scholarly journals The inner product of an automorphic wave form with the pullback of an Eisenstein series

1987 ◽  
Vol 108 ◽  
pp. 93-119
Author(s):  
Shinji Niwa
Keyword(s):  
Eisenstein Series ◽  
Automorphic Forms ◽  
Theta Functions ◽  
Inner Product ◽  
Wave Form ◽  
Prove Theorem ◽  
Wave Forms ◽  
Special Value ◽  
Special Case ◽  
Hilbert Type

In this paper we shall show a relation between a special value of an automorphic wave form and the inner product of the automorphic wave form with the pullback of an Eisenstein series on the upper half space. The main theorem is Theorem 3 in the end of this paper. As is shown in P. B. Garrett [13], pullbacks of Eisenstein series on Siegel upper half spaces have interesting properties as a kernel function of an integral operator. It is natural to try to investigate pullbacks of Eisenstein series of Hilbert type. We can say that Theorem 3 clarifies a property of such pullbacks in a special case. The idea of the proof is a lifting of automorphic forms by theta functions. We discuss a lifting of automorphic wave forms in 1, 2 and 3, and obtain Theorem 2 in the end of 3 as a result. We can prove Theorem 3 without much difficulty by using Theorem 2.

scholarly journals RANKIN’S METHOD AND JACOBI FORMS OF SEVERAL VARIABLES

2010 ◽  
Vol 88 (1) ◽  
pp. 131-143 ◽  
Author(s):  
B. RAMAKRISHNAN ◽  
BRUNDABAN SAHU
Keyword(s):  
Modular Form ◽  
Eisenstein Series ◽  
Cusp Form ◽  
Inner Product ◽  
Jacobi Forms ◽  
Several Variables ◽  
Special Value

AbstractFollowing R. A. Rankin’s method, D. Zagier computed the nth Rankin–Cohen bracket of a modular form g of weight k1 with the Eisenstein series of weight k2, computed the inner product of this Rankin–Cohen bracket with a cusp form f of weight k=k1+k2+2n and showed that this inner product gives, up to a constant, the special value of the Rankin–Selberg convolution of f and g. This result was generalized to Jacobi forms of degree 1 by Y. Choie and W. Kohnen. In this paper, we generalize this result to Jacobi forms defined over ℋ×ℂ(g,1).

scholarly journals Regular and residual Eisenstein series and the automorphic cohomology of Sp(2,2)

2009 ◽  
Vol 146 (1) ◽  
pp. 21-57 ◽  
Author(s):  
Harald Grobner
Keyword(s):  
Algebraic Group ◽  
Eisenstein Series ◽  
Symplectic Group ◽  
Automorphic Forms ◽  
General Theorem ◽  
Simple Algebraic Group ◽  
Ramanujan Conjecture ◽  
Eisenstein Cohomology ◽  
Derivatives Of

AbstractLetGbe the simple algebraic group Sp(2,2), to be defined over ℚ. It is a non-quasi-split, ℚ-rank-two inner form of the split symplectic group Sp8of rank four. The cohomology of the space of automorphic forms onGhas a natural subspace, which is spanned by classes represented by residues and derivatives of cuspidal Eisenstein series. It is called Eisenstein cohomology. In this paper we give a detailed description of the Eisenstein cohomologyHqEis(G,E) ofGin the case of regular coefficientsE. It is spanned only by holomorphic Eisenstein series. For non-regular coefficientsEwe really have to detect the poles of our Eisenstein series. SinceGis not quasi-split, we are out of the scope of the so-called ‘Langlands–Shahidi method’ (cf. F. Shahidi,On certainL-functions, Amer. J. Math.103(1981), 297–355; F. Shahidi,On the Ramanujan conjecture and finiteness of poles for certainL-functions, Ann. of Math. (2)127(1988), 547–584). We apply recent results of Grbac in order to find the double poles of Eisenstein series attached to the minimal parabolicP0ofG. Having collected this information, we determine the square-integrable Eisenstein cohomology supported byP0with respect to arbitrary coefficients and prove a vanishing result. This will exemplify a general theorem we prove in this paper on the distribution of maximally residual Eisenstein cohomology classes.

