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# Triangle A has an area of 12  and two sides of lengths 8  and 7 . Triangle B is similar to triangle A and has a side of length 5 . What are the maximum and minimum possible areas of triangle B?

Jan 23, 2018

Case - Minimum Area :

$D 1 = \textcolor{red}{{D}_{\min}} = \textcolor{red}{1.3513}$

Case - Maximum Area :

$D 1 = \textcolor{g r e e n}{{D}_{\max}} = \textcolor{g r e e n}{370.3704}$

#### Explanation:

Let the two similar triangles be ABC & DEF.

Three sides of the two triangles be a,b,c & d,e,f and the areas A1 & D1.

Since the triangles are similar,

$\frac{a}{d} = \frac{b}{e} = \frac{c}{f}$

Also $\frac{A 1}{D 1} = {a}^{2} / {d}^{2} = {b}^{2} / {e}^{2} = {c}^{2} / {f}^{2}$

Property of a triangle is sum of any two sides must be greater than the third side.

Using this property, we can arrive at the minimum and maximum value of the third side of triangle ABC.

Maximum length of third side $c < 8 + 7$ , say 14.9 (corrected upto one decimal.

When proportional to maximum length, we get minimum area.

Case - Minimum Area :

$D 1 = \textcolor{red}{{D}_{\min}} = A 1 \cdot {\left(\frac{f}{c}\right)}^{2} = 12 \cdot {\left(\frac{5}{14.9}\right)}^{2} = \textcolor{red}{1.3513}$

Minimum length of third side $c > 8 - 7$ , say 0.9 (corrected upto one decimal.

When proportional to minimum length, we get maximum area.

Case - Maximum Area :

$D 1 = \textcolor{g r e e n}{{D}_{\max}} = A 1 \cdot {\left(\frac{f}{c}\right)}^{2} = 12 \cdot {\left(\frac{5}{0.9}\right)}^{2} = \textcolor{g r e e n}{370.3704}$


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# Dictionary Definition

angle

### Noun

1 the space between two lines or planes that intersect; the inclination of one line to another; measured in degrees or radians
2 a biased way of looking at or presenting something [syn: slant]
3 a member of a Germanic people who conquered England and merged with the Saxons and Jutes to become Anglo-Saxons

### Verb

1 move or proceed at an angle; "he angled his way into the room"
2 to incline or bend from a vertical position; "She leaned over the banister" [syn: lean, tilt, tip, slant]
3 seek indirectly; "fish for compliments" [syn: fish]
4 fish with a hook
5 present with a bias; "He biased his presentation so as to please the share holders" [syn: slant, weight]

# User Contributed Dictionary

see Angle

## English

### Etymology

From angle, from angulus.

### Noun

1. A figure formed by two rays which start from a common point (a plane angle) or by three planes that intersect (a solid angle).
the angle between lines A and B
2. The measure of such a figure. In the case of a plane angle, this is the ratio (or proportional to the ratio) of the arc length to the radius of a section of a circle cut by the two rays, centered at their common point. In the case of a solid angle, this is the ratio of the surface area to the square of the radius of the section of a sphere.
The angle between lines A and B is π/4 radians, or 45 degrees.
3. A corner where two walls intersect.
an angle of a building
4. A change in direction.
The horse took off at an angle.
5. A viewpoint.
Look at it from this angle.
6. The focus of a news story, either in print or broadcasting.
7. slang professional wrestling A storyline between two wrestlers, providing the background for and approach to a feud.
8. A scheme; a means of benefitting from a situation, usually hidden, possibly illegal.
His angle is that he gets a percentage, but mostly in trade.

#### Translations

geometrical figure
• Catalan: angle
• Czech: úhel
• Danish: vinkel
• Dutch: hoek
• Esperanto: angulo
• Finnish: kulma
• French: angle
• German: Winkel
• Greek: γωνία
• Hebrew:
• Hungarian: szög
• Icelandic: horn
• Italian: angolo
• Japanese: (かく, kaku), 角度 (かくど, kakudo)
• Korean: (, gak)
• Lithuanian: kampas
• Polish: kąt
• Portuguese: ângulo
• Russian: угол
• Romanian: unghi
• Serbian: ugao, kut
• Slovak: uhol
• Slovene: kot
• Spanish: ángulo
• Swedish: vinkel , plan vinkel , rymdvinkel (solid angle)
• Telugu: కోణం
• Vietnamese: gốc
measure of such a figure
• Catalan: angle
• Danish: vinkel
• Dutch: hoek
• Esperanto: angulo
• French: angle
• German: Winkel
• Greek: γωνία (ɣonía)
• Italian: angolo
• Japanese: 角度 (かくど, kakudo)
• Korean: (, gak)
• Lithuanian: kampas
• Polish: kąt
• Portuguese: ângulo
• Romanian: unghi
• Russian: угол
• Slovene: kot
• Spanish: ángulo
• Swedish: vinkel
• Telugu: కోణం
corner where two walls intersect
• Catalan: racó , cantonada
• Danish: hjørne
• Dutch: hoek
• Esperanto: angulo
• German: Ecke
• Greek: γωνία, κόγχη
• Hebrew: ,
• Icelandic: horn
• Japanese: (かく, kaku), (すみ, sumi)
• Korean: (gwi), 모서리 (moseori)
• Polish: kąt, róg
• Portuguese: esquina
• Romanian: colt
• Russian угол (úgol)
• Spanish: esquina (as seen from the outside), rincón (as seen from the inside)
• Swedish: hörn , knut (as seen from the outside)
change in direction
viewpoint

### Verb

1. To place (something) at an angle.
The roof is angled at 15 degrees.
2. To try to catch fish with a hook and line.
3. intransitive informal To change direction rapidly.
The five ball angled off the nine ball but failed to reach the pocket.

#### Translations

place (something) at an angle
fish with hook and line

# Extensive Definition

In order to measure an angle θ, a circular arc centered at the vertex of the angle is drawn, e.g. with a pair of compasses. The length of the arc s is then divided by the radius of the circle r, and possibly multiplied by a scaling constant k (which depends on the units of measurement that are chosen):
\theta = \frac(k).
The value of θ thus defined is independent of the size of the circle: if the length of the radius is changed then the arc length changes in the same proportion, so the ratio s/r is unaltered.
In many geometrical situations, angles that differ by an exact multiple of a full circle are effectively equivalent (it makes no difference how many times a line is rotated through a full circle because it always ends up in the same place). However, this is not always the case. For example, when tracing a curve such as a spiral using polar coordinates, an extra full turn gives rise to a quite different point on the curve.

### Units

Angles are considered dimensionless, since they are defined as the ratio of lengths. There are, however, several units used to measure angles, depending on the choice of the constant k in the formula above. Of these units, treated in more detail below, the degree and the radian are by far the most common.
With the notable exception of the radian, most units of angular measurement are defined such that one full circle (i.e. one revolution) is equal to n units, for some whole number n. For example, in the case of degrees, A full circle of n units is obtained by setting in the formula above. (Proof. The formula above can be rewritten as One full circle, for which units, corresponds to an arc equal in length to the circle's circumference, which is 2πr, so . Substituting n for θ and 2πr for s in the formula, results in )
• The degree, denoted by a small superscript circle (°) is 1/360 of a full circle, so one full circle is 360°. One advantage of this old sexagesimal subunit is that many angles common in simple geometry are measured as a whole number of degrees. (The goal of having all "interesting" angles measured as whole numbers is of course unattainable.) Fractions of a degree may be written in normal decimal notation (e.g. 3.5° for three and a half degrees), but the following sexagesimal subunits of the "degree-minute-second" system are also in use, especially for geographical coordinates and in astronomy and ballistics:
• The minute of arc (or MOA, arcminute, or just minute) is 1/60 of a degree. It is denoted by a single prime ( ′ ). For example, 3° 30′ is equal to 3 + 30/60 degrees, or 3.5 degrees. A mixed format with decimal fractions is also sometimes used, e.g. 3° 5.72′ = 3 + 5.72/60 degrees. A nautical mile was historically defined as a minute of arc along a great circle of the Earth.
• The second of arc (or arcsecond, or just second) is 1/60 of a minute of arc and 1/3600 of a degree. It is denoted by a double prime ( ″ ). For example, 3° 7′ 30″ is equal to 3 + 7/60 + 30/3600 degrees, or 3.125 degrees.
• The mil is approximately equal to a milliradian. There are several definitions.
• The full circle (or revolution, rotation, full turn or cycle) is one complete revolution. The revolution and rotation are abbreviated rev and rot, respectively, but just r in rpm (revolutions per minute). 1 full circle = 360° = 2π rad = 400 gon = 4 right angles.
• The angle of the equilateral triangle is 1/6 of a full circle. It was the unit used by the Babylonians, and is especially easy to construct with ruler and compasses. The degree, minute of arc and second of arc are sexagesimal subunits of the Babylonian unit. 1 Babylonian unit = 60° = π/3 rad ≈ 1.047197551 rad.
• The grad, also called grade, gradian, or gon is 1/400 of a full circle, so one full circle is 400 grads and a right angle is 100 grads. It is a decimal subunit of the right angle. A kilometer was historically defined as a centi-gon of arc along a great circle of the Earth, so the kilometer is the decimal analog to the sexagesimal nautical mile. The gon is used mostly in triangulation.
• The point, used in navigation, is 1/32 of a full circle. It is a binary subunit of the full circle. Naming all 32 points on a compass rose is called "boxing the compass". 1 point = 1/8 of a right angle = 11.25° = 12.5 gon.
• The astronomical hour angle is 1/24 of a full circle. The sexagesimal subunits were called minute of time and second of time (even though they are units of angle). 1 hour = 15° = π/12 rad = 1/6 right angle ≈ 16.667 gon.
• The binary degree, also known as the binary radian (or brad), is 1/256 of a full circle. The binary degree is used in computing so that an angle can be efficiently represented in a single byte.
• The grade of a slope, or gradient, is not truly an angle measure (unless it is explicitly given in degrees, as is occasionally the case). Instead it is equal to the tangent of the angle, or sometimes the sine. Gradients are often expressed as a percentage. For the usual small values encountered (less than 5%), the grade of a slope is approximately the measure of an angle in radians.

