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# Weighing picogram aerosol droplets with an optical balance

## Abstract

Optical traps are used to isolate and manipulate small objects in air and liquids, enabling the thorough characterization of their properties in situ. However, no broadly applicable technique for mass measurements of optically trapped objects is currently available. Here we propose an optical balance for mass measurements of optically trapped aerosol particles. By analyzing light-induced harmonic oscillations of a particle, its mass is determined non-destructively and with high accuracy on a time scale of seconds. Its performance is demonstrated for aqueous salt droplets, where masses as low as 4 pg (4 × 10−15 kg) have been measured with an accuracy of ~100 fg. The balance is straightforward to implement and broadly applicable.

## Introduction

Optical traps are widely used to isolate small objects in the liquid phase, in the gas phase and in vacuum. The importance such traps have gained has been recognized by the 2018 Nobel prize in Physics awarded to Arthur Ashkin for the invention of optical tweezers and their applications1. Optical traps serve to manipulate small objects in different environments (liquid, gas, vacuum), and for ultrasensitive measurements of forces and torques2,3,4,5,6,7,8,9,10,11,12. With many applications in the liquid phase related to biological systems13,14, such as proteins or living cells, they play a key role in developing and shaping the fields of biophysics and biology4,13,14. Optical trapping has also been realized in low-pressure environments, e.g., for the quantification of the Brownian motion15 or to study fast particle rotations2,16 and the momentum transfer from light to trapped particles17. In the gas phase, optical traps enable the characterization and manipulation of single isolated aerosol particles and droplets in the submicrometer-size and micrometer-size range with attoliter to picoliter volumes4,18,19,20. Studies typically focus on properties such as size, shape, refractive index, chemical composition, viscosity, and surface tension21,22,23,24,25 with the goal to improve our understanding of aerosol processes, such as phase transitions26, diffusion24, evaporation20,27,28, coagulation29, photochemistry30, reactions with gases or radicals20,31,32, and hygroscopic growth20,27.

Despite the fact that the mass is a key quantity, a broadly applicable method for the determination of the mass of optically trapped aerosol droplets and particles has so far not been demonstrated. Currently, the electrodynamic balance (EDB)33,34,35,36,37 is the reference technique to determine the mass of single particles isolated in air. EDBs provide non-destructive, high-accuracy (in the percent range) mass measurements of particles, but they require prior charging of the particles and mass calibration. They usually provide mass measurements for larger particles with radii ranging from a few to many micrometers, not accessing the submicron range. Submicron–sized droplets and particles are particularly relevant in many atmospheric processes, such as scattering and absorption of sunlight38,39 or size-dependent enhanced kinetics22,40,41. Mass measurements of aerosol particles in optical traps are limited to the special case of photophoretic traps42,43. They require the particle to absorb the light of the trapping laser, which heats the particle. In addition to the size limitation to particles of at least a few43 to a few ten42 micrometers in radius, the heating substantially restricts the applicability of these traps to absorbing, non-volatile particles, basically excluding all volatile and semi-volatile particles, in particular also aqueous droplets. With a typical accuracy of 15%42,43, they do currently not reach the accuracy of EDBs.

This study reports an optical balance for the non-destructive mass determination of optically trapped aerosol droplets and particles with sizes down into the submicron range (lower picogram range). Using modulated counter-propagating optical tweezers, the mass of an aqueous NaCl droplet is determined by inducing harmonic oscillations and analyzing the resulting dynamics. The mass is obtained with high accuracy (less than 2%) at different relative humidities (RH), and the hygroscopic mass growth of the droplet is quantified. The optical balance is straightforward to implement, does not require particle charging, and can even be used without prior mass calibration, which makes it broadly applicable for many studies in aerosol science, or other scientific fields which use optical trapping.

## Results

### Principle of mass measurement

Dual beam traps are widely used to trap particles15,44,45,46,47,48,49. Fig. 1a shows a sketch of the counter-propagating optical tweezers30,50 (CPT) used for trapping single aqueous NaCl droplets at ambient pressure (see “Methods” section for droplet generation and control of the relative humidity). Light from a continuous 532 nm laser (Opus 532, Laser Quantum, typical power ≈1 W) passes through a Faraday isolator (Thorlabs, IO-5-532-HP) to avoid back reflections returning to the laser. A half-wave plate is used to orient the polarization of the laser horizontally before the electro-optic modulator (EOM, Conoptics, 350-50-01-RP), which is used to control the polarization of the laser beam. The laser beam is then expanded four times by a two-lens telescope (focal lengths of 50 and 200 mm). A polarization beam-splitter cube (PBC) splits the incident laser beam into two counter propagating beams. The beams are focused with two aspherical lenses (CVI, BFPL-25.4-50.0-UV, 56.6 mm focal length at 532 nm) in the center of a trapping cell, where a single aerosol droplet is immobilized (brown dot). One half-wave plate is used in each trapping beam to align the polarization of the two beams parallel to each other. A path length difference of approximately 50 cm between the two trapping beams is used on purpose to avoid interference between the two beams (the coherence length of the laser is 7 mm). The optical forces are controlled via the power ratio of the two counter-propagating beams with the EOM. By modulating the power ratio of the two beams sinusoidally in time, a harmonic oscillation of the particle is induced along the beam propagation direction (Fig. 1b). The driven harmonic oscillation of the droplet is described by the following equation of motion (see “Methods” section):

$$\ddot z + 2{\uppi}{\it{\varGamma }}_0\dot z + 4{\uppi}^2f_0^2z = 4{\uppi}^2f_0^2Z_0\sin (2{\uppi}ft)$$
(1)

where z is the position of the droplet along the axis of the laser beams, f0 is the natural frequency of the oscillator and Γ0 is its damping rate. The modulation amplitude Z0 (amplitude corresponding to the position change of the minimum of the trapping potential) and the modulation frequency f of the oscillation are controlled by the modulated voltage applied to the EOM. Equation (1) is solved by

$$z(t) = a\sin (2{\uppi}ft - \varphi )$$
(2)

where a is the amplitude of the droplet’s oscillation and φ is its phase. The droplet position is measured by a position sensitive photodiode (PSP) connected to a lock-in amplifier. During the harmonic oscillation of the droplet, the lock-in amplifier directly measures a and φ. For the data analysis, we prefer φ over a because of correlation issues with the latter (see “Discussion” section). A sweep of the modulation frequency provides φ(f), from which f0 and Γ0 are retrieved by fitting the experimental data to Eq. (3):

$$\tan (\varphi ) = \frac{{{\it{\varGamma }}_0f}}{{f_0^2 - f^2}}$$
(3)

The absolute mass m of the droplet is then determined from Eq. (4):

$$m = \frac{{3R\mu }}{{C_{\mathrm{c}}{\it{\varGamma }}_0}}$$
(4)

where R is the particle radius, μ is the dynamic viscosity of the surrounding fluid (nitrogen in our case) and Cc is Cunningham’s correction factor51. Equation (4) shows that an additional measurement of R is sufficient to calculate the mass of the droplet from Γ0. We use broad-band light scattering (BLS)30,50 to determine R (Fig. 1b, “Methods” section and Supplementary Note 1). With our optical balance, m can be retrieved without mass calibration, in contrast to mass measurements with EDBs which require calibration.

### Harmonicity of the droplet oscillation

The harmonicity of the trapping potential and the induced particle oscillation is verified by the results presented in Fig. 2. Fig. 2a shows an example measurement of φ as a function of f for a trapped aqueous NaCl droplet at an RH of 68% for four different EOM modulation voltages between 10 mV to 40 mV. The latter correspond to values for the amplitudes Z0 between ~10 μm and 40 μm. The measured φ(f) are essentially identical for the four different modulation voltages. Fig. 2b illustrates that the values for Γ0 (right ordinate) and f0 (left ordinate) retrieved from the data in panel a using Eq. (3) agree within their uncertainty for the four different modulation voltages. These uncertainties lie below 0.5% for the measurements with the 20, 30, and 40 mV modulation voltage. For the measurement at 10 mV modulation voltage, the uncertainties of Γ0 and f0 are slightly larger (2% and 1%, respectively). This is due to the smaller Z0 resulting in a lower signal-to-noise ratio for φ(f). From Fig. 2 it is clear that φ(f), Γ0 and f0 are virtually independent of Z0 (i.e., the modulation voltage), which demonstrates that the harmonic approximation provides a very good description of the trapping potential and the induced particle oscillation.

