Ordinary Form Research Articles

Estimation of Errors for a POD-based Reduced Order System: Application to Induction Heating

This paper deals with the modeling of a fully dynamic induction heating problem, considered a two-dimensional (2D) axisymmetric problem. The model is described by partial differential equations with nonlinear heat capacity, thermal and electrical conductivities. A discretization technique is applied to obtain a mathematical model in ordinary differential equation form while the nonlinearity is preserved. A model reduction technique is applied to reduce the system size. A small perturbation is given in the initial values of the original model and cumulative errors due to the perturbation and reduction calculated. The errors obtained are very small, thus illustrating the validity of the reduced model. Numerical results are presented and possible extensions are identified.

On Greenberg’s L-invariant of the symmetric sixth power of an ordinary cusp form

AbstractWe derive a formula for Greenberg’s L-invariant of Tate twists of the symmetric sixth power of an ordinary non-CM cuspidal newform of weight ≥4, under some technical assumptions. This requires a ‘sufficiently rich’ Galois deformation of the symmetric cube, which we obtain from the symmetric cube lift to GSp(4)/Q of Ramakrishnan–Shahidi and the Hida theory of this group developed by Tilouine–Urban. The L-invariant is expressed in terms of derivatives of Frobenius eigenvalues varying in the Hida family. Our result suggests that one could compute Greenberg’s L-invariant of all symmetric powers by using appropriate functorial transfers and Hida theory on higher rank groups.

Intermittent theta burst stimulation over ipsilesional primary motor cortex of subacute ischemic stroke patients: A pilot study

BackgroundWe demonstrated that prolonged transcranial magnetic intermittent theta burst stimulation (iTBS, 1200pulses/session=iTBS1200) produces longer-lasting facilitation in corticospinal excitability than ordinary form of iTBS in healthy subjects. ObjectiveHere we show the safety and small-scale efficacy of iTBS1200 over ipsilesional primary motor cortex (M1) in subacute stroke patients. MethodsTwelve patients with first-time, subacute ischemic stroke of the middle cerebral artery were randomized into two groups that received 10 daily-sessions of either iTBS1200 or sham stimulation (n=6/group) over ipsilesional M1 hand region in addition to medical and rehabilitation treatments. The primary outcome measures were safety and tolerability. The secondary outcome measures included the National Institute of Health Stroke Scale (NIHSS), Upper Extremity Fugl-Meyer Test (UE-FMT), Action Research Arm Test (ARAT), active motor thresholds (aMTs)/motor-evoked potentials (MEPs) from the extensor carpi radialis (ECR), and magnetoencephalography on post-intervention day 1 and post-stroke day 60. ResultsiTBS or sham stimulation was well tolerated by all patients without seizures or significant adverse effects. Compared with the matched controls, the iTBS group showed measurable improvements in the NIHSS and the proximal UE-FMT scores on post-intervention day 1 and post-stroke day 60. Nevertheless, the hand ARAT scores, aMT and MEPs from the paretic ECR were not different between groups. Post-movement beta synchronization increased over iTBS-conditioned M1 on post-intervention day 1 in testable patients. ConclusionsRepetitive sessions of iTBS1200 over ipsilesional M1 of subacute stroke patients are safe and the potential benefits encourage a larger trial to determine the efficacy in stroke patients. (ClinicalTrials.gov: NCT-01323881).

Stability and rewiring of nematic braids in chiral nematic colloids

Disclination lines in chiral nematic liquid crystals exhibit larger geometrical diversity than their counterparts in ordinary nematics and form rich entangled structures in systems with dispersed colloidal inclusions. Numerous metastable and stable states separated by low energy barriers allow for simple rewiring of the braids. Introducing a new visualization scheme where variations in splay, bend and twist deformations are followed separately allows us to display the effect of chirality on the director profile of the disclinations, and to uncover the mechanism of their rewiring. We demonstrate the richness of structures by performing a survey of all possible disclination structures that entangle a pair of homeotropic silica microspheres in a π-twisted cholesteric cell using numerical free-energy minimization and laser tweezers assisted microscopy. We analyze their stability against spontaneous rewiring and trace the stability differences to the local effects of chiral environment. We compare the predictions to the experimental results and discuss the differences between both approaches.

