Harmonic Sum Research Articles

The Effect of Advancing Age on Auditory Middle- and Long-Latency Evoked Potentials Using a Steady-State-Response Approach.

The purpose of the study was to objectively detect age-specific changes that occur in equivalent auditory steady-state responses (ASSRs), corresponding to transient middle- and long-latency auditory evoked potentials as a function of repetition rate and advancing age. The study included 48 healthy hearing adults who were equally divided into 3 groups by age: 20-39, 40-59, and 60-79 years. ASSRs were recorded at 7 repetition rates from 40 down to 0.75 Hz, elicited by trains of repeated tone burst stimuli. Temporal analysis of middle- and long-latency equivalent ASSRs revealed no appreciable changes in the magnitudes of the response across the age groups. Likewise, the spectral analysis revealed that advancing age did not substantially affect the spectral content of the response at each repetition rate. Furthermore, the harmonic sum was not significantly different across the 3 age groups, between the younger adults versus the combined Older Group Sample 1 and Sample 2, and between the two extreme age groups (i.e., 20-39 vs. 60-79) for the middle- and long-latency equivalent ASSRs. Advancing age has no effect on the long-latency equivalent ASSRs; however, aging does affect the middle-latency equivalent ASSRs when the mean age difference is ≥ 40 years.

Spectral-Efficient Flexible Optical Multicasting in a Periodically Poled Lithium Niobate Waveguide

We experimentally demonstrate enhanced spectral efficiency (SE) within flexible multicasting operation in a periodically poled Lithium Niobate (PPLN) waveguide. Replicas of quadrature phase-shift keying (QPSK) and 16 quadrature amplitude modulation combs are generated in the C-band exploiting second-order nonlinear effects in a PPLN waveguide as second harmonic generation and sum frequency generation cascaded with difference frequency generation. This technique allows creating equally spaced replicas of the signal to be multicast, being independent of its modulation format. The bit error rate is measured for the signal replicas after transmission through $80\,\text{km}$ of a standard single-mode fiber span. The SE obtained with the multicasting of a $25\,\rm {Gb/s}$ QPSK on $37.5\; {\rm GHz}$ spaced channels is about $1.3\,\rm {b/s/Hz}$ . By introducing the time–frequency packing technique, we demonstrate optical multicasting of a $60\,\rm {Gb/s}$ QPSK signal with $20\,{\rm GHz}$ spacing among the replicas by achieving a SE of $2.7\,\rm {b/s/Hz}$ .

A theoretical study of the critical external pressure for casing collapse

A new theoretical casing collapse model is proposed on the basis of a refined reduced modulus calculation method. The reduced modulus of considers the co-existence of the elastic and plastic regions on the casing cross section. The collapse pressure for an ideal circular casing is expressed as a weighted harmonic sum of the critical elastic pressure and casing yielding pressure. Three kinds of casing imperfections including the wall thickness variation, ovality and non-uniform external pressure on casing collapse are further considered to amend the ideal casing collapse model. The equivalent wall thickness due to wall thickness variation is introduced, and the critical elastic pressure and yielding pressure under the effects of initial ovality and non-uniform external pressure are obtained. On this basis, the collapse pressure for an oval casing with thickness variation under non-uniform external pressure is obtained with superposition principle. The effects of axial force, bending moment and torque on collapse pressure are included with the reduced yielding point. All the above factors are integrated into a unified casing collapse model. At last, the comparison between the new model and API specification is further discussed. The results indicate that the new model gives a more sophisticated description of casing collapse problem. Introducing the casing imperfections improves the consistency between the theoretical and experimental results at the transition between elastic and plastic ranges. The non-uniform external pressure plays the most important role in the casing collapse problem. The existence of non-uniform external pressure, axial force, bending moment and torque also reduces the critical casing collapse pressure a lot.