### Positive and negative angles

A convention universally adopted in mathematical writing is that angles given a sign are positive angles if measured counterclockwise, and negative angles if measured clockwise, from a given line. If no line is specified, it can be assumed to be the x-axis in the Cartesian plane. In many geometrical situations a negative angle of −θ is effectively equivalent to a positive angle of "one full rotation less θ". For example, a clockwise rotation of 45° (that is, an angle of −45°) is often effectively equivalent to a counterclockwise rotation of 360° − 45° (that is, an angle of 315°).
In three dimensional geometry, "clockwise" and "counterclockwise" have no absolute meaning, so the direction of positive and negative angles must be defined relative to some reference, which is typically a vector passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie.
In navigation, bearings are measured from north, increasing clockwise, so a bearing of 45 degrees is north-east. Negative bearings are not used in navigation, so north-west is 315 degrees.

### Approximations

• 1° is approximately the width of a little finger at arm's length
• 10° is approximately the width of a closed fist at arm's length.
• 20° is approximately the width of a handspan at arm's length.

## Identifying angles

In mathematical expressions, it is common to use Greek letters (α, β, γ, θ, φ, ...) to serve as variables standing for the size of some angle. (To avoid confusion with its other meaning, the symbol π is not used for this purpose.) Lower case roman letters (a, b, c, ...) are also used. See the figures in this article for examples.
In geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB and AC (i.e. the lines from point A to point B and point A to point C) is denoted ∠BAC or BÂC. Sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex ("angle A").
Potentially, an angle denoted, say, ∠BAC might refer to any of four angles: the clockwise angle from B to C, the anticlockwise angle from B to C, the clockwise angle from C to B, or the anticlockwise angle from C to B, where the direction in which the angle is measured determines its sign (see Positive and negative angles). However, in many geometrical situations it is obvious from context that the positive angle less than or equal to 180° degrees is meant, and no ambiguity arises. Otherwise, a convention may be adopted so that ∠BAC always refers to the anticlockwise (positive) angle from B to C, and ∠CAB to the anticlockwise (positive) angle from C to B.

## Types of angles

• An angle of 90° (π/2 radians, or one-quarter of the full circle) is called a right angle.
Two lines that form a right angle are said to be perpendicular or orthogonal.
• Angles smaller than a right angle (less than 90°) are called acute angles ("acute" meaning "sharp").
• Angles larger than a right angle and smaller than two right angles (between 90° and 180°) are called obtuse angles ("obtuse" meaning "blunt").
• Angles equal to two right angles (180°) are called straight angles.
• Angles larger than two right angles but less than a full circle (between 180° and 360°) are called reflex angles.
• Angles that have the same measure are said to be congruent.
• Two angles opposite each other, formed by two intersecting straight lines that form an "X" like shape, are called vertical angles or opposite angles. These angles are congruent.
• Angles that share a common vertex and edge but do not share any interior points are called adjacent angles.
• Two angles that sum to one right angle (90°) are called complementary angles.
The difference between an angle and a right angle is termed the complement of the angle.
• Two angles that sum to a straight angle (180°) are called supplementary angles.
The difference between an angle and a straight angle is termed the supplement of the angle.
• Two angles that sum to one full circle (360°) are called explementary angles or conjugate angles.
• An angle that is part of a simple polygon is called an interior angle if it lies in the inside of that the simple polygon. Note that in a simple polygon that is concave, at least one interior angle exceeds 180°.
In Euclidean geometry, the measures of the interior angles of a triangle add up to π radians, or 180°; the measures of the interior angles of a simple quadrilateral add up to 2π radians, or 360°. In general, the measures of the interior angles of a simple polygon with n sides add up to [(n − 2) × π] radians, or [(n − 2) × 180]°.
• The angle supplementary to the interior angle is called the exterior angle. It measures the amount of "turn" one has to make at this vertex to trace out the polygon. If the corresponding interior angle exceeds 180°, the exterior angle should be considered negative. Even in a non-simple polygon it may be possible to define the exterior angle, but one will have to pick an orientation of the plane (or surface) to decide the sign of the exterior angle measure.
In Euclidean geometry, the sum of the exterior angles of a simple polygon will be 360°, one full turn.
• Some authors use the name exterior angle of a simple polygon to simply mean the explementary (not supplementary!) of the interior angle http://mathworld.wolfram.com/ExteriorAngle.html. This conflicts with the above usage.
• The angle between two planes (such as two adjacent faces of a polyhedron) is called a dihedral angle. It may be defined as the acute angle between two lines normal to the planes.
• The angle between a plane and an intersecting straight line is equal to ninety degrees minus the angle between the intersecting line and the line that goes through the point of intersection and is normal to the plane.
• If a straight transversal line intersects two parallel lines, corresponding (alternate) angles at the two points of intersection are congruent; adjacent angles are supplementary (that is, their measures add to π radians, or 180°).

## A formal definition

### Using trigonometric functions

A Euclidean angle is completely determined by the corresponding right triangle. In particular, if \theta is a Euclidean angle, it is true that
\cos \theta = \frac
and
\sin \theta = \frac
for two numbers x and y. So an angle in the Euclidean plane can be legitimately given by two numbers x and y.
To the ratio \frac there correspond two angles in the geometric range 0 , since
\frac = \frac = \frac = \frac = \frac.

### Using rotations

Suppose we have two unit vectors \vec and \vec in the euclidean plane \mathbb^2. Then there exists one positive isometry (a rotation), and one only, from \mathbb^2 to \mathbb^2 that maps u onto v. Let r be such a rotation. Then the relation \vec\mathcal\vec defined by \vec=r(\vec) is an equivalence relation and we call angle of the rotation r the equivalence class \mathbb/\mathcal, where \mathbb denotes the unit circle of \mathbb^2. The angle between two vectors will simply be the angle of the rotation that maps one onto the other. We have no numerical way of determining an angle yet. To do this, we choose the vector (1,0), then for any point M on \mathbb at distance \theta from (1,0) (on the circle), let \vec=\overrightarrow. If we call r_\theta the rotation that transforms (1,0) into \vec, then \left[r_\theta\right]\mapsto\theta is a bijection, which means we can identify any angle with a number between 0 and 2\pi.

## Angles between curves

The angle between a line and a curve (mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the tangents at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—amphicyrtic (Gr. μφί, on both sides, κυρτόσ, convex) or cissoidal (Gr. κισσόσ, ivy), biconvex; xystroidal or sistroidal (Gr. ξυστρίσ, a tool for scraping), concavo-convex; amphicoelic (Gr. κοίλη, a hollow) or angulus lunularis, biconcave.

## The dot product and generalisation

In the Euclidean plane, the angle θ between two vectors u and v is related to their dot product and their lengths by the formula
\mathbf \cdot \mathbf = \cos(\theta)\ \|\mathbf\|\ \|\mathbf\|.
This allows one to define angles in any real inner product space, replacing the Euclidean dot product · by the Hilbert space inner product .

## Angles in Riemannian geometry

In Riemannian geometry, the metric tensor is used to define the angle between two tangents. Where U and V are tangent vectors and gij are the components of the metric tensor G,
\cos \theta = \frac .