Praying the Mass, Vol. II: The Prayers of the Priest by Jeffrey Pinyan

236 Antiphon 16.3 (2012) greater depth to the author’s own voice. A more complete exposition of Sacrosanctum Concilium and its ongoing interpretation would also add authority to the volume’s arguments. The bibliography is narrow and indicates the book’s overall range of sources. These limitations, however, speak more to its audience than its thoughtful content. This book would be a welcome supplement to larger catechetical efforts in a parish, campus ministry, or small faith-sharing setting. The volume’s reflection questions promote discussion of its treatment of the prayers of the people, and making it ideal for group study. Shawn Colberg Saint John’s University Collegeville, Minnesota Jeffrey Pinyan Praying the Mass, Vol. II: The Prayers of the Priest Charleston SC: Jeffrey Pinyan, 2011 iii+239 pp. Paperback $16.00 Treatments of the full, active, and conscious participation of the laity at Mass often focus on the specific roles played by faithful participating in the liturgy. This emphasis can have the unintended consequence of suggesting that the roles played by others, most especially the celebrant , are not worthy of the same attention and study for the laity. Jeffrey Pinyan’s self-published work provides some balance on what it means to “pray the Mass” by unpacking the liturgical action of the celebrant at Mass. This second volume, “The Prayers of the Priest,” follows a first volume that focused on “The Prayers of the People.” Paired with the first, this second volume allows readers to progress more deeply into the fullness of the Divine Liturgy and the mysterium that it conveys by giving useful commentary on the verbal and nonverbal actions of the priest at Mass. The book’s introduction provides a useful orientation to what it means to belong to a “priesthood,” and to the role played by the ordained minister at Mass. It notes that an awareness of these prayers is integral to full participation in the Mass because the priest offers his prayers “‘in the name of the entire holy people and of all present’ (SC 33), so that the whole congregation can pray the Mass with the priest” (4). This sense that the entire people can pray with the celebrant through a proper disposition towards the priest’s prayers offers new ground for formation and catechesis concerning the Mass. Pinyan’s book takes the introduction of the new English translation of the Missale Romanum as an occasion to familiarize readers with 237 Book Reviews the changes they may hear in the priest’s prayers, and form readers through the practice of liturgical catechesis. Pinyan anchors this latter task of catechesis in the ancient practice of mystagogy, which he divides into the work of, first,”interpreting” the liturgical rites in light of salvation history, secondly, “explaining” the signs and symbols used in the rites, and, thirdly, “relating” the rites to all dimensions of Christian life (15). The author carries this tripartite exercise into each of the book’s eight chapters. For example, each chapter culminates in a set of reflection questions arranged into these three categories. The first two chapters, “Preparing for Prayer” and “Ceremonial Actions ,” address specific topics that precede the Mass itself, such as vesting. The remaining six chapters follow the Introductory Rites, Liturgy of the Word, Offertory Prayers, Eucharistic Prayer, Communion Rite, and Concluding Rites, respectively. Throughout these chapters, the volume helpfully provides the texts for all the prayers said by the celebrant in the Ordinary Form of the Mass, including the Latin text for comparison to the English translation. Also helpful are regular scriptural references to both the prayers and the larger liturgical significance behind the prayers. Frequent and apt excurses follow particular prayers, gestures, or other liturgical action in a given chapter. For example, in Chapter Three, Pinyan offers an extended treatment of the ways in which Jesus may be appropriately thought of as an “altar.” These explanations illumine the reason behind the priest’s approach and reverencing of the altar in the opening procession . In these ways, Praying the Mass offers helpful points for reflection as it proceeds through the liturgy’s rites. This book’s strengths lay in its thorough attention to the words of the priest’s...