In the rare case when the droplet size and the wavelength of the trapping laser precisely match a resonance condition, so that a Mie resonance (also called morphology dependent resonance or whispering gallery mode) is excited in the droplet52, its dynamics does no longer follow a harmonic oscillation (see Supplementary Note 2 and Supplementary Fig. 2). Fortunately, Mie resonances are limited to very specific combinations of particle size and wavelength so that they do not pose any practical limitation to the mass measurement. If a resonance is encountered, a minute (sub-percent) change of the relative humidity is enough to restore the harmonic oscillation of the particle.

### Mass measurement and related quantities

Fig. 3a shows the mass m (red triangles) retrieved for aqueous NaCl droplets with the same NaCl content, but varying water content given by the RH. The chosen range of RH covers both supersaturated and undersaturated concentrations (the critical RH of NaCl is 75.3 ± 0.3%53). m is retrieved from Γ0 and R using Eq. (4). The accuracy of m is limited by systematic errors and random errors. The systematic error mainly arises from uncertainties of the spot size of the scattered light on the PSP, and is estimated to lie below 3.5% of the droplet mass (see “Methods” section, Supplementary Note 3 and Supplementary Fig. 3). We further estimate the random error to be less than 2–3% of the droplet mass. With a maximum systematic error of 3.5% and a maximum random error of 2–3%, we specify a maximum total error of 5.5–6.5%. For the specific data in Fig. 3a—which cover droplet sizes in the micron range—we determined a systematic error of 2% (see below) and random errors between 0.5 and 1.9% (Table 1). The maximum random error for submicron droplets (R ~ 800 nm, m of few picograms; not listed in Table 1) is somewhat larger around 3%. A maximum total error of 5.5–6.5% represents a high accuracy, in particular when one considers that this is achieved for very low absolute droplet masses in the lower picogram range without any mass calibration.

Access to the lower picogram mass range combined with this level of accuracy already provides satisfactory results for many applications. Mass calibration with a suitable reference allows us to correct for the systematic error and thus to improve the accuracy even further to the level of the random error of less than 2–3%. The calibration can be performed by comparing the measured density of the aqueous NaCl droplets (from R and m) to the density of corresponding bulk solutions54 (Fig. 3b). We determine a constant calibration factor C = 0.98 for the density (ρc = C · ρ), and use the same factor for mass calibration (mc = C · m). mc and ρc denote the mass and the density, respectively, after calibration (Fig. 3a and b and Table 1). A constant mass calibration factor of 0.98 indicates a systematic bias of 2%, and corrects the systematic error mentioned above. (See the “Discussion” section for a further discussion of Fig. 3b).

In addition to the droplet mass and density, our optical balance also allows us to retrieve hygroscopic mass and size growth factors. The mass growth factor is defined as m*(RH) = m(RH)/mNaCl, where m(RH) is the droplet mass at a given RH and mNaCl is the NaCl mass in the solution droplet (see “Methods” section). Note that m*(RH) does not depend on the calibration. Fig. 3c compares m* from this study with results from a previous investigation with an EDB35. Almost perfect agreement is observed between the two different studies, which demonstrates that mass measurements with the optical balance can easily compete with mass measurements in EDBs. The droplet size growth factor is defined as R*(RH) = R(RH)/Rdry, where R(RH) is the droplet radius at a given RH and Rdry is the radius of a completely dried particle (see Supplementary Note 4 for calculation of R*). The R* values retrieved for the data in Table 1 lie between 1.75 and 2.19 (see Supplementary Note 4 and Supplementary Fig. 4). To allow comparison with previous measurements performed with hygroscopic tandem differential mobility analyzers (HTDMA) at 90% RH, we extrapolate R* to 90% RH (see Supplementary Note 4). The retrieved R*(90%) = 2.40 ± 0.02 agrees within uncertainties with theoretical predictions55 and the HTDMA measurements (R*(90%) = 2.27–2.46)56.

## Discussion

### Range and accuracy of mass measurements

The optical balance permits mass measurements of aerosol particles in the picogram range, down to at least 4 × 10−15 kg. The accuracy lies at 5.5–6.5% without and at 2–3% with mass calibration. The latter corresponds to a sensitivity of ~10–16 kg (100 fg) for the smallest droplet reported here. The optical balance extends particle mass measurements into the submicrometer size range, i.e., to masses that are a few orders of magnitudes lower than typical measurements in EDBs34,35 and photophoretic traps42,43, and hence substantially broadens the range of applications (see below). It is not only the extended mass range that makes it more versatile compared with photophoretic traps and EDBs, but also the fact that the balance is neither limited to charged particles (as EDBs are) nor to non-volatile, light-absorbing solid particles (as photophoretic traps are).

The very high mass accuracy of the optical balance could only be achieved because we use the phase (φ) and not the amplitude (a) in the analysis of the droplet dynamics. The example in Fig. 4 illustrates that in certain ranges the mass cannot be determined unambiguously from a(f) because Γ0 and f0 are strongly correlated. As a consequence, a(f) can become indistinguishable for different combinations of Γ0 and f0 (blue full and dashed line); i.e., for different masses (in Fig. 4 for two masses that differ by a factor of 2.25). This issue does not arise for φ(f) because f0 is the frequency f at which φ(f0) = π/2 (crosses in Fig. 4). Hence, a unique solution is obtained for the particle mass when the phase is used instead of the amplitude. The issue with a(f) especially arises when the particle motion is overdamped. i.e., for smaller aerosol particles and for particles suspended in liquids.

### Applications

The optical balance allows one to access properties of supersaturated solutions, which in general cannot be retrieved from bulk measurements because of precipitation. Among those properties is the density of supersaturated solutions. Comparison with bulk densities is only possible in the undersaturated regime, i. e. for aqueous NaCl solutions above the critical RH of 75.3% (Fig. 3b). Tang et al.35 suggests a polynomial extrapolation for the bulk density down to RHs of ~50%. The corresponding extrapolation (black line in Fig. 3b), however, reveals that a ~4% lower density is obtained at an RH = 68% using this bulk extrapolation compared with our experimental value—a difference that lies outside the experimental uncertainty of ~1.2% (Table 1). The origin of this discrepancy between droplet experiment and bulk extrapolation is not obvious and offers an intriguing subject for further studies.

The knowledge of hygroscopic properties of atmospheric droplets is crucial for determining their radiative forcing57. To fully capture their effects, a comprehensive quantification of RH-dependent droplet properties (e.g., size, mass, and density) is required. An interesting phenomenon is the so far unexplained size-dependence of R* for NaCl and sea salts droplets that was previously reported in droplet studies with HTDMAs56,58,59. Depending on the instrument, HTDMA studies are limited to droplets with corresponding dry salt radii in the range Rdry ~ 100–175 nm60. EDBs, on the other hand, typically cover much larger particle sizes (Rdry of a few micrometers), so that the size range in-between remains unexplored. R* studies with our optical balance will allow us to fill this gap (this study Rdry ~ 500 nm and larger), and to address potential size-dependent effects of R* in this intermediate size range. Similarly, the optical balance also opens up studies of m* in this intermediate size range. To the best of our knowledge, m* has only been reported for supermicrometer-sized particles using EDBs.

These are only few examples of the broad applicability of our optical balance. Aerosol science is expected to be a major area of use of this method. However, we also anticipate the optical balance to be of broad interest for mass measurements of small objects in the liquid phase, such as cells and other biological samples. Many studies exist that determine the mass of objects in liquids61,62,63,64 and many studies apply optical trapping in the liquid phase3,11,12,65, but a combination of both methods has not yet been demonstrated. The optical balance we propose solves the issue of mass determination under overdamped conditions (Fig. 4), so that it might offer a possibility to combine optical trapping applications with mass determination in the liquid phase.

Our optical balance permits mass measurements of aerosol particles with masses in the picogram range, and probably beyond, with a high accuracy of 2–3%, covering a particle size range from micrometers down to submicrometers. The implementation of the balance combines the advantage of a straightforward experimental setup with the applicability to a wide variety of particles (liquid, solid, volatile, non-volatile, charged, uncharged). Its demonstrated performance extends mass measurements of aerosol particles into regions previously inaccessible. Aerosol studies related to atmospheric and technical processes, including pharmaceutical and medical applications, will profit from this extended capability. Beyond aerosol studies, the proposed optical balance holds promise for applications in the liquid phase, enabling the combination of optical trapping and mass measurements with intriguing possibilities, e.g., for the investigation of optically trapped cells and other biological objects.