## Angles in geography and astronomy

In geography we specify the location of any point on the Earth using a Geographic coordinate system. This system specifies the latitude and longitude of any location, in terms of angles subtended at the centre of the Earth, using the equator and (usually) the Greenwich meridian as references.
In astronomy, we similarly specify a given point on the celestial sphere using any of several Astronomical coordinate systems, where the references vary according to the particular system.
Astronomers can also measure the angular separation of two stars by imagining two lines through the centre of the Earth, each intersecting one of the stars. The angle between those lines can be measured, and is the angular separation between the two stars.
Astronomers also measure the apparent size of objects. For example, the full moon has an angular measurement of approximately 0.5°, when viewed from Earth. One could say, "The Moon subtends an angle of half a degree." The small-angle formula can be used to convert such an angular measurement into a distance/size ratio.

## References

• .
• Preliminary">http://www.utexas.edu/cola/centers/lrc/ielex/PokornyMaster-X.html.

angle in Afrikaans: Hoek (meetkunde)
angle in Tosk Albanian: Winkel (Geometrie)
angle in Arabic: زاوية(هندسة)
angle in Asturian: Ángulu
angle in Guarani: Takamby
angle in Aymara: K'uchu
angle in Bengali: সমকোণ
angle in Bosnian: Ugao
angle in Bulgarian: Ъгъл
angle in Catalan: Angle
angle in Czech: Úhel
angle in Danish: Vinkel
angle in German: Winkel
angle in Modern Greek (1453-): Γωνία
angle in Spanish: Ángulo
angle in Esperanto: Angulo
angle in Persian: زاویه
angle in French: Angle
angle in Scottish Gaelic: Ceàrn (Matamataig)
angle in Galician: Ángulo
angle in Classical Chinese: 角
angle in Korean: 각도
angle in Croatian: Kut
angle in Ido: Angulo
angle in Indonesian: Sudut (geometri)
angle in Icelandic: Bogagráða
angle in Italian: Angolo
angle in Hebrew: זווית
angle in Swahili (macrolanguage): Pembe (jiometria)
angle in Haitian: Ang
angle in Latvian: Leņķis
angle in Lithuanian: Kampas
angle in Hungarian: Szög
angle in Macedonian: Агол
angle in Marathi: कोन
angle in Malay (macrolanguage): Sudut
angle in Dutch: Hoek (meetkunde)
angle in Japanese: 角度
angle in Polish: Kąt płaski
angle in Portuguese: Ângulo
angle in Romanian: Ângulo
angle in Quechua: Chhuka
angle in Russian: Угол
angle in Simple English: Angle
angle in Slovak: Uhol
angle in Slovenian: Kot
angle in Serbian: Угао
angle in Finnish: Kulma
angle in Swedish: Vinkel
angle in Tamil: கோணம்
angle in Thai: มุม
angle in Vietnamese: Góc
angle in Turkish: Açı
angle in Ukrainian: Кут
angle in Chinese: 角

# Synonyms, Antonyms and Related Words

Anschauung, L, action, aim, anagnorisis, angle for, angle of vision, angle off, apex, approach, architectonics, architecture, argument, aspect, atmosphere, background, bait the hook, base, basis, be after, bear off, bend, bias, bifurcate, bifurcation, bight, bob, bow, branch, brew, cabal, cant, catastrophe, characterization, chevron, clam, coign of vantage, coin, collude, color, complication, complot, concoct, configuration, connive, conspire, continuity, contrivance, contrive, cook up, corner, countermine, counterplot, crank, crook, crotchet, cusp, dap, deflect, deflection, denouement, design, detail, development, deviate, device, dib, dibble, direction, distance, divagate, diverge, dogleg, drive, edge, effect, eidolon, elbow, ell, engineer, episode, eye, fable, facet, falling action, fashion, feature, figure, finagle, finesse, fish, fish for, flection, flexure, fly-fish, footing, fork, form, frame, frame of reference, frame up, framework, furcate, furcation, gerrymander, gestalt, gig, gimmick, go fishing, grig, ground, guddle, guise, hand, hatch, hatch a plot, hatch up, hook, hunt for, image, imago, impression, incident, inflection, intersection, intrigue, item, jack, jacklight, jig, jockey, knee, lay a plot, light, likeness, line, lineaments, local color, look, look for, machinate, maneuver, manipulate, manner, mental outlook, mood, motif, movement, mythos, net, nook, oblique, oblique angle, operate, outlook, particular, peripeteia, perspective, phase, phasis, place, plan, play games, plot, point, point of view, position, post, projection, pull strings, quoin, recognition, reference, reference system, regard, respect, rig, right angle, rising action, salient angle, scheme, seat, secondary plot, seek, seeming, seine, semblance, shape, sheer, shrimp, side, sight, simulacrum, situation, skew, slant, slue, spin, stand, standing, standpoint, station, status, still-fish, story, structure, style, subject, subplot, sway, swerve, switch, system, thematic development, theme, tone, topic, torch, total effect, trawl, troll, try for, turn, turning, twist, universe, vantage, vantage point, veer, venue, vertex, view, viewpoint, wangle, whale, wise, zag, zig, zigzag

format: plain


text:
請問這句意思 an expectation of break

A global rollout would be even more attractive with an expectation of breakeven within the first few months

with an expectation of breakeven???

Rating

第一季的幾個月中，在收支平衡的期待下，全球首展會更具吸引力。

with an expectation of breakeven

~在收支平衡的期待下

Source(s): 淡大英文系生

全球性首次展示是更加有吸引力的與收支平衡的期望在與收支平衡的期望的最初的少數月内？ ？ ？

有問題可到Yahoo!奇摩翻譯找答案

希望可以幫到ㄋ


text:
paper2.pdf (334.5 kB)

Cyclic Lattices, Ideal Lattices and Bounds for the Smoothing Parameter

preprint
posted on 06.01.2022, 05:43 authored by Zhiyong Zheng, Yunfan Lu
Cyclic lattices and ideal lattices were introduced by Micciancio in \cite{D2}, Lyubashevsky and Micciancio in \cite{L1} respectively, which play an efficient role in Ajtai's construction of a collision resistant Hash function (see \cite{M1} and \cite{M2}) and in Gentry's construction of fully homomorphic encryption (see \cite{G}). Let $R=Z[x]/\langle \phi(x)\rangle$ be a quotient ring of the integer coefficients polynomials ring, Lyubashevsky and Micciancio regarded an ideal lattice as the correspondence of an ideal of $R$, but they neither explain how to extend this definition to whole Euclidean space $\mathbb{R}^n$, nor exhibit the relationship of cyclic lattices and ideal lattices.
In this paper, we regard the cyclic lattices and ideal lattices as the correspondences of finitely generated $R$-modules, so that we may show that ideal lattices are actually a special subclass of cyclic lattices, namely, cyclic integer lattices. In fact, there is a one to one correspondence between cyclic lattices in $\mathbb{R}^n$ and finitely generated $R$-modules (see Theorem \ref{th4} below). On the other hand, since $R$ is a Noether ring, each ideal of $R$ is a finitely generated $R$-module, so it is natural and reasonable to regard ideal lattices as a special subclass of cyclic lattices (see corollary \ref{co3.4} below). It is worth noting that we use more general rotation matrix here, so our definition and results on cyclic lattices and ideal lattices are more general forms. As application, we provide cyclic lattice with an explicit and countable upper bound for the smoothing parameter (see Theorem \ref{th5} below). It is an open problem that is the shortest vector problem on cyclic lattice NP-hard? (see \cite{D2}). Our results may be viewed as a substantial progress in this direction.

History

liu_fx@ruc.edu.cn

Submitting Author's Institution

renmin university of china

China

Exports

figshare. credit for all your research.


text:
# Tag Info

10

You may want to first broadly categorize volatility models before comparing between them within each class, it does not make sense to compare standard deviation models with an implied vol model. I would broadly classify as follows: Historical realized volatility: Those include standard deviation (sum of squared deviations), realized range volatility ...

5

The best paper is probably Relative Volume as a Doubly Stochastic Binomial Point Process - James Mcculloch. In this paper the volume is modelled via a Point Process, and theoretical laws are derived (with confident intervals, etc). And if you can wait few days (it will be available very soon), we put elements about this in Market Microstructure in Practice, ...

4

Working on trigonometric polynomial decomposition, the first step is to take a big look at Fourier transformation. It is very powerfull, well documented and probably well implemented on your favorite language. It will give you the decomposition of your time series. You can remove highest frequencies, which correspond to noise, to have a good estimation.

3

Building upon +Imorin answer, you should have a look specifically at discrete cosine transforms. It's a standard approach when trying to express finite sequences as a sum of cosines. I would start from there, especially as it's implemented in every common language (R, Matlab, Python for starters). Only then evaluate if you need more.

