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## Introduction

The strong penetrating capability of high-energy ionizing radiation, including X-ray (photon with wavelength from 0.01 to 10 nm) or gamma-ray (γ-ray, photon with wavelength shorter than 0.01 nm), make them attractive in non-destructive probing of inside information of condensed matter subjects. After their interaction with electrons in the atoms to ionize the materials, X-rays and γ-rays carry information of the subjects in their variation of the photon intensity (dose rate), direction, phase, and energy. The detection of the X- and γ-ray variation in good faith thus allows accurate retrieval of the subject information. This enables the broad applications of X-ray in many fields including medical imaging, safety screening, shipping container inspection, defect inspection and quality control in material industry, and contamination detection and quality control inspection in food industry, where X-rays penetrate subjects and its attenuation is recorded1,2,3,4,5. The image contrast comes from the difference in absorption or stopping capability of different human tissues or subjects to X-rays. In scientific research, X-rays have been broadly used to enhance the understanding of material sciences in the past few decades by the interaction of X-rays with materials. γ-Rays can be generated in radioactive materials, which presents fingerprints of elements as well as isotopes, and distinguishing these elements or isotopes needs the detection of the energy-dependent photon counts, or spectroscopy. γ-Ray detection is essential for many homeland security applications such as radiological security, nuclear defense, and so on6,7,8,9.

Semiconductor-based ionizing radiation detectors work in a similar way with photodetectors either in current or voltage modes, while more details of specific difference are described in Box 110,11. The current advanced radiation detectors working in current mode include silicon (Si), amorphous selenium (α-Se), mercury(II) iodide (HgI2), cadmium zinc telluride (Cd1 − xZnxTe), while radiation detectors in voltage mode comprise high-purity germanium (HPGe), Cd1 − xZnxTe, thallium(I) bromide (TlBr), and so on. Besides direct detection strategy, X- and γ-ray detection can also be accomplished in an indirect detection strategy through a combination of scintillators and photodetectors. Scintillators convert high-energy X-rays and γ-rays into ultraviolet–visible (UV–Vis) light, which is detected by weak light sensors such as amorphous Si photodiodes, thin film phototransistor (TFT) arrays, photomultiplier tubes (PMTs), complementary metal-oxide semiconductor (CMOS) detectors, silicon avalanche photodiodes and arrays, or charge-coupled devices (CCDs). The dominating commercial X-ray and γ-ray scintillators include thallium-activated cesium iodide (CsI(Tl)), thallium-activated sodium iodide (NaI(Tl)), and so on. Important performance parameters for scintillators are light yield, light decay time, and environmental stability.

The explosion in the use of higher radiation–dose tests, such as computed tomography (CT) imaging, brings concerns on radiation safety. Over 80 million CT scans are performed in the United States, compared with just 3 million in 1980. High-dose X-ray scan increases the risk of cancer to patient later in life due to the damaging of DNA by ionizing radiation12. For example, one-time chest CT scan can have dose equivalent to 2 years accumulation of natural background dose12. There is a report showing evidence of radiation-induced cancer risk at a dose above 100 mSv, as well as an estimation of ~2% of the cancers diagnosed could be related to CT performed in the United States in 200712. The high dose needed for CT scanning comes from the low sensitivity of the present radiation detectors, mainly limited by the detection properties of materials. Therefore, high-performance radiation detectors based on new materials are urgently needed.

In the past few years, a new family of halide perovskites with composition of ABX3 (where A is CH3NH3+ (MA+), HN = CHNH3+ (FA+), or Cs+; B is Pb2+, Sn2+; X is Cl, Br, and I) or A2MM′X6 (where M is Cu+, or Ag+, M′ is Bi3+, Ga3+, or In3+) have attracted lots of attention by their excellent performance in solar cells, light-emitting diodes, lasers, and photodetectors during the past few years13,14,15,16. Halide perovskites are quickly identified to be promising candidate semiconductors for ionizing radiation detection17,18,19. This review exploits the science and application of halide perovskites as a new semiconductor candidate for X-ray imaging and γ-ray spectroscopy. Although there are multiple-stringent requirements on ionizing detector properties, which results in very limited choice on effective materials, halide perovskites show many advantages as promising radiation detector materials, which are discussed in detail in the following sections. There are still outstanding challenges before their commercialization, such as how to stabilize material and interface under large bias in single-photon detectors, which will also be discussed in this review.

## Development status of conventional radiation detectors

### X-ray dosimeter and imaging detectors by direct conversion

For X-ray imaging detectors, a large sensitivity is needed for high-quality images with large contrast at a given X-ray dose rate20, which outputs larger X-ray-induced current10,21. In order to minimize the doses applied to the patient while maintaining image quality, one path is to increase the mobility–lifetime (μτ) product or increase the applied bias to the detectors22.

The μτ product, which characterizes carrier drift length, is a fundamental property of a semiconductor material to predict the capability of extracting the charges deep inside the semiconductor. The μτ product can be determined by material types as well as defects, which is a parameter closely related to the charge carrier diffusion length LD with a relationship of $$L_{\mathrm{D}} = \left( {k_{\mathrm{B}}T\mu \tau /e} \right)^{1/2}$$, where kB, T, and e are the Boltzmann constant, absolute temperature, and elementary charge, respectively. Larger µτ product will result in better charge collection efficiency by increasing the mean drift lengths and/or diffusion lengths in the semiconductor bulk. Applying a high bias can enhance the signal intensity; however, it also increases the device noise.

In addition to a strong signal, a small noise is also important for a large signal-to-noise ratio, which is a figure of merit for many types of sensors18,19,21. The noise level also determines the weakest X-ray dose a detector can measure. The noise of a radiation detector is generally dominated by dark current, though other types of noise might also be present. While some other figures of merit are also important for a high-quality X-ray image, including spatial resolution, response speed, uniformity, and stability. Spatial resolution is a parameter used to characterize image sharpness, which can be quantitatively assessed by the modulation transfer function (MTF)23. Linear dynamic range (LDR) is another important parameter for an X-ray detector to show the responsible range of X-ray dose rate under which sensitivity is a constant. Large LDR will ensure that the detector gives an accurate measurement of X-ray dose rate under a large variation range. Moreover, the X-ray detector needs to have a fast response speed to reduce the exposure time of X-ray irradiation, and also give high frame rate for capturing videos. Frame rate is the inverse of the time needed for the CCD to acquire an image and then completely read an image out, which is not only related to device response time but also the read-out circuitry. A uniform performance of the pixels among large area is also crucial for a high-quality X-ray imager, though it can be partially made-up with imaging processing.

Table 1 summarizes the detailed performance of the current X-ray detectors. They are made by Si, α-Se, HgI2, Cd1 − xZnxTe, and so on. Since Si has low stopping power to X-ray, it is mainly applied in portable X-ray fluorescence spectrometry for soft X-ray detection, especially for photon energy smaller than 10 keV to detect the fingerprints of trace elements as well as isotopes of different bulk materials24. Since α-Se has over ten times stronger stopping power than Si in hard X-ray range25,26, it is the most common semiconductor material for X-ray imaging in direct conversion mode, and it has dominated the direct conversion X-ray imaging market for decades10. Although the cost of α-Se raw material is only ~$0.8 per g, the main flat-panel detector cost comes from thick film deposition process by conventional vacuum thermal deposition. HgI2 and Cd1 − xZnxTe (x is <20%, denoted as CZT) are both promising X-ray detectors. Their sensitivities are good as shown in Table 1, but the performance of HgI2 detector is mainly limited by the large leakage current. While there are also challenges in integrating CZT with read-out circuitry due to the too high temperature that required for high-quality CZT crystal growth. The current state-of-the-art CsI(Tl) scintillator detectors with CMOS sensor have pixel resolution around 50μm, or 10 lpmm−1 (ref. 27), and corresponding X-ray sensitivity28 reaches 5370 μC Gy−1air cm−2. A recent reported scintillator made by terbium-doped gadolinium oxysulfide (GOS:Tb) has been employed for X-ray imaging with spatial resolution of 4.75 lp mm−1 (ref. 29). Although the resolution is not high and the sensitivity is only about 7.3μCGy−1air cm−2, GOS:Tb scintillator are still attractive for its low-cost fabrication from solution. ### γ-Ray spectroscopy detectors As for the γ-ray counting for energy spectrum application, the most important figure of merit is the photopeak energy resolution, which is defined by the ratio of full-width at half-maximum value of the photopeak and its corresponding photon energy. The semiconductor detectors for γ-ray spectrum acquisition actually have been developed for several decades, while there are still only a handful of semiconductors that can be successfully applied for this purpose with satisfying performance30, since γ-ray detection imposes multiple-stringent requirements on semiconductor materials. Due to the much higher energy of γ-ray than X-ray, it needs materials to possess large atomic number (Z) for strong stopping power31; large electron and hole µτEproduct (or Schubweg distance) for efficient charge collection, where E′ is the applied electric field; a low density of trap states to avoid charge trapping or recombination during charge collection for single event analysis; again, a large bulk resistivity over 109 Ωcm is paramount for high signal/noise ratio. Table 2 lists several figures of merit of typical γ-ray semiconductor detectors that can impact the energy spectrum resolution. HPGe-based spectrum detectors now gives the best energy resolution of 0.3%32,33, while it requires cryogenic cooling to suppress its dark current due to its small bandgap of 0.66 eV. CZT materials have a larger bandgap of over 1.6 eV, and its large Z elements ensure a strong stopping power. A significant amount of efforts have been devoted to increase the bulk resistivity to around 109 to 1010 Ω cm by crystal growth. The state-of-the-art CZT detectors achieved an energy resolution of 0.5%, which operates at room temperature34,35. TlBr is another semiconductor that makes detectors with energy resolution of about 1% at room temperature, because of its large stopping power (four to five times stronger than CZT), high resistivity of over 1011 Ω cm, and large bandgap of 2.68 eV, although the best reported μτ product of this material is still one order of magnitude smaller than CZT36, resulting in a very good energy resolution of 1%. Notably, high-quality CZT materials are also not broadly commercially available, and now CZT single crystals cost about$3000 per cm3. While it is possible that the cost of CZT single crystals can be dramatically reduced by scaling up the manufacturing, industry is pushed back by the small demand of CZT materials in the radiation detection market, in strong contrast to silicon.

