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1.
Nobushige Kurokawa Masato Wakayama 《Proceedings of the American Mathematical Society》2004,132(4):935-943
We introduce and study a -analogue of the Euler constant via a suitably defined -analogue of the Riemann zeta function. We show, in particular, that the value is irrational. We also present a -analogue of the Hurwitz zeta function and establish an analogue of the limit formula of Lerch in 1894 for the gamma function. This limit formula can be regarded as a natural generalization of the formula of .
2.
Andreas Weingartner 《Proceedings of the American Mathematical Society》2007,135(9):2677-2681
Let be the sum of the positive divisors of . We show that the natural density of the set of integers satisfying is given by , where denotes Euler's constant. The same result holds when is replaced by , where is Euler's totient function.
3.
In this paper, the inequalities for the weighted mean of -convex functions are established. As applications, inequalities between the two-parameter mean of an -convex function and extended mean values are given.
4.
Jing-Song Huang Marko Tadic 《Transactions of the American Mathematical Society》2005,357(5):2081-2117
Let be the group of rational points of a connected reductive -adic group and let be a maximal compact subgroup satisfying conditions of Theorem 5 from Harish-Chandra (1970). Generalized spherical functions on are eigenfunctions for the action of the Bernstein center, which satisfy a transformation property for the action of . In this paper we show that spaces of generalized spherical functions are finite dimensional. We compute dimensions of spaces of generalized spherical functions on a Zariski open dense set of infinitesimal characters. As a consequence, we get that on that Zariski open dense set of infinitesimal characters, the dimension of the space of generalized spherical functions is constant on each connected component of infinitesimal characters. We also obtain the formula for the generalized spherical functions by integrals of Eisenstein type. On the Zariski open dense set of infinitesimal characters that we have mentioned above, these integrals then give the formula for all the generalized spherical functions. At the end, let as mention that among others we prove that there exists a Zariski open dense subset of infinitesimal characters such that the category of smooth representations of with fixed infinitesimal character belonging to this subset is semi-simple.
5.
Mohammad Abry Jan J. Dijkstra 《Proceedings of the American Mathematical Society》2007,135(8):2623-2628
We find universal functions for the class of lower semi-continuous (LSC) functions with at most -dimensional domain. In an earlier paper we proved that a space is almost -dimensional if and only if it is homeomorphic to the graph of an LSC function with an at most -dimensional domain. We conclude that the class of almost -dimensional spaces contains universal elements (that are topologically complete). These universal spaces can be thought of as higher-dimensional analogues of complete Erdos space.
6.
Brandt Kronholm 《Proceedings of the American Mathematical Society》2005,133(10):2891-2895
In the late 19th century, Sylvester and Cayley investigated the properties of the partition function . This function enumerates the partitions of a non-negative integer into exactly parts. Here we investigate the congruence properties of such functions and we obtain several infinite classes of Ramanujan-type congruences.
7.
A point is covered by a function iff there is a permutation of such that .
By a theorem of Kuratowski, for every infinite cardinal exactly -ary functions are needed to cover all of . We show that for arbitrarily large uncountable it is consistent that the size of the continuum is and is covered by -ary continuous functions.
We study other cardinal invariants of the -ideal on generated by continuous -ary functions and finally relate the question of how many continuous functions are necessary to cover to the least size of a set of parameters such that the Turing degrees relative to this set of parameters are linearly ordered.
8.
Li-Chien Shen 《Transactions of the American Mathematical Society》2005,357(5):2043-2058
Based on properties of the hypergeometric series , we develop a theory of elliptic functions that shares many striking similarities with the classical Jacobian elliptic functions.
9.
B. C. Carlson. 《Mathematics of Computation》2006,75(255):1309-1318
Any product of real powers of Jacobian elliptic functions can be written in the form . If all three 's are even integers, the indefinite integral of this product with respect to is a constant times a multivariate hypergeometric function with half-odd-integral 's and , showing it to be an incomplete elliptic integral of the second kind unless all three 's are 0. Permutations of c, d, and n in the integrand produce the same permutations of the variables }, allowing as many as six integrals to take a unified form. Thirty -functions of the type specified, incorporating 136 integrals, are reduced to a new choice of standard elliptic integrals obtained by permuting , , and in , which is symmetric in its first two variables and has an efficient algorithm for numerical computation.