3

There are tons of quant related blogs out there, some of which contain relatively sophisticated content, others less so. Have a look at the following, which aggregates blogs: MoneyScience Otherwise I could point you to bank/sell-side research. Have a look at the freely available Reuters Messenger (RM), they maintain channels where you can be permissioned ...

3

You could read it like this: The typical change in equity value is equal to the typical change in asset value, adjusted for the probability of the assets surviving. Note that the formula is not specific to Merton models, it's also true for regular options and their underlyings. It's just that volatility of option prices isn't typically a concern in ...

3

Well, the main intuition of the Merton model is that a company's equity can be treated as a call option on its assets, thus allowing for the application of Black-Scholes option pricing methods. Let's consider a company that has assets $A_{t}$ financed by equity $E_{t}$ and a zero-coupon debt $B_{t}$ with face value K, and maturity T. At time of maturity T, ...

2

What you are talking about is called regression using fractional polynomials and it has its merits. The canonical reference is this one: Regression Using Fractional Polynomials of Continuous Covariates: Parsimonious Parametric Modelling by Royston and Altman (1994) From the abstract: The relationship between a response variable and one or more ...

2

See edit and comments, this response might not be applicable to the question: When performing regression you would tend to want your regressors to be of similar type, or at the very least range. Assuming you use log return for price changes I would recommend using the untransformed interest rate. The reason for this is that they are the same type of entity, ...

2

The equation stated in the question is not at the core of Merton's credit model, (Not saying you claimed it is) but is a simple device in helping to solve the system of linear equations. The equation given simply establishes a relationship between the volatility of equity and the volatility of the assets and it follows from the application of Black Scholes ...

2

I deal recently with some analysis of the Volume time series, daily volume in € for European stocks. I found out that an ARIMA model works well. But, some EWMA could also provide good forecast if it's well parameterized. You can also face some seasonality effect due to macroeconomic events, some you may need to clean you data and treat these days in a ...

2

In the Ljung-Box test, the null hypothesis is: $H_0$: The data are independently distributed So, your p-values of 0 indeed indicate that you should reject the null hypothesis, but it means that your data is not independently distributed, and in particular that there is some significant autocorrelation in the process. This is obviously the case, because ...

2

I have a little experience with this. First, NASDAQ has shared a dataset with researchers that flags whether an HFT participated in each trade or not but not the actual MPID - probably less granular than what you want. You generally need a professor to "cosign" your request, write a brief project proposal, and sign an NDA to get it. They also have shared ...

1

From my point of view, dynamic models like the one developped in Relative Volume as a Doubly Stochastic Binomial Point Process - James Mcculloch to provide a dynamic forecast of the volume does not improve significantly the forecasting comparing to a static volume curve forecast using historical data (last month intraday data, and an EWMA algorithm). I've ...

1

Market participant ID data is extremely unlikely to be available without the collaboration of regulators and the exchange itself, as it is a closely guarded information. Even "anonymized" data with no reference to a specific firm could reveal private information to informed market participants. If obtained at all, it is likely to come with draconian ...

1

Saying that you can't analyze something as is does not make it garbage. You can't eat flour "as-is", but that doesn't mean you throw it out. In order to use "standard" analysis tools, you must first transform the series into something compatible. Some examples of such a transformation include k-th order differences or a log transformation. These ...

1

When you ask for "expected economic growth" you need to be specific for what you are asking for. I assume you mean GDP. But what "economic growth" is depends on who you are talking to. In my business (real estate finance, more specifically class A multifamily and industrial in 1st tier cities) economic growth has been spectacular in the past two years with ...

1

I can' read your lm lines but I think this works: mymodel = lm(sales~year+I(year^2)) plot(sales) lines(fitted.values(mymodel)) Or you try just mymodel = lm(sales~I(year^2)) Finally new.data = list(year=2004) predict.lm(mymodel,new.data) gives a useful value.

1

I would not call them "estimators" but rather measures because this is volatility on historical returns. The reason why there are different such measures is because each one represents volatility in different ways in order to measure the volatility of different dynamics, be it price, return, upside price moves only, down side price moves only, intraday ...

1

Economically, the interest rate should be stationary. Unlike a price series, where a price of \$10 may not have had the same meaning for a given stock many years ago as it does today, an interest rate of 10% always means the same thing. Hence I side with Andre's earlier answer that you should use the untransformed interest rate. Also, you need to think more ...

1

It exists several techniques to deal with mixed-frequency data. I believe MIxed DAta Sampling is the best-known. Eg: bridge equation, MIxed DAta Sampling (MIDAS) models Mixed frequency VARs Mixed frequency factor models ... Here is a good document on this topic: A survey of econometric methods for mixed- frequency data

1

Try the following : perform the logarithmic transformation of the volume data. check if the transformed data fits the normal distribution nicely. if you are working with intraday volume, then adjust for the seasonality for time of the day effect, if using daily data, in some cases some special seasonalities like expiry day, etc might be applied but it may ...

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


text:
# Give me a four, give me a two

Algebra Level 5

Let $$x,y,z$$ be positive real numbers such that: \begin{align} xyz&=945 \\ x(y+1)+y(z+1)+z(x+1)&=385. \end{align} If the minimum possible value of $$z+\frac{y}{2}+\frac{x}{4}$$ can be written as $$\frac{a}{b}$$, where $$a$$ and $$b$$ are coprime positive integers. What is the value of $$a+b$$?

×


text:
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# Two-dimensional electronic spectroscopy reveals liquid-like lineshape dynamics in CsPbI3 perovskite nanocrystals

## Abstract

Lead-halide perovskites have attracted tremendous attention, initially for their performance in thin film photovoltaics, and more recently for a variety of remarkable optical properties. Defect tolerance through polaron formation within the ionic lattice is a key aspect of these materials. Polaron formation arises from the dynamical coupling of atomic fluctuations to electronic states. Measuring the properties of these fluctuations is therefore essential in light of potential optoelectronic applications. Here we apply two-dimensional electronic spectroscopy (2DES) to probe the timescale and amplitude of the electronic gap correlations in CsPbI3 perovskite nanocrystals via homogeneous lineshape dynamics. The 2DES data reveal irreversible, diffusive dynamics that are qualitatively inconsistent with the coherent dynamics in covalent solids such as CdSe quantum dots. In contrast, these dynamics are consistent with liquid-like structural dynamics on the 100 femtosecond timescale. These dynamics are assigned to the optical signature of polaron formation, the conceptual solid-state analogue of solvation.

## Introduction

Lead-halide perovskites of the type APbX3, where A is an inorganic or organic cation and X is a halide, have recently attracted significant attention for their potential in optoelectronic applications, in particular photovoltaics1,2,3,4. In addition to a large absorption cross section in the visible range, high carrier mobility, and low non-radiative recombination rates, lead-halide perovskites also possess structural properties that qualitatively distinguish them from other inorganic semiconductors. It is the interplay of these electronic and structural properties that give rise to the performance of lead-halide perovskites5. How the structural properties impact the optoelectronic performance, and the role of dynamic disorder in protecting the optical excitation from scattering processes, is a topic of intense investigation.

In order to understand the relationship between structure and function in lead-halide perovskites, a variety of methods have recently been applied. Collectively, these methods reveal a picture consistent with structural dynamics following photoexcitation6,7,8,9,10,11,12. The dynamics have been assigned to polaron formation and described in terms of a phonon glass with a localized ultrafast response to optical excitation6,7. Polaron formation arises from the dynamical coupling of atomic structure fluctuations to electronic states13,14. This results in a fluctuating electronic gap. Understanding the properties of these fluctuations is essential since it dictates the electronic response of interest for optoelectronic applications. Previous dynamical probes have focused on the terahertz and Raman responses, and have shown how specific phonons are involved in the polaron formation process6,7,10,11,12. For the time-resolved electronic response, one can measure transient-absorption (TA) spectra in the visible region15. With two-dimensional electronic spectroscopy, one can additionally measure lineshape dynamics.

Here we employ 2DES to measure the homogeneous lineshape dynamics of CsPbI3 nanocrystals. These dynamics reveal an isomorphism between the lineshape dynamics of a molecular dye undergoing solvation dynamics and that of CsPbI3 nanocrystals. We assign the observed dynamics in the perovskites to polaron formation, as a conceptual analogue of solvation. By virtue of spreading the optical response on a 2D correlation map, 2DES clearly distinguishes sources of static and dynamic disorder with ~10 fs time resolution. These lineshape dynamics are unique to the 2DES method and cannot be accessed by one-dimensional methods, such as TA spectroscopies. Our measurement yields polaron formation time and binding energy, providing a complement to previous experimental studies and a direct point of comparison for future ab initio works. At low fluences, nanocrystals enable working in the single excitation per particle regime, thereby minimizing effects arising from electron–electron or hole–hole scattering processes16. The use of nanocrystalline perovskites also enables comparison with model systems such as covalent polar nanocrystal quantum dots and molecular dyes in solution. Our data show that the homogeneous linewidth of the CsPbI3 nanocrystals evolves on the 100-fs timescale in a manner consistent with ultrafast polar solvation dynamics, contrasting with the response of covalent CdSe nanocrystal quantum dots. We demonstrate excellent agreement of the experimental perovskite data with modeled data for dissipative, collective reorganization of the lattice, which enables us to assign the observed spectral dynamics to polaron formation.