Similar to X-ray imaging, γ-spectrum can be also collected by a combination of scintillator and a sensitive pixel photodetector. Typical scintillators for γ-application are made of NaI(Tl) crystals and CsI(Tl) crystals. NaI(Tl) crystal emits 415 nm light with efficiency of 11.3%, comparable to the CsI(Tl). The emitting light will be detected by a PMT detector to record light intensity, which is proportional to the γ-ray energy. Recent technology is moving toward γ-detector array so that spectrum can be quickly collected by multiple detectors, which however needs a really good uniformity of performance among all these γ-ray pixels.

## Halide perovskite materials emerge as promising candidates for radiation detectors

Right after the demonstration of efficient solar cell application, halide perovskites were shown to be good photodetector materials due to the large light absorption coefficient, tunable bandgap, large mobility, and long carrier recombination lifetime, even in the form of polycrystalline films made from solution37,38,39,40. All these properties are actually also desired for radiation detection. In this section, we review the properties of perovskites, which make them a new generation of radiation detection materials.

### Strong stopping power

In halide perovskites, the atomic number of Cs+, Pb2+, Ag+, Bi3+, Sn2+, I, and Br ions are as high as 55, 82, 47, 83, 50, 53, and 35, respectively. Some of the atomic numbers are even larger than those of CZT as shown in Table 2. The typical MAPbI3 halide perovskite has a density of around 4 g/cm3, providing a large linear attenuation coefficient of 10 cm−1 to 100 keV X-rays, which is comparable to that of CZT material as shown in Fig. 1a and Table 2. Inorganic halide perovskite CsPbI3 has a larger linear attenuation coefficient of 14 cm−1. The capability to form double perovskites, such as Cs2AgBiI6 and Cs2AgBiBr6, allows the incorporation of many other high Z elements into the perovskite structure.

### Large mu–tau product

Halide perovskites were first applied in dye-sensitized solar cell structure as dye on mesoporous TiO2 surface41, while researchers later found that halide perovskite solar cells with planar heterojunction structure works well due to its long charge carrier diffusion length of over 1000nm42, corresponding to a mu–tau(µτ) product of 3.9 × 10−7 cm2 V−1. This result was confirmed by Yakunin et al.’s17 work using perovskite polycrystalline films for X-ray detectors. Perovskite single crystal presents much lower trap density by removing the grain boundaries of polycrystals as shown in Fig. 1b as well as improved crystallinity. The internal quantum efficiency (IQE) spectrum in Fig. 1c shows that the charge collection efficiency of a 3-mm-thick MAPbI3 single crystal is approaching 100%, indicating an extremely long charge carrier diffusion length43. The mobility and charge recombination lifetime measurement results show a large µτ product of 1.2 × 10−2 cm2 V−1, corresponding to a long diffusion length of 175 µm as the low limit under strong illumination, which is consistent with photoconductivity measurement showing large µτ product of MAPbBr3 perovskite single crystal18, as shown in Fig. 1d. The µτ products of MAPbI3 and MAPbBr3 single crystals are comparable to that of CZT crystals, and are much larger than TlBr at room temperature, as shown in Table 2.

### Large bulk resistivity

The bulk resistivity of the semiconductors is another important figure of merit in radiation detectors for small dark current and noise. One challenge in halide perovskite field is the difficulty in doping them intentionally, although this makes them attractive for radiation detection. Nevertheless, these materials are still subjected to self-doping. Halide perovskites showing excellent photovoltaic performance are mostly intrinsic or weakly p-type44. Replacing partial or all I ions with Br ions or Cl ions can reduce the intrinsic charge carrier concentration due to enlarged bandgap. Typical MAPbI3 perovskite has a resistivity of around 107Ω cm at room temperature45, and the resistivity of MAPbBr3 perovskite can be normally ten-folds higher46. These resistivity are still much smaller than those calculated from intrinsic charge concentration, indicating an unintentional doping in these materials, which was reported44. While the resistivity of inorganic halide perovskites can reach 1010 Ω cm, satisfying the requirement for γ-ray spectroscopy, but the mobility or µτ product is much lower than CZT materials47,48 (See Table 2).

One unique property of halide perovskites is its relatively better tolerance to defects than other semiconductor materials. First-principle calculation shows that the deep charge traps in MAPbI3 such as IPb, IMA, Pbi, and PbI have too large formation energy to form. Most point defects either in the conduction or valence bands are shallow traps, such as MAi, VPb, MAPb, Ii, VI, and VMA49 (see Figs 1e, f). The defect tolerance may explain the reported long electron–hole diffusion length in the single crystalline materials42,43. The defect tolerance also gives rise to the radiation damage tolerance of these materials50,51,52.

Lang et al.53 reported the stability of MAPbI3 solar cells under proton irradiation. A reference device with initial power conversion efficiency (PCE) of 12.1% has only 20% drop in current density, but with no voltage and fill factor loss after irradiation with 68 MeV protons with dose up to 1.02 × 1013 p cm−2. It is also encouraging to find that the device has self-healing capability, that is, the current density of the device recovered to its initial value after shelf storage for 10 days53. Chen et al.54 found that the Frenkel defects were generated by applying bias on perovskite, which may be related to the current density loss under irradiation, since the built-in electric field may also cause Frenkel defects. Self-annihilation of Frenkel defects is also confirmed when releasing the electric field, which is in consistent with the observed self-healing effect. Yang et al.55 also tested the stability of perovskite solar cells to γ-ray irradiation. The PCE barely drops after over 1500 h continuous γ-ray irradiation with total exposed dose of 23 kGy, while glass is significantly colored by γ-ray irradiation55. In striking contrast, CZT detectors lose their performance severely after total 30 kGy γ-ray exposure56, indicating the good γ-ray hardness of perovskite materials.

### Low-cost raw materials and crystal growth

The raw materials for halide perovskite are nature abundant and low cost. The total material cost for 1 cm3 perovskite single crystal is about $2.65 to$3.65 based on raw material prices from suppliers such as Sigma-Aldrich. Considering the material cost generally reduces by 5 to 10 times after scaling up the production, we estimate that material cost of halide perovskite single crystals is <\$0.3 per cm3, which is 3 to 4 orders of magnitude cheaper than CZT crystals. The device fabrication cost by single crystal growth or polycrystalline films deposition process is also low, because they are all done by low-cost solution processes. Large and high-quality perovskite single crystals can be grown from low-cost solution processes57,58,59,60 at relatively low temperature (lower than 150 °C) compared to traditional radiation detector crystals such as HPGe and CZT, which are often grown at temperatures around the materials’ melting points.

In hydrohalic acid, halide perovskites solubility often increases at higher temperatures 57 as shown in Fig. 2a, and single crystals with several millimeter size can be grown by gradually reducing the solution temperature as shown in Fig. 2b. While in some organic solvents such as dimethylformamide and γ-butyrolactone (GBL), halide perovskite has a reduced solubility at higher temperature (Fig. 2c), in striking contrast to many other materials. It was argued that the inverse solubility was caused by the protons as a result of solvent degradation under high temperature61, while exception to this rule can be easily named. The single crystal from such solvents can be grown by increasing the temperature from saturated solution58, and high-quality halide perovskite with dimension size over several millimeters can be obtained at a fast rate of 0.1–0.2 cm h−1 as illustrated in Fig. 2d59. The measured hole carrier mobility of MAPbI3 single crystal grown from GBL solvent is about 67 cm2 V−1 s−1, over two times smaller than mobility of 164cm2 V−1 s−1 of MAPbI3 single crystal grown from HI solvent43. This may be caused by the better quality of crystals grown from hydrohalic acid, since the crystal growth speed in hydrohalic acid is as slow as 0.2 mm per day. In addition, an anti-solvent approach has also been employed to grow perovskite single crystals from mixture solvents, which may give uniform growth rate of single crystals at room temperature62,63,64.

## Advance in the development of perovskite X-ray imaging detectors

The first perovskite radiation detector made by MAPbI3 single crystal showed response to high-energy γ-ray photons under zero bias43. The device worked in γ-voltaic mode with a quantum efficiency of 4% under high flux γ-ray. Later polycrystalline perovskite films made by spray coating were demonstrated to have good X-ray response with imaging capability by moving a single pixel detector along X–Y directions17. By scanning the signal current induced by transmitted X-ray photons, X-ray images of plastic target with clear inner scene were observed, as illustrated in Fig. 3a–d. The imaging time using single pixel detectors is very long, since it takes a lot of time to move the detector or the object along the X–Y plane. Nevertheless, it is still a good conceptual demonstration for perovskite materials for X-ray imaging application. Commercial X-ray detector products for imaging often employ linear detector arrays (LDAs), or, more frequently, flat panel arrays for fast detection. LDAs composed of individual X-ray-sensitive materials are arranged in a line as linear pixels as shown in Fig. 3b, which can scan along one direction to finish imaging (Fig. 3e). Flat panel arrays (Fig. 3c), which are often constructed by integrating X-ray-sensitive semiconductors on application-specific integrated circuits like CMOS read-out circuitry, can be employed to map an object by detecting the spatially dependent transmitted photon flux10. In this section, we review advancements in the development of perovskite single crystals and polycrystalline films for X-ray detection in the form of single pixel, LDAs, and two-dimensional (2D) detector arrays.