Analytical approach to heat and mass transfer in MHD free convection from amoving permeable vertical surface

An analytical study is performed on heat and mass transfer in MHD-free convection from a moving permeable vertical surface and the results are compared with previous works on this phenomenon to test the validity. The coupled equations of boundary layer are transformed from their non-linear form to ordinary form using similarity transformation and then are solved by a newly developed method, homotopy analysis method. Having different base functions, homotopy analysis method provides us with great freedom in choosing the solution of a nonlinear problem. Solving the boundry layer equations, the effects of different parameters such as magnetic field strength parameter (M), Prandtl number (Pr), Schmidt number (Sc), buoyancy ratio and suction/blowing parameter (fw) on velocity, temperature, and concentration profiles are taken into consideration. Obtained results show that increment of magnetic field strength parameter (M) leads to decrease in velocity profile. Copyright © 2011 John Wiley & Sons, Ltd.

The Development of a New Electromagnetic Field Generator for Anti-Scaling

Scaling exists commonly and its harm is huge. Because of this, design an amplitude-modulating electromagnetic field generator. The drive signal is obtained by amplitude modulation and power amplification. Generate two sine signals by the wien bridge and make the two signals through the MC1496 multiplier, the carrier’s amplitude changes with the modulator’s rule. And the frequency of carrier and modulator could be changed with the adjustment of the Oscillation resistance. Then, make use of D class amplifier TDA8920BTH to enlarge the power of modulated signal. At last, load the drive signal on a ferrite of magnetic coil to gain an electromagnetic field whose strength is modulated. Differ from the ordinary form of winding pipe with wire line, this design is making the magnetic coil close to a pipe. That’s more convenient for installing and removing. Of course, the concrete effect of descaling is needed to test by further experiments.

An Inexpensive, Widely Available Material for 4 wt % Reversible Hydrogen Storage Near Room Temperature

The search for cheap, renewable energy sources to replace fossil fuels has identified hydrogen gas (H2) as the most promising, particularly for transportation. However, despite intense research efforts to find reliable storage materials, current practical technologies store only 1.3 wt % H2 at 270 K, far short of the U.S. DOE targets. We report that hexagonal ice, the ordinary form of ice in snow, may be an efficient hydrogen storage material, achieving 3.8 wt % H2 storage and 42 g L–1 at 150K and that after loading at 150 K, the 3.8 wt % H2 can be kept at 270 K and then released upon heating by a few degrees Kelvin. This leads us to propose the ice-fixed melt-triggered (IFMT) strategy for hydrogen storage and utilization with ice as the median.

A NEW CONSTRUCTION OF p-ADIC RANKIN CONVOLUTIONS IN THE CASE OF POSITIVE SLOPE

Given two newforms f and g of respective weights k and l with l < k, Hida constructed a p-adic L-function interpolating the values of the Rankin convolution of f and g in the critical strip l ≤ s ≤ k. However, this construction works only if f is an ordinary form. Using a method developed by Panchishkin to construct p-adic L-function associated with modular forms, we generalize this construction to the case where the slope of f is small.

Analytic solution of natural convection flow of a non-newtonian fluid between two vertical flat plates by using decomposition method

An analysis has been performed to study the natural convection of a non-Newtonian fluid between two infinite parallel vertical flat plates and the effects of the non-Newtonian nature of fluid on the heat transfer are studied. The governing boundary layer and temperature equations for this problem are reduced to an ordinary form and are solved by Adomian decomposition method (ADM) and numerical method. Velocity and temperature profiles are shown graphically. The obtained results are valid for the whole solution domain with high accuracy. These methods can be easily extended to other linear and non-linear equations and so can be found widely applicable in engineering and science. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1384–1395, 2010