## Results

### Static measurements

CsPbI3 nanocrystals were synthesized following previously described methods2,17. Details are provided in the Methods section. Figure 1 displays an overview of the basic properties of these CsPbI3 nanocrystals. The linear absorption spectrum is shown in Fig. 1a, together with the photoluminescence spectrum. The nanocrystals show weakly confined excitons, by virtue of their large size relative to the Bohr radius. For smaller sizes, one finds an excitonic spectrum consistent with the strongly confined quantum dot regime2,18. A typical transmission electron microscope image is shown in the inset, from which we estimate that the cubic-shaped CsPbI3 nanocrystals have an average size of 9.8 nm (Supplementary Fig. 2). A static X-ray diffraction pattern is shown in Fig. 1b. The Bragg peaks are broad due to the small size of the nanocrystals19. These simple properties are consistent with a variety of lead-halide perosvkite nanocrystals1,20,21. Figure 1c, d displays cartoons of phonons (c) and a polaron (d), both of which are expected to play important roles in electron relaxation following photoexcitation.

### 2DES measurements

In order to directly probe the electronic structure and dynamics of the CsPbI3 nanocrystals, we employ the 2DES pulse sequence shown in Fig. 2a. We refer the reader to references for a more detailed overview of the technique22,23. Implementation details are provided in the Methods section and elsewhere24. Figure 2b, c displays two representative correlation maps obtained on the CsPbI3 perovskite nanocrystals, at t2 = 20 and 600 fs. The main qualitative observation is the transformation of the 2D lineshape from elliptical to circular. This behavior is representative of spectral diffusion, a classic example being solvation dynamics25,26,27. The timescale of lineshape dynamics informs on how fast the environment responds to the charge rearrangement resulting from the optical excitation. Figure 2d, e illustrates the basic concepts of spectral diffusion. Figure 2d shows the time-dependent frequency gap, ω + δω(t2), where δω(t2) represents the equilibrium fluctuations from the mean transition frequency. The charge rearrangement due to a single optical excitation is a perturbation to the equilibrium charge distribution. Hence, by the fluctuation–dissipation theorem, the nonequilibrium response of the environment is given by the correlation function of the equilibrium fluctuations28. The equilibration of the electronically excited state is measured via the frequency–frequency correlation function (FFCF), <δω(t2)δω(0)>, which decays with the environment’s response time constant τc, as shown in Fig. 2e. The FFCF relates to the experimental observable of the time-dependent energy gap in 1D spectroscopy. In 2D spectroscopy, one is able to observe diagonal and anti-diagonal linewidths that reflect the process of spectral diffusion with less ambiguity.

### Comparative analysis

In order to place the spectral dynamics of the CsPbI3 perovskites in terms of model systems at two qualitative limits, we conduct a comparative analysis on CdSe nanocrystals and a molecular dye in solution. The key results are summarized in Fig. 3. Figure 3a–c displays a series of spectral projections as a function of t2, obtained from the 2D spectra by selecting a value of E1 (pump energy) and reporting the spectral projection as a function of E3 (probe energy). These datasets can be interpreted in a manner similar to a TA spectrum. Figure 3a shows that the early-time dynamics in the CsPbI3 nanocrystals are dominated by continuous energy relaxation. These data are in stark contrast with the behavior of the CdSe nanocrystals, shown in Fig. 3b, where the spectral dynamics are well known to arise from relaxation within a discrete manifold of states29. Furthermore, the observed dynamics for CsPbI3 are slow compared with the molecule in solution, shown in Fig. 3c, where a similar continuous energy relaxation is expected from solvation but is not captured by our temporal resolution and spectral bandwidth. Despite the high temporal and energy resolution of these spectral projections, their interpretation is complex due to the presence of many overlapping contributions. Indeed, the fast red shift followed by a blue shift observed in Fig. 3a is indicative of competition between mechanisms, such as screening and population cooling.

A closer inspection of the spectral projections shown in Fig. 3a is obtained by reporting pseudo-TA spectra for a set of delays, shown in Fig. 4. These spectra can be analyzed in terms of ground-state bleach, stimulated emission, and excited-state absorption contributions, just like in a TA experiment. The only difference is the additional presence of electronic coherences, which are neglected here. The spectra in Fig. 4a show the rise of a tail to the blue of the main peak, indicated by the black arrow. This tail slightly relaxes at later population times, as can be seen in Fig. 4b. The early-time dynamics of the spectra are inconsistent with previous cooling signatures and confirm that other phenomena dominate the spectral changes15,30. No clear qualitative picture emerges from these one-dimensional results.

In order to unravel the spectral diffusion dynamics in greater detail, we focus on the unique observation enabled by the 2DES method: the time dependence of the anti-diagonal linewidth, represented as the dashed gray line in Fig. 2c. Details on how to extract the anti-diagonal linewidth are presented in Supplementary Note 2 as well as Supplementary Figures 8 and 9. We note that the anti-diagonal linewidth is only a true representation of the homogeneous linewidth in the limit of strong inhomogeneous broadening. Due to the size dispersion of the nanocrystals (see Supplementary Fig. 2), we can expect a large static disorder contribution to the diagonal linewidth of the 2D peak. Here we report the full-width at half-maximum (FWHM) of the anti-diagonal lineshape as a function of t2, which has the advantage to be model free but is an approximation to the true homogeneous linewidth. Figure 3d shows that the anti-diagonal width of the CsPbI3 nanocrystals quickly broadens from its initial homogeneous value to its final value at 400 fs. In contrast, as seen in Fig. 3e, the width of the CdSe nanocrystals stays constant on average, indicating that no spectral diffusion is taking place. Instead, the width is modulated at the LO phonon frequency. Figure 3f shows the lineshape dynamics for Nile Blue in ethanol. By comparing Fig. 3d, f, it becomes apparent that the lineshape behavior of the CsPbI3 nanocrystals qualitatively mimics that of the molecular dye in solution, albeit with a slower timescale and without the strong coherent vibronic modulations arising from the ring distortion mode of Nile Blue31. These data reveal that perovskite nanocrystals undergo spectral dynamics that are consistent with liquids and inconsistent with covalent solids.

## Discussion

Figure 5 rationalizes the observed linewidth behaviors in light of two limits. The first limit corresponds to the diffusive regime, described by irreversible dynamics characteristic of liquids, glasses, and other disordered systems7,25,26. Figure 5a–c shows the dynamics of the CsPbI3 perovskites along with diffusive modeling. The second limit corresponds to reversible wavepacket dynamics, characteristic of vibronic quantum systems. Figure 5d–f shows the dynamics of CdSe nanocrystals along with coherent modeling. In order to reproduce the lineshape behaviors, we performed calculations of the 2D spectra by using the cumulant expansion to second order and the multimode Brownian oscillator model22,23,32. Details can be found in the Supplementary Discussion. The 2D spectra of the CsPbI3 and CdSe nanocrystals were modeled by using a Kubo and a Huang–Rhys lineshape, respectively33. Parameter values are summarized in Supplementary Tables 2 and 3. The CsPbI3 dynamics are well captured by a dissipative, diffusive model, illustrated schematically in Fig. 5b, c.

The Kubo lineshape is derived from the fluctuation–dissipation theorem, with the assumption of Gaussian fluctuations with finite correlation time. It is well known that solvation in a polar solvent can be cast as a fluctuation–dissipation problem25,26,27. The conceptual analogy between solvation in liquids and polaron formation in solids has been established in other studies, such as photoemission works and more recent optical Kerr effect measurements6,34. Indeed, both solvation and polaron formation arise from atomic fluctuations13,14. While the microscopic nature of these fluctuations are not obviously connected with each other in the two cases, both are expected to yield isomorphic diffusive dynamics. For polar solvation dynamics, one has an electronic system coupled to a diffusive solvent bath. For polarons in these bulk-like nanocrystals, this system-bath partitioning is not as straightforward. From liquids, to glasses, to these ionic crystals, the dynamics reflect dynamical disorder in the system. Because properties of the fluctuations are accessed via the FFCF, the 2DES method is uniquely positioned to investigate solvation in polar liquids and polaron formation35,36. Combining the knowledge from previous studies probing structural dynamics in lead-halide perovskites with our main observation of solvation-like electronic response of the CsPbI3 nanocrystals, we assign these diffuse dynamics to the optical signature of polaron formation.