Une notion de multizêtas finis associée au Frobenius du groupe fondamental de [formula omitted

We show that multiple harmonic sums of the form∑apk<n1<…<nd<(a+1)pk1n1s1…ndsd,for a∈Z,k∈N⁎,sd,…,s1∈N⁎, admit a simple canonical expansion in terms of p-adic multiple zeta values. More generally, we interpret geometrically the multiplication by p of the upper bound of a multiple harmonic sum. This is equivalent to the inversion of the sums of series that express p-adic multiple zeta values. The result leads to the definition of a notion of finite multiple zeta values that is of geometric origin; it gives a framework to study the algebraic properties of those multiple harmonic sums whose upper bound is a power of a prime number. The result also has applications to a conjecture of Kaneko and Zagier.

Third harmonic and sum-frequency generation in ZnO quantum dots

Within the framework of the density matrix approach we investigate the third harmonic generation (THG) and sum frequency generation (SFG) associated with the intersublevel transitions in the conduction band of a singly charged ZnO quantum dot (QD). With three optical fields incident on the dot, we obtain the third order nonlinear susceptibility resulting in the THG and the SFG processes. As compared to the bulk enhanced value of the third order nonlinear susceptibility is obtained through the electric dipole interactions of the pump fields with the electron in the QD. Results are obtained for the THG and SFG in the ZnO QD embedded in the HfO2, Al2O3, AlN, and SiO2 matrices. The results show that the dot size and the surrounding matrix significantly influence the third order nonlinear susceptibility, which is found to be relatively higher for the QD embedded in the high-dielectric constant HfO2 matrix as compared to the other matrices.

Ketone Binding at Amino and Ureido Monolayer/Solvent Interfaces Studied by Nonlinear Optical Techniques

Understanding the influence of immobilization is key to advancing efforts in green chemistry based on supported catalyst materials. The binding of a model reactant 4-nitroacetophenone with amino and ureido organocatalytic monolayers has been investigated at the acetonitrile/solid interface using a combination of second harmonic generation (SHG) and sum frequency generation (SFG). By changing the ketone concentration in the bulk solvent, binding isotherms at each interface were determined from SHG measurements. Langmuir fitting of these isotherms yielded binding energies consistent with hydrogen bond formation. Surprisingly, the ketone had a lower binding affinity for ureido monolayers compared with its binding behavior at amino-modified surfaces despite the fact that urea can form two hydrogen bonds with carbonyl groups. This lower binding affinity was attributed to the presence of a hydrogen bond network within the ureido monolayer that must be disrupted to facilitate ketone binding. Vibrational SFG meas...

Sum frequency and second harmonic generation from the surface of a liquid microjet.

The use of a liquid microjet as a possible source of interest for Second Harmonic Generation (SHG) and Sum Frequency Generation (SFG) spectroscopy is examined. We measured non-resonant SHG scattering patterns from the air/water interface of a microjet of pure water and observe a strong enhancement of the SHG signal for certain scattering angles. These enhancements can be explained by the optical properties and the shape of the liquid microjet. SFG experiments at the surface of a liquid microjet of ethanol in air show that it is also possible to measure the coherent vibrational SFG spectrum of the ethanol/air interface in this way. Our findings are useful for future far-UV or X-ray based nonlinear optical surface experiments on liquid jets. In addition, combined X-ray photoelectron spectroscopy and SHG/SFG measurements are feasible, which will be very useful in improving our understanding of the molecular foundations of electrostatic and chemical surface properties and phenomena.

The Brunn-Minkowski type inequalities for mixed brightness-integrals

The mixed brightness-integrals were defined by Li and Zhu. In this paper, we first establish two Brunn-Minkowski inequalities of the mixed brightness-integrals based on the Blaschke sum and Minkowski sum of convex bodies. Further, we also obtain the Beckenbach-Dresher type inequalities of the mixed brightness-integrals combining the harmonic Blaschke sum and the harmonic radial sum of star bodies.

