1,280 research outputs found
On the Asymptotic Existence of Hadamard Matrices
It is conjectured that Hadamard matrices exist for all orders ().
However, despite a sustained effort over more than five decades, the strongest
overall existence results are asymptotic results of the form: for all odd
natural numbers , there is a Hadamard matrix of order ,
where and are fixed non-negative constants. To prove the Hadamard
Conjecture, it is sufficient to show that we may take and . Since
Seberry's ground-breaking result, which showed that we may take and
, there have been several improvements where has been by stages
reduced to 3/8. In this paper, we show that for all , the set of
odd numbers for which there is a Hadamard matrix of order
has positive density in the set of natural numbers.
The proof adapts a number-theoretic argument of Erdos and Odlyzko to show that
there are enough Paley Hadamard matrices to give the result.Comment: Keywords: Hadamard matrices, Asymptotic existence, Cocyclic Hadamard
matrices, Relative difference sets, Riesel numbers, Extended Riemann
hypothesis. (Received 2 August 2008, Available online 18 March 2009
Program in C for studying characteristic properties of two-body interactions in the framework of spectral distribution theory
We present a program in C that employs spectral distribution theory for
studies of characteristic properties of a many-particle quantum-mechanical
system and the underlying few-body interaction. In particular, the program
focuses on two-body nuclear interactions given in a JT-coupled harmonic
oscillator basis and calculates correlation coefficients, a measure of
similarity of any two interactions, as well as Hilbert-Schmidt norms specifying
interaction strengths. An important feature of the program is its ability to
identify the monopole part (centroid) of a 2-body interaction, as well as its
'density-dependent' one-body and two-body part, thereby providing key
information on the evolution of shell gaps and binding energies for larger
nuclear systems. As additional features, we provide statistical measures for
'density-dependent' interactions, as well as a mechanism to express an
interaction in terms of two other interactions. This, in turn, allows one to
identify, e.g., established features of the nuclear interaction (such as
pairing correlations) within a general Hamiltonian. The program handles the
radial degeneracy for 'density-dependent' one-body interactions and together
with an efficient linked list data structure, facilitates studies of nuclear
interactions in large model spaces that go beyond valence-shell applications.Comment: 22 pages, 3 figure
Using Drawing Tasks to Communicate Ideas About Photosynthesis: A Conceptual Change Strategy for Use in the Elementary School Classroom.
It is difficult for a teacher to determine if a learner has acquired an accurate concept of the topic being taught. Most of the children in this study had sufficient language skills to communicate successfully with their teachers even though they held inappropriate concepts of photosynthesis. This study examined the use of drawing tasks to assess children\u27s ideas related to photosynthesis in an elementary-grade classroom. Two research questions guided the study to determine if this strategy was a valid improvement over traditional methods of classroom instruction. The first question asked if elementary-grade students receiving instruction about photosynthesis would acquire and retain more knowledge when facilitated by teacher-analysis of their drawing tasks than students who received didactic instruction. The second question sought to determine if a fifth-grade teacher guided by students\u27 drawing tasks depicting their concepts of photosynthesis could effect more appropriate conceptual change than a teacher using didactic instruction. Two fifth grade treatment groups were used in the study. The teacher in the traditional treatment used didactic methods to instruct and evaluate the learner\u27s concepts. The teacher in the experimental treatment used the learner\u27s drawing tasks to communicate and facilitated activities to challenge and change inaccurate concepts. The quantitative results of a pretest, posttest, and delayed posttest were analyzed by ANCOVA with repeated measures to answer the first question. Clinical interviews, classroom observations, and student artifacts provided data for a qualitative analysis of the second question. These data were examined and analyzed in correspondence with children\u27s written test responses. Students in the experimental treatment were found to acquire a greater amount of content knowledge than those in the traditional treatment. However, retention of knowledge was not significantly different between the two groups. The teacher in the experimental treatment was determined to facilitate a change to an appropriate concept of photosynthesis in more students than the teacher in the traditional treatment. The experimental treatment was found to provide an accurate depiction of the children\u27s concepts while the traditional, didactic-style treatment seemed to influence children to conceal their inaccurate concepts of photosynthesis