The lineshape dynamics of the full 2D dataset is well represented by a simple model with two parameters: a correlation time of the fluctuations τc ~110 fs, and the amplitude of the fluctuations, Δ ~50 meV. We stress that these parameters are not accessible via one-dimensional methods such as TA spectroscopy. From Δ, we can furthermore retrieve a reorganization energy $${\lambda } = {\it{\Delta }}^{{2}}/2k_{\mathrm{B}}T$$ ~50 meV37. This value is a measure of polaron stability, which is expected to play an important role in the competition between polaron formation and hot electron cooling38. Our extracted value is on the same order of magnitude as previous cryo-magneto optical absorption spectroscopy measurements, and also falls within the bounds of calculated binding energies for hybrid organic–inorganic perovskites7,39. As can be seen from Supplementary Fig. 10, our fits are very sensitive to the value of Δ. Having access to the polaron binding energy can help understand why certain lead-halide perovskites feature slower or faster cooling rates, which is ultimately of interest for device fabrication. We also note that the model further shows that the assumption of Gaussian fluctuations in the Kubo lineshape suffices to capture the experimental observation. Our measurement of electronic correlation properties provides an important experimental point of reference for quantum modeling beyond our phenomenological model. Further theoretical work may calculate these electronic fluctuations and hopefully provide a microscopic picture of specific phonons at specific wave vectors coupling to specific electronic states, but this extends beyond the scope of this work.

These 2DES experiments on perovskite nanocrystals reveal the formation of polarons in real time, by directly probing their impact on the optical response. The time dependence of the homogeneous linewidth extracted from the 2DE spectra is used as a sensitive observable to unambiguously distinguish coherent versus diffusive dynamics. The perovskite nanocrystals reveal dynamics consistent with dynamically disordered liquids, and inconsistent with ordered covalent solids. With direct observation of spectral diffusion on the femtosecond timescale, we report that the timescale of polaron formation is ~100 fs, consistent with ultrafast polar solvation dynamics in liquids. These polaron dynamics enable the rapid sampling of electronic configuration space, as revealed by their spectral diffusion dynamics in the electronic regime. This work hints at the importance in controlling the polaron properties for optoelectronic applications, in a similar spirit to previous efforts geared at controlling exciton properties in covalent nanocrystals, such as CdSe quantum dots. Future works may systematically investigate how these properties can be tuned with material parameters, such as chemical composition or quantum confinement, and hopefully provide links between polaron properties and device performance.

## Methods

### Synthesis of the CsPbI3 nanocrystals

The synthesis of these perovskite nanocrystals (NCs) followed previously reported hot-injection methods with some modifications2,17,40. In a three-neck flask, Cs2CO3 (0.60 mol) and diisooctylphosphinic acid (DOPA, 0.20 mol) were injected into octadecene (ODE, 10 mL), which was preheated at 100 °C. The solution was dried for 1 h at 120 °C under vacuum. The temperature was increased to 150 °C under an argon atmosphere, ensuring that all of the DOPA and Cs2CO3 reacted to form a Cs-phosphinate solution. The solution was then kept at 100 °C for future injection.

At the same time, PbX2 (0.38 mmol) was dissolved in ODE (10 mL) in a three-neck flask, and the solution was dried for 1 h at 120 °C under vacuum. After complete solubilization, dried oleylamine (OLA, 1 mL) and dried DOPA (1 mL) were injected under an argon atmosphere. The temperature was then increased to 140–200 °C, and the hot Cs-oleate solution (0.8 mL) was subsequently injected. After 5 s, the flask was cooled down to room temperature by an ice-water bath. Depending on the temperature of injection, a strong light red (140 °C) to dark-red (200 °C) emission was observed. The product was then centrifuged for 5 min at 12,000 rpm, and the excess lead iodide was discarded. The supernatant was mixed with tert-butanol (t-BuOH) (volume ratio of 2:1 with ODE) and centrifuged for 25 min at 12,000 rpm. The supernatant was then discarded in order to remove the excess of ligands, and the NCs were re-dispersed in toluene under an ambient atmosphere and stored, in the absence of light, at 2 °C for characterization.

Fourier-transform infrared spectroscopy (FTIR) data were acquired with a Spectrum II Perkin spectrometer to confirm the lack of unbound ligands. A small amount of the NC solution was deposited onto a KBr substrate, and measured in the wavenumber range of 400–4000 cm−1. Band attribution was made according to previous vibrational analysis41. An example FTIR spectrum is shown in Supplementary Fig. 1. The signal is closely related to a mixture of the toluene and ODE without additional bands. This shows that all the excess ligands have been removed during the second purification step.

Prior to the 2DES experiments, the sample was bubbled with argon for at least 30 min. This was observed to prevent degradation over the timescale of the experiments (2–4 h for a complete set). During the experiments, the sample was flown in a 0.2-mm-thick cell (Starna) with a peristaltic pump to ensure that the laser hits fresh sample.

### X-ray diffraction

Powder X-ray diffraction data (PXRD) were measured with a Bruker D8 diffractometer by using a Cu source (Kα = 1.54˚ A) at 40 kV, 40 mA. Samples were prepared by dropping 50 µL of NCs in toluene on a glass wafer and heating the wafer to evaporate the toluene. Background data from a washed glass wafer were subtracted from raw data, and peaks were then filtered with a numerical method to suppress the noise.

### Transmission electron microscope (TEM)

TEM measurements were performed on a Jeol JEM-2100F analytical transmission electron microscope equipped with energy-selective X-ray spectrometer for chemical analysis. The samples were prepared by placing a drop of diluted CsPbI3 nanocrystal solution on formvar-coated Cu TEM grids. An exemplary TEM image is shown in Supplementary Fig. 2.

### Time-resolved photoluminescence

Time-resolved photoluminescence (t-PL) measurements were conducted by utilizing a streak camera (Axis TRS, Axis Photonique Inc.) in order to estimate the average number of excitations per particle as a function of fluence. The sample, dispersed in toluene, is bubbled with argon for 30 min prior to the experiment and flowed in a 0.5-mm path-length flow cell (Starna Type 48) by using a peristaltic pump (Masterflex 77390-00). The sample is excited with pulse fluences in the range of 1.1–1000 µJ cm−2, at 3.1 eV, with a duration of <100 fs generated by frequency doubling of the output of a Ti:sapphire regenerative amplifier (Coherent Legend Elite Duo HE+, 1 kHz repetition rate) in a 100-µm BBO crystal. The fluorescence is collected at 90˚, collimated, and subsequently focused onto the streak camera slit by using a pair of off-axis parabolic mirrors. The PL is dispersed by a spectrometer (Acton SP-2358i, 150 g/mm, 600-nm blaze), accelerated in a bilamellar streak tube (Photonis P820), and imaged by using an air-cooled CCD (Spectral Instruments 1200 series). The streak tube is electronically triggered with a range of ~50 ns. Overall, the average time and energy resolution of the traces are 0.1 ns and 2.5 meV. The trace and its corresponding background measurement is obtained from an average of ten exposures lasting 10 s each.

The final trace is obtained with minimal post-processing procedures. Namely, a 1% shear correction is applied to correct for electrode and detector alignment. The value for this is obtained from traces collected with a static bias. The intensity is then corrected for uneven time bins, followed by conversion from wavelength to energy (including the Jacobian correction). Due to the simultaneous time and energy resolution of this technique, the kinetic transients extracted at each fluence come from a sum of the intensity values between 1.65 and 2.0 eV. Pump scatter at 3.1 eV is used to determine the instrument response function. Gaussian fitting of this signal results in an IRF of 200 ps. Supplementary Fig. 3 shows a representative kinetic transient with an average number of excitation 〈N〉  1. The decay kinetics are well described by a triexponential, yielding an average lifetime of 17.25 ns, which is consistent with reported values for similar-sized CsPbI3 nanocrystals42,43. Supplementary Note 1 provides details on how the average number of excitations per particle is estimated from these t-PL measurements. Supplementary Fig. 6 shows the results of the analysis.