Elastic displacements in the near-field of a propagating fault

1969 ◽  
Vol 59 (2) ◽  
pp. 865-908
Author(s):  
N. A. Haskell
Keyword(s):  
Fault Plane ◽  
Near Field ◽  
Strong Motion ◽  
Dislocation Motion ◽  
Displacement Discontinuity ◽  
Wave Form ◽  
Strong Motion Station ◽  
Ramp Function ◽  
Wave Forms ◽  
Parkfield Earthquake

abstract Displacement, particle velocity, and acceleration wave forms in the near field of a propagating fault have been computed by numerical integration of the Green's function integrals for an infinite medium. The displacement discontinuity (dislocation) on the fault plane is assumed to have the form of a unilaterally propagating finite ramp function in time. The calculated wave forms in the vicinity of the fault plane are quite similar to those observed at the strong motion station nearest the fault plane at the Parkfield earthquake. The comparison suggests that the propagating ramp time function is roughly representative of the main features of the dislocation motion on the fault plane, but that the actual motion has somewhat more high frequency complexity. Calculated amplitudes indicate that the average final dislocation on the fault at the Parkfield earthquake was more than an order of magnitude greater than the offsets observed on the visible surface trace. Computer generated wave form plots are presented for a variety of locations with respect to the fault plane and for two different assumptions on the relation between fault length and ramp function duration.

Some numerical solutions for Lamb's problem

1974 ◽  
Vol 64 (2) ◽  
pp. 473-491
Author(s):  
Harold M. Mooney
Keyword(s):  
Rayleigh Wave ◽  
Poisson’S Ratio ◽  
Pulse Width ◽  
Numerical Solutions ◽  
Pulse Height ◽  
P Wave ◽  
Poisson's Ratio ◽  
Wave Form ◽  
Lamb’S Problem ◽  
Wave Forms

abstract We consider a version of Lamb's Problem in which a vertical time-dependent point force acts on the surface of a uniform half-space. The resulting surface disturbance is computed as vertical and horizontal components of displacement, particle velocity, acceleration, and strain. The goal is to provide numerical solutions appropriate to a comparison with observed wave forms produced by impacts onto granite and onto soil. Solutions for step- and delta-function sources are not physically realistic but represent limiting cases. They show a clear P arrival (larger on horizontal than vertical components) and an obscure S arrival. The Rayleigh pulse includes a singularity at the theoretical arrival time. All of the energy buildup appears on the vertical components and all of the energy decay, on the horizontal components. The effects of Poisson's ratio upon vertical displacements for a step-function source are shown. For fixed shear velocity, an increase of Poisson's ratio produces a P pulse which is larger, faster, and more gradually emergent, an S pulse with more clear-cut beginning, and a much narrower Rayleigh pulse. For a source-time function given by cos2(πt/T), −T/2 ≦ T/2, a × 10 reduction in pulse width at fixed pulse height yields an increase in P and Rayleigh-wave amplitudes by factors of 1, 10, and 100 for displacement, velocity and strain, and acceleration, respectively. The observed wave forms appear somewhat oscillatory, with widths proportional to the source pulse width. The Rayleigh pulse appears as emergent positive on vertical components and as sharp negative on horizontal components. We show a theoretical seismic profile for granite, with source pulse width of 10 µsec and detectors at 10, 20, 30, 40, and 50 cm. Pulse amplitude decays as r−1 for P wave and r−12 for Rayleigh wave. Pulse width broadens slightly with distance but the wave form character remains essentially unchanged.

The structure of norm Clifford algebras

2012 ◽  
Vol 62 (6) ◽  
Author(s):  
Hans Keller ◽  
Herminia Ochsenius
Keyword(s):  
Clifford Algebra ◽  
Classical Case ◽  
Canonical Representation ◽  
Inner Product ◽  
Infinite Products ◽  
Infinite Dimensional ◽  
Significant Step ◽  
Group Of Isometries ◽  
Inner Automorphisms ◽  
Special Case

