text:
## Main

Molecular systems provide a versatile physical environment in which to study the fundamental symmetries of nature and the interactions and properties of subatomic particles1,2,12,13. Among the four known fundamental forces, the weak force is the only one that is known to violate symmetry with respect to spatial inversion of all particle coordinates (known as parity violation), giving rise to various intriguing phenomena. Some of these parity-violating effects have been measured with high accuracy in atomic systems13,14,15, contributing to the most stringent low-energy tests of the Standard Model of particle physics. In certain molecules, effects resulting from both parity violation (P-odd) and time-reversal violation (T-odd) are considerably enhanced with respect to atomic systems5,7,8,13,16, offering the means to explore unknown aspects of the fundamental laws of physics. The strengths of these interactions scale with atomic number, nuclear spin and nuclear deformation, and so molecular compounds of heavy radioactive nuclei are predicted to exhibit unprecedented sensitivity, with an enhancement of more than two orders of magnitude for effects that are P-odd or simultaneously P- and T-odd5,6,7,8,17,18,19,20.

Since the direct cooling of diatomic molecules with lasers23 was experimentally demonstrated24, there has been a wealth of studies on laser-cooling techniques and applications in molecular physics25,26,27,28,29,30,31. In contrast to other heavy-atom molecules, RaF is predicted to have highly closed excitation and re-emission optical cycles, which would make it ideal for laser cooling and trapping6. Moreover, owing to the recently discovered pear-shaped nuclear deformation of certain radium isotopes11, the interactions of the electrons with the P-odd nuclear anapole moment as well as with the P,T-odd nuclear Schiff and magnetic quadrupole moments are predicted to be enhanced by more than two orders of magnitude4,5,19,32. Hence, these molecules could provide a unique environment in which to measure these symmetry-violating nuclear moments.

## Experimental scheme

Figure 1 shows a diagram of the experimental setup used to produce and study the RaF molecules. As a first step, radium isotopes were produced by diffusion out of an irradiated target (see Methods section ‘Production of RaF molecules’). RaF+ molecular ions were formed upon injection of CF4 gas into the target environment. The molecular ions were extracted from the ion source by applying an electrostatic field, and molecules containing one specific radium isotope were selected with a high-resolution magnetic mass separator (Δm/m ≈ 1/2,000). The ions were collisionally cooled in a radio-frequency quadrupole (RFQ) trap filled with helium gas at room temperature (about 300 K). After up to 10 ms of cooling time, bunches of RaF+ with a 4-μs temporal width were released and accelerated to 39,998(1) eV, before entering into the Collinear Resonance Ionisation Spectroscopy (CRIS) setup22,33,34. At the CRIS beam line, the ions were first neutralized in-flight by passing through a collision cell filled with a sodium vapour, inducing charge exchange according to the reaction RaF+ + Na → RaF + Na+. As the ionization energy of RaF is estimated to be close to that of sodium (5.14 eV)35, the neutralization reaction dominantly populates the RaF X2Σ+ electronic ground state. Molecular pseudo-orbitals obtained from one-component open-shell (neutral) or closed-shell (ion) restricted Hartree–Fock calculations with an energy-consistent effective core potential on radium are shown schematically in Fig. 1 (bottom). The lowest unoccupied molecular orbital in RaF+, which is mainly of non-bonding character, becomes occupied by an unpaired electron (symbolized in Fig. 1 by a red sphere together with an arrow representing the electron spin) upon neutralization. This is shown schematically as an isodensity, with lobes in slightly transparent blue and transparent red indicating different relative phases of the single-electron wavefunction.

After the charge-exchange reaction, non-neutralized RaF+ ions were deflected out of the beam, and the remaining bunch of neutral RaF molecules was overlapped in time and space by several (pulsed) laser beams in a collinear arrangement, along the ultrahigh-vacuum (10−10 mbar) interaction region of 1.2-m length. Laser pulses (step 1) of tunable wavelength were used to resonantly excite the transition of interest, and a high-power 355-nm laser pulse (step 2) was used to subsequently ionize the excited RaF molecules into RaF+ (see Fig. 1, top). The resonantly ionized molecules were then separated from the non-ionized molecules by deflecting the ions onto a particle detector. When the excitation laser is on resonance with a transition in the molecule (step 1 in Fig. 1), the second laser pulse ionizes the molecule, producing a signal at the detector. Molecular excitation spectra were obtained by monitoring the ion counts as a function of the wavenumber of the first laser.

Only theoretical predictions were available for the excitation energies of RaF, and so finding the transition experimentally required scanning a large wavelength range (>1,000 cm−1). The prediction for the A2Π1/2−X2Σ+ (0, 0) transition, for example, was 13,300 cm−1, with an accuracy estimated to be within 1,200 cm−1 (refs. 6,32). Given the bandwidths of the commonly available lasers (<0.3 cm−1), the scan of such a large wavelength region on samples produced at rates below 106 molecules s−1 represented a major experimental challenge. To optimize the search of molecular transitions, three broadband lasers were scanned simultaneously and both collinearly and anti-collinearly (see Methods section ‘Laser setup’).

## Results

The predicted region for the A2Π1/2 ← X2Σ+ transition was scanned at a speed of 0.06 cm−1 s−1, covering a range of 1,000 cm−1 in about 5 h, using the six simultaneously applied scanning regions. After a few hours of scanning on a beam of 226RaF, a clear sequence of vibronic absorption signals was recorded. The measured spectrum assigned to the (v′, v″) vibrational transitions (0, 0), (1, 1), (2, 2), (3, 3) and (4, 4) of the A2Π1/2−X2Σ+ band system is shown in Fig. 2a. Weaker band structures, that were found at about +440 cm−1 and −440 cm−1 with respect to the (0, 0) band, were assigned to the Δv = ±1 transitions (v′, v″) = (1, 0), (2, 1), (3, 2), (4, 3), (5, 4) and (v′, v″) = (0, 1), (1, 2), respectively (Fig. 2b, c). The quantum number assignment for Δv = −1 is tentative, owing to the highly dense structure of overlapping vibronic bands.

In addition to the A2Π1/2−X2Σ+ band system, we found spectroscopic signatures of electronic transitions to higher-lying states. Some examples of recorded spectra are shown in Fig. 2d–f, along with the energy-level scheme. We assign the observed transitions as follows: 1) The band system around 15,325 cm−1 (Fig. 2d) is attributed to the A2Π3/2−X2Σ+ transition, owing to the complex rovibrational structure expected to arise from the intense satellites that are possible in these transitions. Because the bands are comparatively strong, they are assigned to the Δv = 0 band system. Although the individual assignments to vibrational transitions must be considered to be tentative, as per the congested structure of the Franck–Condon profile, the Δv = 0 assignment is substantiated because no additional structure was located within a relative range of −400 to +400 cm−1. The band system located around 15,143 cm−1 (Fig. 2e) is tentatively assigned to the B2Δ3/2−X2Σ+ transition by virtue of the good agreement with the computed excitation energies to the Ω = 3/2 state of mixed Δ/Π character6,32. This mixing provides intensity to the one-photon transition from a Σ state into the Δ manifold. The computed Born–Oppenheimer potentials for this Ω = 3/2 state and the electronic ground state are, however, highly parallel, which would suggest a sparser Franck–Condon profile than was observed experimentally. However, we note that the related B2Δ3/2−X2Σ+ transition in BaH and BaD was reported to have a perturbed character owing to mixing between electronic levels36. Thus, in the present case, a vibrational profile that is richer than expected from adiabatic potentials cannot be ruled out a priori. The band system with origin at 16,175 cm−1 (Fig. 2f) is assigned to the C2Σ+−X2Σ+ transition on the basis of the observed Franck–Condon profile, which is in good agreement with the computed harmonic vibrational energy spacings as well as the expected intensity distribution, and is in a wavenumber region that is only slightly lower than predicted6,32. All measured and assigned vibronic bands of the four electronic transitions are listed in Table 1.

The measured A2Π1/2−X2Σ+ (0, 0) band centre, $${\tilde{{\mathscr{T}}}}_{{\rm{e}}}$$ = 13,287.8(1) cm−1 is in excellent agreement with the ab initio calculated value of 13,300(1,200) cm−1 (ref. 32). In accordance with theoretical predictions6, we found vibronic transitions with Δv = 0 to be much stronger than those with Δv = ±1. For most of the measurements, the power density used for the resonant step was 100(5) μJ cm−2 per pulse, as measured at the entry window of the beam line. Reducing the power by 50% did not reduce the resonant ionization rate, indicating that these transitions were measured well above saturation. The much weaker vibrational transitions with Δv = ±1 were scanned with a pulsed dye laser of 500(5) μJ cm−2 power density per pulse (bandwidth of 0.1 cm−1). The Δv = ±1 transitions were measured well above saturation and with laser beams of different characteristics, and so a precise estimation of the Franck–Condon factors could not be obtained. Instead, a lower limit of 0.97 for the peak intensity ratio I(0, 0)/I(0, 1) was derived, indicating highly diagonal Franck–Condon factors, an essential property for laser cooling6.

By measuring the resonant ionization rate for different time delays between the excitation and ionization laser pulses, we obtained an upper limit for the lifetime of the excited state 2Π1/2 (v′ = 0): T1/2 ≤ 50 ns. The measurements were performed with the wavenumber of the resonant laser fixed at the resonance value of the transition (v′, v″) = (0, 0). The resonant ionization rate dropped by more than 70% for delays above 50 ns. This short lifetime corresponds to a large spontaneous decay rate (>2 × 107 s−1), which would allow for the application of strong optical forces for laser cooling. An additional concern for the suitability of laser cooling is related to the existence of metastable states lying energetically below the 2Π1/2 level, which could prevent the application of a closed optical-cooling loop, a major problem encountered for BaF (ref6.). In contrast to BaF, all other predicted electronic states (2Π3/2, 2Δ3/2 and 2Σ) in RaF were found to be energetically above the 2Π1/2 state, indicating that its electronic structure will allow for efficient optical-cooling cycles.