### Single pixel perovskite detectors

Due to the large atomic number and material density of halide perovskite, the linear attenuation coefficient of MAPbI3 to 120 keV X-rays is about 10 times stronger than α-Se. Nevertheless, MAPbBr3 with slightly weaker X-ray stopping power has been more often used for X-ray detection, because of many advantages including simple single crystal growth, cubic crystal shape, large bandgap, easily crystal quality inspection due to its transparency in visible light range, and less ion migration. One issue frequently encountered is that the MAPbBr3 single crystals are not transparent with visible defects, which can be identified by the naked eye. Highly transparent MAPbBr3 single crystals are grown using precursor solution with non-stoichiometry precursor ratio. Since the PbBr2 has less solubility and precipitates faster than MABr in perovskite solution, a PbBr2/MABr feeding ratio of 0.8 can get equal precursor precipitating speed, which allows the growth of high-quality single crystals. The µτ product of such MAPbBr3 single crystals reached 1.4 × 10−2 cm2 V−1 (ref. 18). Moreover, Cl ions alloying can further improve the hole mobility to 560 cm2 V−1 s−1 (ref. 46), which increased µτ product to 1.8 × 10−2 cm2 V−1. In addition to µτ product, the surface charge recombination rate (s) at perovskite/electrode interface also determines the signal intensity of an X-ray detector, and its dependence can be described by modified Hecht’s equation:

$$I = \frac{{I_0\mu \tau V}}{{L^2}}\frac{{1 - {\mathrm{exp}}( - \frac{{L^2}}{{\mu \tau V}})}}{{1 + \frac{L}{V}\frac{s}{\mu }}},$$
(1)

where I is the measured photocurrent, I0 is the saturated photocurrent, L is the thickness, and V is the applied bias. Clearly, a smaller s can also increase photocurrent. One attractive property of MAPbBr3 in this context lies its automatic surface passivation by oxygen or moisture in air. The charge radiative recombination rate on the surface of MAPbBr3 single crystals varied by over two orders of magnitude upon physisorption or desorption of oxygen and water molecules65. Full physisorption of oxygen on MAPbBr3 surface resulted in a low surface recombination velocity of 4 cm s−1 in this material, even better than that well-passivated single crystalline silicon. In real operational devices, UV–ozone treatment of MAPbBr3 surface is a good strategy to retain the oxygen passivation effect, which successfully suppressed the surface recombination and resulted in a high X-ray sensitivity of 80μCGy−1air cm−2 for the detector under a small bias of 0.1 V18. This sensitivity is already four times higher than α-Se X-ray detectors as shown in Table 121.

After demonstrating the superior detection properties of MAPbBr3 single crystals, a further step was taken to integrate them onto silicon, glass and other metallic substrates, which is touchstone to evaluate integration capability of perovskites with CMOS read-out circuitry. MAPbBr3 can be grown in solution at low temperatures below 120 °C, which is compatible with CMOS read-out circuitry. Wei et al.66 used a small-molecule of (3-aminopropyl)triethoxysilane layer, which has a NH3Br terminate on one end and Si-O bond on the other end, to mechanically and electrically connect the MAPbBr3 single crystals with any surface that can bind with Si-O bond66. The dipole of this bonding molecule at the interface of perovskite and Si also suppress the noise current while maintaining the signal current, which allows the device to operate under a larger optimal bias of −7V. The large gain at larger bias made the Si-integrated MAPbBr3 single crystal detectors to have a much larger sensitivity of 2.1 × 104 μCGy−1air cm−2 under 8 keV soft X-ray. The lowest detectable dose rate was also further reduced down to 36nGyair s−1, much better than the α-Se detectors as shown in Table 1. Cs2AgBiBr6 single crystals have large resistivity over 1010 Ω cm despite small μτ product, which results in a detectable lowest X-ray dose of 59.7nGyair s−1.

In addition to single crystals, polycrystalline perovskite bulk materials are formed by high-pressure sintering process from perovskite powders for X-ray detectors63. A polycrystalline perovskite wafer with diameter of 0.5 in. and thickness of 1 mm was demonstrated. Although the μτ product of the polycrystalline perovskite wafer was only 2 × 10−4 cm2 V−1, about two orders of magnitude lower than that of the single crystals, a reasonably large sensitivity of 2527μCGy−1air cm−2 was obtained under a relatively large electric field of 0.2 Vμm−1 (see Table 1). The challenge will be how to integrate them into CMOS read-out circuitry, because the high-pressure or high temperature process is often not compatible with CMOS technology.

### Linear detector arrays

LDA scanning is more convenient than 2D arrays imaging in some cases where very large area X-ray imaging needs to be performed by portable detectors like cargo inspection. The first perovskite LDA was demonstrated by depositing ten pixel electrodes in a row on a MAPbBr3 single crystal with a pixel electrode size of 200 μm and a pitch of 400 μm. The LDA devices by MAPbBr3 single crystal was demonstrated to possess good X-ray imaging capability at a very low dose rate of 36 nGyair s−1 (ref. 66). Dose–area product (DAP) rate is a figure of merit used to compare absorbed doses in real medical imaging67. DAP rate of around 2.5 Gyair cm2 h−1 is estimated for perovskite detector in interventional cardiology with X-rays continuously on for capturing an image (entrance area of 100 cm2) of the human body, at least an order of magnitude better than that offered by state-of-the-art commercial medical radiographic systems68,69. The resolution of X-ray image reached 10 lp mm−1, which is comparable to those of the best CZT X-ray detectors and CsI(Tl) scintillator detectors as shown in Table 1, and is still lower than α-Se flat panel imagers70. In this case, the lateral diffusion of carriers seemed not to affect the resolution of these detector arrays, despite that the crystals was still thick, maybe due to the relatively large electric field applied.

### 2D detector arrays

Low temperature growth of perovskite single crystals makes it compatible with the CMOS read-out circuitry, and promising for high-sensitivity 2D X-ray detector arrays. The first 2D X-ray detector array was made of perovskite polycrystalline films64. Similar to α-Se, polycrystalline perovskite films are actually attractive options for flat panel imagers, because they can be deposited directly on read-out circuitry with existing deposition techniques.

Doctor blading has been recently established to deposit halide perovskite films for solar cell modules with area close to 225 cm2 with module PCE of ~15%71. Spray coating is another mature technique to deposit thick films, and perovskite X-ray pixel detectors were first demonstrated in 2015 by employing spray-coated MAPbI3 polycrystalline films with thickness of 60 μm, which is thick enough to detect soft 8 keV X-rays17. Fullerene layers are often employed in these cases to block the pin holes to suppress leakage current37, and also passivate the surface charge traps on perovskites32. Recently, Kim et al.64 doctor-bladed 830-μm-thick MAPbI3 films on a 10 cm × 10 cm large TFT substrate64. Under irradiation from a 100kVp bremsstrahlung X-ray source, the 2D detector arrays presented good X-ray response under 50 V bias, showing clear X-ray image of a hand with bones observed inside in Fig. 3f. The image was acquired with 5 ms exposure of X-rays with dose rate of 5mGyair s−1, resulting in an exposed total dose of 25μGyair. The pixel size was around 70μm×70μm, while the resolution was only 3lpmm−1. This indicates that the resolution in these 2D arrays is still limited by the lateral charge diffusion among the crystal grains underneath neighboring pixel arrays, rather than the pixel size.

## Perovskite γ-ray detectors for photon energy spectrum

In contrast to X-rays, γ-rays have much higher energy and stronger penetrating capability. Fortunately, perovskite material has a strong stopping power to γ-rays with linear attenuation coefficient of 0.09 cm−1 at 662 keV (photon energy of 137Cs source), nearly two times larger than that of CdTe material46 (Fig. 1a and Table 2), and high-quality perovskite single crystals with centimeter size can be also easily grown from solution, which together enable sufficient stopping of γ-ray photons. Yakunin et al.72 firstly reported perovskite single crystal detectors toward γ-ray detection made by formamidinium lead triiodide (FAPbI3) single crystals. The crystals are shown in the inset of Fig. 4a, which have a large µτ product of 1 × 10−2 cm2 V−1, and the devices showing good response to 59.6 keV 241Am γ-ray photons at room temperature (Fig. 4b). However, the black phase of FAPbI3 single crystal is not thermodynamically stable at room temperature. Higher energy γ-ray spectrum cannot be acquired in this case, since a larger crystal and larger bulk resistivity for the crystal are needed for stronger stopping power and smaller dark current, respectively.

Good γ-ray detectors should avoid the polarization effect, which describes the phenomenon of continuous dark current drift and associated degradation of detector energy spectrum resolution73. The polarization effect is related to the ion migration in many γ-ray detecting semiconductors materials studied so far74. The dark current stability and energy spectrum stability of device can be monitored to evaluate the degree of the polarization effect, which is also a very common method in traditional radiation detectors, such as CZT and TlBr single crystals73,75,76. Improving the bulk crystal quality and electrode contact can partially suppress the polarization effect at room temperature, although low temperature will also freeze the mobile ions77. Ion migration has been confirmed in halide perovskite materials, and the grain boundaries in polycrystalline films are shown to be the dominating ion migration channels78. Nevertheless, polarization effect (or ion migration) can still occur in single crystals, depending on crystal quality, crystal composition, and applied electric field. For example, ion migration is often more notable in I-based perovskite than Br-based perovskites. Nazarenko et al. improved the FAPbI3 single crystal detector performance by alloying Cs+ ions and Br ions into bulk crystals to get mixture halide perovskite single crystal with composition of CsxFA1 − xPbI3 − yBry (x = 0–0.1, y = 0–0.6)79. Since Cs+ ions and Br ions can partially stabilize the black phase of FAPbI3 single crystal, these have less ion migration compared to FA+ ions and I ions at room temperature, respectively80,81,82. However, despite that the CsxFA1 − xPbI3 − yBry mixture perovskite single crystal showed µτ product of 0.12 cm2 V−1, neither the photopeak or the Compton edge were sharp or clear in the obtained spectrum79. No reason was given in this work, but we suspect the low bulk resistivity and large device noise/dark current in this crystals lead to a low signal-to-noise ratio.