## Methods

### Droplet generation and relative humidity control

The particles were generated from 0.5 M aqueous NaCl solutions using a commercial atomizer (TSI 3076) with pressurized humidified nitrogen (purity 5.0). The use of humidified nitrogen ensures the delivery of liquid droplets to the trapping cell. The concentration of the trapped aqueous NaCl solution droplets is determined by the RH in the trap (and not by the concentration of the initial solution) where the trapped droplets equilibrate quickly to the surrounding RH. The RH in the cell is varied between ~60 and 90% by mixing dry and wet nitrogen with individually adjustable flow rates. The RH in the trapping cell is measured with a sensor (Sensirion, SHT31) placed a few millimeters away from the trapped droplets. The trapping cell is shown in Fig. 5. The cell is equipped with fused silica windows to transmit the light beams of the CPT, the BLS and the scattered light used for the position measurement. Additional apertures are used for the RH sensor and the gas flows through the cell.

### Droplet dynamics

The time dependent position z(t) of the trapped droplet along the trapping axis is described by Newton’s equation of motion, $$m\ddot z = F_{{\mathrm{drag}}} + F_{{\mathrm{opt}}}$$, where Fopt(z) is the optical force, $$F_{{\mathrm{drag}}} = - \gamma \dot z$$ is the drag force due to the surrounding fluid and γ is the drag coefficient. The droplet motion perpendicular to z is negligible in our experiments and is not considered here. Following the harmonic oscillator model, the optical potential Vopt($$F_{{\mathrm{opt}}} = - \partial /\partial zV_{{\mathrm{opt}}}$$) is defined by a second order Taylor expansion Vopt(z,t) = k/2(z − z0(t))2, where k is the trap stiffness and z0(t) is the time dependent position of the minimum of the trapping potential during the modulation. For a sinusoidal modulation of the EOM voltage, the minimum of the trapping potential oscillates as z0(t) = Z0 sin(2πft), where Z0 is the amplitude and f the frequency of the oscillation. Stokes’ law is used to calculate $$\gamma (R) = 6{\uppi}\mu R/C_{\mathrm{c}}(R)$$, where R is the radius of the droplet, μ is the dynamic viscosity of the surrounding fluid (nitrogen) and Cc(R) is the size dependent Cunningham correction factor. As explained in details in ref. 51, the Cunningham correction factor accounts for the nonlinear decrease in the Stoke’s drag force when the particle size approaches the molecular mean free path of the surrounding fluid. It is given by $$C_{\mathrm{c}}(R) = 1 + 1.26\lambda _N/R$$. λN is the mean free path in nitrogen (λN ≈ λair = 68 nm at 20 °C). Using the quantities defined in this paragraph, $$f_0 = 1/2{\uppi}\sqrt {k/m}$$ and $$\varGamma _0 = \gamma /2{\uppi}m$$ in Newton’s equation yields Eq. (1). Substitution of γ in the expression for Γ0 yields Eq. (4).

### Phase measurement and systematic error

z(t) is measured with a commercially available PSP connected to a lock-in amplifier (Zurich Instruments, 500 kHz MFLI). The reference output of the lock-in amplifier is used to drive the EOM (ConOptics) and serves as the reference for the demodulation. During the frequency sweeps of the reference output (f ϵ [0, 2] kHz), the raw phase is obtained directly from the demodulation of the signal of the droplet position along z by the lock-in amplifier. The signal is obtained by dividing the differential photodiode current by the total photodiode current. The measurement of the particle position is slightly delayed by the frequency-dependent impedance of the PSP and the read-out electronics. This delay as a function of modulation frequency, which has to be subtracted from the raw phase to yield φ(f), is determined separately by monitoring the modulation of one of the trapping beams in the frequency range [0, 5] kHz. The phase of this signal is equal to the delay of the PSP and readout electronics because the response time of the EOM to the applied voltage and the light propagation are much faster than the modulation periods used in this study. The PSP and its collecting optics (Fig. 1b) are mounted on a translation stage to adjust the focus of scattered light on the photosensitive area of the PSP. The focusing of the collected light on the PSP sensor affects the accuracy of the particle position tracking. It is mostly responsible for the systematic error of the mass measurement. We have quantified this effect on the accuracy of the mass measurement by repeating phase measurements for different focus sizes. This data is reported in the Supplementary Information (Supplementary Note 3 and Supplementary Fig. 3). Generally, we estimate that the maximal error in the mass measurement caused by the light focus can be up to 3.5% of the absolute droplet mass. For the current measurements, it is 2%. The light focus is not changed between measurements at different RHs, so that the same systematic error applies to all mass measurements reported here.

### Random error

The random error of the mass measurement is calculated by propagating the random error of Γ0 and the error of R (see Supplementary Note 1) using Eq. (4). The random error of Γ0 is determined from the measurements of φ(f) for voltage modulations of 20, 30, and 40 mV (Eq. (3)) as the average error of these three measurements. The measurements at 10 mV are not used for the analysis because of their lower signal-to-noise ratio. The error of R is found by fitting experimental BLS spectra (see below and Supplementary Note 1).

### Broad-band light scattering (BLS)

The principle of the BLS measurements is described in David et al.30. The output of a fiber-coupled broad-band Xe lamp (Energetiq EQ-99XFC LDLS, spectral range 190–2100 nm) is collimated with a reflective collimator (Thorlabs, RC04FC-F01) and then focused on the particle with a reflective objective (Edmund Optics, 89–722). The elastically scattered light is collected over an angular range of 26.6° centered at a scattering angle of θ ≈ 37° with another identical reflective objective. The collected scattered light is then focused on the fiber of the spectrometer (Andor, Kymera 328i with a Newton A-DU940P-BV camera) with a reflective collimator (Thorlabs, RC04SMA-F01). Two shortpass filters (Edmund optics, 49 825, OD > 4) reduce the signal of the elastic scattering of the trapping laser. The wavelength range λ = 350–500 nm is used to retrieve the droplet radius R from fits of the experimental spectra to simulations using Mie theory. R, θ, and n0 and n1 of the wavelength-dependent refractive index of the droplet, n(λ), are fit parameters. A first order Cauchy parametrization is used for n(λ):

$$n(\lambda ) = n_0 + n_1\left( {\left( {\frac{{\lambda _0}}{\lambda }} \right)^2 - 1} \right)$$
(5)

with a reference wavelength of λ0 = 475 nm. Supplementary Fig. 1 shows representative BLS spectra and the values of R and n0 determined in this study.

### Mass growth factor m*

In the present experiments, the mass change of a NaCl droplet caused by variation of the RH is purely due to the uptake or loss of water. Hence, the salt mass (mNaCl) in the droplet remains constant. As a cross-check, we determined mNaCl of the trapped droplet at the different constant RHs using the water activity of bulk solutions with known mass fractions66. The retrieved values of mNaCl at 78%, 81%, and 86% RH (4.56 ± 0.05, 4.50 ± 0.02, and 4.49 ± 0.02 ∙ 10−15 kg, respectively, for one given droplet) agree within 1.5% Below the critical RH (supersaturated solutions) bulk measurements are not possible so that mNaCl cannot be determined in this way. We thus use the average values of mNaCl determined at 78%, 81%, and 86% RH to calculate the mass growth factor m* = m/mNaCl of the droplet for undersaturated and supersaturated solutions (RH = 68%).

## Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Davies, J. F. Mass, charge, and radius of droplets in a linear quadrupole electrodynamic balance. Aerosol Sci. Technol. 53, 309–320 (2019).

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Smith, G. D., Woods, E., DeForest, C. L., Baer, T. & Miller, R. E. Reactive uptake of ozone by oleic acid aerosol particles: application of single-particle mass spectrometry to heterogeneous reaction kinetics. J. Phys. Chem. A 106, 8085–8095 (2002).

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51. 51.

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58. 58.

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65. 65.

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66. 66.

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## Acknowledgements

This work was supported by the Swiss National Science Foundation (SNSF grant no. 200020_172472) and ETH Zurich. We would like to thank David Stapfer and Markus Steger from the ETH mechanical and electronic shop for their help. We would like to acknowledge the contributions of Johannes Cremer in the early stage of this project.

## Author information

Authors

### Contributions

R.S. conceived the project. R.S. and G.D. supervised the project. G.D., O.R., and K.E. implemented the experimental set-up. O.R. performed the experiments and the analysis of the experimental data with assistance from G.D. O.R., G.D., and R.S. wrote the paper. All authors discussed the results and the manuscript.