From combination differences of energetically low-lying vibronic transitions in the band system A2Π1/2−X2Σ+, we have derived experimental values for the harmonic frequency, $${\tilde{\omega }}_{{\rm{e}}}$$, and the dissociation energies, $${\tilde{{\mathscr{D}}}}_{{\rm{e}}}$$, using a Morse potential approximation. Results are given in Table 2, and further details of the analysis can be found in Methods section ‘Spectroscopic analysis’.

Furthermore, we measured the A2Π1/2 ← X2Σ+ vibronic spectra of 226RaF and the short-lived isotopologues 223RaF, 224RaF, 225RaF, and 228RaF (Fig. 3). All vibrational transitions were clearly observed, including those of the molecule with the shortest-lived radium isotope studied, 224RaF (T1/2 = 3.6 d). An on-line irradiation of the target material will enable the study of molecules containing isotopes with lifetimes as short as a few tens of milliseconds. The main limitation is dictated by the release from the target and the time spent in the RFQ trap (>5 ms). Future high-resolution measurements will enable studies of nuclear structure changes resulting from different isotopes and nuclear spins.

## Conclusions and future perspectives

In summary, this Article presents an experimental approach for performing laser spectroscopy studies of molecules containing radioactive nuclei, which are typically produced at rates lower than 106 molecules s−1. Our results have established the energetically low-lying electronic structure of RaF, providing experimental evidence for the suitability of this diatomic molecule in a laser-cooling scheme. These findings are a pivotal step towards precision measurements in this system, which are expected to provide a highly sensitive environment for the exploration of physics beyond the Standard Model of particle physics.

Our experimental scheme can also be used to perform laser spectroscopy of a wide variety of neutral molecules and molecular ions, including those composed of isotopes with lifetimes of a few tens of milliseconds. Radioactive molecules can be precisely tailored to enhance their sensitivity to parity- and time-reversal-violating effects by introducing heavy and octupole-deformed nuclei. Moreover, by systematically replacing their constituent nuclei with different isotopes of the same element, both nuclear-spin-independent and nuclear-spin-dependent effects can be comprehensively studied. In addition, the present technique is applicable to other molecules of interest in studies of fundamental physics that are as yet experimentally unexplored, such as RaOH (ref. 37), RaO (ref. 18), RaH (ref. 17), AcF (ref. 38) and 229ThO (ref. 5).

In addition to the impact of our findings on quantum chemistry, nuclear structure and fundamental physics research, the ability to produce, mass-select and spectroscopically study short-lived radioactive molecules is of importance to other fields of research such as radiochemistry21 and astrophysics39,40. Laboratory measurements of the spectra of radioactive molecules of astrophysical interest will allow their unambiguous identification in future astronomical observations. Furthermore, the possibility of performing spectroscopy on fast molecular beams will enable sub-Doppler spectroscopy to be performed even on molecules created at high temperatures (>600 K). Thus, we expect our results will motivate further avenues of research at the increasingly capable radioactive-ion-beam facilities around the world.

## Methods

### Production of RaF molecules

Ra isotopes were produced 33 d before the laser-spectroscopy measurements by impinging 1.4 GeV protons on the cold UCx target material. The target was exposed to pulses of 1013 protons per pulse over a period of 2 d. After irradiation with a total of 8 × 1017 protons, the target was kept in a sealed chamber filled with Ar gas. After day 33, the target was connected to the High-Resolution Separator (HRS) front-end at ISOLDE. FLUKA41 simulations predicted 2 × 1013 atoms of 226Ra in the target material (7.5 × 10−9 g), following proton irradiation of a cold target. The target was pumped down to pressures below 10−5 mbar, and the target holder and ion source were gradually heated up to about 1,300 °C, in order for the Ra isotopes to diffuse towards the surface of the target material. A leak valve attached to the target was used to inject CF4 into the target environment. The CF4 molecules dissociate and react with atoms and molecules on the target surface until an equilibrium is reached. RaF molecules were formed by reactive collisions of CF4 molecules with Ra atoms present inside the irradiated target material.

According to thermodynamic equilibrium calculations42, RaF2 or RaF are expected to form, depending on the local temperature. Within the temperature gradient between the target (1,300 °C) and the ion source (2,000 °C), RaF2 fully reacts to form RaF. A measured ratio of the ion-beam intensity of Ra+ to RaF+ of less than 0.05 indicates that more than 95% of the Ra isotopes released from the target material are converted and extracted as molecules.

The 226RaF+(A = 245) beam extracted from the ISOLDE target unit was sent to the ISOLTRAP setup43, where the molecular ions were captured, cooled and bunched by a different RFQ trap and subsequently analysed using a multi-reflection time-of-flight mass spectrometer44. A measured mass spectrum is shown in Extended Data Fig. 1. After 1,000 revolutions in the device, a mass resolving power (R = mm) of 1.7 × 105 was achieved, which allowed the isobaric beam composition to be analysed. The only mass peak detected was identified as the signal of 226Ra19F+, confirming the purity of the beam from ISOLDE.

The intensity of RaF+ molecules depends strongly on the target and ion source temperature. For a target temperature of 1,300 °C, a mean value of 2 × 107 molecules s−1 of 226RaF+ was measured after the mass separator. Depending on the molecular mass and beam intensity, the transmission efficiency through the RFQ trap varied from 15% to 30%. The ion-beam transmission from the ion trap to the interaction region was measured to be 25(5)%. The charge exchange cell vapour was heated to produce a measured neutralization rate of 30(5)%. Thus, we estimate that on average 5 × 104 neutral 226RaF molecules s−1 were delivered to be resonantly excited. From the analysis of the measured spectra it was concluded that the neutral molecules populate the low-lying vibrational states ν = 0, 1, 2, 3, 4 following a relative population of 0.47:0.29:0.13:0.05:0.03. Resonantly ionized molecules with rates of the order of 103 counts s−1 at the peak of the 0 ← 0 transition were measured at the particle detector. Future production of RaF+ molecular rates of the order of 109–1010 molecules s−1 is feasible using active proton irradiation45.

### Laser setup

The resonance ionization schemes used for the study of RaF molecules are shown in Fig. 1. Three different laser systems were prepared to cover the scanning range from 12,800 cm−1 to 13,800 cm−1: 1) A dye-laser system (Dye1; Spectrolase 4000, Spectron) provided pulses of 100(5) μJ with a linewidth of 10 GHz (0.3 cm−1). 2) A dye laser (Dye2; Cobra, Sirah) with a narrower linewidth of 2.5 GHz (0.09 cm−1) produced pulses of similar energy. The lasers were loaded with either Styryl 8 or DCM dyes to provide wavenumber ranges 12,800–14,000 cm−1 and 15,150–16,600 cm−1, respectively. Both dye lasers were pumped by 532-nm pulses at 100 Hz, obtained from two different heads of a twin-head Nd:YAG laser (LPY 601 50–100 PIV, Litron). 3) A grating Ti:sapphire laser system with a linewidth of 2 GHz (0.07 cm−1) produced pulses of 20(1) μJ, pumped by 532-nm pulses at 1 kHz from a Nd:YAG laser (LDP-100MQ, LEE Laser). The non-resonant ionization step was obtained by 355-nm pulses of 30 mJ at 100 Hz, produced by the third-harmonic output of a high-power Nd:YAG laser (TRLi 250-100, Litron).

The release of the ion bunch was synchronized with the laser pulses by triggering the flash-lamps and Q-switch of the pulsed lasers with a digital delay pulse generator (Quantum Composers 9528).

The dye-laser wavelengths were measured with a wavelength meter (WS6-600 HighFinesse) and the Ti:sapphire laser wavelengths were measured by a wavelength meter (WSU-2 HighFinesse) calibrated by measuring a reference wavelength provided by a stabilized diode laser (DLC DL PRO 780, Toptica).

### Collinear and anti-collinear excitation

For the initial peak searching, a zero-degree mirror at the end of the beam line was used to reflect the laser light anti-collinearly with respect to the travelling direction of the RaF bunch. Thus, each scanning laser covered two different wavenumber regions in the molecular rest frame, owing to the Doppler shift present for the fast RaF molecules. For a molecule travelling at velocity v, the laser wavenumber in the laboratory frame, $${\tilde{\nu }}_{0}$$, is related to the wavenumber in the molecule rest frame, $$\tilde{\nu }$$, by the expression $$\tilde{\nu }=\tfrac{1+\beta \cos \,\theta }{\sqrt{1-{\beta }^{2}}}{\tilde{\nu }}_{0}$$, with β = v/c (c, speed of light in vacuum) and where θ is the angle between the direction of the laser beam and the velocity of the molecule. For RaF molecules at 39,998(1) eV (v ≈ 0.18 m μs−1), a difference of 15.7 cm−1 is obtained between the laser pulse sent out collinearly (cosθ = 1) and anti-collinearly (cosθ = −1) with respect to the direction of the velocity of the molecule.