New Multiple Harmonic Sum Identities

We consider a special class of binomial sums involving harmonic numbers and we prove three identities by using the elementary method of the partial fraction decomposition. Some applications to infinite series and congruences are given.

High average power, widely tunable femtosecond laser source from red to mid-infrared based on an Yb-fiber-laser-pumped optical parametric oscillator

We report on the highly efficient generation of widely tunable femtosecond pulses based on intracavity second harmonic generation (SHG) and sum frequency generation (SFG) in a MgO-doped periodically poled LiNbO(3) optical parametric oscillator (OPO), which is pumped by a Yb-doped large-mode-area photonics crystal fiber femtosecond laser. Red and near infrared from intracavity SHG and SFG and infrared signals were directly obtained from the OPO. A 2 mm β-BaB(2)O(4) is applied for Type I (oo → e) intracavity SHG and SFG, and then femtosecond laser pulses over 610 nm ~ 668 nm from SFG and 716 nm ~ 970 nm from SHG are obtained with high efficiency. In addition, the oscillator simultaneously generates signal and idler femtosecond pulses over 1450 nm ~ 2200 nm and 2250 nm ~ 4000 nm, respectively.

Geometric approach to the parallel sum of vectors and application to the vector $\varepsilon $ -algorithm

The paper aims at answering the following question, in the scalar as well in the vector case: What do the famous Aitken’s \(\Delta ^{2}\) and Wynn’s \(\varepsilon \)-algorithm exactly do with the terms of the input sequence? Inspecting the rules of these algorithms from a geometric point of view leads to change the question into another one: By what kind of geometric object can the parallel (or harmonic) sum be represented? Thus, the paper begins with geometric considerations on the parallel addition and the parallel subtraction of vectors, including equivalent definition, and properties derived from the new point of view. It is shown how the parallel sum of vectors is related to the Bezier parabola controlled by these vectors and to their interpolating equiangular spiral. In the second part, observing that Aitken’s \(\Delta ^{2}\) and Wynn’s \(\varepsilon \)-algorithm may be defined through hybrid sums, mixing standard and parallel sums, the consequences are drawn regarding the way one can define and analyze these algorithms. New explanatory rules are derived for the \(\varepsilon \)-algorithm, providing a better understanding of its basic step and of its cross-rule.

Halide-Induced Cooperative Acid–Base Behavior at a Negatively Charged Interface

Using second harmonic generation and sum frequency generation spectroscopy, we monitor the influence of sodium and potassium halides on acid–base processes at the negatively charged silica/aqueous electrolyte interface. We find that the two types of acidic silanols at the surface are very sensitive to the presence of halides in the aqueous phase. As the halide size increases, the pH at which half the more acidic sites are deprotonated (pH0.5) shifts to lower pH. Conversely, the pH0.5 of the less acidic sites shifts to higher pH with increasing halide size. We also observe titration curves of increasing sharpness as the halide size increases, indicative of positive cooperativity. Using a simple cooperative model, we find that the cooperative unit for the dissociation of more acidic surface sites is ∼1, 2, and 3 for the chloride, bromide, and iodide electrolytes, respectively, which reveals that the larger anions promote deprotonation among the more acidic silanol groups. We also find that the fraction of m...

A new top boundary condition for modeling surface diffusive exchange of a generic volatile tracer: theoretical analysis and application to soil evaporation

