
Students' Understanding of Chemical Formulae: A review of empirical research
Science.gov (United States)
Taskin, Vahide; Bernholt, Sascha
20140101
The fluent use of the chemical language is a major tool for successfully passing chemistry courses at school or university as well as for working as a chemist, since chemical formulae are both a descriptive and a heuristic tool. However, numerous studies have revealed remarkable difficulties of students with chemical formulae both at school and at university. Although analysed for decades, current studies and practical experiences indicate that the misinterpretation of symbolic representations by students is still an ongoing problem. This review intends not only to summarize but also to categorize students' problems and difficulties when dealing with chemical formulae as reported in empirical studies. For this purpose, two categories of descriptive character were deduced from the empirical data: the type of chemical formulae and the operational activities that were required in the tasks of the studies. All in all, 38 articles were analysed on the basis of these categories. Students' problems and difficulties are then reflected based on three main problem areas: languagebased problems, problems due to conceptual understanding, and problems due to inadequate selection and interpretation of formulae. These three areas call for a broader perspective in the interpretation of students' problems and thus lead to a discussion of implications for further research and changes in teaching practice.

Orbital magnetism of Bloch electrons I. General formula
International Nuclear Information System (INIS)
We derive an exact formula of orbital susceptibility expressed in terms of Bloch wave functions, starting from the exact oneline formula by Fukuyama in terms of Green's functions. The obtained formula contains four contributions: (1) LandauPeierls susceptibility, (2) interband contribution, (3) Fermi surface contribution, and (4) contribution from occupied states. Except for the LandauPeierls susceptibility, the other three contributions involve the crystalmomentum derivatives of Bloch wave functions. Physical meaning of each term is clarified. The present formula is simplified compared with those obtained previously by Hebborn et al. Based on the formula, it is seen first of all that diamagnetism from core electrons and Van Vleck susceptibility are the only contributions in the atomic limit. The band effects are then studied in terms of linear combination of atomic orbital treating overlap integrals between atomic orbitals as a perturbation and the itinerant feature of Bloch electrons in solids are clarified systematically for the first time. (author)

Phenomenological formula for alphadecay halflives
International Nuclear Information System (INIS)
A phenomenological formula is presented for the partial halflife from the Q value for α decay. It is constructed in a conventional way by considering the penetrability of a charged particle in a spherical Coulomb potential. Parameters in the formula are fixed because they are determined by physical constants except for the following three adjustable parameters: the product of the collision frequency of an α particle and the formation probability, N; the distance between the charge radius and the radius of an inner point of the Coulomb barrier, r 0; and the oddmass hindrance, h 0. The values obtained for the three adjustable parameters are reasonable, in contrast with those of conventional models such as the ViolaSeaborg formula. The rootmeansquare deviations from experimental partial halflives for eveneven, oddA, and oddodd nuclei are 0.344, 0.740, and 0.940 (in log10), respectively. The obtained formula gives halflives that are two or three times longer than those obtained using the ViolaSeaborg formula in the superheavy nuclear mass region. (author)

Families of automorphic forms and the trace formula
CERN Document Server
Shin, Sug; Templier, Nicolas
20160101
Featuring the work of twentythree internationallyrecognized experts, this volume explores the trace formula, spectra of locally symmetric spaces, padic families, and other recent techniques from harmonic analysis and representation theory. Each peerreviewed submission in this volume, based on the Simons Foundation symposium on families of automorphic forms and the trace formula held in Puerto Rico in JanuaryFebruary 2014, is the product of intensive research collaboration by the participants over the course of the sevenday workshop. The goal of each session in the symposium was to bring together researchers with diverse specialties in order to identify key difficulties as well as fruitful approaches being explored in the field. The respective themes were counting cohomological forms, padic trace formulas, Hecke fields, slopes of modular forms, and orbital integrals.