### Spectroscopic analysis

The peaks in the different spectra were identified by rebinning the spectra using coarse bin sizes with values up to 1 cm−1. Only groups of data points that were consistently observed with a 5-sigma significance above background were considered as candidates for transitions. The vibrational transitions in Fig. 2 show asymmetric line profiles with a maximum located towards higher wavenumbers. The band centres cannot be determined directly from the measured line profiles, and so we used the wavenumber positions of the maxima in our data analysis. Extended Data Table 1 lists the maximum peak positions and estimated uncertainties are given in parentheses. The wavenumber difference, $$\Delta \tilde{\nu }$$, of vibrational levels in the electronic 2Σ+ ground state and in the 2Π1/2 excited state were derived from combination differences of the recorded 226RaF spectra (see Extended Data Table 1).

In our analysis we used vibrational energy terms Ev/(hc) of a Morse potential according to:

$${E}_{v}/(hc)={\tilde{\omega }}_{{\rm{e}}}\left(v+\frac{1}{2}\right)-\frac{{\tilde{\omega }}_{{\rm{e}}}^{2}}{4{\tilde{{\mathscr{D}}}}_{{\rm{e}}}}{\left(v+\frac{1}{2}\right)}^{2}$$
(1)

Energy-level differences

$$({E}_{v+1}-{E}_{v})/(hc)={\tilde{\omega }}_{{\rm{e}}}-\frac{{\tilde{\omega }}_{{\rm{e}}}^{2}}{2{\tilde{{\mathscr{D}}}}_{{\rm{e}}}}(v+1)$$
(2)

were used to derive the Morse potential parameters $${\tilde{\omega }}_{{\rm{e}}}$$ and $${\tilde{{\mathscr{D}}}}_{{\rm{e}}}$$ from a least-squares fit analysis. The derived energy-level differences are given in Extended Data Table 1, whereas Extended Data Table 2 contains the molecular parameters from the fit. The harmonic vibration frequencies $${\tilde{\omega }}_{{\rm{e}}}$$ of the 2Σ+ and 2Π1/2 states are almost identical and correspond well to the theoretical predictions with a deviation of less than 5%; see Extended Data Table 2. The same holds for the estimated dissociation energy $${\tilde{{\mathscr{D}}}}_{{\rm{e}}}$$, which is in better agreement with the values of ref. 6, as therein also the low-energy part of the potentials was used to estimate the dissociation energy.

In the case of the two low-lying 2Π fine-structure levels, the observed origins T0,0 agree well with the calculated values based on the Relativistic Correlation Consistent – Atomic Natural Orbital (RCC-ANO) basis set. From the energy difference of the fine-structure components the effective spin-orbital coupling parameter A is derived. For the 2Π states, the experimental value of 2,068(5) cm−1 is in good agreement with the calculated value. The band origins are in reasonable agreement with results from the RCC-ANO basis set calculation, if one attributes the Ω = 3/2 levels, which were computationally found to be of mixed Π3/2 and Δ3/2 character in this order of energies. A reverse assignment also gives better agreement with experiment. Calculations of the gas-phase bond lengths, dissociation energies and additional properties of RaF molecules have been reported8,32,35.


text:
# Definition:Polygonal Number

## Definition

Polygonal numbers are those denumerating a collection of objects which can be arranged in the form of an regular polygon.

A polygonal number is an integer defined recursively as follows:

$\forall k \in \Z_{\ge 2}: \forall n \in Z_{\ge 0}: \map P {k, n} = \begin{cases} 0 & : n = 0 \\ \map P {k, n - 1} + \paren {k - 2} \paren {n - 1} + 1 & : n > 0 \end{cases}$

For a given $k$, polygonal numbers are referred to by the name of the appropriate $k$-sided polygon.

For large $k$, they are therefore called (when used) $k$-gonal numbers.

## Also known as

When referring to a $k$-gonal number where $k$ is a more complex expression than just a single number or letter, it may be less unwieldy to refer to it as a polygonal number of order $k$.

## Examples

### Triangular Numbers

When $k = 3$, the recurrence relation is:

$\forall n \in \N: T_n = P \left({3, n}\right) = \begin{cases} 0 & : n = 0 \\ P \left({3, n - 1}\right) + \left({n - 1}\right) + 1 & : n > 0 \end{cases}$

where $P \left({k, n}\right)$ denotes the $k$-gonal numbers.

### Square Numbers

When $k = 4$, the recurrence relation is:

$\forall n \in \N: S_n = \map P {4, n} = \begin{cases} 0 & : n = 0 \\ \map P {4, n - 1} + 2 \paren {n - 1} + 1 & : n > 0 \end{cases}$

where $\map P {k, n}$ denotes the $k$-gonal numbers.

See Square Number.

Square numbers are of course better known as:

$S_n = n^2$

### Pentagonal Numbers

When $k = 5$, the recurrence relation is:

$\forall n \in \N: P_n = \map P {5, n} = \begin {cases} 0 & : n = 0 \\ \map P {5, n - 1} + 3 \paren {n - 1} + 1 & : n > 0 \end {cases}$

where $\map P {k, n}$ denotes the $k$-gonal numbers.

### Hexagonal Numbers

When $k = 6$, the recurrence relation is:

$\forall n \in \N: H_n = P \left({6, n}\right) = \begin{cases} 0 & : n = 0 \\ P \left({6, n - 1}\right) + 4 \left({n - 1}\right) + 1 & : n > 0 \end{cases}$

where $P \left({k, n}\right)$ denotes the $k$-gonal numbers.

See Hexagonal Number.

### Heptagonal Numbers

When $k = 7$, the recurrence relation is:

$\forall n \in \N: H_n = P \left({7, n}\right) = \begin{cases} 0 & : n = 0 \\ P \left({7, n - 1}\right) + 5 \left({n - 1}\right) + 1 & : n > 0 \end{cases}$

where $P \left({k, n}\right)$ denotes the $k$-gonal numbers.

### Octagonal Numbers

When $k = 8$, the recurrence relation is:

$\forall n \in \N: O_n = P \left({8, n}\right) = \begin{cases} 0 & : n = 0 \\ P \left({8, n - 1}\right) + 6 \left({n - 1}\right) + 1 & : n > 0 \end{cases}$

where $P \left({k, n}\right)$ denotes the $k$-gonal numbers.

See Octagonal Number.

## Degenerate Case

Consider the polygonal number $P \left({2, n}\right)$ when $k = 2$.

In this case, the polygon degenerates into a straight line, and the recurrence formula becomes:

$P \left({2, n}\right) = \begin{cases} 0 & : n = 0 \\ P \left({2, n-1}\right) + 0 \times \left({n-1}\right) + 1 & : n > 0 \end{cases}$

Hence:

$P \left({2, n}\right) = P \left({2, n-1}\right) + 1$

and the sequence goes:

$0, 1, 2, 3, \ldots$

which is of course the natural numbers.

## Historical Note

Polygonal numbers were originally studied by the ancient Greeks.

format: plain


text:
Top

# Kinematics Equations

Various representations are used to represent motion of any object like pictorial representation, graphs, and verbal representation etc. The branch of mechanics which studied the motion of an object is called kinematics. It describes the concept of motion with graphical representation of its basic term like displacement, velocity, speed, acceleration etc. and motion equation which gives the relation between these basic terms. It is mainly given the description of relative positions and changes in the position of an object with respect to time.

Here we discuss about four kinematic equations. These are used to detect the unknown value or variable with the use of known information or variables. These equations describe motion either at constant velocity or at constant acceleration. The time period in which the acceleration is changed is not used in these equations. Now we discuss all four kinematic equations and their uses with doing some problem on this topic.

 Related Calculators Kinematics Calculator Calculator Equation 2 Step Equation Calculator Absolute Value Equation Calculator

## What is Kinematics?

In Classical mechanics, we basically define motion in terms of space and time, and ignore the agents that cause that motion. This portion of the classical mechanics is called kinematics.

From our everyday experience, we identify that motion represents a continuous change in the location of an object.
There are three types of motion:

a) Translational Motion: A car moving on a highway is an example of translational motion.

b) Rotational Motion: The Earth’s revolve on its axis, is an example of rotational motion.

c) Vibrational Motion: The motion of the pendulum is an example of vibrational motion.
We could also say that Kinematics is the study of objects in motion.The main concepts of Kinematics include speed, velocity, acceleration, time, distance and displacement.

## Kinematics Formulas

1D Kinematics Equations are as follows:
$d$ = $V_{i}\ t\ +$ $\frac{1}{2}$ $\times\ a\ \times\ t^{2}$

$V_{f}$ = $V_{i}\ +\ a\ \times\ t$

$V_{f}^{2}$ = $V_{i}^{2}\ +\ 2ad$.
Where,

$d$ is displacement and $d$ = $\frac{V_{f}\ +\ V_{i}}{2}$,

$a$ is acceleration,

$t$ is time,

$V_f$ is final velocity,

$V_i$ is initial velocity.

or

There are three equation of motion, which are nothing but kinematics equations, which are :

1) $v$ = $u\ +\ at$

2) $S$ = $ut\ +$ $\frac{1}{2}$ $at^2$

3) $v^2$ = $u^2\ +\ 2as$.

Where,

$v$ = Final Velocity,

$u$ = Initial velocity,

$a$ = acceleration,

$s$ = distance traveled by a body,

$t$ = time taken.

## Inverse Kinematics

Kinematics can be stated as the procedure to calculate the position of the end of a linked structure, if angle of all the joints is already known. Kinematics is easy as there is one solution.

Inverse Kinematics does the reverse of kinematics and in case we have the end point of a particular structure, certain angle values would be needed by the joints to achieve that end point. It is a little difficult and has generally more than one or even infinite solutions.

## Kinetics Vs Kinematics

Below are given the Comparison between Kinetics and Kinematics:

 Kinetics Kinematics Kinetics is a concept of the classical mechanics and it deals with the motion of various objects and the forces which act on the objects either they are in rest or in motion.Here the force for cause of the motion is studied. Kinematics on the other hand, only deals with the motion or movement of various objects, and does not focus on the forces which causes the motion.Here the force for the cause of the motion is not studied.