AbstractOrthomodular Hilbertian spaces are infinite-dimensional inner product spaces (E, 〈·, ·〉) with the rare property that to every orthogonally closed subspace U ⊆ E there is an orthogonal projection from E onto U. These spaces, discovered about 30 years ago, are constructed over certain non-Archimedeanly valued, complete fields and are endowed with a non-Archimedean norm derived from the inner product. In a previous work [KELLER, H. A.—OCHSENIUS, H.: On the Clifford algebra of orthomodular spaces over Krull valued fields. In: Contemp. Math. 508, Amer. Math. Soc., Providence, RI, 2010, pp. 73–87] we described the construction of a new object, called the norm Clifford algebra C̃(E) associated to E. It can be considered a counterpart of the well-established Clifford algebra of a finite dimensional quadratic space. In contrast to the classical case, C̃(E) allows to represent infinite products of reflections by inner automorphisms. It is a significant step towards a better understanding of the group of isometries, which in infinite dimension is complex and hard to grasp.In the present paper we are concerned with the inner structure of these new algebras. We first give a canonical representation of the elements, and we prove that C̃ is always central. Then we focus on an outstanding special case in which C̃ is shown to be a division ring. Moreover, in that special case we completely describe the ideals of the corresponding valuation ring $$\mathcal{A}$$. It turns out, rather unexpectedly, that every left-ideal and every right-ideal of $$\mathcal{A}$$ is in fact bilateral.

STUDIES OF DISTAL COLONIC MOTILITY IN CHILDREN

PEDIATRICS ◽  
1956 ◽  
Vol 17 (6) ◽  
pp. 820-833
Author(s):  
Murray Davidson ◽  
Marvin H. Sleisenger ◽  
Thomas P. Almy ◽  
Samuel Z. Levine
Keyword(s):  
Ulcerative Colitis ◽  
Bowel Movement ◽  
Colonic Motility ◽  
Distal Colon ◽  
Wave Form ◽  
Spontaneous Bowel Movement ◽  
Normal Individuals ◽  
Iced Water ◽  
Wave Forms

A characteristic propulsive wave, previously reported in adults with ulcerative colitis, has been found in infants with acute diarrhea but not in children with ulcerative colitis. The reasons for this are discussed. The induction of propulsive wave forms in normal individuals by administration of magnesium sulfate orally is described and attention is called to its application to therapeutic studies. The relation of this wave form to propulsion of fecal contents and defecation and its role in the production of abdominal discomfort are considered. Observed variations in the responses of individual children to subcutaneous injection of Mecholyl®, to oral administration of iced water, and to rectal distention may explain innate differences in susceptibility to the development of colonic symptoms in different children. A classification of wave forms from the distal colon based on current concepts of their probable physiologic significance is offered. A tracing from the distal colon in a child having a spontaneous bowel movement is presented and commented on.

Bioelectric Regulation of Tentacle Movement in a Dinoflagellate

1967 ◽  
Vol 47 (3) ◽  
pp. 433-446
Author(s):  
ROGER ECKERT ◽  
TAKAO SIBAOKA
Keyword(s):  
Temporal Relationship ◽  
Sea Water ◽  
Outward Current ◽  
Wave Form ◽  
Applied Current ◽  
Inward Current ◽  
Wave Forms ◽  
Noctiluca Miliaris ◽  
Spike Wave ◽  
Peripheral Cytoplasm

1. Recurring extensions and flexions of the food-gathering tentacle of Noctiluca miliaris occur spontaneously. Identical movements can be evoked by appropriate electrical stimulation. 2. Spontaneous recurring potential wave forms (TRPs) were recorded from the vacuole of the luminescent form of Noctiluca during movements of the tentacle. The basic TRP wave form consists of a characteristic negative-going spike which arises at -20 to -30 mV. from the slowly redeveloping negativity of a pre-spike depolarization, and is followed by a quasi-stable post-spike d.c. level of relative vacuolar negativity (-45 to -60 mV.). 3. The TRP complex, similar in shape to that which occurs spontaneously, follows an intracellularly applied current pulse of either polarity if the vacuolar potential is at the post-spike level. The duration of the evoked pre-spike wave is related to the current intensity and duration. During the pre-spike state outward current is ineffective, although a TR spike occurs in response to inward current. 4. The TRP is distinct in its behaviour and wave form from the flash-triggering potential, which can be evoked in the same cell, even though both exhibit all-or-none spikes. 5. Simultaneous recordings of intracellular potentials and movements of the tentacle showed a consistent temporal relationship between potential changes and subsequent movement. Extension of the tentacle begins 1-2 sec. after the spike and flexion begins within 1 sec. after beginning of the pre-spike wave. 6. Tentacle movement ceased in Ca-free sea water even though the cyclic potential changes continued normally. 7. Electron micrographs of the tentacle showed longitudinal aggregations of microtubules near the outer surface of the peripheral cytoplasm. It is proposed that contraction of these microtubules is the immediate cause of tentacle movements.

