Exponential Sums and Congruences with Factorials

We estimate the number of solutions of certain diagonal congruences involving factorials. We use these results to bound exponential sums with products of two factorials $n!m!$ and also derive asymptotic formulas for the number of solutions of various congruences with factorials. For example, we prove that the products of two factorials $n!m!$ with $\max\{n,m\}<p^{1/2+\epsilon}$ are uniformly distributed modulo $p$, and that any residue class modulo $p$ is representable in the form $m!n!+n_1! + ... +n_{49}!$ with $\max \{m,n, n_1, >..., n_{49}\} < p^{8775/8794+ \epsilon}$.Comment: 21 page

Character Sums and Congruences with n!

We estimate character sums with n!, on average, and individually. These bounds are used to derive new results about various congruences modulo a prime p and obtain new information about the spacings between quadratic nonresidues modulo p. In particular, we show that there exists a positive integer $n\ll p^{1/2+\epsilon}, such that n! is a primitive root modulo p. We also show that every nonzero congruence class a \not \equiv 0 \pmod p can be represented as a product of 7 factorials, a \equiv n_1! ... n_7! \pmod p, where$\max \{n_i | i=1,... 7\}=O(p^{11/12+\epsilon}), and we find the asymptotic formula for the number of such representations. Finally, we show that products of 4 factorials $n_1!n_2!n_3!n_4!, with \max\{n_1, n_2, n_3, n_4\}=O(p^{6/7+\epsilon})$ represent ``almost all''residue classes modulo p, and that products of 3 factorials n_1!n_2!n_3! with \max\{n_1, n_2, n_3\}=O(p^{5/6+\epsilon})$ are uniformly distributed modulo p.Comment: 20 pages. Trans. Amer. Math. Soc. (to appear

Self-consistent variational approach to the minimal left-right symmetric model of electroweak interactions

The problem of mass generation is addressed by a Gaussian variational method for the minimal left-right symmetric model of electroweak interactions. Without any scalar bidoublet, the Gaussian effective potential is shown to have a minimum for a broken symmetry vacuum with a finite expectation value for both the scalar Higgs doublets. The symmetry is broken by the fermionic coupling that destabilizes the symmetric vacuum, yielding a self consistent fermionic mass. In this framework a light Higgs is only compatible with the existence of a new high energy mass scale below 2 TeV.Comment: 5 pages, 3 figures. New comments added and typing errors in eq. 8 and 11 correcte

From the triangular to the kagome lattice: Following the footprints of the ordered state

We study the spin-1/2 Heisenberg model in a lattice that interpolates between the triangular and the kagome lattices. The exchange interaction along the bonds of the kagome lattice is J, and the one along the bonds connecting kagome and non-kagome sites is J', so that J'=J corresponds to the triangular limit and J'=0 to the kagome one. We use variational and exact diagonalization techniques. We analyze the behavior of the order parameter for the antiferromagnetic phase of the triangular lattice, the spin gap, and the structure of the spin excitations as functions of J'/J. Our results indicate that the antiferromagnetic order is not affected by the reduction of J' down to J'/J ~ 0.2. Below this value, antiferromagnetic correlations grow weaker, a description of the ground state in terms of a Neel phase renormalized by quantum fluctuations becomes inadequate, and the finite-size spectra develop features that are not compatible with antiferromagnetic ordering. However, this phase does not appear to be connected to the kagome phase as well, as the low-energy spectra do not evolve with continuity for J'-> 0 to the kagome limit. In particular, for any non-zero value of J', the latter interaction sets the energy scale for the low-lying spin excitations, and a gapless triplet spectrum, destabilizing the kagome phase, is expected.Comment: 9 pages, 10 Figures. To be published in PR

Character sums and products of factorials modulo pp

In this paper, we apply character sum estimates to study products of factorials modulo p

Electronic phase separation near the superconductor-insulator transition of Nd1+xBa2−xCu3O7−δ thin films studied by an electric-field-induced doping effect

We report a detailed study of the transport properties of Nd(1+x)Ba(2-x)Cu(3)O(7-delta) thin films with doping changed by field effect. The data cover the whole superconducting to insulating transition and show remarkable Similarities with the effect of chemical doping in high critical temperature superconductors. The results suggest that the add-on of carriers is accompanied by an electronic phase separation, independent on the details of the doping mechanism

Show me the product, show me the model: Effect of picture type on attitudes toward advertising

We suggest that a consideration of consumer selfâ evaluations is fundamental to understanding the conditions under which it is more advantageous to present person or product pictures in print advertisements. We build on the basic human motives of selfâ enhancement and selfâ verification to propose that the specific selfâ esteem level of consumers, in the domain relevant for the category, differentially affects their responses to picture type. Specifically, for consumers with low (high) domainâ specific selfâ esteem, depicting a product (person) in the advertisement enhances attitudes toward the advertisement more than depicting a person (product). In two studies, we demonstrate the proposed matching relationships using two different domains of consumer selfâ evaluation: appearance selfâ esteem and academic selfâ esteem. We also show that increased and more fluent generation of selfâ related mental imagery drives the observed improvement in attitudes toward the advertisement. Our findings suggest direct implications for advertising design.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/141608/1/jcpy506.pd