A WeizsackerBethe type mass formula for hypernuclei
International Nuclear Information System (INIS)
Theoretical estimates of hypernuclear binding energies are generally much larger than the empirical value and the disagreement is rather marked for the binding energy of sub(Λ)5He. Here we try to explain the socalled overbinding problem by way of introducing a WeizsackerBethe type mass formula used for ordinary nuclei. Using the most recent data on binding energies of hypernuclei, parameters of the hypernuclear mass formula are estimated by fitting a leastsquare curve as is the usual practice in nuclear physics. Theoretical predictions for hypernuclear binding energies are then made by using the formula as obtained above and results compared with experimental values. Agreement with experiment is found to be rather good and in particular the result obtained for sub(Λ)5He, although slightly larger than the observed value, has shown significant improvement over earlier estimates. (author)

Analytic flux formulas and tables of shielding functions
Energy Technology Data Exchange (ETDEWEB)
Wallace, O.J.
19810601
Hand calculations of radiation flux and dose rates are often useful in evaluating radiation shielding and in determining the scope of a problem. The flux formulas appropriate to such calculations are almost always based on the point kernel and allow for at most the consideration of laminar slab shields. These formulas often require access to tables of values of integral functions for effective use. Flux formulas and function tables appropriate to calculations involving homogeneous source regions with the shapes of lines, disks, slabs, truncated cones, cylinders, and spheres are presented. Slab shields may be included in most of these calculations, and the effect of a cylindrical shield surrounding a cylindrical source may be estimated. Detector points may be located axially, laterally, or interior to a cylindrical source. Line sources may be tilted with respect to a slab shield. All function tables are given for a wide range of arguments.

Some reference formulas for the generating functions of canonical transformations
Energy Technology Data Exchange (ETDEWEB)
Anselmi, Damiano [Universita di Pisa, Dipartimento di Fisica ' ' Enrico Fermi' ' , Pisa (Italy); INFN, Sezione di Pisa, Pisa (Italy)
20160215
We study some properties of the canonical transformations in classical mechanics and quantum field theory and give a number of practical formulas concerning their generating functions. First, we give a diagrammatic formula for the perturbative expansion of the composition law around the identity map. Then we propose a standard way to express the generating function of a canonical transformation by means of a certain ''componential'' map, which obeys the BakerCampbellHausdorff formula. We derive the diagrammatic interpretation of the componential map, work out its relation with the solution of the HamiltonJacobi equation and derive its timeordered version. Finally, we generalize the results to the BatalinVilkovisky formalism, where the conjugate variables may have both bosonic and fermionic statistics, and describe applications to quantum field theory. (orig.)

Dr. Euler's fabulous formula Cures many mathematical ills
CERN Document Server
Nahin, Paul J
20060101
I used to think math was no fun'Cause I couldn't see how it was doneNow Euler's my heroFor I now see why zeroEquals e[pi] i+1Paul Nahin, electrical engineer In the mideighteenth century, Swissborn mathematician Leonhard Euler developed a formula so innovative and complex that it continues to inspire research, discussion, and even the occasional limerick. Dr. Euler's Fabulous Formula shares the fascinating story of this groundbreaking formulalong regarded as the gold standard for mathematical beautyand shows why it still lies at the heart of complex number theory. This book is the seque

Modular forms and a generalized Cardy formula in higher dimensions
Science.gov (United States)
Shaghoulian, Edgar
20160601
We derive a formula which applies to conformal field theories on a spatial torus and gives the asymptotic density of states solely in terms of the vacuum energy on a parallel plate geometry. The formula follows immediately from global scale and Lorentz invariance, but to our knowledge has not previously been made explicit. It can also be understood from the fact that log Z on T2×Rd 1 transforms as the absolute value of a nonholomorphic modular form of weight d 1 , which we show. The results are extended to theories which violate Lorentz invariance and hyperscaling but maintain a scaling symmetry. The formula is checked for the cases of a free scalar, free Maxwell gauge field, and free N =4 super YangMills. The case of a Maxwell gauge field gives Casimir's original calculation of the electromagnetic force between parallel plates in terms of the entropy of a photon gas.