## 4 Kinematics Equations

The Kinematics Equations are as follows:
$V$ = $V_o\ +\ at$

$X\ -\ Xo$ = $V_o\ t\ +$ $\frac{1}{2}$ $a\ t^{2}$

$V^{2}$ = $V_o^{2}\ +\ 2a(X\ -\ X_o)$

$X\ -\ X_o$ = $\frac{1}{2}$ $(V_o\ +\ V)t$.
Where,

$V$ is final velocity $(m/s)$,

$V_o$ is initial velocity $(m/s)$,

$a$ is acceleration $(m/s^2)$,

$t$ is time $(s)$,

$X$ is final displacement $(m)$,

$X_o$ is initial displacement.

More than one unknowns could be solved for each other by using more than one equation. If any equation needs to be solved for two components, we need to find a common piece (example time), solve for this piece for the rest of the unknown components and set them equal to each other to solve for the other unknowns. It is easier to learn the equations as:
$d_f\ -\ d_i$ = $V_i\ t\ +$ $\frac{1}{2}$ $at^2$

$V^2$ = $V_i^{2}\ +\ 2a(df - di)$

$df\ -\ di$ = $\frac{1}{2}$ $\times\ (V_i\ +\ V_f)t$.

## 2D Kinematics Equations

Since it is 2D equation, we are considering $X$ and $Y$ axis. 2D equations along $x$ and $y$ direction are given below. Considering $x$-direction,
$a_x$ = constant

$V_fx$ = $V_ix\ +\ a_x\ \Delta t$

$X_f$ = $X_i\ +\ V_ix\ \Delta t\ +$ $\frac{1}{2}$ $a_x\ \Delta t^2$

$\Delta t$ = $\frac{V_{fx} - V_{ix}}{a_{x}}$

$V_{fx}^{2}$ = $V_{ix}^{2}\ +\ 2ax\ \Delta x$.

$X_f$ = $X_i\ +$ $\frac{1}{2}$ $(V_fx\ +\ V_ix)\ \Delta\ t$.
and Considering y-direction,
$a_y$ = constant

$V_fy$ = $V_iy\ +\ a_y\ \Delta t$

$y_f$ = $y_i\ +\ V_iy\ \Delta t\ +$ $\frac{1}{2}$ $a_x\ \Delta t^2$

$\Delta t$ = $\frac{V_{fy}\ -\ V_{iy}}{a_{y}}$

$V_{fy}^{2}$ = $V_{iy}^{2}\ +\ 2ay\ \Delta y$

$y_f$ = $y_i\ +$ $\frac{1}{2}$ $(V_fy\ +\ V_iy)\ \Delta t$.
Where,

$V_f$ is final velocity $(m/s)$,

$V_i$ is initial velocity $(m/s)$,

$a$ is acceleration $(m/s^2)$,

$t$ is time $(s)$,

$X$ is final displacement $(m)$,

$X_0$ is initial displacement.

Projectile motion is the best example of the motion of the object in the Two dimension. Here the object has the motion in both the $x$ and the $y$ direction in the Horizontal direction and in the Vertical direction or we can say that the object has the both components of the velocity.

## Kinematics Problems

Below are the problems based on Kinematics Equations:

### Solved Examples

Question 1: Initial velocity of a truck is zero and it is at rest. It experiences a uniform acceleration during the time interval of 5.21 seconds. Distance covered by the truck is 110 m. Find the acceleration?
Solution:

Distance traveled s = 110m,
initial Velocity $v_{i}$ = 0,
time taken t = 5.21 s,
acceleration a = ?
By using the Kinematic Equation we can conclude that,
s = $v_{i}$t + $\frac{1}{2}$ a $t ^{2}$,
110 m = (0) $\times$ (5.21)+ $\frac{1}{2}$ $\times$ a $(5.21)^{2}$,
a = 8.10 m/$s^{2}$.

Question 2: A particle is moving in cm along the x-axis after t seconds of travel is represented by the equation x = 14t2 - t + 10. Find its average Velocity after 3s of its travel?
Solution:

The particles position when t = 0 is x = 10cm.
When t = 3s, x = 133 cm.
Average Velocity, Vav = $\frac{\Delta x}{\Delta t}$
= $\frac{133 - 10}{3 - 0}$
= 41 cm/s.

 Related Topics Physics Help Physics Tutor
*AP and SAT are registered trademarks of the College Board.


text:
How to distinguish between bits?

I read a post about differences between synchronous and asynchronous transmission modes.
That post has a picture about how bits flows from terminal to computer in async mode.

But how computer can distinguish between 1s and determine what 1 does actually mean. Is it parity bit, is it start/stop bit or maybe it's start of idle bits? Or just part of data is being transmitted?

• 1s and 0s, are logic levels, where a 0 refers to a 'low' level and a 1 refers to a 'high' level. Since these are just voltage levels, the computer can recognize and operate on these natively. Also, a computer program is usually stored as 0s and 1s before compiling as well, as everything on your computer will be stored that way. However, after compiling, you get a file filled with what is called machine code. Machine code is a list of binary instructions (as opposed to text like the original source code) interpreted by the circuitry in the processor – quintumnia Jul 27 '18 at 16:59

If the computer starts in the middle of the stream, it has no way to know—it will be completely confused.

Fortunately, that's not how the protocol works. The computer and terminal have to sync up before they can communicate. There are a few different ways of doing that, but once they're in sync (which includes agreeing on how long each 1 or 0 should last), they both follow a set pattern.

In this example:

• The terminal starts out holding the line high (sending constant 1s) to indicate that it has nothing to say.
• When the terminal has something to say, it brings the line low for a moment (sending a 0). That's the "start" bit.
• The terminal then sends seven bits for the character: always exactly seven, since it's using ASCII, which is a seven-bit encoding.
• After those seven bits, it sends one additional bit for error checking. This is a 1 if there were an odd number of 1s in the character, and a 0 otherwise. This is the "parity" bit.
• Finally, it pulls the line high again to indicate that it once more has nothing to say.

The computer keeps track of which "mode" the connection is in at all times. If it sees a zero, it might record it as part of the data, or might switch from "idle" mode to "start listening for data" mode, or check the parity: it all depends on what it's seen before.

• Can you recommend a good book or another resource where it is possible to read about sync/async data transmission? – Kenenbek Arzymatov Jul 30 '18 at 10:48


text:
# Why do we believe local gauge invariance (that tells us photons are exactly massless)? [duplicate]

There are experimental upper limits on the photon's mass, but they are finite. I heard that the reason we know that photons have exactly zero rest mass (and thus travel exactly at the universal invariant speed c) is local gauge invariance.

But why do we believe local gauge invariance?

If the rest mass of the photon were non-zero, the theory of quantum electrodynamics would be "in trouble" primarily through loss of gauge invariance, which would make it non-renormalisable; also, charge conservation would no longer be absolutely guaranteed, as it is if photons have zero rest mass.

But there's no further explanation. The question of local gauge invariance seems to require two explanations:

1. How does a massive photon make QED non-renormalizable?
2. How does a massive photon wreck charge conservation?

I would especially appreciate responses that minimize theoretical abstractions. I'm trying to get to the experimental basis of our belief that light is massless. These two questions connect to empirical basis because (1) QED, which works, needs apparently to be renormalizable and (2) charge is evidently conserved in the world as far as we can tell.

Please note: this is not the same question as Is it experimentally proven that photons travel at speed $c$ in vacuum?, as I'm asking not asking about measurements of the speed of light, but about the theoretical justifications (that have an experimental basis) that lights moves at exactly the universal invariant speed c.

• isn't 1) and 2) related? Finite mass breaks U(1) gauge invariance, which is crucial to charge conservation via Noether's theorem. Because I only took a first semester's worth of field theory, I do not understand how breaking gauge invariance leads to failure of renormalization. However, I think there have been several Physics SE questions on why massless of photon is required by gauge invariance (my understanding is that gauge invariance and massless work together to remove two unphysical degrees of freedom from the four-potential). – wcc Mar 17 at 20:05
• A CERN lecture series linked by anna v in this post seems useful (physics.stackexchange.com/q/328861) – wcc Mar 17 at 20:05
• A critical attitude towards gauge invariance is very well defensible. I showed that classic electromagnetism can be formulated without gauge invariance in this peer reviewed paper: arxiv.org/abs/physics/0106078 – my2cts Mar 17 at 22:39
• Possible duplicates: physics.stackexchange.com/q/31994/2451 , physics.stackexchange.com/q/296975/2451 , physics.stackexchange.com/q/328861/2451 and links therein. – Qmechanic Mar 19 at 21:56


text:
# JEE Main & Advanced Mathematics Trigonometrical Ratios and Identities Trigonometrical Ratios of Allied Angles

## Trigonometrical Ratios of Allied Angles

Category : JEE Main & Advanced

Two angles are said to be allied when their sum or difference is either zero or a multiple of ${{90}^{o}}$.