Abstract. We describe a new top boundary condition (TBC) for representing the air–soil diffusive exchange of a generic volatile tracer. This new TBC (1) accounts for the multi-phase flow of a generic tracer; (2) accounts for effects of soil temperature, pH, solubility, sorption, and desorption processes; (3) enables a smooth transition between wet and dry soil conditions; (4) is compatible with the conductance formulation for modeling air–water volatile tracer exchange; and (5) is applicable to site, regional, and global land models. Based on the new TBC, we developed new formulations for bare-soil resistance and corresponding soil evaporation efficiency. The new soil resistance is predicted as the reciprocal of the harmonic sum of two resistances: (1) gaseous and aqueous molecular diffusion and (2) liquid mass flow resulting from the hydraulic pressure gradient between the soil surface and center of the topsoil control volume. We compared the predicted soil evaporation efficiency with those from several field and laboratory soil evaporation measurements and found good agreement with the typically observed two-stage soil evaporation curves. Comparison with the soil evaporation efficiency equation of Lee and Pielke (1992; hereafter LP92) indicates that their equation can overestimate soil evaporation when the atmospheric resistance is low and underestimate soil evaporation when the soil is dry. Using a synthetic inversion experiment, we demonstrated that using inverted soil resistance data from field measurements to derive empirical soil resistance formulations resulted in large uncertainty because (1) the inverted soil resistance data are always severely impacted by measurement error and (2) the derived empirical equation is very sensitive to the number of data points and the assumed functional form of the resistance. We expect the application of our new TBC in land models will provide a consistent representation for the diffusive tracer exchange at the soil–air interface.

Parametric wave interaction in one-dimensional nonlinear photonic crystal with randomized distribution of second-order nonlinearity

We theoretically study the parametric wave interaction in nonlinear optical media with randomized distribution of the quadratic nonlinearity \(\chi ^{(2)}\). In particular, we discuss the properties of second and cascaded third harmonic generation. We derive analytical formulas describing emission properties of such harmonics in the presence of \(\chi ^{(2)}\) disorder and show that the latter process is governed by the characteristics of the constituent processes, i.e. second harmonic generation and sum frequency mixing. We demonstrate the role of randomness on various second and third harmonic generation regimes such as Raman–Nath and Cerenkov nonlinear diffraction. We show that the randomness-induced incoherence in the wave interaction leads to deterioration of conversion efficiency and angular spreading of harmonic generated in the processes relying on transverse phase matching such as Raman–Nath interaction. On the other hand, the Cerenkov frequency generation is basically insensitive to the domain randomness.

Calculation of the stationary generation power of a solid-state raman laser with intracavity second harmonic generation and sum frequency mixing

Analytical equations for the power of second-harmonic stationary generation or sum frequency mixing in a Raman laser with an intracavity nonlinear medium are developed and analyzed as functions of pumping mode and other parameters of the laser system. Parameter relationships providing the maximum output power and conditions for suppression of both stimulated Raman scattering by sum frequency mixing and; consequently, of sum frequency mixing itself are found.

Steady-state analysis of auditory evoked potentials over a wide range of stimulus repetition rates: Profile in children vs. adults

Objective: Auditory steady-state responses (ASSR) evoked by recurrent brief tones were assessed over a wide range of stimulus repetition rates apropos the traditionally measured obligatory, transient, auditory evoked potentials. Repetition rates of ≤ 10 Hz have received little attention in the context of the ASSR stimulus-response analysis approach, speculated to provide technical advantages/additional information over more traditional transient stimulus-response paradigms. Design: Magnitudes were measured at repetition rates from 0.75 to 80 Hz, using trains of repeated tone-burst stimuli. Study sample: Twelve normal-hearing children and a reference sample of 25 young adults. Results: Results show that response magnitudes were significantly larger in children than adults at repetition rates of ≤ 5 Hz. Magnitudes were largest at the two lowest repetition rates, following the trends expected from the transient auditory evoked potential (AEP) literature. The harmonic sum is proposed as a more appropriate measure of response magnitude than amplitude of the fundamental alone. Conclusions: The analysis methods used in this paper may give information that will have applications for clinical testing. Of pragmatic importance is that the stimulus rate profile could be determined without subjective wave identification and/or interpretation, and thus by a method that is inherently more objective than conventional AEP analysis.

Extensions of Euler harmonic sums

Three new closed-form summation formulae involving harmonic numbers are established using simple arguments and they are very general extensions of Euler's famous harmonic sum identity. Some illustrative special cases as well as immediate consequences of the main results are also considered.