### Two-dimensional electronic spectrometer

The instrument consists of a pulse-shaper-based 2DE spectrometer operating in the pump–probe geometry and has been described in detail elsewhere24,44,45. Briefly, a small fraction of a Ti:Sapph amplifier (Legend Elite Duo, 8 mJ, 1 kHz, pulse duration 130 fs) is focused into a hollow-core fiber filled with argon to produce a broadband continuum. A short-pass filter is used to select the visible part of this continuum, yielding 46-µJ pulses spanning the 550–700-nm range. The visible pulses are then sent into GRISMs (GRatings + prISMs), which pre-compensate for the large dispersion arising from the pulse shapers where the 2DES pulse sequences are produced. As the pulses exit the GRISMs, they are split into a pump arm and a probe arm. The pulse shaper (Dazzler, Fastlite) in the pump arm is used to create the first two pulses in the sequence. A second pulse shaper placed in the probe arm is used to create the third and last pulse in the sequence (the probe). The experiment is carried out by varying time delay t1 and taking the Fourier transform, yielding the first energy axis E1. The Fourier transform of time delay t3 is performed by a spectrometer, yielding the second energy axis E3. Thus, one obtains a 2D correlation map S(E1, t2, and E3), and the experiment is repeated for various population times t2.

The spectral phase of the pulses is measured with a transient-grating frequency-resolved optical gating (TG-FROG) setup in a dispersion-free line46. The spectral phase is optimized by using the phase mask of the pulse shapers until near-Fourier-transform-limited pulses are obtained. An example TG-FROG trace acquired on the same day as the data shown in this paper is displayed in Supplementary Fig. 4.

An advantage of performing 2DES in the pump–probe geometry is that the rephasing and non-rephasing 2D signals are emitted together in the probe direction, enabling one to directly obtain an absorptive 2D spectrum exempt from phase distortions22. However, undesired standard pump–probe contributions are also emitted in the probe direction. By using the technique of phase cycling, as in nuclear magnetic resonance spectroscopy, it is possible to eliminate these pump–probe signals since they do not depend on the relative phase between pulses 1 and 2.

It was shown that a two-step phase cycling is sufficient to isolate the 2D signals in the pump–probe geometry47,48. However, in the case of highly scattering samples, contamination to the 2D spectra can arise from pump–probe or pump–pump scatter terms. In order to minimize scatter contributions from the large CsPbI3 nanocrystals, we performed a four-step phase-cycling scheme to remove pump–probe scatter contributions. The desired signal S2D was then given by49

$$S_{{\mathrm{2D}}} = \ S(\varphi _{\mathrm{1}} = 0,\varphi _{\mathrm{2}} = 0) - S(\varphi _{\mathrm{1}} = 0,\varphi _{\mathrm{2}} = \pi ) + S(\varphi _{\mathrm{1}} = \pi ,\varphi _{\mathrm{2}} = \pi ) \\ \hskip 12pt - S(\varphi _{\mathrm{1}} = \pi ,\varphi _{\mathrm{2}} = 0)$$
(1)

where ϕ1 and ϕ2 are the constant phases of pulses 1 and 2, respectively.

The CsPbI3 data were acquired in a rotating frame at 0.35 PHz, which enabled the sampling of the t1 axis with steps of 3 fs. This enabled the rapid acquisition of the 2D data to ensure that no sample degradation was taking place during the measurements.

The overlap of the laser spectrum with the linear absorption spectra of the samples is shown in Supplementary Fig. 5. The experimental parameters for the three samples are summarized in Supplementary Table 1. The 2DES experiments were repeated on different days with independently synthetized batches of CsPbI3 nanocrystals. The lineshape behavior was reproduced every time, as exemplified in Supplementary Fig. 7.

## Data availability

The data supporting the findings of this study are available upon request.

## Code availability

The code employed for modeling the 2D spectra is open source and available at https://github.com/spalato/Mbo.jl.

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## Acknowledgements

We gratefully acknowledge funding from the Canadian Foundation for Innovation (CFI), the Natural Sciences and Engineering Research Council of Canada (NSERC), and McGill University. Samuel Palato thanks NSERC for financial support. Hélène Seiler thanks the Swiss National Science Foundation for financial support, as well as Pierre-François Duc and Catherine Seiler for help on formatting of figures.

## Author information

Authors

### Contributions

The experiments were conducted by H.S., S.P., C.S., and H.B. Synthesis and characterization was done by E.S. and D.S. Analysis and simulations were done by H.S. and S.P. The paper was written by H.S. and P.K. with input from all authors. The research was supervised by P.K.

### Corresponding author

Correspondence to Patanjali Kambhampati.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature Communications thanks Giulio Cerullo and other anonymous reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

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

## Rights and permissions

Reprints and Permissions

Seiler, H., Palato, S., Sonnichsen, C. et al. Two-dimensional electronic spectroscopy reveals liquid-like lineshape dynamics in CsPbI3 perovskite nanocrystals. Nat Commun 10, 4962 (2019). https://doi.org/10.1038/s41467-019-12830-1

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• DOI: https://doi.org/10.1038/s41467-019-12830-1


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# PNG Images FTW

I had the pleasure to attend a meeting of the LinBox developers this week in Raleigh, NC. One of the question that came up was how to exchange and store matrices. For example, matrices for whose “class” one wants to find/implement dedicated algorithms. To give a more concrete example: matrices appearing during Gröbner basis computations have a special structure which allows to reduce them faster than random matrices (cf. this paper or this code). Hence, we’d like to have some sort of format to store such matrices such that we can work on these dedicated algorithms. Of course, such a format should be flexible, human readable (preferably writable as well) and reasonably simple. I’m not sure how the LinBox crew would think about me blogging about details of their meeting, hence I’ll only use this opportunity to plug my own proposal (at least for reasonably dense matrices): PNG images.

The PNG file format definitely is readable, since there exists many viewers for it (you are probably using one right now). The format can also be edited thanks to GIMP and friends. That is, we can draw matrices! Furthermore, PNG allows to use lossless compression, i.e., it can compress the actual image data using GZIP. Finally, it is pretty flexible: it supports between one (grayscale) and four (RGB + alpha) channels and various bit depths per channel (1,2,4,8,16). Hence, it can store between 1 bit and 64 bits per pixel. Thus, for finite fields up cardinality $2^{64}$ we can store each entry simply as a pixel. If we pick our colour assignment right, the pictures even make sense (using the convention that darker is larger as an integer for example).

Of course, the whole format is fundamentally biased towards dense matrices. In fact, we’ve been using 1-bit PNG images as data storage for M4RI matrices for a while now: both Sage and PolyBoRi use the GD library to write M4RI matrices to disk. However, using GD has some shortcomings such as high memory requirements (the whole image is constructed in RAM before being written to disk). This week, I implemented reading/writing of PNG 1-bit images feature directly in the M4RI library using libpng directly. This allows to save a lot of memory and for some cool other features such as control over the GZIP compression level, custom comments, “unknown chunks” as attachments, etc.

I’ve also conducted some experiments to get an impression how well this format works in terms of storage space and loading time:


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# Failure of the Hahn-Banach in ℂₚ

The Hahn-Banach is known to be a crucial tool in functional analysis. The following are lecture notes written a while ago for my functional analysis course on the failure of Hahn-Banach in the $p$-adic completion of $\bar{\mathbb{Q}}_p$ (denoted by $\mathbb{C}_p$). For those who are well-versed in the basics of $\mathbb{Q}_p$, I would recommend skipping to around the middle of this page.

Since these are lecture notes, I present as much proof as I can without going into too much gritty detail.

Definition. The $p$-adic norm of $x\in\mathbb{Q}\setminus \{0\}$ is defined as $|x|_p=p^{-r}$ where $r$ is the unique power of $p$ dividing $x$. We define $|0|_p:=0$.

Let $x=75/2=5^2\cdot 3\cdot 2^{-1}$. Then

$|x|_5=5^2, |x|_3=3, |x|_2=2^{-1}$

Definition. A norm $|\cdot|:X\to \mathbb{R}$ is called a non-archimedean norm if

$|x+y|\leq \max\{|x|,|y|\}$

If a norm does not satisfy the above inequality, then $|\cdot|$ is a archimedean norm.

This inequality is called the strong triangle inequality. As a metric, this norm induces the ultrametric.

Proposition. The $p$-adic norm is a non-archimedean norm of the field $\mathbb{Q}$.

Proof. By definition, $|x|_p=0\iff x=0$ and $|x|_p\geq 0$ for all $x$. Since $\mathbb{Z}$ is UFD, $|xy|_p=|x|_p|y|_p$. Finally, if $x,y\in\mathbb{Q}$,

$x+y=p^r\frac{u}{v}+p^s\frac{w}{z},\quad\gcd(u,v)=1,\quad\gcd(w,z)=1$

Without loss of generality, let $s\geq r$. Then

$x+y=p^{r}\frac{uz+p^{s-r}vw}{vz}\implies |x+y|_{p}\leq|x|_p.$

where the implication follows since $\gcd(vz,p)=1$ and $\gcd(uz+p^{s-r}vw,p)=1$.
Thus in full generality,

$|x+y|_p\leq \max\{|x|_p,|y|_p\}\leq |x|_p+|y|_p.$

$\blacksquare$

Definition. The metric completion of $\mathbb{Q}$ with respect to the metric $|\cdot|_p$ is called the $p$-adic numbers, denoted by $\mathbb{Q}_p$.