 Allied angles $\to$ $\sin \theta$ $cos\theta$ $tan\theta$ Trigo. Ratio $\downarrow \,\,(-\theta )$ $-\sin \theta$ $cos\theta$ $-tan\theta$ $(90-\theta )$ or $\left( \frac{\pi }{2}-\theta \right)$ $cos\theta$ $\sin \theta$ $\cot \,\theta$ $(90-\theta )$ or $\left( \frac{\pi }{2}-\theta \right)$ $\cos \theta$ $-\,\sin \theta$ $-\cot \,\theta$ $(180-\theta )$ or$(\pi -\theta )$ $\sin \theta$ $-\,\cos \theta$ $-tan\theta$ $(180+\theta )$ or $(\pi -\theta )$ $-\,\sin \theta$ $-\,\cos \theta$ $tan\theta$ $(270-\theta )$or $\left( \frac{3\pi }{2}-\theta \right)$ $-\,\cos \theta$ $-\,\sin \theta$ $\cot \,\theta$ $(270+\theta )$ or $\left( \frac{3\pi }{2}-\theta \right)$ $-\,\cos \theta$ $\sin \theta$ $-\cot \,\theta$ $(360-\theta )$ or $(2\pi -\theta )$ $-\,\sin \theta$ $cos\theta$ $-tan\theta$

You need to login to perform this action.
You will be redirected in 3 sec


text:
## ?= Positive Lookahead

Asserts that the contents of this group exists after the preceding portion of the regex, without being a part of the matched text. For example, given the text “dynamically negotiate multifunctional external sources”:

• negotiate: matches the word “negotiate”
• n(?=egotiate): matches “n”
• n(?=egotia)te: matches nothing
• n(?=egotiate)egotiate: matches the word “negotiate”

## ?! Negative Lookahead

Asserts that the contents of this group does not exist after the preceding portion of the regex. For example, given the text “assertively deliver error-free niche delivery markets”:

• deliver: matches the word “deliver” and all but the last character of “delivery”
• deliver(?!y): only matches the word “deliver”

## ?<= Positive Lookbehind

Asserts that the contents of this group exists before the subsequent portion of the regex, without being a part of the matched text. Semantics match positive lookahead.

## ?<! Negative Lookbehind

Asserts that the contents of this group does not exist before the subsequent portion of the regex. Semantics match negative lookbehind.

Edit


text:
# A hard integral identity on MATH.SE

$$\int_0^{1}\arctan\left(\frac{\mathrm{arctanh}\ x-\arctan{x}}{\pi+\mathrm{arctanh}\ x-\arctan{x}}\right)\frac{dx}{x}=\frac{\pi}{8}\log\frac{\pi^2}{8}$$

seems to be very difficult to prove.

Question: I worked on this identity for several days without any success. Is there any clue how to prove this integral identity?

-
I'm waiting for Noam Elkies's solution :-) –  Suvrit Jan 18 at 19:20
A related problem can be found here. –  Lucian Jan 21 at 15:08
What happens if $\pi$ in the integral is replaced by some other real number? Can one give a closed-form formula for the integral in this case? –  Zurab Silagadze Jan 23 at 5:32
Where does this problem come from and why do you care about it? –  Lev Borisov Jan 24 at 3:46
@LevBorisov: This identity is a long-lasting problem(which was raised by another user) on Math.SE, and I think it is just the type of identity which may appear in Ramanujan's notebooks. Although I don't know what this identity may related to, it would be better to draw more attention to this elegent identity. –  zy_ Jan 28 at 12:15

## 1 Answer

I have proved this equality by means of Cauchy’s Theorem applied to an adequate function. Since my solution is too long to post it here, I posted it in arXiv, you can get it at http://arxiv.org/abs/1402.3830

The function $$G(z)=\frac{\log(1+(1+i)\,f(z)\,)}z$$ where $$f(x)=\frac{\operatorname{arctanh}(x)-\arctan(x)}{\pi}$$ extended analytically.

-
That's great. I suggest that you at least copy the definitions of the key functions $f(z)$ and $G(z)$ into your answer. –  Neil Strickland Feb 18 at 8:57
@larry, I would like to know how this question was generated. I have seen so much structure in your integrand that I will be surprised if there is not something hidden. –  juan Feb 18 at 9:49
Nice observation! Thanks for your answer to this identity! –  zy_ Feb 19 at 14:42
Thank you for the fantastic contribution. –  rlgordonma Feb 19 at 14:55
Awesome answer! --- it's more than just an observation, I'd say :-) –  Suvrit Feb 19 at 16:32


text:
Math Central - mathcentral.uregina.ca
Problem of the Month
 Currentproblem Recent problemswith solutions
 Older problems from 2000 a 2005 2005/2006 06/07 07/08 08/09 09/10 10/11 11/12

Solution for February 2012

 The Problem: Does there exist a bounded real-valued function $f(x)$ with $f(1) > 0$ such that for all real numbers $x$ and $y$, $$\left(f(x+y)\right)^2 \ge \left(f(x)\right)^2 + 2f(xy) +\left(f(y)\right)^2 .$$

Solution: No, no such function exists.

Correct solutions were submitted by
 Lamis Alsheikh (Syria) Lou Cairoli (USA) Bernard Collignon (France) Hubert Desprez (France) Mei-Hui Fang (Austria) Frank Feys Verena Haider (Austria) Benoît Humbert (France) Omran Kouba (Syria) Marc Lichtenberg (France) Matthew Lim (USA) Albert Stadler (Switzerland) Alessandro Ventullo (Italy)

We also received three incomplete submissions.

Our correspondents established the nonexistence in two different ways.

Method 1. Let $x_0=1$ and define $x_{n+1}=x_n + \frac1{x_n}$. From the given inequality we have
\begin{eqnarray*}
f^2(x_{n+1}) &=& f^2\left(x_n + \frac1{x_n}\right)\\
&\ge& f^2(x_n) +2f(1) + f^2\left(\frac1{x_n}\right) \ge
f^2(x_n) + 2f(1).
\end{eqnarray*}
Applying this result recursively we deduce that
$$f^2(x_n) \ge f^2(1) + 2nf(1),$$
which grows without bound.

Method 2. Suppose to the contrary that $f$ is bounded; then $f^2$ would be bounded above for all real numbers $x$. Denote by $s$ the supremum of $f^2$ over all nonzero $x$; that is, $s=\sup_{x\ne 0}f^2(x)$ is the smallest number for which $f^2(x) \le s$ for all nonzero real numbers $x$. (Recall that the supremum for bounded real-valued functions exists for any nonempty subset of the domain and is unique.) Since the functional inequality implies that for nonzero $x$,
$$f^2\left(x+\frac1x\right) \ge f^2(x) + 2f(1) +f^2\left(\frac1x\right)\ge f^2(x) +2f(1),$$
it follows that for all $x \ne 0$, $x+\frac1x = \frac{x^2+1}{x} \ne 0$, whence
$$s=\sup_{x\ne 0}f^2(x) \ge \sup_{x\ne 0}f^2\left(x+\frac1x\right) \ge \sup_{x \ne 0}\left((f^2(x) + 2f(1)\right) = s + f(1).$$
That is, $s \ge s + f(1)$, which implies that $f(1) \le 0$, contrary to the given inequality for $f(1)$.

Comments. February's problem was $\#5$ in the 11th form's final round of the XXXI Russian Mathematical Olympiad (2005). The proposer, N. Agakhanov, perhaps chose "2" to be the coefficient of $f(xy)$ (in the problem's inequality) for aesthetic reasons. As Hubert Desprez pointed out in his soulution, it could as well have been 2012, or any other positive number. (He also provided an easy argument for when the problem is modified so that there is no $f(xy)$ term on the right side of the inequality.) He added, as did Lim, that the number 1 likewise plays no essential role in the problem --- the condition $f(1) \ge 0$ could be replaced by $f(c) > 0$ for your favorite constant $c$. Some positivity condition is critical though: Lim observed that the constant function $f(x) = -1$ (for all $x$) is bounded and satisfies the functional inequality (but, of course, fails to satisfy $f(1) > 0$); for a nonconstant example Kouba suggests the bounded function $f(x) = -\min(1, |x|)$.

 Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.
 about math central :: site map :: links :: notre site français


text:
The start of the current financial year heralds what the Government has called “the most significant reforms to superannuation since the introduction of compulsory superannuation in 1992”, so maybe it’s time to review your own superannuation arrangements. And while you are at it, why not take a little time to review other aspects of your personal finances as well?

In the current financial year, the Government is introducing the following key changes to superannuation:

• When you change jobs any time from 1 November 2021, in the absence of an explicit choice by you, your new employer will contribute to your existing ‘stapled’ superannuation fund rather than to your new employer’s own ‘default’ fund.

• The Government will require superannuation products to meet an annual objective performance test. Those that fail will be required to inform members and persistently underperforming products will be prevented from taking on new members. Members will be notified by 1 October 2021 if their fund has failed this test.

• The Government will make it easier to choose a better performing fund[1], by providing access to a new interactive online YourSuper comparison tool. This will commence from 1 July 2021.

Also, many of the dollar limits relating to superannuation will change in the new financial year (for example, the Concessional Contributions Cap will be increased from $25,000 to$27,500). Non-Concessional Contribution limits will also increase as will the Transfer Balance Cap and the Total Superannuation Balance limit.

Given these changes, you might like to consider the following actions early in the financial year:

• are you in a ‘well-performing fund’? Take a look at your fund’s performance in the government’s new YFYS comparison tool. It’s true that past performance isn’t necessarily indicative of future performance, but if your existing fund appears to under-perform chroinically, perhaps it may be time to consider switching to a strongly performing fund;

• are you one of the estimated 4.4 million Australians who are paying fees on multiple superannuation accounts? Unless there is a sound reason to maintain a second account (for example, to retain access to your preferred insurance arrangements), having multiple accounts is likely to result in multiple administration fees and a sub-optimal outcome at retirement;

• do the higher superannuation limits leave you with any opportunity to take additional advantage of the tax-favoured superannuation tax environment (e.g. via additional concessional contributions)?; and

• is it time to consider switching to a self-managed super fund (SMSF)? A ‘rule of thumb is that the ‘breakeven’ size of a fund, above which the fixed costs of running an SMSF are likely to be less than the largely variable costs of membership of a public fund, is somewhere between $200,000 and$500,000. But running your own SMSF also requires substantially more time and effort than membership of a public fund.