Molecular Mie model for second harmonic generation and sum frequency generation

A theoretical model to simulate second harmonic and sum frequency generation from stratified spherical particles of arbitrary material is presented and compared with the widely used Rayleigh-Gans-Debye approximation and to experimental results from polystyrene particles with adsorbed malachite green molecules. In this model, the nonlinear polarization is caused by individual dipoles placed in the vicinity of the sphere and is simulated on a molecular basis. This offers greater flexibility to model more sophisticated systems.

Collinear phase-matching loci of LBO crystal in three-wave interactions

By choice of a reliable dispersion relation of LBO crystal, the loci of LBO crystal in second harmonic generation and sum-, difference-frequency generation (three wave interactions) are presented over its entire transparency region for the first time, on the basis of a new topological classification of phase-matching direction loci in biaxial acentric crystals. The result shows that there are totally 7 loci and 9 transitions for second harmonic generation, 12 loci and 15 transitions for three wave interactions at the room temperature 20 °C. The temperature dependence of LBO yields that the noncritical phase-matching loci of second harmonic generation varies slowly in the range of 20–300 °C, while the phase-matching loci in three wave interactions demonstrate varying locus distribution in the spectral region with temperature.

Steady-state analysis of auditory evoked potentials over a wide range of stimulus repetition rates: Profile in adults

Objective: Quasi-steady-state responses were assessed over a wide range of stimulus repetition rates embracing well the traditionally measured transient AEPs (obligatory auditory evoked potentials of all latencies). Repetition rates of ≤10 Hz have received little attention in the context of the ASSR stimulus-response analysis approach which is speculated to provide technical advantages, if not additional information, over more traditional transient stimulus-response paradigms. Design: A measure introduced and defined as the sum of the response at the stimulus frequency and its harmonics. The magnitude of steady-state responses were measured at repetition rates from 0.75 to 80 Hz, using trains of repeated tone-burst stimuli. Study sample: Twenty-five normal-hearing adults. Results: Results show that the magnitudes of the response across repetition rates are largest at the two lowest rates, following trends expected from the transient AEP literature. Good reliability overall was observed for the harmonic sum. Conclusions: The analysis methods used in this paper may give information that will have application for clinical testing. Of pragmatic importance is that the rate profile could be determined without subjective wave identification and/or interpretation, and thus by a method that is inherently more objective than conventional AEP tests.SumarioObjetivo: Se evaluaron respuestas de cuasi-estado estable en un amplio rango de tasas de repetición del estímulo, que correspondían a los AEP tradicionalmente medidos (potenciales evocados auditivos obligatorios de todas las latencias). Las tasas de repetición de ≤10 Hz han recibido poca atención en el contexto del enfoque de análisis de estímulo-respuesta para ASSR, que se especula proporciona ventajas técnicas, además de información adicional, sobre paradigmas más tradicionales de estímulo-respuesta transitorios. Diseño: Una medida introducida y definida como la suma (armónica) de la respuesta, a la frecuencia de estímulo y de sus armónicos. La magnitud de las respuestas de estado estable fueron medidas a tasas de repetición de 0.75 a 80 Hz, usando series de estímulos de ráfagas tonales repetidas. Muestra del Estudio: Veinticinco adultos con audición normal. Resultados: Los resultados muestran que las magnitudes de las respuestas en las tasas de repetición son mayores en las dos tasas más bajas, siguiendo la tendencia esperada de la literatura de AEP transitorios. Se observó una buena confiabilidad para la suma armónica. Conclusiones: Los métodos de análisis usados en este trabajo pueden dar información que tendrá aplicaciones para evaluación clínica. De importancia pragmática es que el perfil de la tasa podría ser determinado sin identificación y/o interpretación subjetiva de ondas, y por tanto, por un método que es inherentemente más objetivo que las pruebas convencionales de AEP.