10.
Vladimir Derkach Seppo Hassi 《Proceedings of the American Mathematical Society》2003,131(12):3795-3806
A new model for generalized Nevanlinna functions will be presented. It involves Bezoutians and companion operators associated with certain polynomials determined by the generalized zeros and poles of . The model is obtained by coupling two operator models and expressed by means of abstract boundary mappings and the corresponding Weyl functions.
11.
Barbara F. Csima Antonio Montalbá n 《Proceedings of the American Mathematical Society》2006,134(5):1499-1502
We construct a minimal pair of -degrees. We do this by showing the existence of an unbounded nondecreasing function which forces -triviality in the sense that is -trivial if and only if for all , .
12.
We study the boundary regularity in the Dirichlet problem of the differential operators
Our main result is: if -1/2$"> is neither an integer nor a half-integer not less than , one cannot expect global smooth solutions of ; if satisfies , then must be either a polynomial of degree at most or a polyharmonic function of degree .
Our main result is: if -1/2$"> is neither an integer nor a half-integer not less than , one cannot expect global smooth solutions of ; if satisfies , then must be either a polynomial of degree at most or a polyharmonic function of degree .
13.
Eric Mortenson 《Transactions of the American Mathematical Society》2003,355(3):987-1007
Fernando Rodriguez-Villegas has conjectured a number of supercongruences for hypergeometric Calabi-Yau manifolds of dimension . For manifolds of dimension , he observed four potential supercongruences. Later the author proved one of the four. Motivated by Rodriguez-Villegas's work, in the present paper we prove a general result on supercongruences between values of truncated hypergeometric functions and Gaussian hypergeometric functions. As a corollary to that result, we prove the three remaining supercongruences.
14.
Richard Aron V. I. Gurariy J. B. Seoane 《Proceedings of the American Mathematical Society》2005,133(3):795-803
We show that there is an infinite-dimensional vector space of differentiable functions on every non-zero element of which is nowhere monotone. We also show that there is a vector space of dimension of functions every non-zero element of which is everywhere surjective.
15.
In this note, we show that one can derive from any -spline function of order ( ) an MRA tight wavelet frame in that is generated by the dyadic dilates and integer shifts of three compactly supported real-valued symmetric wavelet functions with vanishing moments of the highest possible order .
16.
Masayoshi Takeda Kaneharu Tsuchida 《Transactions of the American Mathematical Society》2007,359(8):4031-4054
Let be a signed Radon measure in the Kato class and define a Schrödinger type operator on . We show that its spectral bound is differentiable if and is Green-tight.
17.
Yong Zhou 《Proceedings of the American Mathematical Society》2006,134(1):149-156
In this paper we establish a Serrin-type regularity criterion on the gradient of pressure for the weak solutions to the Navier-Stokes equations in . It is proved that if the gradient of pressure belongs to with , , then the weak solution is actually regular. Moreover, we give a much simpler proof of the regularity criterion on the pressure, which was showed recently by Berselli and Galdi (Proc. Amer. Math. Soc. 130 (2002), no. 12, 3585-3595).
18.
Stamatis Koumandos. 《Mathematics of Computation》2008,77(264):2261-2275
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.
19.
We define a class of generic CR submanifolds of of real codimension , , called the Bloom-Graham model graphs, whose graphing functions are partially decoupled in their dependence on the variables in the real directions. We prove a global version of the Baouendi-Treves CR approximation theorem for Bloom-Graham model graphs with a polynomial growth assumption on their graphing functions.
20.
N. Bejhaj Rhouma 《Proceedings of the American Mathematical Society》2003,131(12):3747-3755
We show the existence of principal eigenvalues of the problem in where is an indefinite weight function. The existence of a continuous family of principal eigenvalues is demonstrated. Also, we prove the existence of a principal eigenvalue for which the principal eigenfunction at .