The Ritus Sacer of the Sacrament of Marriage in the Byzantine Churches

THE RITUS SACER OF THE SACRAMENT OF MARRIAGE IN THE BYZANTINE CHURCHES George Gallaro1 and Dimitri Salachas2 The Codex Canonum Ecclesiarum Orientalium (CCEO) canon 828, in line with the previous legislation of the Eastern Catholic churches establishes that: “§1 Only those marriages are valid that are celebrated with a sacred rite, in the presence of the local hierarch, local pastor, or a priest who has been given the faculty of blessing the marriage by either of them, and at least two witnesses . . . ”§2.The very intervention of a priest who assists and blesses is regarded as a sacred rite for the present purpose.”3 In drafting this canon, the consultors of the Commission for the Revision of the Code of Eastern Canon Law held “that this fundamental norm has to be maintained, since this makes the necessary condition in the Eastern law for the validity of the ordinary form of the celebration of marriage beside the active assistance of the priest, the sacred rite or priestly blessing.”4 Moreover, the same consultors, looking at the possibility of extending to deacons the faculty of blessing marriages, “reject that a deacon, in conformity with the Eastern tradition and for ecumenical reasons, should be considered as minister before whom a marriage could take place in its ordinary form.”5 The Latin Church regulation in this matter is different. In fact, the Codex Iuris Canonici (CIC) canon 1108 establishes that:§1.Only those marriages are valid which are contracted before the local ordinary, pastor, or a priest or deacon delegated by either of them, who assist, and before two witnesses . . . The Jurist 70 (2010) 206–234 206 1 Byzantine Catholic Seminary, Pittsburgh, PA. 2 Urbaniana University and Pontifical Oriental Institute, Rome. 3 Code of Canons of the Eastern Churches, Latin-English Edition (Washington, DC: CLSA, 2001). 4 See Nuntia 8 (1979) 21. 5 Ibid. §2. The person who assists at a marriage is understood to be only that person who is present, asks for the manifestation of the consent of the contracting parties, and receives it in the name of the Church.”6 Furthermore, Latin canon 1112, §1 establishes that: “Where there is a lack of priests and deacons, the diocesan bishop can delegate lay persons to assist at marriages, with the previous favorable vote of the conference of bishops and after he has obtained the permission of the Holy See.”7 As a corroboration of this delegation to the deacon, the canon refers to Lumen gentium 29: “ . . . To the extent that a deacon has been authorized by competent authority, he is . . . to assist at and bless marriages in the name of the Church . . . ”8 This different canonical discipline between the Eastern churches and the Church of Rome reflects their diverse theological perspective on the sacrament of matrimony. Their terminology on the matter also differs; for instance, where the Latin code has “matrimonia quae contrahuntur, matrimoniis assistere,” the Eastern code has “matrimonia quae celebrantur , matrimonia benedicere.”9 Eastern Theology on Marriage and Sacred Rite In his Letter to the Ephesians, Saint Paul reflecting on Genesis, chapter two—“. . . that is why a man leaves his father and mother and clings to his wife, and the two of them become one body ”—affirms that “this is a great mystery, but I speak in reference to Christ and the Church.” Saint Paul sees in the text of Genesis a prophetic image of the union of Christ and his Church, a mystery hidden throughout the centuries and revealed now, as the mystery of salvation of the nations. This Pauline teaching has affected the theology, polity, and liturgy of the sacrament of marriage of the Christian East. The rites and symbols of marriage in the tradition of the various Eastern churches are to be understood in the mystical and the RITUS SACER of the sacrament of marriage 207 6 Code of Canon Law, Latin-English Edition (Washington, DC: CLSA, 1998). 7 See 1983 code, c. 1112; see also Leges Ecclesiae, ed. Xavier Ochoa (Rome: Commentarium pro Religiosis, 1980) 5: 6954–6956. 8 Decrees of the Ecumenical Councils, ed. Norman Tanner, 2 vols. (Washington, DC: Georgetown University Press, 1990) 2: 874. 9 See 1983 code, cc. 1058 and 1108; CCEO...

QUANTUM CANONICAL TRANSFORMATIONS IN WEYL–WIGNER–GROENEWOLD–MOYAL FORMALISM

A conjecture in quantum mechanics states that any quantum canonical transformation can decompose into a sequence of three basic canonical transformations; gauge, point and interchange of coordinates and momenta. It is shown that if one attempts to construct the three basic transformations in star-product form, while gauge and point transformations are immediate in star-exponential form, interchange has no correspondent, but it is possible in an ordinary exponential form. As an alternative approach, it is shown that all three basic transformations can be constructed in the ordinary exponential form and that in some cases this approach provides more useful tools than the star-exponential form in finding the generating function for given canonical transformation or vice versa. It is also shown that transforms of c-number phase space functions under linear–nonlinear canonical transformations and intertwining method can be treated within this argument.