### Other aspects of your personal finances…

But why stop at re-thinking your super arrangements? There are other aspects of your personal finances which may also be worth reviewing.

For example, home lending rates are at historical lows in this current, ultra-low interest rate environment. Whilst banks have reduced borrowing rates for variable rate borrowers, it sometimes happens that the best borrowing terms are offered to new rather than existing borrowers. To avoid the ‘lazy tax’, you might consider the following:

• consult the websites that provide mortgage comparisons such as Canstar, Mozo, RateCity or Infochoice, and look for the best currently available borrowing rates;

• perhaps put a call through to one or two of the apparently most competitive options to confirm whether you would be eligible to refinance to their offering and investigate any conditions that they might impose; and

• then pick up the telephone to your existing lender and tell them “I’ve noticed that I qualify for a borrowing rate of x% with ‘ABC Lending Co’, but I’m paying y% on my current loan. What is the best rate that you can offer on my existing loan?”

Anecdotally (and in the author’s own experience), existing lenders may go at least some way towards matching a lower rate that you have located elsewhere. If your existing lender matches the alternative offer, or at least ‘goes close’, you might decide to accept their offer. The alternative is to commence the process of applying for a refinancing loan from the alternative lender. Refinancing is likely to involve some imposition on your time (perhaps half a day or a day, in total) to deal with the application process, even if you have the assistance of a competent and supportive mortgage broker.

Another option to consider may be to fix your interest rate for a two to five-year period.  You may have just missed the absolute nadir of the fixed-rate offerings, as some banks have recently increased their fixed rates in response to concerns about sooner-than-expected rises in the cash rate. Even so, some borrowers may find the appeal of repayments that are guaranteed not to increase over the next three, four or five years quite attractive.

There are many other aspects of your personal finances that you might also consider reviewing. Many of those aspects are covered in much more detail in my book Slow and Steady: 100 wealth-building strategies for all ages, including the following:

• Save systematically

• Establish an emergency fund

• Reduce unnecessary expenditure

• Pay down your mortgage fast

• Invest in managed funds

• Get your asset allocation right

• Diversify

• Keep your investment costs low

• Put assets in the name of the lower earning partner

• Consider private health insurance

• Buy only the types of insurance you need

• Write a will

• Establish Enduring Powers of Attorney and Guardianships

It may not take too much effort to produce a new, financially fitter you. Good luck!

 [1] Based on historical performance, which might not be a good indication of future performance

CPD: Actuaries Institute Members can claim two CPD points for every hour of reading articles on Actuaries Digital.

### Comment on the article (Be kind)

Your comment will be revised by the site if needed.

Previous

Next


text:
2021年1月21日

# Almost nef regular foliations and Fujita's decomposition of reflexive sheaves

to appear in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze.
• Masataka Iwai

In this paper, we study almost nef regular foliations. We give a structure
theorem of a smooth projective variety $X$ with an almost nef regular foliation
$\mathcal{F}$: $X$ admits a smooth morphism $f: X \rightarrow Y$ with
rationally connected fibers such that $\mathcal{F}$ is a pullback of a
numerically flat regular foliation on $Y$. Moreover, $f$ is characterized as a
relative MRC fibration of an algebraic part of $\mathcal{F}$. As a corollary,
an almost nef tangent bundle of a rationally connected variety is generically
ample. For the proof, we generalize Fujita's decomposition theorem. As a
by-product, we show that a reflexive hull of $f_{*}(mK_{X/Y})$ is a direct sum
of a hermitian flat vector bundle and a generically ample reflexive sheaf for
any algebraic fiber space $f : X \rightarrow Y$. We also study foliations with
nef anti-canonical bundles.

リンク情報
arXiv
http://arxiv.org/abs/arXiv:2007.13954
URL
http://arxiv.org/abs/2007.13954v1
URL
http://arxiv.org/pdf/2007.13954v1 本文へのリンクあり

エクスポート


text:
# Doubling voltage and power in a light bulb

1. Jun 22, 2010

### daysrunaway

1. The problem statement, all variables and given/known data
A light bulb is designed for use in a 120 V circuit. If by mistake it is plugged into a 240 V circuit, what will happen to the power of the light bulb? Explain.

2. Relevant equations
I = V/R
P = VI

3. The attempt at a solution
Since I = V/R and the resistance of the lightbulb does not change, doubling V also doubles I. Pf = 2V2I = 4P0. Alternatively, P = V2/R, so doubling V quadruples P. Where I'm getting stuck is on explaining this. What are the practical implications of quadrupling the power? What is the physical representation of quadrupling it - I mean, aside from the equations, WHY is it quadrupled? I'm really stuck on this.

2. Jun 22, 2010

### poor mystic

Tungsten actually has a pronounced positive change in resistance with increasing heat, so that a cold lamp suffers an 'inrush' current as it comes to its working temperature and resistance.
If the voltage is doubled from the rated voltage, less than 4 times the rated power is dissipated in the filament. However, experience shows that commercial tungsten filament lamps are not designed to withstand such treatment: we know that the bulb will fail due to melting of the filament and consequent lack of a conduction path.

I hate to leave you feeling stuck on the matter of why. All I can say is that if you had to carry twice as much water, twice as high, to generate the potential to run some machine, you'd soon understand what 'quadrupling' the power means.

Last edited: Jun 22, 2010
3. Jun 22, 2010

### wwshr87

Ok, so lets say that the light bulb is 100 ohms. At 120V it would consume 1.2A and 144 Watts. We connected by mistake to 240V, then it consumes 2.4A and 576 Watts. Most likely the light bulb is not designed to take that much current and it fries.

4. Jun 22, 2010

### Dickfore

Light bulbs are heated to a high temperature and their resistance is much greater than the resistance of a "cold" light bulb. Therefore, assuming R = const. is not valid.

Let us assume that the resistance is proportional to the aboslute temperature and that the temperature determines the radiated power according to Stefan's Law, i.e. the radiated power is proportional to the 4th power of the absolute temperature. These two proportionalities would mean that the resistance is proporional to the fourth root of the radiated power.

In equilibrium, the radiated power is equal to the power consumed. Using the two above equations to eliminate I, you should get:

$$I = \frac{V}{R}$$

$$P = V I = \frac{V^{2}}{R}$$

or

$$P R = V^{2}$$

But, $R \propto P^{1/4}$, so:

$$P^{5/4} \propto V^{2}$$

$$P \propto V^{8/5}$$

So, if you double the voltage, this "improved" model would predict that the power should increase by a factor of:

$$2^{8/5} = 2 \, 2^{3/5} = 3.03$$

One should perform an experiment to see if the above scaling arguments holds.


text:
# List All Files in a Directory in Python

This tutorial shows how to list files with an extension in a directory and its subdirectories. This is an example directory which consists of files and directories.

## Using a working directory

If you want to use the working directory, instead of the specified one, you can use the following code.

## List all files and directories in the directory non-recursively

This code will get all filenames + extensions and directories from the directory without entering other directories that are inside this one.

And this is the output:

['data', 'pic1.jpg', 'pic2.jpg', 'pic3.jpg', 'text', 'text.txt']

It may be useful if you want to create a program that will print the current directory tree. If you write such a script, you can then convert it to a single exe file.

## List only files in the directory non-recursively

In the previous example, both, files and directories are listed.

Let’s display only files.

output

['pic1.jpg', 'pic2.jpg', 'pic3.jpg', 'text.txt']

## List all files recursively (directory and subdirectories)

In order to list all files in the directory and its subdirectories, we will use os.walk. It will check each directory recursively and display all files.

output

['pic1.jpg', 'pic2.jpg', 'pic3.jpg', 'text.txt', 'data.txt', 'data.xlsx', 'simple text.docx']

## List all files and directories recursively (directory and subdirectories)

output

['pic1.jpg', 'pic2.jpg', 'pic3.jpg', 'text.txt', 'data.txt', 'data.xlsx', 'simple text.docx', 'data', 'text', 'spreadsheet', 'notepad', 'word']

## List all files recursively using a wildcard and display the full path

So far, we were displaying only files and directories names. This time let’s display the full path. The wildcard (*.*) means that it displays all types of files.

output

['D:\\mydir\\pic1.jpg', 'D:\\mydir\\pic2.jpg', 'D:\\mydir\\pic3.jpg', 'D:\\mydir\\text.txt', 'D:\\mydir\\data\\data.txt', 'D:\\mydir\\data\\spreadsheet\\data.xlsx', 'D:\\mydir\\text\\word\\simple text.docx']

## List all files recursively with a specific type of file

You can also display only one type of file. Let’s display only text files. Change (*.*) to (*.txt).

And this is the output.

['D:\\mydir\\text.txt', 'D:\\mydir\\data\\data.txt']

## List all files and directories recursively and display the full path

Only a slight modification (*.* >> *) in the script to display both files and directories.


text:
# Relationships when moving abroad for graduate study.

1. Oct 1, 2012

### JesseC

I have put this post in academic guidance because I am not looking for relationship advice, but for info about how easy it is keep friend and family ties when moving to study somewhere completely new, as well as the practicalities involved.

I'm considering graduate study in the USA(I'm from the UK). I've gone so far as to sit the GRE and start the applications so I'm quite serious. My main concern with US PhDs is that they take a long time, during which I would lose contact with a lot of people. I realise that facebook and skype makes it much easier to keep in touch over long distances, but it would still be impossible to pop over for tea on the weekend for example.

