Multiple Sine Research Articles

Multiple elliptic gamma functions associated to cones

We define generalizations of the multiple elliptic gamma functions and the multiple sine functions, associated to good rational cones. We explain how good cones are related to collections of SLr(Z)-elements and prove that the generalized multiple sine and multiple elliptic gamma functions enjoy infinite product representations and modular properties determined by the cone. This generalizes the modular properties of the elliptic gamma function studied by Felder and Varchenko, and the results about the usual multiple sine and elliptic gamma functions found by Narukawa.

Elliptical Fourier analysis: fundamentals, applications, and value for forensic anthropology.

The numerical description of skeletal morphology enables forensic anthropologists to conduct objective, reproducible, and structured tests, with the added capability of verifying morphoscopic-based analyses. One technique that permits comprehensive quantification of outline shape is elliptical Fourier analysis. This curve fitting technique allows a form's outline to be approximated via the sum of multiple sine and cosine waves, permitting the profile perimeter of an object to be described in a dense (continuous) manner at a user-defined level of precision. A large amount of shape information (the entire perimeter) can thereby be collected in contrast to other methods relying on sparsely located landmarks where information falling in between the landmarks fails to be acquired. First published in 1982, elliptical Fourier analysis employment in forensic anthropology from 2000 onwards reflects a slow uptake despite large computing power that makes its calculations easy to conduct. Without hurdles arising from calculation speed or quantity, the slow uptake may partly reside with the underlying mathematics that on first glance is extensive and potentially intimidating. In this paper, we aim to bridge this gap by pictorially illustrating how elliptical Fourier harmonics work in a simple step-by-step visual fashion to facilitate universal understanding and as geared towards increased use in forensic anthropology. We additionally provide a short review of the method's utility for osteology, a summary of past uses in forensic anthropology, and software options for calculations that largely save the user the trouble of coding customized routines.

Fourier Transform using Spring-Mass System

Fourier transform converts an amplitude over time data set into an amplitude over frequency data set. An example of its application is the equalizer in a sound system. The equalizer takes a sample of the sound and converts it into an amplitude over frequency bar graph. Every sample of an analog wave form can be broken down into a set of sine waves. It is even possible to make something as complex as a square wave by combining multiple sine waves together. There are many mathematical models and algorithms to accomplish this task, however there aren’t many mechanical analogies for the process.

Sinuous meander patterns: a family of multi-frequency spatial rhythms

This paper presents a family of two-dimensional parametric curves that produce frieze patterns with a range of symmetries, organic variations and spatial rhythms. The curves are described by Whewell equations that relate the tangential angle of the curve to its arc length through a series of sine terms. When the low-order harmonics of two sine components are comparable, the pattern develops rhythmic variations (i.e., beats) that may be thought of as a spatial analogue of combination tones in music theory. Inclusion of multiple sine components creates complex, polyphonic patterns with looping forms on a range of spatial scales. The family of curves is based on a historical description of meander bends and rivers, and extends that model to include multiple frequency components.

Touch Screen Sensing Circuit with Rotating Auto-Zeroing Offset Cancellation

In this paper, we present a rotating auto-zeroing offset cancellation technique, which can improve the performance of touch screen sensing circuits. Our target touch screen detection method employs multiple continuous sine waves to achieve a high speed for large touch screens. While conventional auto-zeroing schemes cannot handle such continuous signals properly, the proposed scheme does not suffer from switching noise and provides effective offset cancellation for continuous signals. Experimental results show that the proposed technique improves the signal-to-noise ratio by 14 dB compared to a conventional offset cancellation scheme. For the realistic simulation results, we used Cadence SPECTRE with an accurate TSP model and noise source. We also applied an asymmetric device size (10% MOS size mismatch) to the OP Amp design in order to measure the effectiveness of offset cancellation. We implemented the proposed circuit as part of a touch screen controller system-on-chip by using a Magnachip/SK Hynix 0.18-μm complementary metal-oxide semiconductor (CMOS) process.

