we deduce an infinite family of curves for which has at least four nontrivial points.