Dopant compensation is a good strategy to increase the resistivity of MAPbBr3 by alloying n-type MAPbCl3 into p-type MAPbBr3 to reduce its self-doping level46. The MAPbBr3 − xClx perovskite single crystal gradually changed from p-type to n-type as increased Cl ions amount. MAPbBr2.94Cl0.06 perovskite single crystal is almost intrinsic and has the lowest charge carrier concentration among all the alloys. The alloying technique also increased the µτ product from 1.2×10−2 to 1.8×10−2 cm2 V−1. By suppressing the crystal surface/edge leakage current with a guard ring electrode83 (Fig. 4c), a ten-fold improved bulk resistivity to 3.6 × 109 Ω cm is revealed for the MAPbBr2.94Cl0.06 perovskite single crystals46. 137Cs γ-ray energy spectrum with all features was then acquired at room temperature, benefiting from the high signal-to-noise ratio. The best photopeak energy resolution reached 6.5% as shown in Fig. 4d, as the first demonstration of γ-ray energy spectrum collection capability by halide perovskite single crystals46. Recently, He and co-workers improved the quality of CsPbBr3 single crystals grown from melt (Fig. 4e) by slowing down the crystal growth speed with a two-stage cooling process, first 20 K h−1 for 20 h and then 5 K h−1 for another 35 h, which is critical parameter for crystal quality grown by high temperature Bridgman method48. Asymmetry electrodes of gold (Au) and gallium (Ga) serving as working electrodes on opposite surfaces of perovskite single crystals not only efficiently collect charges due to their work function difference43, the Schottky junction also suppressed device noise48. The crystals maintained a good µτ product of 1.34 × 10−3 cm2 V−1. 137Cs γ-ray spectrum with 662 keV photopeak was obtained with energy resolution of 3.9% using a thin crystal as shown in Fig. 4f, comparable to, or even better than, the commercial NaI(Tl) scintillator (See Table 2).

## Challenges and outlook

Halide perovskites have been demonstrated as a new generation of room temperature radiation detection materials in the past 5 years. Efforts are still needed to improve the sensitivity of X-ray detectors for imaging, because the lowest detectable dose rate by the best reported perovskite detectors is still much higher than the average background irradiation of 0.1nGyair s−1 in United States. Although increasing the working bias can enhance charge extraction in the detectors, it also increases noise. The dark current density of halide perovskite X-ray detectors is still higher than that of the commercial α-Se detectors. Therefore, the resistivity of perovskites needs further improvement based on new material design and composition. The background charge carrier concentration is still dominated by the self-doping effect in halide perovskites, which limits the lowest detectable X-ray dose rate. Further improvement can be expected if deep mechanism of self-doping is understood. Although these hybrid perovskite materials are more prone to degradation after exposure to heat or moisture than conventional detectors, it remains to be determined whether the environmental stability would impact the X-ray detectors, because perovskites will be encapsulated by X-ray penetration materials.

For spectroscopy applications, the state-of-the-art room temperature γ-ray detectors are made of CZT single crystals, which can achieve an energy resolution of around 0.5%, although it takes several decades for them to reach this high resolution. The performance of halide perovskite γ-ray detectors is far from this level. The relatively lower bulk resistivity is among one of the challenges to be addressed. The size of high-quality single crystals is not large yet, though there should be no fundamental limit for achieving larger size. The competition between charge collection and charge recombination in these big crystals is not understood yet. Long-term continuous operation of halide perovskite devices under large field to detect X-rays and γ-rays is still a major concern, since the polarization effect may be still notable at room temperature. Designing a good perovskite X-ray imager or spectrometer should consider all the required material properties as discussed that lead to a high signal-to-noise ratio and fast response. Considering the nature-abundant and low-cost raw materials with all the exciting discoveries in such a short time, halide perovskites still hold a big promise to compete with traditional ionizing radiation detection materials.


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# If k is an integer and x(x – k) = k + 1, what is the value of x?

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Re: If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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18 Jul 2016, 20:41
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Bunuel wrote:
If k is an integer and x(x – k) = k + 1, what is the value of x?

(1) x < k
(2) x = 3 – k

Kudos for a correct solution.

Given: x(x-k) = k + 1
x^2 - xk -k - 1 = 0
x^2 - 1 -k(x+1) = 0
(x-1)(x+1) - k (x+1) = 0
(x+1)(x-1-k) = 0 - (i)

Hence x = - 1 or x = 1+k

Required: x = ?

Statement 1: x < k
From this, we can say that ≠ k+1
Hence x can only take the value -1
SUFFICIENT

Statement 2: x = 3-k
On substituting the value of k in the solutions for the equation, we get
3-k = - 1 and 3-k = 1+k
Hence k = 4 and k = 1
INSUFFICIENT

Correct Option: A

A very good question which can trap you in choosing the option C

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Re: If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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18 Nov 2016, 08:48
Bunuel wrote:
If k is an integer and x(x – k) = k + 1, what is the value of x?

(1) x < k
(2) x = 3 – k

Kudos for a correct solution.

I picked A...
x<k
so we have few options:
1)
x negative
k negative
x-k = negative -> x*(x-k) = positive.
since k+1 must be positive, k must be either zero, or a non-integer. but since we assumed k is negative, then it doesn't work...

2)
x positive
k positive
x-k - negative
negative * positive = negative
it means that k+1 must be negative, meaning that k must be < -1. but we assumed k is positive. doesn't work...

3) we can't assume x positive and k negative, as we are told that x<k

4.
x negative
k positive

x-k - negative
multiply by x (negative) = positive
k+1 is positive.

let's test some values:
x=-1
k=1
-1(-1-1) = -1*-2 = 2 = 1+1 - works

x=-2
k=1
-2(-2-1) = 6
but k=1. so doesn't work

we can test more options, but it will not work unless x=-1. A is sufficient.

2. x=3-k
x(x – k) = k + 1
k=3-x

x(x-3+x) = 3-x+1
x(2x-3) = 4-x
2x^2 -3x = 4-x
2x^2 -2x -4 = 0 | divide by 2
x^2 -x -2 = 0
(x+1)(x-2)=0
x=2; x=-1

1st option
x=2
x = 3 – k
2=3-k
k=1
x(x – k) = k + 1 = 2(2-1) = 1+1
2=2

2nd option:
x=-1
x = 3 – k
-1 = 3-k
k=4
x(x – k) = k + 1 = 2(2-1) = 1+1
-1(-1-4) = 4+1 = 5=5
works
2 outcomes for x - not sufficient.

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If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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18 Nov 2016, 09:11
Bunuel wrote:
If k is an integer and x(x – k) = k + 1, what is the value of x?

(1) x < k
(2) x = 3 – k

Kudos for a correct solution.

From statement 1>
x(x-k) = k+1 => x^2-xk-k-1 = 0
=> x = (k+(k^2+4k+2^2)^1/2)/2 or x = (k- (k^2+4k+2^2)^1/2)/2
Since, k^2+4k+2^2 = (k+2)^2
By solving this we get
either x= k+1 or x= -1; but x <k, therefore x can't be k+1
Hence, x = -1 => Sufficient

From statement 2 >
we can solve quadratic for k and get values 2 values for x and k each. Therefore it is insufficient

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Re: If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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18 Nov 2016, 23:42
(x-1)*(x+1)-k(x+1)=0

(x+1)*(x-1-k)=0

x+1=0, x=-1
OR
x-1-k=0, x=k+1

St.1 says that second scenario is impossible, so x=-1

A

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If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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31 Mar 2017, 23:27
Bunuel wrote:
If k is an integer and x(x – k) = k + 1, what is the value of x?

(1) x < k
(2) x = 3 – k

Kudos for a correct solution.

statement 1:
x(x-k)=k+1
x^2-xk=k+1
x^2-1=k+xk
(x-1)(x+1)-(x+1)(k)=0
(x+1)[x-1-k]=0

this means x=-1; or x=k+1 not valid since x<k(given)
in sufficient as we cannot say anything about k when x=-1

statement 2:
putting x=3-k given 2 values x=-1;k=4
x=2;k=1
insufficient

taking both together:
x=-1;k=4
solution

give kudos if you liked this explanation

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If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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31 Mar 2017, 23:35
Engr2012 wrote:
Swaroopdev wrote:
Hi Bunuel,

None of the above solutions show A as the answer, is the OA wrong ? If not, could you please explain the solution ?

Thanks.

I think everyone is missing out on a very important information.

Lets analyse the question:

Given equation: x(x-k) = k+1 ----> $$x^2-kx-k-1=0$$ --->$$x = \frac {k\pm \sqrt{k^2-4(-k-1)}}{2}$$

Thus you get 2 values of x as =$$\frac {k\pm(k+2)}{2}$$ ---> x = k+1 or x=-1

Per statement 1, x<k ---> x $$\neq$$k+1 ---> x = -1 . Thus this statement IS SUFFICIENT

Per statement 2, x = 3-k ---> susbtituting in the above given equation, you get, k = 4 or 1 ---> x = -1 or 2. Thus this statement is NOT sufficient.

This question is a classic "C-Trap" question.

how can we conclude through statement A when x=-1 x<k
say x=-1 and k=-2 then also they satisfy the equation and
when x=-1 k=2 they also satisfy the equation

then how can we find the values where k>x always????????

thus answer is C when x=-1 k also takes unique value k=4

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Re: If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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31 Mar 2017, 23:38
OptimusPrepJanielle wrote:
Bunuel wrote:
If k is an integer and x(x – k) = k + 1, what is the value of x?

(1) x < k
(2) x = 3 – k

Kudos for a correct solution.

Given: x(x-k) = k + 1
x^2 - xk -k - 1 = 0
x^2 - 1 -k(x+1) = 0
(x-1)(x+1) - k (x+1) = 0
(x+1)(x-1-k) = 0 - (i)

Hence x = - 1 or x = 1+k

Required: x = ?

Statement 1: x < k
From this, we can say that ≠ k+1
Hence x can only take the value -1
SUFFICIENT

Statement 2: x = 3-k
On substituting the value of k in the solutions for the equation, we get
3-k = - 1 and 3-k = 1+k
Hence k = 4 and k = 1
INSUFFICIENT

Correct Option: A

A very good question which can trap you in choosing the option C

how can we conclude through statement A when x=-1 x<k
say x=-1 and k=-2 then also they satisfy the equation and
when x=-1 k=2 they also satisfy the equation

then how can we find the values where k>x always????????

thus answer is C when x=-1 k also takes unique value k=4

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If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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12 Apr 2017, 00:15
Bunuel, I was wondering about Statement 2. Using x = 3-k, we could calculate as follows:

(3-k) (3-k-k) = k+1 ==> k^2 - 5k + 4 = 0 ; k= 4, 1

Using K=4 in the original equation x(x-k) = k+1; we get:

x^2 - 4x - 5 =0 ==> x= 5 , -1

Using K=1 in the original equation x(x-k) = k+1; we get:

x^2 - x - 2 = 0 ==> x = 2 , -1

Since -1 is the only solution of x that satisfies both values of k in the original equation, can we not claim that statement 2 is sufficient?