Dualities for absolute zeta functions and multiple gamma functions

We study absolute zeta functions from the view point of a canonical normalization. We introduce the absolute Hurwitz zeta function for the normalization. In particular, we show that the theory of multiple gamma and sine functions gives good normalizations in cases related to the Kurokawa tensor product. In these cases, the functional equation of the absolute zeta function turns out to be equivalent to the simplicity of the associated non-classical multiple sine function of negative degree.

Formal group laws for multiple sine functions and applications

We investigate addition relations for multiple sine functions from the view point of formal group laws. We find that the functions which appear in the coefficients are related to classical Eisenstein serires. As application we obtain a limit formula for automorphic forms.

The Study and Experiment of Advance Control of Chaotic Vibration Mill with High Vibration Intensity Based on SCM (II) — The System Construction and the Controlling Methods

A intelligent variable frequency amplitude reduction advance control system whose single-chip-microcomputer act as the core controller is built, and the vibration frequency curve with multiple wave variable sine increase and decrease periodic cycle is applied as the input frequency curve, which made the vibration mill have multiple frequency and amplitude. The control methods are studied, and the operation program is debugged. By using the sensor signal amplification, A/D conversion, on-line monitoring, data mining, and the forecast of the extreme value point of the vibration intensity and its distribution, and then by conducting the judgment, detection, correction and real-time feedback of constraint condition to achieve advance control of vibration intensity over-limit and over-time.

Suppression of Heave-Induced Pressure Fluctuations in MPD

A model describing the flow and pressure fluctuations in the bore-hole due to drillstring movement has been presented. It consists of a pair of coupled nonlinear partial differential equations modelling the distributed pressure and flow in the well, and a superposition of multiple sine waves for the disturbance. Considering only top-side flow and pressure as measurements, it is shown that the model can be represented by a linear time invariant finite-dimensional system with output delay. This result is achieved by linearization and de-coupling using Riemann invariants. An infinite-dimensional observer is designed that estimates the disturbance, and the estimate is used in a controller that rejects the effect of the disturbance on the down-hole pressure. A model reduction technique based on the Laguerre series representation of the transfer function is used to derive a finite-dimensional, rational transfer function for the controller. The performance of the full-order and reduced-order controllers are compared in simulations, which show satisfactory attenuation of the heave disturbance for both controllers.

A Novel Image Encryption Technique using Multi-Wave Based Carrier Image

Information security is a very important topic in fast growing communication technologies. Image security is also a part of this information security. In this paper we propose a novel technique for image encryption based on carrier image. Here, pixels of a carrier image are created by amplitude values of multiple sine waves generated by alphanumeric password. Finally this carrier image is added with original image to obtain encrypted image. Decryption process involves the same procedure to create carrier image and it is added with the encrypted image to get original (decrypted) image. Experiment is conducted for various combinations of multiple waves to generate various patterns in carrier image

The number of full-sine cycles per pulse influences the efficacy of multicycle transcranial magnetic stimulation

Previous studies have shown that the efficacy of transcranial magnetic stimulation (TMS) to excite corticospinal neurons depends on pulse waveform. In this study, we examined whether the effectiveness of polyphasic TMS can be increased by using a pulse profile that consists of multiple sine cycles. In eight subjects, single-pulse TMS was applied to the left primary motor hand area through a round coil attached to a stimulator device that generated polyphasic pulses consisting of one to six full-sine cycles with a cycle length of 86 μs. In different blocks, we varied the number of sine cycles per pulse and recorded the motor-evoked potential (MEP) from the right first dorsal interosseus muscle. For each stimulus type, we determined resting motor threshold (RMT), stimulus-response curve (SRC), and mean MEP amplitude evoked at maximal stimulator output to assess the efficacy of stimulation. Multicycle pulses were more effective than a single full-sine cycle in exciting corticospinal neurons. TMS with multicycle pulses resulted in lower RMT, larger MEP amplitudes at maximal stimulator output and a steeper slope of the SRC relative to a TMS pulse consisting of a single-sine cycle. The increase in efficacy was already evident when two full-sine cycles were used and did not increase further by adding more cycles to the TMS pulse. Increasing the number of full-sine cycles per pulse can improve the efficacy of TMS to excite corticospinal neurons, but there is no simple linear relationship between the number of cycles and TMS efficacy.