As a PhD student on an average stipend, how often would I be able to fly home and visit?

Does it sound reasonable to set aside $2k per year? If flights were$700 return, that would be about three trips home per year.

How much time off could I expect to take?

I would also be interested in hearing about how other people have dealt with these issues.

2. Oct 2, 2012

### Someone1987

Hi JesseC,

I completely understand your concern about moving so far from home. I moved about an 8 hour drive away from home (not the same as overseas but just far enough where it doesn't make sense to just hop in my car on a whim and drive home for the weekend). There are times where I really miss being able to see my family on a regular basis and wish I hadn't moved so far away. I find the best way to deal with it is to remind myself that graduate school is not forever and that when it is over I can make every effort to move closer to home.

Anyway, to answer your questions, on an average stipend plane tickets will be quite expensive anytime of the year. However, I think it would be doable to make it home a few times a year if you plan/save up ahead of time like you intend. Perhaps you could sign up for frequent flyer miles so that every now and then you can get a free trip?

In terms of how much time you can take off, that depends on your research and your advisor. Most advisors I have seen are pretty good about this but some are more uptight so this is something you will have to gauge before you start working for a particular professor.

Best of luck with your decision.

3. Oct 2, 2012

### JesseC

Thanks Someone, air/sky miles are a good idea. My google search tells me that flights to the north east coast cost at least £400 and to the west coast cost at least £600. That translates to $650 and$1000 dollars. So I wouldn't be able to afford as many trips home from the west coast, which is a shame because I am looking at the UoWashington.

As an undergraduate I got 3 weeks off for Christmas, 3 off for Easter and about 14 off for the summer, making a total of 20 weeks or just under 5 months. (Actually surprised by this, it felt like nothing at the time since Christmas and Easter were usually spent revising for exams). I understand that PhD students are expect to work through the summer and I don't want to sound like I'm planning to skyve off all the time!

So, does 2 months off per year for a PhD student sound reasonable?

4. Oct 2, 2012

### gravenewworld

It's terrible. This is coming from something with tons of experience doing this. 2 months per year? Yeah right. We get 3 weeks here. You're also poor, so flying is expensive. Hopefully your BF has money to fly to see you instead.

You're not in college anymore, so get that mindset completely out of your head. Treat grad school more like a full time job. How many full time jobs do you know of that give someone 2 months off every year?

I just bought a ticket to see the sig other after 9 months away. I expect it to take me practically all of next year to pay it off (or at least a major portion of it).

5. Oct 2, 2012

### JesseC

School teachers do... :p. So you're telling me that taking 2 weeks off over Christmas, 2 weeks off at Easter and say 3 weeks during the summer is unheard of? I get that the US has some of the longest working hours and least holiday time of any country, but seriously...?

May I ask how far away your significant other is and what your stipend is?

6. Oct 2, 2012

### TMFKAN64

School teachers are a special situation and of course get the entire summer off. Their paychecks reflect this though.

Where I work is not atypical. Here, people get 2-4 weeks of vacation (depending on seniority), plus 11 holiday days (one of which "floats" and can be used on any day). That would be a total of 4-6 weeks off, not the 7 you are discussing here.

This is also a job, not graduate school. You will probably work more in graduate school.

As a graduate student, your bigger problem will be money though.

Last edited: Oct 2, 2012
7. Oct 2, 2012

### Nano-Passion

Here are some other things you can try.

This one is pretty obvious. Technology has made it so you can view the other person in real time as you communicate with them, use this.

Try to hang out with her in a virtual world such as second life.

Wait until tele-emerging technolgy develops.

Soon distance will not be an issue. Think of how far we have come since the 1600s with respect to long distance communication. A hundred years from now there will also be virtual kinesthetic communication in real time, and later virtual (or not so virtual) sex.

Last edited by a moderator: Sep 25, 2014
8. Oct 2, 2012

### bromden

I live about 1800 miles from my hometown, even farther from where I spent my years as an undergraduate, which was my most recent home before this. It is difficult in the beginning, but if you have a good group of friends and find the folks in your research bearable, I think it eases the burden a lot.

My first semester was tough, but now, I don't necessarily equate time off with going home. I'd rather travel other places or just explore my local area. I guess the point is that it's possible for the place where you go to graduate school to become your home. Once that happens, you'll enjoy time with your family when you see them, but your life will be wherever you are.

If you save money and budget wisely, you should be able to afford a few flights per year. I've found I would rather be more social most of the time than be a stickler throughout the year just so I can see family more often. Maybe that sounds harsh, but most of the time you are at your graduate instiution. It makes more sense to me to maximize your day-to-day happiness. Your mileage may vary, though.

I'm going to have to agree with gravenewworld. Two months is a lot. But it really is such a function of your advisor that it's hard to say. My advisor has the work hard/play hard mentality, so graduate students in our group typically get 4 to 5 weeks off per year. He also is from Europe, though, so he understands what it's like to live very far from your family.

I have friends who complain that their advisors only give them off two weeks per year, but most of them only work forty hours a week. I just don't see how you can expect to put in the bare minimum that is required of you but then be rewarded with extradinary vacation packages.

The key here is to speak with graduate students in your potential research groups. They'll be able to honestly tell you what it's like to work with that Professor: how time off works, how their social lifes are, etc.

I agree up to the point that technologies like Skype and mobile phones have made it easier to stay in contact with family and friends, but it is still not the same as spending time with loved ones in person. The dynamic is mostly different when you're face-to-face for an extended period of time versus staring at eachother on computers for an hour or so a week. I posit that technology will never replace real, face-to-face interaction.

It's much more difficult to do this with a significant other. Long distance relationships have varying degrees of success. But I will say that my friends that are in long distance relationships spend a substantial amount of time on Skype every night, to the point where they break social obligations in person just to "be" with their SO. A friend of mine from undergrad, also attending graduate school on the East Coast (who is dating someone halfway across the country) recently told me that he feels he has no friends, just colleagues. Why? When everyone else was getting to know each other, he was staying at home, talking to his girlfriend.

My point is that distance is hard, and it's even harder when you're a graduate student who puts in a lot of hours per week.

The main takeaway is this:

Graduate school is something that you really have to want, pretty much for its own sake. I think it's awesome that I get paid to do science, even if that means working insane hours. Just realize you will have to make sacrifices, you will drift apart from some friends, and you may miss out on some things with your family. To expect otherwise would be naive. The great part is there is nothing stopping you from making new friends, being fulfilled by your work, and having a new home.

9. Oct 2, 2012

### Nano-Passion

Click on my link that I posted to see what I meant by tele emerging technology, if you already haven't that is. Your right, skype and other mobile apps are not as good as the real thing. However, in the future the word long-distance will be almost redundant as technology progresses.

10. Oct 2, 2012

### Jorriss

A lot of this entirely depends on your adviser. I know of advisers who expect students to take effectively no vacation time and others who effectively do not care how much you take at all.

11. Oct 3, 2012

### chill_factor

How long is a manageable commute? Is driving 5 hours manageable every few months and taking 3 days off sometimes to see friends OK? I've never driven so long before. The longest I've driven was 3 hours at a time stuck in traffic.

Has anyone had similar experiences?

12. Oct 3, 2012

### Someone1987

I agree with the other posters that 2 months is probably too much to count on. As a domestic student who normally works 50-60 hours a week, the only time off I usually plan for is 1-2 weeks for Christmas and New Year's and 1 week off during the summer.

Sometimes, if there is a special occasion (like my mother's 60th birthday this February) I will ask my advisor if it is alright to take a day or two off here and there and she usually has no problem with this.

Another thing you may want to consider is how being so far from home will impact what you can do during a family emergency. For example, my mother was very sick for a month and a half earlier this year. She recovered but it required major surgery. During this time, I appreciated the fact that I could get home pretty quickly and easily (just hop in my car go) but if you are living overseas this may be more difficult.

13. Oct 3, 2012

### JesseC

OK, so you're suggesting cryogenic storage until technology becomes more advanced? ;)

Yeah I realise that moving somewhere for 5 years means making a new life and being prepared to leave a lot behind. Some parents see this as essential for their children, for example I know a lot of Chinese students who go without seeing their families for years just to get good qualifications abroad. However, from my parents point of view it is not necessary for me to go so far since there are very good universities here too.

I don't really understand this issue about being poor. I am a naturally frugal person and I've looked at the fellowships at the universities I'm interested in and they range from $25k to$30k per year, which seems like a lot to me. Here the highest stipend you can get is £16k ($25k) per year and that is special to London because the living expenses are so much higher. Are average living expenses in the US higher than London?! Or is the money not just to live on but also for academic expenses? I would do this, but some of the programs I'm looking state that prof.-student assignment doesn't happen until after you arrive. I would also feel awkward emailing random grad students about their social lives XD. Thanks for the advice though, it is helping me make more sense of the whole thing. 14. Oct 3, 2012 ### TMFKAN64$25-30K is about $2000-$2500 per month. If you wanted to go to Stanford, for example, you could easily spend all of that on a simple studio apartment. But you'd probably also want to eat, buy textbooks, and get around...

It's not desperate poverty, but you'll think twice about taking too many expensive flights home!

15. Oct 3, 2012

### Mute

There is one, maybe two, days off for Easter. There is a week off for spring break, typically. I don't think you're going to get 2 weeks off around Easter anywhere, especially if you have to TA.

I typically go home (US to Canada) twice a year, once at christmas for 2-3 weeks and once in the summer, typically for about two weeks. I have taken week-long trips during Thanksgiving week and/or spring break week before, but I have not gone further than US->Canada for those. So, I suppose in early grad school I got around 2 months of vacation scattered about the year (though I still did some work on vacation), but lately I haven't really gone anywhere for thanksgiving and my last spring break trip was only about 5 days (2 of which were travel).