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Re: If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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12 Apr 2017, 01:13
dg88074 wrote:
Bunuel, I was wondering about Statement 2. Using x = 3-k, we could calculate as follows:

(3-k) (3-k-k) = k+1 ==> k^2 - 5k + 4 = 0 ; k= 4, 1

Using K=4 in the original equation x(x-k) = k+1; we get:

x^2 - 4x - 5 =0 ==> x= 5 , -1

Using K=1 in the original equation x(x-k) = k+1; we get:

x^2 - x - 2 = 0 ==> x = 2 , -1

Since -1 is the only solution of x that satisfies both values of k in the original equation, can we not claim that statement 2 is sufficient?

We have: x(x – k) = k + 1 and x = 3 – k.
k = 1 and x = 2 OR k = 4 and x = -1. Both sets satisfy the given inequalities.
_________________

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Re: If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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12 Apr 2017, 02:13
Bunuel wrote:
dg88074 wrote:
Bunuel, I was wondering about Statement 2. Using x = 3-k, we could calculate as follows:

(3-k) (3-k-k) = k+1 ==> k^2 - 5k + 4 = 0 ; k= 4, 1

Using K=4 in the original equation x(x-k) = k+1; we get:

x^2 - 4x - 5 =0 ==> x= 5 , -1

Using K=1 in the original equation x(x-k) = k+1; we get:

x^2 - x - 2 = 0 ==> x = 2 , -1

Since -1 is the only solution of x that satisfies both values of k in the original equation, can we not claim that statement 2 is sufficient?

We have: x(x – k) = k + 1 and x = 3 – k.
k = 1 and x = 2 OR k = 4 and x = -1. Both sets satisfy the given inequalities.

Oh it makes sense now. Thanks a lot Bunuel

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Re: If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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21 Apr 2017, 15:18
Bunuel wrote:
If k is an integer and x(x – k) = k + 1, what is the value of x?

(1) x < k
(2) x = 3 – k

Kudos for a correct solution.

st1: x^2-xk=k+1
(x-1)(x+1)=K(x+1)
x+1= 0
x=-1, k=0
x=k+1( irrelevant since we know k>x)

st2: x=3-k
x=2, k=1 or x=-1, k=4 (insufficient on its own)

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Re: If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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07 Jul 2017, 05:34
smanujahrc wrote:
OptimusPrepJanielle wrote:
Bunuel wrote:
If k is an integer and x(x – k) = k + 1, what is the value of x?

(1) x < k
(2) x = 3 – k

Kudos for a correct solution.

Given: x(x-k) = k + 1
x^2 - xk -k - 1 = 0
x^2 - 1 -k(x+1) = 0
(x-1)(x+1) - k (x+1) = 0
(x+1)(x-1-k) = 0 - (i)

Hence x = - 1 or x = 1+k

Required: x = ?

Statement 1: x < k
From this, we can say that ≠ k+1
Hence x can only take the value -1
SUFFICIENT

Statement 2: x = 3-k
On substituting the value of k in the solutions for the equation, we get
3-k = - 1 and 3-k = 1+k
Hence k = 4 and k = 1
INSUFFICIENT

Correct Option: A

A very good question which can trap you in choosing the option C

how can we conclude through statement A when x=-1 x<k
say x=-1 and k=-2 then also they satisfy the equation and
when x=-1 k=2 they also satisfy the equation

then how can we find the values where k>x always????????

thus answer is C when x=-1 k also takes unique value k=4

You are given that x=-1 OR x=1+k,

Statement 1 says that x<k, so how can x=1+k be true? x=1+k will ALWAYS make x>k, thus going against the information mentioned in statement 1. Thus, you MUST reject x=1+k, leaving you with x=-1 as the only solution from statement 1. Hence this statement is sufficient.

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Re: If k is an integer and x(x – k) = k + 1, what is the value of x? [#permalink]

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16 Dec 2017, 05:20
Finally some sane question.
x(x-k) = k+1
=> x^2 - xk = k+1
=> X^2-1 - xk - k = 0
=> (x-1)(x+1) - k(x+1) = 0
=> (x+1((x-1-k)=0
POssible values x=-1, or, x=k+1

(1) x<k. Therefore, the only value is x=-1. Sufficient
(2) x=3-k.
Checking two possible values of x, x= k+1 => 3-k =k+1 => k=1 & x=2.
if x=-1, => 3-k=-1, => k=4, x=-1.
Both values of x are possible. Ambiguous. Insufficient

Hence A

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Re: If k is an integer and x(x – k) = k + 1, what is the value of x?   [#permalink] 16 Dec 2017, 05:20

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## Difference calculus with applications to difference equations.(English)Zbl 0592.39001

General inequalities 4, Mem. E. F. Beckenbach, 4th Int. Conf., Oberwolfach/Ger. 1983, ISNM 71, 95-110 (1984).
[For the entire collection see Zbl 0573.00004.]
The paper is primarily concerned with n-th order nonlinear difference equations of the form $\Delta^ nx(t)+\sum^{m}_{i=1}f_ i(t)F_ i(x(t),\Delta x(t),...,\Delta^{n-1}(t))=0,$ t being a nonnegative integer. Discrete analogues are presented of some known oscillation theorems for differential equations. The proofs are based on discrete versions of Taylor’s formula, l’Hospital’s rule and a theorem of Kneser which are derived in this paper.
Reviewer: B.Aulbach

### MSC:

 39A10 Additive difference equations 39A11 Stability of difference equations (MSC2000)

Zbl 0573.00004


text:
# How to use GLSL texelFetch?

I have read the OpenGl documentation of texelFetch, which is:

gvec4 texelFetch(gsampler2D sampler, ivec2 P, int lod).

I know the first argument "sampler" is just a texture. But I am very confused about the second and third arguments.

According to the documentation, the second argument "P" is a texture coordinate. But it has the type ivec2, which means that it is a vector of integers. Shouldn't a texture coordinate be in the range [0, 1]?

And for the third argument "lod", in the documentation, it said "If present, specifies the level-of-detail within the texture from which the texel will be fetched." I have no idea how to set this argument...

Could anyone explain this and give a real example of how to use texelFetch?

Thank you!

## 1 Answer

OK, I have figured out this problem.

For the texture coordinates, there are two kind of texture coordinates. One is normalized texture coordinates, which is in the range [0, 1]. The other is texel space, which is in the range [0, size), where size is the size of the texture. For texelFetch(), the texel space is used.

For lod, it means the level of detail in the mipmap. We can simply use 0 for the base level (the original size).

Thus, an example of using texelFetch() should be:

vec4 texelValue= texelFetch(texture2D, ivec2(gl_FragCoord.xy), 0);

• Many thanks to @NicolBolas who helped editting the format. I am new to StackOverflow. After doing another editting, I noticed that your name was gone... Didn't mean to do that – Yuchen Jun 5 '18 at 15:06
• Don't forget to also accept your answer! – bernie Jun 11 '18 at 14:56


text:
# A basic, modern, idiomatic GNU makefile

I have prepared a small makefile for a small project, but I am trying to find what would be the best practices; up to now I have gathered these:

• relatively idiomatic (typical structures, flows, var names for Least Astonishment)
• define the products in terms of the inputs (if I have .c files and want an exec, why should I think about intermediate files?)
• up-to-date with GNU make's and gcc's capabilities (auto-dependency tracking)
• silent by default
• but can be verbose (for Eclipse integration for example)
• keeps object files in directories separate from sources
• avoids implicit rules (explicit is clearer and anyway I wanted to mute the output)

Also a couple of make-quirk-addressing details like immediate assignment :=, order-only prerequisites…

So here's the makefile:

WARNINGS := -Wall -Wextra
CFLAGS ?= -std=gnu99 -g $(WARNINGS) OBJDIR := obj SRCDIR := src # silent by default ifeq ($(VERBOSE),1)
SILENCER :=
else
SILENCER := @
endif

# pass this environment variable to the C source
ifeq ($(DEBUG_BUILD),1) CFLAGS +=-DDEBUG_BUILD endif # typical way to list files and build full paths # list the sources, not the object files (nor includes) _SRCS := uthreads.c main.c SRCS :=$(patsubst %,$(SRCDIR)/%,$(_SRCS))
OBJS := $(patsubst %,$(OBJDIR)/%,$(_SRCS:c=o)) # generate phony deps during compilation CFLAGS += -MMD -MP DEPS :=$(patsubst %,$(OBJDIR)/%,$(_SRCS:c=d))
# can't include the deps before the first, default target has appeared!