Perspectives on an Organic Development of the Liturgy

Perspectives on an Organic Development of the Liturgy László Dobszay With Pope Benedict XVI’s 2007 apostolic letter Summorum Pontificum, issued motu proprio, the coexistence of two rites – or, in official terms, two forms of the same Roman Rite – has begun. As it turns out, the widespread apprehension that some kind of schism would follow and compromise the unity of the Church was unjustified. While the traditional Roman liturgy (as celebrated according to the liturgical books in force in 1962), now termed the “extraordinary form,” has gained greater freedom, the worthy celebration of the novus ordo of Pope Paul VI (the “ordinary form”) has also received guidelines in papal encyclicals, directives, and statements. The long-lasting debate has finally abated. I am of the opinion, however, that this mode of coexistence is satisfactory only for the present and immediate future. It has been left to the more-distant future to resolve the anomalies that remain in the Roman liturgy, and it is not at all premature to start thinking about the possible solutions. Summorum Pontificum itself seems to This essay was delivered as a paper at the international conference “Pope Benedict XVI and the Sacred Liturgy,” hosted by the St. Augustine Liturgical Atelier of Hungary at the Liszt Ferenc Academy of Music in Budapest, Hungary, 21-24 August 2008. It appears in Antiphon by the kind permission of the conference committee and without prejudice to the copyright. Benedict XVI, Apostolic Letter Summorum pontificum, 7 July 2007, Acta Apostolicae Sedis [henceforth AAS] 99 (2007) 779: “Art. 1. Missale RomanumaPauloVIpromulgatumordinariaexpressio‘Legisorandi’Ecclesiæ catholicæ ritus Latini est. Missale autem Romanum a S. Pio V promulgatum et a beato Ioanne XXIII denuo editum habeatur uti extraordinaria expressio eiusdem ‘Legis orandi’ Ecclesiæ et ob venerabilem et antiquum eius usum debito gaudeat honore. Hæ duæ expressiones ‘legis orandi’ Ecclesiæ, minime vero inducent in divisionem ‘legis credendi’ Ecclesiæ; sunt enim duo usus unici rutus romani.” Also Benedict XVI, Letter (Epistula) Con grande fiducia accompanying the Apostolic Letter Summorum pontificum, 7 July 2007, AAS 99 (2007) 795: “Non è appropriato parlare di queste due stesure del Messale Romano come se fossero ‘due Riti’. Si tratta, piuttosto, di un uso duplice dell’unico e medesimo Rito.” Antiphon 13.1 (2009): 18-27 19 Perspectives on an Organic Development of the Liturgy allude to this eventuality, suggesting that this has not been the final word on the matter. What anomalies are we talking about? On the one hand, a part of the criticism regarding the ordinary form touches upon the way it is celebrated in many places¸ but this really is only one part of the problem. Even if the instructions of the Pope shall correct many failures and abuses (for which there is no guarantee without some disciplinary steps), the break in the liturgy remains palpable in its actual contents. On the other hand, even though the liberalization of the usus antiquior of the Latin Mass has undoubtedly brought many blessings and joy to its supporters, in its present state it will not be able to effect the life of the whole Church as deeply as many hope. The extraordinary form remains extraordinary, reserved to relatively small and isolated groups; in a good case, perhaps, to one among fifty churches in a given area. If you visit the Mass of an average town or village church, you will perceive immediately the unfeasibility of superimposing the extraordinary form of Mass upon the community or its priest; they want to adhere to the true or supposed advantages of the ordinary form. One may be enthusiastic about majestic liturgies accurately performed with a fine schola, well-trained assistants, and properly instructed congregation; but in the present state of the Church is it realistic to suppose that the same can be accomplished also in smaller churches? Even if the Mass of 1962 were introduced (or reintroduced) in every church (although I expect it shall not be), it is not this splendid liturgical form that would become the typical experience, but rather its most simplified mode of celebration, that is, what we may call a private Low Mass open to the public. Surely, we can find examples in the history of the Church when...