### Corresponding authors

Correspondence to Grégory David or Ruth Signorell.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Reich, O., David, G., Esat, K. et al. Weighing picogram aerosol droplets with an optical balance. Commun Phys 3, 223 (2020). https://doi.org/10.1038/s42005-020-00496-x

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Chapter 1 - Introduction to Algebraic Expressions - 1.7 Multiplication and Division of Real Numbers - 1.7 Exercise Set - Page 58: 42

$-6$

Work Step by Step

The product of two numbers with opposite signs is negative while the product of two numbers with the same sign is positive. Hence, the given expression, $-\dfrac{8}{3}\cdot\dfrac{9}{4}$, simplifies to \begin{array}{l}\require{cancel} -\dfrac{72}{12} \\\\= -\dfrac{\cancel{12}\cdot6}{\cancel{12}} \\\\= -6 .\end{array}

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×

# Countable and (un)countable

Find an example of a set $$S$$ such that $$S$$ is countable, yet $$S'$$ (the set of all limit points of $$S$$) is uncountable.

Note by Austin Stromme
2 years, 8 months ago

Sort by:

The rationals. More generally, if $$A$$ is an uncountable compact subset of a metric space then, for any $$n \in \mathbb{N}$$ there is a finite subset $$S_n$$ of $$A$$ such that every element of $$A$$ is within $$n^{-1}$$ of some element of $$S_n$$. Then $S=\bigcup_{n\in\mathbb{N}}S_n$ Is a countable subset of $$A$$ whose closure is $$A$$. · 2 years, 8 months ago

One example is the set $$S:= \{ x \colon x\in \mathbb{R}^2 \land \| x\| <1 \}\cap \mathbb{Q}^2$$. · 2 years, 8 months ago


text:
# Thread: Variation of parameters Diff EQ

1. ## Variation of parameters Diff EQ

y" + 2y' + y = 4(e^-x)*ln(x)

Here's my work:

1. r^2 + 2r + 1 = 0
2. (r + 1)^2 = 0; r= -1, -1
3. yc = c1e^(-x) + c2xe^(-x)
4. yp = f1e^(-x) + f2xe^(-x)

So:

f1'e^(-x) + f2'xe^(-x) = 0
-f1'e^(-x) + f2'(e^(-x) - xe^(-x)) = 4(e^-x)*ln(x)

This system simplifies to:

f1' + f2'x = 0
-f1' + f2'(1 - x) = 4*ln(x)

Then I used Cramer's Rule to solve for f1' and f2'. I end up with:

f1' = 4x*ln(x)
f2' = 4*ln(x)

Then I integrate these to find f1 and f2 and plug back into yp = f1e^(-x) + f2xe^(-x). Then the answer is y = yc + yp. However, I don't get the answer in the back of the book. My yc is correct; so my error has to lie in my calculation of the yp. Are my f1' and f2' correct? Any suggestions?

Thanks a bunch,

Jim

2. I used maple to check your work.
I agree you are very close but have lost a minus sign.

I hope you can follow the maple screen shot below.

$f_1'=-4x\ln(x)$

I hope this helps.
Good luck.

3. Thanks Empty Set! Are you a math teacher?

JN


text:
# how to quickly determine whether a given natural number is a power of any other natural number?

We have a natural number $n>1$. We want to determine whether exists any natural numbers $a, k>1$ such that $n = a^k$.

-
Naive answer: approximate the logarithm to about the same number of places as digits in n (there exist algorithms polynomial time in the number of digits), then check if kth roots are integers for k < log n. Each step takes polynomial time, so the the algorithm terminates in polynomial time. –  S. Carnahan Feb 2 '10 at 18:12
Simply compute $n^{1/k}$ for $k=2,\ldots,\lfloor \log\_2 n \rfloor$. Arbitrary roots can be computed in polynomial time (directly or by using logarithms, as Scott Carnahan said), so this is a polynomial-time algorithm. –  Darsh Ranjan Feb 2 '10 at 18:26
I am ignorant in CS .. so this may be a dumb question. Expressing a number to the base p is not a polynomial time problem? –  Anweshi Feb 2 '10 at 18:56
Oops, sorry, then one would need to determine the prime factors of a number too.. Which complicates the issue. Anyway I am totally ignorant of this type of things, as I said. –  Anweshi Feb 2 '10 at 19:34

This can be done in "essentially linear time." Check out Daniel Bernstein's website: http://cr.yp.to/arith.html

Especially note his papers labeled [powers] and [powers2].

-

In order to test whether or not a natural number $n$ is a perfect power, we can conduct a binary search of the integers {1,2,...,n} for a number a number $m$ such that $n = m^b$ for some $b>1$. Let $b>1$. If a solution $m$ to $m^b =n$ exists,then it must lie in some interval $[c_i,d_i]$. When $i = 0$ we may take $[c_0,d_0] = [1,n]$. To define $[c_{i+1},d_{i+1}]$, consider $\alpha:= \left\lfloor \frac{(ci+di)}{2}\right\rfloor$. If $\alpha^b = n$ then we’re done. If $\alpha^b > n$, let $[c_{i+1}, d_{i+1}] = [c_i, \alpha]$; otherwise $\alpha^b < n$ and we let $[c_{i+1}, d_{i+1}] = [\alpha, d_i]$. We continue in this manner until $|c_i − d_i| \leq 1$. We then increase the value stored in variable $b$ and start the loop again. Performing this loop for all $b \leq log(n)$ completes the algorithm.

A pseudocode implementation of this algorithm can be found on page 21 of Dietzelbinger's Primality Testing in Polynoial Time. Its complexity is about $O(log^3(n))$.

-

For each $k \le \log n/\log 2$, compute an approximation to the positive real $k$-th root of $n$ using Newton's method to enough precision to check if it is an integer. Alternatively, use $p$-adic roots for a suitable $p$, with Newton turning into Hensel.

-

The computer algebra system GAP performs this test and determines a smallest root $a$ of a given integer $n$ quite efficiently. The following is copied directly from its source code (file gap4r6/lib/integer.gi), and should be self-explaining:

#############################################################################
##
#F  SmallestRootInt( <n> )  . . . . . . . . . . . smallest root of an integer
##
InstallGlobalFunction(SmallestRootInt,

function ( n )

local   k, r, s, p, l, q;

# check the argument
if   n > 0  then k := 2;  s :=  1;
elif n < 0  then k := 3;  s := -1;  n := -n;
else return 0;
fi;

# exclude small divisors, and thereby large exponents
if n mod 2 = 0  then
p := 2;
else
p := 3;  while p < 100  and n mod p <> 0  do p := p+2;  od;
fi;
l := LogInt( n, p );

# loop over the possible prime divisors of exponents
# use Euler's criterion to cast out impossible ones
while k <= l  do
q := 2*k+1;  while not IsPrimeInt(q)  do q := q+2*k;  od;
if PowerModInt( n, (q-1)/k, q ) <= 1  then
r := RootInt( n, k );
if r ^ k = n  then
n := r;
l := QuoInt( l, k );
else
k := NextPrimeInt( k );
fi;
else
k := NextPrimeInt( k );
fi;
od;

return s * n;
end);
-


text:
# $T(n) = 2T(n-1)$ Recurrences

What would be the complexity of the following recurrence?

$$T(n) = 2T(n-1)$$

• What do you mean by complexity of the recurrence? What is the complexity of the recurrence $T(n) = T(n-1) + 1$, for example? – Yuval Filmus Mar 28 '18 at 21:18
• Your recurrence $T(n) = 2T(n-1)$ isn't satisfied by your initial values. That's like asking "if $x = 0$ and $x = 1$, then which is true, $x = 0$ or $x = 1$?" – Yuval Filmus Mar 28 '18 at 21:22
• Possible duplicate of Solving or approximating recurrence relations for sequences of numbers – David Richerby Mar 29 '18 at 10:26

Note: The original version of the post included the initial values $T(0) = 0$ and $T(1) = 1$.

If $T(n) = 2T(n-1)$ for all $n > C$ then, as you can prove by induction, $$T(n) = 2^{n-C} T(C).$$ In particular, in your case $T(n) = 2^{n-1}$. Note that given your initial values, the recurrence $T(n) = 2T(n-1)$ cannot hold for $n=1$; presumably it holds for all larger $n$.