# "all" is the classical default target
all: main

createdir:
$(SILENCER)mkdir -p$(OBJDIR)

main: $(OBJS) #@echo " LINK$^"
$(SILENCER)$(CC) $(CFLAGS) -o$@ $^ # put object and dependency files away from the sources # create the dir before building into it$(OBJDIR)/%.o: $(SRCDIR)/%.c | createdir #@echo " CC$<"
$(SILENCER)$(CC) $(CFLAGS) -c -o$@ $< clean:$(SILENCER)$(RM) -f *~ core main$(SILENCER)$(RM) -r$(OBJDIR)

.PHONY: clean all

# dependencies won't turn into a default target here
-include $(DEPS)  I'll be grateful for any criticism or addition to either the list of "best practices" or to the makefile itself. Thanks! ## 2 Answers You should be able to simplify the definition of DEPS to DEPS :=$(OBJS:.o=.d)

For small projects that don't require complicated configuration schemes, it's often times easiest to get the list of source files by wildcard instead of manually maintaining an explicit list:

SRCS := $(wildcard$(SRCDIR)/*.c)
OBJS := $(patsubst$(SRCDIR)/%.c,$(OBJDIR)/%.o,$(SRCS))
DEPS := $(OBJS:.o=.d)  That way, you won't have to change the makefile when you add, delete, or rename source files. Since all of your object files are going into the same output folder, you only need to create the output folder when it does not exist. You are currently calling mkdir for every C file that gets compiled. This won't break your build, but it's unnecessary overhead and doesn't scale well. You should be able to simplify to something like this: $(OBJDIR)/%.o: $(SRCDIR)/%.c #@echo " CC$<"
$(SILENCER)$(CC) $(CFLAGS) -c -o$@ $<$(OBJS): | $(OBJDIR)$(OBJDIR):
$(SILENCER)mkdir -p$(OBJDIR)

The second line of your recipe for "clean" should add the "-f" flag to avoid error messages when $(OBJDIR) doesn't exist. Generally good, clean and straightforward. Few notes however: • Too many comments. • Group CFLAGS assignments together. • I am kind of surprised that -g is set regardless of DEBUG_BUILD. • Speaking of debug, it usually helps to separate debug and release targets (different directories, different CFLAGS). • I recommend to abstain from relying on environment variables (or make command line arguments): you cannot express that the target depends on them. In other words, $ make
$export DEBUG_BUILD=1$ make
make: all is up to date

• The comments are mostly a "teaching aid", they won't stay - probably. It's interesting that your 3 points about debug builds do reinforce each other. I'll fix it. Thank you! – hmijail Sep 16 '15 at 18:28


text:
# [OS X TeX] Landscape mode

David Watson dewatson at mac.com
Sat May 24 20:12:49 CEST 2008

I seem to remember something about using
\usepackage{graphicx}
in order to get the driver set appropriately for page size and
orientation.

David

On May 24, 2008, at 12:33 PM, Holger Schulz wrote:

> I'm using TeXShop with TeXLive as a backend.
>
> I need to use landscape mode in a document. I do not use pdflatex
> with that but rather the way via dvi, ps since I use PSTricks in
> this document.
>
> In this constellation the class option landscape seems not to have
> an effect. Thpugh my document starts with
>
> \documentclass[12pt,landscape]{article}
>
> It is still displayed in portrait mode.
>
> Is the an easy way to use lnadscape mode?
>
> Thanks
>
> hs
>


text:
## Intermediate Algebra for College Students (7th Edition)

The set of rational numbers is the set of all numbers that can be expressed as the ratio of two integers, with the denominator never being $0$.


text:
## What Is the Exchange Ratio?

The exchange ratio is the relative number of new shares that will be given to existing shareholders of a company that has been acquired or that has merged with another. After the old company shares have been delivered, the exchange ratio is used to give shareholders the same relative value in new shares of the merged entity.

## The Formula for the Exchange Ratio Is

\begin{aligned} &\text{Exchange Ratio} = \frac{ \text{Target Share Price} }{ \text{Acquirer Share Price} } \\ \end{aligned}

## What Does the Exchange Ratio Tell You?

An exchange ratio is designed to give shareholders the amount of stock in an acquirer company that maintains the same relative value of the stock the shareholder held in the target, or acquired company. The target company share price is typically increased by the amount of a "takeover premium," or an additional amount of money an acquirer pays for the right to buy 100% of the company's outstanding shares and have a 100% controlling interest in the company.

Relative value does not mean, however, that the shareholder receives the same number of shares or same dollar value based on current prices. Instead, the intrinsic value of the shares and the underlying value of the company are considered when coming up with an exchange ratio.

### Key Takeaways

• The exchange ratio calculates how many shares an acquiring company needs to issue for each share an investor owns in a target or acquired, company, to provide the same relative value to the investor.
• The target company purchase price often includes a price premium paid by the acquirer due to buying control of 100% of the target's stock shares.

## Example of How to Use the Exchange Ratio

The exchange ratio in a merger and acquisition is the opposite of a fixed value deal in which a buyer offers a dollar amount to the seller, meaning that the number of shares or other assets backing the dollar value can fluctuate in an exchange ratio.

For example, imagine that the buyer offers the seller two shares of the buyer's company in exchange for 1 share of the seller's company. Prior to the announcement of the deal, the buyer's or acquirer's shares may be trading at $10, while the seller's or target's shares trade at$15. Due to the 2 to 1 exchange ratio, the buyer is effectively offering $20 for a seller share that is trading at$15.

Fixed exchange ratios are usually limited by caps and floors to reflect extreme changes in stock prices. Caps and floors prevent the seller from receiving significantly less consideration than anticipated, and they likewise prevent the buyer from giving up significantly more consideration than anticipated. Exchange ratios can also be accompanied by a cash component in a merger or acquisition, depending on the preferences of the companies involved in the deal.

### Investment Implications

Post announcement of a deal, there is usually a gap in valuation between the seller's and buyer's shares to reflect the time value of money and risks. Some of these risks include the deal being blocked by the government, shareholder disapproval, or extreme changes in markets or economies.

Taking advantage of the gap, believing that the deal will go through, is referred to as merger arbitrage and is practiced by hedge funds and other investors. Leveraging the example above, assume that the buyer's shares stay at $10 and the seller's shares jump to$18. There will be a $2 gap that investors can secure by buying one seller share for$18 and shorting two buyer shares for $20. If the deal closes, investors will receive two buyer shares in exchange for one seller share, closing out the short position and leaving investors with$20 in cash. Minus the initial outlay of $18, investors will net$2.


text:
# Tag Info

## Hot answers tagged total

6

Don't subtract dividends; add them. Add-back the dividends as if they had not been paid out. That will ensure that you have a positive price when deriving the returns. For example, MSFT paid a 0.16 dividend on 2011-15-02. Here are the raw prices, according to Yahoo: date close ---------------- 2011.02.01 27.99 2011.02.02 27.94 2011.02.03 27.65 ...

3

Not sure I understand your question. If I have a fixed stream of payments it has some value $V_{fixed}$ I can always solve for a spread to LIBOR by simply adding the spread $S$ to my calculated stream of LIBOR. That is the value of the LIBOR + spread leg is $$V_{LIBOR}(S) = \sum_{n=1}^{N} D(t_{n}) \alpha(t_{n-1},t_{n}) [L(t_{n-1},t_{n}) + S]$$ where ...

3

I think the answer is in your question. Yahoo uses a percentage adjustment for adjusted close prices. So this is the procedure I would do. 1) Calculate the proper return for each day (taking into account splits and divs). 2) Apply the returns going backwards from the current price. By doing it this way it is impossible to get a negative adjusted price, ...

3

The problem is, you're calculating the "return on investment" or "return on original investment". Anytime you are given back ALL or MORE THAN your original investment, you are no longer "invested". As a result, calculating "return on investment" no longer applies. For example with a 100% return of capital, the "original investment" drops to zero. So, ...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
# Automated procedure for selecting subset of data points w/ strongest correlation?

Is there some standard procedure (such that one might cite it as a reference) for selecting the subset of data points from a larger pool with the strongest correlation (along just two dimensions)?

For instance, say you have 100 data points. You want a subset of 40 points with the strongest correlation possible along the X and Y dimensions.

I realize that writing code to do this would be relatively straightforward, but I'm wondering if there's any source to cite for it?

• "I realize that writing code to do this would be relatively straightforward". Ah? And how would you do that? – user603 Jul 10 '12 at 23:03
• I suppose she meant something like "best subset correlation"; select subsets of $k$ ($k=40$ in her example) data points out of your $N$ ($N=100$ in her example) and calculate the estimate of the correlation $\rho(X,Y)$ (assuming that she meant to know a subset of points with the best linear correlation). However, this process seems computationally expensive for large $N$, because you have to calculate $\binom{N}{k}$ times the coefficient. – Néstor Jul 10 '12 at 23:48
• If you're willing to look at linear combinations of the $X$ variables, canonical correlations is what you're looking for. Otherwise, correlation feature selection might be of interest. – MånsT Jul 11 '12 at 6:15
• I think some may be misunderstanding me. @Néstor seems to have it right. There are 100 items, each with an X value and a Y value. I want to find the subset of 40 that have the strongest correlation possible (w/ linear regression) between the X and Y values. I can write code to explore the entire search space, but what would I cite to support such a method? What is it called to find the optimal correlation among all possible subsets? – Julie Jul 13 '12 at 14:27
• Are you interested in maximizing the correlation or getting the best fit regression line as, for example, measured by minimum residual variance? The two are not the same when you get to choose your data points. – jbowman Jul 13 '12 at 22:10

I would say that your method fits into the general category described in this wikipedia article which also has other references if you need something more than just wikipedia. Some of the links within that article would also apply.

Other terms that could apply (if you want to do some more searching) include "Data Dredging" and "Torturing the data until it confesses".

Note that you can always get a correlation of 1 if you just choose 2 points that don't have identical x or y values. There was an article in Chance magazine a few years back that showed when you have an x and y variable with essentially no correlation you can find a way to bin the x's and average the y's within the bins to show either an increasing or decreasing trend (Chance 2006, Visual Revelations: Finding What Is Not There through the Unfortunate binning of Results: The Mendel Effect, pp. 49-52). Also with a full dataset showing a moderate positive correlation it is possible to choose a subset that shows a negative correlation. Given these, even if you have a legitimate reason for doing what you propose, you are giving any skeptics a lot of arguments to use against any conclusions that you come up with.

• What's the name of the article from The American Statistician? – assumednormal Jul 16 '12 at 21:07
• I misremembered where I saw the article, it was actually in Chance Magazine rather than The American Statistician. I have corrected that above and included the year, title, and page numbers so that interested parties should be able to find copies easily. – Greg Snow Jul 17 '12 at 17:03

The RANSAC algorithm sounds like what you want. Basically, it assumes your data consists of a mix of inliers and outliers, and tries to identify the inliers by repeatedly sampling subsets of the data, fitting a model to it, then trying to fit every other data point to the model. Here's the wikipedia article about it.

In your case, you can just keep repeating the algorithm while saving the current best model that fits at least 40 points, so it won't guarantee you the absolute best correlation, but it should get close.