Proposition. Any $x\in \mathbb{Q}_p$ has the form

$x=\sum_{i=k}^\infty a_ip^i$

where $a_k\neq 0$ and $a_i\in\{0,…,p-1\}$ for $i>k$.

Consider $\mathbb{Q}_2$ and sums of the form

$x=\sum_{i=0}^\infty a_ip^i$

where $a_i\in\{0,1\}$. Consider the infinite coordinate $\tilde{x}=(a_0,a_1,a_2,…)$. Then $\tilde{x}$ is the binary representation of $x$.

Proposition. $\zeta_n\in\mathbb{Q}_p$ if and only if $p\equiv 1\mod n$.

Proof. F.Q. Gouvêa, $p$-adic Numbers, 1997

$\blacksquare$

Corollary. $\mathbb{Q}_p$ is not algebraically closed.

Proof. For every $p$, $\zeta_{p+1}\not\in\mathbb{Q}_p$. Thus $\mathbb{Q}_p$ does not contain the root of the polynomial $x^{p+1}-1=0$.

$\blacksquare$

Definition. The algebraic closure of $\mathbb{Q}_p$ is denoted by $\overline{\mathbb{Q}}_p$.

Proposition. The norm $|\cdot|_{p}$ extends to $\overline{\mathbb{Q}}_p$.

Proof. Take $x\in\overline{\mathbb{Q}}_p$. Then $\mathbb{Q}_p(x)$ is a finite extension, of say degree $n$, since $x$ is the root of a polynomial of finite degree. Define $\psi:\overline{\mathbb{Q}}_p(x)\to\overline{\mathbb{Q}}_p(x)$ by

$w\mapsto xw$

Then $\psi$ is $\mathbb{Q}_p$-linear map, thus define

$|x|_{p}=\sqrt[n]{|\det \psi|_p}$

and check axioms.

$\blacksquare$

Proposition. $\overline{\mathbb{Q}}_p$ is not complete with respect to $|\cdot|_p$.

Sketch of Proof. Define

$a_i:=\begin{cases} \zeta_i,\quad\mbox{if } (i,p)=1,\\ 1,\quad \mbox{otherwise}. \end{cases}$

Define

$x_{n,m}:=\sum_{i=n}^m a_ip^i$

Then $\{x_{n,m}\}$ is Cauchy under the $p$-norm, but the limit is not in $\overline{\mathbb{Q}}_p$.

$\blacksquare$

Definition. The metric completion of $\overline{\mathbb{Q}}_p$ with respect to $|\cdot|_p$ is called the $p$-adic complex numbers, denoted by $\mathbb{C}_p$.

Definition. A metric space $T$ is called spherically complete if for any nested sequence of closed balls $B_1\supseteq B_2\supseteq…$ in $T$,

$\bigcap_{n\in\mathbb{N}}B_n\neq\emptyset$

Some examples of spherically complete fields are $\mathbb{R}$  and $\mathbb{C}$ by Cantor’s intersection theorem. In fact, these fields are complete with respect to nested sequences of compact sets.

Lemma. $\mathbb{Q}_p$ is spherically complete.

Lemma. $\mathbb{C}_p$ is not spherically complete.

Proof. Since $\mathbb{C}_p$ is separable (See A Course in p-adic Analysis, A. Robert.), there exists a countable dense subset $A$ which can be written as a sequence $(a_1,a_2,…)$ where $a_i\in\mathbb{C}_p$. Fix a sequence $(\epsilon_n)_{n=1}^\infty$ where $1>\epsilon_1>\epsilon_2>…>1/2$. On $\mathbb{C}_p$, define a relation $a\sim b$ iff $|a-b|_p\leq\epsilon_1$. This is an equivalence relation since

$a\sim b\sim c\implies |a-c|_p=|a-b+b-c|_p\leq \max\{|a-b|,|b-c|\}\leq \epsilon_1\implies a\sim c$

Since the other properties are trivially satisfied, we have our claim. Visually, this equivalence relation partitions $\mathbb{C}_p$ as the set of all $\epsilon_1$ balls where each ball is in fact disjoint! (For $\mathbb{R}$, note that $(-1,1)\cap (0,2)=(0,1)$, so the disjointness is indeed very surprising.)

Now observe that $|\cdot|_p$ is a norm mapping to $\mathbb{R}$, but it ONLY takes on values $p^\mathbb{Z}$ when considered under $\mathbb{Q}$. It can be shown that for $\mathbb{C}_p$, the norm takes on values $p^\mathbb{Q}$. Moreover, one may notice these values are dense in $\mathbb{R}_+^\times$. It follows that the diameter of every equivalence class is actually $\epsilon_1$.

We can now iterate. Take $B_1$ to be an equivalency class such that $a_1\notin B_1$. Then on $B_1$, define an equivalence relation $a\sim b$ iff $|a-b|_p\leq\epsilon_2$. Then take $B_2\subseteq B_1$ to be an equivalency class such that $a_2\notin B_2$. Inductively, we have a nested sequence

$B_1\supseteq B_2\supseteq…\mbox{ such that }\operatorname{diam}(B_i)=\epsilon_i\mbox{ and }a_i\not\in B_i \forall i\in\mathbb{N}$

I claim $\cap_{n\in\mathbb{N}} B_n=\emptyset$. Indeed, if not, then there exists $b\in \cap_{n\in\mathbb{N}} B_n$. Since $b$ is in each equivalency class, we have $B_i=B_{\epsilon_i}(b)$ for all $i$, so $B_{1/2}(b)\subset\cap_{n\in\mathbb{N}} B_n$ since $\epsilon_i>1/2$. By construction, no $a_i$ can be contained in $B_{1/2}(b)$, but this contradicts the density of $\overline{\mathbb{Q}}_p$ in $\mathbb{C}_p$.

$\blacksquare$

Definition. Let $X$ be any set, and let $k$ be a complete field with norm $|\cdot|_k$. Then we denote by $l^\infty(X)$ the set of all bounded functions on $X$ to $k$. That is

$l^\infty(X):=\left\{\varphi:X\to k\mid \sup_{x\in X}|\varphi(x)|_k<\infty\right\}$

Lemma. Over $\mathbb{C}_p$, $l^\infty(X)$ is spherically complete and $c_0$ is non-spherically complete.

Proof. Non-Archimedean Functional Analysis, A.C.M. Van Rooij: If $k$ has a dense norm, then $l^\infty$ is spherically complete. Since the norm of $\mathbb{C}_p$ is dense in $\mathbb{R}^+_{\geq 0}$, we are done.

$\blacksquare$

Recall

Theorem. [Hahn-Banach] Let $V$ be a vector-space over a complete archimedian field. Let $p$ be a sub-linear functional on $V$. For every linear subspace $E\subseteq V$, for all $f\in E'$ dominated by $p$, there exists a $g\in V'$ such that $f=g|_{E}$ and $||g||\leq ||p||$.

This theorem fails very harshly when the Banach space is non-archimedean; in fact, we can find examples that do not even satisfy the following properties:

Definition. A Banach space $V$ is said to satisfy the Hahn-Banach extension property (HBEP) if for every linear subspace $E\subseteq V$, for all $f\in E'$, there exists a $g\in V'$ such that $f=g|_{E}$ and $||g||=||f||$.

Theorem. HBEP does not hold for Banach spaces over $\mathbb{C}_p$.

Proof. Consider the closed subspace of $l^\infty(\mathbb{N})$

$\Delta=\left\{(a,a,…):a\in\mathbb{C}_p\right\}$

Consider the linear functional

$\varphi:\Delta\to\mathbb{C}_p\mbox{ defined by }(a,a,…)\mapsto a$

Clearly $||\varphi||=1$. Now since $\mathbb{C}_p$ is not spherically complete, there is a nested sequence of balls $\{B_{r_i}(x_i)\}$ with empty intersection in $\mathbb{C}_p$. Consider the nested sequence $B_i^*:=B_{r_i}((x_i,x_i,…))\subseteq l^\infty(\mathbb{N})$. The intersection of this family is non-empty since $l^\infty(\mathbb{N})$ is spherically complete. Thus let $x$ be in the intersection. Now suppose $\varphi$ extends to a linear map $\psi:l^\infty(\mathbb{N})\to \mathbb{C}_p$ with norm preserved. Then

$|\psi(x)-x_i|_\infty = |\psi(x)-\varphi(x_i)|_\infty \leq ||\psi||||x-x_i||_{A}=|x-x_i|_{p}\leq\max\{|x|_p,|x_i|_p\}=\max\{r,r_i\}$

so $\psi(x)\in\cap_n B_n$, a contradiction.

$\blacksquare$

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