You are asking for the complexity of the recurrence. Recurrences have no such thing as complexity. A recurrence might describe the running time of an algorithm, and then its solution is the time complexity of the algorithm. But a recurrence can describe any number of things, or even be completely abstract.

i don't know what do you mean about complexity of reccurence but if you want to find the closed form you can use the backward substitution method:
$T(n) = 2T(n-1)$
and then expand the $T(n-1)$:
$T(n) = 2(2T(n-2))$
and after that expand $T(n-2)$:
$T(n) = 2(2(2T(n-3)))$
...
$T(n) = 2^nT(0)$
and if you have $T(0) = C$ as a base case, your closed form would be: $T(n) = C2^n$

• What if $T(0) = 0$, does that make $T(n)=0$ and therefore $T(n)=O(1)$ instead of $T(n)=O(2^n)$? This is what confuses me the most – Josh3248 Mar 29 '18 at 11:22
• in practice, when you have a base case like: $T(0) = 0$, you mean the cost of the algorithm with the size of ZERO is ZERO and after that for the size of ONE is ZERO too. because the closed form constructed by the base case and the equation doesn't have a heterogeneous part, it is strictly based on the base case and i think it doesn't true because of the wrong base case value. – Danial Bidekani Bakhtiarvand Mar 29 '18 at 14:20


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## The Sacred Imperial State of Estonul

You're the owner of a country and want to negotiate with other countries? Here is the right place to do so.

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## Which flags looks best?

Flag 1
6
55%
Flag 2
5
45%

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### The Sacred Imperial State of Estonul

"Estonul, The land blessed and granted by Terdisu"

Flag 1

Flag 2

General Information

Official Name: The Sacred Imperial State of Estonul
Form of Government: Absolutist Monarchical Empire
Maximum Figuration (Emperor): Diego Sierra II
Independence: December 9, 1960
Formation of the Empire: January 17, 1985
Official Religion: Seigopresmism
Surface3 089,402 km2, 7th World
GDP: $106'428 trillions of dollars GDP percapita:$26,369 USD
Currency: Tersu

History

A town located in what would be Etia is an old text of religious appearance, the curiosity of knowing the content is read since it was the same language, it contained stories about a supreme god who was called Terdisu, the small town easily believes in that book that is called as the "Sacred Word", the spread attracted the attention of the Osneraldians because according to that book, this land was owned by Terdisu, this land can not be dominated by another nation that do not show allegiance to the supreme god.
This new religion spread through Etha, New Ambaria and some parts of Anentova, following the orders that the terdisu land should be free the capital and several peoples rose up against the Osnerald Empire, the support for independence was huge, the royal army could to face the war but the crown preferred to avoid losses and guaranteed the freedom but only of the region of Two Whales.
When the Osnerald Empire desingrated , Estonul sought to extend his religion by means of a war against his neighbors, in this case the Old Franxha
Estonul intended to declare war on Anetova but the another nations intervened if it did not stop its expansion.
Estonul has declared a free country so this nation will not tolerate leftist countries or those in which it considers religion is constantly attacked.
Last edited by kimTeg on Mon Oct 01, 2018 6:23, edited 1 time in total.
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### Re: The Sacred Imperial State of Estonul

As stated by commander, kindly refrain using irl places and countries except UN since we had a UN in the forums, other than that everything seems to be in order
Hon. Ryan Jaminal
President of the United Federation of Arizelon
IFIJ|SAITO member
With Courage and Optimism, we shall endure

Mabuhay ang Pilipino!

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### Re: The Sacred Imperial State of Estonul

Ok thanks for letting me know, I'm going to change the content.
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### Re: The Sacred Imperial State of Estonul

Greetings from UHAE , Emerlian Kingdon and COK , we are all willing to open embassies , start trades routes and open several consulate in your major cities .
Emperor of Atlantean federated empire

Founder state of Internationxal council
Leader of AEOY and steel pact

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### Re: The Sacred Imperial State of Estonul

As a reason for international security the Republic of Estonul asks you to check certain requirements to make known their intentions and if they meet national expectations.
The following requirements will be applied to any nation that wishes to establish a relationship, be it political, religious, economic or military, without any exclusion.

What is the objective of approaching a relationship with the nation?
What do you hope to consolidate with it?
What can your nation offer to interest us?

Now in relation to other issues
Does your nation have the capacity to tolerate our policies towards religion?
Would your nation accept military practices?

The next questions are for security between both nations, if your nation wants to know other things or ask for something else, it is fully accepted.

Secretary State
Fervane Reavsak
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### Re: The Sacred Imperial State of Estonul

Greetings
The United Federation of Arizelon would like to establish formal diplomatic and trade relations with Estonsul due to the following reasons:

A. It is the policy of our nation to have contact to all nations of theoearth to expand our diplomatic horizons and assist in any way.
B. To open each other economically and to allow trade to grow
C. We can offer mostly monetary grants, trade deals(if possible via the TTO or an Individual Free Trade Agreement), technological and military support and such.

We are a democratic nation and very religously diverse, a majority Christian nation, we protect the right to worship any religion as long as it conforms to our laws.
Our military is a very important aspect of our country. We see it as a vital part of our foreign policy. We have a vast military industrial complex(not specifically a big industrial zone, it refers to our Vast Defense Sector with many defense companies and manufacturers).

For now we would like to establish an embassy and your nation could also do the same on our capital. And we would like to open trade routes as well and a possibility of a trade deal between our two countries.

Signed
Secretary of State Ranjie Abne
On the authority vested by the President
Hon. Ryan Jaminal
President of the United Federation of Arizelon
IFIJ|SAITO member
With Courage and Optimism, we shall endure

Mabuhay ang Pilipino!

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### Re: The Sacred Imperial State of Estonul

kimTeg wrote:
Wed Oct 03, 2018 3:04
As a reason for international security the Republic of Estonul asks you to check certain requirements to make known their intentions and if they meet national expectations.
The following requirements will be applied to any nation that wishes to establish a relationship, be it political, religious, economic or military, without any exclusion.

a)
UHAE , Emerlian kingdom and Oceanians kingdoms consider diplomacy a very important thing which mean we try to have good relations with every state .
b)
We hope to create and consolidate good relations between us , economics and diplomaties
c)
Trading with us will allow you to import many material and ressources , we are quite strong and we are also an influential ally , and if in the need of help , UHAE economy proven to be ... usefull to some states , and this could finally grant you possibility to access steel pact which would protect you a lot from potentials enemy states .

Now in relation to other issues

Leading religions are christianism in our 3 states, Emerlian army got an powerfull army , Oceanians kingdoms got a capable army and UHAE got a huge army with cutting edge technologies able to take out strongest states in few days , however , all our states are pacifist and we are willing to stay at peace and UHAE army is not deployed on earth .

Every religions in UHAE, Emerlian and Oceanians kingdoms are protected and equals , and while all states are christians, being from an other religion is not a problem as long as you respect our state constitution

The next questions are for security between both nations, if your nation wants to know other things or ask for something else, it is fully accepted.

Secretary State
Fervane Reavsak
Emperor of Atlantean federated empire

Founder state of Internationxal council
Leader of AEOY and steel pact

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### Re: The Sacred Imperial State of Estonul

Aisatsu,
The Socialist Union of Svostine would humbly requests for the establishment formal diplomatic and trade relations with Estonsul for the following reasons;

What is the objective of approaching a relationship with the nation?
Our objective is to to have formal diplomatic relations and trade relations with every nation in the planet, as part of our world peace initiative and bringing the world closer to each other. Our another objective is to invite your nation to join the TheoEarth Trade Organization.

What do you hope to consolidate with it?
We hope for the prosperity of both our nations and for trade to flourish between us.

What can your nation offer to interest us?
As being the leading in science and technology, our nation can offer your nation great technologies that will benefit your nation and we can even help your nation's space program from launching a space station to landing man in a different world.

Does your nation have the capacity to tolerate our policies towards religion?
My nation, Svostine, is one of the biggest number of Atheist in the planet, basically 97% of our population are atheist. We don't really mind "policies" and beliefs of nations but of course, we highly respect it.

Would your nation accept military practices?
It will depend. Our nation keeps our military discreet and hidden from the world as we want to protect our national security and to avoid being considered a threat.

The military is very important in our nation as it protects us from external threats and is a deterrent to others nations from thinking of taking away our sovereignty to our lands. Our military is state of the art and our defense industries are very big that further boost our military might. Religion on the other hands, is not important for the 97% of our population.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
BonitaBenito
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BEING LOVED BACK FEELS SO GREAT!
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### Re: The Sacred Imperial State of Estonul

Federal Monarchy of Freetradezonia was happily to open the relationship between us,this is our answers responds to your questions

What is the objective of approaching a relationship with the nation?
From Us,which is entitled to having a relationship on all country which is exist on earth,if they aren't too overpowered

What do you hope to consolidate with it?
We are hoping to create both in more and better in economy and also more benefitable in earth

What can your nation offer to interest us?
We Can offer your in economics which is helping you into better nation which is became advanced economics

Does your nation have the capacity to tolerate our policies towards religion?
Hmm... This is ok,Bascially we are accepting all religions including Atheist(which is no religions) if they aren't too overpowered in "policy" and "national beliefs" issue

Would your nation accept military practices?
This will depend on the issue,which is under own national security on first behalf and also basically this was out of national security threat

Basically it was both important,On Freetradezonia,we are try to balancing the power on both part which military can't be overpowered and also same to religions and last part which to free from our sovergnity and indepence be taken by others.
KING HENDERION
Chairman/Founder of
TTNMF(Theotown Nations Monetary Fund)
CEO of
OKPhones Inc.
Musicmain International Co.
We are one and united!