The Parkland formula under fire: is the criticism justified?
Science.gov (United States)
Blumetti, Jennifer; Hunt, John L; Arnoldo, Brett D; Parks, Jennifer K; Purdue, Gary F
20080101
Controversy has continued regarding the practicality and accuracy of the Parkland burn formula since its introduction over 35 years ago. The best guide for adequacy of resuscitation is urine output (UOP) per hour. A retrospective study of patients resuscitated with the Parkland formula was conducted to determine the accuracy (calculated vs. actual volume) based on UOP. A review of burn resuscitation from a single institution over 15 years was conducted. The Parkland formula was defined as fluid resuscitation of 3.7 to 4.3 ml/kg/% total body surface area (TBSA) burn in the first 24 hours. Adequate resuscitation was defined as UOP of 0.5 to 1.0 ml/kg/hr. Overresuscitation was defined as UOP > 1.0 ml/kg/hr. Patients were stratified according to UOP. Burns more than 19% TBSA were included. Electrical burns, trauma, and children (UOP is the important parameter.

Inversion formula for the growth function of a cancellative monoid
CERN Document Server
Saito, Kyoji
20120101
We consider any cancellative monoid $M$ equipped with a discrete degree map $deg:M\\to R_{\\ge0}$ and associated generating function $P(t)=\\sum_{m\\in M}t^{deg(m)}$, called the growth function of $M$. We also introduce, using some towers of minimal common multiple sets in $M$, another signed generating function $N(t)$, called the skewgrowth function of $M$. We show that these functions satisfy the inversion formula $P(t)N(t)=1$. In case the monoid is the set of positive integers with ordinary product structure and the degree map is logarithm function, using the coordinate change $t=exp(s)$, the inversion formula turns out to be the Euler product formula for the Riemann's zeta function.

Crosstalk Model and Estimation Formula for VLSI Interconnect Wires
Institute of Scientific and Technical Information of China (English)
无
20020101
We develop an interconnect crosstalk estimation model on the assumption of linearity for CMOS device. First, we analyze the terminal response of RC model on the worst condition from the S field to the time domain. The exact 3 order coefficients in S field are obtained due to the interconnect tree model. Based on this, a crosstalk peak estimation formula is presented. Unlike other crosstalk equations in the literature, this formula is only used coupled capacitance and grand capacitance as parameter. Experimental results show that, compared with the SPICE results, the estimation formulae are simple and accurate. So the model is expected to be used in such fields as layoutdriven logic and high level synthesis, performancedriven floorplanning and interconnect planning.

A Gamma Class Formula for Open GromovWitten Calculations
CERN Document Server
Mahowald, Matthew
20160101
For toric CalabiYau threefolds, open GromovWitten invariants associated to Riemann surfaces with one boundary component can be written as the product of a disk factor and a closed invariant. When the Lagrangian boundary cycle is preserved by the torus action and can be locally described as the fixed locus of an antiholomorphic involution, we prove a formula that expresses the disk factor in terms of a gamma class and combinatorial data about the image of the Lagrangian cycle in the moment polytope. As a corollary, we construct a generating function for these invariants using Givental's $J$ function. We then verify that this formula encodes the expected invariants obtained from localization by comparing with several examples. Finally, motivated by large $N$ duality, we show that this formula also unexpectedly applies to Lagrangian cycles on $\\mathcal{O}_{\\mathbb{P}^1}(1,1)$ constructed from torus knots.

New integral formula and its applications to light nucleus reactions
CERN Document Server
Sun, Xiaojun
20150101
A new integral formula, which has not been compiled in any integral tables or mathematical softwares, is proposed to obtain the analytical energyangular spectra of the particles that are sequentially emitted from the discrete energy levels of the residual nuclei in the statistical theory of light nucleus reaction (STLN). In the cases of the neutron induced light nucleus reactions, the demonstration of the kinetic energy conservation in the sequential emission processes becomes straightforward thanks to this new integral formula and it is also helpful to largely reduce the volume of file6 in nuclear reaction databases. Furthermore, taking p+$^9$Be reaction at 18 MeV as an example, this integral formula is extended to calculate the energyangular spectra of the sequentially emitted neutrons for proton induced light nucleus reactions in the frame of STLN.

Analytic study of the MigdalKadanoff recursion formula
International Nuclear Information System (INIS)
After proposing lattice gauge field models in which the Migdal renormalization group recursion formulas are exact, we study the recursion formulas analytically. If D is less than 4, it is shown that the effective actions of Ddimensional U(1) lattice gauge models are uniformly driven to the high temperature region no matter how low the initial temperature is. If the initial temperature is large enough, this holds for any D and gauge group G. These are also the cases for the recursion formulas of Kadanoff type. It turns out, however, that the string tension for D=3 obtained by these methods is rather big compared with the one already obtained by Mack, Goepfert and by the present author. The reason is clarified. (orig.)