Effect of Forcing Terms and General Characteristics of Torsional Vibrations of Marine Engine Systems with Variable Inertia

1974 ◽  
Vol 18 (02) ◽  
pp. 131-138
Author(s):  
W. D. Carnegie ◽  
M. S. Pasricha
Keyword(s):  
Equations Of Motion ◽  
Mean Value ◽  
Wave Form ◽  
Actual System ◽  
Computer Methods ◽  
Reciprocating Engine ◽  
Complex Wave ◽  
Wave Forms ◽  
Crankshaft Rotation ◽  
Engine Systems

The torsional vibration phenomenon in the running gear of reciprocating engine systems is usually dealt with by considering a series of constant inertias connected by sections of massless shafting. Such a simplified model does not reproduce the exact dynamic characteristics of the actual system. In recent years several cases of marine crankshaft failures have been attributed to the phenomenon of secondary resonance, which is explained by the fact that the effective inertia of each slider crankmechanism varies about a mean value in relation to the position of the crank. When the variableinertia effect is allowed for, the equations of motion taking into account the effect are nonlinear. Assuming small displacements, the equations can be linearized to predict important characteristics of the motion. The motions in the form of complex wave forms are studied at different speeds of engine rotation and some of the wave form solutions are analyzed in the range of present investigations. Computer methods making use of numerical analysis processes, namely, the modifiedEuler's equations and the Runge-Kutta constants, have been applied in the investigations. A study of the effect on the motion of the system due to variation of inertia ratio is carried out at a particular speed of the crankshaft rotation; also investigated are the variations in the motions due to the action of external excitations with respect to changes in phase angle and inertia ratio. General comments on Draminsky's work in the light of the present investigations are included.

Uterine work in parturition

1976 ◽  
Vol 41 (3) ◽  
pp. 316-322 ◽  
Author(s):  
T. J. Kriewall
Keyword(s):  
Viscoelastic Model ◽  
Geometrical Model ◽  
Clinical Findings ◽  
Cervical Dilatation ◽  
Wave Form ◽  
Intrauterine Pressure ◽  
Wave Forms ◽  
Internal Volume ◽  
Synergistic Relationship ◽  
Incremental Volume

This paper presents a theoretical analysis relating work expended by the uterus to the synergistic relationship between intrauterine pressure (IUP) and cervical dilatation (CD). By utilizing a geometrical model for the fetal presenting part which is assumed to be in contact with the cervix, the changes in internal volume of the uterus with each contraction can be shown to be functionally related to cervical dilatation. Thus work, expressed in terms of foot-pounds, can be calculated using the integral of pressure times incremental volume. By simulating intrauterine pressure and cervical dilatation with continuous analytical wave forms, the alterations required in uterine work to dilate the cervix are calculated for various wave-form aberrations which are seen in clinical situations. The wave-form aberrations are applied to an elastic cervical model as well as a viscoelastic model. Using the principles of thermodynamics the areas of the fetouterine complex which absorb the work generated by the contractions are defined. It is shown that the efficiency of the contractions to dilate the cervix can be calculated by evaluating the work expended in these various areas of energy absorption. The purpose of this paper is to present the theory upon which clinical findings in obstetrics can be based so that conclusions drawn will be technically sound.

Polynomial identities between Hecke eigenforms

2019 ◽  
Vol 15 (10) ◽  
pp. 2135-2150
Author(s):  
Dianbin Bao
Keyword(s):  
Eisenstein Series ◽  
Product Formula ◽  
Inner Product ◽  
Polynomial Identities ◽  
Number Of Solutions ◽  
Hecke Eigenforms

In this paper, we study solutions to [Formula: see text], where [Formula: see text] are Hecke newforms with respect to [Formula: see text] of weight [Formula: see text] and [Formula: see text]. We show that the number of solutions is finite for all [Formula: see text]. Assuming Maeda’s conjecture, we prove that the Petersson inner product [Formula: see text] is nonzero, where [Formula: see text] and [Formula: see text] are any nonzero cusp eigenforms for [Formula: see text] of weight [Formula: see text] and [Formula: see text], respectively. As a corollary, we obtain that, assuming Maeda’s conjecture, identities between cusp eigenforms for [Formula: see text] of the form [Formula: see text] all are forced by dimension considerations. We also give a proof using polynomial identities between eigenforms that the [Formula: see text]-function is algebraic on zeros of Eisenstein series of weight [Formula: see text].

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