Group theoretical approach to pairing and non-linear phenomena in atomic nuclei
The symplectic sp(4) algebra provides a natural framework for studying proton-neutron (pn) and like-nucleon pairing correlations as well as higher-J pn interactions in nuclei when protons and neutrons occupy the same shell. While these correlations manifest themselves most clearly in the binding energies of 0+ ground states, they also have a large effect on the spectra of excited isobaric analog 0+ states. With a view towards nuclear structure applications, a fermion realization of sp(4) is explored and its q-deformed extension, sp(4)q, is constructed for single and multiple shells. The su(2)(q) substructures that enter are associated with isospin symmetry and with identical-particle and pn pairing. We suggest a non-deformed as well as a q-deformed algebraic descriptions of pairing for even-A nuclei of the mass 32 \u3c A \u3c 164 region. A Hamiltonian with a symplectic dynamical symmetry is constructed and its eigenvalues are fit to the relevant Coulomb corrected experimental 0+ state energies in both the “classical” and “deformed” cases. While the non-deformed microscopic theory yields results that are comparable to other models for light nuclei, the present approach succeeds in providing a reasonable estimate for interaction strength parameters as well as a detailed investigation of isovector pairing, symmetry energy and symmetry breaking effects. It also reproduces the relevant ground and excited 0+ state energies and predicts some that are not yet measured. The model successfully interprets fine features driven by pairing correlations and higher-J nuclear interactions. In a classification scheme that is inherent to the sp(4) algebraic approach, a finite energy difference technique is used to investigate two-particle separation energies, irregularities found around the N = Z region, and like-particle and pn isovector pairing gaps. The analysis identifies a prominent staggering behavior between groups of even-even and odd-odd nuclides that is due to discontinuities in the pairing and symmetry terms. While the “classical” limit of the theory provides good overall results, the analysis also shows that q-deformation can be used to gain a better understanding of higher-order effects in the interaction within each individual nucleus
Hoyle state and rotational features in Carbon-12 within a no-core shell model framework
By using only a fraction of the model space extended beyond current no-core
shell-model limits and a schematic effective many-nucleon interaction, we gain
additional insight within a symmetry-guided shell-model framework, into the
many-body dynamics that gives rise to the ground state rotational band together
with phenomena tied to alpha-clustering substructures in the low-lying states
in C-12, and in particular, the challenging Hoyle state and its first 2+
excitation. For these states, we offer a novel perspective emerging out of
no-core shell-model considerations, including a discussion of associated
nuclear shapes and matter radii. This, in turn, provides guidance for ab initio
shell models by informing key features of nuclear structure and the
interaction.Comment: 5 pages, 4 figure
Ab initio Translationally Invariant Nonlocal One-body Densities from No-core Shell-model Theory
[Background:] It is well known that effective nuclear interactions are in
general nonlocal. Thus if nuclear densities obtained from {\it ab initio}
no-core-shell-model (NCSM) calculations are to be used in reaction
calculations, translationally invariant nonlocal densities must be available.
[Purpose:] Though it is standard to extract translationally invariant one-body
local densities from NCSM calculations to calculate local nuclear observables
like radii and transition amplitudes, the corresponding nonlocal one-body
densities have not been considered so far. A major reason for this is that the
procedure for removing the center-of-mass component from NCSM wavefunctions up
to now has only been developed for local densities. [Results:] A formulation
for removing center-of-mass contributions from nonlocal one-body densities
obtained from NCSM and symmetry-adapted NCSM (SA-NCSM) calculations is derived,
and applied to the ground state densities of He, Li, C, and
O. The nonlocality is studied as a function of angular momentum
components in momentum as well as coordinate space [Conclusions:] We find that
the nonlocality for the ground state densities of the nuclei under
consideration increases as a function of the angular momentum. The relative
magnitude of those contributions decreases with increasing angular momentum. In
general, the nonlocal structure of the one-body density matrices we studied is
given by the shell structure of the nucleus, and can not be described with
simple functional forms.Comment: 13 pages, 11 Figure
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