So, I think it might be reasonable to go back to the UK twice a year, but 3 is probably going to be pushing it (and if you're not able to do work on vacation it may also extend your time in grad school!). Also, when I traveled home, it was usually my parents who bought tickets with their airmiles. If I paid for it out of pocket every time, I might only go home once a year.

16. Oct 3, 2012

### Choppy

When I was a graduate student (in Canada) we officially got 2 weeks of vacation, but no one really enforced this. As long as we progressed on our projects, kept our supervisors happy and fulfilled our TA duties, we weren't really monitored all that closely. Really the big issue is that the more time you take off, the longer it takes to complete your project.

As fas as relationships go, you're basically starting a new life in my opinion. Sure you'll keep in touch with old friends - particularly the important ones. But if you do it more than a couple of years, you'll find that you'll make new friends and develop new relationships.

I travelled to the other side of the country for graduate school (4 hour flight) and as a student managed to get home twice a year at first, later once a year. I quickly met an awesome girl and eventually we got married. Now the this is our home.

17. Oct 3, 2012

### ParticleGrl

When I was in gradschool, we got 10 vacation days (2 weeks a year) and how much it was enforced depended on the advisor. I generally took three weeks (one at Christmas, two in August) and my advisor was fine with that. I probably couldn't have taken much more.

Stipend at my university was about a bit under 20k a year. 2k isn't an impossible goal to set aside, though for much of the US that will equate to one trip transatlantic a year.

18. Oct 3, 2012

### Nano-Passion

Well, I was just trying to get your hopes up. Cryogenic storage hasn't been made available yet, sorry for the tease!


text:
# Figure on an even page and caption on the following page

I would like to place a figure on an even page, then the caption on the following page.

The reason for that is that the figure + caption is too big to fit in one page, but I want the reader to be able to see both image and caption.

Here is a MWE to compile with xelatex:

\documentclass[12pt,english]{extbook}
\usepackage{fontspec}
\usepackage{xunicode}
\usepackage[a4paper]{geometry}
\usepackage[english]{babel}
\usepackage{caption}
\usepackage{afterpage}
\usepackage{ifoddpage}
\usepackage{blindtext}

% Create big picture
\renewcommand{\includegraphics}[2][]{%
\setlength{\fboxsep}{-\fboxrule}%
\framebox{\rule{0pt}{15cm}\rule{10cm}{0pt}}%
}

% Add figure on even page and caption on odd page, plus a ruler to separate caption from text
\newcommand{\doublesidefigurewithruler}[5][1]{%
\afterpage{
\newpage
\checkoddpage
\ifoddpage
\doublesidefigurewithruler[#1]{#2}{#3}{#4}{#5}
\else
\doublesidefigurecontent{#1}{#2}{#3}{#4}{#5}{1}
\fi
}
}%

% Add the figure and the caption
\newcommand{\doublesidefigurecontent}[6]{%
\begin{center}
\vspace*{\fill}
\includegraphics[width=#1\textwidth]{#2}
\captionof{figure}[]{#4}
\label{#3}
\vspace*{\fill}
\end{center}

\newpage

\captionof{figure}[#4]{#4 #5}

\ifnum \numexpr#6=1
\noindent\rule{\textwidth}{0.2pt}
\fi
}%

\begin{document}
\Blindtext

\doublesidefigurewithruler[1]{%
bigfigure}{%
mylabel}{%
Mytitle}{%
\blindtext
\blindtext
}

\Blindtext
\Blindtext
\Blindtext
\end{document}

It does work, but there is probably a better way to do it, right? In addition, figure numbering is sometimes wrong if I mix this with regular figure environment, and I have to add \clearpage between two \doublesidefigurewithruler....

Besides, I would prefer to place the caption at the bottom of the next page if possible, and I would rather prefer to have an environment:

\begin{doublesidefigure}
\centering
\includegraphics...
\caption[for the toc]{My content}
\label{mylabel}
\end{doublesidefigure}

What I did work, but it is not very robust. How can I do it properly?

• Do you want the caption centered in an otherwise empty page, or just at the top before the text continues? Also, the lisr of figures will use the page the caption is on, not the figure. (Technically it should be called the list of captions.) Dec 3 '15 at 5:14
• I would like the caption to be at the top of the page (or even better at the bottom), and the rest of the page filled with normal text. Dec 3 '15 at 12:29

I take it you plan on having a really long caption, as it looks a little weird right now. Are you sure you don't want to put the caption in the margin instead?

BTW, since the odd page test is at the start of a new page, the page counter is accurate.

\documentclass[12pt,english]{extbook}
%\usepackage{fontspec}% don't need them myself
%\usepackage{xunicode}
\usepackage[a4paper]{geometry}
\usepackage[english]{babel}
\usepackage{caption}
\usepackage{afterpage}
\usepackage{environ}
\usepackage{mwe}
\usepackage{showframe}

\newsavebox{\doublepagecaption}
\newsavebox{\doublepageimage}

\NewEnviron{doublepagefigure}[2][]% #1=short caption (optional), #2=caption
{\global\setbox\doublepagecaption=\hbox{%
\parbox{\textwidth}{\captionof{figure}[#1]{#2}}}%
\global\setbox\doublepageimage=\hbox{\parbox{\textwidth}{\BODY}}%
\afterpage{\clearpage
\ifodd\value{page}%
\afterpage{%
\vspace*{\fill}\vspace{-\baselineskip}%
\noindent\usebox{\doublepageimage}\vfill\pagebreak
\begin{figure}[b]\usebox{\doublepagecaption}\end{figure}}%
\else
\vspace*{\fill}\vspace{-\baselineskip}%
\noindent\usebox{\doublepageimage}\vfill\pagebreak
\begin{figure}[b]\usebox{\doublepagecaption}\end{figure}%
\fi}}

\renewcommand{\bottomfraction}{0.5}% fraction of page allowed for bottom floats

\begin{document}

\listoffigures
\bigskip
Reference to Figure \ref{fig:test}.
\bigskip

\lipsum[1]

\begin{doublepagefigure}[short caption]{long caption\label{fig:test}}
\centering\fbox{\includegraphics[width=.95\textwidth,height=.95\textheight]{example-image}}
\end{doublepagefigure}

\lipsum[2-16]

\end{document}

\afterpage waits until the page break to execute. If a paragraph is split, it will insert text BEFORE the second half. Calling \afterpage inside \afterpage waits two pages, and so on.

\clearpage forces any outstanding floats to be printed now, even if it adds a page. (You probably don't need it, but in general you want floats to be printed in the correct order.)

I use box registers to implement the figure and caption, just like normal floats. The system is smart enough to delay writing to the TOC until the box is displayed. It also increments the figure counter NOW.

All local definitions are lost at the end of an environment, and \savebox is a local definition. Hence the \global\setbox\name=\hbox{...} form.

Here is a stylistically updated version. It does not require the environ or caption packages.

\documentclass[12pt,english]{extbook}
%\usepackage{fontspec}% don't need them myself
%\usepackage{xunicode}
\usepackage[a4paper]{geometry}
\usepackage[english]{babel}
\usepackage{afterpage}
\usepackage{graphicx,lipsum}
\usepackage{showframe}

\newsavebox{\doublepagecaption}
\newsavebox{\doublepageimage}

\makeatletter
\newenvironment{doublepagefigure}[2][]% #1=short caption (optional), #2=caption
{\def\@captype{figure}%
\global\setbox\doublepagecaption=\vbox{\caption[#1]{#2}}%
\begin{lrbox}{\doublepageimage}}% body goes here
{\end{lrbox}%
\global\setbox\doublepageimage=\box\doublepageimage
\afterpage{\clearpage
\ifodd\value{page}%
\afterpage{%
\vspace*{\fill}\vspace{-\baselineskip}%
\noindent\usebox{\doublepageimage}\vfill\pagebreak
\begin{figure}[b]\usebox{\doublepagecaption}\end{figure}}%
\else
\vspace*{\fill}\vspace{-\baselineskip}%
\noindent\usebox{\doublepageimage}\vfill\pagebreak
\begin{figure}[b]\usebox{\doublepagecaption}\end{figure}%
\fi}}
\makeatother

\renewcommand{\bottomfraction}{0.5}% fraction of page allowed for bottom floats

\begin{document}

\listoffigures
\bigskip
Reference to Figure \ref{fig:test}.
\bigskip

\lipsum[1]

\begin{doublepagefigure}[short caption]{long caption\label{fig:test}}
\centering\fbox{\includegraphics[width=.95\textwidth,height=.95\textheight]{example-image}}
\end{doublepagefigure}

\lipsum[2-16]

\end{document}

• Wow, that's a difficult thing for me. Could you explain it a little bit? It does work btw, thanks! The problem with scientific figure is that we are limited to a couple of figure per article, so we gather every related results into one figure. Then we describe the results in the caption which can be very long. Dec 3 '15 at 12:25
• I moved the caption to the bottom of the second page. I experimented with allowing the caption to go on the first page if it fit, but decided not to. If you can fit the caption on the same page, you should use a regular figure. Dec 3 '15 at 16:13
• BTW, do not create a second doublepagefigure until the first is displayed, as you will overwrite the box registers. (Didn't want to complicate matters with dynamic box allocation. See tex.stackexchange.com/questions/185140/reusing-saveboxes) Dec 3 '15 at 16:42
• thanks! It works perfectly and explanation are clear enough! Dec 5 '15 at 7:53