Analytic flux formulas and tables of shielding functions
International Nuclear Information System (INIS)
Hand calculations of radiation flux and dose rates are often useful in evaluating radiation shielding and in determining the scope of a problem. The flux formulas appropriate to such calculations are almost always based on the point kernel and allow for at most the consideration of laminar slab shields. These formulas often require access to tables of values of integral functions for effective use. Flux formulas and function tables appropriate to calculations involving homogeneous source regions with the shapes of lines, disks, slabs, truncated cones, cylinders, and spheres are presented. Slab shields may be included in most of these calculations, and the effect of a cylindrical shield surrounding a cylindrical source may be estimated. Detector points may be located axially, laterally, or interior to a cylindrical source. Line sources may be tilted with respect to a slab shield. All function tables are given for a wide range of arguments

Some reference formulas for the generating functions of canonical transformations
International Nuclear Information System (INIS)
We study some properties of the canonical transformations in classical mechanics and quantum field theory and give a number of practical formulas concerning their generating functions. First, we give a diagrammatic formula for the perturbative expansion of the composition law around the identity map. Then we propose a standard way to express the generating function of a canonical transformation by means of a certain ''componential'' map, which obeys the BakerCampbellHausdorff formula. We derive the diagrammatic interpretation of the componential map, work out its relation with the solution of the HamiltonJacobi equation and derive its timeordered version. Finally, we generalize the results to the BatalinVilkovisky formalism, where the conjugate variables may have both bosonic and fermionic statistics, and describe applications to quantum field theory. (orig.)

Explorations of two empirical formulas for fermion masses
International Nuclear Information System (INIS)
Two empirical formulas for the lepton and quark masses (i.e. Kartavtsev's extended Koide formulas), Kl = (sum lml)/(sum l√(ml))2 = 2/3 and Kq = (sum qmq)/(sum q√(mq))2 = 2/3, are explored in this paper. For the lepton sector, we show that Kl = 2/3, only if the uncertainty of the tauon mass is relaxed to about 2σ confidence level, and the neutrino masses can consequently be extracted with the current experimental data. For the quark sector, the extended Koide formula should only be applied to the running quark masses, and Kq is found to be rather insensitive to the renormalization effects in a large range of energy scales from GeV to 1012 GeV. We find that Kq is always slightly larger than 2/3, but the discrepancy is merely about 5%. (orig.)

Conditions for predicting quasistationary states by rearrangement formula
Science.gov (United States)
Yamaguchi, Yoshiyuki Y.; Ogawa, Shun
20151001
Predicting the longlasting quasistationary state for a given initial state is one of central issues in Hamiltonian systems having longrange interaction. A recently proposed method is based on the Vlasov description and uniformly redistributes the initial distribution along contours of the asymptotic effective Hamiltonian, which is defined by the obtained quasistationary state and is determined selfconsistently. The method, to which we refer as the rearrangement formula, was suggested to give precise prediction under limited situations. Restricting initial states consisting of a spatially homogeneous part and small perturbation, we numerically reveal two conditions that the rearrangement formula prefers: One is a no Landau damping condition for the unperturbed homogeneous part, and the other comes from the Casimir invariants. Mechanisms of these conditions are discussed. Clarifying these conditions, we validate to use the rearrangement formula as the response theory for an external field, and we shed light on improving the theory as a nonequilibrium statistical mechanics.

Some reference formulas for the generating functions of canonical transformations
Science.gov (United States)
Anselmi, Damiano
20160201
We study some properties of the canonical transformations in classical mechanics and quantum field theory and give a number of practical formulas concerning their generating functions. First, we give a diagrammatic formula for the perturbative expansion of the composition law around the identity map. Then we propose a standard way to express the generating function of a canonical transformation by means of a certain "componential" map, which obeys the BakerCampbellHausdorff formula. We derive the diagrammatic interpretation of the componential map, work out its relation with the solution of the HamiltonJacobi equation and derive its timeordered version. Finally, we generalize the results to the BatalinVilkovisky formalism, where the conjugate variables may have both bosonic and fermionic statistics, and describe applications to quantum field theory.