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### Re: The Sacred Imperial State of Estonul

Special Statement: Arizelon

In response to the answers stated above, the central government of the Sacred Imperial State of Estonul took the following actions:
The United Federation of Arizelon will be granted the right to discuss and declare opinions about the nation.
The Embassy will be built in the national territory, this will be seen as an opening to the world and use this medium as an international communication.
For reasons that will not be declared, commercial, political and military treaties will not be elaborated for the time being.

The foregoing declared will only be temporary, exposed to modifications in what consists of time.

Imperial Representative
Fervane Reavsak
Maximum Representation of G.E.M.A

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### Re: The Sacred Imperial State of Estonul

Declaration to:UHAE, Emerlian kingdom and Oceanians kingdoms

The nation wants to know more about the nations declared above in order to understand the interests of each nation.
The embassies will be built jointly if so desired in the face of ignorance of the relations of the nations that request them.
Maximum Representation of G.E.M.A

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### Re: The Sacred Imperial State of Estonul

Declaration to: Svostine

Due to several inequalities in our ideals, the State of Estonul will only be seen with the intention of building embassies.

Communicate of
Imperial Representative
Fervane Reavsak
Maximum Representation of G.E.M.A

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### Re: The Sacred Imperial State of Estonul

Declaration to: Federal Monarchy of Freetradezonia

The Sacred Imperial State considers the Federal Monarchy of Freetrazedonia possibly as an essential ally for the improvement and development of the nation.
The answers have been appropriate for national interests except for one but not the least, this is about religion and the state, we hope this is not a problem in the future.
To consolidate the beginning of relations and conversations, we will build an embassy in the nation, we hope that our nations continue to be friendly, if not, we regret it.

Imperial Representative
Fervane Reavsak
Maximum Representation of G.E.M.A

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### Re: The Sacred Imperial State of Estonul

Mabuhay from Avalon!

We have officially and formally declared our recognition for the state of Estonul, and due to that, we would like to begin and formalize relations between our two nations for the following reasons:

1. Avalon's (United Nations of Avalon) foreign policy is to have diplomatic relations with each and every state even if there are ideological differences.
2. Avalon has a strong export policy, and our companies viewtopic.php?f=74&t=7636 have managed to reach nearly every nation around the globe, thus improving the business sectors of both our countries.
3. Avalon is an emerging market and superpower, we would be able to help Estonul through any economic means.
4. Avalon has a vast arsenal of weaponry and has ramped up cyber defense and warfare. We could easily trade some of our military tech, of which is extremely valued and is currently and ever developing.
5. Avalon could offer monetary and trade deals (under the TheoEarth Trade Organization, which Avalon has created) which could fuel Estonul's expansionist policy
6. Avalon is an agricultural powerhouse, thus we would be able to supply and provide sources of food for Estonul

Avalon is composed of six independent nations, with a strong democracy and government. We are also a secular state. We have laws that protect any minority of our population, which includes religion as we are a Catholic majority. As long as those policies do not compromise the national security, which most likely won't, we will tolerate it.

Our nation tolerates joint military practices and is in fact open to it. It would however depend on the issue, however we are open to any talks regarding it as it could improve relations.

Religion and military is a very important aspect of Avalon. For religion, while we are largely and mostly Catholic, a lot of our population are open to accepting other issues such as the LGBT, divorce, and such. For military, our military is state of the art, and is ever improving, a large portion of our military is developed by Nexus Inc.

We would like to begin our newfound relationship by constructing embassies. We are open to also any trade routes, talks or deals between our two nations.

If there is any concern that the Estonul government would like us to address, do not be afraid to clarify.

Regards,
Diamond Palace

President Damien Scott De Guzman
President of the United Nations of Avalon
Founder of the TTO
HIC, EPMU, STEEL PACT, RMC MEMBER
For the Greater Glory of God

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### Re: The Sacred Imperial State of Estonul

Mabuhay yet again!

Regarding any additional info as seen from previous declarations from the Estonul government,

The United Nations of Avalon is composed of six independent nations, namely:

Federal Republic of Avista
United States of Rhinestone
Union of Eastern Oceanic Republics
United Western Oceanic States
Republic of Rhinean Timor
Republic of Rhinean Palaos

Together they form the United Nations of Avalon

President Damien Scott De Guzman
President of the United Nations of Avalon
Founder of the TTO
HIC, EPMU, STEEL PACT, RMC MEMBER
For the Greater Glory of God

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### Re: The Sacred Imperial State of Estonul

Ace_TFC wrote:
Sun Oct 07, 2018 10:50
Mabuhay from Avalon!

We have officially and formally declared our recognition for the state of Estonul, and due to that, we would like to begin and formalize relations between our two nations for the following reasons:

1. Avalon's (United Nations of Avalon) foreign policy is to have diplomatic relations with each and every state even if there are ideological differences.
2. Avalon has a strong export policy, and our companies viewtopic.php?f=74&t=7636 have managed to reach nearly every nation around the globe, thus improving the business sectors of both our countries.
3. Avalon is an emerging market and superpower, we would be able to help Estonul through any economic means.
4. Avalon has a vast arsenal of weaponry and has ramped up cyber defense and warfare. We could easily trade some of our military tech, of which is extremely valued and is currently and ever developing.
5. Avalon could offer monetary and trade deals (under the TheoEarth Trade Organization, which Avalon has created) which could fuel Estonul's expansionist policy
6. Avalon is an agricultural powerhouse, thus we would be able to supply and provide sources of food for Estonul

Avalon is composed of six independent nations, with a strong democracy and government. We are also a secular state. We have laws that protect any minority of our population, which includes religion as we are a Catholic majority. As long as those policies do not compromise the national security, which most likely won't, we will tolerate it.

Our nation tolerates joint military practices and is in fact open to it. It would however depend on the issue, however we are open to any talks regarding it as it could improve relations.

Religion and military is a very important aspect of Avalon. For religion, while we are largely and mostly Catholic, a lot of our population are open to accepting other issues such as the LGBT, divorce, and such. For military, our military is state of the art, and is ever improving, a large portion of our military is developed by Nexus Inc.

We would like to begin our newfound relationship by constructing embassies. We are open to also any trade routes, talks or deals between our two nations.

If there is any concern that the Estonul government would like us to address, do not be afraid to clarify.

Regards,
Diamond Palace
We humbly accept your proposal, except for what we will say below.
The Sacred Imperial State of Estonul is not available at the moment to be part of TheoEarth Trade Organization, in what consists of other proposals, we accept without any problem and we hope to develop these issues in what remains of the time.
Maximum Representation of G.E.M.A

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# Is there a criterion for a link complement to have a hyperbolic structure with finite volume

For many links in $S^3$, the link complement can be equipped with a Riemannian structure which is complete, of constant sectional curvature -1, and has finite volume (i.e., a hyperbolic structure with finite volume). However, not every link complement can be endowed with such a structure. Does anybody know if there is a criterion for a link to have a corresponding hyperbolic structure? Since this problem is fundamental to the definition of hyperbolic invariants of knots and links, I guess people have already studied it thoroughly. However, I couldn't find such results. Could you please help me? Thank you very much!

There is a topological criterion due to Thurston. Using the JSJ machine (and work of many others) this criterion can also be phrased algebraically. I'll essay these below. Please note that the situation is much simpler for knots. To answer your question most directly, here is the desired reference to Wikipedia.

Now, here is Thurston's criterion. (EDIT: exposition improved after reading Bruno Martelli's answer.)

Suppose that $L$ is the link and $X$ is the link complement. Suppose $\pi = \pi_1(X)$. We assume the following properties (and each property assumes the proceeding ones). $\newcommand{\ZZ}{\mathbb{Z}}$

• $L$ is not a split link. Equivalently, $X$ is contains no essential two-sphere. Equivalently, $\pi$ is not a free product.
• $L$ is not the unknot. Equivalently, $X$ contains no essential disk. Equivalently, $\pi$ is not $\ZZ$.
• $L$ has no component that is an "undisturbed satellite knot". Equivalently, $X$ contains no essential torus.
• $L$ is not a torus knot. Equivalently, $X$ contains no essential annulus. These last two topological properties are equivalent to $\pi$ not containing a copy of $\ZZ^2$.