Lifting Galois representations of number fields

Let F be a number field. Given a continuous representation ρ ¯ : G F → GL 2 ( F ¯ ℓ ) with insoluble image we show, under moderate assumptions at primes dividing ℓ∞, that ρ ¯ ∼ ρ mod ℓ for some continuous representation ρ : G F → GL 2 ( Q ¯ ℓ ) which is unramified outside finitely many primes. We also establish level lowering when F is totally real, ρ ¯ is the reduction of a nearly ordinary Hilbert modular form and is distinguished at ℓ.

Analytic solution of natural convection flow of a non-Newtonian fluid between two vertical flat plates using homotopy analysis method

An analysis has been performed to study the natural convection of a non-Newtonian fluid between two infinite parallel vertical flat plates and the effects of the non-Newtonian nature of fluid on the heat transfer are studied. The governing boundary layer and temperature equations for this problem are reduced to an ordinary form and are solved by homotopy analysis method (HAM), and numerical method. Velocity and temperature profiles are shown graphically. The obtained results are valid for the whole solution domain with high accuracy. These methods can be easily extended to other linear and non-linear equations and so can be found widely applicable in engineering and sciences.

In-plane free vibration of a single-crystal silicon ring

In this paper the natural frequencies and the associated mode shapes of in-plane free vibration of a single-crystal silicon ring are analyzed. It is found that the Si(1 1 1) ring is two-dimensionally isotropic in the (1 1 1) plane for elastic constants but three-dimensionally anisotropic, while the Si(1 0 0) ring is fully anisotropic. Hamilton’s principle is used to derive the equations of vibration, which is a set of partial differential equations with coefficients being periodic in polar variable. Expressing the radial and tangential displacements in sinusoidal form with non-predetermined amplitudes, and through the integration with respect to the circumferential variable, the original governing equations in partial differential form can be converted into the amplitude equations in ordinary differential form. The exact expressions for frequencies and mode shapes are obtained. It is found that for Si(1 0 0) rings the frequencies of a pair of modes, which are equal for an isotropic ring, split due to the anisotropic effect only for the second in-plane vibration mode. The phenomena of frequency splitting and degenerate modes can be proved either based on the conservation of averaged mechanical energy or by the concept of crystallographic symmetry groups. When the single-crystal silicon is replaced by the polycrystalline silicon, which is isotropic in elastic constants, the derived equations for frequencies correctly predict the vanishing of the phenomenon of frequency splitting.

Long-term variations and trends in the simulation of the middle atmosphere 1980–2004 by the chemistry-climate model of the Meteorological Research Institute

Abstract. A middle-atmosphere simulation of the past 25 years (from 1980 to 2004) has been performed with a chemistry-climate model (CCM) of the Meteorological Research Institute (MRI) under observed forcings of sea-surface temperature, greenhouse gases, halogens, volcanic aerosols, and solar irradiance variations. The dynamics module of MRI-CCM is a spectral global model truncated triangularly at a maximum wavenumber of 42 with 68 layers extending from the surface to 0.01 hPa (about 80 km), wherein the vertical spacing is 500 m from 100 to 10 hPa. The chemistry-transport module treats 51 species with 124 reactions including heterogeneous reactions. Transport of chemical species is based on a hybrid semi-Lagrangian scheme, which is a flux form in the vertical direction and an ordinary semi-Lagrangian form in the horizontal direction. The MRI-CCM used in this study reproduced a quasi-biennial oscillation (QBO) of about a 20-month period for wind and ozone in the equatorial stratosphere. Multiple linear regression analysis with time lags for volcanic aerosols was performed on the zonal-mean quantities of the simulated result to separate the trend, the QBO, the El Chichón and Mount Pinatubo, the 11-year solar cycle, and the El Niño/Southern Oscillation (ENSO) signals. It is found that MRI-CCM can more or less realistically reproduce observed trends of annual mean temperature and ozone, and those of total ozone in each month. MRI-CCM also reproduced the vertical multi-cell structures of tropical temperature, zonal-wind, and ozone associated with the QBO, and the mid-latitude total ozone QBO in each winter hemisphere. Solar irradiance variations of the 11-year cycle were found to affect radiation alone (not photodissociation) because of an error in making the photolysis lookup table. Nevertheless, though the heights of the maximum temperature (ozone) in the tropics are much higher (lower) than observations, MRI-CCM could reproduce the second maxima of temperature and ozone in the lower stratosphere, demonstrating that the dynamic effect outweighs the photochemical effect there. The ENSO signals of annual mean temperature, zonal wind, and ozone are generally reproduced in the troposphere and below the middle stratosphere. The volcanic signals of temperature increase and ozone decrease are much overestimated for both El Chichón and Mount Pinatubo.