I have a hard time imagining a context in which this would be good practice, but lets assume for a moment that you indeed have a good reason for doing this.

A brute force algorithm could be something like this:

1. You calculate all possible sub-samples of n out of your overall sample of N. Most statistical packages have functions for calculating combinations without replacements that will do this for you.

2. You estimate the correlation between x and y for each one of the sub-samples and select the maximum out of that set.

I just saw the original poster's comment regarding a reference for this procedure. I am not sure that someone has a specific name for this procedure after all you are simply generating an empirical distribution of all possible correlation in your dataset and selecting the maximum. Similar approaches are used when doing bootstraping, but in that case you are interested in the empirical variability, you DO NOT use them to pick a specific sub-sample associated with the max.

• I presume you have access to the $10^{32}$ or so CPU cycles needed to solve the problem for $N=100$ and $n=40$? (That would be only around a million years if you could harness every PC in the world full time. :-) – whuber Jul 16 '12 at 19:54
• No need to be snarky about it :-p. Fair point. – David Jul 16 '12 at 20:15
• Sorry... I like those figures, though, because they give us lots of room for an improved algorithm :-). – whuber Jul 16 '12 at 20:21


text:
MSc research

This page lists various projects during my Masters degrees (2010-2012) at University of Warwick.

### RSK correspondence and Pitman’s Theorem

My MASDOC MSc dissertation, supervised by Professor Neil O’Connell, is a survey focused on a variety of RSK correspondences in combinatorics and Pitman’s theorems in probability and their interconnections. The dissertation can be downloaded here.

### Simulating spiral waves on spherical surfaces using surface finite element method

This is the RSG implementation stage project I worked on. This is joint work with Amal Alphonse and Simon Bignold supervised by Professor Dwight Barkley and Dr Andreas Dedner. The objective is to simulate the following equation:

\begin{align*} \dot u + u\nabla_\Gamma\cdot\mathbf{v} - a\Delta_\Gamma u &= \frac{1}{\epsilon}u(1-u)(u-\frac{v+b}{c}); \text{ in }\mathcal{G}_t:= \bigcup _{t\in[0,T]}\Gamma_t\times\{t\}\\ \dot v+v\nabla_\Gamma\cdot\mathbf{v}&=u-v, \text{ in }\mathcal{G}_t \\ u(\cdot,0)&=u_0,\quad v(\cdot,0)=v_0,\text{ in }\Gamma_0.\end{align*}

where $$\dot u$$ and $$\dot v$$ denotes the material derivative of $$u$$ and $$v$$ and $$\mathbf{v}$$ is the surface velocity. We implement (using Distributed and Unified Numerics Environment, DUNE) this equation on both fixed and deforming surfaces. Below is a simulation on a deforming surface and a simulation of pulse,

The details of these simulation can be found on the project webpage. You can also download the research report here.

### Path-dependent option pricing using Monte Carlo method

This is the RSG (Research Study Group) proposal stage topic I worked on. For definition and description of RSG see here and here (Warwick login required). This is joint work with Owen Daniel and Barnaby Garrod, supervised by Dr Aleksandar Mijatović. The objective is to compute the price $$V(0)$$ of European path-dependent options like Asian, lookback or Parisian options:

\begin{equation*} V(0)=\mathbb{E}_xe^{-rT}V(X,T),\end{equation*}

where the $$V(\cdot,T)$$ is a functional on $$C([0,T])$$ and $$\mathbb{E}_x$$ is the risk-neutral measure starting with $$x$$. In finance, $$X$$ stands for underlying price of the option, $$V(X,T)$$ is the payoff of the option at expiry $$T$$. We proposed direct sampling and an importance sampling method to approximate the law of $$(X_{t})_{0\le t\le T}$$, and hence to approximate the expectation. The proposal can be downloaded here, whose implementation was passed to another group whose members are Chin Lun, Abhishek Shukla and Matthew Thorpe.

### Stochastic travelling wave solutions of stochastic reaction-diffusion equations

This is my first masters project, supervised by Dr Roger Tribe. Consider two similar 1-dimensional stochastic parabolic equations:

$du_{t}=\frac{d^2u_{t}}{dx^2}dt+f(u_{t})dt+g(u_{t})\circ dB_{t}.\qquad(1)$ $du_{t}=\frac{d^2u_{t}}{dx^2}dt+f(u_{t})dt+g(u_{t})dW_{t}(x).\qquad(2)$

Where $$f,g\in C^3[0,1]$$ with $$f(0)=f(1)=g(0)=g(1)=0$$, the noise $$dB_{t}$$ is a Brownian motion noise and $$dW_{t}(x)$$ is space-time white noise. Equation (1) has stochastic travelling wave solutions (Tribe and Woodward 2011) when $$f$$ is stable(which means $$f>0$$ on $$(0,1)$$), bistable($$f<0$$ on $$(0,a)$$ and $$f>0$$ on $$(a,1)$$ for some $$a$$) or unstable($$f>0$$ on $$(0,a)$$ and $$f<0$$ on $$(a,1)$$ for some $$a$$). My objective was to find the asymptotic wave speeds of stochastic travelling wave solutions of (1) and (2) and to prove they are independent of $$g$$.


text:
# Given two angle and a segment, can you find $h$?

by b00n heT   Last Updated September 11, 2019 18:20 PM - source

While designing another problem I came up with the following question:

Considering the picture below, can you determine $$h$$ based only on the information of $$\alpha$$, $$\beta$$ and $$x$$ given the fact that the segment $$x$$ is the continuation of the height?

My guess is that the problem is not solvable (in the sense that one can construct multiple $$h$$ with the same $$\alpha,\beta$$ and $$x$$), but cannot see how to show this.

So far I gave the problem a try by splitting the angle and combining some trigonometric identities but couldn't conclude.

So far I only obtained the following \begin{align*} \tan(\alpha)=\frac{(h_1+h_2)(x+y)}{(x+y)^2-h_1h_2} \end{align*} and $$\tan(\beta)=\frac{(h_1+h_2)\cdot y}{y^2-h_1h_2}.$$


text:
An inverse of the evaluation functional for typed lambda-calculus
Ulrich Berger and Helmut Schwichtenberg

Abstract
In any model of the typed lambda-calculus contaning some basic arithmetic, a functional is defined which inverts the evaluation functional for typed lambda-terms. Combined with the evaluation functional, this yields an efficient normalization algorithm. The method is extended to lambda-calculi with constants and is used to normalize (the lambda-representations of) natural deduction proofs of (higher order) arithmetic. A consequence of theoretical interest is a strong completeness theorem for beta-eta-reduction, generalizing results of Friedman and Statman: If two lambda-terms have the same value in some model containing representations of the primitive recursive functions (of level 1) then they are provably equal in the beta-eta-calculus.

Bibtex entry

@Inproceedings{Berger91,
Author = "Ulrich Berger and Helmut Schwichtenberg",
Title = "An inverse of the evaluation functional for
typed $\lambda$--calculus",
Booktitle = "Proceedings of the Sixth Annual IEEE
Symposium on Logic in Computer Science",
Editor = "R. Vemuri",
Pages = "203--211",
Publisher = "IEEE Computer Society Press, Los Alamitos",
Year = 1991}


text:
refit

Refit neighborhood component analysis (NCA) model for classification

Syntax

mdlrefit = refit(mdl,Name,Value)

Description

mdlrefit = refit(mdl,Name,Value) refits the model mdl, with modified parameters specified by one or more Name,Value pair arguments.

Input Arguments

expand all

Neighborhood component analysis model or classification, specified as a FeatureSelectionNCAClassification object.

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Fitting Options

expand all

Method for fitting the model, specified as the comma-separated pair consisting of 'FitMethod' and one of the following.

• 'exact' — Performs fitting using all of the data.

• 'none' — No fitting. Use this option to evaluate the generalization error of the NCA model using the initial feature weights supplied in the call to fscnca.

• 'average' — The function divides the data into partitions (subsets), fits each partition using the exact method, and returns the average of the feature weights. You can specify the number of partitions using the NumPartitions name-value pair argument.

Example: 'FitMethod','none'

Regularization parameter, specified as the comma-separated pair consisting of 'Lambda' and a non-negative scalar value.

For n observations, the best Lambda value that minimizes the generalization error of the NCA model is expected to be a multiple of 1/n

Example: 'Lambda',0.01

Data Types: double | single

Solver type for estimating feature weights, specified as the comma-separated pair consisting of 'Solver' and one of the following.

• 'lbfgs' — Limited memory BFGS (Broyden-Fletcher-Goldfarb-Shanno) algorithm (LBFGS algorithm)

• 'sgd' — Stochastic gradient descent

• 'minibatch-lbfgs' — Stochastic gradient descent with LBFGS algorithm applied to mini-batches

Example: 'solver','minibatch-lbfgs'

Initial feature weights, specified as the comma-separated pair consisting of 'InitialFeatureWeights' and a p-by-1 vector of real positive scalar values.

Data Types: double | single

Indicator for verbosity level for the convergence summary display, specified as the comma-separated pair consisting of 'Verbose' and one of the following.

• 0 — No convergence summary

• 1 — Convergence summary including iteration number, norm of the gradient, and objective function value.

• >1 — More convergence information depending on the fitting algorithm

When using solver 'minibatch-lbfgs' and verbosity level >1, the convergence information includes iteration log from intermediate minibatch LBFGS fits.

Example: 'Verbose',2

Data Types: double | single

LBFGS or Mini-Batch LBFGS Options

expand all

Relative convergence tolerance on the gradient norm for solver lbfgs, specified as the comma-separated pair consisting of 'GradientTolerance' and a positive real scalar value.

Data Types: double | single

SGD or Mini-Batch LBFGS Options

expand all

Initial learning rate for solver sgd, specified as the comma-separated pair consisting of 'InitialLearningRate' and a positive scalar value.

When using solver type 'sgd', the learning rate decays over iterations starting with the value specified for 'InitialLearningRate'.