Then $X$ admits a hyperbolic structure.

• Thank you so much! Your answer is really helpful. However I have another question: even if X admits a hyperbolic structure, does it always have finite volume? (Or, to put it in another way, is the hyperbolic volume of a link always finite?) I have this question because we need the uniqueness of the hyperbolic structure in order to define hyperbolic invariants, but the Mostow rigidity theorem only works when the volume is finite. Do you know any related results? I might be asking very basic questions... So thank you for your help and patience! Jan 2 '14 at 0:35
• Yes, hyperbolic link complements always have finite hyperbolic volume. (There is a special case when we discuss the unknot, but I will ignore that for the present.) Here is a proof sketch. First, the finitely many cusp neighborhoods $C_i$ of $X$ each contain finite volume. (This is a standard exercise - see for example Peter Scott's article in the Bulletin of the AMS.) Next, notice $X - \cup C_i$ is compact. So this also has finite volume and we are done. Jan 2 '14 at 8:24


text:
# SMatrix1DSweep¶

class ipkiss3.all.circuit_sim.SMatrix1DSweep(**kwargs)

A 3D matrix. The first two dimensions are related to the terms, the last dimension is related to a 1D sweep (i.e., wavelength sweep).

Parameters: sweep_parameter_name: str, required Name of the parameter that is swept, e.g. ‘frequency’, ‘wavelength’ sweep_parameter_values: required Values that are swept sweep_parameter_unit: ( str ), optional, *None allowed* Unit of the sweep parameter, e.g. ‘GHz’, ‘um’. None = unknown. n_ports: int and number > 0, optional The number of simulated ports symmetric: ( bool, bool_, bool or int ), optional When True, the matrix is assumed to be symmetrical, and the associated component is a reciprocal component. This means the transmission from port x to y is equal to the transmission from port y to x. term_modes: OrderedTypedDict, optional dictionary with { name (type: string) : (term_name(type:string), mode (type: int/str)) } pairs These are the (term-mode) pairs associated with the SMatrix. The dimension of the SMatrix equals len(term_modes) x len(term_modes). This dict is ordered: the order depends on the order they were added. This is useful if you want to ‘bulk’ set the S-matrix, S[:, :] = [[0, 1, 0], [0, 0, 1], [1, 0, 0]] : without the order information, setting these values is ambiguous. terms: ( OrderedNamedTypedDict ), optional, *None allowed* When term_modes is not given, we extract it from the the terms The term_modes pairs are added in consequetive blocks for each mode. The term_mode name is the following: - term_name, for the first mode with index 0: in - term_name_mode# for the modes with index higher than 0: in_1, in_2, in_3 data: optional Stores the raw data of the simulation results dtype: type, optional Type of the numpy array. By default, this is a complex number, because the scatter matrix contains both amplitude and phase information. term_mode_map: optional, *None allowed* A dictionary that maps term,mode tuples to indices in the data array
static from_touchstone(filename, term_mode_map=None, unit=None)

Imports an S matrix frequency sweep from a touchstone file. More information on the touchstone file format can be found in this document.

Parameters: filename: str filename to import term_mode_map: dict Maps (term, modenr) to TouchStone port_mode numbers. Use None for first attempting to derive from tool-specific parsing of the file header and when that fails assign default port names unit: str When specified, the sweep values will be converted to this unit. e.g. when the unit is ‘um’ the sweep_parameter_values will be in micrometer instead of the unit used in the touchstone file. smatrix: SMatrix1DSweep 1D S matrix sweep

Examples

In this example we import a Touchstone file named ‘smatrix_fdtd.s6p’, made using the IPKISS-Lumerical link. You can download the Touchstone file and try the example below yourself. It represents a component with one input port and two output ports, where each port has 2 modes assigned to it. This configuration produces the number 6 (= 3 ports x 2 modes) in the extension. Also note how the file has a header containing metadata about the ports:

! Touchstone file export IPKISS-Lumerical FDTD link
! Luceda port in:mode 0:0
! Luceda port in:mode 1:1
! Luceda port out2:mode 0:2
! Luceda port out2:mode 1:3
! Luceda port out1:mode 0:4
! Luceda port out1:mode 1:5

This metadata is used by our importer if term_mode_map is set to None. It represents a mapping between ports in IPKISS/Lumerical and the Touchstone port numbers. For example, the second line maps the IPKISS port in:0 (= port in:mode 1) to Touchstone port 0.

import ipkiss3.all as i3
from ipkiss3.io.touchstone import import_touchstone_smatrix
import matplotlib.pyplot as plt
import numpy as np

S = import_touchstone_smatrix('smatrix_fdtd.s6p')

plt.figure()
plt.plot(S.sweep_parameter_values, np.abs(S["out1", "in"])**2, 'o--', label='in->out1', markersize=5)
plt.plot(S.sweep_parameter_values, np.abs(S["out2", "in"])**2, 'o--', label='in->out2', markersize=5)
plt.xlabel('frequency [GHz]')
plt.ylabel('magnitude')
plt.title('S-parameters')
plt.legend()
plt.show()

The next example shows how to set your own term_mode_map:

import ipkiss3.all as i3
from ipkiss3.io.touchstone import import_touchstone_smatrix
import matplotlib.pyplot as plt
import numpy as np
touchstone_4port = '''! 4-port S-parameter data, taken at three frequency points
# GHZ S MA R 50
5.00000 0.60 161.24 0.40 -42.20 0.42 -66.58 0.53 -79.34 !row 1
0.40 -42.20 0.60 161.20 0.53 -79.34 0.42 -66.58 !row 2
0.42 -66.58 0.53 -79.34 0.60 161.24 0.40 -42.20 !row 3
0.53 -79.34 0.42 -66.58 0.40 -42.20 0.60 161.24 !row 4
6.00000 0.57 150.37 0.40 -44.34 0.41 -81.24 0.57 -95.77 !row 1
0.40 -44.34 0.57 150.37 0.57 -95.77 0.41 -81.24 !row 2
0.41 -81.24 0.57 -95.77 0.57 150.37 0.40 -44.34 !row 3
0.57 -95.77 0.41 -81.24 0.40 -44.34 0.57 150.37 !row 4
7.00000 0.50 136.69 0.45 -46.41 0.37 -99.09 0.62 -114.19 !row 1
0.45 -46.41 0.50 136.69 0.62 -114.19 0.37 -99.09 !row 2
0.37 -99.09 0.62 -114.19 0.50 136.69 0.45 -46.41 !row 3
0.62 -114.19 0.37 -99.09 0.45 -46.41 0.50 136.69 !row 4
'''
filename = "touchstone_4port_example.s4p"

with open(filename, 'w') as f:
f.write(touchstone_4port)

term_mode_map = {
("opt1", 0): 0,
("opt2", 0): 1,
("opt3", 0): 2,
("opt4", 0): 3
}

S = import_touchstone_smatrix(filename, term_mode_map=term_mode_map)

plt.figure()
plt.plot(S.sweep_parameter_values, np.abs(S["opt2", "opt1"])**2, 'o--', label='opt1->opt2', markersize=5)
plt.plot(S.sweep_parameter_values, np.abs(S["opt3", "opt1"])**2, 'o--', label='opt1->opt3', markersize=5)
plt.plot(S.sweep_parameter_values, np.abs(S["opt4", "opt1"])**2, 'o--', label='opt1->opt4', markersize=5)
plt.xlabel('frequency [GHz]')
plt.ylabel('magnitude')
plt.title('S-parameters')
plt.legend()
plt.show()

passivity_check()

Check if the SMatrix1DSweep is passive.

Parameters: smatrix1dsweep: SMatrix1DSweep stores the SMatrices from a sweep indices: A numpy array that contains the indices of the sweep values that are not passive

Examples

>>> smat = i3.device_sim.SMatrix1DSweep.from_touchstone('mydc.s4p')
>>> indices = smat.passivity_check()
>>> non_passivity_wls = smat.sweep_parameter_values[indices]
>>> is_passive = len(indices) == 0

reciprocity_check()

Check if the SMatrix1DSweep is reciprocal.