Witt’s theorems for Galois Ring valued quadratic forms

We prove Witt’s cancelation and extension theorems for Galois Ring valued quadratic forms. The proof is based on the properties of the invariant I , previously defined by the authors, that classifies, together with the type of the corresponding bilinear form (alternating or not), nonsingular Galois Ring valued quadratic forms. Our results extend the Witt’s theorem for mod four valued quadratic forms. On the other hand, the known relation between the invariant I and the Arf invariant of an ordinary quadratic form (if the associated nonsingular bilinear form is alternating) is extended to the nonalternating case by explaining the invariant I in terms of Clifford algebras.

Rotational-vibrational energy spectra of triatomic molecules near relative equilibria

Complete Hamiltonian operators have been obtained in terms of internal coordinates on the basis of the fiber bundle theory in geometry [J. Math Phys. 44, 4411 (2003)]. In this article, the full Hamiltonian is specialized for a rigid and for a semirigid molecule. For the rigid molecule, all internal coordinates are fixed at constants, so that the Hamiltonian operator comes to take an ordinary matrix form, and accordingly, the Schrödinger equation becomes an algebraic eigenvalue equation. The eigenvalues then provide rotational energy spectra of the rigid molecule. For the semirigid molecule, the full Hamiltonian is expanded in the vicinity of an equilibrium position into a power series in an infinitesimal parameter, to which the perturbation method is applied to obtain energy spectra in the form of a power series in the infinitesimal parameter. Indeed, the energy spectra are calculated to the second order term in the infinitesimal parameter in both the cases where the unperturbed energy spectra are nondegenerate and where those are degenerate. It then turns out that the amount of the energy shift caused by the rotation, which is the sum of the pure rotational and the rotation-vibration coupling terms, is proportional to the total angular momentum eigenvalue. It is also observed that a vibrational energy shift occurs simultaneously, which is caused by the metric defined on the internal space. No symmetry is assumed on the shape of the molecule in equilibrium.

Equivariant Intersection Theory and Surgery Theory for Manifolds with Middle Dimensional Singular Sets

AbstractLet G denote a finite group and n = 2k 6 an even integer. Let X denote a simply connected, compact, oriented, smooth G-manifold of dimension n. Let L denote a union of connected, compact, neat submanifolds in X of dimension k. We invoke the hypothesis that L is a G-subcomplex of a G-equivariant smooth triangulation of X and contains the singular set of the action of G on X. If the dimension of the G-singular set is also k then the ordinary equivariant self-intersection form is not well defined, although the equivariant intersection form is well defined. The first goal of the paper is to eliminate the deficiency above by constructing a new, well defined, equivariant, self-intersection form, called the generalized (or doubly parametrized) equivariant self-intersection form. Its value at a given element agrees with that of the ordinary equivariant self-intersection form when the latter value is well defined. Let denote a finite family of immersions withtrivial normal bundle of k-dimensional, connected, closed, orientable, smooth manifolds into X. Assume that the integral (and mod 2) intersection forms applied to members of and to orientable (and nonorientable) k-dimensional members of L are trivial. Then the vanishing of the equivariant intersection form on × and the generalized equivariant self-intersection form on is a necessary and sufficient condition that is regularly homotopic to a family of disjoint embeddings, each of which is disjoint from L. This property, when is a finite family of immersions of the k-dimensional sphere Sk into X, is just what is needed for constructing an equivariant surgery theory for G-manifolds X as above whose G-singular set has dimension less than or equal to k. What is new for surgery theory is that the equivariant surgery obstruction is defined for an almost arbitrary singular set of dimension k and in particular, the k-dimensional components of the singular set can be nonorientable.