Example: 'InitialLearningRate',0.8

Data Types: double | single

Maximum number of passes for solver 'sgd' (stochastic gradient descent), specified as the comma-separated pair consisting of 'PassLimit' and a positive integer. Every pass processes size(mdl.X,1) observations.

Example: 'PassLimit',10

Data Types: double | single

SGD or LBFGS or Mini-Batch LBFGS Options

expand all

Maximum number of iterations, specified as the comma-separated pair consisting of 'IterationLimit' and a positive integer.

Example: 'IterationLimit',250

Data Types: double | single

Output Arguments

expand all

Neighborhood component analysis model for classification, returned as a FeatureSelectionNCAClassification object. You can either save the results as a new model or update the existing model as mdl = refit(mdl,Name,Value).

Examples

expand all

Generate checkerboard data using the generateCheckerBoardData.m function.

rng(2016,'twister'); % For reproducibility
pps = 1375;
[X,y] = generateCheckerBoardData(pps);
X = X + 2;

Plot the data.

figure
plot(X(y==1,1),X(y==1,2),'rx')
hold on
plot(X(y==-1,1),X(y==-1,2),'bx')

[n,p] = size(X)
n =

22000

p =

2

Add irrelevant predictors to the data.

Q = 98;
Xrnd = unifrnd(0,4,n,Q);
Xobs = [X,Xrnd];

This piece of code creates 98 additional predictors, all uniformly distributed between 0 and 4.

Partition the data into training and test sets. To create stratified partitions, so that each partition has similar proportion of classes, use y instead of length(y) as the partitioning criteria.

cvp = cvpartition(y,'holdout',2000);

cvpartition randomly chooses 2000 of the observations to add to the test set and the rest of the data to add to the training set. Create the training and validation sets using the assignments stored in the cvpartition object cvp .

Xtrain = Xobs(cvp.training(1),:);
ytrain = y(cvp.training(1),:);

Xval = Xobs(cvp.test(1),:);
yval = y(cvp.test(1),:);

Compute the misclassification error without feature selection.

nca = fscnca(Xtrain,ytrain,'FitMethod','none','Standardize',true, ...
'Solver','lbfgs');
loss_nofs = loss(nca,Xval,yval)
loss_nofs =

0.5165

'FitMethod','none' option uses the default weights (all 1s), which means all features are equally important.

This time, perform feature selection using neighborhood component analysis for classification, with .

w0 = rand(100,1);
n = length(ytrain)
lambda = 1/n;
nca = refit(nca,'InitialFeatureWeights',w0,'FitMethod','exact', ...
'Lambda',lambda,'solver','sgd');
n =

20000

Plot the objective function value versus the iteration number.

figure()
plot(nca.FitInfo.Iteration,nca.FitInfo.Objective,'ro')
hold on
plot(nca.FitInfo.Iteration,movmean(nca.FitInfo.Objective,10),'k.-')
xlabel('Iteration number')
ylabel('Objective value')

Compute the misclassification error with feature selection.

loss_withfs = loss(nca,Xval,yval)
loss_withfs =

0.0115

Plot the selected features.

figure
semilogx(nca.FeatureWeights,'ro')
xlabel('Feature index')
ylabel('Feature weight')
grid on

Select features using the feature weights and a relative threshold.

tol = 0.15;
selidx = find(nca.FeatureWeights > tol*max(1,max(nca.FeatureWeights)))
selidx =

1
2

Feature selection improves the results and fscnca detects the correct two features as relevant.


text:
Benford's Law

Very similar to Zipf’s Law, which focused on the relationship of letters and words. Benford’s Law looks at the frequency of numbers starting with a particular number.

$P_{D}=\frac{\int_{D}^{D+1}P(x)dx}{\int_{1}^{10}P(x)dx}=log_{10}\left ( 1+\frac{1}{D} \right )$

Distribution:

Written on January 10, 2023


text:
# $\alpha$ decay to more than one nuclear state

I do not understand how $\alpha$ decay can be a probabilistic process such that there are multiple products from the decay. For example:

$^{241}\mathrm{Cm}$ decays to the excited states of $^{237}\mathrm{Pu}$, *

0.202MeV, spin: 5/2 with probability 13%,

0.125MeV, spin 3.2 with probability 17%

and 0.145MeV, spin 1/2 with probability 70%

Also there is no decay to the $^{237}\mathrm{Pu}$ ground state.

I dont not understand how the decay has this choice?

* class notes

-

The simplest way to think of it is to note that the nucleons in a heavy nucleus are arranged in a shell structure not unlike the electrons in an atom, and when you want to form that alpha particle you have to grab two protons and two neutrons to do it.

They do not have to be the highest energy ones.

If you pull some of them out of low lying shells you leave a hole in the remnant nucleus which means that it is in an excited state. This process is not unlike what happens in the production of characteristic atomic x-rays.

Now, there are some limits. In particular energy conservation set an absolute limit on how excited the remnant can be.

There are also correlations between the nucleons in the parent nucleus that contribute to the selection process (imprecisely some pairs of nucleons are "near" each other more often than others).

-
Ok I think I understand, all that I'm unsure of now is for the example decay, (which is only from class notes, but I will assume to be true) why the ground state cannot be reached by the decay. When you grab two protons and neutrons, why can you not take the highest energy ones and thus be in the $^{237}Pm$ ground state –  user57142 Feb 20 '13 at 21:26
Hmmm. Curium-241 has spin-parity of $\frac{1}{2}^{+}$; the ground state of Pu-237 has $\frac{7}{2}^-$, and the alpha is $0^+$. To make the angular momentum work the alpha would need to be emitted with a orbital angular momentum of 3 which is parity negative and would seem to be allowed. Can't say. –  dmckee Feb 20 '13 at 21:39
BTW, the daughter nucleus has got to be plutonium and not promethium. –  dmckee Feb 20 '13 at 21:41


text:
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# A square region has an area of b square inches and a perimeter of n in

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A square region has an area of b square inches and a perimeter of n in  [#permalink]

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31 Oct 2018, 01:06
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A square region has an area of b square inches and a perimeter of n inches. If n = b/3 what is the length, in inches, of the side of the square?

A. 12
B. 12√2
C. 48
D. 48√2
E. 144

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Re: A square region has an area of b square inches and a perimeter of n in  [#permalink]

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31 Oct 2018, 01:16
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Let the side of a square be 'a'. Given the area of square $$a^2 = b$$

Now perimeter of the square is $$4a = n$$

Given, $$n = \frac{b}{3}$$

$$4a = \frac{a^2}{3}$$

$$a = 12$$

OPTION : A
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A square region has an area of b square inches and a perimeter of n in  [#permalink]

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Updated on: 31 Oct 2018, 02:01
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side of square = $$\sqrt{b}$$

Perimeter of a square = 4*side = 4 * $$\sqrt{b}$$

Given the perimeter = n

then 4*$$\sqrt{b}$$ = $$\frac{b}{3}$$

then it becomes $$12 = \frac{b}{\sqrt{b}}$$

squaring both sides we get $$144 = \frac{b^2}{b}$$

this then equals 144 = b

b = 144 is the area

Taking the square root of 144 = 12

Side is then 12.

Thanks to eswarchethu135 for pointing out

Originally posted by Salsanousi on 31 Oct 2018, 01:21.
Last edited by Salsanousi on 31 Oct 2018, 02:01, edited 1 time in total.
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Re: A square region has an area of b square inches and a perimeter of n in  [#permalink]

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31 Oct 2018, 01:24
hi eswarchethu135

I think you started off with the wrong equation $$a^2 =$$b it should be $$area = b$$

so it means area = $$\sqrt{b} * \sqrt{b}$$ = b
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Re: A square region has an area of b square inches and a perimeter of n in  [#permalink]

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31 Oct 2018, 01:39
Hi Salsanousi

I took the side of a square as 'a', then the area will be $$a^2$$. Now the area is equal to 'b'. So $$a^2 = b$$. Question said the area of a square is 'b'. However, I don't find any mistake in your approach too. Will wait for OA
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Re: A square region has an area of b square inches and a perimeter of n in  [#permalink]

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31 Oct 2018, 01:58
Hi eswarchethu135 You're right! what I found was actually the area not the side ^_^

So I should take the square root of b to find the side.

Thank you!
Re: A square region has an area of b square inches and a perimeter of n in &nbs [#permalink] 31 Oct 2018, 01:58
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True or false?

Algebra Level 2

Consider the function $$f(x) = \ln(x+1)$$ on the range $$0 < x < 1$$.

True or false: $$f(x) > x$$ for all $$x$$ in this range.

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# Intersection of $\left(A_i\cup B_i\right)$ for two disjoint sets of events

Let $\{A_1,A_2\dots A_n\}$ be a mutually disjoint set of events. Let $\{B_1,B_2\dots B_n\}$ be another mutually disjoint set of events. I am trying to transform the expression $$C=\bigcap\limits_{i\in\{1,2,\dots,n\}}\left(A_i\cup B_i\right)$$ to something equivalent by somehow taking the union symbol out. I think that for $n=2$ it's $$C=(A_1\cap B_2)\cup(B_1\cap A_2)$$ but I cannot generalize it for any $n$.

• @layman That's true but as I stated $\{A_1,A_2\dots A_n\}$ are mutually disjoint. This holds for $\{B_1,B_2\dots B_n\}$, as well. So $(A_1\cap A_2)=\emptyset$ and $(B_1\cap B_2)=\emptyset$
– mgus
Nov 2 '17 at 19:44

I renamed your variables $A_{1,i}$ and $A_{2,i}$ $$C=\bigcap\limits_{i\in\{1,2,\dots,n\}}\left(A_{1,i}\cup A_{2,i}\right)$$
By generalized distributivity $$C=\bigcup\limits_{i_1\in\{1,2\}} \cdots \bigcup\limits_{i_n\in\{1,2\}} \left(\bigcap\limits_{j\in \{1,2,\cdots,n\}} A_{i_j,j} \right)$$
$$C=\bigcup\limits_{(i_1,\cdots,i_n)\in\{1,2\}^n} \left(\bigcap\limits_{j\in \{1,2,\cdots,n\}} A_{i_j,j} \right)$$