Parameters: smatrix1dsweep: SMatrix1DSweep stores the SMatrices from a sweep indices: A numpy array that contains the indices of the sweep values that are not reciprocal

Examples

>>> smat = i3.device_sim.SMatrix1DSweep.from_touchstone('mydc.s4p')
>>> indices = smat.reciprocity_check()
>>> non_reciprocal_wls = smat.sweep_parameter_values[indices]
>>> is_reciprocal = len(indices) == 0


text:
Search by Topic

Resources tagged with Mathematical reasoning & proof similar to Integral Sandwich:

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Broad Topics > Using, Applying and Reasoning about Mathematics > Mathematical reasoning & proof

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Follow the hints and prove Pick's Theorem.

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Have a go at being mathematically negative, by negating these statements.

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Stage: 2, 3, 4 and 5

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Stage: 5 Challenge Level:

This problem is a sequence of linked mini-challenges leading up to the proof of a difficult final challenge, encouraging you to think mathematically. Starting with one of the mini-challenges, how. . . .

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text:
# Precision measurement noise asymmetry and its annual modulation as a dark matter signature

B. M. Roberts and A. Derevianko [arXiv:1803.00617]

Monopole-like DM objects incident upon the Earth, and the induced shift and asymmetry in the noise distribution.

Dark matter may be composed of ultralight quantum fields that form macroscopic objects. As the Earth moves through the galaxy, interactions with such objects may leave transient signatures in terrestrial experiments. These signatures may be sought by analyzing correlations between multiple devices in a distributed network. However, if the objects are small ($$\lesssim10^3\,{\rm km}$$) it becomes unlikely that more than one device will be affected in a given event. Such models may, however, induce an observable asymmetry in the noise distributions of precision measurement devices, such as atomic clocks. Further, an annual modulation in this asymmetry is expected. Such an analysis may be performed very simply using existing data, and would be sensitive to models with a high event rate, even if individual events cannot be resolved. For certain models, our technique extends the discovery reach beyond that of existing experiments by many orders of magnitude.


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# Performing Operations?

• Dec 18th 2009, 12:03 PM
deekay930
Performing Operations?
Something so different from what I taught. I have no idea how to solve this.

$24x^2z^2-2x^2z^2+x^2+z^2$

$(x+y+xy)-(3x+4x^2)$

NOTE: the exponents behind the z's are suppose to be -2.
• Dec 18th 2009, 12:05 PM
Dinkydoe
Solve what actually?

I see no equation or a problem. Only two strings of variables I suppose?
• Dec 18th 2009, 12:09 PM
deekay930
Quote:

Originally Posted by Dinkydoe
Solve what actually?

I see no equation or a problem. Only two strings of variables I suppose?

It says to perform each of the operation and put the answers in descending order with regard to the x variable.
• Dec 19th 2009, 02:49 AM
HallsofIvy
Quote:

Originally Posted by deekay930
Something so different from what I taught. I have no idea how to solve this.

$24x^2z^{-2}-2x^2z^{-2}+x^2+z^{-2}$

Did you notice that the x and z dependence of the first two terms are exactly the same? Combine them. If you were told "put the answers in descending order with regard to the x variable", since all three terms have $x^2$, they can be in any order

Quote:

$(x+y+xy)-(3x+4x^2)$
x- 3x= -2x.

Quote:

NOTE: the exponents behind the z's are suppose to be -2.
So use "z^{-2}".


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# Power-law density spherical potential with an exponential cut-off¶

class galpy.potential.PowerSphericalPotentialwCutoff(amp=1.0, alpha=1.0, rc=1.0, normalize=False, r1=1.0, ro=None, vo=None)[source]

Class that implements spherical potentials that are derived from power-law density models

$\rho(r) = \mathrm{amp}\,\left(\frac{r_1}{r}\right)^\alpha\,\exp\left(-(r/rc)^2\right)$
__init__(amp=1.0, alpha=1.0, rc=1.0, normalize=False, r1=1.0, ro=None, vo=None)[source]

NAME:

__init__

PURPOSE:

initialize a power-law-density potential

INPUT:

amp= amplitude to be applied to the potential (default: 1); can be a Quantity with units of mass density or Gxmass density

alpha= inner power

rc= cut-off radius (can be Quantity)

r1= (1.) reference radius for amplitude (can be Quantity)

normalize= if True, normalize such that vc(1.,0.)=1., or, if given as a number, such that the force is this fraction of the force necessary to make vc(1.,0.)=1.

ro=, vo= distance and velocity scales for translation into internal units (default from configuration file)

OUTPUT:

(none)

HISTORY:

2013-06-28 - Written - Bovy (IAS)


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# How to include the right reference in C# using the “Browser” tab

I am working on VS2010 C# and I want to add a .dll reference compiled under VS2008; this .dll exists in 2 versions: Debug and Release.

The .dll is not under .NET, COM or projects tabs, so I only have the "browser" tab to add the .dll to the reference.

My question is: How can I indicate to VS to take the release .dll version when I compile in the release mode and to take the debug .dll version when I compile in the debug mode?

Thanks.

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You'll probably need to manually edit the underlying .csproj file. Approximately thusly

<Reference Condition=" '$Configuration'=='Debug' " Include="path\to\Debug\Foo.dll" /> <Reference Condition=" '$Configuration'=='Release' "
Include="path\to\Release\Foo.dll" />

(May be easy to add a reference to debug version via browse, then right click project in solution explorer, click 'Unload project', then right click again, 'Edit your.csproj', make the edit suggested above to the Foo.dll that you just added, then right click, 'Reload project'.)

EDIT

To prevent seeing two copies inside VS, maybe something like

<Reference debugstuff as before>
<Visible Condition=debugcond>true</Visible>
<Visible Condition=releasecond>false</Visible>
</Reference>
and same for release

That is, conditionally set Visible metadata under the reference node to true/false based on the condition. I haven't tried to to know if it works.

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Thanks Brian, it is working now. Please, let me ask a one more question related to your answer. When I see the reference option, two lines referred to both the path\to\Debug\Foo.dll and the path\to\Release\Foo.dll are displayed; how can I avoid this behavior? I mean, if I am in Debug mode only the line related to path\to\Debug\Foo.dll should be displayed and so on. Thanks!! –  vizcayno May 18 '10 at 20:18
I added to my answer something to try. –  Brian May 18 '10 at 21:24
Brian, seems it doesn't work, the "Visible Condition" is not been recognized by VS. Anyway, your first solution works for me. My best regards. –  vizcayno May 19 '10 at 5:02

A better way to implement Brian's solution is to do it this way:

<Reference Include="AjaxControlToolkit">
<SpecificVersion>False</SpecificVersion>
<HintPath>path/to/folder/bin/$(Configuration)/AjaxControlToolkit.dll</HintPath> </Reference>  This will switch nicely between Release and Debug without it showing up twice in your References view. - Sweet. This should've been the accepted answer IMHO. – Mark Green Jul 10 '12 at 15:54 The standard approach is to add the project to the solution. It is then entirely automatic. - Thanks Hans, my problem here is that the .dll components I want to add in the reference were compiled in VS2008. When I try to add the project into the VS2010 solution, it forces me to upgrade the project. I would not like to have duplicated projects (one in 2008 and another in 2010). – vizcayno May 18 '10 at 20:42 You've already got some projects that only load in 2008, some in 2010. I recommend you don't let that linger on for too long. SCCSs are built to make that not a problem. – Hans Passant May 18 '10 at 20:52 Hello Hans, I appreciate your recommendation very much. In this moment I am using a trial version of VS2010 and I am not sure if my company is going to pay for it so, I do not want to touch the VS2008 components. Thanks!! – vizcayno May 19 '10 at 5:05 we found some ideal way to deal with this situation. first, publish your library file to three special folders:$(Configuration)\MyLibrary.dll Debug\MyLibrary.dll Release\MyLibrary.dll

then in your application, reference the '$(Configuration)\MyLibrary.dll' one. done! now you change the solution configuration, the dll will be updated automatically. - Going off of @Kevin Yang's answer, I did the following: 1. Build into path\to\Debug\MyCode.dll and path\to\Release\MyCode.dll. 2. Reference one of them in VS (using the 'Browse' tab). 3. Unload the project that is referencing your DLL. 4. Right-click on the unloaded project and click Edit MyReferencingProject.csproj... 5. Find the reference, and in the HintPath element, change Debug/Release to $(Configuration) (so it should be path\to\$(Configuration)\MyCode.dll). 6. Rinse/repeat (with any other similar DLL's). When you're done, should look like: <Reference Include="MyCode"> <HintPath>C:\path\to\$(Configuration)\MyCode.dll</HintPath>
</Reference>

-