Determinants and conformal anomalies of GJMS operators on spheres

The conformal anomalies and functional determinants of the Branson-GJMS operators, P2k, on the d-dimensional sphere are evaluated in explicit terms for any d and k such that k ⩽ d/2 (if d is even). The determinants are given in terms of multiple gamma functions and a rational multiplicative anomaly, which vanishes for odd d. Taking the mode system on the sphere as the union of Neumann and Dirichlet ones on the hemisphere is a basic part of the method and leads to a heuristic explanation of the non-existence of ‘super-critical’ operators, 2k > d for even d. Significant use is made of the Barnes zeta function. The results are given in terms of ratios of determinants of operators on a (d + 1)-dimensional bulk dual sphere. For odd dimensions, the log determinant is written in terms of multiple sine functions and agreement is found with holographic computations, yielding an integral over a Plancherel measure. The N–D determinant ratio is also found explicitly for even dimensions. Ehrhart polynomials are encountered.

G100052 弾性体の自由振動を利用した振動減衰の測定([G10005]機械力学・計測制御部門一般セッション(5):粘弾性・ダンピング)

For a measurement of very small damping, we propose a method of identifying damping by free vibration. The free vibration response is measured by using sound with the vibration. Two methods, method of using the adaptive notch filter and method by curve fit through synthesizing multiple sine wave, are tried. As a result, valid values of damping factor can be obtained from the method of using adaptive notch filter.

Generalized log sine integrals and the Mordell-Tornheim zeta values

We introduce certain integrals of a product of the Bernoulli polynomials and logarithms of Milnor’s multiple sine functions. It is shown that all the integrals are expressed by the Mordell-Tornheim zeta values at positive integers and that the converse is also true. Moreover, we apply the theory of the integral to obtain various new results for the Mordell-Tornheim zeta values.

Extremal values of multiple gamma and sine functions

We study the extremal values of multiple gamma and sine functions in the fundamental intervals. We show the number and locations of the extremal points, and prove that all the local maximum and minimum values are greater and less than one, respectively.

ITERATED EULER'S INTEGRALS AND NORMALIZED MULTIPLE SINE FUNCTIONS

We introduce iterated Euler's integrals, and we give expressions using zeta functions. Moreover, we prove that normalized multiple sine functions are expressed via iterated Euler's integrals. Then, we show basic properties of multiple sine functions from these results containing Kummer type formula.

Weierstrass product representations of multiple gamma and sine functions

It is well known that the Weierstrass product representation of the Barnes multiple gamma function Γr(z) can be calculated concretely. However, there has been no study on its explicit formulation. In this paper, its simple formulation is achieved. It is applicable to the Weierstrass product representation of the Vignéras multiple gamma function also. Moreover, the Weierstrass product representation of the Kurokawa multiple sine function Sr(z) is also formulated explicitly.

ASYMPTOTIC EXPANSIONS FOR PERIOD DEFORMATIONS OF MULTIPLE HURWITZ ZETA FUNCTIONS AND THEIR ASSOCIATED FUNCTIONS

We study the asymptotic behavior of multiple Hurwitz zeta functions and generalized multiple gamma and sine functions as some period parameters tend to 0. We obtain the perfect asymptotic expansions and get some applications in refining a previous paper by the author.

Algebraic relations between special values of multiple sine functions

Algebraic relations between special values of multiple sine functions

Central values of generalized multiple sine functions

We give a nontrivial estimate of the central value of the generalized multiple sine function by a new integral expression of its logarithm. Moreover, when all the periods coincide with 1, we show relations between the central values of the generalized multiple sine functions and the special values of the Riemann zeta function.